Educational Measurement Solutions Rasch Model Guide

Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
APPLYING THE RASCH MODEL TO
PSYCHO-SOCIAL MEASUREMENT
A PRACTICAL APPROACH
Margaret Wu
&
Ray Adams
Documents supplied on behalf of the authors by
Educational Measurement Solutions
1
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
TABLE OF CONTENT
CHAPTER ONE: WHAT IS MEASUREMENT?
MEASUREMENTS IN THE PHYSICAL WORLD
MEASUREMENTS IN THE PSYCHO-SOCIAL SCIENCE CONTEXT
PSYCHOMETRICS
FORMAL DEFINITIONS OF PSYCHO-SOCIAL MEASUREMENT
LEVELS OF MEASUREMENT
4
4
4
4
5
5
CHAPTER TWO: AN IDEAL MEASUREMENT
AN IDEAL MEASUREMENT
ABILITY ESTIMATES BASED ON RAW SCORES
LINKING PEOPLE TO TASKS
ESTIMATING ABILITY USING ITEM RESPONSE THEORY
IRT VIEWED AS A TRANSFORMATION OF RAW SCORES
10
10
10
12
13
16
HOW ABOUT OTHER TRANSFORMATIONS OF RAW SCORES, FOR EXAMPLE,
STANDARDISED SCORE (Z-SCORE) AND PERCENTILE RANKS? DO THEY
PRESERVE “DISTANCES” BETWEEN PEOPLE?
17
CHAPTER THREE: DEVELOPING TESTS FROM IRT PERSPECTIVES –
CONSTRUCT AND FRAMEWORK
WHAT IS A CONSTRUCT?
LINKING VALIDITY TO CONSTRUCT
CONSTRUCT AND ITEM RESPONSE THEORY (IRT)
UNI-DIMENSIONALITY
SUMMARY
18
18
18
19
21
23
CHAPTER FOUR: THE RASCH MODEL (THE DICHOTOMOUS CASE)
THE RASCH MODEL
PROPERTIES OF THE RASCH MODEL
28
28
29
CHAPTER FIVE: THE RASCH MODEL (THE POLYTOMOUS CASE)
INTRODUCTION
THE DERIVATION OF THE PARTIAL CREDIT MODEL
PCMPROBABILITIES FOR ALL RESPONSE CATEGORIES
SOME OBSERVATIONS
THE INTERPRETATION OF k !
TAU’S AND DELTA DOT
THURSTONIAN THRESHOLDS, OR GAMMAS (" )
USING EXPECTED SCORES AS MEASURES OF ITEM DIFFICULTY
SUM OF DICHOTOMOUS ITEMS AND THE PARTIAL CREDIT MODEL
39
39
39
40
41
41
47
50
51
54
CHAPTER SIX: PREPARING DATA FOR RASCH ANALYSIS
CODING
SCORING AND CODING
DATA ENTRY
56
56
57
58
CHAPTER SEVEN: ITEM ANALYSIS STEPS
GENERAL PRINCIPLES OF ESTIMATION PROCEDURES
TYPICAL OUTPUT OF IRT PROGRAMS
EXAMINE ITEM STATISTICS
CHECKING FOR DIFFERENTIAL ITEM FUNCTIONING
62
62
63
64
69
2
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
CHAPTER EIGHT: HOW WELL DO THE DATA FIT THE MODEL?
FIT STATISTICS
RESIDUAL BASED FIT STATISTICS
INTERPRETATIONS OF FIT MEAN SQUARE
74
74
75
76
THE FIT t STATISTIC
SUMMARY
82
85
3
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Chapter One: What is Measurement?
Measurements in the physical world
Most of us are familiar with Measurement in the physical world, whether it is
measuring today's maximum temperature, or the height of a child, or the dimensions
of a house, where numbers are given to represent "quantities" of some kind, on some
scales, to convey properties of some attributes that are of interest to us. For example,
if yesterday's maximum temperature in London was 12!C, one gets a sense of how
cold (or warm) it was, without actually having to go to London in person to know the
weather there. If a house is situated 1.5 km from the nearest train station, one gets a
sense of how far away that is, and how long it might take to walk to the train station.
Measurement in the physical world is all around us, and there are well-established
measuring instruments and scales that provide us with useful information about the
world around us.
Measurements in the psycho-social science context
Measurements in the psycho-social world are also abound, but perhaps less well
established universally as temperature and distance measures. A doctor may provide
a score for a measure of the level of depression. These scores may provide
information to the patients, but the scores may not necessarily be meaningful to
people who are not familiar with these measures. A teacher may provide a score of
student achievement in mathematics. These may provide the students and parents
with some information about progress in learning. But the scores will generally not
provide much information beyond the classroom. The difficulty with Measurement in
the psycho-social world is that the attributes of interest are generally not directly
visible to us as objects of the physical world are. It is only through observable
indicators of the attributes that measurements can be made. For example,
sleeplessness and eating disorders may be symptoms of depression. Through the
observation of the symptoms of depression, one can then develop a measuring
instrument, and a scale of levels of depression. Similarly, to provide a measurement
of student academic achievement, one needs to find out what a student knows and can
do academically. A test in a subject domain may provide us with some information
about a student's academic achievement. That is, one cannot "see" academic
achievement as one sees the dimensions of a house. One can only measure academic
achievement through indicator variables such as the tasks students can perform.
Psychometrics
From the above discussion, it can be seen that not only is the measurement of psychosocial attributes difficult, but often the attributes themselves are some "concepts" or
"notions" which lack clear definitions. Typical, these psycho-social attributes need
clarification before measurements can take place.
For example, "academic
achievement" needs to be defined before any measurement can be taken. In the
following, psycho-social attributes that are of interest to be measured are referred to
as "latent traits" or "constructs". The science of measuring the latent traits is referred
to as psychometrics.
In general, psychometrics deals with the measurement of all "latent traits", and not
just those in the psycho-social context. For example, the quality of wine has been an
attribute of interest, and researchers have applied psychometric methodologies in
4
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
establishing a measuring scale for it. One can regard "the quality of wine" as a latent
trait because it is not directly visible (therefore "latent"), and it is a concept that can
have ratings from low to high (therefore "trait" to be measured).
In general,
psychometrics is about measuring latent traits, where the attribute of interest is not
directly visible so that the measurement is achieved through collecting information on
indicator variables associated with the attribute. In addition, the attribute of interest to
be measured varies in levels from low to high so that it is meaningful to provide
"measures" of the attribute.
Formal definitions of psycho-social measurement
Various formal definitions of psycho-social measurement can be found in the
literature. The following are four different definitions of measurement. It is
interesting to compare the scope of measurement covered by each definition.
"
(Measurement is) a procedure for the assignment of numbers to specified
properties of experimental units in such a way as to characterise and preserve
specified relationships in the behavioural domain. Lord, F., & Novick, M. (1968) Statistical Theory of Mental Test Scores
"
(Measurement is) the assigning of numbers to individuals in a systematic way
as a means of representing properties of the individuals.
Allen, M.J. and Yen, W. M. (1979.) Introduction to Measurement Theory
"
Measurement consists of rules for assigning numbers to objects in such a way
as to represent quantities of attributes.
Nunnally, J.C. (1978) Psychometric Theory
"
A measure is a location on a line. Measurement is the process of constructing
lines and locating individuals on lines.
Wright, D. N. and M. H. Stone (1979). Best Test Design.
All four definitions relate measurement to assigning numbers to objects. The third
and fourth definitions also bring in a notion of representing quantities, while the first
and second merely state the assignment of numbers in some well-defined ways. The
fourth definition goes further than the third in specifying that the quantity represented
by the measurement is a continuous variable (i.e., on a real-number line), and not just
a discrete rank ordering of objects.
So it can be seen that the first and second definitions are broader than the third and the
fourth. Measurements under the first and second definitions may not be very useful,
if the numbers are simply labels for the objects. These provide "low" levels of
measurement. The fourth definition provides the highest level of measurement, in
that the assignment of numbers can be called measurement only if the numbers
represent the distances between objects in terms of the level of the attribute being
measured (i.e., locations on a line). This kind of measurement will provide us with
more information in discriminating between objects in terms of the levels of the
attribute the objects possess.
Levels of Measurement
More formally, there are definitions for four levels of measurement (nominal, ordinal,
interval and ratio) in terms of the way the numbers are assigned and in terms of the
5
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
inference that can be drawn from the numbers assigned. Each of these levels is
discussed below.
Nominal
When numbers are assigned to objects simply as labels for the objects, the numbers
are said to be nominal. For example, each player in a basketball team is assigned a
number. The numbers do not mean anything other than for the identification of the
players. Similarly, codes assigned for categorical variables such as gender (male=1;
female=2) are all nominal. In this course, the use of nominal numbers is not
considered as measurement, because there is no notion of "more" or "less" in the
representation of the numbers. The kind of measurement described in this course
refers to methodologies for finding out "more" or "less" of some attribute of interest.
Ordinal
When numbers are assigned to objects to indicate ordering among the objects, the
numbers are said to be ordinal. For example, in a car race, numbers are used to
represent the order in which the cars finish the race. In a survey where respondents
are asked to rate their responses, the numbers 0 to 3 are used to represent strongly
disagree, disagree, agree, strongly agree. In this case, the numbers represent an
ordering of the responses. Ordinal measurements are often used, such as for ranking
students, or for ranking candidates in an election, or for arranging a list of objects in
order of preference.
Interval
When numbers are assigned to objects to indicate the amount of an attribute, the
numbers are said to represent interval measurement. For example, clock time
provides an interval measure in that 7 o'clock is two hours away from 5 o'clock, and
four hours from 3 o'clock. In this example, the numbers not only represent ordering,
but also represent an "amount" of the attribute so that distances between the numbers
are meaningful and can be compared. Interval measurements do not necessarily have
an absolute zero, or an origin.
Ratio
In contrast, measurements are at the ratio level when numbers represent interval
measures with an absolute zero. For example, the number of votes a candidate
receives in an election is a ratio measurement. If one candidate receives 300 votes
and another receives 600 votes, one can say that the first candidate obtained half the
number of votes as that obtained by the second candidate. In this case, not only
distances between numbers can be compared, the numbers can form ratios and the
ratios are meaningful for comparison.
Increasing levels of measurement
6
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Ratio
Interval
Ordinal
Nominal
It can be seen that the four levels of measurement from nominal to ratio provides
increasing power in the meaningfulness of the numbers used for measurement. If a
measurement is at the ratio level, then comparisons between numbers both in terms of
differences and in terms of ratios are meaningful. If a measurement is at the interval
level, then comparisons between the numbers in terms of differences are meaningful.
For ordinal measurements, only ordering can be inferred from the numbers, and not
the actual distances between the numbers. Nominal level numbers do not provide
much information in terms of "measurement" as defined in this course.
Clearly, when one is developing a scale for measuring latent traits, it will be best if
the numbers on the scale represent the highest level of measurement. In general,
latent traits do not have an absolute zero. That is, it is difficult to define the point
where there is no latent trait. But if one can achieve interval measurement for the
scale constructed to measure a latent trait, then the scale can provide more
information than an ordinal scale where only rankings of objects can be made.
Bearing these points in mind, the next Chapter examines the properties of an ideal
measurement in the psycho-social context.
References
Allen, M. J., & Yen, W. M. (1979). Introduction to Measurement Theory. Monterey,
California: Brooks/Cole Publishing Company.
Lord, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores. Reading,
MA: Addison-Wesley.
Nunnally, J.C. (1978). Psychometric theory. New York: McGraw-Hill Book Company.
UNESCO-IIEP (2004). Southern and Eastern Africa Consortium for monitoring educational
quality (SACMEQ) Data Archive.
Wright, B.D., & Stone, M.H. (1979). Best test design. Chicago, IL: Mesa Press.
7
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Exercises
The following are some data collected in SACMEQ (Southern and Eastern Africa
Consortium for Monitoring Educational Quality, UNESCO-IIEP, 2004). For each
variable, state whether the numerical coding provides nominal, ordinal, interval or
ratio measures?
(1) PENGLISH
Do you speak English outside school?
(Please tick only one box.)
Never
(1)
Sometimes
(2)
All of the time
(3)
(2) XEXPER
How many years altogether have you been teaching?
(Please round to '1' if it is less than 1 year.)
years
(3) PCLASS
Which Standard 6 class are you in this term?
(Please tick only one box.)
6A
6B
6C
6D
6E
6F
6G
6H
6I
6J
6K
6L
(01)
(02)
(03)
(04)
(05)
(06)
(07)
(08)
(09)
(10)
(11)
(12)
8
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
(4) PSTAY
Where do you stay during the school week?
(Please tick only one box.)
(1)
(2)
(3)
(4)
In my parents’/legal guardian’s home
With other relatives or another family
In a hostel/boarding school accommodation
Somewhere by myself or with other children
9
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Chapter Two: An Ideal Measurement
An Ideal Measurement
Consider an example where one is interested in measuring students' academic ability
in a subject domain. Suppose a test is developed to measure students’ ability in this
subject domain, one would like the test scores to be accurate and useful.
By accurate, we mean that the score a student obtains can be trusted. If Tom gets 12
out of 20 on a geometry test, we hope that this score provides a measure of what Tom
can do on this test, and that if the test could be administered again, he is likely to get
12 out of 20 again. This notion of “accuracy” relates to the concept of “reliability” in
educational jargon.
We would also like the test scores to be useful for some purpose we have in mind.
For example, if we want to select students for a specialist course, we would want our
test scores to reflect students’ suitability for doing this course. This notion of
“usefulness” relates to the concept of “validity” in educational jargon.
Furthermore, we would like the test scores to provide us with a stable frame of
reference in comparing different students. For example, if the test scores from one
test tell us that, on a scale of geometry ability from low to high, Tom, Bev and Ed are
located as follows:
Tom
Low ability
Bev
Ed
Geometry ability scale
High ability
Figure 1 Locations of Tom, Bev and Ed on the Geometry Ability Scale
If we give Tom, Bev and Ed another test on geometry, we hope that they will be
placed on the geometry ability scale in the same way as that shown in Figure 1. That
is, no matter which geometry test we administer, we will find that Bev is a little better
than Tom in geometry, but Ed is very much better than both Tom and Bev. In this
way, the measurement is at the interval level, where statements about the distances
between students can be made, and not just rank ordering.
Ability Estimates Based on Raw Scores
Now let us consider using raw scores on a test (number of items correct) as a measure
of ability. Suppose two geometry tests are administered to a group of students, where
test 1 is easy and test 2 is hard. Suppose A, B, C and D are four students with
differing ability in geometry. A is an extremely able student in geometry, B is an
extremely poor student in geometry, and C and D are somewhat average students in
geometry.
If the scores of students A, B C and D on the two tests are plotted, one may get the
following picture.
10
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
100
A
!A
Raw Score on Hard Test
90
80
70
60
50
"C
"D
40 C
D
30
20
10
0
B #B
B
0
10
D C
20
30
40
50
60
70
80
A
90 100
Raw Score on Easy Test
Figure 2 Plot of Student Raw Scores on an Easy Test and a Hard Test
That is, A, being an excellent student in geometry, is likely to score high on both the
easy test and the hard test. B, being a rather poor student at geometry, is likely to
score low on both tests. C and D are likely to score somewhat higher on the easy test,
and somewhat lower on the hard test.
On the horizontal axis where the scores on the easy test are placed, it can be seen that
A and C are closer together than B and C in terms of their raw scores. However, on
the vertical axis where the scores on the hard test are placed, A and C are further apart
than C and B. If both the easy test and the hard test measure the same ability, one
would expect to see the same distance between A and C, irrespective of which test is
administered. From this point of view, we can see that raw scores do not provide us
with a stable frame of reference in terms of the distances between students on the
ability scale. However, raw scores do provide us with a stable frame of reference in
terms of ordering students on the ability scale.
In more technical terms, one can say that raw scores provide ordinal measurement,
and not interval measurement. This is not entirely true, as raw scores provide
measures somewhere in-between ordinal and interval measurement. For example,
from Figure 2, one can still make the judgement that C and D are closer together in
terms of their ability than B and C, say.
Another important observation is that the relationship between the scores on the two
tests is not linear (not a straight line). That is, to map the scores of the hard test onto
scores of the easy test, there is not a simple linear transformation such as a constant
shift or a constant scaling factor.
Consequently, the ability estimates based on raw scores are dependent on the
particular test administered. This is certainly not a desirable characteristic of an ideal
measurement.
11
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Linking People to Tasks
Another characteristic of an ideal measurement is that “meanings” can be given to
scores. That is, we would like to know what the student can actually do if a student
obtained a score of, say, 55 out of 100, on a test. Therefore if student scores can be
linked to the items in someway, then substantive meanings can be given to scores in
terms of the underlying skills or proficiencies. For example, one would like to make
statements such as
“Students who obtained 55 out of 100 on this test are likely to be able to carry
out two-digit multiplications and solve arithmetic change problems”.
When raw percentages are used to measure students’ abilities and item difficulties, it
is not immediately obvious how one can link student scores to item scores. For
example, Figure 3 shows two scales, one for item difficulty, and one for person
ability. The item difficulty scale on the left shows that word problems have an
average percentage correct of 25%. That is 25% of the students obtained the correct
answers on these items. In contrast, 90% of the students correctly carried out single
digit additions.
Link Raw Scores on Items and Persons
Item Difficulties
Student Scores
word problems
25%
arithmetic with
vulgar fractions
50%
multi-step
arithmetic
70%
single digit
addition
?
?
?
?
90%
90%
70%
50%
25%
Figure 3 Link Raw Scores on Items and Persons
On the other hand, the person ability scale shows students who obtained 90% on the
test, and those who obtained 70%, 50% and 25% on the test. The percentages on the
two scales are not easily matched in any way. Can the students who obtained 70% on
the test perform arithmetic with vulgar fractions? We cannot make any inference
because we do not know what proportions of items are single digit addition, multistep arithmetic, or other types. It may be the case that 70% of the items are singledigit addition items, so that the students who obtained 70% correct on the test cannot
perform tasks much more difficult than single-digit addition.
Even if we have information on the composition of the test in terms of the number of
items for each type of problems, it is still a difficult job to match student scores with
12
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
tasks. The underlying skills for each student score will need to be studies separately,
and descriptions written for each student score. No systematic approach can be taken.
When a different test is administered, a new set of descriptions will need to be
developed, as there is no simple and direct relationship between student scores and
item scores.
Estimating Ability Using Item Response Theory
The main idea of item response theory is to use a mathematical model for predicting
the probability of success of a person on an item, depending on the person’s “ability”
and the item “difficulty”. Typically, the probability of success on an item for people
with varying ability is plotted as an “item characteristic curve” (ICC), as shown in
Figure 4.
Item Characteristic Curve
Probability of Success
1.0
"
!
0.5
#
0.0
!
Very low achievement
Very high achievement
Figure 4 An Example Item Characteristic Curve
Figure 4 shows that, for a high achiever (!), the probability of success on this item is
close to 1. For a low achiever (#), the probability of success on this item is close to
zero. For an average ability student ("), the probability of success is 0.5. The dotted
blue line shows the probability of success on this item at each ability level.
Under this model, the item difficulty for an item is defined as the level of ability at
which the probability of success on the item is 0.5. In the example given in Figure 4,
the ability level of the average person (!) is also the item difficulty of this item.
Defined in this way, the notion of item difficulty relates to the difficulty of the task
“on average”. Obviously for a very able person, the item in Figure 4 is very easy, and
for a low ability person, the item is difficult. But the item difficulty (!) is defined in
relation to the ability level of a person who has a 50-50 percent chance of being
successful on the item.
Figure 5 shows three item characteristic curves with varying item difficulty. It can be
seen that the item with the green ICC is the easiest item among the three, while the
13
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
item with the blue ICC is the most difficult. The item difficulties for the three items
are denoted by !1, !2, !3.
Variation in item difficulty
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
!3 !0 1 !1 2
-1
2
3
4
Figure 5 Three ICCs with Varying Item Difficulty
As the item difficulties are defined in relation to ability levels, both the item difficulty
and person ability are defined on the same scale. If we know a person’s ability, we
can predict how that person is likely to perform on an item, without administering the
item to the person. This is an advantage of using a mathematical function to model
the probability of success. Figure 6 shows an example of finding the probabilities of
success on three items if the ability of the person (") is known.
By defining item difficulty and person ability on the same scale, we can easily
construct interpretations for person ability “scores” in terms of the task demands.
Figure 7 shows an example. The person ability scale on the left and the item
difficulty scale on the right are linked through the mathematical function of
probability of success. If a student has an ability of ", one can easily compute this
student’s chances of success on items 1 to 6, with item difficulty !1, !2, …, !6,
respectively. As one can describe the underlying skills required to answer each item
correctly, one can easily describe a student’s level of proficiency once we have
located the student on the scale.
14
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Probabilities of success for a person
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
"1
2
3
4
Figure 6 Probabilities of Success for a Person at an Ability Level
Comparing Students and Items
Task Difficulties
More able
Location of a student
(")
(!1)
1
(!2)
2
(!3)
3
arithmetic with
vulgar fractions
(!4)
(!5)
4
5
multi-step
arithmetic
(!6)
6
single digit
addition
word problems
Less able
Figure 7 Linking Students and Items through an IRT scale
15
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Additional Notes
IRT Viewed as a Transformation of Raw Scores
The Rasch model is a particular IRT model. The Rasch model can be viewed as
applying a transformation to the raw scores so that distances between the locations of
two people can be preserved, independent of the particular items administered. The
curved line in Figure 2 will be “straightened” through this transformation. Figure 8
shows an example. Note that the distance between A and C on the easy test
(horizontal axis) is the same as the distance between A and C on the hard test (vertical
axis). However, the absolute values of the Rasch scores for an individual may not be
the same for the easy test and the hard test, but the relative distances between people
are constant.
4
Rasch Score on Hard Test
3
2
A
!A
1
0
"C
"D
C
-1 D
-2
-3
#B
-4 B
B
-5
-4
-3
A
DC
-2
-1
0
1
2
3
4
Rasch Score on Easy Test
Figure 8 Plot of Student Rasch Scores on an Easy Test and a Hard Test
A number of points can be made about IRT (Rasch) transformation of raw scores:
#
The transformation preserves the order of raw scores. That is, Rasch scores do
not alter the ranking of people by their raw scores. Technically, the
transformation is said to be monotonic. If one is only interested in ordering
students in ability, or items in difficulty, then raw scores will serve just as well.
No IRT is needed.
#
There is a one-to-one correspondence between raw scores and Rasch scores.
That is the pattern of correct/incorrect responses does not play a role in
determining the Rasch score.
#
The correlation between raw score and Rasch score will be close to 1, as a
result of the property of the Rasch model.
16
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
How about other transformations of raw scores, for example,
standardised score (z-score) and percentile ranks? Do they preserve
“distances” between people?
Using classical test theory approach, raw scores are sometimes transformed to zscores or percentile ranks. Some people have raised the question whether these
transformations have the property of preserving “distances” between the locations of
people on an achievement scale.
For z-scores, a transformation is applied to make the mean of the raw scores zero, and
the standard deviation 1. This transformation is linear, so that the relative distance
between two points will be the same whether raw scores or z-scores are used. For
example, if A and C are further apart than C and B in raw scores, then the z-scores
will also reflect the same relative difference. Consequently, z-scores suffer from the
same problem as raw scores. That is, z-scores on an easy test and a hard test will not
necessarily preserve the same relative distances between students.
Transforming raw scores to percentile ranks will solve the problem of producing
differing distances between two people on two different tests. This is because
percentile ranks have relinquished the actual distances between people, and turned the
scores to ranks (ordering) only. So, on the one hand, the percentile ranks of people on
two different tests may indeed be the same, on the other hand, we have lost the actual
distances between people! Raw scores, while not quite providing an interval scale,
offer more than just ordinal scales.
Exercises
In SACMEQ, item response modelling was used to produce student ability estimates.
Suppose that the data fit the item response model, do you agree or disagree with each
of the following statements:
(1) Students with the same ability estimate are likely to have similar patterns of
correct/incorrect answers.
(2) The ability estimates have the property of interval measurement. That is, the
difference in ability estimates between two students provides an estimate of how far
apart the two students are in ability.
(3) A transformation was applied to the IRT ability estimates so that the mean score
across all countries was 500 and the standard deviation was 100. This transformation
preserved the interval property of IRT scores.
17
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Chapter Three: Developing Tests From IRT Perspectives –
Construct and Framework
What is a Construct?
In Chapter One, the terms "latent trait" and "construct" are used to refer to the psychosocial attributes that are of interest to be measured. How are "constructs" conceived
and defined? Can a construct be any arbitrarily defined concept, or does a construct
need to have specific properties in terms of measurement? First, let's discuss what a
construct is. Consider the following example.
I am a regular listener of the radio station RPH (Radio for the Print Handicapped).
The listeners of RPH are constantly reminded that “1 in 10 in our population cannot
read print”. This statement raises an interesting question for me. That is, if I want to
measure people’s ability to read print, how would I go about it? And how does this
differ from the ‘reading abilities’ we are accustomed to measure through achievement
tests?
To address these questions, the starting point is to clearly explicate the “construct” of
such a test. Loosely speaking, the construct can be defined as “what we are trying to
measure”. We need to be clear about what it is that we are trying to measure, before
we start developing a test instrument.
In the case of RPH radio station, my first impression is that this radio station is for
vision-impaired people. Therefore to measure the ability to read print, for the purpose
of assessing the targeted listeners of RPH, is to measure the degree of vision
impairment of people. This, no doubt, is an over simplified view of the services of
RPH. In fact, RPH can also serve those who have low levels of reading ability and do
not necessarily have vision impairment. Furthermore, people with low levels of
reading achievement but also a low level of the English language would not benefit
from RPH. For example, migrants may have difficulties to read newspapers, but they
will also have difficulties in listening to broadcasts in English. There are also people
like me, who spend a great deal of time in traffic jams, and who find it easier to
“listen” to newspapers than to “read” newspapers.
Thus the definition of “the ability to read print”, for RPH, is not straightforward to
define. If ever a test instrument is developed to measure this, the construct needs to
be carefully examined.
Linking Validity to Construct
From the above example, it is clear that the definition of the construct is closely
linked to validity issues. That is, the inferences made from test scores and the use of
test scores should reflect the definition of the construct. In the same way, when
constructs are defined, one should clearly anticipate the way test scores are intended
to be used, or at least make clear to test users the inferences that can be drawn from
test scores.
There are many different purposes for measurement. A class teacher may set a test to
measure the extent to which students have learned two science topics taught in a
semester. In this case, the test items will be drawn from the material that was taught,
and the test scores will be used to report the proportion of knowledge/skills the
students have acquired from class instructions in that semester. In this case, the
18
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
construct of the test will be the material that was taught in class. The test scores will
not be used to reflect general science ability of the students.
In developing state-wide achievement tests, it is often the case that the content, or
curriculum coverage, is used to define test construct. Therefore one might develop a
mathematics test based on the Curriculum Standards Framework. That is, what is
tested is the extent to which students have attained the intended mathematics
curriculum. Any other inferences made about the test scores such as the suitability for
course entry, employment, or general levels of mathematics literacy, will need to be
treated with caution.
What if one does want to make inferences about students’ abilities beyond the set of
items in a test? What assumptions will need to be made about the test and test items
so one can provide some generalisations of students’ scores? Consider the PISA
(Programme for International Student Assessment) tests, where the constructs were
not based on school curricula, can one make statements that the PISA scores reflect
the levels of general mathematics, reading and science literacy? What are the
conditions under which one can make inferences beyond the set of items in a test?
The short answer is that item response theory helps us to link the construct to the kind
of inferences that we can make.
Construct and Item Response Theory (IRT)
The notion of a construct has a special meaning in item response theory. Under the
approach of the classical test theory, all inferences are made about a student’s true test
score on a test. There is no generalisation about the level of any “trait” that a person
might possess. Under the approaches of IRT, a test sets out to measure the level of a
latent trait in each individual. The item responses and the test scores reflect the level
of this trait. The trait is “latent”, because it is not directly observable. Figure 9 shows
a latent trait model under the IRT approach.
Latent
Variable
1
!
2
!
3
!
Other stuff
A Big Idea
4
!
5
!
6
!
Little Ideas
Figure 9 Latent Variables and Manifest (Observable) Variables 19
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
In Figure 9, the latent variable is the construct to be measured. Some examples of the
latent variable could be proficiency in geometry, asthma severity, professional status
of teachers, familiarity with sport, etc. Since one cannot directly measure a latent
variable, “items” will need to be devised to tap into the latent variable. A person’s
response on an item is observable. In this sense the items are sometimes known as
“manifest variables”. Through a person’s item response patterns, we can make some
inferences about a person’s level on the latent variables. The items represent little
ideas based on the bigger idea of the latent variable. For example, if the latent
variable is proficiency in geometry, then the items are individual questions about
specific knowledge or skills in geometry.
The arrows in Figure 9 indicate that the level of the latent variable determines the
likely responses to the items. It is important to note the direction of the arrows. That
is, the item response pattern is driven by the level of the latent variable. It is not the
case that the latent variable is defined by the item responses. For example, the
consumer price index (CPI) is defined as the average price of a fixed number of
goods. If the prices of these goods are regarded as items, then the average of the
prices of these items defines CPI. In this case, CPI should not be regarded as a latent
variable. Rather, it is an index defined by a fixed set of some observable entities. We
cannot change the set of goods and still retain the same meaning of CPI. In the case
of IRT, since the level of the latent variable determines the likelihood of the item
responses, the items can be changed, for as long as all items tap into the same latent
variable, and we will still be able to measure the level of the latent variable.
Another way to distinguish between classical test theory and item response theory is
that, under classical test theory, we only consider the right-hand side of the picture
(little ideas) of Figure 9 as shown in Figure 10.
1
2
3
Total
score
4
5
6
Little Ideas
Figure 10 Model of Classical Test Theory
Consequently, under classical test theory, we can only make inferences about the
score on this set of items. We cannot make inferences about any latent trait
20
!
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
underlying these items, since the model does not make any assumptions about latent
trait. As a result, we cannot make inferences beyond the set of items being tested.
In contrast, under item response theory, the set of items are meant to tap into one
latent trait. For as long as we use items that tap into this latent trait, we can exchange
items in the test and still measure the same latent trait. Of course, this relies on the
assumption that the items used indeed all tap into the same latent trait. This
assumption needs to be tested before we can claim that the total test score reflects the
level of the latent trait. That is, we need to establish whether arrows in Figure 9 can
be placed from the latent variable to the items. It may be the case that some items do
not tap into the latent variable, as shown in Figure 11.
bad “fit”
Latent
Variable
1
!
2
!
3
!
Other stuff
A Big Idea
4
!
5
!
6
!
Little Ideas
Figure 11 Test Whether Items Tap into the Latent Variable
Uni-dimensionality
The IRT model shown in Figure 9 shows that there is one latent variable and all items
tap into this latent variable. We say that this model is uni-dimensional, in that there is
ONE latent variable of interest, and the level of this latent variable is the focus of the
measurement. If there are multiple latent variables to be measured in one test, and the
items tap into different latent variables, we say that the IRT model is multidimensional. Whenever test scores are computed as the sum of individual item
scores, there is an implicit assumption of uni-dimensionality. That is, for aggregated
item scores to be meaningful, all items should tap into the same latent variable.
Otherwise, an aggregated score is un-interpretable, because the same total score for
students A and B could mean that student A scored high on latent variable X, and low
on latent variable Y, and vice versa for student B.
The Nature of the Construct – Psychological Trait or Arbitrary Construct?
The theoretical notion of latent traits as shown in Figure 9 seems to suggest that there
exists distinct “abilities” (latent traits) within each person, and the construct must
reflect one of these distinct abilities for the item response model to hold. This is not
necessarily the case.
21
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Consider the following example. Reading and mathematics are considered as
different latent variables in most cases. That is, a student who is good at reading is
not necessarily also good at mathematics. So in general, one would not administer
one test containing both reading and mathematics items and compute a total score for
each student. Such a total score would be difficult to interpret.
However, consider the case of mathematical problem solving, where each problem
requires a certain amount of reading and mathematics proficiencies to arrive at an
answer. If a test consists of problem solving items where each item requires the same
“combination” of reading ability and mathematics ability, the test can still be
considered “uni-dimensional”, with a single latent variable called “problem solving”.
From this point of view, whether a test is “uni-dimensional” depends on the extent to
which the items are testing the same construct, where the construct can be defined as a
composite of abilities (Reckase, Ackerman & Carlson, 1988).
In short, latent variables do not have to correspond to the physical existence of
distinct “traits” or “abilities”. Latent variables are, in general, arbitrary constructs.
Practical Considerations of Uni-dimensionality
In practice, one is not likely to find two items that test exactly the same construct. As
all items require different, composite, abilities. So all tests with more than one item
are “multi-dimensional”, to different degrees. For example, the computation of “7 "
9” may involved quite different cognitive processes to the computation of “27 + 39”.
To compute “7 " 9”, it is possible that only recall is required for those students who
were drilled on the “Times Table”. To compute “27 + 39”, some procedural
knowledge is required. However, one would say that these two computational items
are still closer to each other for testing the same construct as, say, solving a crossword
puzzle. So in practice, the dimensionality of a test should be viewed in terms of the
practical utility of the use of the test scores. For example, if the purpose of a test is to
select students for entering into a music academy, then a test of “music ability” may
be constructed. If one is selecting an accompanist for a choir, then the specific ability
of piano playing may be the primary focus. Similarly, if an administrative position is
advertised, one may administer a test of “general abilities” including both numeracy
AND literacy items. If a company public relations officer is required, one may focus
only on literacy skills. That is, the degree of specificity of a test depends on the
practical utility of the test scores.
Theoretical and Practical Considerations in Reporting Sub-scale Scores
In achievement tests, there is still the problem of how test scores should be reported in
terms of cognitive domains. Typically, it is perceived to be more informative if a
breakdown of test scores is given, so that one can report on students’ achievement
levels in sub-areas of cognitive domains. For example, a mathematics test is often
reported by an overall performance on the whole test, and also by performances on
mathematics sub-strands such as Number, Measurement, Space, Data, etc. Few
people query about the appropriateness of such reporting, as this matches with
curriculum classifications of mathematics. However, when one considers reporting
from an IRT point of view, there is an implicit assumption that whenever sub-scales
are reported, the sub-scales relate to different latent traits. Curriculum classifications,
in general, take no consideration of latent traits. Furthermore, since sub-scale level
reporting implies that the sub-scales cannot be regarded as measuring the same latent
22
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
trait, it will be theoretically incorrect to combine the sub-scales as one measure of
some latent trait. This theoretical contradiction, however, is generally ignored in
practice. One may argue that, since most cognitive dimensions are highly correlated
(e.g., Adams & Wu, 2002), one may still be able to justify the combination of subscales within a subject domain.
Summary
In summary, the development of a framework is essential before test construction. It
is not only for satisfying protocols. It is a step to establish clearly in our minds what
we are trying to measure. Furthermore, if we want to make inferences beyond
students’ performances on the set of items in a test, we need to make more
assumptions about the construct. In the case of IRT, we begin by relating the
construct of a test to some latent trait, and we develop a framework to provide a clear
explication of this latent trait.
It should be noted that there are two sides of the coin that we need to keep in mind.
First, no two items are likely to measure exactly the same construct. If the sample
size is large enough, all items will show misfit when tested for unidimensionality.
Second, while it is impossible to find items that measure the same construct, cognitive
abilities are highly correlated so that, in practice, what we should be concerned with is
not whether a test is unidimensional, but whether a test is sufficiently unidimensional
for our purposes. Therefore, it is essential to link the construct to validity issues in
justifying the fairness of the items, and the meaningfulness of test scores.
References
Adams, R. J., & Wu, M. L. (2002). PISA 2000 technical report. Paris: OECD.
Reckase, M. D., Ackerman, T. A., & Carlson, J. E. (1988). Building a unidimensional test
using multidimensional items. Journal of Educational Measurement, 25, 193-203.
Discussion Points
(1) In many cases, the clients of a project provide a pre-defined framework,
containing specific test blueprints, such as the one shown in Figure 12.
23
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
FINAL FORM MATRIX
Yr 3
Links
3/5
Yr 5
Links
5/7
Yr 7
14
5
16
5
17
Space
8
2
9
2
10
Measurement
8
2
9
2
10
Chance & Data
4
2
6
2
6
34
11
40
11
43
Number
Total
Figure 12 Example Client Specifications for a Test
These frameworks and test blueprints were usually developed with no consideration
of the latent trait model. So when we assess items from the perspective of item
response models, we often face a dilemma whether to reject an item because the item
does not fit the latent trait model, but yet the item belongs to part of the blueprint
specified by the clients. How do we reconcile the ideals of measurement against
client demands?
(2) To what extent do we make our test “uni-dimensional”? Consider a spelling test.
Spelling words generally have different discriminating power, as shown in the
following examples.
Spelling word:
(heart)
Categories
Count
Percent (%)
Pt-Biserial
Mean Ability
Infit MNSQ = 0.85
Disc = 0.82
0 [0]
1 [1]
13
39
25.0
75.0
-0.82
0.82
-0.08
3.63
Spelling word:
(discuss)
Categories
Count
Percent (%)
Pt-Biserial
Mean Ability
Infit MNSQ = 1.29
Disc = 0.49
0 [0]
1 [1]
40
42
48.8
51.2
-0.49
0.49
0.76
2.40
Can we select only spelling words that have the same discriminating power to ensure
we have “unidimensionality”, and call that a spelling test? If we include a random
sample of spelling words with varying discriminating power, what are the
consequences in terms of the departure from the ideals of measurement?
24
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
(3) Can we assume that the developmental stages from K to 12 form one
unidimensional scale? If not, how do we carry out equating across the year levels?
Exercises
In SACMEQ, some variables were combined to form a composite variable. For
example, the following seven variables were combined to derive a composite score,
ZPHINT:
24.
How often does a person other than your teacher make sure that you
have done your homework?
(Please tick only one box.)
PHMWKDON
(1)
(2)
(3)
(4)
25.
Never
Sometimes
Most of the time
How often does a person other than your teacher usually help you
with your homework?
(Please tick only one box.)
PHMWKHLP
(1)
(2)
(3)
(4)
26.
I do not get any homework.
I do not get any homework.
Never
Sometimes
Most of the time
How often does a person other than your teacher ask you to read to
him/her?
(Please tick only one box.)
PREAD
Never
(1)
(2)
(3)
Sometimes
Most of the time
25
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
27.
How often does a person other than your teacher ask you to do
mathematical calculations?
(Please tick only one box.)
PCALC
(1)
(2)
(3)
28.
(2)
(3)
Most of the time
Never
Sometimes
Most of the time
How often does a person other than your teacher ask you questions
about what you have been doing in Mathematics?
(Please tick only one box.)
PQUESTM
(1)
(2)
(3)
30.
Sometimes
How often does a person other than your teacher ask you questions
about what you have been reading?
(Please tick only one box.)
PQUESTR
(1)
29.
Never
Never
Sometimes
Most of the time
How often does a person other than your teacher look at the work that
you have completed at school?
(Please tick only one box.)
PLOOKWK
(1)
(2)
(3)
Never
Sometimes
Most of the time
26
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
The composite score, ZPHINT, is an aggregate of the above seven variables.
Q1. In the context of IRT, the value of ZPHINT can be regarded as reflecting the
level of a construct, where the seven individual variables are manifest variables. In a
few lines, describe what this construct is.
Q2. For the score of the composite variable to be meaningful and interpretable in the
context of IRT, what are the underlying assumptions regarding the seven manifest
variables?
27
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Chapter Four: The Rasch Model (the dichotomous case)
The Rasch Model
Item response models typically apply a mathematical function to model the
probability of a student’s response to an item, as a function of the student’s “ability”
level. This probability function, known as item characteristic curve, typically has an
“S” shape as shown in Figure 13.
Item Characteristic Curve for An Item
Probability of Success
1.0
"
!
0.5
#
0.0
!
Very low achievement
"
Very high achievement
Figure 13 An Example Item Characteristic Curve
In the case of the Rasch model (Rasch, 1960), the mathematical function of the item
characteristic curve for a dichotomous1 item is given by
p ' P# X ' 1$ '
exp#" % ! $
1 & exp#" % ! $
(4.1)
where X is a random variable indicating success or failure on the item. X=1 indicates
success (or a correct response) on the item, and X=0 indicates failure (or an incorrect
response) on the item.
" is a person-parameter denoting the person’s ability on the latent variable scale, and
! is an item-parameter, generally called the item difficulty, on the same latent
variable scale.
Eq. (4.1) shows that the probability of success is a function of the difference between
a person’s ability and the item difficulty. When the ability equals the item difficulty,
the probability of success is 0.5.
Re-arranging Eq. (4.1), it is easy to demonstrate that
1
A dichotomous item is one where there are only two response categories (correct and incorrect).
28
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
- p *
log++
(( ' " % !
,1 % p )
(4.2)
Equation (4.2) shows that, " % ! , the distance between a person’s ability and the item
difficulty, is expressed as the logarithm of the odds2 of success of the person on the
item. This is the reason that the meaurement unit of the scale for ability and item
difficulty is generally known as “logit”, a contraction of “log odds unit”. More
generally, one can think of the ability score in logits as a transformation of the
percentage correct, in much the same way as other scaled scores which are
transformations of the raw scores.
Additional Notes
Many IRT models use the logistic item response function (e.g., Embretson & Reise,
2000; van der Linden & Hambleton, 1997). The choice of the item response function
is not simply for mathematical convenience. There are sound theoretical reasons why
item response data may follow the logistic model (e.g., Rasch, 1960; Wright, 1977).
It has also been shown empirically that item response data do generally fit the logistic
model (e.g., Thissen & Wainer, 2001). In addition to logistic functions, the normal
ogive function has also been used (Lord & Novick, 1968; Samejima, 1977). In
general, the normal ogive model can be approximated by the logistic item response
model (Birnbaum, 1968).
Properties of the Rasch Model
Specific Objectivity
Rasch (1977) pointed out that the model specified by Eq. (4.1) has a special property
called specific objectivity. The principle of specific objectivity is that comparisons
between two objects must be free from the conditions under which the comparisons
are made. For example, the comparison between two persons should not be
influenced by the specific items used for the comparison. To demonstrate this
principle, consider the log odds for two persons with abilities " 1 and " 2 on an item
with difficulty ! . Let p1 be the probability of success of person 1 on the item, and
p2 be the probability of success of person 2 on the item.
- p *
log++ 1 (( ' " 1 % !
, 1 % p1 )
- p *
log++ 2 (( ' " 2 % !
, 1 % p2 )
2
(4.3)
Odds ratio is the ratio of the probability of success over the probability of failure.
29
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
The difference between the log odds for the two persons is given by
- p *
- p *
log++ 1 (( % log++ 2 (( ' " 1 % ! % #" 2 % ! $ ' " 1 % " 2
, 1 % p2 )
, 1 % p1 )
(4.4)
Eq. (4.4) shows that the difference between the log odds ratios for two persons
depends only on the ability parameters and not on the item parameter. That is,
irrespective of which items are used to compare two persons, the difference between
the log odds for the two persons is the same.
Similarly, it can be demonstrated that the comparison between two items is personfree. That is, the difference between the log odds ratios for two items is the same
regardless of which person took the two items.
Some psychometricians regard this sample-free property of the Rasch model as most
important for constructing sound measurements, because statements can be made
about relative item difficulties without reference to specific persons, and similarly
statements can be made about relative proficiencies of people without reference to
specific items. This item- and person-invariance property does not hold for other IRT
models.
Indeterminacy of An Absolute Location of Ability
Eq (4.1) shows that the probability of success of a person on an item depends on the
difference between ability and item difficulty, " % ! . If one adds a constant to
ability " , and one adds the same constant to item difficulty ! , the difference " % !
will remain the same, so that the probability will remain the same. Consequently, the
logit scale does not determine an absolute location of ability and item difficulty. The
logit scale only determines relative differences between abilities, between item
difficulties, and between ability and item difficulty. This means that, in scaling a set
of items to estimate item difficulties and abilities, one can choose an arbitrary origin
for the logit scale, and that the resulting estimates are subject to a location shift
without changing the fit to the model.
To emphasise further this indeterminacy of the absolute location of ability and item
difficulty estimates, one must not associate any interpretation to the logit value
without making some reference to the nature of the origin of the scale, however it was
set. For example, if an item has a difficulty value of 1.2 logits from one scaling, and a
different item has a difficulty value of 1.5 logits from another scaling, one cannot
make any inference about the relative difficulties of the two items without examining
how the two scalings were performed in terms of setting the origins of the scales.
30
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Additional Notes
I cannot stress this point more, as problems have occurred in the past such as in the
use of benchmark logits. If a benchmark logit was set at, say –1.2 logits, from one
scaling of item response data, this benchmark logit cannot be applied to any future
scalings of item response data unless these scalings adopt the same origin as the one
when the benchmark logit was derived. This can be achieved through linking the
instruments and equating processes. That is, a benchmark logit value does not have
any absolute meaning.
Equal Discrimination
Under the Rasch model, the theoretical item characteristic curves for a set of items in
a test are all parallel, in the sense that they do not cross, and that they all have the
same shape except for a location shift, as shown in Figure 14. This property is known
as equal discrimination or equal slope parameter. That is, each item provides the
same discriminating power in measuring the latent trait of the objects.
Theoretical ICCs for Three Items
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
!3 !0 1 !1 2
-1
2
3
4
Figure 14 Three Example ICCs with Varying Item Difficulty
Indeterminacy of An Absolute Discrimination
While the Rasch model models all items in a test with the same “discrimination” (or
the same “slope”), the Rasch model does not specify an absolute value for the
discrimination. For example, Figure 15 shows two sets of items with different
discriminating power. While items within each set have the same “slope”, Set 2 items
31
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
are more discriminating than Set 1 items when administered to the same group of
people.
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
Figure 15 Two Sets of Items with Different Discriminating Power
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.2
0.3
0.1
0.2
0
0.1
0
Figure 16 Two Sets of Items, after Rasch Scaling
When each set of items is scaled using the Rasch model, the slope parameter of the
item characteristic curve is set to a “1”, so that the two sets of items appear to have
the same slope pictorially (Figure 16). However, students taking Set 2 items will have
ability estimates that are more spread out. (See the change in the scale of the
horizontal axes of the ICCs from Figure 15 to Figure 16). That is, the variance of the
ability distribution using Set 2 items will be larger than the variance of the ability
distribution when Set 1 items are used. Consequently, the reliability of a test using
Set 2 items will be higher.
However, Set 1 items fit the Rasch model equally as well as Set 2 items. But if the
two sets are combined in one test, the items will show misfit.
32
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Additional Notes
A Simulation Study on the Effect of Varying Item Discrimination
Data Set
Abilities for 1000 persons were drawn from a normal distribution with mean 0 and
standard deviation 1. Item responses to 22 items were generated for each of the 1000
persons. The first set of 11 items had item difficulty values of –2, -1.6, -1.2, -0.8,
-0.4, 0, 0.4, 0.8, 1.2, 1.6, 2.0 respectively, and a slope parameter of 1. The second set
of items had the same item difficulty values as for Set 1, but had a slope parameter of
2. More specifically, the generating probabilities of success for the two sets of items
are given by Equations (4.5) and (4.6) respectively.
p ' P# X ' 1$ '
exp#" % ! $
1 & exp#" % ! $
(4.5)
p ' P # X ' 1$ '
exp#2#" % ! $$
1 & exp#2#" % ! $$
(4.6)
That is, the items in the second set are more discriminating than the items in the first
set.
Results of Simulation
Two analyses were carried out, one using the first set of 11 items, and one using the
second set of 11 items. The results are summarized in Table 1 and Table 2.
Table 1 Mean, Variance and Reliability
Estimate of population mean
Estimate of population variance
Reliability of the 11-item test
Item Set 1 (less
discriminating items)
0.015
0.979
0.60
Item Set 2 (more
discriminating items)
0.049
4.006
0.79
Table 2 Item Parameters and Infit t statistics
Generating
item difficulty
value
-2
-1.6
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
Item Set 1 (less discriminating
items)
Estimate of
Infit t
item difficulty
-1.990
-1.538
-1.205
-0.773
-0.406
-0.026
0.323
0.826
1.281
0.2
0.5
-0.1
0.1
-0.2
-1.0
-0.9
0.5
0.6
33
Item Set 2 (more
discriminating items)
Estimate of
Infit t
item difficulty
-4.078
-3.214
-2.320
-1.654
-0.823
-0.063
0.762
1.610
2.558
0.3
-0.8
0.8
1.8
0.3
-2.1
0.1
1.3
0.1
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
1.6
2.0
1.595
1.913
-0.8
0.6
3.177
4.045
-0.5
0.3
From Table 1, it can be seen that, when a set of more discriminating items are used,
person abilities are spread out more than when less discriminating items are used.
The magnitudes of item difficulty estimates for Set 1 and Set 2 items also reflect this
difference. It is also interesting to note that, despite the differing slope parameters in
Sets 1 and 2, the infit t values showed no misfit in both sets.
Length of a logit
The above results show that the length of one unit “logit” does not have an absolute
meaning. Two people can be close together in terms of their abilities estimated from
one calibration of a test, and be further apart from the calibration of another test. How
far apart two people are on the ability scale depends on the discriminating power of
the items used. Clearly, less discriminating items have less power in separating
people in terms of their abilities, even when the items fit the Rasch model well.
It should be noted that, under the assumptions of the Rasch model, two sets of items
with differing discrimination power as shown in Figure 15 cannot be testing the same
construct, since, by definition, all items testing the same construct should have the
same discriminating power, if they were to fit the Rasch model.
However, in practice, the notion of equal discriminating is only approximate, and
items in a test often have varying discriminating power. For example, open-ended
items are often more discriminating then multiple-choice items. Therefore, we should
be aware of the implications of issues regarding the length of a logit, particularly
when we select items for equating purposes.
Raw scores as sufficient statistics
Under the Rasch model, there is a one-to-one correspondence between a person’s
estimated ability in logits and his/her raw score on the test. That is, people with the
same raw score will be given the same ability estimate in logits, irrespective of which
items they answered correctly. An explanation for this may be construed as follows:
if all items have the same discriminating power, then each item should have the same
weight in determining ability, whether they are easy or difficult items.
However, if two persons were administered different sets of items, raw scores will no
longer be sufficient statistics for their ability estimates. This occurs when rotated test
booklets are used, where different sets of items are placed in different booklets. It is
also the case when items with missing responses are treated as if the items were notadministered, so that people with different missing response patterns are regarded as
being administered different tests. Under these circumstances, the raw score will no
longer be sufficient statistic for the ability estimate.
So if you have found that the correlation between the raw scores and Rasch ability
estimates is close to 1 in a test, do not get over excited that you are onto some new
discovery. The Rasch model dictates this relationship! It does not show anything
about how well your items worked!
34
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Fit of Data to the Rasch Model
The nice properties of the Rasch model discussed so far only hold if the data fit the
model. That is, if the data do not fit the Rasch model, by applying a Rasch scaling,
the items will not work any better. Therefore, to claim the benefit of using the Rasch
model, the data must fit the model to begin with. Applying the Rasch model cannot
“fix” problematic items! From this point of view, the use of the Rasch model in the
pilot stage for selecting items is most important. If the item response data from the
final form of a test do not fit the Rasch model, the scale construction will not be valid
even when the Rasch model is applied.
References
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s
ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores
(pp.395-479). Reading, MA: Addison-Wesley.
Embretson, S. E., & Reise, S. P. (2000). Item response theory for psychologists. Mahwah,
NJ: Lawrence Erlbaum Associates.
Lord, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores. Reading,
MA: Addison-Wesley.
Rasch, G. (1960).
Probabilistic models for some intelligence and attainment tests.
Copenhagen: Danish Institute for Educational Research.
Samejima, F. (1977).
The use of the information function in tailored testing. Applied
Psychological Measurement, 1, 233-247.
Thissen, D., & Wainer, H. (2001). Test scoring. NJ: Lawrence Erlbaum Associates.
van der Linden, W. J., & Hambleton, R. K. (1997). Handbook of modern item response
theory. New York: Springer-Verlag.
Wright, B. D. (1977). Solving measurement problems with the Rasch model. Journal of
Educational Measurement, 14, 97-115.
Exercises
Task
In EXCEL, compute the probability of success under the Rasch model, given an
ability measure and an item difficulty measure. Plot the item characteristic curve.
Follow the steps below.
35
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Step 1
In EXCEL, create a spreadsheet with the first column showing abilities from -3 to 3,
in steps of 0.1. In Cell B2, type in a value for an item difficulty, say 0.8, as shown
below.
Step 2
In Cell B4, compute the probability of success: Type the following formula, as shown
=exp($A4-B$2)/(1+ exp($A4-B$2))
Step 3
Autofill the rest of column B, for all ability values, as shown
Step 4
Make a XY (scatter) plot of ability against probability of success, as shown below.
36
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
This graph shows the probability of success (Y axis) against ability (X axis), for an
item with difficulty 0.8.
Q1. When the ability equals the item difficulty (0.8 in this case), what is the
probability of success?
Step 5
Add another item in the spreadsheet, with item difficulty -0.3. In Cell C2, enter -0.3.
Autofill cell C4 from cell B4. Then autofill the column of C for the other ability
values.
Step 6
Plot the probability of success on both items, as a function of ability (hint: plot
columns A, B and C).
1.2
1
0.8
0.6
0.4
0.2
0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
37
3.0
4.0
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Q2. A person with ability -1.0 has a probability of 0.1418511 of getting the first item
right. At what ability does a person have the same probability of getting the second
item right?
Q3. What is the difference between the abilities of the two persons with the same
probability of getting the first and second item right?
Q4. How does this difference relate to the item difficulties of the two items?
Q5. If there is a very difficult item (say, with difficulty value of 2), can you sketch
the probability curves on the above graph (without computing it in EXCEL)? Check
your graph with an actual computation and plot in EXCEL.
38
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Chapter Five: The Rasch Model (the polytomous case)
Introduction
In some cases, item responses may reflect a degree of correctness in the answer to a
question, rather than simply correct/incorrect. To model these item responses, the
Partial Credit Model (PCM) (Masters, 1982) can be applied where item scores have
more than two ordered categories (polytomous items).
The partial credit model has been applied to a wide range of item types. Some
examples include the following
!
!
Likert type questionnaire items, such as strongly agree, agree, disagree, strongly
disagree.
Essay ratings, for example, on a scale from 0-5.
!
Items requiring multiple steps, such as a problem-solving item requiring students to
perform 2 separate steps.
!
Items where some answers are more correct than others. For example, if one is asked
who won the AFL (Australian Football League) grand final in 2004, then the answer
“Brisbane” is probably a better answer than “Richmond”, even both are incorrect3.
!
A “testlet” or “item bundle” consisting of a number of questions. The total number
correct for the testlet is modelled with the PCM.
Are all of the above item types appropriate for applying the PCM? How does one
interpret the PCM item parameters in relation to the different item types?
To make life more difficult, there are a number of different ways for the
parameterisation of PCM, and for constructing measures of “difficulty” in relation to
a partial credit item. A clear understanding of the “item difficulty” parameters in
PCM is important when described proficiency scales are constructed where meanings
are associated with the levels on the scale according to the “item locations” on the
scale.
The Derivation of the Partial Credit Model
It will be helpful to first describe the derivation of the PCM, to clarify the underlying
assumptions in a PCM.
Masters (1982) derived the PCM by applying the dichotomous Rasch model to
adjacent pairs of score categories. That is, given that a student’s score is k-1 or k, the
probability of being in score category k has the form of the simple Rasch model.
Consider a 3-category partial credit item, with 0, 1 and 2 as possible scores for the
item.
The PCM specifies that, conditional on scoring a 0 or 1, the probability of X=0 and
the probability of X=1 are given by
3
For those who are not familiar Aussie rule football, Brisbane played Port Adelaide in the grand final,
and lost. Richmond was at the bottom of the ladder for the 2004 season.
39
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
p 0 / 0,1 ( Pr " X ( 0 / X ( 0 or X ( 1# (
Pr " X ( 0 #
1
(
Pr " X ( 0 # ' Pr " X ( 1# 1 ' exp"% & $ 1 #
(5.1)
exp"% & $ 1 #
Pr " X ( 1#
(
Pr " X ( 0 # ' Pr " X ( 1# 1 ' exp"% & $ 1 #
(5.2)
Eq. (5.1) and Eq. (5.2) are in the form of the dichotomous Rasch probabilities.
p1 / 0,1 ( Pr " X ( 1 / X ( 0 or X ( 1# (
Similarly, conditional on scoring a 1 or 2, the probability of X=1 and the probability
of X=2 are given by
p1 / 1, 2 ( Pr " X ( 1 / X ( 1 or X ( 2 # (
Pr " X ( 1#
1
(
Pr " X ( 1# ' Pr " X ( 2 # 1 ' exp"% & $ 2 #
(5.3)
p 2 / 1, 2 ( Pr " X ( 2 / X ( 1 or X ( 2 # (
exp"% & $ 2 #
Pr " X ( 2 #
(
Pr " X ( 1# ' Pr " X ( 2 # 1 ' exp"% & $ 2 #
(5.4)
Eq. (5.3) and Eq. (5.4) are in the form of the dichotomous Rasch probabilities.
PCM Probabilities for All Response Categories
While the derivation of the PCM is based on specifying probabilities for adjacent
score categories, the probability for each score, when all score categories are
considered collectively, can be derived. The following gives the probability of each
score category for a 3-category (0, 1, 2) PCM.
p 0 ( Pr " X ( 0 # (
p1 ( Pr " X ( 1# (
1
1 ' exp"% & $ 1 # ' exp"2% & "$ 1 ' $ 2 ##
exp"% & $ 1 #
1 ' exp"% & $ 1 # ' exp"2% & "$ 1 ' $ 2 ##
p 2 ( Pr " X ( 2 # (
exp"2% & "$ 1 ' $ 2 ##
1 ' exp"% & $ 1 # ' exp"2% & "$ 1 ' $ 2 ##
(5.5)
(5.6)
(5.7)
More generally, if item i is a polytomous item with score categories 0, 1, 2, …, mi ,
the probability of person n scoring x on item i is given by
x
Pr " X ni ( x # (
exp ) "% n & $ ik #
mi
k (0
h
) exp ) "%
h (0
(5.8)
n
& $ ik #
k (0
0
where we define exp ) "% n & $ ik # ( 1 .
k (0
40
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Some Observations
Dichotomous Rasch model is a special case
Note that the simple dichotomous Rasch model is a special case of the PCM. For this
reason, software programs that can fit the PCM can generally fit the dichotomous
model without special instructions to distinguish between the dichotomous model and
PCM. Dichotomous and partial credit items can generally be “mixed” in one analysis.
The score categories of PCM are “ordered”
The score categories 0, 1, 2, …, m, of a PCM item should be “ordered” to reflect
increasing competence of some trait. Under the PCM, there is an assumption that
students with higher abilities are more likely to score higher for the item.
Consider the lowest two score categories: 0 and 1. Since the simple dichotomous
Rasch model applies if we consider the case where the score categories are only 0 and
1. Then students with higher abilities are more likely to achieve a score of 1 than 0.
By the same token, if we consider scores 1 and 2, then higher ability students are
more likely to achieve a score of 2 than 1. Consequently, when we consider all score
categories for a partial credit item, higher ability students are expected to score higher
than low ability students.
PCM is not a sequential steps model
The derivation of PCM simply specifies the “conditional probability” of two adjacent
score categories. The PCM does not make any assumption that there is an underlying
sequential step process to achieve a score. That is, there is no assumption that a
student must be successful in all tasks for lower score categories to achieve success in
tasks for a higher score. In fact, strictly speaking, the Steps model (Verhelst, Glas and
de Vries, 1997) should be used for items where students cannot achieve a higher score
unless tasks for lower scores are successfully completed (a sequential step process).
This observation is important for the interpretation of the item parameters, $ k . In the
above example where there are 3 score categories, the parameter, $ 2 , does not reflect
the item difficulty of being successful in both “steps”, or for achieving a score of 2.
Nor does $ 2 reflect the item difficulty for the second “step” as an independent step.
The interpretation of $ k
The derivation of the PCM, based on the simple Rasch model for adjacent score
categories, leads to the misconception that $ k is the difficulty parameter for step k,
had step k been administered as an independent item. The interpretation of $ k can be
clarified graphically through the item characteristic curves.
Item Characteristic Curves (ICC) for PCM
Item characteristic curves for a partial credit item are plots of the probabilities of
being in each score category, as a function of the ability, % . Figure 17 shows
example item characteristic curves for a 3-category partial credit item.
41
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Category 2
Category 0
Category 1
Figure 17 Theoretical Item Characteristic Curves for a 3-category Partial Credit Item
From Figure 17, it can be seen that as ability increases, the probability of being in a
higher score category also increases.
Graphical interpretation of the delta ( $ ) parameters
Score 2
Score 0
Score 1
$1
$2
Figure 18 Graphical representations of the delta ( $ ) parameters
Mathematically, it can be shown that the delta ( $ ) parameters in Eq. (5.1) to (5.4)
are the abilities at which adjacent ICCs intersect. That is, $ k is the point at which the
42
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
probability of being in category k-1 and category k is equal4. This mathematical fact
provides an interpretation for the delta ( $ ) parameters. Figure 18 shows a 3category partial credit item. It can be seen that the two delta parameters, $ 1 and $ 2 ,
divide the ability continuum into three regions. From & * to $ 1 , the most likely
single score category is “0”. Between $ 1 and $ 2 , the most likely single score
category is “1”. When the ability of a student is above $ 2 , the most likely single
score category is “2”.
The phrase “the most likely single score category” is used to stress that it is the most
likely score category when each individual score category is considered. For
example, in Figure 18, between $ 1 and $ 2 , score 1 has a higher probability than score
0 or score 2. However, the combined probability of scores 0 and 2 is higher than the
probability of score 1. Since the probability of score 1 is less than 0.5 between $ 1 and
$ 2 , so the combined probability of scores 0 and 2 must be more than 0.5, in this
example.
Consequently, if the delta ( $ ) parameters are used as indicators of “item difficulty”,
one might say that $ 1 is a point such that, beyond this point, the probability of
achieving a score of 1 is higher than the probability of achieving a score of 0.
Similarly, beyond $ 2 , the probability of achieving a score of 2 is higher than the
probability of achieving a score of 0 or 1.
Problems with the interpretation of the delta ( $ ) parameters
For some items, the delta ( $ ) parameters may not be ordered. Figure 19 shows an
example.
4
This probability is not 0.5, but less than 0.5, because the probability of being in categories other than
k-1 and k is not zero.
43
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Score 2
Score 0
Score 1
$2
$1
Figure 19 ICC for PCM where the delta parameters are dis-ordered
Figure 19 shows that the probability curve for the middle category, score 1, is very
flat, indicating that there are few students who are likely to score 1. On might say that
score 1 is not a very “popular” category. In this case, the interpretation of the ICCs
becomes more difficult, as score 1 is never the most likely single category for any
ability level, and that the parameters $ 1 and $ 2 are not ordered ( $ 1 > $ 2 ). This
phenomenon was one disadvantage of using the delta ( $ ) parameters to interpret
item responses in relation to ability.
Linking the graphical interpretation of $ to the derivation of PCM
Masters and Wright (1997) pointed out that the dis-ordering of the delta ( $ )
parameters was not necessarily an indication of a problematic item, since the
derivation of the partial credit model did not place any restriction on the ordering of
item parameters, $ . More specifically, the derivation of the PCM states that,
considering only students in score categories k-1 and k, the probability of being in
category k follows the Rasch model. Figure 20 shows an example ICC for the
conditional probability of score category k, given the score is either k-1 or k.
44
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
$
Figure 20 An example ICC of conditional probability between two adjacent score
categories
$ is the ability at which there is an equal probability of being in category k-1 or k. In
this case, the probability is 0.5, because we are only considering students with score
categories of k-1 and k.
When all score categories are considered in an ICC plot, such as that shown in Figure
18, the $ parameter is still the value at which adjacent score categories have equal
probability. However, the probably is no longer 0.5, since there is the possibility of
being in score categories other than k-1 and k. It can be seen from Figure 18 and
Figure 19 that the point of intersection of two adjacent categories will be dependent
on the relative chances of being in all categories. For example, in Figure 19, if the
probability of being in category 1 is small throughout the whole ability range (may be
due to an easy step “2”), then the point of intersection (equal probability) between
category 0 and 1 is likely to be a high value, and the intersection point between
category 1 and 2 is likely to be a low value.
It is clear then that the delta ( $ ) parameters are dependent on the number of students
in each category, and so $ cannot reflect “independent” step difficulty. Rather, the
values of $ will depend on the difficulties of all “steps”.
See Verhelst and
Verstralen (1997) for an example about the dependence between the delta ( $ )
parameters.
Delta ( $ ) parameters and different types of item responses
When the PCM is applied to items where score categories correspond to sequential
“steps” to solve a problem, the problem of dis-ordering of $ is likely to occur. This
is because that, very often, later steps are easy steps as compared to earlier steps. For
example, an item involving a first step of conceptualising the formulation and a
45
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
second step of carrying out computation will often result in most students being in the
0 category or the 2 category. That is, few students who successfully conceptualised
the formulation will make a computational mistake (Figure 21).
On the other hand, when the PCM is applied to holistic scoring rubrics such as those
used for essay marking, the problem of dis-ordering of $ is less likely to occur
(Figure 22).
Item 5 - pharm
Item analysis (Item 5 – pharm)
In the Pharmochem company, there are 57 employees.
Each employee speaks either German or English, or both.
25 employees can speak German and 48 employees can
speak English. How many employees can speak both
German and English? Show how you found your
answer.
------------------------------------------Response Score Count
% of tot Pt Bis
------------------------------------------16*
2
293
61.68
0.43
comp err
1
18
3.79
0.01
Other
0
117
24.63
-0.36
Discrimination=0.44
Infit=1.27
Comments:
Fully correct answer was given a score of 2. For
responses with correct method but incorrect
computation, a score of 1 was awarded.
*Correct answer
Figure 21 An item and corresponding ICC where two-steps are involved for PCM
scoring
46
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Figure 22 ICC for an essay marking criterion, “Cohesion”, using PCM on a 6-point
scale
Tau’s and Delta Dot
A variation of the parameterisation of the PCM is the use of + ’s (tau’s) and $ ! (delta
dot). Mathematically, the delta ( $ ik ) parameters in Eq. (5.8) can be re-written in the
following way:
Using the notations as in Eq. (5.8) but dropping the index i for simplicity, let
mi
$ ! ( )$ k m
(5.9)
k (1
That is, $ ! is the average of the delta ( $ k ) parameters.
Define + k as the difference between $ ! and $ k . That is,
+ k ( $! &$k
Graphically, the relationships between + k , $ ! and $ k
(Adams, 2002).
47
(5.10)
are illustrated in Figure 23
Probability
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
+2
Pr(0)
+4
+1
Pr(4)
+3
Pr(3)
Pr(1)
Pr(2)
-4
-3
-2
$-11
$02
$! $13
$24
3
4
Ability (logits)
Figure 23 Item Characteristic Curves for a Five-Category Item with Taus and Deltas
A worked example is given in Adams (2002).
The parameterisation of the PCM using $ ! and + k is mathematically equivalent to the
parameterisation using $ k . Using Eq. (5.9) and (5.10), one can compute $ ! and + k
from $ k . Conversely, given + k , and $ ! , one can compute $ k as
$ k ( $ ! &+ k
(5.11)
Interpretation of $ ! and + k
The parameter $ ! may be thought of as a kind of “average” item difficulty for a
partial credit item. This may be useful, if one wishes to have one indicative difficulty
parameter for a partial credit item as a whole. Otherwise, to describe the difficulty of
a partial credit item, one needs to describe the difficulties of individual steps, or
individual scores, within the item.
The + k parameters are more difficult to interpret as stand-alone values. These need to
be interpreted in conjunction with $ ! . That is, + k , as a “step parameter”, shows the
distance of a partial credit score category from the “average” item difficulty. The + k
parameters suffer from the same problem as $ k ’s, in that the + k ’s can be dis-ordered.
48
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Additional Notes
Mathematically, $ ! is the intersection point of the probability curves for the first and
last score categories of a partial credit item. For example, if there are 5 score
categories as shown in Figure 23, $ ! is the intersection point of the curves Pr(0) and
Pr(4).
In the case of a 3-category partial credit item, the curves Pr(0) and Pr(2) are
symmetrical about $ ! . That is, the curve Pr(0) is a reflection of the curve Pr(2) about
the line % ( $ ! , and the curve Pr(1) is symmetrical about the line % ( $ ! . This is not
usually the case when the number of score categories is more than 3. Some examples
are given below.
49
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Thurstonian Thresholds, or Gammas ( , )
As was discussed in previous sections, the delta ( $ ) parameters do not reflect the
difficulty of achieving a score point in a partial credit item. For partial credit items, to
achieve a score of 2, students would generally need to accomplish more tasks than for
achieving a score of 1. To reflect this “cumulative achievement”, the Thurstonian
thresholds are sometimes used as indicators of “score difficulties”.
The Thurstonian threshold for a score category is defined as the ability at which the
probability of achieving that score or higher reaches 0.50. Graphically, the
Thurstonian thresholds are shown in Figure 24.
Pr(-1)
Pr(-3)
Pr(-4)
Pr(-2)
,1
,3
,2
,4
Figure 24 Cumulative Probability Curves to show Thurstonian thresholds
Figure 24 shows cumulative probability curves for a 5-category partial credit item.
The blue curve shows the probability of achieving a score of 1 or more, as a function
of ability. The green curve shows the probability of achieving a score of 2 or more,
and so on.
Interpretation of Thurstonian thresholds
Consider Figure 24. Moving along the horizontal ability scale from & * to , 1 , the
probability of achieving a score of 1 or more is less than 0.5 (The blue curve is less
than 0.5 in this range). The probability of achieving a score of 0 is more than 0.5.
Therefore one might label the region from & * to , 1 as the “score 0” region. As the
ability increases from , 1 to , 2 , the probability of achieving a score of 1 or more is
more than 0.5 (the blue curve), but the probability of achieving 2 or more is less than
0.5 (the green curve). So one might label the region from , 1 to , 2 as “score 1”
region. In the same manner, we can label “score 2”, “score 3” and “score 4” regions.
From this point of view, Thurstonian thresholds can be viewed as cutpoints for
dividing up the ability continuum into “score regions”.
50
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
So, how do Thurstonian thresholds represent item score difficulties? Is , 1 a suitable
measure for the difficulty of score 1, or is the region between , 1 to , 2 a better
indication of score 1 “difficulty”? Should we use the mid-point between , 1 to , 2 as a
measure of score 1 difficulty?
Comparing with the dichotomous case regarding the notion of item difficulty
In the dichotomous case, item difficulty is defined as the ability at which the
probability of success on the item is 0.5. From this point of view, item difficulty for
the dichotomous case is also a threshold, and it divides the ability continuum into two
regions: score 0 and score 1 regions, and the item difficulty is the point where score 1
region starts. Extending this notion to the PCM, the Thurstonian thresholds can also
be regarded as “score difficulties”. That is, , 1 is a measure of score 1 difficulty, and
, 2 is a measure of score 2 difficulty, and so on. For example, if the Thurstonian
thresholds (in logits) for a 3-category item are –1.2 and 2.3, this suggests that it is
relatively easy to receive a score of 1, but relatively difficult to receive a score of 2.
In this case, the “score 1 region” is very wide.
Using Expected Scores as Measures of Item Difficulty
Another measure of item difficulty can be derived by computing the expected score
on an item, as a function of ability. Consider an item with 3 score categories. The
probabilities of scoring a 0, 1 or 2 are given by Eq. (5.5) to (5.7). The expected score,
E, on this item, as a function of the ability % and delta parameters $ 1 and $ 2 , is given
by
E ( 0 . Pr " X ( 0 # ' 1 . Pr " X ( 1# ' 2 . Pr " X ( 2 # ,
(5.12)
using the general formula for computing expectations. Computing E as a function of
% , one can construct an Expected Score Curve, similar to the item characteristic
curve. Figure 25 shows an example.
E1
E2
Figure 25 Expected Score Curve for a 3-Category Partial Credit Item
51
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Let E1 be the ability at which the expected score on this item is 0.5. Let E 2 be the
ability at which the expected score is 1.5. One might regard the region between E1
and E 2 as the “score 1 region”, and the ability continuum below E1 as the “score 0
region”, and the ability continuum above E 2 as the “score 2 region”. In this way, E1
could be regarded as an item difficulty parameter for score 1, and E 2 could be
regarded as an item difficulty parameter for score 2.
The advantage of using E1 and E 2 as indicators of difficulty is that the notion of
expected scores is readily comprehensible to the layman. In the case of Thurstonian
thresholds, the notion of cumulative probability is more difficulty to explain.
The problem is that, depending on which item difficulty measure you choose to use,
you get different values. Figure 26 and Figure 27 show the delta parameters and
Thurstonian thresholds for the item shown in Figure 25.
Figure 26 Delta Parameters for Item 6
52
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Figure 27 Thurstonian Thresholds for Item 6
In the case of 3-categories, it can be shown mathematically that the Thurstonian
thresholds are always "wider" than the deltas, if there are no reversals of the delta
values (For an example, see Figure 28).
3
2
1
delta1
delta2
0
threshold1
threshold2
-1
-2
-3
Figure 28 Comparisons of threshold and delta values for 25 items
53
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Additional Notes
Sum of Dichotomous Items and the Partial Credit Model
Verhelst and Verstralen (1997) showed that if a set of dichotomous items fit the Rasch
model, then the sum of individual item scores can be modeled using the partial credit
model. However, the converse is not true. Polytomous item scores fitting the partial
credit model cannot always be decomposed into individual Rasch item scores.
Verhelst and Verstralen made the following statement regarding using sum scores for
testlets5:
If the main purpose of the model construction is to determine % as accurate as
possible, no information with respect to % is lost if local independence is not violated;
if it is violated, the embarrassing implications are avoided by considering sums of
item scores. (p.12)
That is, if there is a reason to think that there is dependency between a set of items,
then a better way is to model the set of items as one partial credit item. The
dependency will be taken into account then. However, it will not be possible to match
the item parameters to individual items in the set.
5
A testlet is a set of items
54
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
References
Adams, R. J. (2002). Some Comments on Rasch Model Terminology. Notes distributed to the
Benchmarking Equating Steering Committee (BESC), June, 2002.
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149174.
Masters, G. N., & Wright, B. D. (1997). The partial credit model. In W. J. van der Linden &
R. K. Hambleton (Eds.), Handbook of modern item response theory (pp.101-121). New
York: Springer-Verlag.
Verhelst, N. D., Glas, C. A. W., & de Vries, H. H. (1997). A steps model to analyse partial
credit. In W. J. van der Linden & R. K. Hambleton (Eds.), Handbook of modern item
response theory (pp.123-138). NY: Springer-Verlag.
Verhelst, N. D., & Verstralen, H. H. F. M. (1997). Modeling sums of binary responses by the
partial credit model. Cito Measurement and Research Department Reports, 97-7. Cito.
55
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Chapter Six: Preparing data for Rasch analysis
After data have been collected, they need to be coded and entered into computer files
before they can be analysed.
Coding
It is really important to capture all responses in the data collected. In general, it is
easier to deal with numerical codes than text data. So a "codebook" needs to be
prepared to record all the coding schemes applied to the raw data. For example, the
following shows an excerpt of the codebook used in SACMEQ.
95
PSIT
PQ22 p/sitting place
96
PWRITE
PQ23 p/writing place
97
p/homeworkPHMWKDON PQ24
make sure
98
PHMWKHLP PQ25
99
PREAD
PQ26
100 PCALC
PQ27
101 PQUESTR
PQ28
102 PQUESTM
PQ29
103 PLOOKWK
PQ30
PQ31
.1
PQ31
.2
PQ31
.3
104 PEXTENG
105 PEXTMAT
106 PEXTOTH
107 PEXTPAY
PQ32
p/homeworkhelp
p/ask to read
p/ask to
calculate
p/questionreading
p/questionmath
p/look at work
p/extra tuitionsubject
p/extra tuitionsubject
p/extra tuitionsubject
p/extra tuitionpayment
1=floor, 2=log/stone/box/tin, 3=chair/bench/seat.
1=nowhere, 2=chair/bench/log/stone/box/tin, 3=desk/table.
1=no homework, 2=never, 3=sometimes, 4=most of time.
1=no homework, 2=never, 3=sometimes, 4=most of time.
1=never, 2=sometimes, 3=most of the time.
1=never, 2=sometimes, 3=most of the time.
1=never, 2=sometimes, 3=most of the time.
1=never, 2=sometimes, 3=most of the time.
1=never, 2=sometimes, 3=most of the time.
1=do not take, 2=take.
1=do not take, 2=take.
1=do not take, 2=take.
1=do not take extra tuition, 2=pay, 3=do not pay, 4=do not
know.
Missing and invalid responses
A code should be designated for missing responses. For example, "9" may be used
for missing, and "8" may be used for invalid response.
Multiple-choice items
For multiple-choice items, raw responses should be captured, not scores. For
example, in the following item, there are five options. Numerical codes from 1 to 5
If March 5th is a Wednesday, what day of the week is March 22nd?
A.
B.
C.
D.
E.
Monday
Thursday
Friday
Saturday
Sunday
56
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
can be used to record responses A to E respectively. "9" can be used to record
missing response. "8" can be used to record invalid responses. The correct answer
for this item is D. So students who answered D will score 1 and the others will score
0. But the scoring of multiple-choice items can be carried out within the item
response modelling program.
Open-ended items
In the case of open-ended test items, coding may be necessary. For example, consider
the following item.
John has 54 marbles and Peter has 28 marbles. How many marbles should John give Peter so that they
have the same number of marbles? Show your work.
Since student answers could cover a wide range of numbers, a coding scheme such as
the following could be devised:
Code 2: 13
Code 1: 26
Code 0: other numbers
Code 9: missing response
In this case, one has the opportunity to award partial credit score later. For example,
Code 2 could receive a score of 2, Code 1 a score of 1, and Code 0 a score of 0.
The treatment of missing responses varies. Sometimes these are treated as incorrect
responses, and sometimes as not-administered items. The decision on how to treat
missing responses depends on the test length and the purpose of the test. For
example, if the test construct is one where the time taken to complete a task is
relevant, then one may choose to score missing responses as incorrect. In other cases,
missing responses are distinguished between 'embedded missing' (skipped items), and
'not-reached' items (missing items at the end of a test). "Embedded missing" items
are always treated as incorrect.
"Not-reached " items are treated as "notadministered" for item calibration, but as "incorrect" for the calibration of abilities.
Scoring and coding
While scoring may be carried out later using other software programs, there are a
number of issues relating to scoring which may be taken into account when codes are
designed.
(1) If in doubt, create more coding categories than fewer categories to distinguish
between different responses. Categories can always be collapsed later. This allows
for the implementation of different possible scoring schemes to determine which
scoring scheme provides the best fit.
(2) Scores should reflect the level of the "latent trait" being measured, and not reflect
technical correctness of the answer. Consider the following example.
A rectangular room is 5m wide and 3.5 m long. What is the floor area?
57
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Sample student answers: 17.5m2; 17.5m, 17.5, 17.0m2 (=2!(5+3.5)), 8.5m2
Of the five sample student answers, the first three are correct in the computation of
the area. The first has the correct unit. The second has incorrect unit, so the answer is
technically incorrect. The third omits the unit, so the answer is still technically
correct. If we were to follow the rule of technical correctness, we would score the
first and third correct, and the second and others incorrect. Does this scoring scheme
reflect the level of mathematical competency in the students? In other words, we are
saying that students who answered 17.5m have as low mathematical competency as
those who answered 17.0 m2 or 8.5m2. A better scoring scheme would be to regard
the first three as equally correct, and the others as incorrect. Remember that the items
are used to estimate a student's level on the latent trait, so the scoring of the responses
should reflect that level.
Weighting of scores
In general, an item should not receive more scores just because it is a difficult item.
That is, the weighting of scores should not be dependent on the difficulty level of an
item. Rather, the weighting should depend on the discriminating power of an item.
For example, if an item does not discriminate well between low and high achievers,
then it should receive a lower score (less weight). If an item discriminates well
between objects with low and high levels on the latent trait, then it should receive a
higher score (more weight).
Data entry
The coded responses will need to be entered into computers. In general, item
response modelling software programs require the data to be in text files (ASCII
format), where each variable is entered at fixed columns in the file.
For large-scale surveys, specialised data entry programs are prepared to enable
efficient and accurate data entry. These programs typically have validation checks for
the range of values entered. For small-scaled surveys, data can be prepared directly in
a text editor (e.g., Notepad, Wordpad, or other more sophisticated text editors), or via
EXCEL or SPSS and then exported as text files.
!
A tutorial on the preparation of text files for the IRT software ConQuest can
be found through the link tutorial2\index.html. The preparation of data files
for other IRT software will be similar.
Exercises
Q1. The following shows a question and some sample answers. Based on the sample
answers, how would you design a codebook for this question?
What are the outside walls of the place (home) where you stay during the school week mostly made
of?
Sample answers: wood, cut stone, stones, concrete blocks, cardboard, canvas, reeds,
mudbricks, metal sheets, timber, bricks, Plastic sheeting, grass thatch, planks.
58
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Q2. The following shows a question and some sample answers. How would you
score the sample answers?
What is the capital city of the United States?
Sample answers: Washington; Washington DC; New York; Los Angeles; Washington CD;
London; Washington Columbia; Washington Maryland.
59
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Q3A. Administer, in class, the following questionnaire on job satisfaction and
prepare a data file in text format.
There are many things that measure a person's satisfaction with his/her work. How
satisfied are you with each of the following?
(Please tick the appropriate box for each statement.)
Not
satisfactory
Your travel distance to office
Satisfactory
Extremely
satisfactory
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
Quality of office furniture
(1)
(2)
(3)
Level of salary
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
Location of office
Quality of the office buildings
Availability of office furniture
Quality of office management and
administration
Amicable working relationships
with other staff members
Stimulating work
Opportunities for promotion
Opportunities for professional
development
60
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Q3B. Administer, in class, the following questionnaire on "the effort to have a
healthy diet", and prepare a data file in text format.
How often do you do the following?
(Please tick the appropriate box for each statement.)
Hardly ever
Sometimes
Most of the
time
Buy organically grown food
(1)
(2)
(3)
Eat fast food
(1)
(2)
(3)
Prepare vegetable with each meal
(1)
(2)
(3)
(1)
(2)
(3)
Buy low fat food
(1)
(2)
(3)
Buy low carbo-hydrate food
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
yes
(1)
(2)
no
(3)
<25
(1)
(2)
>40
(3)
Weigh yourself
Eat fruit each day
Look for healthy food recipes
Read ingredients on food
packaging
Drink alcohol
Have soft drink
Does your religion forbid you to
eat meat?
What is your age group?
61
2640
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Chapter Seven: Item Analysis Steps
After an instrument (a questionnaire or a test paper) has been developed and
administered, and data file prepared, the next step is to run item response modelling
software to estimate the level of latent trait for each subject and the difficulty of each
item. While IRT programs will produce estimates of modelled parameters, it is also
important to check whether the data fit the IRT model, so that the results produced by
the program are valid. Typically, there will be an iterative process of (1) running IRT
program (2) checking model fit (3) revising data or model (4) re-running IRT
program.
General principles of estimation procedures
From the item responses collected through a questionnaire or a test, two sets of
parameters are estimated: "person parameters" and "item parameters". Most IRT
programs use the following principles for estimating the parameters.
Recall that, in Chapter Four, "person parameters" ( ! ) and "item parameters" ( " ) are
defined on the same scale, as shown below.
Item Characteristic Curve for An Item
Probability of Success
1.0
!
0.5
"
#
0.0
!
Very low achievement
"
Very high achievement
Figure 29 An Example Item Characteristic Curve
That is, the item difficulty, " , is the ability at which there is a 50% chance for a
person with that ability to get the correct answer on the item. Therefore, if the
abilities of people are known, then it is easy to determine the item difficulty, " , by
examining groups of people at different ability levels, and finding the group that has
approximately 50% of the people getting the item right.
Similarly, if item difficulties are known, then one can look through the scores of a
person on all the items, arranged in difficulty order, and determine the location at
which the person is likely to get more items wrong than right. This could be an
estimate of the person's ability.
62
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Since neither person nor item parameters are known, one could go about estimating
the parameters in an iterative way. First, make some initial guesses for the item
parameters. Pretending that these are the actual item parameters, one could proceed
to estimate person parameters. Once the person parameters are obtained, take these as
the true person parameters and estimate item parameters again. In this way, the
parameters will improve with each iteration, until they converge to stable values.
$
A practical session will be conducted at this point to show how an IRT
program is run. A tutorial on how to run ConQuest can be found through the
link tutorial1\index.html
Typical output of IRT programs
IRT programs will generally provide a table of estimated item parameters (see Figure
30), and a table of estimated person parameters, as well as information about the fit of
the data to the model, and other characteristics of the items.
=========================================================================================
ConQuest: Generalised Item Response Modelling Software
Wed Sep 17 17:55:15
TABLES OF RESPONSE MODEL PARAMETER ESTIMATES
=========================================================================================
TERM 1: item
----------------------------------------------------------------------------------------VARIABLES
UNWEIGHTED FIT
WEIGHTED FIT
----------------------------------------------------------item
ESTIMATE ERROR^
MNSQ
CI
T
MNSQ
CI
T
----------------------------------------------------------------------------------------1
BSMMA01
0.363
0.050
0.85 ( 0.91, 1.09) -3.4
0.88 ( 0.94, 1.06) -4.2
2
BSMMA02
-0.178
0.052
1.07 ( 0.91, 1.09) 1.6
0.97 ( 0.92, 1.08) -0.8
3
BSMMA03
-0.025
0.051
0.93 ( 0.91, 1.09) -1.7
0.95 ( 0.93, 1.07) -1.4
4
BSMMA04
0.838
0.049
0.95 ( 0.91, 1.09) -1.2
0.96 ( 0.95, 1.05) -1.8
5
BSMMA05
1.182
0.049
1.15 ( 0.91, 1.09) 3.2
1.09 ( 0.95, 1.05) 3.6
6
BSMMA06
-0.314
0.052
1.08 ( 0.91, 1.09) 1.8
1.03 ( 0.92, 1.08) 0.7
7
BSMSA07
-0.392
0.053
1.14 ( 0.91, 1.09) 3.1
1.06 ( 0.92, 1.08) 1.4
8
BSMSA08
-0.327
0.053
1.18 ( 0.91, 1.09) 3.7
1.11 ( 0.92, 1.08) 2.7
9
BSMSA09
-0.963
0.056
0.92 ( 0.91, 1.09) -1.7
1.00 ( 0.89, 1.11) 0.0
10 BSMSA10
-0.392
0.053
1.14 ( 0.91, 1.09) 3.0
1.08 ( 0.92, 1.08) 1.8
11 BSMSA11
-0.499
0.053
0.87 ( 0.91, 1.09) -2.9
0.95 ( 0.91, 1.09) -1.1
12 BSMSA12
0.707* 0.172
0.96 ( 0.91, 1.09) -0.9
0.99 ( 0.95, 1.05) -0.4
----------------------------------------------------------------------------------------An asterisk next to a parameter estimate indicates that it is constrained
Separation Reliability = 0.993
Chi-square test of parameter equality = 1503.366, df = 11, Sig Level = 0.000
^ Quick standard errors have been used
=========================================================================================
Figure 30 Example Table of Item Parameters from ConQuest
Item statistics should be examined to identify problematic items. The following
presents a description of the steps that should be carried out in item analysis. The
examples used relate to a questionnaire constructed to measure the level of women's
autonomy (Demographic and Health Surveys, Women's Questionnaire:
www.measuredhs.com).
63
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Examine Item Statistics.
A number of item statistics can help us assess how well each item "works" in
measuring the latent variable, women's autonomy, in this case. For example, one
might ask the following questions:
#
#
#
Does the item do a good job in discriminating persons located low and high on the
autonomy scale?
Are the response categories scored in the correct order?
Is the item measuring the same latent variable (autonomy) as other items in the
questionnaire?
Figure 31 shows an example of item analysis display
Item 4 Who decides on contraception?
-----Cases for this item
3746
Discrimination 0.05
Item Threshold(s):
-3.93 -1.69 1.18
Weighted MNSQ
1.05
Item Delta(s):
-3.82 -1.74 1.12
-----------------------------------------------------------------------------Label
Score
Count
% of tot Pt Bis
t
PV1Avg
PV1 SD
-----------------------------------------------------------------------------0 (other)
0.00
11
0.29
-0.04
-2.42
-0.12
0.33
1 (partner) 1.00
441
11.77
-0.08
-4.77
-0.04
0.29
2 (joint)
2.00
2445
65.27
0.05
3.27
0.04
0.32
3 (self)
3.00
849
22.66
0.00
0.26
0.04
0.33
==============================================================================
Figure 31 An Example Item Analysis
Check the Classical Test Theory Discrimination Index
In Figure 31, this index is labelled "Discrimination 0.05". This discrimination
index is the correlation between a person's score on this item and her total score on
the questionnaire. If this item reflects well the level of autonomy (for which the total
score on the questionnaire is a surrogate measure), then one would expect a high
correlation between the score on this item and the total score on the questionnaire. A
discrimination value of 0 indicates that there is no relationship between the item score
and the total score. A positive discrimination indicates a positive relationship.
Clearly, the higher the discrimination index, the better the item is able to discriminate
between people according to their autonomy level. For the item in Figure 31, the
discrimination index is 0.05. This is an extremely low value for discrimination. In
general, one would not accept any item with discrimination index less than 0.2. It
would be preferable to select items with high discrimination index such as those
above 0.4.
However, before rejecting the item straight away, check that the scoring of the
response categories is correct, and "category disordering" has not occurred6.
6
For achievement tests, a close-to-zero or negative discrimination is often an indication that the "key"
for a multiple-choice item is incorrectly specified.
64
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Check for the Scoring of Response Categories
If the scores assigned to each response category do not correspond to the level of
autonomy, then "category disordering" (Linacre, Rasch Measurement Transaction
13:1) is said to have occurred.
Category disordering is not the same as Steps disordering (see Chapter 4 on the
disordering of the delta parameters). The disordering of the delta parameters (Steps)
is not an indication that the scoring of categories is disordered. Category disordering
may be reflected in three measures: fit mean square, point-biserial correlation, and
average measure value, shown in Figure 31 under headings "Weighted MNSQ", " Pt
Bis", and " PV1Avg" respectively.
When category disordering occurs, the fit mean square will tend to be larger than one,
showing that the item misfits the model. The point-biserial correlation values may not
be in increasing order with increasing category scores, showing that some higher
category scores may be associated with low levels of autonomy, or vice versa.
Similarly, the average measure values may not be in increasing order, showing that
for a lower score category, the average autonomy level is higher than that for a higher
score category.
In the case of the example in Figure 31, the point-biserial values for all four categories
are very close to zero. The average measure values are also not greatly different
across the four categories. These two observations indicate that there is not a strong
relationship between the category scores and increasing level of autonomy, and rearranging the scoring of the categories will not help as no category shows high
positive point-biserial correlation with the total score, nor does any category show
high average measure.
Checking the Item Characteristic Curve
The item characteristic curve can also provide useful information about the behaviour
of an item. Figure 32 shows the ICC for the item in Figure 31. It can be seen that, as
the level of autonomy increases along the horizontal axis, the observed probability of
being in response category 2 (blue dots on top of graph) does not decrease while the
theoretical probability decreases (blue line on top of graph), and the observed
probability of being in response category 3 (red dots) does not increase, while the
theoretical probability of being in category 3 (red line) increases with increasing level
of autonomy. In summary, all observed curves are rather flat, indicating that there is
little relationship between the response categories and the level of autonomy.
All these observations indicate that Item 4 in Figure 31 is not a very good item for
measuring the level of autonomy, and it is a candidate for deletion from the
instrument.
65
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Figure 32 Item characteristic curve for Item 4
Checking the Fit Indices
Fit indices (see Chapter 8 for a detailed explanation of fit indices) indicate the extent
to which the item fits the item response model. In Figure 31, the heading "Weighted
MNSQ 1.05" shows a fit index. Typically, a fit index close to 1 shows that the item
fits the model well, and an index away from 1 shows poor fit to the model.
When the fit index is greater than 1, the item is generally less discriminating than the
model predicts. Figure 33, Figure 34 and Figure 35 show three expected score curves
where the horizontal axis shows the level of autonomy, and the vertical axis shows the
expected score on the item.
It can be seen that the fit index reflects the "slope" of the curve. When the fit index is
greater than one, the observed curve is flatter than the theoretical curve. When the fit
index is less than one, the observed curve is steeper than the theoretical curve.
While both Figure 33 and Figure 35 show some misfit, items with "steep" observed
curves are more discriminating items than items with "flat" curves. Further, a set of
more discriminating items, in general, have more power in separating people on the
latent variable scale than a set of less discriminating items. For this reason, items
showing fit index much greater than one are potential candidates for deletion. But
items showing fit index less than one should be kept unless there are good reasons
why these items should be removed.
It should be noted that the (asymptotic) variance of the fit mean square statistic is
2
. That is, if the sample size is large, then the fit mean square statistic will be
N
closer to 1. If the sample size is small, then the fit mean square statistic will be
further away from 1. From this point of view, it is difficult to set an absolute range of
values for acceptable item fit.
66
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Figure 33 An item with fit index greater than one
Figure 34 An item with fit index close to one
67
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Figure 35 An item with fit index less than one
Item 1 Relationship to household head
Cases for this item 12826
Discrimination 0.33
Item Threshold(s):
-1.73 -0.64 -0.63 -0.53 -0.53 1.77
Weighted MNSQ
1.19
Item Delta(s):
-1.69 4.63 -4.59 5.05 -7.63 1.77
---------------------------------------------------------------------------------Label
Score
Count
% of tot Pt Bis
t
PV1Avg
PV1 SD
---------------------------------------------------------------------------------0 (daughter-in-law)
0
317
2.47
-0.12
-14.16
-0.21
0.31
1 (daughter)
1
1222
9.53
-0.22
-26.11
-0.18
0.34
2 (mother)
2
9
0.07
-0.02
-1.90
-0.12
0.46
3 (granddaughter)
3
754
5.88
-0.06
-6.62
-0.06
0.34
4 (mother-in-law)
4
3
0.02
-0.00
-0.03
0.08
0.37
5 (wife)
5
8870
69.16
-0.01
-1.27
-0.01
0.31
6 (head)
6
1651
12.87
0.31
37.22
0.20
0.33
==================================================================================
Figure 36 Categories with few respondents
Checking the Frequencies
The frequency counts for the response categories may reflect the effective usefulness
of the category. Figure 36 shows an item where the interviewee's relationship with
the household head is ranked in terms of autonomy status. Seven categories are
recorded and scored from 0 to 6. The frequency counts show that categories 2 and 4
have only 9 and 3 respondents respectively. These two categories with so few
respondents will not provide much useful information, and they could be combined
with neighbouring categories so there is a total of five categories. To combine
68
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
response categories, one could compare the point-biserial correlations and the average
measures between adjacent categories, and make a decision whether to combine an
category with the category below or above it. Note, however, by collapsing the
response categories, the maximum score for the item is reduced. This changes the
weight of the item, since the maximum score of an item provides the relative weight
for the item in the questionnaire.
Considering the Maximum Score for an Item
Since the weight of an item is determined by the maximum score of the item, it is
important to consider the number of response categories specified for an item,
particularly when each category is designated a score. An item should not have a
large maximum score simply because the item lends itself to many categories, such as
the example in Figure 36. The weight (or the maximum score) of an item should be
determined by the discriminating power of the item. If an item has more
discriminating power in terms of separating people on the measured variable, then the
item should carry more weight in the questionnaire. If an item does not have much
discriminating power, then it should carry less weight.
Summary Characteristics of a "Good" Item
In summary, an item that is "working well" in an instrument may have the following
characteristics:
#
The (classical test theory) discrimination index is high, say above 0.4.
#
The fit mean square index is close to one.
#
The point-biserial correlation increases with increasing score. For the highest
score category, the point-biserial correlation should be positive.
#
The average measure increases with increasing score.
#
The observed item characteristic curve is close to the theoretical one.
Checking the Reliability of the Instrument
The reliability of an instrument is often used to judge the overall quality of the
instrument. It is important to check the reliability each time some recoding or
collapsing of categories is made, or when items are removed, to assess the impact of
these changes on the reliability of the instrument.
Iterative Process
Note that item selection is an iterative process. Each time some changes are made to
the instrument, check item characteristics for all items as well as the reliability of the
whole instrument, to ensure that changes have brought about an overall improvement
to the quality of the instrument. In particular, check the fit of the items. The
goodness-of-fit of an item is a relative measure, since the fit index measures how well
an item "fits" with the rest of the items. When one item has changed, the fit of the
other items will likely to change as well.
Checking for Differential Item Functioning
A scale constructed from an instrument should be valid for all subgroups of
respondents. In this example, for people at the same level of autonomy, there should
69
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
be no difference in the way the subgroups respond to an item. If differences are
observed, then the item is said to exhibit Differential Item Functioning (DIF), and the
item should not be treated as the same item for the subgroups. In the example dataset,
respondents are from two countries: Zambia and Kenya. The items in the
questionnaire should be checked for differential item functioning in these two
countries.
A good visual way to check for DIF is to plot the observed average score of a
response category at each level of autonomy, for the two countries separately, and
compare the average scores. In this way, the comparison of the scores is made for
people at the same level of autonomy in each country.
Figure 37 An item exhibiting DIF: Zambia scores higher than Kenya
Figure 38 An item showing no DIF between Kenya and Zambia
70
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Figure 39 An item showing large DIF: Kenya scores higher than Zambia
Figure 37 shows a comparison between Zambia and Kenya's average scores on an
item at each autonomy level. The item is "Who decides on large household
purchases?" It can be seen that, at all levels of autonomy, respondents in Zambia
have a slightly higher score than respondents in Kenya. That is, at the same level of
overall autonomy, women in Zambia have higher autonomy in deciding on large
household purchases.
Figure 38 shows that, for the item "Who has the final say on health care?", there is no
difference between respondents in Zambia and Kenya. But, for the question "Is wife
beating justified if she goes out without telling him?", respondents in Kenya score a
great deal higher than respondents in Zambia (Figure 39). That is, at the same level of
overall autonomy, more women in Kenya answered no to this question.
While the presence of DIF reveals interesting differences between countries, the items
that show DIF could not be used in the questionnaire as common items for all
countries, since the DIF items behave differently in different countries.
Dealing with DIF Items
While visual comparisons of country differences is helpful, there is generally a formal
test of statistical significance provided by item response software. However, it should
be noted that, when sample size is large, a small difference in the way an item
functions across sub-groups will result in a significant statistical test, because a large
sample provides sufficient power to detect small differences. Since, in real life, all
items will likely to behave in (at least slightly) different ways for all subgroups, the
majority of items will show differential item functioning when the sample is large
enough. Consequently, the decision to accept or reject an item based on DIF will still
need to be made somewhat subjectively.
Once a decision is made that an item exhibits unacceptable amount of DIF, there are a
few options to deal with the item.
#
The item could be removed from the item pool. In the case where there are a large
number of items in the item pool, this may be the easiest option.
71
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
#
The item could be treated as different items for different groups. In this way, the item
will still contribute to the estimation of the level of autonomy for each group. However,
when developing a described autonomy scale for all groups, this item needs to be treated
carefully as it has different difficulty values for different groups.
#
DIF can be treated as an item-by-group interaction term, and be modelled and estimated
in the item response model. This approach is more general, and it can avoid the decision
to set a cut-point for deciding whether DIF is present. However, for the construction of
described autonomy scale, care should be taken that different groups have different item
difficulty values, as for the option of splitting the items in the previous option.
Exercises
In the pupil questionnaire of SACMEQ, the following question was asked among a
group of items measuring students' socio-economic status:
What are the outside walls of the place (home) where you stay during the school
week mostly made of?
(Please tick only one box.)
(1)
(2)
(3)
(4)
(5)
(6)
PWALL
Cardboard/ Plastic sheeting/ Canvas
Reeds/ Sticks/ Grass thatch
Stones/ Mudbricks
Metal sheets / Asbestos sheets
Wood (planks or timber)
Cut stone/ Concrete blocks/ Burned bricks
The following shows the item analysis and the item characteristic curves of this item
when scaled with other items tapping into the SES meaure.
item:21 (wall2)
Cases for this item 11200
Discrimination 0.63
Item Threshold(s):
-3.09 -1.72 -0.48 -0.34 -0.23
Weighted MNSQ
1.38
Item Delta(s):
-2.82 -1.91 0.92 -0.03 -2.16
-----------------------------------------------------------------------------Label
Score
Count
% of tot Pt Bis
t
PV1Avg:1 PV1 SD:1
-----------------------------------------------------------------------------1
0.00
289
2.58
-0.12
-13.28
-0.95
0.77
2
1.00
1330
11.88
-0.36
-40.86
-1.11
0.60
3
2.00
3391
30.28
-0.35
-39.65
-0.73
0.66
4
3.00
695
6.21
0.02
2.18
-0.26
0.69
5
4.00
520
4.64
-0.01
-1.59
-0.35
0.69
6
5.00
4975
44.42
0.60
78.34
0.25
0.78
==============================================================================
72
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
From the item analysis table and the item characteristic curves shown, comment on
the performance of the items with regard to the discrimination index, fit index, pointbiserial correlations, category ordering, category average measure.
Make
recommendations of how the categories may be recoded to improve the item
characteristics. How could the item be improved if it is to be administered in another
survey?
73
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Chapter Eight: How Well Do the Data Fit the Model?
While the Rasch model has many good measurement properties, there is no guarantee that the
item response data collected will conform to the mathematical formulation of the Rasch model.
If the data collected do not “fit” the Rasch model, the application of the Rasch model will not
improve the measurement properties of the data. That is, unless the data actually fit the Rasch
model, there is little point in using the Rasch model. Therefore, it is important to assess the
extent to which the data fit the Rasch model.
The key feature of the Rasch model is that the probability of success on an item can be
completely determine by two values: an item difficulty  and a person ability  . Equation
(4.1) shows the Rasch model for the probability of success for a person on an item.
p  P X  1 
exp    
1  exp    
(4.1)
If there are factors, other than the item difficulty and person ability, that influence the
probability of success for a person on an item, then the assumptions of the Rasch model are
violated. Some of these factors may include the following:

Guessing. Guessing can occur, particularly for difficult multiple choice items. In
general, we often find that open-ended items are more “discriminating” than multiplechoice items.

Item Dependency. The “local independence” assumption of the Rasch model is violated
when the probability of success on an item depends on the response(s) on other item(s).
For example, an item requires information from the answer of a previous item, or, one
item provides clues to the answer of another item.

Differential Item Functioning (DIF). DIF occurs when different groups of students
respond to an item in different ways. For example, boys may perform better than girls on
an item about football because boys are more engaged with the sport.

Other Traits. An item may tap into a number of “traits”. For example, a mathematics
item may be testing both conceptual understanding and computational accuracy. These
two “traits” may be different for different individuals. That is, a person may be high on
one trait, but low on the other.
Fit Statistics
The extent to which the Rasch model assumptions are violated can be tested through “fit
statistics”. However, since there are many factors that can affect the assumptions of the Rasch
model, different fit statistics have been designed to detect different kinds of violations. This is
an important point to remember, as too often we make judgements based on a single fit statistic
about whether data fit the Rasch model. It should be noted that each fit statistic is sensitive
only to specific violations of the model, and not sensitive to other violations of the model.
74
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Residual Based Fit Statistics
In this section, we will focus our attention on one type of fit statistics: the residual based fit
statistics. This type of fit statistics is reported in a number of IRT software packages such as
Winsteps (Linacre & Wright, 2000), RUMM (2001), Quest (Adams & Khoo, 1996) and
ConQuest (Wu, Adams & Wilson, 1998).
Wright (1977) proposed several item fit and person fit statistics based on standardised residuals
for the Rasch model. Let xni be the observed score for person n on item i , and Pni be the
probability of obtaining a correct response for person n on item i . Then the standardised
residual is defined as
z ni 
xni  E xni 
1
Var xni  2
(4.2)
In the case of the dichotomous Rasch model, E  xni   Pni and Var xni   Pni 1  Pni  . These
residuals have served as general diagnostic tools in the assessment of model fit. They are
mostly presented as graphical displays to draw attention to problem items/persons, rather than
used as vigorous statistical tests for the fit of the model.
Squaring z ni and summing over n , a statistic is derived that can be used as a fit index for item
i . Squaring z ni and summing over i , a statistic is derived that can be used as a fit index for
person n (Wright and Masters, 1982). For item fit, Wright and Masters proposed an
unweighted and a weighted statistic (sometimes called outfit and infit, or unweighted total fit
and weighted total fit). The unweighted fit mean-square (outfit) is defined as
z
n
Unweighted mean-square =
N
2
ni
=
1
N
xni  E xni 2
n Var x 
ni
(4.3)
where N is the total number of respondents. The weighted fit mean-square (infit) is defined as
 z Var x   x  E x 
=
Var x 
Var x 
2
2
ni
Weighted mean-square =
ni
ni
ni
n
n
ni
(4.4)
ni
n
n
When certain assumptions are made, it can be shown that both the unweighted mean-square
and the weighted mean-square have expectations of one. The variances of the mean-square can
also be computed. Wright and Panchapakesan (1969) indicated that both the weighted and the
unweighted mean-square can be treated as chi-square variates. They also suggested the use of a
cube root transformation (the Wilson-Hilferty transformation) of the mean-square to obtain a t
statistic that has an approximate normal distribution so that a frame of reference can be
established in testing the fit of the model.
75
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Additional Notes
The term “weighted mean-square” is used to indicate that the square of the standardised
residuals are weighted by the variance of the item response (See Eq. (4.4)). Each z ni2 is
multiplied by Var xni  in the numerator of Eq. (4.4). The denominator is the sum of the
weights. In contrast, for unweighted mean-square (Eq. (4.3)), each z ni2 can be considered to
have a weight of one (equal weight), and the denominator, N , is the sum of the weights.
There is a common sense justification for the weight, Var xni  , used in weighted mean-square.
Essentially, when the item difficulty of an item is close to the ability of a person, Var xni  is
relatively large. When an item is “off-target” (too easy or too hard), Var xni  is relatively
small. So one uses a larger weight when an item provides more “information” about an item or
student (an on-target item), and one uses a smaller weight when an item does not provide much
“information” about the item or person (an off-target item).
Example Display of Fit Statistics from ConQuest
---------------------------------------------------VARIABLES
UNWEIGHTED FIT
WEIGHTED FIT
---------------------- ------------item ESTIMATE ERROR
MNSQ
T
MNSQ
T
---------------------------------------------------1
0.639
0.072
1.08
1.3
1.07
1.9
2
-0.323
0.072
0.83 -2.9
0.87 -3.6
3
-1.806
0.081
1.01
0.1
0.99 -0.2
4
-0.492
0.072
1.00 -0.0
0.98 -0.4
5
0.529
0.072
1.09
1.4
1.04
1.1
6
0.679
0.072
1.02
0.4
1.03
0.7
7
-0.442
0.072
0.92 -1.2
0.95 -1.3
8
1.890
0.080
0.92 -1.2
1.01
0.2
9
1.185
0.075
1.11
1.7
1.03
0.6
10
-1.493
0.078
0.91 -1.4
1.00
0.0
Figure 1 Example output from IRT software showing residual based fit indices
Figure 1 shows an example output from ConQuest showing values of fit mean-square and t
statistics for each item. It can be seen that the mean-square values are centred around one, and
the t values are centred around zero.
Interpretations of Fit Mean-square
While it is stated that the fit mean-square value has an expectation of one, we need to make an
assessment of how far away the mean-square value is from one before we conclude that an item
is regarded as a misfitting item. Further, when an item shows misfit, we need to understand the
76
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
meaning of “over-fit” (mean-square value less than one) and “under-fit” (mean-square value
greater than one).
Equal Slope Parameter
The mean-square statistic defined in Eq. (4.3) tests whether the item has the same “slope” as
the other items in the test, since the Rasch model makes the assumption that all items have the
same slope, or the same “ discrimination” parameter value. It can be shown that, when the
observed item characteristic curve (ICC) is “steeper” than the expected ICC, the fit meansquare value is less than one. When the observed ICC is flatter than the expected ICC, the fit
mean-square value is greater than one. Figure 2 and Figure 3 show two examples where the
observed ICC is flatter, and steeper, than the expected ICC, respectively. (See Additional
Notes at the end of this Topic for more detailed mathematical explanations).
Figure 2 Observed ICC is “flatter” than expected ICC
77
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Figure 3 Observed ICC is “steeper” than expected ICC
Not About the Amount of Noise Around the Line
Contrary to common belief, the residual based fit statistics do not provide an indication of how
far away the observed ICC is from the theoretical ICC. That is, provided that the “slope” of the
observed ICC is the same as the slope of the theoretical ICC, the fit mean-square will not show
misfit whether the observed ICC is close or far away from the theoretical ICC.
Figure 4 shows an item where the observed ICC is very close to the theoretical ICC for all
ability groups. The weighted fit mean-square is 1.00. By contrast, Figure 5 shows an item
where the observed ICC has a number of points “far away” from the theoretical ICC,
particularly for ability groups in the middle range. Yet the weighted fit mean-square is also
1.00. These two examples show that the fit mean-square statistic is not about the amount of
“noise” of the observed ICC as compared to the theoretical ICC. Rather, the fit mean-square
statistic is testing whether the “slope” of the observed ICC is the same as the theoretical ICC.
It is worth stressing the point that the Rasch model does not specify an absolute value for the
discrimination parameter. Therefore, when an item is identified as a misfitting item, it shows
that the item is different from the other items. It does not say anything about whether this item
is a good or bad item in terms of its discriminating power. So from this point of view, the “fit”
index shows “relative” fit, and not absolute “fit”. An item showing misfit in one test may very
well fit with items in another test.
78
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Figure 4 Observed ICC is close to the theoretical ICC
Figure 5 Observed ICC “far away” from the theoretical ICC
Assessing the Noise in the Data
As an aside, if one is interested in assessing the amount of noise in the data, the discrimination
index and reliability index from classical test theory will provide better information. The
discrimination index is defined as the correlation between a student’s score on an item with the
79
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
student’s total score on the test. For example, Table 1 shows a fictitious set of student
responses on three items, sorted according to the total scores of students.
Table 1 A fictitious set of student responses on three items
Student
Number
1
Total score
on test
25
Score on
Item 1
0
Score on
Item 2
0
Score on
Item 3
0
2
28
0
0
0
3
36
0
1
1
4
45
0
0
0
5
52
0
0
1
6
57
1
0
0
7
66
1
0
0
8
67
1
1
1
9
73
1
0
1
10
76
1
1
0
11
80
1
1
1
12
92
1
1
1
13
95
1
1
0
Student scores on Item 1 follow a Guttman pattern, where all students scoring a “1” have
higher total scores than students scoring a “0”. This item is a highly discriminating item. Item
2 scores are close to a Guttman pattern, with some randomness in the student scores. Item 3
scores appear not to have much association with the total score. The discrimination is low for
this item. In general, the discrimination index reflects more about the amount of “noise” in the
data than fit statistics do.
Distributional Properties of Fit Mean-square
In the section about the derivation of the fit mean-square statistic (Eq. (4.3) and Eq. (4.4)), it
was stated that the expectation of these two statistics is one. That is, when the data fit the
model, we expect the fit mean-square to be close to one. But “how close to one” is a
judgement call. To assess “how close to one is close enough”, we will need to know the
amount of variation of the mean-square. More formally, the asymptotic variance of the fit
mean-square is given by 2/N, where N is the sample size of students. This means that if a test is
given to a small group of students, we would expect the fit mean-square for each item to
fluctuate quite widely around one, even when the items fit the Rasch model. For example, if
the sample size is 200, we would expect the mean-square values to be between 0.8 and 1.2
2
 0.1 ). When the same test is given to a large group of students, the fit
(standard error =
200
mean-square will be very close to one. For example, if the sample size is 2000, we would
2
 0.03 ). Since the
expect the mean-square values to be between 0.94 and 1.06 (
2000
80
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
variance of the mean-square statistic depends on the sample size, we need to be careful about
applying fixed limits around one to make an assessment of the fit of an item.
Figure 6 shows a fit map of 20 items administered to 100 students for a simulated data set. It
can be seen that the fit mean-square values are generally between 0.8 and 1.2.
In contrast, Figure 7 shows a fit map of the same 20 items administered to 500 students. It can
be seen that the fit mean-square values are generally between 0.9 and 1.10. The only difference
between the two analyses is the sample size. The same items were used for both analyses.
Since the data were simulated according to the Rasch model, all items were expected to fit the
model. These two examples demonstrated that an assessment of the magnitude of the fit meansquare statistic should take into account of the sample size of the test administration.
QUEST: The Interactive Test Analysis System
-------------------------------------------------------------------------------Item Fit
8/10/2004 12:49
all on all (N = 101 L = 20 Probability Level=0.50)
-------------------------------------------------------------------------------INFIT
MNSQ
0.63
0.71
0.83
1.00
1.20
1.40
1.60
--------------+---------+---------+---------+---------+---------+---------+----1 item 1
*
|
2 item 2
|*
3 item 3
*|
4 item 4
* |
5 item 5
|*
6 item 6
|*
7 item 7
*
|
8 item 8
| *
9 item 9
| *
10 item 10
|*
11 item 11
*
|
12 item 12
* |
13 item 13
| *
14 item 14
|*
15 item 15
*
16 item 16
|
*
17 item 17
* |
18 item 18
* |
19 item 19
| *
20 item 20
|
*
================================================================================
Figure 6 Fit map when sample size = 100
81
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
QUEST: The Interactive Test Analysis System
-------------------------------------------------------------------------------Item Fit
8/10/2004 12:49
all on all (N = 500 L = 20 Probability Level=0.50)
-------------------------------------------------------------------------------INFIT
MNSQ
0.63
0.71
0.83
1.00
1.20
1.40
1.60
--------------+---------+---------+---------+---------+---------+---------+----1 item 1
* |
2 item 2
|*
3 item 3
*|
4 item 4
*|
5 item 5
*|
6 item 6
*|
7 item 7
* |
8 item 8
| *
9 item 9
| *
10 item 10
|*
11 item 11
* |
12 item 12
* |
13 item 13
| *
14 item 14
|*
15 item 15
*|
16 item 16
| *
17 item 17
* |
18 item 18
*|
19 item 19
| *
20 item 20
| *
================================================================================
Figure 7 Fit map when sample size=500
The Fit t Statistic
The fit t statistic, however, does take sample size into account. The fit t statistic can be
regarded as a normal deviate with a mean of zero and a standard deviation of one. It is a
transformation of the fit mean-square value, taking into account of the mean and variance of the
fit mean-square statistic.
Additional Notes
To transform the fit mean squares to a standardised normal statistic so that one can look up the
level
of
significance
easily,
the
Wilson-Hilferty
transformation
12
13
tunwtt  Fit unwtt  1  2 9 N  2 9 N  is often used, where Fit is the mean-square value.
An alternative transformation is given in Wright and Masters (1982) that uses a cube root
transformation of the fit mean-square and its variance:


13
t unwtt  Fit unwt
1 
3
Var  Fit unwtt 

Var  Fit unwt 
3
Since the fit t statistic can be regarded as a normal deviate, a t value outside the range of –2.0 to
2.0 (or –1.96 to 1.96, to be more precise) can be regarded as an indication of misfit, at the 95%
confidence level.
Thus, our problem regarding the lack of a stable frame of reference for the fit mean-square
values seems to have been solved. Unfortunately, life is not meant to be so simple.
82
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
The problem is, in real-life, no item fits the Rasch model perfectly! When items do not fit the
Rasch model, any misfit, however small, can be detected when the sample size is large enough.
This means that the fit t values will invariably show significance when the sample size is very
large. In some sense, the t values are telling the “truth”, that there are indeed differences
between items, and the items do not tap into the same construct. However, some of these
differences between items may be minute.
The following shows an example of how sample size affects the fit t values.
Item response data from a large international comparative survey was scaled using ConQuest,
first selecting just 300 students at random, and then selecting 2500, and 15000 students at
random. That is, the items scaled in all three samples were exactly the same, but the sample
included increased in size. Figure 8 to Figure 10 show the fit t values for these three samples.
------------------------------------------------VARIABLES
UNWEIGHTED FIT
WEIGHTED FIT
---------------------- ------------item ESTIMATE
MNSQ
T
MNSQ
T
------------------------------------------------1
-1.336
0.94 -0.4
1.07
0.6
2
0.571
1.12
0.8
1.08
0.8
3
0.758
0.80 -1.5
0.90 -0.9
4
1.606
0.94 -0.4
1.02
0.2
5
-0.560
0.99 -0.1
1.03
0.3
6
1.578
0.88 -0.9
0.92 -0.6
7
-1.354
0.76 -1.8
0.91 -0.6
8
0.738
0.87 -0.9
0.98 -0.2
9
-0.942
0.99 -0.0
0.95 -0.4
10
-0.590
1.05
0.4
1.00
0.0
11
-1.332
1.26
1.6
1.05
0.3
12
0.329
1.01
0.1
1.07
0.7
13
-1.003
0.87 -0.8
0.99 -0.1
14
-0.403
1.02
0.2
1.22
1.5
15
1.895
0.63 -2.8
0.95 -0.2
Figure 8 Fit t values for a sample of 300 students
83
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
------------------------------------------------VARIABLES
UNWEIGHTED FIT
WEIGHTED FIT
---------------------- ------------item ESTIMATE
MNSQ
T
MNSQ
T
------------------------------------------------1
-1.510
1.09
1.6
0.99 -0.1
2
0.446
0.99 -0.2
1.00
0.1
3
0.865
0.77 -4.9
0.84 -4.5
4
1.489
0.83 -3.5
0.93 -1.1
5
-0.576
1.01
0.2
1.02
0.5
6
1.262
1.11
2.1
1.01
0.2
7
-1.401
0.77 -5.1
0.88 -2.6
8
0.575
1.00
0.0
1.03
0.8
9
-1.043
1.04
0.9
0.98 -0.3
10
-0.725
0.99 -0.2
0.99 -0.3
11
-0.983
1.24
4.4
1.11
2.1
12
0.368
0.94 -1.1
0.95 -1.4
13
-0.818
1.02
0.4
0.95 -1.2
14
-0.399
1.26
4.6
1.20
3.6
15
1.536
0.85 -3.0
1.00 -0.0
Figure 9 Fit t values for a sample of 2500 students
------------------------------------------------VARIABLES
UNWEIGHTED FIT
WEIGHTED FIT
---------------------- ------------item ESTIMATE
MNSQ
T
MNSQ
T
------------------------------------------------1
-1.471
1.15
7.0
1.06
2.7
2
0.436
0.96 -2.0
0.99 -0.5
3
0.807
0.80 -10.2
0.87 -8.7
4
1.490
0.92 -3.8
0.96 -1.4
5
-0.641
1.00
0.0
1.02
1.1
6
1.260
1.04
2.0
1.03
2.0
7
-1.466
0.77 -11.8
0.90 -5.1
8
0.661
0.97 -1.3
1.00
0.0
9
-0.911
0.91 -4.2
0.92 -4.8
10
-0.902
0.99 -0.4
0.99 -0.8
11
-0.972
1.30 12.7
1.12
5.5
12
0.329
0.94 -3.0
0.96 -2.6
13
-0.872
0.98 -0.7
0.97 -2.1
14
-0.464
1.50 20.8
1.20
9.2
15
1.633
0.85 -7.3
0.96 -1.7
Figure 10 Fit t values for a sample of 15000 students
From Figure 8 to Figure 10, it can be seen that as sample size increases, the fit t values became
progressively larger, so that many items showed misfit.
84
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Summary
These results place us in a dilemma. If we use fit mean-square values to set criteria for
accepting or rejecting items on the basis of fit, we are likely to declare that all items fit well
when the sample size is large enough. On the other hand, if we set limits to fit t values as a
criterion for detecting misfit, we are likely to reject most items when the sample size is large
enough.
Many textbooks or other resources make recommendations on the range of acceptable meansquare values or t values for residual based fit statistics. There are probably no right or wrong
answers. You will need to understand the issues with these fit statistics when you apply rules
of thumb.
More importantly, fit statistics should serve as an indication for detecting problematic items
rather than for setting concrete rules for accepting or rejecting items. Based on the fit statistics,
one should examine the items and look for sources of misfit. Improve or reject items if sources
of misfit can be identified. The fit statistics should not be used blindly to reject items,
particularly those that “over-fit”, as you may remove the best items in your test because the rest
of the items are not as “good” as these items.
Furthermore, when residual based fit statistics show that items fit the Rasch model, this is not
sufficient to conclude that you have the best test. The reliability of the test and item
discrimination indices should also be considered in making an overall assessment.
Additional Notes
Item Characteristic Curve
Probability of Success
 A
 B
0.5
 C
 D
Very high
Very low

achievement
achievementFigure 11 Expected ICC and observed
ICC points
Figure 11 shows the theoretical, or expected, item characteristic curve for an item, with four
points, A, B, C, and D denoting four regions where the observed ICC may fall. Point A
denotes the region above the theoretical ICC, and to the right of the vertical line where  = ,
the ability at which there is a 50% chance of obtaining the correct answer. Point B denotes the
region below the theoretical ICC and to the right of the vertical line  = . Point C denotes the
85
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
region above the theoretical ICC but to the left of the  =  line. Point D denotes the region
below the theoretical ICC and to the left of the  =  line. It can be shown mathematically that
the contribution of observed points in the A and D region to the outfit mean-square,
x  E xni 2 , has an expectation less than one, while the expectation of z 2 for points in
z ni2  ni
ni
Var xni 
the C and B regions is greater than one (see Appendix 1). It is clear then the fit mean-square
value provides a test of whether the “slope” of the observed ICC is the same as the theoretical
one. Given that the theoretical one can be regarded as an “average” of all items, the fit meansquare value tests whether the observed ICC for this item is the same as the slopes of the other
items.
When residual based fit statistics show that items fit the Rasch model, this is not sufficient to
conclude that you have the best test.
References
Adams, R. J., & Khoo, S. (1996). Quest: The interactive test analysis system. Camberwell: Australian
Council for Educational Research.
Linacre, J. M., & Wright, B. D. (2000). WINSTEPS: A Rasch computer program. Chicago: MESA
Press.
RUMM Laboratory. (2001). Rasch Unified Measurement Models. Perth.
Wright, B.D. (1977). Solving measurement problems with the Rasch Model. Journal of Educational
Measurement, 14, 97-116.
Wright, B.D., & Masters, G.N. (1982). Rating scale analysis. Chicago: MESA Press.
Wright, B.D., & Panchapakesan, N. (1969). A procedure for sample-free item analysis. Educational and
Psychological Measurement, 29, 23-48.
Wu, M. L., Adams, R. J., & Wilson, M. R. (1998). ConQuest: Generalised item response modelling
software. Camberwell: Australian Council for Educational Research
86
Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach.
Educational Measurement Solutions, Melbourne.
_____________________________________________________________________________________________________
Appendix 1: Derivation of the expectation of z ni2 when the observation does not
follow the Rasch model
z
2
ni
2

xni  E xni 

Var xni 
Let p be the theoretical probability of success according to the Rasch model.
Let p  be the expectation of the observed probability of success.
Let   p  p .
Then
z
2
ni
xni  p 2

p1  p 
E zni2  
xni  p  p  p 2

p1  p 
xni  p2  2xni  p p  p    p  p 2

p1  p 
2
  p 1    p   2  1  2 p   p  p 2
p1  p    p  p 

p1  p 
p1  p 
p  p2
Therefore, E zni2  will be greater than one if 1  2 p  is positive,
and E zni2  will be less than one if 1  2 p  is negative.
The following is a table showing the four cases corresponding to the regions defined by A, B, C
and D in Figure 11.
Table 2 Mean-square in four regions of the ICC
Region

p
1  2 p 
1  2 p 
A
>0
>0.5
<0
<0
Meansquare
<1
B
<0
>0.5
<0
>0
>1
C
>0
<0.5
>0
>0
>1
D
<0
<0.5
>0
<0
<1
It should also be noted that when p = 0.5, 1  2 p  = 0, so that 1  2 p  = 0. That is, when
the ability is equal to the item difficulty, any observed misfit at this ability will not contribute to
the deviation of the mean-square from one. In general, when ability is close to the item
difficulty, the amount of misfit,  , will not contribute much to the deviation of the meansquare. Thus, it is the amount of misfit near the lower- and upper- ends of the ability scale that
determines the size of the deviation of the mean-square from one. Figure 2 and Figure 5
support this observation.
87
">