The INBREED Procedure SAS/STAT 13.1 User’s Guide ®

The INBREED Procedure SAS/STAT 13.1 User’s Guide ®
®
SAS/STAT 13.1 User’s Guide
The INBREED Procedure
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Chapter 50
The INBREED Procedure
Contents
Overview: INBREED Procedure . . . . . . . . . . . . .
Getting Started: INBREED Procedure . . . . . . . . . .
The Format of the Input Data Set . . . . . . . . .
Performing the Analysis . . . . . . . . . . . . . .
Syntax: INBREED Procedure . . . . . . . . . . . . . .
PROC INBREED Statement . . . . . . . . . . . .
BY Statement . . . . . . . . . . . . . . . . . . .
CLASS Statement . . . . . . . . . . . . . . . . .
GENDER Statement . . . . . . . . . . . . . . . .
MATINGS Statement . . . . . . . . . . . . . . .
VAR Statement . . . . . . . . . . . . . . . . . . .
Details: INBREED Procedure . . . . . . . . . . . . . .
Missing Values . . . . . . . . . . . . . . . . . . .
DATA= Data Set . . . . . . . . . . . . . . . . . .
Computational Details . . . . . . . . . . . . . . .
OUTCOV= Data Set . . . . . . . . . . . . . . . .
Displayed Output . . . . . . . . . . . . . . . . . .
ODS Table Names . . . . . . . . . . . . . . . . .
Examples: INBREED Procedure . . . . . . . . . . . . .
Example 50.1: Monoecious Population Analysis .
Example 50.2: Pedigree Analysis . . . . . . . . .
Example 50.3: Pedigree Analysis with BY Groups
References . . . . . . . . . . . . . . . . . . . . . . . .
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3977
3978
3978
3979
3983
3983
3985
3985
3985
3986
3986
3987
3987
3987
3988
3994
3996
3996
3997
3997
3999
4001
4002
Overview: INBREED Procedure
The INBREED procedure calculates the covariance or inbreeding coefficients for a pedigree. PROC INBREED is unique in that it handles very large populations.
The INBREED procedure has two modes of operation. One mode carries out analysis on the assumption that
all the individuals belong to the same generation. The other mode divides the population into nonoverlapping
generations and analyzes each generation separately, assuming that the parents of individuals in the current
generation are defined in the previous generation.
PROC INBREED also computes averages of the covariance or inbreeding coefficients within sex categories
if the sex of individuals is known.
3978 F Chapter 50: The INBREED Procedure
Getting Started: INBREED Procedure
This section demonstrates how you can use the INBREED procedure to calculate the inbreeding or covariance
coefficients for a pedigree, how you can control the analysis mode if the population consists of nonoverlapping
generations, and how you can obtain averages within sex categories.
For you to use PROC INBREED effectively, your input data set must have a definite format. The following
sections first introduce this format for a fictitious population and then demonstrate how you can analyze this
population by using the INBREED procedure.
The Format of the Input Data Set
The SAS data set used as input to the INBREED procedure must contain an observation for each individual.
Each observation must include one variable identifying the individual and two variables identifying the
individual’s parents. Optionally, an observation can contain a known covariance coefficient and a character
variable defining the gender of the individual.
For example, consider the following data:
data Population;
input Individual $ Parent1 $ Parent2 $
Covariance Sex $ Generation;
datalines;
Mark
George Lisa
.
M 1
Kelly Scott Lisa
.
F 1
Mike
George Amy
.
M 1
.
Mark
Kelly 0.50 . 1
David Mark
Kelly
.
M 2
Merle Mike
Jane
.
F 2
Jim
Mark
Kelly 0.50 M 2
Mark
Mike
Kelly
.
M 2
;
It is important to order the pedigree observations so that individuals are defined before they are used as
parents of other individuals. The family relationships between individuals cannot be ascertained correctly
unless you observe this ordering. Also, older individuals must precede younger ones. For example, ‘Mark’
appears as the first parent of ‘David’ at observation 5; therefore, his observation needs to be defined prior
to observation 5. Indeed, this is the case (see observation 1). Also, ‘David’ is older than ‘Jim’, whose
observation appears after the observation for ‘David’, as is appropriate.
In populations with distinct, nonoverlapping generations, the older generation (parents) must precede the
younger generation. For example, the individuals defined in Generation=1 appear as parents of individuals
defined in Generation=2.
PROC INBREED produces warning messages when a parent cannot be found. For example, ‘Jane’ appears
as the second parent of the individual ‘Merle’ even though there are no previous observations defining her
own parents. If the population is treated as an overlapping population, that is, if the generation grouping
is ignored, then the procedure inserts an observation for ‘Jane’ with missing parents just before the sixth
observation, which defines ‘Merle’ as follows:
Performing the Analysis F 3979
Jane
Merle
.
Mike
.
Jane
.
.
F
F
2
2
However, if generation grouping is taken into consideration, then ‘Jane’ is defined as the last observation in
Generation=1, as follows:
Mike
Jane
George Amy
.
.
.
.
M
F
1
1
In this latter case, however, the observation for ‘Jane’ is inserted after the computations are reported for
the first generation. Therefore, she does not appear in the covariance/inbreeding matrix, even though her
observation is used in computations for the second generation (see Figure 50.2).
If the data for an individual are duplicated, only the first occurrence of the data is used by the procedure, and
a warning message is displayed to note the duplication. For example, individual ‘Mark’ is defined twice, at
observations 1 and 8. If generation grouping is ignored, then this is an error and observation 8 is skipped.
However, if the population is processed with respect to two distinct generations, then ‘Mark’ refers to two
different individuals, one in Generation=1 and the other in Generation=2.
If a covariance is to be assigned between two individuals, then those individuals must be defined prior to
the assignment observation. For example, a covariance of 0.50 can be assigned between ‘Mark’ and ‘Kelly’
since they are previously defined. Note that assignment statements must have different formats depending
on whether the population is processed with respect to generations (see the section “DATA= Data Set” on
page 3987 for further information). For example, while observation 4 is valid for nonoverlapping generations,
it is invalid for a processing mode that ignores generation grouping. In this latter case, observation 7 indicates
a valid assignment, and observation 4 is skipped.
The latest covariance specification between any given two individuals overrides the previous one between the
same individuals.
Performing the Analysis
To compute the covariance coefficients for the overlapping generation mode, use the following statements:
proc inbreed data=Population covar matrix init=0.25;
run;
Here, the DATA= option names the SAS data set to be analyzed, and the COVAR and MATRIX options tell
the procedure to output the covariance coefficients matrix. If you omit the COVAR option, the inbreeding
coefficients are output instead of the covariance coefficients.
Note that the PROC INBREED statement also contains the INIT= option. This option gives an initial
covariance between any individual and unknown individuals. For example, the covariance between any
individual and ‘Jane’ would be 0.25, since ‘Jane’ is unknown, except when ‘Jane’ appears as a parent (see
Figure 50.4).
3980 F Chapter 50: The INBREED Procedure
Figure 50.1 Analysis for an Overlapping Population
The INBREED Procedure
Covariance Coefficients
Individual
George
Lisa
Mark
Scott
Kelly
Amy
Mike
David
Jane
Merle
Jim
Parent1
Parent2
George
Lisa
Scott
Lisa
George
Mark
Amy
Kelly
Mike
Mark
Jane
Kelly
George
Lisa
Mark
Scott
Kelly
1.1250
0.2500
0.6875
0.2500
0.2500
0.2500
0.6875
0.4688
0.2500
0.4688
0.4688
0.2500
1.1250
0.6875
0.2500
0.6875
0.2500
0.2500
0.6875
0.2500
0.2500
0.6875
0.6875
0.6875
1.1250
0.2500
0.5000
0.2500
0.4688
0.8125
0.2500
0.3594
0.8125
0.2500
0.2500
0.2500
1.1250
0.6875
0.2500
0.2500
0.4688
0.2500
0.2500
0.4688
0.2500
0.6875
0.5000
0.6875
1.1250
0.2500
0.2500
0.8125
0.2500
0.2500
0.8125
Covariance Coefficients
Individual
George
Lisa
Mark
Scott
Kelly
Amy
Mike
David
Jane
Merle
Jim
Parent1
Parent2
George
Lisa
Scott
Lisa
George
Mark
Amy
Kelly
Mike
Mark
Jane
Kelly
Amy
Mike
David
Jane
Merle
0.2500
0.2500
0.2500
0.2500
0.2500
1.1250
0.6875
0.2500
0.2500
0.4688
0.2500
0.6875
0.2500
0.4688
0.2500
0.2500
0.6875
1.1250
0.3594
0.2500
0.6875
0.3594
0.4688
0.6875
0.8125
0.4688
0.8125
0.2500
0.3594
1.2500
0.2500
0.3047
0.8125
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
1.1250
0.6875
0.2500
0.4688
0.2500
0.3594
0.2500
0.2500
0.4688
0.6875
0.3047
0.6875
1.1250
0.3047
Covariance Coefficients
Individual
George
Lisa
Mark
Scott
Kelly
Amy
Mike
David
Jane
Merle
Jim
Parent1
Parent2
George
Lisa
Scott
Lisa
George
Mark
Amy
Kelly
Mike
Mark
Jane
Kelly
Number of Individuals
Jim
0.4688
0.6875
0.8125
0.4688
0.8125
0.2500
0.3594
0.8125
0.2500
0.3047
1.2500
11
Performing the Analysis F 3981
In the previous example, PROC INBREED treats the population as a single generation. However, you might
want to process the population with respect to distinct, nonoverlapping generations. To accomplish this, you
need to identify the generation variable in a CLASS statement, as shown by the following statements:
proc inbreed data=Population covar matrix init=0.25;
class Generation;
run;
Note that, in this case, the covariance matrix is displayed separately for each generation (see Figure 50.5).
Figure 50.2 Analysis for a Nonoverlapping Population
The INBREED Procedure
Generation = 1
Covariance Coefficients
Individual
Parent1
Parent2
Mark
Kelly
Mike
George
Scott
George
Lisa
Lisa
Amy
Mark
Kelly
Mike
1.1250
0.5000
0.4688
0.5000
1.1250
0.2500
0.4688
0.2500
1.1250
Number of Individuals
3
The INBREED Procedure
Generation = 2
Covariance Coefficients
Individual
Parent1
Parent2
David
Merle
Jim
Mark
Mark
Mike
Mark
Mike
Kelly
Jane
Kelly
Kelly
David
Merle
Jim
Mark
1.2500
0.3047
0.8125
0.5859
0.3047
1.1250
0.3047
0.4688
0.8125
0.3047
1.2500
0.5859
0.5859
0.4688
0.5859
1.1250
Number of Individuals
4
You might also want to see covariance coefficient averages within sex categories. This is accomplished
by indicating the variable defining the gender of individuals in a GENDER statement and by adding the
AVERAGE option to the PROC INBREED statement. For example, the following statements produce the
covariance coefficient averages shown in Figure 50.3:
3982 F Chapter 50: The INBREED Procedure
proc inbreed data=Population covar average init=0.25;
class Generation;
gender Sex;
run;
Figure 50.3 Averages within Sex Categories for a Nonoverlapping Generation
The INBREED Procedure
Generation = 1
Averages of Covariance Coefficient Matrix in Generation 1
Male X Male
Male X Female
Female X Female
Over Sex
On Diagonal
Below Diagonal
1.1250
.
1.1250
1.1250
0.4688
0.3750
0.0000
0.4063
Number of Males
Number of Females
Number of Individuals
2
1
3
The INBREED Procedure
Generation = 2
Averages of Covariance Coefficient Matrix in Generation 2
Male X Male
Male X Female
Female X Female
Over Sex
On Diagonal
Below Diagonal
1.2083
.
1.1250
1.1875
0.6615
0.3594
0.0000
0.5104
Number of Males
Number of Females
Number of Individuals
3
1
4
PROC INBREED Statement F 3983
Syntax: INBREED Procedure
The following statements are available in the INBREED procedure:
PROC INBREED < options > ;
BY variables ;
CLASS variable ;
GENDER variable ;
MATINGS individual-list1 / mate-list1 < , . . . , individual-listn / mate-listn > ;
VAR variables ;
The PROC INBREED statement is required. Items within angle brackets (< >) are optional. The syntax of
each statement is described in the following sections.
PROC INBREED Statement
PROC INBREED < options > ;
The PROC INBREED statement invokes the INBREED procedure. Table 50.1 summarizes the options
available in the PROC INBREED statement.
Table 50.1
Option
PROC INBREED Statement Options
Description
Specify Data Sets
DATA=
Names the SAS data set
OUTCOV= Names an output data set to contain the inbreeding coefficients
Control Type of Coefficient
COVAR
Specifies that all coefficients output consist of covariance coefficients
SELFDIAG Includes an individual’s self-mating kinship coefficient
Control Displayed Tables
AVERAGE Produces a table of averages of coefficients
IND
Displays the individuals’ inbreeding coefficients
MATRIX
Displays the inbreeding coefficient matrix
Specify Default Covariance Value
INIT=
Specifies the covariance value
Suppress Output
INDL
Displays individuals’ coefficients for only the last generation
MATRIXL Displays coefficients for only the last generation
NOPRINT
Suppresses the display of all output
3984 F Chapter 50: The INBREED Procedure
AVERAGE
A
produces a table of averages of coefficients for each pedigree of offspring. The AVERAGE option is
used together with the GENDER statement to average the inbreeding/covariance coefficients within
sex categories.
COVAR
C
specifies that all coefficients output consist of covariance coefficients rather than inbreeding coefficients.
DATA=SAS-data-set
names the SAS data set to be used by PROC INBREED. If you omit the DATA= option, the most
recently created SAS data set is used.
IND
I
displays the individuals’ inbreeding coefficients (diagonal of the inbreeding coefficients matrix) for
each pedigree of offspring.
If you also specify the COVAR option, the individuals’ covariance coefficients (diagonal of the
covariance coefficients matrix) are displayed.
INDL
displays individuals’ coefficients for only the last generation of a multiparous population.
INIT=cov
specifies the covariance value cov if any of the parents are unknown; a value of 0 is assumed if you do
not specify the INIT= option.
MATRIX
M
displays the inbreeding coefficient matrix for each pedigree of offspring.
If you also specify the COVAR option, the covariance matrices are displayed instead of inbreeding
coefficients matrices.
MATRIXL
displays coefficients for only the last generation of a multiparous population.
NOPRINT
suppresses the display of all output. Note that this option temporarily disables the Output Delivery
System (ODS). For more information on ODS, see Chapter 20, “Using the Output Delivery System.”
OUTCOV=SAS-data-set
names an output data set to contain the inbreeding coefficients. When the COVAR option is also
specified, covariance estimates are output to the OUTCOV= data set instead of inbreeding coefficients.
SELFDIAG
includes an individual’s self-mating kinship coefficient instead of the individual’s inbreeding coefficient
on the diagonal of the matrix in the OUTCOV= data set when the COVAR option is not specified.
BY Statement F 3985
BY Statement
BY variables ;
You can specify a BY statement with PROC INBREED to obtain separate analyses of observations in groups
that are defined by the BY variables. When a BY statement appears, the procedure expects the input data
set to be sorted in order of the BY variables. If you specify more than one BY statement, only the last one
specified is used.
If your input data set is not sorted in ascending order, use one of the following alternatives:
• Sort the data by using the SORT procedure with a similar BY statement.
• Specify the NOTSORTED or DESCENDING option in the BY statement for the INBREED procedure.
The NOTSORTED option does not mean that the data are unsorted but rather that the data are arranged
in groups (according to values of the BY variables) and that these groups are not necessarily in
alphabetical or increasing numeric order.
• Create an index on the BY variables by using the DATASETS procedure (in Base SAS software).
For more information about BY-group processing, see the discussion in SAS Language Reference: Concepts.
For more information about the DATASETS procedure, see the discussion in the Base SAS Procedures Guide.
CLASS Statement
CLASS variable ;
To analyze the population within nonoverlapping generations, you must specify the variable that identifies
generations in a CLASS statement. Values of the generation variable, called generation numbers, must be
integers, but generations are assumed to occur in the order of their input in the input data set rather than in
numerical order of the generation numbers. The name of an individual needs to be unique only within its
generation.
When the MATRIXL option or the INDL option is specified, each generation requires a unique generation
number in order for the specified option to work correctly. If generation numbers are not unique, all the
generations with a generation number that is the same as the last generation’s are output.
GENDER Statement
GENDER variable ;
The GENDER statement specifies a variable that indicates the sex of the individuals. Values of the sex
variable must be character beginning with ‘M’ or ‘F’, for male or female. The GENDER statement is needed
only when you specify the AVERAGE option to average the inbreeding/covariance coefficients within sex
categories or when you want to include a gender variable in the OUTCOV= data set.
PROC INBREED makes the following assumptions regarding the gender of individuals:
3986 F Chapter 50: The INBREED Procedure
• The first parent is always assumed to be the male. See the section “VAR Statement” on page 3986.
• The second parent is always assumed to be the female. See the section “VAR Statement” on page 3986.
• If the gender of an individual is missing or invalid, this individual is assumed to be a female unless the
population is overlapping and this individual appears as the first parent in a later observation.
Any contradictions to these rules are reported in the SAS log.
MATINGS Statement
MATINGS individual-list1 / mate-list1 < , . . . , individual-listn / mate-listn > ;
You can specify the MATINGS statement with PROC INBREED to specify selected matings of individuals.
Each individual given in individual-list is mated with each individual given in mate-list. You can write
multiple mating specifications if you separate them by commas or asterisks. The procedure reports the
inbreeding coefficients or covariances for each pair of mates. For example, you can use the following
statement to specify the mating of an individual named ‘David’ with an individual named ‘Jane’:
matings david / jane;
VAR Statement
VAR individual parent1 parent2 < covariance > ;
The VAR statement specifies three or four variables: the first variable contains an individual’s name, the
second variable contains the name of the individual’s first parent, and the third variable contains the name of
the individual’s second parent. An optional fourth variable assigns a known value to the covariance of the
individual’s first and second parents in the current generation.
The first three variables in the VAR statement can be either numeric or character; however, only the first 12
characters of a character variable are recognized by the procedure. The fourth variable, if specified, must be
numeric.
If you omit the VAR statement, then the procedure uses the first three unaddressed variables as the names of
the individual and its parents. (Unaddressed variables are those that are not referenced in any other PROC
INBREED statement.) If the input data set contains an unaddressed fourth variable, then it becomes the
covariance variable.
Details: INBREED Procedure F 3987
Details: INBREED Procedure
Missing Values
A missing value for a parent implies that the parent is unknown. Unknown parents are assumed to be
unrelated and not inbred unless you specify the INIT= option.
When the value of the variable identifying the individual is missing, the observation is not added to the list of
individuals. However, for a multiparous population, an observation with a missing individual is valid and is
used for assigning covariances.
Missing covariance values are determined from the INIT=cov option, if specified. Observations with missing
generation variables are excluded.
If the gender of an individual is missing, it is determined from the order in which it is listed on the first
observation defining its progeny for an overlapping population. If it appears as the first parent, it is set to
‘M’; otherwise, it is set to ‘F’. When the gender of an individual cannot be determined, it is assigned a default
value of ‘F’.
DATA= Data Set
Each observation in the input data set should contain necessary information such as the identification of an
individual and the first and second parents of an individual. In addition, if a CLASS statement is specified,
each observation should contain the generation identification; and, if a GENDER statement is specified, each
observation should contain the gender of an individual. Optionally, each observation might also contain the
covariance between the first and the second parents. Depending on how many statements are specified with
the procedure, there should be enough variables in the input data set containing this information.
If you omit the VAR statement, then the procedure uses the first three unaddressed variables in the input data
set as the names of the individual and his or her parents. Unaddressed variables in the input data set are those
variables that are not referenced by the procedure in any other statements, such as CLASS, GENDER, or BY
statements. If the input data set contains an unaddressed fourth variable, then the procedure uses it as the
covariance variable.
If the individuals given by the variables associated with the first and second parents are not in the population,
they are added to the population. However, if they are in the population, they must be defined prior to the
observation that gives their progeny.
When there is a CLASS statement, the functions of defining new individuals and assigning covariances
must be separated. This is necessary because the parents of any given individual are defined in the previous
generation, while covariances are assigned between individuals in the current generation.
3988 F Chapter 50: The INBREED Procedure
Therefore, there could be two types of observations for a multiparous population:
• one to define new individuals in the current generation whose parents have been defined in the previous
generation, as in the following, where the missing value is for the covariance variable:
Mark
Kelly
George Lisa
Scott Lisa
.
.
M
F
1
1
• one to assign covariances between two individuals in the current generation, as in the following, where
the individual’s name is missing, ‘Mark’ and ‘Kelly’ are in the current generation, and the covariance
coefficient between these two individuals is 0.50:
.
Mark
Kelly
0.50
.
1
Note that the observations defining individuals must precede the observation assigning a covariance value
between them. For example, if a covariance is to be assigned between ‘Mark’ and ‘Kelly’, then both of them
should be defined prior to the assignment observation.
Computational Details
This section describes the rules that the INBREED procedure uses to compute the covariance and inbreeding
coefficients. Each computational rule is explained by an example referring to the fictitious population
introduced in the section “Getting Started: INBREED Procedure” on page 3978.
Coancestry (or Kinship Coefficient)
To calculate the inbreeding coefficient and the covariance coefficients, use the degree of relationship by
descent between the two parents, which is called coancestry or kinship coefficient (Falconer and Mackay
1996, p.85), or coefficient of parentage (Kempthorne 1957, p.73). Denote the coancestry between individuals
X and Y by fXY . For information on how to calculate the coancestries among a population, see the section
“Calculation of Coancestry” on page 3989.
Covariance Coefficient (or Coefficient of Relationship)
The covariance coefficient between individuals X and Y is defined by
Cov.X; Y/ D 2fXY
where fXY is the coancestry between X and Y. The covariance coefficient is sometimes called the coefficient
of relationship or the theoretical correlation (Falconer and Mackay (1996, p.153); Crow and Kimura (1970,
p.134)). If a covariance coefficient cannot be calculated from the individuals in the population, it is assigned
to an initial value. The initial value is set to 0 if the INIT= option is not specified or to cov if INIT=cov.
Therefore, the corresponding initial coancestry is set to 0 if the INIT= option is not specified or to 12 cov if
INIT=cov.
Computational Details F 3989
Inbreeding Coefficients
The inbreeding coefficient of an individual is the probability that the pair of alleles carried by the gametes that
produced it are identical by descent (Falconer and Mackay (1996, Chapter 5), Kempthorne (1957, Chapter
5)). For individual X, denote its inbreeding coefficient by FX . The inbreeding coefficient of an individual
is equal to the coancestry between its parents. For example, if X has parents A and B, then the inbreeding
coefficient of X is
FX D fAB
Calculation of Coancestry
Given individuals X and Y, assume that X has parents A and B and that Y has parents C and D. For
nonoverlapping generations, the basic rule to calculate the coancestry between X and Y is given by the
following formula (Falconer and Mackay 1996, p.86):
fXY D
1
.fAC C fAD C fBC C fBD /
4
And the inbreeding coefficient for an offspring of X and Y, called Z, is the coancestry between X and Y:
FZ D fXY
Figure 50.4 Inbreeding Relationship for Nonoverlapping Population
For example, in Figure 50.4, ‘Jim’ and ‘Mark’ from Generation 2 are progenies of ‘Mark’ and ‘Kelly’ and of
‘Mike’ and ‘Kelly’ from Generation 1, respectively. The coancestry between ‘Jim’ and ‘Mark’ is
3990 F Chapter 50: The INBREED Procedure
fJim;Mark D
1
fMark;Mike C fMark;Kelly C fKelly;Mike C fKelly;Kelly
From the covariance matrix for Generation=1 in Figure 50.4 and the relationship that coancestry is half of
the covariance coefficient,
fJim;Mark
1
D
4
0:4688 0:5 0:25 1:125
C
C
C
2
2
2
2
D 0:29298
For overlapping generations, if X is older than Y, then the basic rule can be simplified to
FZ D fXY D
1
.fXC C fXD /
2
That is, the coancestry between X and Y is the average of coancestries between older X with younger Y’s
parents. For example, in Figure 50.5, the coancestry between ‘Kelly’ and ‘David’ is
fKelly;David D
1
fKelly;Mark C fKelly;Kelly
2
Computational Details F 3991
Figure 50.5 Inbreeding Relationship for Overlapping Population
This is so because ‘Kelly’ is defined before ‘David’; therefore, ‘Kelly’ is not younger than ‘David’, and
the parents of ‘David’ are ‘Mark’ and ‘Kelly’. The covariance coefficient values Cov(Kelly,Mark) and
Cov(Kelly,Kelly) from the matrix in Figure 50.5 yield that the coancestry between ‘Kelly’ and ‘David’ is
fKelly;David
1
D
2
0:5 1:125
C
2
2
D 0:40625
The numerical values for some initial coancestries must be known in order to use these rule. Either the
parents of the first generation have to be unrelated, with f = 0 if the INIT= option is not specified in the
PROC INBREED statement, or their coancestries must have an initial value of 12 cov, where cov is set by the
INIT= option. Then the subsequent coancestries among their progenies and the inbreeding coefficients of
their progenies in the rest of the generations are calculated by using these initial values.
Special rules need to be considered in the calculations of coancestries for the following cases.
3992 F Chapter 50: The INBREED Procedure
Self-Mating
The coancestry for an individual X with itself, fXX , is the inbreeding coefficient of a progeny that is produced
by self-mating. The relationship between the inbreeding coefficient and the coancestry for self-mating is
fXX D
1
.1 C FX /
2
The inbreeding coefficient FX can be replaced by the coancestry between X’s parents A and B, fAB , if A and
B are in the population:
fXX D
1
.1 C fAB /
2
If X’s parents are not in the population, then FX is replaced by the initial value 12 cov if cov is set by the
INIT= option, or FX is replaced by 0 if the INIT= option is not specified. For example, the coancestry of
‘Jim’ with himself is
fJim;Jim D
1
1 C fMark;Kelly
2
where ‘Mark’ and ‘Kelly’ are the parents of ‘Jim’. Since the covariance coefficient Cov(Mark,Kelly) is 0.5 in
Figure 50.5 and also in the covariance matrix for GENDER=1 in Figure 50.4, the coancestry of ‘Jim’ with
himself is
fJim;Jim
1
0:5
D
1C
D 0:625
2
2
When INIT=0.25, then the coancestry of ‘Jane’ with herself is
fJane;Jane
1
0:25
D
1C
D 0:5625
2
2
because ‘Jane’ is not an offspring in the population.
Offspring and Parent Mating
Assuming that X’s parents are A and B, the coancestry between X and A is
fXA D
1
.fAB C fAA /
2
The inbreeding coefficient for an offspring of X and A, denoted by Z, is
Computational Details F 3993
FZ D fXA D
1
.fAB C fAA /
2
For example, ‘Mark’ is an offspring of ‘George’ and ‘Lisa’, so the coancestry between ‘Mark’ and ‘Lisa’ is
fMark;Lisa D
1
fLisa;George C fLisa;Lisa
2
From the covariance coefficient matrix in Figure 50.5, fLisa;George D 0:25=2 D 0:125, fLisa;Lisa D
1:125=2 D 0:5625; so that
fMark;Lisa D
1
.0:125 C 0:5625/ D 0:34375
2
Thus, the inbreeding coefficient for an offspring of ‘Mark’ and ‘Lisa’ is 0.34375.
Full Sibs Mating
This is a special case for the basic rule given at the beginning of the section “Calculation of Coancestry” on
page 3989. If X and Y are full sibs with same parents A and B, then the coancestry between X and Y is
fXY D
1
.2fAB C fAA C fBB /
4
and the inbreeding coefficient for an offspring of A and B, denoted by Z, is
FZ D fXY D
1
.2fAB C fAA C fBB /
4
For example, ‘David’ and ‘Jim’ are full sibs with parents ‘Mark’ and ‘Kelly’, so the coancestry between
‘David’ and ‘Jim’ is
fDavid;Jim D
1
2fMark;Kelly C fMark;Mark C fKelly;Kelly
4
Since the coancestry is half of the covariance coefficient, from the covariance matrix in Figure 50.5,
fDavid;Jim
1
0:5 1:125 1:125
D
2
C
C
D 0:40625
4
2
2
2
3994 F Chapter 50: The INBREED Procedure
Unknown or Missing Parents
When individuals or their parents are unknown in the population, their coancestries are assigned by the value
1
2 cov if cov is set by the INIT= option or by the value 0 if the INIT= option is not specified. That is, if either
A or B is unknown, then
1
fAB D cov
2
For example, ‘Jane’ is not in the population, and since ‘Jane’ is assumed to be defined just before the
observation at which ‘Jane’ appears as a parent (that is, between observations 4 and 5), then ‘Jane’ is not
older than ‘Scott’. The coancestry between ‘Jane’ and ‘Scott’ is then obtained by using the simplified basic
rule (see the section “Calculation of Coancestry” on page 3989):
fScott;Jane D
1
fScott; C fScott;
2
Here, dots () indicate Jane’s unknown parents. Therefore, fScott; is replaced by 12 cov, where cov is set by
the INIT= option. If INIT=0.25, then
fScott;Jane D
1
2
0:25 0:25
C
2
2
D 0:125
For a more detailed discussion on the calculation of coancestries, inbreeding coefficients, and covariance
coefficients, see Falconer and Mackay (1996); Kempthorne (1957); Crow and Kimura (1970).
OUTCOV= Data Set
The OUTCOV= data set has the following variables:
• a list of BY variables, if there is a BY statement
• the generation variable, if there is a CLASS statement
• the gender variable, if there is a GENDER statement
• _Type_, a variable indicating the type of observation. The valid values of the _Type_ variable are
‘COV’ for covariance estimates and ‘INBREED’ for inbreeding coefficients.
• _Panel_, a variable indicating the panel number used when populations delimited by BY groups
contain different numbers of individuals. If there are n individuals in the first BY group and if any
subsequent BY group contains a larger population, then its covariance/inbreeding matrix is divided into
panels, with each panel containing n columns of data. If you put these panels side by side in increasing
_Panel_ number order, then you can reconstruct the covariance or inbreeding matrix.
OUTCOV= Data Set F 3995
• _Col_, a variable used to name columns of the inbreeding or covariance matrix. The values of this
variable start with ‘COL’, followed by a number indicating the column number. The names of the
individuals corresponding to any given column i can be found by reading the individual’s name across
the row that has a _Col_ value of ‘COLi’. When the inbreeding or covariance matrix is divided into
panels, all the rows repeat for the first n columns, all the rows repeat for the next n columns, and so on.
• the variable containing the names of the individuals, that is, the first variable listed in the VAR statement
• the variable containing the names of the first parents, that is, the second variable listed in the VAR
statement
• the variable containing the names of the second parents, that is, the third variable listed in the VAR
statement
• a list of covariance variables Col1–Coln, where n is the maximum number of individuals in the first
population
The functions of the variables _Panel_ and _Col_ can best be demonstrated by an example. Assume that
there are three individuals in the first BY group and that, in the current BY group (Byvar=2), there are five
individuals with the following covariance matrix.
COV
1
2
3
4
5
1
2
3
4
5
Cov(1,1)
Cov(2,1)
Cov(3,1)
Cov(4,1)
Cov(5,1)
Cov(1,2)
Cov(2,2)
Cov(3,2)
Cov(4,2)
Cov(5,2)
Cov(1,3)
Cov(2,3)
Cov(3,3)
Cov(4,3)
Cov(5,3)
Cov(1,4)
Cov(2,4)
Cov(3,4)
Cov(4,4)
Cov(5,4)
Cov(1,5)
Cov(2,5)
Cov(3,5)
Cov(4,5)
Cov(5,5)
Panel 1
Panel 2
Then the OUTCOV= data set appears as follows.
Byvar
_Panel_
_Col_
Individual
2
2
2
2
2
1
1
1
1
1
COL1
COL2
COL3
2
2
2
2
2
2
2
2
2
2
COL1
COL2
Parent
Parent2
Col1
Col2
Col3
1
2
3
4
5
Cov(1,1)
Cov(2,1)
Cov(3,1)
Cov(4,1)
Cov(5,1)
Cov(1,2)
Cov(2,2)
Cov(3,2)
Cov(4,2)
Cov(5,2)
Cov(1,3)
Cov(2,3)
Cov(3,3)
Cov(4,3)
Cov(5,3)
1
2
3
4
5
Cov(1,4)
Cov(2,4)
Cov(3,4)
Cov(4,4)
Cov(5,4)
Cov(1,5)
Cov(2,5)
Cov(3,5)
Cov(4,5)
Cov(5,5)
.
.
.
.
.
Notice that the first three columns go to the first panel (_Panel_=1), and the remaining two go to the second
panel (_Panel_=2). Therefore, in the first panel, ‘COL1’, ‘COL2’, and ‘COL3’ correspond to individuals
1, 2, and 3, respectively, while in the second panel, ‘COL1’ and ‘COL2’ correspond to individuals 4 and 5,
respectively.
3996 F Chapter 50: The INBREED Procedure
Displayed Output
The INBREED procedure can output either covariance coefficients or inbreeding coefficients. Note that the
following items can be produced for each generation if generations do not overlap.
The output produced by PROC INBREED can be any or all of the following items:
• a matrix of coefficients
• coefficients of the individuals
• coefficients for selected matings
ODS Table Names
PROC INBREED assigns a name to each table it creates. You can use these names to reference the table
when using the Output Delivery System (ODS) to select tables and create output data sets. These names are
listed in Table 50.2. For more information on ODS, see Chapter 20, “Using the Output Delivery System.”
Table 50.2 ODS Tables Produced by PROC INBREED
ODS Table Name
Description
Statement
Option
AvgCovCoef
Averages of covariance
coefficient matrix
Averages of inbreeding
coefficient matrix
Covariance coefficient table
Inbreeding coefficient table
Covariance coefficients
of individuals
Inbreeding coefficients
of individuals
Covariance coefficients
of matings
Inbreeding coefficients
of matings
Number of observations
GENDER
COVAR and AVERAGE
GENDER
AVERAGE
PROC
COVAR and MATRIX
PROC
MATRIX
PROC
IND and COVAR
PROC
IND
MATINGS
COVAR
AvgInbreedingCoef
CovarianceCoefficient
InbreedingCoefficient
IndividualCovCoef
IndividualInbreedingCoef
MatingCovCoef
MatingInbreedingCoef
NumberOfObservations
MATINGS
PROC
Example 50.1: Monoecious Population Analysis F 3997
Examples: INBREED Procedure
Example 50.1: Monoecious Population Analysis
The following example shows a covariance analysis within nonoverlapping generations for a monoecious
population. Parents of generation 1 are unknown and therefore assumed to be unrelated. The following
statements produce Output 50.1.1 through Output 50.1.3:
data Monoecious;
input Generation Individual
datalines;
1 1 . . .
1 2 . . .
2 1 1 1 .
2 2 1 2 .
3 1 1 2 .
3 2 1 3 .
3 4 1 3 .
3 . 2 3 0.50
;
Parent1 Parent2 Covariance @@;
1
2
3
3
3
3
3
.
.
2
2
4
. .
3 .
1 .
3 1.135
title 'Inbreeding within Nonoverlapping Generations';
proc inbreed ind covar matrix data=Monoecious;
class Generation;
run;
Output 50.1.1 Monoecious Population Analysis, Generation 1
Inbreeding within Nonoverlapping Generations
The INBREED Procedure
Generation = 1
Covariance Coefficients
Individual
Parent1
Parent2
1
2
3
1
2
3
1.0000
.
.
.
1.0000
.
.
.
1.0000
Covariance Coefficients of Individuals
Individual
Parent1
Parent2
Coefficient
1
2
3
1.0000
1.0000
1.0000
Number of Individuals
3
3998 F Chapter 50: The INBREED Procedure
Output 50.1.2 Monoecious Population Analysis, Generation 2
Inbreeding within Nonoverlapping Generations
The INBREED Procedure
Generation = 2
Covariance Coefficients
Individual
Parent1
Parent2
1
2
3
1
1
2
1
2
3
1
2
3
1.5000
0.5000
.
0.5000
1.0000
0.2500
.
0.2500
1.0000
Covariance Coefficients of Individuals
Individual
Parent1
Parent2
1
2
3
1
1
2
1
2
3
Coefficient
1.5000
1.0000
1.0000
Number of Individuals
3
Output 50.1.3 Monoecious Population Analysis, Generation 3
Inbreeding within Nonoverlapping Generations
The INBREED Procedure
Generation = 3
Covariance Coefficients
Individual
Parent1
Parent2
1
2
3
4
1
1
2
1
2
3
1
3
1
2
3
4
1.2500
0.5625
0.8750
0.5625
0.5625
1.0000
1.1349
0.6250
0.8750
1.1349
1.2500
1.1349
0.5625
0.6250
1.1349
1.0000
Covariance Coefficients of Individuals
Individual
Parent1
Parent2
1
2
3
4
1
1
2
1
2
3
1
3
Coefficient
1.2500
1.0000
1.2500
1.0000
Example 50.2: Pedigree Analysis F 3999
Output 50.1.3 continued
Number of Individuals
4
Note that, since the parents of the first generation are unknown, off-diagonal elements of the covariance
matrix are all 0s and on-diagonal elements are all 1s. If there is an INIT=cov value, then the off-diagonal
elements would be equal to cov, while on-diagonal elements would be equal to 1 C cov =2.
In the third generation, individuals 2 and 4 are full siblings, so they belong to the same family. Since PROC
INBREED computes covariance coefficients between families, the second and fourth columns of inbreeding
coefficients are the same, except that their intersections with the second and fourth rows are reordered. Notice
that, even though there is an observation to assign a covariance of 0.50 between individuals 2 and 3 in the
third generation, the covariance between 2 and 3 is set to 1.135, the same value assigned between 4 and 3.
This is because families get the same covariances, and later specifications override previous ones.
Example 50.2: Pedigree Analysis
In the following example, an inbreeding analysis is performed for a complicated pedigree. This analysis
includes computing selective matings of some individuals and inbreeding coefficients of all individuals. Also,
inbreeding coefficients are averaged within sex categories. The following statements produce Output 50.2.1:
data Swine;
input Swine_Number $ Sire $ Dam $ Sex $;
datalines;
3504 2200 2501 M
3514 2521 3112 F
3519 2521 2501 F
2501 2200 3112 M
2789 3504 3514 F
3501 2521 3514 M
3712 3504 3514 F
3121 2200 3501 F
;
title 'Least Related Matings';
proc inbreed data=Swine ind average;
var Swine_Number Sire Dam;
matings 2501 / 3501 3504 ,
3712 / 3121;
gender Sex;
run;
Note the following from Output 50.2.1:
• Observation 4, which defines Swine_Number=2501, should precede the first and third observations
where the progeny for 2501 are given. PROC INBREED ignores observation 4 since it is given out of
order. As a result, the parents of 2501 are missing or unknown.
4000 F Chapter 50: The INBREED Procedure
• The first column in the “Inbreeding Averages” table corresponds to the averages taken over the ondiagonal elements of the inbreeding coefficients matrix, and the second column gives averages over the
off-diagonal elements.
Output 50.2.1 Pedigree Analysis
Least Related Matings
The INBREED Procedure
Inbreeding Coefficients of Individuals
Swine_
Number
Sire
2200
2501
3504
2521
3112
3514
3519
2789
3501
3712
3121
Dam
2200
2501
2521
2521
3504
2521
3504
2200
3112
2501
3514
3514
3514
3501
Coefficient
.
.
.
.
.
.
.
.
0.2500
.
.
Inbreeding Coefficients of Matings
Sire
Dam
Coefficient
2501
2501
3712
3501
3504
3121
.
0.2500
0.1563
Averages of Inbreeding Coefficient Matrix
Male X Male
Male X Female
Female X Female
Over Sex
Inbreeding
Coancestry
0.0625
.
0.0000
0.0227
0.1042
0.1362
0.1324
0.1313
Number of Males
Number of Females
Number of Individuals
4
7
11
Example 50.3: Pedigree Analysis with BY Groups F 4001
Example 50.3: Pedigree Analysis with BY Groups
This example demonstrates the structure of the OUTCOV= data set created by PROC INBREED. Note that
the first BY group has three individuals, while the second has five. Therefore, the covariance matrix for the
second BY group is broken up into two panels. The following statements produce Output 50.3.1.
data Swine;
input Group Swine_Number $ Sire $ Dam $ Sex $;
datalines;
1 2789 3504 3514 F
2 2501 2200 3112 .
2 3504 2501 3782 M
;
proc inbreed data=Swine covar noprint outcov=Covariance
init=0.4;
var Swine_Number Sire Dam;
gender Sex;
by Group;
run;
title 'Printout of OUTCOV= data set';
proc print data=Covariance;
format Col1-Col3 4.2;
run;
Output 50.3.1 Pedigree Analysis with BY Groups
Printout of OUTCOV= data set
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
Group
Sex
_TYPE_
_PANEL_
_COL_
1
1
1
2
2
2
2
2
2
2
2
2
2
M
F
F
M
F
M
F
M
M
F
M
F
M
COV
COV
COV
COV
COV
COV
COV
COV
COV
COV
COV
COV
COV
1
1
1
1
1
1
1
1
2
2
2
2
2
COL1
COL2
COL3
COL1
COL2
COL3
COL1
COL2
Swine_
Number
3504
3514
2789
2200
3112
2501
3782
3504
2200
3112
2501
3782
3504
Sire
Dam
3504
3514
2200
3112
2501
3782
2200
3112
2501
3782
COL1
COL2
COL3
1.20
0.40
0.80
1.20
0.40
0.80
0.40
0.60
0.40
0.40
0.40
1.20
0.80
0.40
1.20
0.80
0.40
1.20
0.80
0.40
0.60
0.60
0.60
0.80
0.80
1.20
0.80
0.80
1.20
0.80
0.80
1.20
0.40
0.80
.
.
.
.
.
4002 F Chapter 50: The INBREED Procedure
References
Crow, J. F. and Kimura, M. (1970), An Introduction to Population Genetics Theory, New York: Harper &
Row.
Falconer, D. S. and Mackay, T. F. C. (1996), Introduction to Quantitative Genetics, 4th Edition, London:
Longman.
Kempthorne, O. (1957), An Introduction to Genetic Statistics, New York: John Wiley & Sons.
Subject Index
coefficient
of relationship (INBREED), 3988
covariance coefficients, see INBREED procedure
full sibs mating
INBREED procedure, 3993
generation (INBREED)
nonoverlapping, 3977, 3981
number, 3985
overlapping, 3977, 3979
variable, 3985
INBREED procedure
coancestry, computing, 3989
coefficient of relationship, computing, 3988
covariance coefficients, 3977, 3979, 3981, 3984,
3986, 3988
covariance coefficients matrix, output, 3984
first parent, 3986
full sibs mating, 3993
generation number, 3985
generation variable, 3985
generation, nonoverlapping, 3977, 3981
generation, overlapping, 3977, 3979
inbreeding coefficients, 3977, 3979, 3984, 3986,
3989
inbreeding coefficients matrix, output, 3984
individuals, outputting coefficients, 3984
individuals, specifying, 3981, 3985, 3986
initial covariance value, 3987
initial covariance value, assigning, 3984
initial covariance value, specifying, 3979
kinship coefficient, 3988
last generation’s coefficients, output, 3984
mating, offspring and parent, 3992, 3993
mating, self, 3992
matings, output, 3986
monoecious population analysis, example, 3997
offspring, 3984, 3991
ordering observations, 3978
OUTCOV= data set, 3984, 3994
output table names, 3996
panels, 3994, 4001
pedigree analysis, 3977, 3978
pedigree analysis, example, 3999, 4001
population, monoecious, 3997
population, multiparous, 3984, 3988
population, nonoverlapping, 3985
population, overlapping, 3978, 3979, 3990
progeny, 3987, 3989, 3992, 3999
second parent, 3986
selective matings, output, 3986
specifying gender, 3981
theoretical correlation, 3988
unknown or missing parents, 3994
variables, unaddressed, 3986, 3987
initial covariance value
assigning (INBREED), 3984
INBREED procedure, 3987
specifying (INBREED), 3979
mating
offspring and parent (INBREED), 3992, 3993
self (INBREED), 3992
monoecious population analysis
example (INBREED), 3997
offspring
INBREED procedure, 3984, 3991
ordering observations
INBREED procedure, 3978
output data sets
OUTCOV= data set (INBREED), 3984, 3994
output table names
INBREED procedure, 3996
panels
INBREED procedure, 3994, 4001
pedigree analysis
example (INBREED), 3999, 4001
INBREED procedure, 3977, 3978
population (INBREED)
monoecious, 3997
multiparous, 3984, 3988
nonoverlapping, 3985
overlapping, 3978, 3979, 3990
progeny
INBREED procedure, 3987, 3989, 3992, 3999
theoretical correlation
INBREED procedure, 3988
unknown or missing parents
INBREED procedure, 3994
variables, unaddressed
INBREED procedure, 3986, 3987
Syntax Index
AVERAGE option
PROC INBREED statement, 3984
BY statement
INBREED procedure, 3985
PROC INBREED statement, 3984
OUTCOV= option
PROC INBREED statement, 3984
PROC INBREED statement, see INBREED procedure
CLASS statement
INBREED procedure, 3985
COVAR option
PROC INBREED statement, 3984
DATA= option
PROC INBREED statement, 3984
GENDER statement, INBREED procedure, 3985
INBREED procedure
syntax, 3983
INBREED procedure, BY statement, 3985
INBREED procedure, CLASS statement, 3985
INBREED procedure, GENDER statement, 3985
INBREED procedure, MATINGS statement, 3986
INBREED procedure, PROC INBREED statement,
3983
AVERAGE option, 3984
COVAR option, 3984
DATA= option, 3984
IND option, 3984
INDL option, 3984
INIT= option, 3984
MATRIX option, 3984
MATRIXL option, 3984
NOPRINT option, 3984
OUTCOV= option, 3984
SELFDIAG option, 3984
INBREED procedure, VAR statement, 3986
IND option
PROC INBREED statement, 3984
INDL option
PROC INBREED statement, 3984
INIT= option
PROC INBREED statement, 3984
MATINGS statement, INBREED procedure, 3986
MATRIX option
PROC INBREED statement, 3984
MATRIXL option
PROC INBREED statement, 3984
NOPRINT option
SELFDIAG option
PROC INBREED statement, 3984
VAR statement
INBREED procedure, 3986
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