SPSS Regression 22

SPSS Regression 22
IBM SPSS Regression 22
Note
Before using this information and the product it supports, read the information in “Notices” on page 31.
Product Information
This edition applies to version 22, release 0, modification 0 of IBM® SPSS® Statistics and to all subsequent releases
and modifications until otherwise indicated in new editions.
Contents
Chapter 1. Choosing a Procedure for
Binary Logistic Regression . . . . . . 1
Chapter 2. Logistic Regression . . . . . 3
Logistic Regression Set Rule . . . . . . .
Logistic Regression Variable Selection Methods .
Logistic Regression Define Categorical Variables
Logistic Regression Save New Variables . . .
Logistic Regression Options . . . . . . .
LOGISTIC REGRESSION Command Additional
Features . . . . . . . . . . . . . .
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Chapter 3. Multinomial Logistic
Regression . . . . . . . . . . . . . 9
Multinomial Logistic Regression . . . .
Build Terms . . . . . . . . .
Multinomial Logistic Regression Reference
Multinomial Logistic Regression Statistics
Multinomial Logistic Regression Criteria.
Multinomial Logistic Regression Options
Multinomial Logistic Regression Save .
NOMREG Command Additional Features
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Category 10
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Chapter 4. Probit Analysis . . . . . . 15
Probit Analysis Define Range . . . .
Probit Analysis Options . . . . . .
PROBIT Command Additional Features .
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Nonlinear Regression Loss Function . . .
Nonlinear Regression Parameter Constraints
Nonlinear Regression Save New Variables .
Nonlinear Regression Options . . . . .
Interpreting Nonlinear Regression Results .
NLR Command Additional Features . . .
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Chapter 6. Weight Estimation . . . . . 23
Weight Estimation Options . . . .
WLS Command Additional Features .
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Chapter 7. Two-Stage Least-Squares
Regression . . . . . . . . . . . . . 25
Two-Stage Least-Squares Regression Options .
2SLS Command Additional Features . . . .
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Chapter 8. Categorical Variable Coding
Schemes . . . . . . . . . . . . . . 27
Deviation .
Simple . .
Helmert .
Difference .
Polynomial
Repeated .
Special . .
Indicator .
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Notices . . . . . . . . . . . . . . 31
Chapter 5. Nonlinear Regression . . . 17
Trademarks .
Conditional Logic (Nonlinear Regression) .
Nonlinear Regression Parameters . . . .
Nonlinear Regression Common Models . .
Index . . . . . . . . . . . . . . . 35
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iii
iv
IBM SPSS Regression 22
Chapter 1. Choosing a Procedure for Binary Logistic
Regression
Binary logistic regression models can be fitted using the Logistic Regression procedure and the
Multinomial Logistic Regression procedure. Each procedure has options not available in the other. An
important theoretical distinction is that the Logistic Regression procedure produces all predictions,
residuals, influence statistics, and goodness-of-fit tests using data at the individual case level, regardless
of how the data are entered and whether or not the number of covariate patterns is smaller than the total
number of cases, while the Multinomial Logistic Regression procedure internally aggregates cases to form
subpopulations with identical covariate patterns for the predictors, producing predictions, residuals, and
goodness-of-fit tests based on these subpopulations. If all predictors are categorical or any continuous
predictors take on only a limited number of values—so that there are several cases at each distinct
covariate pattern—the subpopulation approach can produce valid goodness-of-fit tests and informative
residuals, while the individual case level approach cannot.
Logistic Regression provides the following unique features:
v Hosmer-Lemeshow test of goodness of fit for the model
v Stepwise analyses
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v
v
v
v
Contrasts to define model parameterization
Alternative cut points for classification
Classification plots
Model fitted on one set of cases to a held-out set of cases
Saves predictions, residuals, and influence statistics
Multinomial Logistic Regression provides the following unique features:
v Pearson and deviance chi-square tests for goodness of fit of the model
v
v
v
v
v
Specification of subpopulations for grouping of data for goodness-of-fit tests
Listing of counts, predicted counts, and residuals by subpopulations
Correction of variance estimates for over-dispersion
Covariance matrix of the parameter estimates
Tests of linear combinations of parameters
v Explicit specification of nested models
v Fit 1-1 matched conditional logistic regression models using differenced variables
Note: Both of these procedures fit a model for binary data that is a generalized linear model with a
binomial distribution and logit link function. If a different link function is more appropriate for your
data, then you should use the Generalized Linear Models procedure.
Note: If you have repeated measurements of binary data, or records that are otherwise correlated, then
you should consider the Generalized Linear Mixed Models or Generalized Estimating Equations
procedures.
© Copyright IBM Corporation 1989, 2013
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IBM SPSS Regression 22
Chapter 2. Logistic Regression
Logistic regression is useful for situations in which you want to be able to predict the presence or absence
of a characteristic or outcome based on values of a set of predictor variables. It is similar to a linear
regression model but is suited to models where the dependent variable is dichotomous. Logistic
regression coefficients can be used to estimate odds ratios for each of the independent variables in the
model. Logistic regression is applicable to a broader range of research situations than discriminant
analysis.
Example. What lifestyle characteristics are risk factors for coronary heart disease (CHD)? Given a sample
of patients measured on smoking status, diet, exercise, alcohol use, and CHD status, you could build a
model using the four lifestyle variables to predict the presence or absence of CHD in a sample of
patients. The model can then be used to derive estimates of the odds ratios for each factor to tell you, for
example, how much more likely smokers are to develop CHD than nonsmokers.
Statistics. For each analysis: total cases, selected cases, valid cases. For each categorical variable:
parameter coding. For each step: variable(s) entered or removed, iteration history, –2 log-likelihood,
goodness of fit, Hosmer-Lemeshow goodness-of-fit statistic, model chi-square, improvement chi-square,
classification table, correlations between variables, observed groups and predicted probabilities chart,
residual chi-square. For each variable in the equation: coefficient (B), standard error of B, Wald statistic,
estimated odds ratio (exp(B)), confidence interval for exp(B), log-likelihood if term removed from model.
For each variable not in the equation: score statistic. For each case: observed group, predicted probability,
predicted group, residual, standardized residual.
Methods. You can estimate models using block entry of variables or any of the following stepwise
methods: forward conditional, forward LR, forward Wald, backward conditional, backward LR, or
backward Wald.
Logistic Regression Data Considerations
Data. The dependent variable should be dichotomous. Independent variables can be interval level or
categorical; if categorical, they should be dummy or indicator coded (there is an option in the procedure
to recode categorical variables automatically).
Assumptions. Logistic regression does not rely on distributional assumptions in the same sense that
discriminant analysis does. However, your solution may be more stable if your predictors have a
multivariate normal distribution. Additionally, as with other forms of regression, multicollinearity among
the predictors can lead to biased estimates and inflated standard errors. The procedure is most effective
when group membership is a truly categorical variable; if group membership is based on values of a
continuous variable (for example, "high IQ" versus "low IQ"), you should consider using linear regression
to take advantage of the richer information offered by the continuous variable itself.
Related procedures. Use the Scatterplot procedure to screen your data for multicollinearity. If
assumptions of multivariate normality and equal variance-covariance matrices are met, you may be able
to get a quicker solution using the Discriminant Analysis procedure. If all of your predictor variables are
categorical, you can also use the Loglinear procedure. If your dependent variable is continuous, use the
Linear Regression procedure. You can use the ROC Curve procedure to plot probabilities saved with the
Logistic Regression procedure.
Obtaining a Logistic Regression Analysis
1. From the menus choose:
Analyze > Regression > Binary Logistic...
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2. Select one dichotomous dependent variable. This variable may be numeric or string.
3. Select one or more covariates. To include interaction terms, select all of the variables involved in the
interaction and then select >a*b>.
To enter variables in groups (blocks), select the covariates for a block, and click Next to specify a new
block. Repeat until all blocks have been specified.
Optionally, you can select cases for analysis. Choose a selection variable, and click Rule.
Logistic Regression Set Rule
Cases defined by the selection rule are included in model estimation. For example, if you selected a
variable and equals and specified a value of 5, then only the cases for which the selected variable has a
value equal to 5 are included in estimating the model.
Statistics and classification results are generated for both selected and unselected cases. This provides a
mechanism for classifying new cases based on previously existing data, or for partitioning your data into
training and testing subsets, to perform validation on the model generated.
Logistic Regression Variable Selection Methods
Method selection allows you to specify how independent variables are entered into the analysis. Using
different methods, you can construct a variety of regression models from the same set of variables.
Enter. A procedure for variable selection in which all variables in a block are entered in a single step.
Forward Selection (Conditional). Stepwise selection method with entry testing based on the significance
of the score statistic, and removal testing based on the probability of a likelihood-ratio statistic based
on conditional parameter estimates.
v Forward Selection (Likelihood Ratio). Stepwise selection method with entry testing based on the
significance of the score statistic, and removal testing based on the probability of a likelihood-ratio
statistic based on the maximum partial likelihood estimates.
v Forward Selection (Wald). Stepwise selection method with entry testing based on the significance of the
score statistic, and removal testing based on the probability of the Wald statistic.
v Backward Elimination (Conditional). Backward stepwise selection. Removal testing is based on the
probability of the likelihood-ratio statistic based on conditional parameter estimates.
v
v
Backward Elimination (Likelihood Ratio). Backward stepwise selection. Removal testing is based on the
probability of the likelihood-ratio statistic based on the maximum partial likelihood estimates.
v Backward Elimination (Wald). Backward stepwise selection. Removal testing is based on the probability
of the Wald statistic.
v
The significance values in your output are based on fitting a single model. Therefore, the significance
values are generally invalid when a stepwise method is used.
All independent variables selected are added to a single regression model. However, you can specify
different entry methods for different subsets of variables. For example, you can enter one block of
variables into the regression model using stepwise selection and a second block using forward selection.
To add a second block of variables to the regression model, click Next.
Logistic Regression Define Categorical Variables
You can specify details of how the Logistic Regression procedure will handle categorical variables:
Covariates. Contains a list of all of the covariates specified in the main dialog box, either by themselves
or as part of an interaction, in any layer. If some of these are string variables or are categorical, you can
use them only as categorical covariates.
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IBM SPSS Regression 22
Categorical Covariates. Lists variables identified as categorical. Each variable includes a notation in
parentheses indicating the contrast coding to be used. String variables (denoted by the symbol <
following their names) are already present in the Categorical Covariates list. Select any other categorical
covariates from the Covariates list and move them into the Categorical Covariates list.
Change Contrast. Allows you to change the contrast method. Available contrast methods are:
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Indicator. Contrasts indicate the presence or absence of category membership. The reference category
is represented in the contrast matrix as a row of zeros.
Simple. Each category of the predictor variable (except the reference category) is compared to the
reference category.
Difference. Each category of the predictor variable except the first category is compared to the
average effect of previous categories. Also known as reverse Helmert contrasts.
Helmert. Each category of the predictor variable except the last category is compared to the average
effect of subsequent categories.
Repeated. Each category of the predictor variable except the first category is compared to the category
that precedes it.
Polynomial. Orthogonal polynomial contrasts. Categories are assumed to be equally spaced.
Polynomial contrasts are available for numeric variables only.
v Deviation. Each category of the predictor variable except the reference category is compared to the
overall effect.
v
If you select Deviation, Simple, or Indicator, select either First or Last as the reference category. Note
that the method is not actually changed until you click Change.
String covariates must be categorical covariates. To remove a string variable from the Categorical
Covariates list, you must remove all terms containing the variable from the Covariates list in the main
dialog box.
Logistic Regression Save New Variables
You can save results of the logistic regression as new variables in the active dataset:
Predicted Values. Saves values predicted by the model. Available options are Probabilities and Group
membership.
v Probabilities. For each case, saves the predicted probability of occurrence of the event. A table in the
output displays name and contents of any new variables. The "event" is the category of the dependent
variable with the higher value; for example, if the dependent variable takes values 0 and 1, the
predicted probability of category 1 is saved.
v Predicted Group Membership. The group with the largest posterior probability, based on discriminant
scores. The group the model predicts the case belongs to.
Influence. Saves values from statistics that measure the influence of cases on predicted values. Available
options are Cook's, Leverage values, and DfBeta(s).
v Cook's. The logistic regression analog of Cook's influence statistic. A measure of how much the
residuals of all cases would change if a particular case were excluded from the calculation of the
regression coefficients.
v Leverage Value. The relative influence of each observation on the model's fit.
v DfBeta(s). The difference in beta value is the change in the regression coefficient that results from the
exclusion of a particular case. A value is computed for each term in the model, including the constant.
Residuals. Saves residuals. Available options are Unstandardized, Logit, Studentized, Standardized, and
Deviance.
Chapter 2. Logistic Regression
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Unstandardized Residuals. The difference between an observed value and the value predicted by the
model.
v Logit Residual. The residual for the case if it is predicted in the logit scale. The logit residual is the
residual divided by the predicted probability times 1 minus the predicted probability.
v Studentized Residual. The change in the model deviance if a case is excluded.
v Standardized Residuals. The residual divided by an estimate of its standard deviation. Standardized
residuals, which are also known as Pearson residuals, have a mean of 0 and a standard deviation of 1.
v
v
Deviance. Residuals based on the model deviance.
Export model information to XML file. Parameter estimates and (optionally) their covariances are
exported to the specified file in XML (PMML) format. You can use this model file to apply the model
information to other data files for scoring purposes.
Logistic Regression Options
You can specify options for your logistic regression analysis:
Statistics and Plots. Allows you to request statistics and plots. Available options are Classification plots,
Hosmer-Lemeshow goodness-of-fit, Casewise listing of residuals, Correlations of estimates, Iteration
history, and CI for exp(B). Select one of the alternatives in the Display group to display statistics and
plots either At each step or, only for the final model, At last step.
v Hosmer-Lemeshow goodness-of-fit statistic. This goodness-of-fit statistic is more robust than the traditional
goodness-of-fit statistic used in logistic regression, particularly for models with continuous covariates
and studies with small sample sizes. It is based on grouping cases into deciles of risk and comparing
the observed probability with the expected probability within each decile.
Probability for Stepwise. Allows you to control the criteria by which variables are entered into and
removed from the equation. You can specify criteria for Entry or Removal of variables.
v Probability for Stepwise. A variable is entered into the model if the probability of its score statistic is less
than the Entry value and is removed if the probability is greater than the Removal value. To override
the default settings, enter positive values for Entry and Removal. Entry must be less than Removal.
Classification cutoff. Allows you to determine the cut point for classifying cases. Cases with predicted
values that exceed the classification cutoff are classified as positive, while those with predicted values
smaller than the cutoff are classified as negative. To change the default, enter a value between 0.01 and
0.99.
Maximum Iterations. Allows you to change the maximum number of times that the model iterates before
terminating.
Include constant in model. Allows you to indicate whether the model should include a constant term. If
disabled, the constant term will equal 0.
LOGISTIC REGRESSION Command Additional Features
The command syntax language also allows you to:
v Identify casewise output by the values or variable labels of a variable.
v Control the spacing of iteration reports. Rather than printing parameter estimates after every iteration,
you can request parameter estimates after every nth iteration.
v Change the criteria for terminating iteration and checking for redundancy.
v Specify a variable list for casewise listings.
v Conserve memory by holding the data for each split file group in an external scratch file during
processing.
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IBM SPSS Regression 22
See the Command Syntax Reference for complete syntax information.
Chapter 2. Logistic Regression
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IBM SPSS Regression 22
Chapter 3. Multinomial Logistic Regression
Multinomial Logistic Regression is useful for situations in which you want to be able to classify subjects
based on values of a set of predictor variables. This type of regression is similar to logistic regression, but
it is more general because the dependent variable is not restricted to two categories.
Example. In order to market films more effectively, movie studios want to predict what type of film a
moviegoer is likely to see. By performing a Multinomial Logistic Regression, the studio can determine the
strength of influence a person's age, gender, and dating status has upon the type of film they prefer. The
studio can then slant the advertising campaign of a particular movie toward a group of people likely to
go see it.
Statistics. Iteration history, parameter coefficients, asymptotic covariance and correlation matrices,
likelihood-ratio tests for model and partial effects, –2 log-likelihood. Pearson and deviance chi-square
goodness of fit. Cox and Snell, Nagelkerke, and McFadden R 2. Classification: observed versus predicted
frequencies by response category. Crosstabulation: observed and predicted frequencies (with residuals)
and proportions by covariate pattern and response category.
Methods. A multinomial logit model is fit for the full factorial model or a user-specified model.
Parameter estimation is performed through an iterative maximum-likelihood algorithm.
Multinomial Logistic Regression Data Considerations
Data. The dependent variable should be categorical. Independent variables can be factors or covariates. In
general, factors should be categorical variables and covariates should be continuous variables.
Assumptions. It is assumed that the odds ratio of any two categories are independent of all other
response categories. For example, if a new product is introduced to a market, this assumption states that
the market shares of all other products are affected proportionally equally. Also, given a covariate pattern,
the responses are assumed to be independent multinomial variables.
Obtaining a Multinomial Logistic Regression
1. From the menus choose:
Analyze > Regression > Multinomial Logistic...
2. Select one dependent variable.
3. Factors are optional and can be either numeric or categorical.
4. Covariates are optional but must be numeric if specified.
Multinomial Logistic Regression
By default, the Multinomial Logistic Regression procedure produces a model with the factor and
covariate main effects, but you can specify a custom model or request stepwise model selection with this
dialog box.
Specify Model. A main-effects model contains the covariate and factor main effects but no interaction
effects. A full factorial model contains all main effects and all factor-by-factor interactions. It does not
contain covariate interactions. You can create a custom model to specify subsets of factor interactions or
covariate interactions, or request stepwise selection of model terms.
Factors & Covariates. The factors and covariates are listed.
© Copyright IBM Corporation 1989, 2013
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Forced Entry Terms. Terms added to the forced entry list are always included in the model.
Stepwise Terms. Terms added to the stepwise list are included in the model according to one of the
following user-selected Stepwise Methods:
v Forward entry. This method begins with no stepwise terms in the model. At each step, the most
significant term is added to the model until none of the stepwise terms left out of the model would
have a statistically significant contribution if added to the model.
v Backward elimination. This method begins by entering all terms specified on the stepwise list into the
model. At each step, the least significant stepwise term is removed from the model until all of the
remaining stepwise terms have a statistically significant contribution to the model.
v Forward stepwise. This method begins with the model that would be selected by the forward entry
method. From there, the algorithm alternates between backward elimination on the stepwise terms in
the model and forward entry on the terms left out of the model. This continues until no terms meet the
entry or removal criteria.
v Backward stepwise. This method begins with the model that would be selected by the backward
elimination method. From there, the algorithm alternates between forward entry on the terms left out
of the model and backward elimination on the stepwise terms in the model. This continues until no
terms meet the entry or removal criteria.
Include intercept in model. Allows you to include or exclude an intercept term for the model.
Build Terms
For the selected factors and covariates:
Interaction. Creates the highest-level interaction term of all selected variables.
Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.
All 4-way. Creates all possible four-way interactions of the selected variables.
All 5-way. Creates all possible five-way interactions of the selected variables.
Multinomial Logistic Regression Reference Category
By default, the Multinomial Logistic Regression procedure makes the last category the reference category.
This dialog box gives you control of the reference category and the way in which categories are ordered.
Reference Category. Specify the first, last, or a custom category.
Category Order. In ascending order, the lowest value defines the first category and the highest value
defines the last. In descending order, the highest value defines the first category and the lowest value
defines the last.
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IBM SPSS Regression 22
Multinomial Logistic Regression Statistics
You can specify the following statistics for your Multinomial Logistic Regression:
Case processing summary. This table contains information about the specified categorical variables.
Model. Statistics for the overall model.
v Pseudo R-square. Prints the Cox and Snell, Nagelkerke, and McFadden R 2 statistics.
v Step summary. This table summarizes the effects entered or removed at each step in a stepwise
method. It is not produced unless a stepwise model is specified in the Model dialog box.
v Model fitting information. This table compares the fitted and intercept-only or null models.
v Information criteria. This table prints Akaike’s information criterion (AIC) and Schwarz’s Bayesian
information criterion (BIC).
v Cell probabilities. Prints a table of the observed and expected frequencies (with residual) and
proportions by covariate pattern and response category.
v Classification table. Prints a table of the observed versus predicted responses.
v Goodness of fit chi-square statistics. Prints Pearson and likelihood-ratio chi-square statistics. Statistics
are computed for the covariate patterns determined by all factors and covariates or by a user-defined
subset of the factors and covariates.
v Monotinicity measures. Displays a table with information on the number of concordant pairs,
discordant pairs, and tied pairs. The Somers' D, Goodman and Kruskal's Gamma, Kendall's tau-a, and
Concordance Index C are also displayed in this table.
Parameters. Statistics related to the model parameters.
v Estimates. Prints estimates of the model parameters, with a user-specified level of confidence.
v Likelihood ratio test. Prints likelihood-ratio tests for the model partial effects. The test for the overall
model is printed automatically.
v Asymptotic correlations. Prints matrix of parameter estimate correlations.
v Asymptotic covariances. Prints matrix of parameter estimate covariances.
Define Subpopulations. Allows you to select a subset of the factors and covariates in order to define the
covariate patterns used by cell probabilities and the goodness-of-fit tests.
Multinomial Logistic Regression Criteria
You can specify the following criteria for your Multinomial Logistic Regression:
Iterations. Allows you to specify the maximum number of times you want to cycle through the
algorithm, the maximum number of steps in the step-halving, the convergence tolerances for changes in
the log-likelihood and parameters, how often the progress of the iterative algorithm is printed, and at
what iteration the procedure should begin checking for complete or quasi-complete separation of the
data.
v Log-likelihood convergence. Convergence is assumed if the absolute change in the log-likelihood
function is less than the specified value. The criterion is not used if the value is 0. Specify a
non-negative value.
v Parameter convergence. Convergence is assumed if the absolute change in the parameter estimates is
less than this value. The criterion is not used if the value is 0.
Delta. Allows you to specify a non-negative value less than 1. This value is added to each empty cell of
the crosstabulation of response category by covariate pattern. This helps to stabilize the algorithm and
prevent bias in the estimates.
Chapter 3. Multinomial Logistic Regression
11
Singularity tolerance. Allows you to specify the tolerance used in checking for singularities.
Multinomial Logistic Regression Options
You can specify the following options for your Multinomial Logistic Regression:
Dispersion Scale. Allows you to specify the dispersion scaling value that will be used to correct the
estimate of the parameter covariance matrix. Deviance estimates the scaling value using the deviance
function (likelihood-ratio chi-square) statistic. Pearson estimates the scaling value using the Pearson
chi-square statistic. You can also specify your own scaling value. It must be a positive numeric value.
Stepwise Options. These options give you control of the statistical criteria when stepwise methods are
used to build a model. They are ignored unless a stepwise model is specified in the Model dialog box.
v Entry Probability. This is the probability of the likelihood-ratio statistic for variable entry. The larger
the specified probability, the easier it is for a variable to enter the model. This criterion is ignored
unless the forward entry, forward stepwise, or backward stepwise method is selected.
v Entry test. This is the method for entering terms in stepwise methods. Choose between the
likelihood-ratio test and score test. This criterion is ignored unless the forward entry, forward stepwise,
or backward stepwise method is selected.
v Removal Probability. This is the probability of the likelihood-ratio statistic for variable removal. The
larger the specified probability, the easier it is for a variable to remain in the model. This criterion is
ignored unless the backward elimination, forward stepwise, or backward stepwise method is selected.
v Removal Test. This is the method for removing terms in stepwise methods. Choose between the
likelihood-ratio test and Wald test. This criterion is ignored unless the backward elimination, forward
stepwise, or backward stepwise method is selected.
v Minimum Stepped Effects in Model. When using the backward elimination or backward stepwise
methods, this specifies the minimum number of terms to include in the model. The intercept is not
counted as a model term.
v Maximum Stepped Effects in Model. When using the forward entry or forward stepwise methods,
this specifies the maximum number of terms to include in the model. The intercept is not counted as a
model term.
v Hierarchically constrain entry and removal of terms. This option allows you to choose whether to
place restrictions on the inclusion of model terms. Hierarchy requires that for any term to be included,
all lower order terms that are a part of the term to be included must be in the model first. For
example, if the hierarchy requirement is in effect, the factors Marital status and Gender must both be in
the model before the Marital status*Gender interaction can be added. The three radio button options
determine the role of covariates in determining hierarchy.
Multinomial Logistic Regression Save
The Save dialog box allows you to save variables to the working file and export model information to an
external file.
Saved variables. The following variables can be saved:
Estimated response probabilities. These are the estimated probabilities of classifying a
factor/covariate pattern into the response categories. There are as many estimated probabilities as there
are categories of the response variable; up to 25 will be saved.
v Predicted category. This is the response category with the largest expected probability for a
factor/covariate pattern.
v Predicted category probabilities. This is the maximum of the estimated response probabilities.
v Actual category probability. This is the estimated probability of classifying a factor/covariate pattern
into the observed category.
v
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IBM SPSS Regression 22
Export model information to XML file. Parameter estimates and (optionally) their covariances are
exported to the specified file in XML (PMML) format. You can use this model file to apply the model
information to other data files for scoring purposes.
NOMREG Command Additional Features
The command syntax language also allows you to:
v Specify the reference category of the dependent variable.
v Include cases with user-missing values.
v Customize hypothesis tests by specifying null hypotheses as linear combinations of parameters.
See the Command Syntax Reference for complete syntax information.
Chapter 3. Multinomial Logistic Regression
13
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IBM SPSS Regression 22
Chapter 4. Probit Analysis
This procedure measures the relationship between the strength of a stimulus and the proportion of cases
exhibiting a certain response to the stimulus. It is useful for situations where you have a dichotomous
output that is thought to be influenced or caused by levels of some independent variable(s) and is
particularly well suited to experimental data. This procedure will allow you to estimate the strength of a
stimulus required to induce a certain proportion of responses, such as the median effective dose.
Example. How effective is a new pesticide at killing ants, and what is an appropriate concentration to
use? You might perform an experiment in which you expose samples of ants to different concentrations
of the pesticide and then record the number of ants killed and the number of ants exposed. Applying
probit analysis to these data, you can determine the strength of the relationship between concentration
and killing, and you can determine what the appropriate concentration of pesticide would be if you
wanted to be sure to kill, say, 95% of exposed ants.
Statistics. Regression coefficients and standard errors, intercept and standard error, Pearson
goodness-of-fit chi-square, observed and expected frequencies, and confidence intervals for effective levels
of independent variable(s). Plots: transformed response plots.
®
This procedure uses the algorithms proposed and implemented in NPSOL by Gill, Murray, Saunders &
Wright to estimate the model parameters.
Probit Analysis Data Considerations
Data. For each value of the independent variable (or each combination of values for multiple
independent variables), your response variable should be a count of the number of cases with those
values that show the response of interest, and the total observed variable should be a count of the total
number of cases with those values for the independent variable. The factor variable should be categorical,
coded as integers.
Assumptions. Observations should be independent. If you have a large number of values for the
independent variables relative to the number of observations, as you might in an observational study, the
chi-square and goodness-of-fit statistics may not be valid.
Related procedures. Probit analysis is closely related to logistic regression; in fact, if you choose the logit
transformation, this procedure will essentially compute a logistic regression. In general, probit analysis is
appropriate for designed experiments, whereas logistic regression is more appropriate for observational
studies. The differences in output reflect these different emphases. The probit analysis procedure reports
estimates of effective values for various rates of response (including median effective dose), while the
logistic regression procedure reports estimates of odds ratios for independent variables.
Obtaining a Probit Analysis
1. From the menus choose:
Analyze > Regression > Probit...
2. Select a response frequency variable. This variable indicates the number of cases exhibiting a response
to the test stimulus. The values of this variable cannot be negative.
3. Select a total observed variable. This variable indicates the number of cases to which the stimulus was
applied. The values of this variable cannot be negative and cannot be less than the values of the
response frequency variable for each case.
Optionally, you can select a Factor variable. If you do, click Define Range to define the groups.
© Copyright IBM Corporation 1989, 2013
15
4. Select one or more covariate(s). This variable contains the level of the stimulus applied to each
observation. If you want to transform the covariate, select a transformation from the Transform
drop-down list. If no transformation is applied and there is a control group, then the control group is
included in the analysis.
5. Select either the Probit or Logit model.
v Probit Model. Applies the probit transformation (the inverse of the cumulative standard normal
distribution function) to the response proportions.
v Logit Model. Applies the logit (log odds) transformation to the response proportions.
Probit Analysis Define Range
This allows you to specify the levels of the factor variable that will be analyzed. The factor levels must be
coded as consecutive integers, and all levels in the range that you specify will be analyzed.
Probit Analysis Options
You can specify options for your probit analysis:
Statistics. Allows you to request the following optional statistics: Frequencies, Relative median potency,
Parallelism test, and Fiducial confidence intervals.
v Relative Median Potency. Displays the ratio of median potencies for each pair of factor levels. Also
shows 95% confidence limits for each relative median potency. Relative median potencies are not
available if you do not have a factor variable or if you have more than one covariate.
v
v
Parallelism Test. A test of the hypothesis that all factor levels have a common slope.
Fiducial Confidence Intervals. Confidence intervals for the dosage of agent required to produce a certain
probability of response.
Fiducial confidence intervals and Relative median potency are unavailable if you have selected more than
one covariate. Relative median potency and Parallelism test are available only if you have selected a
factor variable.
Natural Response Rate. Allows you to indicate a natural response rate even in the absence of the
stimulus. Available alternatives are None, Calculate from data, or Value.
v Calculate from Data. Estimate the natural response rate from the sample data. Your data should contain
a case representing the control level, for which the value of the covariate(s) is 0. Probit estimates the
natural response rate using the proportion of responses for the control level as an initial value.
v Value. Sets the natural response rate in the model (select this item when you know the natural
response rate in advance). Enter the natural response proportion (the proportion must be less than 1).
For example, if the response occurs 10% of the time when the stimulus is 0, enter 0.10.
Criteria. Allows you to control parameters of the iterative parameter-estimation algorithm. You can
override the defaults for Maximum iterations, Step limit, and Optimality tolerance.
PROBIT Command Additional Features
The command syntax language also allows you to:
v Request an analysis on both the probit and logit models.
v Control the treatment of missing values.
v Transform the covariates by bases other than base 10 or natural log.
See the Command Syntax Reference for complete syntax information.
16
IBM SPSS Regression 22
Chapter 5. Nonlinear Regression
Nonlinear regression is a method of finding a nonlinear model of the relationship between the dependent
variable and a set of independent variables. Unlike traditional linear regression, which is restricted to
estimating linear models, nonlinear regression can estimate models with arbitrary relationships between
independent and dependent variables. This is accomplished using iterative estimation algorithms. Note
that this procedure is not necessary for simple polynomial models of the form Y = A + BX**2. By defining
W = X**2, we get a simple linear model, Y = A + BW, which can be estimated using traditional methods
such as the Linear Regression procedure.
Example. Can population be predicted based on time? A scatterplot shows that there seems to be a strong
relationship between population and time, but the relationship is nonlinear, so it requires the special
estimation methods of the Nonlinear Regression procedure. By setting up an appropriate equation, such
as a logistic population growth model, we can get a good estimate of the model, allowing us to make
predictions about population for times that were not actually measured.
Statistics. For each iteration: parameter estimates and residual sum of squares. For each model: sum of
squares for regression, residual, uncorrected total and corrected total, parameter estimates, asymptotic
standard errors, and asymptotic correlation matrix of parameter estimates.
®
Note: Constrained nonlinear regression uses the algorithms proposed and implemented in NPSOL by
Gill, Murray, Saunders, and Wright to estimate the model parameters.
Nonlinear Regression Data Considerations
Data. The dependent and independent variables should be quantitative. Categorical variables, such as
religion, major, or region of residence, need to be recoded to binary (dummy) variables or other types of
contrast variables.
Assumptions. Results are valid only if you have specified a function that accurately describes the
relationship between dependent and independent variables. Additionally, the choice of good starting
values is very important. Even if you've specified the correct functional form of the model, if you use
poor starting values, your model may fail to converge or you may get a locally optimal solution rather
than one that is globally optimal.
Related procedures. Many models that appear nonlinear at first can be transformed to a linear model,
which can be analyzed using the Linear Regression procedure. If you are uncertain what the proper
model should be, the Curve Estimation procedure can help to identify useful functional relations in your
data.
Obtaining a Nonlinear Regression Analysis
1. From the menus choose:
Analyze > Regression > Nonlinear...
2. Select one numeric dependent variable from the list of variables in your active dataset.
3. To build a model expression, enter the expression in the Model field or paste components (variables,
parameters, functions) into the field.
4. Identify parameters in your model by clicking Parameters.
A segmented model (one that takes different forms in different parts of its domain) must be specified by
using conditional logic within the single model statement.
© Copyright IBM Corporation 1989, 2013
17
Conditional Logic (Nonlinear Regression)
You can specify a segmented model using conditional logic. To use conditional logic within a model
expression or a loss function, you form the sum of a series of terms, one for each condition. Each term
consists of a logical expression (in parentheses) multiplied by the expression that should result when that
logical expression is true.
For example, consider a segmented model that equals 0 for X<=0, X for 0<X<1, and 1 for X>=1. The
expression for this is:
(X<=0)*0 + (X>0 & X<1)*X + (X>=1)*1.
The logical expressions in parentheses all evaluate to 1 (true) or 0 (false). Therefore:
If X<=0, the above reduces to 1*0 + 0*X + 0*1 = 0.
If 0<X<1, it reduces to 0*0 + 1*X + 0*1 = X.
If X>=1, it reduces to 0*0 + 0*X + 1*1 = 1.
More complicated examples can be easily built by substituting different logical expressions and outcome
expressions. Remember that double inequalities, such as 0<X<1, must be written as compound
expressions, such as (X>0 & X<1).
String variables can be used within logical expressions:
(city='New York')*costliv + (city='Des Moines')*0.59*costliv
This yields one expression (the value of the variable costliv) for New Yorkers and another (59% of that
value) for Des Moines residents. String constants must be enclosed in quotation marks or apostrophes, as
shown here.
Nonlinear Regression Parameters
Parameters are the parts of your model that the Nonlinear Regression procedure estimates. Parameters
can be additive constants, multiplicative coefficients, exponents, or values used in evaluating functions.
All parameters that you have defined will appear (with their initial values) on the Parameters list in the
main dialog box.
Name. You must specify a name for each parameter. This name must be a valid variable name and must
be the name used in the model expression in the main dialog box.
Starting Value. Allows you to specify a starting value for the parameter, preferably as close as possible to
the expected final solution. Poor starting values can result in failure to converge or in convergence on a
solution that is local (rather than global) or is physically impossible.
Use starting values from previous analysis. If you have already run a nonlinear regression from this
dialog box, you can select this option to obtain the initial values of parameters from their values in the
previous run. This permits you to continue searching when the algorithm is converging slowly. (The
initial starting values will still appear on the Parameters list in the main dialog box.)
Note: This selection persists in this dialog box for the rest of your session. If you change the model, be
sure to deselect it.
18
IBM SPSS Regression 22
Nonlinear Regression Common Models
The table below provides example model syntax for many published nonlinear regression models. A
model selected at random is not likely to fit your data well. Appropriate starting values for the
parameters are necessary, and some models require constraints in order to converge.
Table 1. Example model syntax
Name
Model expression
Asymptotic Regression
b1 + b2 * exp(b3 * x)
Asymptotic Regression
b1 – (b2 * (b3 ** x))
Density
(b1 + b2 * x) ** (–1 / b3)
Gauss
b1 * (1 – b3 * exp(–b2 * x ** 2))
Gompertz
b1 * exp(–b2 * exp(–b3 * x))
Johnson-Schumacher
b1 * exp(–b2 / (x + b3))
Log-Modified
(b1 + b3 * x) ** b2
Log-Logistic
b1 – ln(1 + b2 * exp(–b3 * x))
Metcherlich Law of Diminishing Returns
b1 + b2 * exp(–b3 * x)
Michaelis Menten
b1 * x / (x + b2)
Morgan-Mercer-Florin
(b1 * b2 + b3 * x ** b4) / (b2 + x ** b4)
Peal-Reed
b1 / (1+ b2 * exp(–(b3 * x + b4 * x **2 + b5 * x ** 3)))
Ratio of Cubics
(b1 + b2 * x + b3 * x ** 2 + b4 * x ** 3) / (b5 * x ** 3)
Ratio of Quadratics
(b1 + b2 * x + b3 * x ** 2) / (b4 * x ** 2)
Richards
b1 / ((1 + b3 * exp(–b2 * x)) ** (1 / b4))
Verhulst
b1 / (1 + b3 * exp(–b2 * x))
Von Bertalanffy
(b1 ** (1 – b4) – b2 * exp(–b3 * x)) ** (1 / (1 – b4))
Weibull
b1 – b2 * exp(–b3 * x ** b4)
Yield Density
(b1 + b2 * x + b3 * x ** 2) ** (–1)
Nonlinear Regression Loss Function
The loss function in nonlinear regression is the function that is minimized by the algorithm. Select either
Sum of squared residuals to minimize the sum of the squared residuals or User-defined loss function to
minimize a different function.
If you select User-defined loss function, you must define the loss function whose sum (across all cases)
should be minimized by the choice of parameter values.
v Most loss functions involve the special variable RESID_, which represents the residual. (The default
Sum of squared residuals loss function could be entered explicitly as RESID_**2.) If you need to use the
predicted value in your loss function, it is equal to the dependent variable minus the residual.
v It is possible to specify a conditional loss function using conditional logic.
You can either type an expression in the User-defined loss function field or paste components of the
expression into the field. String constants must be enclosed in quotation marks or apostrophes, and
numeric constants must be typed in American format, with the dot as a decimal delimiter.
Chapter 5. Nonlinear Regression
19
Nonlinear Regression Parameter Constraints
A constraint is a restriction on the allowable values for a parameter during the iterative search for a
solution. Linear expressions are evaluated before a step is taken, so you can use linear constraints to
prevent steps that might result in overflows. Nonlinear expressions are evaluated after a step is taken.
Each equation or inequality requires the following elements:
v An expression involving at least one parameter in the model. Type the expression or use the keypad,
which allows you to paste numbers, operators, or parentheses into the expression. You can either type
in the required parameter(s) along with the rest of the expression or paste from the Parameters list at
the left. You cannot use ordinary variables in a constraint.
v One of the three logical operators <=, =, or >=.
v A numeric constant, to which the expression is compared using the logical operator. Type the constant.
Numeric constants must be typed in American format, with the dot as a decimal delimiter.
Nonlinear Regression Save New Variables
You can save a number of new variables to your active data file. Available options are Residuals,
Predicted values, Derivatives, and Loss function values. These variables can be used in subsequent
analyses to test the fit of the model or to identify problem cases.
v Residuals. Saves residuals with the variable name resid.
v Predicted Values. Saves predicted values with the variable name pred_.
Derivatives. One derivative is saved for each model parameter. Derivative names are created by
prefixing 'd.' to the first six characters of parameter names.
v Loss Function Values. This option is available if you specify your own loss function. The variable name
loss_ is assigned to the values of the loss function.
v
Nonlinear Regression Options
Options allow you to control various aspects of your nonlinear regression analysis:
Bootstrap Estimates. A method of estimating the standard error of a statistic using repeated samples from
the original data set. This is done by sampling (with replacement) to get many samples of the same size
as the original data set. The nonlinear equation is estimated for each of these samples. The standard error
of each parameter estimate is then calculated as the standard deviation of the bootstrapped estimates.
Parameter values from the original data are used as starting values for each bootstrap sample. This
requires the sequential quadratic programming algorithm.
Estimation Method. Allows you to select an estimation method, if possible. (Certain choices in this or
other dialog boxes require the sequential quadratic programming algorithm.) Available alternatives
include Sequential quadratic programming and Levenberg-Marquardt.
v Sequential Quadratic Programming. This method is available for constrained and unconstrained models.
Sequential quadratic programming is used automatically if you specify a constrained model, a
user-defined loss function, or bootstrapping. You can enter new values for Maximum iterations and
Step limit, and you can change the selection in the drop-down lists for Optimality tolerance, Function
precision, and Infinite step size.
v Levenberg-Marquardt. This is the default algorithm for unconstrained models. The LevenbergMarquardt method is not available if you specify a constrained model, a user-defined loss function, or
bootstrapping. You can enter new values for Maximum iterations, and you can change the selection in
the drop-down lists for Sum-of-squares convergence and Parameter convergence.
20
IBM SPSS Regression 22
Interpreting Nonlinear Regression Results
Nonlinear regression problems often present computational difficulties:
v The choice of initial values for the parameters influences convergence. Try to choose initial values that
are reasonable and, if possible, close to the expected final solution.
v Sometimes one algorithm performs better than the other on a particular problem. In the Options dialog
box, select the other algorithm if it is available. (If you specify a loss function or certain types of
constraints, you cannot use the Levenberg-Marquardt algorithm.)
v When iteration stops only because the maximum number of iterations has occurred, the "final" model
is probably not a good solution. Select Use starting values from previous analysis in the Parameters
dialog box to continue the iteration or, better yet, choose different initial values.
v Models that require exponentiation of or by large data values can cause overflows or underflows
(numbers too large or too small for the computer to represent). Sometimes you can avoid these by
suitable choice of initial values or by imposing constraints on the parameters.
NLR Command Additional Features
The command syntax language also allows you to:
v Name a file from which to read initial values for parameter estimates.
v Specify more than one model statement and loss function. This makes it easier to specify a segmented
model.
v Supply your own derivatives rather than use those calculated by the program.
v Specify the number of bootstrap samples to generate.
v Specify additional iteration criteria, including setting a critical value for derivative checking and
defining a convergence criterion for the correlation between the residuals and the derivatives.
Additional criteria for the CNLR (constrained nonlinear regression) command allow you to:
v Specify the maximum number of minor iterations allowed within each major iteration.
v Set a critical value for derivative checking.
v Set a step limit.
v Specify a crash tolerance to determine if initial values are within their specified bounds.
See the Command Syntax Reference for complete syntax information.
Chapter 5. Nonlinear Regression
21
22
IBM SPSS Regression 22
Chapter 6. Weight Estimation
Standard linear regression models assume that variance is constant within the population under study.
When this is not the case--for example, when cases that are high on some attribute show more variability
than cases that are low on that attribute--linear regression using ordinary least squares (OLS) no longer
provides optimal model estimates. If the differences in variability can be predicted from another variable,
the Weight Estimation procedure can compute the coefficients of a linear regression model using
weighted least squares (WLS), such that the more precise observations (that is, those with less variability)
are given greater weight in determining the regression coefficients. The Weight Estimation procedure tests
a range of weight transformations and indicates which will give the best fit to the data.
Example. What are the effects of inflation and unemployment on changes in stock prices? Because stocks
with higher share values often show more variability than those with low share values, ordinary least
squares will not produce optimal estimates. Weight estimation allows you to account for the effect of
share price on the variability of price changes in calculating the linear model.
Statistics. Log-likelihood values for each power of the weight source variable tested, multiple R,
R-squared, adjusted R-squared, ANOVA table for WLS model, unstandardized and standardized
parameter estimates, and log-likelihood for the WLS model.
Weight Estimation Data Considerations
Data. The dependent and independent variables should be quantitative. Categorical variables, such as
religion, major, or region of residence, need to be recoded to binary (dummy) variables or other types of
contrast variables. The weight variable should be quantitative and should be related to the variability in
the dependent variable.
Assumptions. For each value of the independent variable, the distribution of the dependent variable
must be normal. The relationship between the dependent variable and each independent variable should
be linear, and all observations should be independent. The variance of the dependent variable can vary
across levels of the independent variable(s), but the differences must be predictable based on the weight
variable.
Related procedures. The Explore procedure can be used to screen your data. Explore provides tests for
normality and homogeneity of variance, as well as graphical displays. If your dependent variable seems
to have equal variance across levels of independent variables, you can use the Linear Regression
procedure. If your data appear to violate an assumption (such as normality), try transforming them. If
your data are not related linearly and a transformation does not help, use an alternate model in the
Curve Estimation procedure. If your dependent variable is dichotomous--for example, whether a
particular sale is completed or whether an item is defective--use the Logistic Regression procedure. If
your dependent variable is censored--for example, survival time after surgery--use Life Tables,
Kaplan-Meier, or Cox Regression, available in the Advanced Statistics option. If your data are not
independent--for example, if you observe the same person under several conditions--use the Repeated
Measures procedure, available in the Advanced Statistics option.
Obtaining a Weight Estimation Analysis
1. From the menus choose:
Analyze > Regression > Weight Estimation...
2. Select one dependent variable.
3. Select one or more independent variables.
4. Select the variable that is the source of heteroscedasticity as the weight variable.
© Copyright IBM Corporation 1989, 2013
23
v
Weight Variable. The data are weighted by the reciprocal of this variable raised to a power. The
regression equation is calculated for each of a specified range of power values and indicates the power
that maximizes the log-likelihood function.
v
Power Range. This is used in conjunction with the weight variable to compute weights. Several
regression equations will be fit, one for each value in the power range. The values entered in the
Power range test box and the through text box must be between -6.5 and 7.5, inclusive. The power
values range from the low to high value, in increments determined by the value specified. The total
number of values in the power range is limited to 150.
Weight Estimation Options
You can specify options for your weight estimation analysis:
Save best weight as new variable. Adds the weight variable to the active file. This variable is called
WGT_n, where n is a number chosen to give the variable a unique name.
Display ANOVA and Estimates. Allows you to control how statistics are displayed in the output.
Available alternatives are For best power and For each power value.
WLS Command Additional Features
The command syntax language also allows you to:
v Provide a single value for the power.
v Specify a list of power values, or mix a range of values with a list of values for the power.
See the Command Syntax Reference for complete syntax information.
24
IBM SPSS Regression 22
Chapter 7. Two-Stage Least-Squares Regression
Standard linear regression models assume that errors in the dependent variable are uncorrelated with the
independent variable(s). When this is not the case (for example, when relationships between variables are
bidirectional), linear regression using ordinary least squares (OLS) no longer provides optimal model
estimates. Two-stage least-squares regression uses instrumental variables that are uncorrelated with the
error terms to compute estimated values of the problematic predictor(s) (the first stage), and then uses
those computed values to estimate a linear regression model of the dependent variable (the second stage).
Since the computed values are based on variables that are uncorrelated with the errors, the results of the
two-stage model are optimal.
Example. Is the demand for a commodity related to its price and consumers' incomes? The difficulty in
this model is that price and demand have a reciprocal effect on each other. That is, price can influence
demand and demand can also influence price. A two-stage least-squares regression model might use
consumers' incomes and lagged price to calculate a proxy for price that is uncorrelated with the
measurement errors in demand. This proxy is substituted for price itself in the originally specified model,
which is then estimated.
Statistics. For each model: standardized and unstandardized regression coefficients, multiple R, R 2,
adjusted R 2, standard error of the estimate, analysis-of-variance table, predicted values, and residuals.
Also, 95% confidence intervals for each regression coefficient, and correlation and covariance matrices of
parameter estimates.
Two-Stage Least-Squares Regression Data Considerations
Data. The dependent and independent variables should be quantitative. Categorical variables, such as
religion, major, or region of residence, need to be recoded to binary (dummy) variables or other types of
contrast variables. Endogenous explanatory variables should be quantitative (not categorical).
Assumptions. For each value of the independent variable, the distribution of the dependent variable
must be normal. The variance of the distribution of the dependent variable should be constant for all
values of the independent variable. The relationship between the dependent variable and each
independent variable should be linear.
Related procedures. If you believe that none of your predictor variables is correlated with the errors in
your dependent variable, you can use the Linear Regression procedure. If your data appear to violate one
of the assumptions (such as normality or constant variance), try transforming them. If your data are not
related linearly and a transformation does not help, use an alternate model in the Curve Estimation
procedure. If your dependent variable is dichotomous, such as whether a particular sale is completed or
not, use the Logistic Regression procedure. If your data are not independent--for example, if you observe
the same person under several conditions--use the Repeated Measures procedure, available in the
Advanced Models option.
Obtaining a Two-Stage Least-Squares Regression Analysis
1. From the menus choose:
Analyze > Regression > 2-Stage Least Squares...
2. Select one dependent variable.
3. Select one or more explanatory (predictor) variables.
4. Select one or more instrumental variables.
v Instrumental. These are the variables used to compute the predicted values for the endogenous
variables in the first stage of two-stage least squares analysis. The same variables may appear in both
© Copyright IBM Corporation 1989, 2013
25
the Explanatory and Instrumental list boxes. The number of instrumental variables must be at least as
many as the number of explanatory variables. If all explanatory and instrumental variables listed are
the same, the results are the same as results from the Linear Regression procedure.
Explanatory variables not specified as instrumental are considered endogenous. Normally, all of the
exogenous variables in the Explanatory list are also specified as instrumental variables.
Two-Stage Least-Squares Regression Options
You can select the following options for your analysis:
Save New Variables. Allows you to add new variables to your active file. Available options are Predicted
and Residuals.
Display covariance of parameters. Allows you to print the covariance matrix of the parameter estimates.
2SLS Command Additional Features
The command syntax language also allows you to estimate multiple equations simultaneously. See the
Command Syntax Reference for complete syntax information.
26
IBM SPSS Regression 22
Chapter 8. Categorical Variable Coding Schemes
In many procedures, you can request automatic replacement of a categorical independent variable with a
set of contrast variables, which will then be entered or removed from an equation as a block. You can
specify how the set of contrast variables is to be coded, usually on the CONTRAST subcommand. This
appendix explains and illustrates how different contrast types requested on CONTRAST actually work.
Deviation
Deviation from the grand mean. In matrix terms, these contrasts have the form:
mean
df(1)
df(2)
.
.
df(k-1)
( 1/k
1/k
(1-1/k
-1/k
( -1/k 1-1/k
.
.
( -1/k -1/k
...
1/k
1/k)
... -1/k -1/k)
... -1/k -1/k)
... 1-1/k
-1/k)
where k is the number of categories for the independent variable and the last category is omitted by
default. For example, the deviation contrasts for an independent variable with three categories are as
follows:
( 1/3
( 2/3
(-1/3
1/3
-1/3
2/3
1/3)
-1/3)
-1/3)
To omit a category other than the last, specify the number of the omitted category in parentheses after the
DEVIATION keyword. For example, the following subcommand obtains the deviations for the first and
third categories and omits the second:
/CONTRAST(FACTOR)=DEVIATION(2)
Suppose that factor has three categories. The resulting contrast matrix will be
( 1/3
( 2/3
(-1/3
1/3
-1/3
-1/3
1/3)
-1/3)
2/3)
Simple
Simple contrasts. Compares each level of a factor to the last. The general matrix form is
mean
df(1)
df(2)
.
.
df(k-1)
(1/k 1/k
( 1
0
( 0
1
.
.
( 0
0
...
...
...
1/k
0
0
1/k)
-1)
-1)
...
1
-1)
where k is the number of categories for the independent variable. For example, the simple contrasts for
an independent variable with four categories are as follows:
(1/4
( 1
( 0
( 0
1/4
0
1
0
1/4 1/4)
0
-1)
0
-1)
1
-1)
To use another category instead of the last as a reference category, specify in parentheses after the SIMPLE
keyword the sequence number of the reference category, which is not necessarily the value associated
with that category. For example, the following CONTRAST subcommand obtains a contrast matrix that omits
the second category:
/CONTRAST(FACTOR) = SIMPLE(2)
Suppose that factor has four categories. The resulting contrast matrix will be
© Copyright IBM Corporation 1989, 2013
27
(1/4
( 1
( 0
( 0
1/4
-1
-1
-1
1/4 1/4)
0
0)
1
0)
0
1)
Helmert
Helmert contrasts. Compares categories of an independent variable with the mean of the subsequent
categories. The general matrix form is
mean
df(1)
df(2)
.
.
df(k-2)
df(k-1)
(1/k
( 1
( 0
(
(
0
0
1/k
-1/(k-1)
1
.
.
0
0
...
1/k
1/k
1/k)
... -1/(k-1) -1/(k-1) -1/(k-1))
... -1/(k-2) -1/(k-2) -1/(k-2))
...
...
1
0
-1/2
1
-1/2)
-1)
where k is the number of categories of the independent variable. For example, an independent variable
with four categories has a Helmert contrast matrix of the following form:
(1/4 1/4 1/4
1/4)
( 1 -1/3 -1/3 -1/3)
( 0
1 -1/2 -1/2)
( 0
0
1
-1)
Difference
Difference or reverse Helmert contrasts. Compares categories of an independent variable with the mean
of the previous categories of the variable. The general matrix form is
mean
df(1)
df(2)
.
.
df(k-1)
(
(
(
1/k
-1
-1/2
1/k
1/k ... 1/k)
1
0 ...
0)
-1/2
1 ...
0)
.
.
(-1/(k-1) -1/(k-1) -1/(k-1) ...
1)
where k is the number of categories for the independent variable. For example, the difference contrasts
for an independent variable with four categories are as follows:
( 1/4 1/4 1/4
( -1
1
0
(-1/2 -1/2
1
(-1/3 -1/3 -1/3
1/4)
0)
0)
1)
Polynomial
Orthogonal polynomial contrasts. The first degree of freedom contains the linear effect across all
categories; the second degree of freedom, the quadratic effect; the third degree of freedom, the cubic; and
so on, for the higher-order effects.
You can specify the spacing between levels of the treatment measured by the given categorical variable.
Equal spacing, which is the default if you omit the metric, can be specified as consecutive integers from 1
to k, where k is the number of categories. If the variable drug has three categories, the subcommand
/CONTRAST(DRUG)=POLYNOMIAL
is the same as
/CONTRAST(DRUG)=POLYNOMIAL(1,2,3)
Equal spacing is not always necessary, however. For example, suppose that drug represents different
dosages of a drug given to three groups. If the dosage administered to the second group is twice that
given to the first group and the dosage administered to the third group is three times that given to the
first group, the treatment categories are equally spaced, and an appropriate metric for this situation
consists of consecutive integers:
/CONTRAST(DRUG)=POLYNOMIAL(1,2,3)
28
IBM SPSS Regression 22
If, however, the dosage administered to the second group is four times that given to the first group, and
the dosage administered to the third group is seven times that given to the first group, an appropriate
metric is
/CONTRAST(DRUG)=POLYNOMIAL(1,4,7)
In either case, the result of the contrast specification is that the first degree of freedom for drug contains
the linear effect of the dosage levels and the second degree of freedom contains the quadratic effect.
Polynomial contrasts are especially useful in tests of trends and for investigating the nature of response
surfaces. You can also use polynomial contrasts to perform nonlinear curve fitting, such as curvilinear
regression.
Repeated
Compares adjacent levels of an independent variable. The general matrix form is
mean
df(1)
df(2)
.
.
df(k-1)
(1/k 1/k 1/k ... 1/k 1/k)
( 1 -1
0 ...
0
0)
( 0
1
-1 ...
0
0)
.
.
( 0
0
0 ...
1
-1)
where k is the number of categories for the independent variable. For example, the repeated contrasts for
an independent variable with four categories are as follows:
(1/4
( 1
( 0
( 0
1/4
-1
1
0
1/4 1/4)
0
0)
-1
0)
1
-1)
These contrasts are useful in profile analysis and wherever difference scores are needed.
Special
A user-defined contrast. Allows entry of special contrasts in the form of square matrices with as many
rows and columns as there are categories of the given independent variable. For MANOVA and LOGLINEAR,
the first row entered is always the mean, or constant, effect and represents the set of weights indicating
how to average other independent variables, if any, over the given variable. Generally, this contrast is a
vector of ones.
The remaining rows of the matrix contain the special contrasts indicating the comparisons between
categories of the variable. Usually, orthogonal contrasts are the most useful. Orthogonal contrasts are
statistically independent and are nonredundant. Contrasts are orthogonal if:
v For each row, contrast coefficients sum to 0.
v The products of corresponding coefficients for all pairs of disjoint rows also sum to 0.
For example, suppose that treatment has four levels and that you want to compare the various levels of
treatment with each other. An appropriate special contrast is
(1
1
1
1)
(3 -1 -1 -1)
(0 2 -1 -1)
(0 0 1 -1)
weights
compare
compare
compare
for
1st
2nd
3rd
mean
with
with
with
calculation
2nd through 4th
3rd and 4th
4th
which you specify by means of the following CONTRAST subcommand for MANOVA, LOGISTIC REGRESSION,
and COXREG:
/CONTRAST(TREATMNT)=SPECIAL( 1 1 1 1
3 -1 -1 -1
0 2 -1 -1
0 0 1 -1 )
Chapter 8. Categorical Variable Coding Schemes
29
For LOGLINEAR, you need to specify:
/CONTRAST(TREATMNT)=BASIS SPECIAL( 1 1 1 1
3 -1 -1 -1
0 2 -1 -1
0 0 1 -1 )
Each row except the means row sums to 0. Products of each pair of disjoint rows sum to 0 as well:
Rows 2 and 3:
Rows 2 and 4:
Rows 3 and 4:
(3)(0) + (–1)(2) + (–1)(–1) + (–1)(–1) = 0
(3)(0) + (–1)(0) + (–1)(1) + (–1)(–1) = 0
(0)(0) + (2)(0) + (–1)(1) + (–1)(–1) = 0
The special contrasts need not be orthogonal. However, they must not be linear combinations of each
other. If they are, the procedure reports the linear dependency and ceases processing. Helmert, difference,
and polynomial contrasts are all orthogonal contrasts.
Indicator
Indicator variable coding. Also known as dummy coding, this is not available in LOGLINEAR or MANOVA.
The number of new variables coded is k–1. Cases in the reference category are coded 0 for all k–1
variables. A case in the i th category is coded 0 for all indicator variables except the i th, which is coded 1.
30
IBM SPSS Regression 22
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31
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32
IBM SPSS Regression 22
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Notices
33
34
IBM SPSS Regression 22
Index
A
asymptotic regression
in Nonlinear Regression
deviance function
for estimating dispersion scaling
value 12
DfBeta
in Logistic Regression 5
dispersion scaling value
in Multinomial Logistic
Regression 12
19
B
backward elimination
in Logistic Regression 4
binary logistic regression 1
F
fiducial confidence intervals
in Probit Analysis 16
forward selection
in Logistic Regression 4
full factorial models
in Multinomial Logistic Regression
C
categorical covariates 4
cell probabilities tables
in Multinomial Logistic
Regression 11
cells with zero observations
in Multinomial Logistic
Regression 11
classification
in Multinomial Logistic Regression
classification tables
in Multinomial Logistic
Regression 11
confidence intervals
in Multinomial Logistic
Regression 11
constant term
in Linear Regression 6
constrained regression
in Nonlinear Regression 20
contrasts
in Logistic Regression 4
convergence criterion
in Multinomial Logistic
Regression 11
Cook's D
in Logistic Regression 5
correlation matrix
in Multinomial Logistic
Regression 11
covariance matrix
in Multinomial Logistic
Regression 11
covariates
in Logistic Regression 4
Cox and Snell R-square
in Multinomial Logistic
Regression 11
custom models
in Multinomial Logistic Regression
D
delta
as correction for cells with zero
observations 11
density model
in Nonlinear Regression 19
G
9
Gauss model
in Nonlinear Regression
Gompertz model
in Nonlinear Regression
goodness of fit
in Multinomial Logistic
Regression 11
19
19
H
Hosmer-Lemeshow goodness-of-fit
statistic
in Logistic Regression 6
I
intercept
include or exclude 9
iteration history
in Multinomial Logistic
Regression 11
iterations
in Logistic Regression 6
in Multinomial Logistic
Regression 11
in Probit Analysis 16
J
9
Johnson-Schumacher model
in Nonlinear Regression 19
L
leverage values
in Logistic Regression
5
9
likelihood ratio
for estimating dispersion scaling
value 12
goodness of fit 11
Linear Regression
Two-Stage Least-Squares
Regression 25
weight estimation 23
log-likelihood
in Multinomial Logistic
Regression 11
in Weight Estimation 23
log-modified model
in Nonlinear Regression 19
Logistic Regression 3
binary 1
categorical covariates 4
classification cutoff 6
coefficients 3
command additional features 6
constant term 6
contrasts 4
define selection rule 4
display options 6
example 3
Hosmer-Lemeshow goodness-of-fit
statistic 6
influence measures 5
iterations 6
predicted values 5
probability for stepwise 6
residuals 5
saving new variables 5
set rule 4
statistics 3
statistics and plots 6
string covariates 4
variable selection methods 4
M
main-effects models
in Multinomial Logistic Regression 9
McFadden R-square
in Multinomial Logistic
Regression 11
Metcherlich law of diminishing returns
in Nonlinear Regression 19
Michaelis Menten model
in Nonlinear Regression 19
Morgan-Mercer-Florin model
in Nonlinear Regression 19
Multinomial Logistic Regression 9, 11
command additional features 13
criteria 11
exporting model information 12
models 9
reference category 10
save 12
statistics 11
35
N
Nagelkerke R-square
in Multinomial Logistic
Regression 11
nonlinear models
in Nonlinear Regression 19
Nonlinear Regression 17
bootstrap estimates 20
command additional features 21
common nonlinear models 19
conditional logic 18
derivatives 20
estimation methods 20
example 17
interpretation of results 21
Levenberg-Marquardt algorithm 20
loss function 19
parameter constraints 20
parameters 18
predicted values 20
residuals 20
save new variables 20
segmented model 18
sequential quadratic
programming 20
starting values 18
statistics 17
P
parallelism test
in Probit Analysis 16
parameter constraints
in Nonlinear Regression 20
parameter estimates
in Multinomial Logistic
Regression 11
Peal-Reed model
in Nonlinear Regression 19
Pearson chi-square
for estimating dispersion scaling
value 12
goodness of fit 11
Probit Analysis
command additional features 16
criteria 16
define range 16
example 15
fiducial confidence intervals 16
iterations 16
natural response rate 16
parallelism test 16
relative median potency 16
statistics 15, 16
R
ratio of cubics model
in Nonlinear Regression
ratio of quadratics model
in Nonlinear Regression
reference category
in Multinomial Logistic
Regression 10
relative median potency
in Probit Analysis 16
36
19
19
IBM SPSS Regression 22
Richards model
in Nonlinear Regression
19
S
separation
in Multinomial Logistic
Regression 11
singularity
in Multinomial Logistic
Regression 11
step-halving
in Multinomial Logistic
Regression 11
stepwise selection
in Logistic Regression 4
in Multinomial Logistic Regression
string covariates
in Logistic Regression 4
9
T
Two-Stage Least-Squares Regression 25
command additional features 26
covariance of parameters 26
example 25
instrumental variables 25
saving new variables 26
statistics 25
V
Verhulst model
in Nonlinear Regression
Von Bertalanffy model
in Nonlinear Regression
19
19
W
Weibull model
in Nonlinear Regression 19
Weight Estimation 23
command additional features 24
display ANOVA and estimates 24
example 23
iteration history 24
log-likelihood 23
save best weights as new variable 24
statistics 23
Y
yield density model
in Nonlinear Regression
19
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