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.

4 Chapter 2

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
E From the menus choose: Analyze Regression Binary Logistic.
5 Logistic Regression Figure 2-1 Logistic Regression dialog box
E Select one dichotomous dependent variable. This variable may be numeric or short

string.

E 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.

6 Chapter 2

Logistic Regression Set Rule
Figure 2-2 Logistic Regression Set Rule dialog box
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.
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.
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.

7 Logistic Regression

Backward Elimination (Conditional). Backward stepwise selection. Removal
testing is based on the probability of the likelihood-ratio statistic based on conditional parameter estimates.
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.
Backward Elimination (Wald). Backward stepwise selection. Removal testing is
based on the probability of the Wald statistic. 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

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.
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 SPSS command language also allows you to: Identify casewise output by the values or variable labels of a variable. Control the spacing of iteration reports. Rather than printing parameter estimates after every iteration, you can request parameter estimates after every nth iteration.

12 Chapter 2

Change the criteria for terminating iteration and checking for redundancy. Specify a variable list for casewise listings. Conserve memory by holding the data for each split file group in an external scratch file during processing. See the SPSS Command Syntax Reference for complete syntax information.
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 persons 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 R2. 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.
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.

14 Chapter 3

Obtaining a Multinomial Logistic Regression
E From the menus choose: Analyze Regression Multinomial Logistic. Figure 3-1 Multinomial Logistic Regression dialog box
E Select one dependent variable. E Factors are optional and can be either numeric or categorical. E Covariates are optional but must be numeric if specified.
15 Multinomial Logistic Regression
Multinomial Logistic Regression Models
Figure 3-2 Multinomial Logistic Regression Model dialog box
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 and Covariates. The factors and covariates are listed with (F) for factor and

Asymptotic correlations. Prints matrix of parameter estimate correlations. 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.

20 Chapter 3

Multinomial Logistic Regression Criteria
Figure 3-5 Multinomial Logistic Regression Convergence Criteria dialog box
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. 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.
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.
Singularity tolerance. Allows you to specify the tolerance used in checking for

singularities.

21 Multinomial Logistic Regression
Multinomial Logistic Regression Options
Figure 3-6 Multinomial Logistic Regression Options dialog box
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.

Saved variables: 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.
Predicted category. This is the response category with the largest expected
probability for a factor/covariate pattern.
Predicted category probabilities. This is the maximum of the estimated response

probabilities.

Actual category probability. This is the estimated probability of classifying a
factor/covariate pattern into the observed category.
Export model information to XML file. Parameter estimates and (optionally) their

24 Chapter 3

NOMREG Command Additional Features
The SPSS command language also allows you to: Specify the reference category of the dependent variable. Include cases with user-missing values. Customize hypothesis tests by specifying null hypotheses as linear combinations of parameters. See the SPSS Command Syntax Reference for complete syntax information.
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.
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.

26 Chapter 4

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
E From the menus choose: Analyze Regression Probit. Figure 4-1 Probit Analysis dialog box
27 Probit Analysis E 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.
E 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.

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 SPSS 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.

35 Nonlinear Regression

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 5-1 Example model syntax
Name Asymptotic Regression Asymptotic Regression Density Gauss Gompertz Johnson-Schumacher Log-Modified Log-Logistic Metcherlich Law of Diminishing Returns Michaelis Menten Morgan-Mercer-Florin Peal-Reed Ratio of Cubics Ratio of Quadratics Richards Verhulst Von Bertalanffy Weibull Yield Density
Model expression b1 + b2 *exp( b3 * x ) b1 ( b2 *( b3 ** x )) ( b1 + b2 * x )**(1/ b3 ) b1 *(1 b3 *exp( b2 * x **2)) b1 *exp( b2 * exp( b3 * x )) b1 *exp( b2 / ( x + b3)) ( b1 + b3 * x ) ** b2 b1 ln(1+ b2 *exp( b3 * x )) b1 + b2 *exp( b3 * x ) b1* x /( x + b2 ) ( b1 * b2 + b3 * x ** b4 )/( b2 + x ** b4 ) b1 /(1+ b2 *exp(( b3 * x + b4 * x **2+ b5 * x **3))) ( b1 + b2 * x + b3 * x **2+ b4 * x **3)/( b5 * x **3) ( b1 + b2 * x + b3 * x **2)/( b4 * x **2) b1 /((1+ b3 *exp( b2 * x ))**(1/ b4 )) b1 /(1 + b3 * exp( b2 * x )) ( b1 ** (1 b4 ) b2 * exp( b3 * x )) ** (1/(1 b4 )) b1 b2 *exp( b3 * x ** b4 ) (b1 + b2 * x + b3 * x **2)**(1)

36 Chapter 5

Nonlinear Regression Loss Function
Figure 5-3 Nonlinear Regression Loss Function dialog box
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. 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. 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.

37 Nonlinear Regression

Nonlinear Regression Parameter Constraints
Figure 5-4 Nonlinear Regression Parameter Constraints dialog box
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: 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. One of the three logical operators <=, =, or >=. 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.

38 Chapter 5

endogenous variables in the first stage of two-stage least squares analysis. The same variables may appear in both 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
Figure 7-2 2-Stage Least Squares Options dialog box
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 SPSS command language also allows you to estimate multiple equations simultaneously. See the SPSS Command Syntax Reference for complete syntax information.

Appendix

Categorical Variable Coding Schemes
In many SPSS 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(k1) ( 1/k ( 1/k ( 11/k ( 1/k 1/k 1/k 11/k. 1/k. 11/k 1/k ). 1/k 1/k 1/k 1/k ) 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 )

50 Appendix A

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(k1) (0 ( 1/k (1 (0 1/k 0 1. 0. ). 1/k 1/k ) 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 1/1/1/4 ) 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
51 Categorical Variable Coding Schemes
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
( 1/4 (1 (0 (0 1/1/1/4 ) 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(k2) df(k1) (0 (0 ( 1/k (1 (0 1/k 1/(k1) 1. 1. 1/1/). 1/k 1/(k1) 1/(k2) 1/k ) 1/(k1) ) 1/(k2) )
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 (0 (0 1/4 1/0 1/4 1/3 1/1/4 ) 1/3 ) 1/2 ) 1 )

52 Appendix A

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(k1) ( 1/(k1) ( 1/k ( 1 ( 1/2 1/k 1 1/2. 1/(k1) 1/(k1). 1) 1/k 0 1. 1/k ) 0) 0)
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 ( 1/2 ( 1/3 1/1/2 1/3 1/1 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)
53 Categorical Variable Coding Schemes
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:

(1 (3 (0 (1 1) 1 ) 1 ) 1 ) weights for mean calculation compare 1st with 2nd through 4th compare 2nd with 3rd and 4th compare 3rd with 4th
55 Categorical Variable Coding Schemes
which you specify by means of the following CONTRAST subcommand for MANOVA, LOGISTICREGRESSION, and COXREG:
/CONTRAST(TREATMNT)=SPECIAL( 3 -1 -1 -2 -1 --1 )
For LOGLINEAR, you need to specify:
/CONTRAST(TREATMNT)=BASIS SPECIAL( 3 -1 -1 -2 -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 k1. Cases in the
reference category are coded 0 for all k1 variables. A case in the ith category is coded 0 for all indicator variables except the ith, which is coded 1.
asymptotic regression in Nonlinear Regression, 35
Cox and Snell R-square in Multinomial Logistic Regression, 18 custom models in Multinomial Logistic Regression, 15
backward elimination in Logistic Regression, 6 binary logistic regression, 1
categorical covariates, 7 cell probabilities tables in Multinomial Logistic Regression, 18 cells with zero observations in Multinomial Logistic Regression, 20 classification in Multinomial Logistic Regression, 13 classification tables in Multinomial Logistic Regression, 18 confidence intervals in Multinomial Logistic Regression, 18 constant term in Linear Regression, 10 constrained regression in Nonlinear Regression, 37 contrasts in Logistic Regression, 7 convergence criterion in Multinomial Logistic Regression, 20 Cooks D in Logistic Regression, 9 correlation matrix in Multinomial Logistic Regression, 18 covariance matrix in Multinomial Logistic Regression, 18 covariates in Logistic Regression, 7
delta as correction for cells with zero observations, 20 density model in Nonlinear Regression, 35 deviance function for estimating dispersion scaling value, 21 DfBeta in Logistic Regression, 9 dispersion scaling value in Multinomial Logistic Regression, 21
fiducial confidence intervals in Probit Analysis, 28 forward selection in Logistic Regression, 6 full factorial models in Multinomial Logistic Regression, 15

SPSS Advanced Models 15.0
For more information about SPSS software products, please visit our Web site at http://www.spss.com or contact SPSS Inc. 233 South Wacker Drive, 11th Floor Chicago, IL 60606-6412 Tel: (312) 651-3000 Fax: (312) 651-3668 SPSS is a registered trademark and the other product names are the trademarks of SPSS Inc. for its proprietary computer software. No material describing such software may be produced or distributed without the written permission of the owners of the trademark and license rights in the software and the copyrights in the published materials. The SOFTWARE and documentation are provided with RESTRICTED RIGHTS. Use, duplication, or disclosure by the Government is subject to restrictions as set forth in subdivision (c) (1) (ii) of The Rights in Technical Data and Computer Software clause at 52.227-7013. Contractor/manufacturer is SPSS Inc., 233 South Wacker Drive, 11th Floor, Chicago, IL 60606-6412. Patent No. 7,023,453 General notice: Other product names mentioned herein are used for identication purposes only and may be trademarks of their respective companies. TableLook is a trademark of SPSS Inc. Windows is a registered trademark of Microsoft Corporation. DataDirect, DataDirect Connect, INTERSOLV, and SequeLink are registered trademarks of DataDirect Technologies. Portions of this product were created using LEADTOOLS 19912000, LEAD Technologies, Inc. ALL RIGHTS RESERVED. LEAD, LEADTOOLS, and LEADVIEW are registered trademarks of LEAD Technologies, Inc. Sax Basic is a trademark of Sax Software Corporation. Copyright 19932004 by Polar Engineering and Consulting. All rights reserved. A portion of the SPSS software contains zlib technology. Copyright 19952002 by Jean-loup Gailly and Mark Adler. The zlib software is provided as is, without express or implied warranty. A portion of the SPSS software contains Sun Java Runtime libraries. Copyright 2003 by Sun Microsystems, Inc. All rights reserved. The Sun Java Runtime libraries include code licensed from RSA Security, Inc. Some portions of the libraries are licensed from IBM and are available at http://www-128.ibm.com/developerworks/opensource/. SPSS Advanced Models 15.0 Copyright 2006 by SPSS Inc. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. 06
ISBN-13: 978-1-56827-384-6 ISBN-10: 1-56827-384-3

Preface

SPSS 15.0 is a comprehensive system for analyzing data. The SPSS Advanced Models optional add-on module provides the additional analytic techniques described in this manual. The Advanced Models add-on module must be used with the SPSS 15.0 Base system and is completely integrated into that system.

Installation

To install the SPSS Advanced Models add-on module, run the License Authorization Wizard using the authorization code that you received from SPSS Inc. For more information, see the installation instructions supplied with the SPSS Advanced Models add-on module.

Compatibility

SPSS is designed to run on many computer systems. See the installation instructions that came with your system for specic information on minimum and recommended requirements.

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Technical Support

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Example. Twelve students are assigned to a high- or low-anxiety group based on their scores on
an anxiety-rating test. The anxiety rating is called a between-subjects factor because it divides the subjects into groups. The students are each given four trials on a learning task, and the number of errors for each trial is recorded. The errors for each trial are recorded in separate variables, and a within-subjects factor (trial) is dened with four levels for the four trials. The trial effect is found to be signicant, while the trial-by-anxiety interaction is not signicant.

15 GLM Repeated Measures

Statistics. Post hoc range tests and multiple comparisons (for between-subjects factors): least
signicant difference, Bonferroni, Sidak, Scheff, Ryan-Einot-Gabriel-Welsch multiple F, Ryan-Einot-Gabriel-Welsch multiple range, Student-Newman-Keuls, Tukeys honestly signicant difference, Tukeys b, Duncan, Hochbergs GT2, Gabriel, Waller Duncan t test, Dunnett (one-sided and two-sided), Tamhanes T2, Dunnetts T3, Games-Howell, and Dunnetts C. Descriptive statistics: observed means, standard deviations, and counts for all of the dependent variables in all cells; the Levene test for homogeneity of variance; Boxs M; and Mauchlys test of sphericity.
Plots. Spread-versus-level, residual, and prole (interaction). Data. The dependent variables should be quantitative. Between-subjects factors divide the sample
into discrete subgroups, such as male and female. These factors are categorical and can have numeric values or string values of up to eight characters. Within-subjects factors are dened in the Repeated Measures Dene Factor(s) dialog box. Covariates are quantitative variables that are related to the dependent variable. For a repeated measures analysis, these should remain constant at each level of a within-subjects variable. The data le should contain a set of variables for each group of measurements on the subjects. The set has one variable for each repetition of the measurement within the group. A within-subjects factor is dened for the group with the number of levels equal to the number of repetitions. For example, measurements of weight could be taken on different days. If measurements of the same property were taken on ve days, the within-subjects factor could be specied as day with ve levels. For multiple within-subjects factors, the number of measurements for each subject is equal to the product of the number of levels of each factor. For example, if measurements were taken at three different times each day for four days, the total number of measurements is 12 for each subject. The within-subjects factors could be specied as day(4) and time(3).

E From the menus choose: Analyze General Linear Model Repeated Measures. Figure 3-1 Repeated Measures Define Factor(s) dialog box
E Type a within-subject factor name and its number of levels. E Click Add. E Repeat these steps for each within-subjects factor.
To dene measure factors for a doubly multivariate repeated measures design:
E Type the measure name. E Click Add.
After dening all of your factors and measures:

E Click Define.

17 GLM Repeated Measures Figure 3-2 Repeated Measures dialog box
E Select a dependent variable that corresponds to each combination of within-subjects factors
(and optionally, measures) on the list. To change positions of the variables, use the up and down arrows. To make changes to the within-subjects factors, you can reopen the Repeated Measures Dene Factor(s) dialog box without closing the main dialog box. Optionally, you can specify between-subjects factor(s) and covariates.
GLM Repeated Measures Define Factors
GLM Repeated Measures analyzes groups of related dependent variables that represent different measurements of the same attribute. This dialog box lets you dene one or more within-subjects factors for use in GLM Repeated Measures. See Figure 3-1 on p. 16. Note that the order in which you specify within-subjects factors is important. Each factor constitutes a level within the previous factor. To use Repeated Measures, you must set up your data correctly. You must dene within-subjects factors in this dialog box. Notice that these factors are not existing variables in your data but rather factors that you dene here.
Example. In a weight-loss study, suppose the weights of several people are measured each week
for ve weeks. In the data le, each person is a subject or case. The weights for the weeks are recorded in the variables weight1, weight2, and so on. The gender of each person is recorded in another variable. The weights, measured for each subject repeatedly, can be grouped by dening a within-subjects factor. The factor could be called week, dened to have ve levels. In the main dialog box, the variables weight1,., weight5 are used to assign the ve levels of week. The variable in the data le that groups males and females (gender) can be specied as a between-subjects factor to study the differences between males and females.

18 Chapter 3

Measures. If subjects were tested on more than one measure at each time, dene the measures. For example, the pulse and respiration rate could be measured on each subject every day for a week. These measures do not exist as variables in the data le but are dened here. A model with more than one measure is sometimes called a doubly multivariate repeated measures model.

57 Generalized Linear Models
Generalized Linear Models Statistics
Figure 6-8 Generalized Linear Models: Statistics tab
Model Effects. Analysis Type. Specify the type of analysis to produce. Type I analysis is generally appropriate
when you have a priori reasons for ordering predictors in the model, while Type III is more generally applicable. Wald statistics are produced in any case.
Confidence intervals. Specify a condence level greater than 50 and less than 100. Wald
intervals are based on the assumption that parameters have an asymptotic normal distribution.
Log-likelihood function. This controls the display format of the log-likelihood function. The
full function includes an additional term that is constant with respect to the parameter estimates; it has no effect on parameter estimation and is left out of the display in some software products.
Print. The following output is available. Case processing summary. Displays the number and percentage of cases included and
excluded from the analysis.
Descriptive statistics. Displays descriptive statistics and summary information about the
dependent variable, covariates, and factors.

58 Chapter 6

Model information. Displays the dataset name, dependent variable or events and trials
variables, offset variable, scale weight variable, probability distribution, and link function.
Goodness of fit statistics. Displays deviance and scaled deviance, Pearson chi-square and
scaled Pearson chi-square, log likelihood, Akaikes information criterion (AIC), nite sample corrected AIC (AICC), Bayesian information criterion (BIC), consistent AIC (CAIC).
Model summary statistics. Displays model t tests, including likelihood ratio statistics for the
model t omnibus test and statistics for the Type I or Type III contrasts for each effect.
Parameter estimates. Displays parameter estimates and corresponding test statistics and
condence intervals. You can optionally display exponentiated parameter estimates in addition to the raw parameter estimates.
Covariance matrix for parameter estimates. Displays the estimated parameter covariance

matrix.

Correlation matrix for parameter estimates. Displays the estimated parameter correlation
Contrast coefficient (L) matrices. Displays contrast coefcients for the default effects and for
the estimated marginal means if requested on the EM Means tab.
General estimable functions. Displays the matrices for generating the contrast coefcient (L)

matrices.

Iteration history. Displays the iteration history for the parameter estimates and log-likelihood,
and prints the last evaluation of the gradient vector and the Hessian matrix. The iteration history table displays parameter estimates for every n iterations beginning with the 0th iteration (the initial estimates), where n is the value of the print interval. If the iteration history is requested, then the last iteration is always displayed regardless of n.
Lagrange multiplier test. Displays Lagrange multiplier test statistics for assessing the validity
of a scale parameter that is computed using the deviance or Pearson chi-square, or set at a xed number, for the normal, gamma, and inverse Gaussian distributions. For the negative binomial distribution, this tests the xed ancillary parameter.
59 Generalized Linear Models
Generalized Linear Models EM Means
Figure 6-9 Generalized Linear Models: EM Means tab
This tab allows you to display the estimated marginal means for levels of factors and factor interactions. You can also request that the overall estimated mean be displayed.
Factors and Interactions. This list contains factors specied on the Predictors tab and factor
interactions specied on the Model tab. Covariates are excluded from this list. Terms can be selected directly from this list or combined into an interaction term using the By * button.
Display Means For. Estimated means are computed for the selected factors and factor interactions. The contrast determines how hypothesis tests are set up to compare the estimated means. The simple contrast requires a reference category or factor level against which the others are compared. Pairwise. Pairwise comparisons are computed for all-level combinations of the specied or
implied factors. This is the only available contrast for factor interactions.
Simple. Compares the mean of each level to the mean of a specied level. This type of
contrast is useful when there is a control group.
Deviation. Each level of the factor is compared to the grand mean. Deviation contrasts are

not orthogonal.

60 Chapter 6
Difference. Compares the mean of each level (except the rst) to the mean of previous levels.
They are sometimes called reverse Helmert contrasts.
Helmert. Compares the mean of each level of the factor (except the last) to the mean of

subsequent levels.

Repeated. Compares the mean of each level (except the last) to the mean of the subsequent

level.

Polynomial. Compares the linear effect, quadratic effect, cubic effect, and so on. The
rst degree of freedom contains the linear effect across all categories; the second degree of freedom, the quadratic effect; and so on. These contrasts are often used to estimate polynomial trends.

Scale. Estimated marginal means can be computed for the response, based on the original
scale of the dependent variable, or for the linear predictor, based on the dependent variable as transformed by the link function.
Adjustment for Multiple Comparisons. When performing hypothesis tests with multiple contrasts, the overall signicance level can be adjusted from the signicance levels for the included contrasts. This group allows you to choose the adjustment method. Least significant difference. This method does not control the overall probability of rejecting
the hypotheses that some linear contrasts are different from the null hypothesis values.
Bonferroni. This method adjusts the observed signicance level for the fact that multiple
contrasts are being tested.
Sequential Bonferroni. This is a sequentially step-down rejective Bonferroni procedure that is
much less conservative in terms of rejecting individual hypotheses but maintains the same overall signicance level.
Sidak. This method provides tighter bounds than the Bonferroni approach. Sequential Sidak. This is a sequentially step-down rejective Sidak procedure that is much less
conservative in terms of rejecting individual hypotheses but maintains the same overall signicance level.
61 Generalized Linear Models
Generalized Linear Models Save
Figure 6-10 Generalized Linear Models: Save tab
Checked items are saved with the specied name; you can choose to overwrite existing variables with the same name as the new variables or avoid name conicts by appendix sufxes to make the new variable names unique.
Predicted value of mean of response. Saves model-predicted values for each case in the
original response metric.
Confidence interval for mean of response. Saves the upper and lower bounds of the condence
interval for the mean of the response.
Predicted value of linear predictor. Saves model-predicted values for each case in the metric of
the linear predictor (transformed response via the specied link function).
Estimated standard error of predicted value of linear predictor. Cooks distance. A measure of how much the residuals of all cases would change if a particular
Leverage value. Measures the inuence of a point on the t of the regression. The centered
leverage ranges from 0 (no inuence on the t) to (N-1)/N.

62 Chapter 6

Raw residual. The difference between an observed value and the value predicted by the model. Pearson residual. The square root of the contribution of a case to the Pearson chi-square
statistic, with the sign of the raw residual.
Standardized Pearson residual. The Pearson residual multiplied by the square root of the
inverse of the product of the scale parameter and 1leverage for the case.
Deviance residual. The square root of the contribution of a case to the Deviance statistic,

between any two elements is equal to for adjacent elements, 2 for elements that are separated by a third, and so on. is constrained so that 1<<1.
Exchangeable. This structure has homogenous correlations between elements. It is also
known as a compound symmetry structure.
M-dependent. Consecutive measurements have a common correlation coefcient, pairs of
measurements separated by a third have a common correlation coefcient, and so on, through pairs of measurements separated by m1 other measurements. Measurements with greater separation are assumed to be uncorrelated. When choosing this structure, specify a value of m less than the order of the working correlation matrix.
Unstructured. This is a completely general correlation matrix.
By default, the procedure will adjust the correlation estimates by the number of nonredundant parameters. Removing this adjustment may be desirable if you want the estimates to be invariant to subject-level replication changes in the data.
Maximum iterations. The maximum number of iterations the generalized estimating equations
algorithm will execute. Specify a non-negative integer. This specication applies to the parameters in the linear model part of the generalized estimating equations, while the specication on the Estimation tab applies only to the initial generalized linear model.
Update matrix. Elements in the working correlation matrix are estimated based on the
parameter estimates, which are updated in each iteration of the algorithm. If the working correlation matrix is not updated at all, the initial working correlation matrix is used throughout the estimation process. If the matrix is updated, you can specify the iteration
67 Generalized Estimating Equations
interval at which to update working correlation matrix elements. Specifying a value greater than 1 may reduce processing time.
Convergence criteria. These specications apply to the parameters in the linear model part of the
generalized estimating equations, while the specication on the Estimation tab applies only to the initial generalized linear model.
Parameter convergence. When selected, the algorithm stops after an iteration in which the
Hessian convergence. Convergence is assumed if a statistic based on the Hessian is less than
the value specied, which must be positive.

The General Loglinear Analysis procedure displays model information and goodness-of-t statistics. In addition, you can choose one or more of the following:
Display. Several statistics are available for displayobserved and expected cell frequencies;
raw, adjusted, and deviance residuals; a design matrix of the model; and parameter estimates for the model.
Plot. Plots, available for custom models only, include two scatterplot matrices (adjusted residuals
or deviance residuals against observed and expected cell counts). You can also display normal probability and detrended normal plots of adjusted residuals or deviance residuals.
Confidence Interval. The condence interval for parameter estimates can be adjusted. Criteria. The Newton-Raphson method is used to obtain maximum likelihood parameter estimates. You can enter new values for the maximum number of iterations, the convergence criterion, and delta (a constant added to all cells for initial approximations). Delta remains in the cells for saturated models.
General Loglinear Analysis Save
Figure 9-4 General Loglinear Analysis Save dialog box
91 General Loglinear Analysis
Select the values you want to save as new variables in the active dataset. The sufx n in the new variable names increments to make a unique name for each saved variable. The saved values refer to the aggregated data (cells in the contingency table), even if the data are recorded in individual observations in the Data Editor. If you save residuals or predicted values for unaggregated data, the saved value for a cell in the contingency table is entered in the Data Editor for each case in that cell. To make sense of the saved values, you should aggregate the data to obtain the cell counts. Four types of residuals can be saved: raw, standardized, adjusted, and deviance. The predicted values can also be saved.
Residuals. Also called the simple or raw residual, it is the difference between the observed
cell count and its expected count.
Standardized residuals. The residual divided by an estimate of its standard error. Standardized
residuals are also known as Pearson residuals.
Adjusted residuals. The standardized residual divided by its estimated standard error. Since
the adjusted residuals are asymptotically standard normal when the selected model is correct, they are preferred over the standardized residuals for checking for normality.
Deviance residuals. The signed square root of an individual contribution to the likelihood-ratio
chi-square statistic (G squared), where the sign is the sign of the residual (observed count minus expected count). Deviance residuals have an asymptotic standard normal distribution.

Example. Is a new nicotine patch therapy better than traditional patch therapy in helping people to quit smoking? You could conduct a study using two groups of smokers, one of which received the traditional therapy and the other of which received the experimental therapy. Constructing life tables from the data would allow you to compare overall abstinence rates between the two groups to determine if the experimental treatment is an improvement over the traditional therapy. You can also plot the survival or hazard functions and compare them visually for more detailed information. Statistics. Number entering, number leaving, number exposed to risk, number of terminal events,
proportion terminating, proportion surviving, cumulative proportion surviving (and standard error), probability density (and standard error), and hazard rate (and standard error) for each time interval for each group; median survival time for each group; and Wilcoxon (Gehan) test for comparing survival distributions between groups. Plots: function plots for survival, log survival, density, hazard rate, and one minus survival.
Data. Your time variable should be quantitative. Your status variable should be dichotomous or
categorical, coded as integers, with events being coded as a single value or a range of consecutive values. Factor variables should be categorical, coded as integers.
Assumptions. Probabilities for the event of interest should depend only on time after the initial eventthey are assumed to be stable with respect to absolute time. That is, cases that enter the study at different times (for example, patients who begin treatment at different times) should behave similarly. There should also be no systematic differences between censored and uncensored cases. If, for example, many of the censored cases are patients with more serious conditions, your results may be biased.

98 Chapter 11

Related procedures. The Life Tables procedure uses an actuarial approach to this kind of analysis
(known generally as Survival Analysis). The Kaplan-Meier Survival Analysis procedure uses a slightly different method of calculating life tables that does not rely on partitioning the observation period into smaller time intervals. This method is recommended if you have a small number of observations, such that there would be only a small number of observations in each survival time interval. If you have variables that you suspect are related to survival time or variables that you want to control for (covariates), use the Cox Regression procedure. If your covariates can have different values at different points in time for the same case, use Cox Regression with Time-Dependent Covariates.

113 Computing Time-Dependent Covariates
In the Compute Time-Dependent Covariate dialog box, you can use the function-building controls to build the expression for the time-dependent covariate, or you can enter it directly in the Expression for T_COV_ text area. Note that string constants must be enclosed in quotation marks or apostrophes, and numeric constants must be typed in American format, with the dot as the decimal delimiter. The resulting variable is called T_COV_ and should be included as a covariate in your Cox Regression model.
Computing a Time-Dependent Covariate
E From the menus choose: Analyze Survival Cox w/ Time-Dep Cov. Figure 14-1 Compute Time-Dependent Covariate dialog box
E Enter an expression for the time-dependent covariate. E Click Model to proceed with your Cox Regression.
Note: Be sure to include the new variable T_COV_ as a covariate in your Cox Regression model. For more information, see Cox Regression Analysis in Chapter 13 on p. 106.
Cox Regression with Time-Dependent Covariates Additional Features
The SPSS command language also allows you to specify multiple time-dependent covariates. Other command syntax features are available for Cox Regression with or without time-dependent covariates. See the SPSS Command Syntax Reference for complete syntax information.

Appendix

Categorical Variable Coding Schemes
In many SPSS 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(k1) ( 1/k ( 1/k ( 11/k ( 1/k 1/k 1/k 11/k. 1/k. 11/k 1/k ). 1/k 1/k 1/k 1/k ) 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 rst 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 )
115 Categorical Variable Coding Schemes

Simple

Simple contrasts. Compares each level of a factor to the last. The general matrix form is

mean df(1) df(2). df(k1) (0 ( 1/k (1 (0 1/k 0 1. 0. ). 1/k 1/k ) 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 1/1/1/4 ) 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
( 1/4 (1 (0 (0 1/1/1/4 ) 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). ( 1/k (1 (0 1/k 1/(k1) 1. 1/k 1/(k1) 1/(k2) 1/k ) 1/(k1) ) 1/(k2) )

116 Appendix A

df(k2) df(k1)
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 (0 (0 1/4 1/0 1/4 1/3 1/1/4 ) 1/3 ) 1/2 ) 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(k1) ( 1/(k1) ( 1/k ( 1 ( 1/2 1/k 1 1/2. 1/(k1) 1/(k1). 1) 1/k 0 1. 1/k ) 0) 0)
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 ( 1/2 ( 1/3 1/1/2 1/3 1/1 1/3 1/4 ) 0) 0) 1)

Polynomial

Orthogonal polynomial contrasts. The rst 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 specied 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

117 Categorical Variable Coding Schemes /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 rst group and the dosage administered to the third group is three times that given to the rst group, the treatment categories are equally spaced, and an appropriate metric for this situation consists of consecutive integers:
/CONTRAST(DRUG)=POLYNOMIAL(1,2,3)
If, however, the dosage administered to the second group is four times that given to the rst group, and the dosage administered to the third group is seven times that given to the rst group, an appropriate metric is
/CONTRAST(DRUG)=POLYNOMIAL(1,4,7)
In either case, the result of the contrast specication is that the rst 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 tting, such as curvilinear regression.

Repeated

Compares adjacent levels of an independent variable. The general matrix form is
mean df(1) df(2). df(k1) (0 ( 1/k (1 (0 1/k 1 1. 0 0. ) 1/k 0 1. 1/k 1/k ) 0) 0)
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 1/1/1/4 ) 0) 0) 1 )
These contrasts are useful in prole analysis and wherever difference scores are needed.

118 Appendix A

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 rst row entered is always the mean, or constant, effect and represents the

ARMA(1,1). This is a rst-order autoregressive moving average structure. It has homogenous
variances. The correlation between two elements is equal to * for adjacent elements, *(2) for elements separated by a third, and so on. and are the autoregressive and moving average parameters, respectively, and their values are constrained to lie between 1 and 1, inclusive.
121 Covariance Structures
Compound Symmetry. This structure has constant variance and constant covariance.
Compound Symmetry: Correlation Metric. This covariance structure has homogenous variances and homogenous correlations between elements.
Compound Symmetry: Heterogenous. This covariance structure has heterogenous variances and constant correlation between elements.
Diagonal. This covariance structure has heterogenous variances and zero correlation between

elements.

Factor Analytic: First-Order. This covariance structure has heterogenous variances that are
composed of a term that is heterogenous across elements and a term that is homogenous across elements. The covariance between any two elements is the square root of the product of their heterogenous variance terms.

122 Appendix B

Factor Analytic: First-Order, Heterogenous. This covariance structure has heterogenous variances that are composed of two terms that are heterogenous across elements. The covariance between any two elements is the square root of the product of the rst of their heterogenous variance terms.
Huynh-Feldt. This is a circular matrix in which the covariance between any two elements
is equal to the average of their variances minus a constant. Neither the variances nor the covariances are constant.
Scaled Identity. This structure has constant variance. There is assumed to be no correlation

between any elements.

Toeplitz. This covariance structure has homogenous variances and heterogenous correlations
between elements. The correlation between adjacent elements is homogenous across pairs of adjacent elements. The correlation between elements separated by a third is again homogenous, and so on.