The LOGISTIC Procedure - SAS

1 SAS/STAT User s GuideThe LOGISTIC ProcedureThis document is an individual chapter fromSAS/STAT User s correct bibliographic citation for the complete manual is as follows: SAS Institute Inc. User s , NC: SAS Institute 2013, SAS Institute Inc., Cary, NC, USAAll rights reserved. Produced in the United States of a hard-copy book: No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or byany means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the publisher, SAS a web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the timeyou acquire this scanning, uploading, and distribution of this book via the Internet or any other means without the permission of the publisher isillegal and punishable by law. Please purchase only authorized electronic editions and do not participate in or encourage electronicpiracy of copyrighted materials.

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3 Formore information about our offerings, call and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in theUSA and other countries. indicates USA brand and product names are trademarks of their respective and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are trademarks of their respective companies. 2013 SAS Institute Inc. All rights reserved. all that you need on your journey to knowledge and additional books and Greater Insight into Your SAS Software with SAS 58 The LOGISTIC ProcedureContentsOverview: LOGISTIC Procedure ..4485 Getting Started: LOGISTIC Procedure ..4488 Syntax: LOGISTIC Procedure ..4496 PROC LOGISTIC Statement ..4497BY Statement ..4509 CLASS Statement.

4 4510 CODE Statement ..4513 CONTRAST Statement ..4513 EFFECT Statement ..4516 EFFECTPLOT Statement ..4518 ESTIMATE Statement ..4519 EXACT Statement ..4520 EXACTOPTIONS Statement ..4522 FREQ Statement ..4525ID Statement ..4526 LSMEANS Statement ..4526 LSMESTIMATE Statement ..4527 MODEL Statement ..4529 NLOPTIONS Statement ..4544 ODDSRATIO Statement ..4544 OUTPUT Statement ..4546 ROC Statement ..4551 ROCCONTRAST Statement ..4551 SCORE Statement ..4552 SLICE Statement ..4555 STORE Statement ..4555 STRATA Statement ..4555 TEST Statement ..4557 UNITS Statement ..4557 WEIGHT Statement ..4558 Details: LOGISTIC Procedure ..4559 Missing Values ..4559 Response Level Ordering ..4559 Link Functions and the Corresponding Distributions ..4561 Determining Observations for Likelihood Contributions ..4562 Iterative Algorithms for Model Fitting ..45634484 FChapter 58: The LOGISTIC ProcedureConvergence Criteria.

5 4565 Existence of Maximum Likelihood Estimates ..4565 Effect-Selection Methods ..4566 Model Fitting Information ..4567 Generalized Coefficient of Determination ..4568 Score Statistics and Tests ..4569 Confidence Intervals for Parameters ..4571 Odds Ratio Estimation ..4572 Rank Correlation of Observed Responses and Predicted Probabilities ..4575 Linear Predictor, Predicted Probability, and Confidence Limits ..4576 Classification Table ..4577 Overdispersion ..4579 The Hosmer-Lemeshow Goodness-of-Fit Test ..4581 Receiver Operating Characteristic Curves ..4582 Testing Linear Hypotheses about the Regression Coefficients ..4585 Regression Diagnostics ..4586 Scoring Data Sets ..4589 Conditional LOGISTIC Regression ..4594 Exact Conditional LOGISTIC Regression ..4597 Input and Output Data Sets ..4601 Computational Resources ..4606 Displayed Output ..4609 ODS Table Names ..4614 ODS Graphics.

6 4616 Examples: LOGISTIC Procedure ..4618 Example : Stepwise LOGISTIC Regression and Predicted Values ..4618 Example : LOGISTIC Modeling with Categorical Predictors ..4635 Example : Ordinal LOGISTIC Regression ..4644 Example : Nominal Response Data: Generalized Logits Model ..4650 Example : Stratified Sampling ..4657 Example : LOGISTIC Regression Diagnostics ..4659 Example : ROC Curve, Customized Odds Ratios, Goodness-of-Fit Statistics, R-Square, and Confidence Limits ..4668 Example : Comparing Receiver Operating Characteristic Curves ..4672 Example : Goodness-of-Fit Tests and Subpopulations ..4681 Example : Overdispersion ..4684 Example : Conditional LOGISTIC Regression for Matched Pairs Data ..4688 Example : Firth s Penalized Likelihood Compared with Other Approaches ..4693 Example : Complementary Log-Log Model for Infection Rates ..4696 Example : Complementary Log-Log Model for Interval-Censored Survival Times 4701 Example : Scoring Data Sets.

7 4706 Example : Using the LSMEANS Statement ..4712 Example : Partial Proportional Odds Model ..4719 References ..4724 Overview: LOGISTIC ProcedureF4485 Overview: LOGISTIC ProcedureBinary responses (for example, success and failure), ordinal responses (for example, normal, mild, andsevere), and nominal responses (for example, major TV networks viewed at a certain hour) arise in manyfields of study. LOGISTIC regression analysis is often used to investigate the relationship between these discreteresponses and a set of explanatory variables. Texts that discuss LOGISTIC regression include Agresti (2002);Allison (1999); Collett (2003); Cox and Snell (1989); Hosmer and Lemeshow (2000); Stokes, Davis, andKoch (2012).For binary response models, the response,Y, of an individual or an experimental unit can take on one of twopossible values, denoted for convenience by 1 and 2 (for example,Y= 1 if a disease is present, otherwiseY=2).

8 Supposexis a vector of explanatory variables and the response probability to bemodeled. The linear LOGISTIC model has the formlogit. / log 1 D C 0xwhere is the intercept parameter and D. 1;:::; s/0is the vector ofsslope parameters. Notice that theLOGISTIC Procedure , by default, models the probability of thelowerresponse LOGISTIC model shares a common feature with a more general class of linear models: a functiongDg. /of the mean of the response variable is assumed to be linearly related to the explanatory variables. Sincethe mean implicitly depends on the stochastic behavior of the response, and the explanatory variables areassumed to be fixed, the functiongprovides the link between the random (stochastic) component and thesystematic (deterministic) component of the response variableY. For this reason, Nelder and Wedderburn(1972) refer tog. /as a link function. One advantage of the logit function over other link functions is thatdifferences on the LOGISTIC scale are interpretable regardless of whether the data are sampled prospectivelyor retrospectively (McCullagh and Nelder 1989, Chapter 4).

9 Other link functions that are widely used inpractice are the probit function and the complementary log-log function. The LOGISTIC Procedure enablesyou to choose one of these link functions, resulting in fitting a broader class of binary response models of theformg. /D C 0xFor ordinal response models, the response,Y, of an individual or an experimental unit might be restricted toone of a (usually small) number of ordinal values, denoted for convenience by1;:::;k;kC1. For example,the severity of coronary disease can be classified into three response categories as 1=no disease, 2=anginapectoris, and 3=myocardial infarction. The LOGISTIC Procedure fits a common slopes cumulative model,which is a parallel lines regression model based on the cumulative probabilities of the response categoriesrather than on their individual probabilities. The cumulative model has the ijx//D iC 0x; iD1;:::;kwhere 1;:::; karekintercept parameters, and is the vector of slope parameters.

10 This model has beenconsidered by many researchers. Aitchison and Silvey (1957) and Ashford (1959) employ a probit scaleand provide a maximum likelihood analysis; Walker and Duncan (1967) and Cox and Snell (1989) discussthe use of the log odds scale. For the log odds scale, the cumulative logit model is often referred to as theproportional 58: The LOGISTIC ProcedureFor nominal response LOGISTIC models, where thekC1possible responses have no natural ordering, the logitmodel can also be extended to amultinomialmodel known as ageneralizedorbaseline-categorylogit model,which has the formlog D iC 0ix; iD1;:::;kwhere the 1;:::; karekintercept parameters, and the 1;:::; karekvectors of slope parameters. Thesemodels are a special case of thediscrete choiceorconditional logitmodels introduced by McFadden (1974).The LOGISTIC Procedure fits linear LOGISTIC regression models for discrete response data by the method ofmaximum likelihood.