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Overview of Logistics Regression and its SAS implementation.

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Presentation Transcript

• Logistics regression is widely used nowadays in finance, marketing research and clinical studies when the dependent variable is dichotomous, representing an event or a non-event. However, because ordinary linear regression was routinely used before we had the modern statistical packages for analyzing logit, we will compare the statistical assumptions of logistic regression with that of ordinary least square linear regression.

• Next we will examine PROC LOGISTICS implemented in SAS and discuss the basic statistic output for understanding the logistic regression results.

• We will then discuss how to setup and understand logistics regression when the dependent variable has more than two outcomes.

• We will conclude the presentation by comparing PROC LOGISTICS with other SAS procedures that can also perform logistics regression.

• Examples of discrete responses:

• Getting decease vs. not getting decease

• Good, medium and bad credit risks

• Responders vs. non – responders (both in marketing or clinical trial studies)

• Married vs. unmarried

• Guilty vs. not guilty

• Linear Probability Model

• Logit Model

Why can linear regression work reasonable well on binary dependent variables ?

• If 1) and 2) are true, it can be shown that 3) and 5) are necessarily false. However, the consequences may not be as serious as you expect.

Logistic regression for binary response variables dependent variables ?

Basic Syntax:

• proclogisticdata=chdage1 outest=parmsdescending;

model chd = age /

selection = stepwise

ctablepprob = (0 to 1 by 0.1)

outroc=roc1;

• procscoredata=chdage1 score = parms out=scored type=parms;

var age;

run;

In the events/trials syntax, you specify two variables that contain count data for a binomial experiment. These two variables are separated by a slash. The value of the first variable, events, is the number of positive responses (or events). The value of the second variable, trials, is the number of trials.

Interpretation of SAS output - dependent variables ?continued

• Model Selection Criteria:

• Convergence - difference in parameter estimates is small enough.

• Model Fit Statistics Criteria:

• Likelihood Function:

• – 2 * log (likelihood )

• AIC = – 2 * log ( max likelihood ) + 2 * k

• SIC = – 2 * log ( max likelihood ) + log (N) * k

• Testing Global Null Hypothesis: BETA=0

• Likelihood ratio: ln(L intercept)- ln(L int + covariates),

• Score: 1st and 2nd derivative of Log(L)

• Wald: (coefficient / std error)2

Interpretation of SAS output - dependent variables ?continued

• Analysis of Maximum Likelihood Estimates

• Parameter estimates and significance test

• Odds Ratio Estimates

• Odds:

• Odds ratio: Oi / Oj per unit change in covariate.

• Association of Predicted Probabilities and Observed Responses

• Pairs: 43 (event) * 57 (non event) = 2451

• Concordant (0- lower prob vs. 1- higher prob)

• Discordant (0- higher prob vs. 1- lower prob)

• Tie – all other

• ROC used to visualize model model prediction strength.

Interpretation of SAS output - dependent variables ?continued

Classification Table:

• The model classifies an observation as an event if its estimated probability is greater than or equal to a given probability cutpoints.

Logistic regression for polychotomous response variables dependent variables ?

• Example: Three outcomes

• The cumulative probability model

• The assumption:

• A common slope parameter associated with the predictor.

Logistic regression for polychotomous response variables dependent variables ?

Examples:

• proclogisticdata=diabetes descending;

model group=glutest;

outputout=probs predicted=prob xbeta=logit;

format group gp.;

run;

References dependent variables ?

• Hosmer, D.W, Jr. and Lemeshow, S. (1989), Applied Logistic Regression, New York: John Wiley & Sons, Inc.

• SAS Institute Inc. (1995), Logistic Regression Examples Using the SAS System, Cary, NC: SAS Institute Inc.

• Paul D. Allison (1999)Logistic Regression Using the SAS System: Theory and Application,BBU Press and John Wiley Sons Inc.