The PROBIT Procedure

1 Chapter 54 The PROBIT ProcedureChapter Table of the Natural Response Threshold Parameter ..2843 OUTPUTS tatement ..2846 WEIGHTS tatement ..2847 Response Level Ordering ..2847 ComputationalMethod ..2851 Tolerance Distribution ..2852 InverseConfidenceLimits ..2852 OUTEST= Data ..2854 ODST ableNames .. Multilevel Response ..2865 Example Logistic regression .. Chapter 54. The PROBIT ProcedureSAS OnlineDoc : Version 8 Chapter 54 The PROBIT ProcedureOverviewThe PROBIT Procedure calculates maximum likelihood estimates of regression pa-rameters and the natural (or threshold) response rate for quantal response data frombiological assays or other discrete event data.

2 This includes PROBIT , logit, ordinallogistic, and extreme value (or gompit) regression analysis developed from the need to analyze qualitative (dichotomous or poly-tomous) dependent variables within the regression framework. Many response vari-ables are binary by nature (yes/no), while others are measured ordinally rather thancontinuously (degree of severity). Ordinary least squares (OLS) regression has beenshown to be inadequate when the dependent variable is discrete (Collett, 1991 andAgresti, 1990). PROBIT or logit analyses are more appropriate in this PROBIT Procedure computes maximum likelihood estimates of the parameters andCof the PROBIT equation using a modified Newton-Raphson algorithm.

3 Whenthe response Y is binary, with values 0 and 1, the PROBIT equation isp=Pr(Y=0)=C+(1 C)F(x0 )where is a vector of parameter estimatesFis a cumulative distribution function (the normal, logistic, or extreme value)xis a vector of explanatory variablespis the probability of a responseCis the natural (threshold) response rateNotice that PROC PROBIT , by default, models the probability of thelowerresponselevels. The choice of the distribution functionF(normal for the PROBIT model, logisticfor the logit model, and extreme value or Gompertz for the gompit model) determinesthe type of analysis. For most problems, there is relatively little difference betweenthe normal and logistic specifications of the model.

4 Both distributions are symmetricabout the value zero. The extreme value (or Gompertz) distribution, however, is notsymmetric, approaching 0 on the left more slowly than it approaches 1 on the can use the extreme value distribution where such asymmetry is ordinal response models, the response, Y, of an individual or an experimental unitmay be restricted to one of a (usually small) number,k+1(k 1), of ordinal values,denoted for convenience by1;:::;k;k+1. For example, the severity of coronary2834 Chapter 54. The PROBIT Proceduredisease can be classified into three response categories as 1=no disease, 2=anginapectoris, and 3=myocardial infarction.

5 The PROBIT Procedure fits a common slopescumulative model, which is a parallel lines regression model based on the cumulativeprobabilities of the response categories rather than on their individual cumulative model has the formPr(Y 1jx)=F(x0 )Pr(Y ijx)=F( i+x0 );2 i kwhere 2;:::; karek 1intercept parameters. By default, the covariate vectorxcontains an overall intercept can set or estimate the natural (threshold) response rateC. Estimation ofCcan begin either from an initial value that you specify or from the rate observed in acontrol group. By default, the natural response rate is fixed at observation in the data set analyzed by the PROBIT Procedure may contain theresponse and explanatory values for one subject.

6 Alternatively, it may provide thenumber of observed events from a number of subjects at a particular setting of theexplanatory variables. In this case, PROC PROBIT models the probability of StartedThe following example illustrates how you can use the PROBIT Procedure to com-pute the threshold response rate and regression parameter estimates for quantal re-sponse the Natural Response Threshold ParameterSuppose you want to test the effect of a drug at 12 dosage levels. You randomlydivide 180 subjects into 12 groups of 15 one group for each dosage level. You thenconduct the experiment and, for each subject, record the presence or absence of apositive response to the drug.

7 You summarize the data by counting the number ofsubjects responding positively in each dose group. Your data set is as follows:data study;input Dose Respond;Number = 15;Observed=Respond/Number;datalines; 5 SAS OnlineDoc : Version 8 Estimating the Natural Response Threshold Parameter 13;run;The variabledoserepresents the amount of drug administered. The first group, re-ceiving a dose level of 0, is the control group. The variablenumberrepresents thenumber of subjects in each group. All groups are equal in size; hence,numberhasthe value 15 for all observations. The variablerespondrepresents the number ofsubjects responding to the associated drug dosage.

8 The variableobservedis usedin subsequent statements for comparison with the predicted probabilities output fromthe PROBIT can model the probability of positive response as a function of dosage using thefollowing statements:proc PROBIT data=study log10 optc;model respond/number=dose;output out=new p=p_hat;run;The DATA= option specifies that PROC PROBIT analyze the SAS data LOG10 option replaces the first continuous independent variable (dose) by itscommon logarithm. The OPTC option estimates the natural response rate. When youuse the LOG10 option with the OPTC option, any observations with a dose value lessthan or equal to zero are used in the estimation as a control OUTPUT statement creates a new data set,new, that contains all the variablesin the original data set, and a new variable,p hat, that represents the predicted MODEL statement specifies a proportional response using the variablesrespondandnumberinevents/trials syntax.

9 The variabledoseis the stimulus or explanatoryvariable. The results from this analysis are displayed in the following OnlineDoc : Version 82836 Chapter 54. The PROBIT ProcedureThe SAS SystemProbit ProcedureModel InformationData Set Variable RespondTrials Variable NumberNumber of Observations 12 Number of Events 81 Number of Trials 180 Number of Events In Control Group 3 Number of Trials In Control Group 15 Name of Distribution NORMALLog Likelihood Fitting Information for the PROBIT ProcedureFigure displays background

10 Information about the model fit. Included are thename of the input data set, the response variables used, and the number of obser-vations, events, and trials. The last line in Figure shows the final value of thelog-likelihood displays the table of parameter estimates for the model. The parameterC,which is the natural response threshold or the proportion of individuals responding atzero dose, is estimated to be Since both the intercept and the slope coefficienthave significantp-values ( , ), you can write the model forPr(response)=C+(1 C)F(x0 )asPr(response)=0:2409+0:7591( ( 4:1439+6:2308 log10(dose)))where is the normal cumulative distribution ProcedureAnalysis of Parameter EstimatesStandardVariable DF Estimate Error Chi-Square Pr > ChiSq LabelIntercept 1 InterceptLog10(Dose) 1 Lower thresholdFigure Parameter Estimates for the PROBIT ProcedureSAS OnlineDoc.