Example 72.1 Dosage Levels

In this example, Dose is a variable representing the level of a stimulus, N represents the number of subjects tested at each level of the stimulus, and Response is the number of subjects responding to that level of the stimulus. Both probit and logit response models are fit to the data. The LOG10 option in the PROC PROBIT statement requests that the log base 10 of Dose is used as the independent variable. Specifically, for a given level of Dose, the probability of a positive response is modeled as

The probabilities are estimated first by using the normal distribution function (the default) and then by using the logistic distribution function. Note that, in this model specification, the natural rate is assumed to be zero.

The LACKFIT option specifies lack-of-fit tests and the INVERSECL option specifies inverse confidence limits.

In the DATA step that reads the data, a number of observations are generated that have a missing value for the response. Although the PROBIT procedure does not use the observations with the missing values to fit the model, it does give predicted values for all nonmissing sets of independent variables. These data points fill in the plot of fitted and observed values in the logistic model displayed in Output 72.1.7. The plot, requested with the PLOT=PREDPPLOT option, displays the estimated logistic cumulative distribution function and the observed response rates.

The following statements produce Output 72.1.1:

data a;
   infile cards eof=eof;
   input Dose N Response @@;
   Observed= Response/N;
   output;
   return;
eof: do Dose=0.5 to 7.5 by 0.25;
        output;
     end;
   datalines;
1 10 1  2 12 2  3 10 4  4 10 5
5 12 8  6 10 8  7 10 10
;

proc probit log10;
   model Response/N=Dose / lackfit inversecl itprint;
   output out=B p=Prob std=std xbeta=xbeta;
run;

The -values in the goodness-of-fit table of 0.6009 for the Pearson’s chi-square and 0.4616 for the likelihood ratio chi-square indicate an adequate fit for the model fit with the normal distribution.

Tolerance distribution parameter estimates for the normal distribution indicate a mean tolerance for the population of 0.5303.

Output 72.1.2 displays probit analysis with the logarithm of dose levels. The LD50 (ED50 for log dose) is 0.5303, the dose corresponding to a probability of 0.5. This is the same as the mean tolerance for the normal distribution.

Output 72.1.3 displays probit analysis with dose levels. The ED50 for dose is 3.39 with a 95% confidence interval of (2.61, 4.27).

The following statements request probit analysis of dosage levels with the logistic distribution:

ods graphics on;
 
proc probit log10 plot=predpplot;
   model Response/N=Dose / d=logistic inversecl;
   output out=B p=Prob std=std xbeta=xbeta;
run;
 
ods graphics off;

The regression parameter estimates in Output 72.1.4 for the logistic model of 3.22 and 5.97 are approximately times as large as those for the normal model.

The PLOT=PREDPPLOT option together with the ODS GRAPHICS statement creates the plot of observed and fitted probabilities in Output 72.1.7. The dashed line represent pointwise confidence bands for the probabilities.

Output 72.1.7 Plot of Observed and Fitted Probabilities