The PROBIT Procedure

Estimating the Natural Response Threshold Parameter

Suppose you want to test the effect of a drug at 12 dosage levels. You randomly divide 180 subjects into 12 groups of 15—one group for each dosage level. You then conduct the experiment and, for each subject, record the presence or absence of a positive response to the drug. You summarize the data by counting the number of subjects responding positively in each dose group. Your data set is as follows:

data study;
   input Dose Respond @@;
   Number = 15;
0     3   1.1   4   1.3   4   2.0   3   2.2   5   2.8   4
3.7   5   3.9   9   4.4   8   4.8  11   5.9  12   6.8  13

The variable dose represents the amount of drug administered. The first group, receiving a dose level of 0, is the control group. The variable number represents the number of subjects in each group. All groups are equal in size; hence, number has the value 15 for all observations. The variable respond represents the number of subjects responding to the associated drug dosage.

You can model the probability of positive response as a function of dosage by using the following statements:

ods graphics on;

proc probit data=study log10 optc plots=(predpplot ippplot);
   model respond/number=dose;
   output out=new p=p_hat;

The DATA= option specifies that PROC PROBIT analyze the SAS data set study. The LOG10 option replaces the first continuous independent variable (dose) with its common logarithm. The OPTC option estimates the natural response rate. When you use the LOG10 option with the OPTC option, any observations with a dose value less than or equal to zero are used in the estimation as a control group.

The PLOTS= option in the PROC PROBIT statement, together with the ODS GRAPHICS statement, requests two plots for the estimated probability values and dosage levels. For general information about ODS Graphics, see Chapter 21: Statistical Graphics Using ODS. For specific information about the graphics available in the PROBIT procedure, see the section ODS Graphics.

The MODEL statement specifies a proportional response by using the variables respond and number in events/trials syntax. The variable dose is the stimulus or explanatory variable.

The OUTPUT statement creates a new data set, new, that contains all the variables in the original data set, and a new variable, p_hat, that represents the predicted probabilities.

The results from this analysis are displayed in the following figures.

Figure 81.1 displays background information about the model fit. Included are the name of the input data set, the response variables used, and the number of observations, events, and trials. The last line in Figure 81.1 shows the final value of the log-likelihood function.

Figure 81.2 displays the table of parameter estimates for the model. The parameter C, which is the natural response threshold or the proportion of individuals responding at zero dose, is estimated to be 0.2409. Since both the intercept and the slope coefficient have significant p-values (0.0020, 0.0010), you can write the model for

\[ \mbox{Pr}(\Variable{response}) = C + (1 - C) F(\mb{x}^{\prime }\beta ) \]


\[ \mbox{Pr}(\Variable{response}) = 0.2409 + 0.7591 (\Phi (-4.1439 + 6.2308 \times \log _{10}{(\Variable{dose})})) \]

where $\Phi $ is the normal cumulative distribution function.

Finally, PROC PROBIT specifies the resulting tolerance distribution by providing the mean MU and scale parameter SIGMA as well as the covariance matrix of the distribution parameters in Figure 81.3.

Figure 81.1: Model Fitting Information for the PROBIT Procedure

The Probit Procedure

Model Information
Events Variable Respond
Trials Variable Number
Number 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 Normal
Log Likelihood -104.3945783

Figure 81.2: Model Parameter Estimates for the PROBIT Procedure

Analysis of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard
95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 -4.1438 1.3415 -6.7731 -1.5146 9.54 0.0020
Log10(Dose) 1 6.2308 1.8996 2.5076 9.9539 10.76 0.0010
_C_ 1 0.2409 0.0523 0.1385 0.3433    

Figure 81.3: Tolerance Distribution Estimates for the PROBIT Procedure

Estimated Covariance Matrix for Tolerance Parameters
MU 0.001158 -0.000493 0.000954
SIGMA -0.000493 0.002394 -0.000999
_C_ 0.000954 -0.000999 0.002731

The PLOT=PREDPPLOT option creates the plot in Figure 81.4, showing the relationship between dosage level, observed response proportions, and estimated probability values. The dashed lines represent pointwise confidence bands for the fitted probabilities, and a reference line is plotted at the estimated threshold value of 0.24.

The PLOT=IPPPLOT option creates the plot in Figure 81.5, showing the inverse relationship between dosage level and observed response proportions/estimated probability values. The dashed lines represent pointwise fiducial limits for the predicted values of the dose variable, and a reference line is also plotted at the estimated threshold value of 0.24.

The two plot options can be put together with the PLOTS= option, as shown in the PROC PROBIT statement.

Figure 81.4: Plot of Observed and Fitted Probabilities versus Dose Level

Plot of Observed and Fitted Probabilities versus Dose Level

Figure 81.5: Inverse Predicted Probability Plot with Fiducial Limits

Inverse Predicted Probability Plot with Fiducial Limits