Getting Started

The following example illustrates how you can use the PROBIT procedure to compute the threshold response rate and regression parameter estimates for quantal response data.

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;   datalines;   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   ;   run;  

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 using the following statements:

  proc probit data=study log10 optc;   model respond/number=dose;   output    out = new p = p_hat;   predpplot var = dose cfit = blue cframe=ligr inborder;   inset;   ippplot   var = dose cfit = blue cframe=ligr inborder;   inset;   run;  

The DATA= option specifies that PROC PROBIT analyze the SAS data set study . The LOG10 option replaces the first continuous independent variable ( dose ) by 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 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 MODEL statement specifies a proportional response using the variables respond and number in events/trials syntax. The variable dose is the stimulus or explanatory variable. The results from this analysis are displayed in the following figures.

Figure 60.1 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 60.1 shows the final value of the log- likelihood function.

  Probit Procedure   Model Information   Data Set                               WORK.STUDY   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   Algorithm converged.  

Figure 60.1: Model Fitting Information for the PROBIT Procedure

Figure 60.2 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

  Probit Procedure   Analysis of Parameter Estimates   Standard   95% Confidence     Chi   Parameter   DF Estimate    Error       Limits       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 60.2: Model Parameter Estimates for the PROBIT Procedure

as

where 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.

  Probit Procedure   Probit Model in Terms of Tolerance Distribution   MU         SIGMA   0.66506312    0.16049411   Estimated Covariance Matrix for Tolerance Parameters   MU             SIGMA               _C_   MU             0.001158   0.000493          0.000954   SIGMA   0.000493          0.002394   0.000999   _C_            0.000954   0.000999          0.002731  

Figure 60.3: Tolerance Distribution Estimates for the PROBIT Procedure

The PREDPPLOT statement creates the plot in Figure 60.4, 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 .24.

Figure 60.4: Plot of Observed and Fitted Probabilities versus Dose Level

The IPPPLOT statement creates the plot in Figure 60.5, 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 .24.

Figure 60.5: Inverse Predicted Probability Plot with Fiducial Limits

The INSET statement after each of these plot statements draws a box within the plot. In the inset box, summary information about the model fitting is printed.