Estimating the Natural Response Threshold Parameter

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;
      Observed=Respond/Number;
      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. The variable observed is used in subsequent statements for comparison with the predicted probabilities output from the PROBIT procedure.

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;
   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.

The SAS System

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 54.1: Model Fitting Information for the PROBIT Procedure

Figure 54.1 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 54.1 shows the final value of the log-likelihood function.

Figure 54.2 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

${Pr({\hv response})} = C + (1 - C) F({x^'\beta})$

${Pr({\hv response})} = 0.2409 + 0.7591 (\Phi(-4.1439 + 6.2308 x \log_{10}{({\hv dose})}))$

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

Probit Procedure

Analysis of Parameter Estimates
Variable	DF	Estimate	Standard Error	Chi-Square	Pr > ChiSq	Label
Intercept	1	-4.14385	1.34149	9.5419	0.0020	Intercept
Log10(Dose)	1	6.23076	1.89958	10.7588	0.0010
_C_		0.24088	0.05226			Lower threshold

Figure 54.2: Model Parameter Estimates for the PROBIT Procedure

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 54.3: Tolerance Distribution Estimates for the PROBIT Procedure

The following PROC GPLOT statements request a plot of the fitted probabilities and observed proportions versus the variable dose.

   symbol1 i=spline v=dot c=white;
   symbol2 i=none   v=dot c=yellow;
   legend1 frame cframe=ligr cborder=black position=center
      label=none value=(justify=center);
   axis1 minor=none color=black label=(angle=90 rotate=0);
   axis2 minor=none color=black;

   proc gplot data=new;
      plot (p_hat observed)*dose/overlay
           frame cframe=ligr vaxis=axis1 haxis=axis2 legend=legend1;
   run;

The SYMBOL statements determine line type, plotting symbol, and color. The AXIS and LEGEND statements determine settings for the plot axes and legend.

The OVERLAY option in the PLOT statement specifies that both plots (p hat versus dose and observed versus dose) are displayed on the same axes. The VAXIS, HAXIS, and LEGEND options direct the GPLOT procedure to use the settings defined in the previous AXIS and LEGEND statements.

Figure 54.4: Plot of Observed and Fitted Probabilities versus Dose Level

The plot in Figure 54.4 shows the relationship between dosage level, observed response proportions, and estimated probability values.

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