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 The PROBIT Procedure

Example 54.1: Dosage Levels

In this example, Dose is the variable representing the level of the 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 statement requests that the log base 10 of Dose is used as the independent variable. Specifically, for a given level of Dose, the probability p of a positive response is modeled as

p = Pr( Response) = F ( b0 + b1 ×log10( Dose) )

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

Lack-of-fit tests and inverse confidence limits are also requested.

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. The plot displays the estimated logistic cumulative distribution function and the observed response rates.

The following statements produce Output 54.1.1:

   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;
model Response/N=Dose / d=logistic inversecl;
output out=B p=Prob std=std xbeta=xbeta;
title 'Output from Probit Procedure';
run;

legend1 label=none frame cframe=ligr cborder=black
position=center value=(justify=center);
axis1 minor=none color=black label=(angle=90 rotate=0) ;
axis2 minor=none color=black;
proc gplot;
plot Observed*Dose Prob*Dose / overlay frame cframe=ligr
vaxis=axis1 haxis=axis2 legend=legend1;
title 'Plot of Observed and Fitted Probabilities';
run;


Output 54.1.1: Dosage Levels: PROC PROBIT

 Output from Probit Procedure

 Probit Procedure

 Iteration History for Parameter Estimates Iter Ridge Loglikelihood Intercept Log10(Dose) 0 0 -51.292891 0 0 1 0 -37.881166 -1.355817008 2.635206083 2 0 -37.286169 -1.764939171 3.3408954936 3 0 -37.280389 -1.812147863 3.4172391614 4 0 -37.280388 -1.812704962 3.418117919

 Output from Probit Procedure

 Probit Procedure

 Model Information Data Set WORK.B Events Variable Response Trials Variable N Number of Observations 7 Number of Events 38 Number of Trials 74 Missing Values 29 Name of Distribution NORMAL Log Likelihood -37.28038802

 Last Evaluation of the Negativeof the Gradient Intercept Log10(Dose) 3.4349069E-7 -2.09809E-8

 Last Evaluation of the Negative of the Hessian Intercept Log10(Dose) Intercept 36.005280383 20.152675982 Log10(Dose) 20.152675982 13.078826305

 Goodness-of-Fit Tests Statistic Value DF Pr > ChiSq Pearson Chi-Square 3.6497 5 0.6009 L.R. Chi-Square 4.6381 5 0.4616

 Response-Covariate Profile Response Levels 2 Number of Covariate Values 7

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

 Output from Probit Procedure

 Probit Procedure

 Analysis of Parameter Estimates Variable DF Estimate Standard Error Chi-Square Pr > ChiSq Label Intercept 1 -1.81270 0.44934 16.2743 <.0001 Intercept Log10(Dose) 1 3.41812 0.74555 21.0196 <.0001

 Probit Model in Terms ofTolerance Distribution MU SIGMA 0.53032254 0.29255866

 Estimated Covariance Matrix for ToleranceParameters MU SIGMA MU 0.002418 -0.000409 SIGMA -0.000409 0.004072

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

 Output from Probit Procedure

 Probit Procedure

 Probit Analysis on Log10(Dose) Probability Log10(Dose) 95% Fiducial Limits 0.01 -0.15027 -0.69520 0.07710 0.02 -0.07052 -0.55768 0.13475 0.03 -0.01992 -0.47066 0.17157 0.04 0.01814 -0.40535 0.19941 0.05 0.04911 -0.35235 0.22218 0.06 0.07546 -0.30733 0.24165 0.07 0.09857 -0.26794 0.25882 0.08 0.11926 -0.23275 0.27426 0.09 0.13807 -0.20081 0.28837 0.10 0.15539 -0.17148 0.30142 0.15 0.22710 -0.05087 0.35631 0.20 0.28410 0.04368 0.40124 0.25 0.33299 0.12342 0.44116 0.30 0.37690 0.19348 0.47857 0.35 0.41759 0.25658 0.51505 0.40 0.45620 0.31428 0.55183 0.45 0.49356 0.36754 0.58999 0.50 0.53032 0.41693 0.63057 0.55 0.56709 0.46296 0.67451 0.60 0.60444 0.50618 0.72271 0.65 0.64305 0.54734 0.77603 0.70 0.68374 0.58745 0.83551 0.75 0.72765 0.62776 0.90265 0.80 0.77655 0.66999 0.98009 0.85 0.83354 0.71675 1.07280 0.90 0.90525 0.77313 1.19192 0.91 0.92257 0.78645 1.22098 0.92 0.94139 0.80083 1.25266 0.93 0.96208 0.81653 1.28760 0.94 0.98519 0.83394 1.32673 0.95 1.01154 0.85367 1.37150 0.96 1.04250 0.87669 1.42425 0.97 1.08056 0.90479 1.48930 0.98 1.13116 0.94189 1.57603 0.99 1.21092 0.99987 1.71322

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 from Probit Procedure

 Probit Procedure

 Probit Analysis on Dose Probability Dose 95% Fiducial Limits 0.01 0.70750 0.20174 1.19428 0.02 0.85012 0.27690 1.36381 0.03 0.95517 0.33833 1.48445 0.04 1.04266 0.39323 1.58275 0.05 1.11971 0.44428 1.66794 0.06 1.18976 0.49280 1.74444 0.07 1.25478 0.53959 1.81474 0.08 1.31600 0.58513 1.88043 0.09 1.37427 0.62978 1.94253 0.10 1.43019 0.67379 2.00182 0.15 1.68696 0.88948 2.27148 0.20 1.92353 1.10582 2.51907 0.25 2.15276 1.32868 2.76162 0.30 2.38180 1.56126 3.01001 0.35 2.61573 1.80541 3.27375 0.40 2.85893 2.06198 3.56308 0.45 3.11573 2.33096 3.89040 0.50 3.39096 2.61173 4.27141 0.55 3.69051 2.90372 4.72622 0.60 4.02199 3.20757 5.28094 0.65 4.39594 3.52649 5.97082 0.70 4.82770 3.86764 6.84712 0.75 5.34134 4.24384 7.99198 0.80 5.97787 4.67723 9.55182 0.85 6.81617 5.20898 11.82500 0.90 8.03992 5.93102 15.55685 0.91 8.36704 6.11581 16.63355 0.92 8.73752 6.32162 17.89203 0.93 9.16385 6.55428 19.39079 0.94 9.66463 6.82242 21.21933 0.95 10.26925 7.13946 23.52336 0.96 11.02811 7.52812 26.56140 0.97 12.03830 8.03145 30.85292 0.98 13.52585 8.74757 37.67327 0.99 16.25233 9.99702 51.66816

The ED50 for dose is 3.39 with a 95% confidence interval of (2.61, 4.27).

 Output from Probit Procedure

 Probit Procedure

 Model Information Data Set WORK.B Events Variable Response Trials Variable N Number of Observations 7 Number of Events 38 Number of Trials 74 Missing Values 29 Name of Distribution LOGISTIC Log Likelihood -37.11065336

 Algorithm converged.

 Output from Probit Procedure

 Probit Procedure

 Analysis of Parameter Estimates Variable DF Estimate Standard Error Chi-Square Pr > ChiSq Label Intercept 1 -3.22464 0.88606 13.2447 0.0003 Intercept Log10(Dose) 1 5.97018 1.44917 16.9721 <.0001

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

 Output from Probit Procedure

 Probit Procedure

 Probit Analysis on Log10(Dose) Probability Log10(Dose) 95% Fiducial Limits 0.01 -0.22955 -0.97443 0.04234 0.02 -0.11175 -0.75160 0.12404 0.03 -0.04212 -0.62020 0.17266 0.04 0.00780 -0.52620 0.20771 0.05 0.04693 -0.45266 0.23533 0.06 0.07925 -0.39207 0.25827 0.07 0.10686 -0.34039 0.27796 0.08 0.13103 -0.29522 0.29530 0.09 0.15259 -0.25503 0.31085 0.10 0.17209 -0.21876 0.32498 0.15 0.24958 -0.07553 0.38207 0.20 0.30792 0.03091 0.42645 0.25 0.35611 0.11742 0.46451 0.30 0.39820 0.19143 0.49933 0.35 0.43644 0.25684 0.53275 0.40 0.47221 0.31587 0.56619 0.45 0.50651 0.36985 0.60090 0.50 0.54013 0.41957 0.63807 0.55 0.57374 0.46559 0.67895 0.60 0.60804 0.50846 0.72475 0.65 0.64381 0.54895 0.77673 0.70 0.68205 0.58815 0.83638 0.75 0.72414 0.62752 0.90583 0.80 0.77233 0.66915 0.98877 0.85 0.83067 0.71631 1.09243 0.90 0.90816 0.77561 1.23344 0.91 0.92766 0.79014 1.26932 0.92 0.94922 0.80607 1.30913 0.93 0.97339 0.82378 1.35392 0.94 1.00100 0.84384 1.40524 0.95 1.03332 0.86713 1.46548 0.96 1.07245 0.89511 1.53866 0.97 1.12237 0.93053 1.63230 0.98 1.19200 0.97952 1.76331 0.99 1.30980 1.06166 1.98571

 Output from Probit Procedure

 Probit Procedure

 Probit Analysis on Dose Probability Dose 95% Fiducial Limits 0.01 0.58945 0.10606 1.10241 0.02 0.77312 0.17717 1.33059 0.03 0.90757 0.23977 1.48818 0.04 1.01813 0.29772 1.61328 0.05 1.11413 0.35264 1.71923 0.06 1.20018 0.40545 1.81245 0.07 1.27896 0.45668 1.89655 0.08 1.35218 0.50673 1.97380 0.09 1.42100 0.55586 2.04573 0.10 1.48625 0.60428 2.11340 0.15 1.77656 0.84036 2.41031 0.20 2.03199 1.07377 2.66962 0.25 2.27043 1.31043 2.91417 0.30 2.50152 1.55391 3.15737 0.35 2.73172 1.80650 3.40997 0.40 2.96627 2.06954 3.68293 0.45 3.21006 2.34343 3.98929 0.50 3.46837 2.62766 4.34580 0.55 3.74746 2.92137 4.77469 0.60 4.05546 3.22449 5.30576 0.65 4.40366 3.53960 5.98046 0.70 4.80891 3.87389 6.86087 0.75 5.29836 4.24153 8.05054 0.80 5.92009 4.66819 9.74470 0.85 6.77126 5.20363 12.37174 0.90 8.09391 5.96506 17.11758 0.91 8.46559 6.16797 18.59179 0.92 8.89644 6.39834 20.37650 0.93 9.40575 6.66466 22.59024 0.94 10.02317 6.97974 25.42373 0.95 10.79732 7.36425 29.20649 0.96 11.81534 7.85434 34.56649 0.97 13.25466 8.52168 42.88406 0.98 15.55972 9.53935 57.98471 0.99 20.40815 11.52540 96.76344

Both the ED50 and the LD50 are similar to those for the normal model.

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