### Example 75.1 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 p 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 75.1.7. 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 75.1.1:

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
;

ods graphics on;

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


Output 75.1.1: Probit Analysis with Normal Distribution

The 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
5 0 -37.280388 -1.812704962 3.418117919

Model Information
Data Set WORK.A
Events Variable Response
Trials Variable N
Number of Observations 7
Number of Events 38
Number of Trials 74
Name of Distribution Normal
Log Likelihood -37.28038802

Last Evaluation of the Negative
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 Value/DF Pr > ChiSq
Pearson Chi-Square 3.6497 5 0.7299 0.6009
L.R. Chi-Square 4.6381 5 0.9276 0.4616

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

Analysis of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard Error 95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 -1.8127 0.4493 -2.6934 -0.9320 16.27 <.0001
Log10(Dose) 1 3.4181 0.7455 1.9569 4.8794 21.02 <.0001

Probit Model in Terms of
Tolerance Distribution
MU SIGMA
0.53032254 0.29255866

Estimated Covariance Matrix for Tolerance
Parameters
MU SIGMA
MU 0.002418 -0.000409
SIGMA -0.000409 0.004072

The p-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 75.1.2 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 75.1.2: Probit Analysis with Normal Distribution

The Probit Procedure

Probit Analysis on Log10(Dose)
Probability Log10(Dose) 95% Fiducial Limits
0.01 -0.15027 -0.69518 0.07710
0.02 -0.07052 -0.55766 0.13475
0.03 -0.01992 -0.47064 0.17156
0.04 0.01814 -0.40534 0.19941
0.05 0.04911 -0.35233 0.22218
0.06 0.07546 -0.30731 0.24165
0.07 0.09857 -0.26793 0.25881
0.08 0.11926 -0.23273 0.27425
0.09 0.13807 -0.20080 0.28837
0.10 0.15539 -0.17147 0.30142
0.15 0.22710 -0.05086 0.35631
0.20 0.28410 0.04369 0.40124
0.25 0.33299 0.12343 0.44116
0.30 0.37690 0.19348 0.47857
0.35 0.41759 0.25658 0.51504
0.40 0.45620 0.31429 0.55182
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 0.83550
0.75 0.72765 0.62776 0.90265
0.80 0.77655 0.66999 0.98008
0.85 0.83354 0.71675 1.07279
0.90 0.90525 0.77313 1.19191
0.91 0.92257 0.78646 1.22098
0.92 0.94139 0.80083 1.25265
0.93 0.96208 0.81653 1.28759
0.94 0.98519 0.83394 1.32672
0.95 1.01154 0.85367 1.37149
0.96 1.04250 0.87669 1.42424
0.97 1.08056 0.90480 1.48928
0.98 1.13116 0.94189 1.57602
0.99 1.21092 0.99987 1.71321

Output 75.1.3 displays probit analysis with dose levels. The ED50 for dose is 3.39 with a 95% confidence interval of (2.61, 4.27).

Output 75.1.3: Probit Analysis with Normal Distribution

The Probit Procedure

Probit Analysis on Dose
Probability Dose 95% Fiducial Limits
0.01 0.70750 0.20175 1.19427
0.02 0.85012 0.27691 1.36380
0.03 0.95517 0.33834 1.48444
0.04 1.04266 0.39324 1.58274
0.05 1.11971 0.44429 1.66793
0.06 1.18976 0.49282 1.74443
0.07 1.25478 0.53960 1.81473
0.08 1.31600 0.58515 1.88042
0.09 1.37427 0.62980 1.94252
0.10 1.43019 0.67380 2.00181
0.15 1.68696 0.88950 2.27147
0.20 1.92353 1.10584 2.51906
0.25 2.15276 1.32870 2.76161
0.30 2.38180 1.56128 3.01000
0.35 2.61573 1.80543 3.27374
0.40 2.85893 2.06200 3.56306
0.45 3.11573 2.33098 3.89038
0.50 3.39096 2.61175 4.27138
0.55 3.69051 2.90374 4.72619
0.60 4.02199 3.20759 5.28090
0.65 4.39594 3.52651 5.97077
0.70 4.82770 3.86765 6.84706
0.75 5.34134 4.24385 7.99189
0.80 5.97787 4.67724 9.55169
0.85 6.81617 5.20900 11.82480
0.90 8.03992 5.93105 15.55653
0.91 8.36704 6.11584 16.63320
0.92 8.73752 6.32165 17.89163
0.93 9.16385 6.55431 19.39034
0.94 9.66463 6.82245 21.21881
0.95 10.26925 7.13949 23.52275
0.96 11.02811 7.52816 26.56066
0.97 12.03830 8.03149 30.85201
0.98 13.52585 8.74763 37.67206
0.99 16.25233 9.99709 51.66627

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

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


The regression parameter estimates in Output 75.1.4 for the logistic model of –3.22 and 5.97 are approximately times as large as those for the normal model.

Output 75.1.4: Probit Analysis with Logistic Distribution

The 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
Name of Distribution Logistic
Log Likelihood -37.11065336

Analysis of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard Error 95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 -3.2246 0.8861 -4.9613 -1.4880 13.24 0.0003
Log10(Dose) 1 5.9702 1.4492 3.1299 8.8105 16.97 <.0001

Output 75.1.5 and Output 75.1.6 show that both the ED50 and the LD50 are similar to those for the normal model.

Output 75.1.5: Probit Analysis with Logistic Distribution

The Probit Procedure

Probit Analysis on Log10(Dose)
Probability Log10(Dose) 95% Fiducial Limits
0.01 -0.22955 -0.97441 0.04234
0.02 -0.11175 -0.75158 0.12404
0.03 -0.04212 -0.62018 0.17265
0.04 0.00780 -0.52618 0.20771
0.05 0.04693 -0.45265 0.23533
0.06 0.07925 -0.39205 0.25826
0.07 0.10686 -0.34037 0.27796
0.08 0.13103 -0.29521 0.29530
0.09 0.15259 -0.25502 0.31085
0.10 0.17209 -0.21875 0.32498
0.15 0.24958 -0.07552 0.38207
0.20 0.30792 0.03092 0.42645
0.25 0.35611 0.11742 0.46451
0.30 0.39820 0.19143 0.49932
0.35 0.43644 0.25684 0.53275
0.40 0.47221 0.31588 0.56619
0.45 0.50651 0.36986 0.60089
0.50 0.54013 0.41957 0.63807
0.55 0.57374 0.46559 0.67894
0.60 0.60804 0.50846 0.72474
0.65 0.64381 0.54896 0.77673
0.70 0.68205 0.58815 0.83637
0.75 0.72414 0.62752 0.90582
0.80 0.77233 0.66915 0.98876
0.85 0.83067 0.71631 1.09242
0.90 0.90816 0.77562 1.23343
0.91 0.92766 0.79014 1.26931
0.92 0.94922 0.80607 1.30912
0.93 0.97339 0.82378 1.35391
0.94 1.00100 0.84384 1.40523
0.95 1.03332 0.86713 1.46546
0.96 1.07245 0.89511 1.53864
0.97 1.12237 0.93053 1.63228
0.98 1.19200 0.97952 1.76329
0.99 1.30980 1.06166 1.98569

Output 75.1.6: Probit Analysis with Logistic Distribution

The Probit Procedure

Probit Analysis on Dose
Probability Dose 95% Fiducial Limits
0.01 0.58945 0.10607 1.10241
0.02 0.77312 0.17718 1.33058
0.03 0.90757 0.23978 1.48817
0.04 1.01813 0.29773 1.61327
0.05 1.11413 0.35266 1.71922
0.06 1.20018 0.40546 1.81244
0.07 1.27896 0.45670 1.89654
0.08 1.35218 0.50675 1.97379
0.09 1.42100 0.55588 2.04572
0.10 1.48625 0.60430 2.11339
0.15 1.77656 0.84038 2.41030
0.20 2.03199 1.07379 2.66961
0.25 2.27043 1.31046 2.91416
0.30 2.50152 1.55393 3.15736
0.35 2.73172 1.80652 3.40996
0.40 2.96627 2.06957 3.68292
0.45 3.21006 2.34345 3.98927
0.50 3.46837 2.62768 4.34578
0.55 3.74746 2.92138 4.77466
0.60 4.05546 3.22451 5.30573
0.65 4.40366 3.53961 5.98041
0.70 4.80891 3.87391 6.86079
0.75 5.29836 4.24155 8.05044
0.80 5.92009 4.66820 9.74455
0.85 6.77126 5.20365 12.37149
0.90 8.09391 5.96508 17.11715
0.91 8.46559 6.16800 18.59129
0.92 8.89644 6.39837 20.37592
0.93 9.40575 6.66469 22.58957
0.94 10.02317 6.97977 25.42292
0.95 10.79732 7.36428 29.20549
0.96 11.81534 7.85438 34.56521
0.97 13.25466 8.52173 42.88232
0.98 15.55972 9.53941 57.98207
0.99 20.40815 11.52549 96.75820

The PLOT=PREDPPLOT option together with the ODS GRAPHICS statement creates the plot of observed and fitted probabilities in Output 75.1.7. The dashed line represent pointwise confidence bands for the probabilities.

Output 75.1.7: Plot of Observed and Fitted Probabilities