Example 22.3: Logistic Regression, Standard Response Function

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The CATMOD Procedure

Example 22.3: Logistic Regression, Standard Response Function

In this data set, from Cox and Snell (1989), ingots are prepared with different heating and soaking times and tested for their readiness to be rolled. The response variable Y has value 1 for ingots that are not ready and value 0 otherwise. The explanatory variables are Heat and Soak.

   title 'Maximum Likelihood Logistic Regression';
   data ingots;
      input Heat Soak nready ntotal @@;
      Count=nready;
      Y=1;
      output;
      Count=ntotal-nready;
      Y=0;
      output;
      drop nready ntotal;
      datalines;
   7 1.0 0 10   14 1.0 0 31   27 1.0 1 56   51 1.0 3 13
   7 1.7 0 17   14 1.7 0 43   27 1.7 4 44   51 1.7 0  1
   7 2.2 0  7   14 2.2 2 33   27 2.2 0 21   51 2.2 0  1
   7 2.8 0 12   14 2.8 0 31   27 2.8 1 22   51 4.0 0  1
   7 4.0 0  9   14 4.0 0 19   27 4.0 1 16
   ;

Logistic regression analysis is often used to investigate the relationship between discrete response variables and continuous explanatory variables. For logistic regression, the continuous design-effects are declared in a DIRECT statement. The following statements produce Output 22.3.1 through Output 22.3.7.

   proc catmod data=ingots;
      weight Count;
      direct Heat Soak;
      model Y=Heat Soak / freq covb corrb;
   quit;

Output 22.3.1: Maximum Likelihood Logistic Regression

Maximum Likelihood Logistic Regression

The CATMOD Procedure

Response	Y	Response Levels	2
Weight Variable	Count	Populations	19
Data Set	INGOTS	Total Frequency	387
Frequency Missing	0	Observations	25

Population Profiles
Sample	Heat	Soak	Sample Size
1	7	1	10
2	7	1.7	17
3	7	2.2	7
4	7	2.8	12
5	7	4	9
6	14	1	31
7	14	1.7	43
8	14	2.2	33
9	14	2.8	31
10	14	4	19
11	27	1	56
12	27	1.7	44
13	27	2.2	21
14	27	2.8	22
15	27	4	16
16	51	1	13
17	51	1.7	1
18	51	2.2	1
19	51	4	1

You can verify that the populations are defined as you intended by looking at the "Population Profiles" table in Output 22.3.1.

Output 22.3.2: Response Summaries

Maximum Likelihood Logistic Regression

The CATMOD Procedure

Response Profiles
Response	Y
1	0
2	1

Response Frequencies
Sample	Response Number
Sample	1	2
1	10	0
2	17	0
3	7	0
4	12	0
5	9	0
6	31	0
7	43	0
8	31	2
9	31	0
10	19	0
11	55	1
12	40	4
13	21	0
14	21	1
15	15	1
16	10	3
17	1	0
18	1	0
19	1	0

Since the "Response Profiles" table shows the response level ordering as 0, 1, the default response function, the logit, is defined as log([(p_{Y = 0})/(p_{Y = 1})]).

Output 22.3.3: Iteration History

Maximum Likelihood Logistic Regression

The CATMOD Procedure

Maximum Likelihood Analysis
Iteration	Sub Iteration	-2 Log Likelihood	Convergence Criterion	Parameter Estimates
Iteration	Sub Iteration	-2 Log Likelihood	Convergence Criterion	1	2	3
0	0	536.49592	1.0000	0	0	0
1	0	152.58961	0.7156	2.1594	-0.0139	-0.003733
2	0	106.76066	0.3003	3.5334	-0.0363	-0.0120
3	0	96.692171	0.0943	4.7489	-0.0640	-0.0299
4	0	95.383825	0.0135	5.4138	-0.0790	-0.0498
5	0	95.345659	0.000400	5.5539	-0.0819	-0.0564
6	0	95.345613	4.8289E-7	5.5592	-0.0820	-0.0568
7	0	95.345613	7.73E-13	5.5592	-0.0820	-0.0568

Maximum likelihood computations converged.

Seven Newton-Raphson iterations are required to find the maximum likelihood estimates.

Output 22.3.4: Analysis of Variance Table

Maximum Likelihood Logistic Regression

The CATMOD Procedure

Maximum Likelihood Analysis of Variance
Source	DF	Chi-Square	Pr > ChiSq
Intercept	1	24.65	<.0001
Heat	1	11.95	0.0005
Soak	1	0.03	0.8639
Likelihood Ratio	16	13.75	0.6171

The analysis of variance table (Output 22.3.4) shows that the model fits since the likelihood ratio goodness-of-fit test is nonsignificant. It also shows that the length of heating time is a significant factor with respect to readiness but that length of soaking time is not.

Output 22.3.5: Maximum Likelihood Estimates

Maximum Likelihood Logistic Regression

The CATMOD Procedure

Analysis of Maximum Likelihood Estimates
Effect	Parameter	Estimate	Standard Error	Chi- Square	Pr > ChiSq
Intercept	1	5.5592	1.1197	24.65	<.0001
Heat	2	-0.0820	0.0237	11.95	0.0005
Soak	3	-0.0568	0.3312	0.03	0.8639

Output 22.3.6: Covariance Matrix

Maximum Likelihood Logistic Regression

The CATMOD Procedure

Covariance Matrix of the Maximum Likelihood Estimates
	1	2	3
1	1.2537133	-0.0215664	-0.2817648
2	-0.0215664	0.0005633	0.0026243
3	-0.2817648	0.0026243	0.1097020

Output 22.3.7: Correlation Matrix

Maximum Likelihood Logistic Regression

The CATMOD Procedure

Correlation Matrix of the Maximum Likelihood Estimates
	1	2	3
1	1.00000	-0.81152	-0.75977
2	-0.81152	1.00000	0.33383
3	-0.75977	0.33383	1.00000

From the table of maximum likelihood estimates (Output 22.3.5), the fitted model is

E( logit(p)) = 5.559 - 0.082( Heat) - 0.057( Soak)

For example, for Sample 1 with Heat =7 and Soak =1, the estimate is

E( logit(p)) = 5.559 - 0.082(7) - 0.057(1) = 4.9284

Predicted values of the logits, as well as the probabilities of readiness, could be obtained by specifying PRED=PROB in the MODEL statement. For the example of Sample 1 with Heat =7 and Soak =1, PRED=PROB would give an estimate of the probability of readiness equal to 0.9928 since

$4.9284 = \log ( \frac{\hat{p}}{1 - \hat{p}} )$

implies that

$\hat{p} = \frac{e^{4.9284}}{1 + e^{4.9284}} = 0.9928$

As another consideration, since soaking time is nonsignificant, you could fit another model that deleted the variable Soak.

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