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 The LOGISTIC Procedure

## Example 39.3: Logistic Modeling with Categorical Predictors

Consider a study of the analgesic effects of treatments on elderly patients with neuralgia. Two test treatments and a placebo are compared. The response variable is whether the patient reported pain or not. Researchers recorded age and gender of the patients and the duration of complaint before the treatment began. The data, consisting of 60 patients, are contained in the data set Neuralgia.

   Data Neuralgia;
input Treatment $Sex$ Age Duration Pain \$ @@;
datalines;
P  F  68   1  No   B  M  74  16  No  P  F  67  30  No
P  M  66  26  Yes  B  F  67  28  No  B  F  77  16  No
A  F  71  12  No   B  F  72  50  No  B  F  76   9  Yes
A  M  71  17  Yes  A  F  63  27  No  A  F  69  18  Yes
B  F  66  12  No   A  M  62  42  No  P  F  64   1  Yes
A  F  64  17  No   P  M  74   4  No  A  F  72  25  No
P  M  70   1  Yes  B  M  66  19  No  B  M  59  29  No
A  F  64  30  No   A  M  70  28  No  A  M  69   1  No
B  F  78   1  No   P  M  83   1  Yes B  F  69  42  No
B  M  75  30  Yes  P  M  77  29  Yes P  F  79  20  Yes
A  M  70  12  No   A  F  69  12  No  B  F  65  14  No
B  M  70   1  No   B  M  67  23  No  A  M  76  25  Yes
P  M  78  12  Yes  B  M  77   1  Yes B  F  69  24  No
P  M  66   4  Yes  P  F  65  29  No  P  M  60  26  Yes
A  M  78  15  Yes  B  M  75  21  Yes A  F  67  11  No
P  F  72  27  No   P  F  70  13  Yes A  M  75   6  Yes
B  F  65   7  No   P  F  68  27  Yes P  M  68  11  Yes
P  M  67  17  Yes  B  M  70  22  No  A  M  65  15  No
P  F  67   1  Yes  A  M  67  10  No  P  F  72  11  Yes
A  F  74   1  No   B  M  80  21  Yes A  F  69   3  No
;


The data set Neuralgia contains five variables: Treatment, Sex, Age, Duration, and Pain. The last variable, Pain, is the response variable. A specification of Pain=Yes indicates there was pain, and Pain=No indicates no pain. The variable Treatment is a categorical variable with three levels: A and B represent the two test treatments, and P represents the placebo treatment. The gender of the patients is given by the categorical variable Sex. The variable Age is the age of the patients, in years, when treatment began. The duration of complaint, in months, before the treatment began is given by the variable Duration. The following statements use the LOGISTIC procedure to fit a two-way logit with interaction model for the effect of Treatment and Sex, with Age and Duration as covariates. The categorical variables Treatment and Sex are declared in the CLASS statement.

   proc logistic data=Neuralgia;
class Treatment Sex;
model Pain= Treatment Sex Treatment*Sex Age Duration / expb;
run;


In this analysis, PROC LOGISTIC models the probability of no pain (Pain=No). By default, effect coding is used to represent the CLASS variables. Two dummy variables are created for Treatment and one for Sex, as shown in Output 39.3.1.

Output 39.3.1: Effect Coding of CLASS Variables

 The LOGISTIC Procedure
 Class Level Information Class Value Design Variables 1 2 Treatment A 1 0 B 0 1 P -1 -1 Sex F 1 M -1

PROC LOGISTIC displays a table of the Type III analysis of effects based on the Wald test (Output 39.3.2). Note that the Treatment*Sex interaction and the duration of complaint are not statistically significant (p=0.9318 and p=0.8752, respectively). This indicates that there is no evidence that the treatments affect pain differently in men and women, and no evidence that the pain outcome is related to the duration of pain.

Output 39.3.2: Wald Tests of Individual Effects

 The LOGISTIC Procedure
 Type III Analysis of Effects Effect DF WaldChi-Square Pr > ChiSq Treatment 2 11.9886 0.0025 Sex 1 5.3104 0.0212 Treatment*Sex 2 0.1412 0.9318 Age 1 7.2744 0.0070 Duration 1 0.0247 0.8752

Parameter estimates are displayed in Output 39.3.3. The Exp(Est) column contains the exponentiated parameter estimates. These values may, but do not necessarily, represent odds ratios for the corresponding variables. For continuous explanatory variables, the Exp(Est) value corresponds to the odds ratio for a unit increase of the corresponding variable. For CLASS variables using the effect coding, the Exp(Est) values have no direct interpretation as a comparison of levels. However, when the reference coding is used, the Exp(Est) values represent the odds ratio between the corresponding level and the last level. Following the parameter estimates table, PROC LOGISTIC displays the odds ratio estimates for those variables that are not involved in any interaction terms. If the variable is a CLASS variable, the odds ratio estimate comparing each level with the last level is computed regardless of the coding scheme. In this analysis, since the model contains the Treatment*Sex interaction term, the odds ratios for Treatment and Sex were not computed. The odds ratio estimates for Age and Duration are precisely the values given in the Exp(Est) column in the parameter estimates table.

Output 39.3.3: Parameter Estimates with Effect Coding

 The LOGISTIC Procedure
 Analysis of Maximum Likelihood Estimates Parameter DF Estimate StandardError Chi-Square Pr > ChiSq Exp(Est) Intercept 1 19.2236 7.1315 7.2661 0.0070 2.232E8 Treatment A 1 0.8483 0.5502 2.3773 0.1231 2.336 Treatment B 1 1.4949 0.6622 5.0956 0.0240 4.459 Sex F 1 0.9173 0.3981 5.3104 0.0212 2.503 Treatment*Sex A F 1 -0.2010 0.5568 0.1304 0.7180 0.818 Treatment*Sex B F 1 0.0487 0.5563 0.0077 0.9302 1.050 Age 1 -0.2688 0.0996 7.2744 0.0070 0.764 Duration 1 0.00523 0.0333 0.0247 0.8752 1.005
 Odds Ratio Estimates Effect Point Estimate 95% WaldConfidence Limits Age 0.764 0.629 0.929 Duration 1.005 0.942 1.073

The following PROC LOGISTIC statements illustrate the use of forward selection on the data set Neuralgia to identify the effects that differentiate the two Pain responses. The option SELECTION=FORWARD is specified to carry out the forward selection. Although it is the default, the option RULE=SINGLE is explicitly specified to select one effect in each step where the selection must maintain model hierarchy. The term Treatment|Sex@2 illustrates another way to specify main effects and two-way interaction as is available in other procedures such as PROC GLM. (Note that, in this case, the "@2" is unnecessary because no interactions besides the two-way interaction are possible).

   proc logistic data=Neuralgia;
class Treatment Sex;
model Pain=Treatment|Sex@2 Age Duration/selection=forward
rule=single
expb;
run;


Results of the forward selection process are summarized in Output 39.3.4. The variable Treatment is selected first, followed by Age and then Sex. The results are consistent with the previous analysis (Output 39.3.2) in which the Treatment*Sex interaction and Duration are not statistically significant.

Output 39.3.4: Effects Selected into the Model

 The LOGISTIC Procedure
 Forward Selection Procedure
 Summary of Forward Selection Step EffectEntered DF NumberIn ScoreChi-Square Pr > ChiSq 1 Treatment 2 1 13.7143 0.0011 2 Age 1 2 10.6038 0.0011 3 Sex 1 3 5.9959 0.0143

Output 39.3.5 shows the Type III analysis of effects, the parameter estimates, and the odds ratio estimates for the selected model. All three variables, Treatment, Age, and Sex, are statistically significant at the 0.05 level (p=0.0011, p=0.0011, and p=0.0143, respectively). Since the selected model does not contain the Treatment*Sex interaction, odds ratios for Treatment and Sex are computed. The estimated odds ratio is 24.022 for treatment A versus placebo, 41.528 for Treatment B versus placebo, and 6.194 for female patients versus male patients. Note that these odds ratio estimates are not the same as the corresponding values in the Exp(Est) column in the parameter estimates table because effect coding was used. From Output 39.3.5, it is evident that both Treatment A and Treatment B are better than the placebo in reducing pain; females tend to have better improvement than males; and younger patients are faring better than older patients.

Output 39.3.5: Type III Effects and Parameter Estimates with Effect Coding

 The LOGISTIC Procedure
 Forward Selection Procedure
 Type III Analysis of Effects Effect DF WaldChi-Square Pr > ChiSq Treatment 2 12.6928 0.0018 Sex 1 5.3013 0.0213 Age 1 7.6314 0.0057
 Analysis of Maximum Likelihood Estimates Parameter DF Estimate StandardError Chi-Square Pr > ChiSq Exp(Est) Intercept 1 19.0804 6.7882 7.9007 0.0049 1.9343E8 Treatment A 1 0.8772 0.5274 2.7662 0.0963 2.404 Treatment B 1 1.4246 0.6036 5.5711 0.0183 4.156 Sex F 1 0.9118 0.3960 5.3013 0.0213 2.489 Age 1 -0.2650 0.0959 7.6314 0.0057 0.767
 Odds Ratio Estimates Effect Point Estimate 95% WaldConfidence Limits Treatment A vs P 24.022 3.295 175.121 Treatment B vs P 41.528 4.500 383.262 Sex F vs M 6.194 1.312 29.248 Age 0.767 0.636 0.926

Finally, PROC LOGISTIC is invoked to refit the previously selected model using reference coding for the CLASS variables. Two CONTRAST statments are specified. The one labeled 'Pairwise' specifies three rows in the contrast matrix, L, for all the pairwise comparisons between the three levels of Treatment. The contrast labeled 'Female vs Male' compares female to male patients. The option ESTIMATE=EXP is specified in both CONTRAST statements to exponentiate the estimates of .With the given specification of contrast coefficients, the first row of the 'Pairwise' CONTRAST statement corresponds to the odds ratio of A versus P, the second row corresponds to B versus P, and the third row corresponds to A versus B. There is only one row in the 'Female vs Male' CONTRAST statement, and it corresponds to the odds ratio comparing female to male patients.

   proc logistic data=Neuralgia;
class Treatment Sex /param=ref;
model Pain= Treatment Sex age;
contrast 'Pairwise' Treatment 1 0 -1,
Treatment 0 1 -1,
Treatment 1 -1 0 / estimate=exp;
contrast 'Female vs Male' Sex 1 -1 / estimate=exp;
run;


Output 39.3.6: Reference Coding of CLASS Variables

 The LOGISTIC Procedure
 Class Level Information Class Value Design Variables 1 2 Treatment A 1 0 B 0 1 P 0 0 Sex F 1 M 0

The reference coding is shown in Output 39.3.6. The Type III analysis of effects, the parameter estimates for the reference coding, and the odds ratio estimates are displayed in Output 39.3.7. Although the parameter estimates are different (because of the different parameterizations), the "Type III Analysis of Effects" table and the "Odds Ratio" table remain the same as in Output 39.3.5. With effect coding, the treatment A parameter estimate (0.8772) estimates the effect of treatment A compared to the average effect of treatments A, B, and placebo. The treatment A estimate (3.1790) under the reference coding estimates the difference in effect of treatment A and the placebo treatment.

Output 39.3.7: Type III Effects and Parameter Estimates with Reference Coding

 The LOGISTIC Procedure
 Type III Analysis of Effects Effect DF WaldChi-Square Pr > ChiSq Treatment 2 12.6928 0.0018 Sex 1 5.3013 0.0213 Age 1 7.6314 0.0057
 Analysis of Maximum Likelihood Estimates Parameter DF Estimate StandardError Chi-Square Pr > ChiSq Intercept 1 15.8669 6.4056 6.1357 0.0132 Treatment A 1 3.1790 1.0135 9.8375 0.0017 Treatment B 1 3.7264 1.1339 10.8006 0.0010 Sex F 1 1.8235 0.7920 5.3013 0.0213 Age 1 -0.2650 0.0959 7.6314 0.0057
 Odds Ratio Estimates Effect Point Estimate 95% WaldConfidence Limits Treatment A vs P 24.022 3.295 175.121 Treatment B vs P 41.528 4.500 383.262 Sex F vs M 6.194 1.312 29.248 Age 0.767 0.636 0.926

Output 39.3.8 contains two tables: the "Contrast Test Results" table and the "Contrast Rows Estimation and Testing Results" table. The former contains the overall Wald test for each CONTRAST statement. Although three rows are specifed in the 'Pairwise' CONTRAST statement, there are only two degrees of freedom, and the Wald test result is identical to the Type III analysis of Treatment in Output 39.3.7. The latter table contains estimates and tests of individual contrast rows. The estimates for the first two rows of the 'Pairwise' CONTRAST statement are the same as those given in the "Odds Ratio Estimates" table (in Output 39.3.7). Both treatments A and B are highly effective over placebo in reducing pain. The third row estimates the odds ratio comparing A to B. The 95% confidence interval for this odds ratio is (0.0932, 3.5889), indicating that the pain reduction effects of these two test treatments are not that different. Again, the 'Female vs Male' contrast shows that female patients fared better in obtaining relief from pain than male patients.

Output 39.3.8: Results of CONTRAST Statements

 The LOGISTIC Procedure
 Contrast Test Results Contrast DF WaldChi-Square Pr > ChiSq Pairwise 2 12.6928 0.0018 Female vs Male 1 5.3013 0.0213
 Contrast Rows Estimation and Testing Results Contrast Type Row Estimate StandardError Alpha Lower Limit Upper Limit WaldChi-Square Pr > ChiSq Pairwise EXP 1 24.0218 24.3473 0.05 3.2951 175.1 9.8375 0.0017 Pairwise EXP 2 41.5284 47.0877 0.05 4.4998 383.3 10.8006 0.0010 Pairwise EXP 3 0.5784 0.5387 0.05 0.0932 3.5889 0.3455 0.5567 Female vs Male EXP 1 6.1937 4.9053 0.05 1.3116 29.2476 5.3013 0.0213

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