The LOGISTIC Procedure

Getting Started: LOGISTIC Procedure

The LOGISTIC procedure is similar in use to the other regression procedures in the SAS System. To demonstrate the similarity, suppose the response variable y is binary or ordinal, and x1 and x2 are two explanatory variables of interest. To fit a logistic regression model, you can specify a MODEL statement similar to that used in the REG procedure. For example:

proc logistic;
   model y=x1 x2;
run;

The response variable y can be either character or numeric. PROC LOGISTIC enumerates the total number of response categories and orders the response levels according to the response variable option ORDER= in the MODEL statement.

You can also input binary response data that are grouped. In the following statements, n represents the number of trials and r represents the number of events:

proc logistic;
   model r/n=x1 x2;
run;

The following example illustrates the use of PROC LOGISTIC. The data, taken from Cox and Snell (1989, pp. 10–11), consist of the number, r, of ingots not ready for rolling, out of n tested, for a number of combinations of heating time and soaking time.

data ingots;
   input Heat Soak r n @@;
   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
;

The following invocation of PROC LOGISTIC fits the binary logit model to the grouped data. The continuous covariates Heat and Soak are specified as predictors, and the bar notation (|) includes their interaction, Heat*Soak. The ODDSRATIO statement produces odds ratios in the presence of interactions, and a graphical display of the requested odds ratios is produced when ODS Graphics is enabled.

ods graphics on;
proc logistic data=ingots;
   model r/n = Heat | Soak;
   oddsratio Heat / at(Soak=1 2 3 4);
run;
ods graphics off;

The results of this analysis are shown in the following figures. PROC LOGISTIC first lists background information in Figure 54.1 about the fitting of the model. Included are the name of the input data set, the response variable(s) used, the number of observations used, and the link function used.

Figure 54.1: Binary Logit Model

The LOGISTIC Procedure

Model Information
Data Set WORK.INGOTS
Response Variable (Events) r
Response Variable (Trials) n
Model binary logit
Optimization Technique Fisher's scoring

Number of Observations Read 19
Number of Observations Used 19
Sum of Frequencies Read 387
Sum of Frequencies Used 387


The Response Profile table (Figure 54.2) lists the response categories (which are Event and Nonevent when grouped data are input), their ordered values, and their total frequencies for the given data.

Figure 54.2: Response Profile with Events/Trials Syntax

Response Profile
Ordered
Value
Binary Outcome Total
Frequency
1 Event 12
2 Nonevent 375

Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.


The Model Fit Statistics table (Figure 54.3) contains Akaike’s information criterion (AIC), the Schwarz criterion (SC), and the negative of twice the log likelihood (–2 Log L) for the intercept-only model and the fitted model. AIC and SC can be used to compare different models, and the ones with smaller values are preferred. Results of the likelihood ratio test and the efficient score test for testing the joint significance of the explanatory variables (Soak, Heat, and their interaction) are included in the Testing Global Null Hypothesis: BETA=0 table (Figure 54.3); the small p-values reject the hypothesis that all slope parameters are equal to zero.

Figure 54.3: Fit Statistics and Hypothesis Tests

Model Fit Statistics
Criterion Intercept Only Intercept and Covariates
Log Likelihood Full Log Likelihood
AIC 108.988 103.222 35.957
SC 112.947 119.056 51.791
-2 Log L 106.988 95.222 27.957

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 11.7663 3 0.0082
Score 16.5417 3 0.0009
Wald 13.4588 3 0.0037


The Analysis of Maximum Likelihood Estimates table in Figure 54.4 lists the parameter estimates, their standard errors, and the results of the Wald test for individual parameters. Note that the Heat*Soak parameter is not significantly different from zero (p=0.727), nor is the Soak variable (p=0.6916).

Figure 54.4: Parameter Estimates

Analysis of Maximum Likelihood Estimates
Parameter DF Estimate Standard
Error
Wald
Chi-Square
Pr > ChiSq
Intercept 1 -5.9901 1.6666 12.9182 0.0003
Heat 1 0.0963 0.0471 4.1895 0.0407
Soak 1 0.2996 0.7551 0.1574 0.6916
Heat*Soak 1 -0.00884 0.0253 0.1219 0.7270


The Association of Predicted Probabilities and Observed Responses table (Figure 54.5) contains four measures of association for assessing the predictive ability of a model. They are based on the number of pairs of observations with different response values, the number of concordant pairs, and the number of discordant pairs, which are also displayed. Formulas for these statistics are given in the section Rank Correlation of Observed Responses and Predicted Probabilities.

Figure 54.5: Association Table

Association of Predicted Probabilities and
Observed Responses
Percent Concordant 70.9 Somers' D 0.537
Percent Discordant 17.3 Gamma 0.608
Percent Tied 11.8 Tau-a 0.032
Pairs 4500 c 0.768


The ODDSRATIO statement produces the Odds Ratio Estimates and Wald Confidence Intervals table (Figure 54.6), and a graphical display of these estimates is shown in Figure 54.7. The differences between the odds ratios are small compared to the variability shown by their confidence intervals, which confirms the previous conclusion that the Heat*Soak parameter is not significantly different from zero.

Figure 54.6: Odds Ratios of Heat at Several Values of Soak

Odds Ratio Estimates and Wald Confidence Intervals
Label Estimate 95% Confidence Limits
Heat at Soak=1 1.091 1.032 1.154
Heat at Soak=2 1.082 1.028 1.139
Heat at Soak=3 1.072 0.986 1.166
Heat at Soak=4 1.063 0.935 1.208


Figure 54.7: Plot of Odds Ratios of Heat at Several Values of Soak

Plot of Odds Ratios of Heat at Several Values of Soak


Since the Heat*Soak interaction is nonsignificant, the following statements fit a main-effects model:

proc logistic data=ingots;
   model r/n = Heat Soak;
run;

The results of this analysis are shown in the following figures. The model information and response profiles are the same as those in Figure 54.1 and Figure 54.2 for the saturated model. The Model Fit Statistics table in Figure 54.8 shows that the AIC and SC for the main-effects model are smaller than for the saturated model, indicating that the main-effects model might be the preferred model. As in the preceding model, the Testing Global Null Hypothesis: BETA=0 table indicates that the parameters are significantly different from zero.

Figure 54.8: Fit Statistics and Hypothesis Tests

The LOGISTIC Procedure

Model Fit Statistics
Criterion Intercept Only Intercept and Covariates
Log Likelihood Full Log Likelihood
AIC 108.988 101.346 34.080
SC 112.947 113.221 45.956
-2 Log L 106.988 95.346 28.080

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 11.6428 2 0.0030
Score 15.1091 2 0.0005
Wald 13.0315 2 0.0015


The Analysis of Maximum Likelihood Estimates table in Figure 54.9 again shows that the Soak parameter is not significantly different from zero (p=0.8639). The odds ratio for each effect parameter, estimated by exponentiating the corresponding parameter estimate, is shown in the Odds Ratios Estimates table (Figure 54.9), along with 95% Wald confidence intervals. The confidence interval for the Soak parameter contains the value 1, which also indicates that this effect is not significant.

Figure 54.9: Parameter Estimates and Odds Ratios

Analysis of Maximum Likelihood Estimates
Parameter DF Estimate Standard
Error
Wald
Chi-Square
Pr > ChiSq
Intercept 1 -5.5592 1.1197 24.6503 <.0001
Heat 1 0.0820 0.0237 11.9454 0.0005
Soak 1 0.0568 0.3312 0.0294 0.8639

Odds Ratio Estimates
Effect Point Estimate 95% Wald
Confidence Limits
Heat 1.085 1.036 1.137
Soak 1.058 0.553 2.026

Association of Predicted Probabilities and
Observed Responses
Percent Concordant 64.4 Somers' D 0.460
Percent Discordant 18.4 Gamma 0.555
Percent Tied 17.2 Tau-a 0.028
Pairs 4500 c 0.730


Using these parameter estimates, you can calculate the estimated logit of $\pi $ as

\[  -5.5592+0.082 \times \Variable{Heat}+0.0568 \times \Variable{Soak}  \]

For example, if Heat=7 and Soak=1, then logit$({\widehat{\pi }})=-4.9284$. Using this logit estimate, you can calculate ${\widehat{\pi }}$ as follows:

\[  {\widehat{\pi }}=1 / (1 + \mbox{e}^{4.9284})=0.0072  \]

This gives the predicted probability of the event (ingot not ready for rolling) for Heat=7 and Soak=1. Note that PROC LOGISTIC can calculate these statistics for you; use the OUTPUT statement with the PREDICTED= option, or use the SCORE statement.

To illustrate the use of an alternative form of input data, the following program creates the ingots data set with the new variables NotReady and Freq instead of n and r. The variable NotReady represents the response of individual units; it has a value of 1 for units not ready for rolling (event) and a value of 0 for units ready for rolling (nonevent). The variable Freq represents the frequency of occurrence of each combination of Heat, Soak, and NotReady. Note that, compared to the previous data set, NotReady=1 implies Freq=r, and NotReady=0 implies Freq=nr.

data ingots;
   input Heat Soak NotReady Freq @@;
   datalines;
7 1.0 0 10  14 1.0 0 31  14 4.0 0 19  27 2.2 0 21  51 1.0 1  3
7 1.7 0 17  14 1.7 0 43  27 1.0 1  1  27 2.8 1  1  51 1.0 0 10
7 2.2 0  7  14 2.2 1  2  27 1.0 0 55  27 2.8 0 21  51 1.7 0  1
7 2.8 0 12  14 2.2 0 31  27 1.7 1  4  27 4.0 1  1  51 2.2 0  1
7 4.0 0  9  14 2.8 0 31  27 1.7 0 40  27 4.0 0 15  51 4.0 0  1
;

The following statements invoke PROC LOGISTIC to fit the main-effects model by using the alternative form of the input data set:

proc logistic data=ingots;
   model NotReady(event='1') = Heat Soak;
   freq Freq;
run;

Results of this analysis are the same as the preceding single-trial main-effects analysis. The displayed output for the two runs are identical except for the background information of the model fit and the Response Profile table shown in Figure 54.10.

Figure 54.10: Response Profile with Single-Trial Syntax

The LOGISTIC Procedure

Response Profile
Ordered
Value
NotReady Total
Frequency
1 0 375
2 1 12

Probability modeled is NotReady=1.



By default, Ordered Values are assigned to the sorted response values in ascending order, and PROC LOGISTIC models the probability of the response level that corresponds to the Ordered Value 1. There are several methods to change these defaults; the preceding statements specify the response variable option EVENT= to model the probability of NotReady=1 as displayed in Figure 54.10. See the section Response Level Ordering for more details.