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
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=1E8) 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 interceptonly 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 pvalues 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  ChiSquare  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 ChiSquare 
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  Taua  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
Since the Heat
*Soak
interaction is nonsignificant, the following statements fit a maineffects 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 maineffects model are smaller than for the saturated model, indicating that the maineffects 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
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  ChiSquare  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 ChiSquare 
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  Taua  0.028 
Pairs  4500  c  0.730 
Using these parameter estimates, you can calculate the estimated logit of as

For example, if Heat
=7 and Soak
=1, then logit. Using this logit estimate, you can calculate as follows:

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
=n
–r
.
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 maineffects 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 singletrial maineffects 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 SingleTrial Syntax
Response Profile  

Ordered Value 
NotReady  Total Frequency 
1  0  375 
2  1  12 
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.