The LOGISTIC Procedure
- Overview
- Getting Started
-
Syntax
PROC LOGISTIC Statement BY Statement CLASS Statement CONTRAST Statement EFFECT Statement EFFECTPLOT Statement ESTIMATE Statement EXACT Statement EXACTOPTIONS Statement FREQ Statement LSMEANS Statement LSMESTIMATE Statement MODEL Statement ODDSRATIO Statement OUTPUT Statement ROC Statement ROCCONTRAST Statement SCORE Statement SLICE Statement STORE Statement STRATA Statement TEST Statement UNITS Statement WEIGHT Statement -
Details
Missing Values Response Level Ordering Link Functions and the Corresponding Distributions Determining Observations for Likelihood Contributions Iterative Algorithms for Model Fitting Convergence Criteria Existence of Maximum Likelihood Estimates Effect-Selection Methods Model Fitting Information Generalized Coefficient of Determination Score Statistics and Tests Confidence Intervals for Parameters Odds Ratio Estimation Rank Correlation of Observed Responses and Predicted Probabilities Linear Predictor, Predicted Probability, and Confidence Limits Classification Table Overdispersion The Hosmer-Lemeshow Goodness-of-Fit Test Receiver Operating Characteristic Curves Testing Linear Hypotheses about the Regression Coefficients Regression Diagnostics Scoring Data Sets Conditional Logistic Regression Exact Conditional Logistic Regression Input and Output Data Sets Computational Resources Displayed Output ODS Table Names ODS Graphics
-
Examples
Stepwise Logistic Regression and Predicted Values Logistic Modeling with Categorical Predictors Ordinal Logistic Regression Nominal Response Data: Generalized Logits Model Stratified Sampling Logistic Regression Diagnostics ROC Curve, Customized Odds Ratios, Goodness-of-Fit Statistics, R-Square, and Confidence Limits Comparing Receiver Operating Characteristic Curves Goodness-of-Fit Tests and Subpopulations Overdispersion Conditional Logistic Regression for Matched Pairs Data Firth’s Penalized Likelihood Compared with Other Approaches Complementary Log-Log Model for Infection Rates Complementary Log-Log Model for Interval-Censored Survival Times Scoring Data Sets Using the LSMEANS Statement
- References
| Score Statistics and Tests |
To understand the general form of the score statistics, let 
be the vector of first partial derivatives of the log likelihood with respect to the parameter vector 
, and let 
be the matrix of second partial derivatives of the log likelihood with respect to . That is, is the gradient vector, and is the Hessian matrix. Let 
be either 
or the expected value of . Consider a null hypothesis 
. Let 
be the MLE of under . The chi-square score statistic for testing is defined by
 |
and it has an asymptotic 
distribution with 
degrees of freedom under , where is the number of restrictions imposed on by .
Residual Chi-Square
When you use SELECTION=FORWARD, BACKWARD, or STEPWISE, the procedure calculates a residual chi-square score statistic and reports the statistic, its degrees of freedom, and the p-value. This section describes how the statistic is calculated.
Suppose there are 
explanatory effects of interest. The full cumulative response model has a parameter vector
 |
where 
are intercept parameters, and 
are the common slope parameters for the explanatory effects. The full generalized logit model has a parameter vector
 |
 |
 |
|||
 |
|
 |
where 
is the slope parameter for the 
th effect in the 
th logit.
Consider the null hypothesis 
, where 
for the cumulative response model, and 
, for the generalized logit model. For the reduced model with 
explanatory effects, let 
be the MLEs of the unknown intercept parameters, let 
be the MLEs of the unknown slope parameters, and let 
, be those for the generalized logit model. The residual chi-square is the chi-square score statistic testing the null hypothesis ; that is, the residual chi-square is
|
where for the cumulative response model 
, and for the generalized logit model 
, where 
denotes a vector of 
zeros.
The residual chi-square has an asymptotic chi-square distribution with degrees of freedom (
for the generalized logit model). A special case is the global score chi-square, where the reduced model consists of the 
intercepts and no explanatory effects. The global score statistic is displayed in the "Testing Global Null Hypothesis: BETA=0" table. The table is not produced when the NOFIT option is used, but the global score statistic is displayed.
Testing Individual Effects Not in the Model
These tests are performed when you specify SELECTION=FORWARD or STEPWISE, and are displayed when the DETAILS option is specified. In the displayed output, the tests are labeled "Score Chi-Square" in the "Analysis of Effects Not in the Model" table and in the "Summary of Stepwise (Forward) Selection" table. This section describes how the tests are calculated.
Suppose that intercepts and explanatory variables (say 
) have been fit to a model and that 
is another explanatory variable of interest. Consider a full model with the intercepts and 
explanatory variables (
) and a reduced model with excluded. The significance of adjusted for 
can be determined by comparing the corresponding residual chi-square with a chi-square distribution with one degree of freedom ( degrees of freedom for the generalized logit model).
Testing the Parallel Lines Assumption
For an ordinal response, PROC LOGISTIC performs a test of the parallel lines assumption. In the displayed output, this test is labeled "Score Test for the Equal Slopes Assumption" when the LINK= option is NORMIT or CLOGLOG. When LINK=LOGIT, the test is labeled as "Score Test for the Proportional Odds Assumption" in the output. For small sample sizes, this test might be too liberal (Stokes, Davis, and Koch; 2000). This section describes the methods used to calculate the test.
For this test the number of response levels, 
, is assumed to be strictly greater than 2. Let 
be the response variable taking values 
. Suppose there are explanatory variables. Consider the general cumulative model without making the parallel lines assumption
 |
where 
is the link function, and 
is a vector of unknown parameters consisting of an intercept 
and slope parameters 
. The parameter vector for this general cumulative model is
 |
Under the null hypothesis of parallelism 
, there is a single common slope parameter for each of the explanatory variables. Let 
be the common slope parameters. Let and 
be the MLEs of the intercept parameters and the common slope parameters. Then, under , the MLE of is
 |
and the chi-square score statistic 
has an asymptotic chi-square distribution with 
degrees of freedom. This tests the parallel lines assumption by testing the equality of separate slope parameters simultaneously for all explanatory variables.