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
| Regression Diagnostics |
For binary response data, regression diagnostics developed by Pregibon (1981) can be requested by specifying the INFLUENCE option. For diagnostics available with conditional logistic regression, see the section Regression Diagnostic Details. These diagnostics can also be obtained from the OUTPUT statement.
This section uses the following notation:
is the number of event responses out of 
trials for the 
th observation. If events/trials syntax is used, is the value of events and is the value of trials. For single-trial syntax, 
, and 
if the ordered response is 1, and 
if the ordered response is 2. 
is the weight of the
th observation. 
is the probability of an event response for the
th observation given by 
, where 
is the inverse link function. 
is the maximum likelihood estimate (MLE) of
. 
is the estimated covariance matrix of
. 
is the estimate of evaluated at , and 
.
Pregibon (1981) suggests using the index plots of several diagnostic statistics to identify influential observations and to quantify the effects on various aspects of the maximum likelihood fit. In an index plot, the diagnostic statistic is plotted against the observation number. In general, the distributions of these diagnostic statistics are not known, so cutoff values cannot be given for determining when the values are large. However, the IPLOTS and INFLUENCE options in the MODEL statement and the PLOTS option in the PROC LOGISTIC statement provide displays of the diagnostic values, allowing visual inspection and comparison of the values across observations. In these plots, if the model is correctly specified and fits all observations well, then no extreme points should appear.
The next five sections give formulas for these diagnostic statistics.
Hat Matrix Diagonal (Leverage)
The diagonal elements of the hat matrix are useful in detecting extreme points in the design space where they tend to have larger values. The th diagonal element is
 |
where
 |
 |
 |
|||
 |
|
 |
and 
and 
are the first and second derivatives of the link function 
, respectively.
For a binary response logit model, the hat matrix diagonal elements are
 |
If the estimated probability is extreme (less than 0.1 and greater than 0.9, approximately), then the hat diagonal might be greatly reduced in value. Consequently, when an observation has a very large or very small estimated probability, its hat diagonal value is not a good indicator of the observation’s distance from the design space (Hosmer and Lemeshow; 2000, p. 171).
Residuals
Residuals are useful in identifying observations that are not explained well by the model. Pearson residuals are components of the Pearson chi-square statistic and deviance residuals are components of the deviance. The Pearson residual for the th observation is
 |
The Pearson chi-square statistic is the sum of squares of the Pearson residuals.
The deviance residual for the th observation is
 |
 |
where the plus (minus) in 
is used if 
is greater (less) than 
. The deviance is the sum of squares of the deviance residuals.
The STDRES option in the INFLUENCE and PLOTS=INFLUENCE options computes three more residuals (Collett; 2003). The Pearson and deviance residuals are standardized to have approximately unit variance:
 |
 |
 |
|||
 |
|
 |
The likelihood residuals, which estimate components of a likelihood ratio test of deleting an individual observation, are a weighted combination of the standardized Pearson and deviance residuals
 |
|
 |
DFBETAS
For each parameter estimate, the procedure calculates a DFBETAS diagnostic for each observation. The DFBETAS diagnostic for an observation is the standardized difference in the parameter estimate due to deleting the observation, and it can be used to assess the effect of an individual observation on each estimated parameter of the fitted model. Instead of reestimating the parameter every time an observation is deleted, PROC LOGISTIC uses the one-step estimate. See the section Predicted Probability of an Event for Classification. For the th observation, the DFBETAS are given by
 |
where 
is the standard error of the 
th component of , and 
is the th component of the one-step difference
 |
is the approximate change (
) in the vector of parameter estimates due to the omission of the th observation. The DFBETAS are useful in detecting observations that are causing instability in the selected coefficients.
C and CBAR
C and CBAR are confidence interval displacement diagnostics that provide scalar measures of the influence of individual observations on . These diagnostics are based on the same idea as the Cook distance in linear regression theory (Cook and Weisberg; 1982), but use the one-step estimate. C and CBAR for the th observation are computed as
 |
and
 |
respectively.
Typically, to use these statistics, you plot them against an index and look for outliers.
DIFDEV and DIFCHISQ
DIFDEV and DIFCHISQ are diagnostics for detecting ill-fitted observations; in other words, observations that contribute heavily to the disagreement between the data and the predicted values of the fitted model. DIFDEV is the change in the deviance due to deleting an individual observation while DIFCHISQ is the change in the Pearson chi-square statistic for the same deletion. By using the one-step estimate, DIFDEV and DIFCHISQ for the th observation are computed as
 |
and
 |