The LOGISTIC Procedure
- Overview
- Getting Started
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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
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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
| The Hosmer-Lemeshow Goodness-of-Fit Test |
Sufficient replication within subpopulations is required to make the Pearson and deviance goodness-of-fit tests valid. When there are one or more continuous predictors in the model, the data are often too sparse to use these statistics. Hosmer and Lemeshow (2000) proposed a statistic that they show, through simulation, is distributed as chi-square when there is no replication in any of the subpopulations. This test is available only for binary response models.
First, the observations are sorted in increasing order of their estimated event probability. The event is the response level specified in the response variable option EVENT=, or the response level that is not specified in the REF= option, or, if neither of these options was specified, then the event is the response level identified in the "Response Profiles" table as "Ordered Value 1". The observations are then divided into approximately 10 groups according to the following scheme. Let 
be the total number of subjects. Let 
be the target number of subjects for each group given by
 |
where 
represents the integral value of 
. If the single-trial syntax is used, blocks of subjects are formed of observations with identical values of the explanatory variables. Blocks of subjects are not divided when being placed into groups.
Suppose there are 
subjects in the first block and 
subjects in the second block. The first block of subjects is placed in the first group. Subjects in the second block are added to the first group if
 |
Otherwise, they are placed in the second group. In general, suppose subjects of the (
)th block have been placed in the 
th group. Let 
be the total number of subjects currently in the th group. Subjects for the 
th block (containing 
subjects) are also placed in the th group if
 |
Otherwise, the subjects are put into the next group. In addition, if the number of subjects in the last group does not exceed 
(half the target group size), the last two groups are collapsed to form only one group.
Note that the number of groups, 
, can be smaller than 10 if there are fewer than 10 patterns of explanatory variables. There must be at least three groups in order for the Hosmer-Lemeshow statistic to be computed.
The Hosmer-Lemeshow goodness-of-fit statistic is obtained by calculating the Pearson chi-square statistic from the 
table of observed and expected frequencies, where is the number of groups. The statistic is written
 |
where 
is the total frequency of subjects in the 
th group, 
is the total frequency of event outcomes in the th group, and 
is the average estimated predicted probability of an event outcome for the th group. (Note that the predicted probabilities are computed as shown in the section Linear Predictor, Predicted Probability, and Confidence Limits and are not the cross validated estimates discussed in the section Classification Table.) The Hosmer-Lemeshow statistic is then compared to a chi-square distribution with 
degrees of freedom, where the value of 
can be specified in the LACKFIT option in the MODEL statement. The default is 
. Large values of 
(and small p-values) indicate a lack of fit of the model.