The LOGISTIC Procedure
- Overview
- Getting Started
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Syntax
PROC LOGISTIC StatementBY StatementCLASS StatementCODE StatementCONTRAST StatementEFFECT StatementEFFECTPLOT StatementESTIMATE StatementEXACT StatementEXACTOPTIONS StatementFREQ StatementID StatementLSMEANS StatementLSMESTIMATE StatementMODEL StatementNLOPTIONS StatementODDSRATIO StatementOUTPUT StatementROC StatementROCCONTRAST StatementSCORE StatementSLICE StatementSTORE StatementSTRATA StatementTEST StatementUNITS StatementWEIGHT Statement -
DetailsMissing ValuesResponse Level OrderingLink Functions and the Corresponding DistributionsDetermining Observations for Likelihood ContributionsIterative Algorithms for Model FittingConvergence CriteriaExistence of Maximum Likelihood EstimatesEffect-Selection MethodsModel Fitting InformationGeneralized Coefficient of DeterminationScore Statistics and TestsConfidence Intervals for ParametersOdds Ratio EstimationRank Correlation of Observed Responses and Predicted ProbabilitiesLinear Predictor, Predicted Probability, and Confidence LimitsClassification TableOverdispersionThe Hosmer-Lemeshow Goodness-of-Fit TestReceiver Operating Characteristic CurvesTesting Linear Hypotheses about the Regression CoefficientsJoint Tests and Type 3 TestsRegression DiagnosticsScoring Data SetsConditional Logistic RegressionExact Conditional Logistic RegressionInput and Output Data SetsComputational ResourcesDisplayed OutputODS Table NamesODS Graphics
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ExamplesStepwise Logistic Regression and Predicted ValuesLogistic Modeling with Categorical PredictorsOrdinal Logistic RegressionNominal Response Data: Generalized Logits ModelStratified SamplingLogistic Regression DiagnosticsROC Curve, Customized Odds Ratios, Goodness-of-Fit Statistics, R-Square, and Confidence LimitsComparing Receiver Operating Characteristic CurvesGoodness-of-Fit Tests and SubpopulationsOverdispersionConditional Logistic Regression for Matched Pairs DataExact Conditional Logistic RegressionFirth’s Penalized Likelihood Compared with Other ApproachesComplementary Log-Log Model for Infection RatesComplementary Log-Log Model for Interval-Censored Survival TimesScoring Data SetsUsing the LSMEANS StatementPartial Proportional Odds Model
- References
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 N be the total number of subjects. Let M be the target number of subjects for each group given by
![\[ M = [0.1 \times N + 0.5] \]](images/statug_logistic0447.png){kind=small}
where ![$[x]$](images/statug_logistic0448.png){kind=small}
represents the integral value of x. 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
![\[ n_1 < M \quad \mbox{and} \quad n_1 + [0.5 \times n_2] \leq M \]](images/statug_logistic0449.png){kind=small}
Otherwise, they are placed in the second group. In general, suppose subjects of the (j – 1) block have been placed in the kth group. Let c be the total number of subjects currently in the kth group. Subjects for the jth block (containing 
subjects) are also placed in the kth group if
![\[ c < M \quad \mbox{and} \quad c + [0.5 \times n_ j] \leq M \]](images/statug_logistic0450.png){kind=small}
Otherwise, the subjects are put into the next group. In addition, if the number of subjects in the last group does not exceed ![$[0.05 \times N]$](images/statug_logistic0451.png){kind=small}
(half the target group size), the last two groups are collapsed to form only one group.
Note that the number of groups, g, 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 g is the number of groups. The statistic is written
![\[ \chi ^2_{HL} = \sum ^{g}_{i=1} \frac{(O_ i - N_ i {\bar{\pi }}_ i)^2}{N_ i {\bar{\pi }}_ i (1 - {\bar{\pi }}_ i)} \]](images/statug_logistic0453.png){kind=small}
where 
is the total frequency of subjects in the ith group, 
is the total frequency of event outcomes in the ith group, and 
is the average estimated predicted probability of an event outcome for the ith 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 n can be specified in the LACKFIT
option in the MODEL
statement. The default is n = 2. Large values of 
(and small p-values) indicate a lack of fit of the model.