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 CoefficientsRegression 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 DataFirth’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
Rank Correlation of Observed Responses and Predicted Probabilities
The predicted mean score of an observation is the sum of the Ordered Values (shown in the “Response Profile” table) minus one, weighted by the corresponding predicted probabilities for that observation; that is, the predicted means
score 
, where 
is the number of response levels and 
is the predicted probability of the ith (ordered) response.
A pair of observations with different observed responses is said to be concordant if the observation with the lower ordered response value has a lower predicted mean score than the observation with the higher
ordered response value. If the observation with the lower ordered response value has a higher predicted mean score than the
observation with the higher ordered response value, then the pair is discordant. If the pair is neither concordant nor discordant, it is a tie. Enumeration of the total numbers of concordant and discordant pairs is carried out by categorizing the predicted mean score
into intervals of length 
and accumulating the corresponding frequencies of observations. Note that the length of these intervals can be modified by
specification of the BINWIDTH= option in the MODEL statement.
Let N be the sum of observation frequencies in the data. Suppose there are a total of t pairs with different responses: 
of them are concordant, 
of them are discordant, and 
of them are tied. PROC LOGISTIC computes the following four indices of rank correlation for assessing the predictive ability
of a model:
![\[ \begin{array}{lcl} \mbox{\textit{c} } & =& (n_ c+0.5(t-n_ c-n_ d))/t \\ \mbox{Somers \textit{D} (Gini coefficient) } & =& (n_ c-n_ d)/t \\ \mbox{Goodman-Kruskal Gamma } & =& (n_ c-n_ d)/(n_ c+n_ d) \\ \mbox{Kendalls Tau-\textit{a} } & =& (n_ c-n_ d)/(0.5N(N-1)) \end{array} \]](images/statug_logistic0371.png) |
If there are no ties, then Somers’ D (Gini’s coefficient) 
. Note that the concordance index, c, also gives an estimate of the area under the receiver operating characteristic (ROC) curve when the response is binary (Hanley
and McNeil, 1982). See the section ROC Computations for more information about this area.
For binary responses, the predicted mean score is equal to the predicted probability for Ordered Value 2. As such, the preceding definition of concordance is consistent with the definition used in previous releases for the binary response model.
These statistics are not available when the STRATA statement is specified.