The MI Procedure
- Overview
- Getting Started
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Syntax
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DetailsDescriptive StatisticsEM Algorithm for Data with Missing ValuesStatistical Assumptions for Multiple ImputationMissing Data PatternsImputation MethodsMonotone Methods for Data Sets with Monotone Missing PatternsMonotone and FCS Regression MethodsMonotone and FCS Predictive Mean Matching MethodsMonotone Propensity Score MethodMonotone and FCS Discriminant Function MethodsMonotone and FCS Logistic Regression MethodsFCS Methods for Data Sets with Arbitrary Missing PatternsChecking Convergence in FCS MethodsMCMC Method for Arbitrary Missing Multivariate Normal DataProducing Monotone Missingness with the MCMC MethodMCMC Method SpecificationsChecking Convergence in MCMCInput Data SetsOutput Data SetsCombining Inferences from Multiply Imputed Data SetsMultiple Imputation EfficiencyImputer’s Model Versus Analyst’s ModelParameter Simulation versus Multiple ImputationSummary of Issues in Multiple ImputationODS Table NamesODS Graphics
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ExamplesEM Algorithm for MLEMonotone Propensity Score MethodMonotone Regression MethodMonotone Logistic Regression Method for CLASS VariablesMonotone Discriminant Function Method for CLASS VariablesFCS Method for Continuous VariablesFCS Method for CLASS VariablesFCS Method with Trace PlotMCMC MethodProducing Monotone Missingness with MCMCChecking Convergence in MCMCSaving and Using Parameters for MCMCTransforming to NormalityMultistage Imputation
- References
Monotone and FCS Regression Methods
The regression method is the default imputation method in the MONOTONE and FCS statements for continuous variables.
In the regression method, a regression model is fitted for a continuous variable with the covariates constructed from a set
of effects. Based on the fitted regression model, a new regression model is simulated from the posterior predictive distribution
of the parameters and is used to impute the missing values for each variable (Rubin, 1987, pp. 166–167). That is, for a continuous variable 
with missing values, a model
![\[ Y_{j} = {\beta }_{0} + {\beta }_{1} \, X_{1} + {\beta }_{2} \, X_{2} + \ldots + {\beta }_{k} \, X_{k} \]](images/statug_mi0099.png){kind=small} |
is fitted using observations with observed values for the variable and its covariates 
, 
, …, 
.
The fitted model includes the regression parameter estimates 
and the associated covariance matrix 
, where 
is the usual 
inverse matrix derived from the intercept and covariates , , …, .
The following steps are used to generate imputed values for each imputation:
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New parameters
and 
are drawn from the posterior predictive distribution of the parameters. That is, they are simulated from 
, 
, and . The variance is drawn as
![\[ {\sigma }_{*j}^2 = \hat{\sigma }_{j}^2 (n_{j}-k-1) / g \]](images/statug_mi0111.png)
where g is a
random variate and 
is the number of nonmissing observations for . The regression coefficients are drawn as
![\[ \bbeta _{*} = \hat{\bbeta } + {\sigma }_{*j} \mb {V}_{hj}’ \mb {Z} \]](images/statug_mi0113.png)
where
is the upper triangular matrix in the Cholesky decomposition, 
, and 
is a vector of 
independent random normal variates.
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The missing values are then replaced by
![\[ {\beta }_{*0} + {\beta }_{*1} \, x_{1} + {\beta }_{*2} \, x_{2} + \ldots + {\beta }_{*(k)} \, x_{k} + z_{i} \, {\sigma }_{*j} \]](images/statug_mi0118.png)
where
are the values of the covariates and 
is a simulated normal deviate.