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SAS，

Procedure

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Chapter 9

The MI Procedure

Chapter Table of Contents

OVERVIEW ...................................131

GETTING STARTED ..............................133

SYNTAX .....................................137

PROCMIStatement ..............................138

BYStatement..................................141

EMStatement..................................141

FREQStatement ................................142

MCMCStatement................................143

MONOTONEStatement ............................149

TRANSFORMStatement............................150

VARStatement .................................151

DETAILS .....................................152

DescriptiveStatistics ..............................152

EM Algorithm for Data with Missing Values . . ................153

Statistical Assumptions for Multiple Imputation ................154

MissingDataPatterns..............................155

ImputationMechanisms ............................156

Regression Method for Monotone Missing Data ................157

Propensity Score Method for Monotone Missing Data . . ...........158

MCMC Method for Arbitrary Missing Data . . ................159

Producing Monotone Missingness with the MCMC Method ..........164

MCMCMethodSpeciﬁcations.........................166

ConvergenceinMCMC.............................167

Input Data Sets .................................170

OutputDataSets ................................171

Combining Inferences from Multiply Imputed Data Sets ...........173

Multiple Imputation Efﬁciency . . . ......................174

Imputer’sModelVersusAnalyst’sModel ...................174

Parameter Simulation Versus Multiple Imputation ...............175

ODSTableNames ...............................176

EXAMPLES ...................................177

Example 9.1 EM Algorithm for MLE .....................177

Chapter 9

The MI Procedure

Overview

The experimental MI procedure performs multiple imputation of missing data. Miss-

ing values are an issue in a substantial number of statistical analyses. Most SAS

statistical procedures exclude observations with any missing variable values from

the analysis. These observations are called incomplete cases. While analyzing only

complete cases has its simplicity, the information contained in the incomplete cases

is lost. This approach also ignores possible systematic differences between the com-

plete cases and the incomplete cases, and the resulting inference may not be appli-

cable to the population of all cases, especially with a smaller number of complete

cases.

Some SAS procedures use all the available cases in an analysis, that is, cases with

available information. For example, the CORR procedure estimates a variable mean

by using all cases with nonmissing values for this variable, ignoring the possible

missing values in other variables. PROC CORR also estimates a correlation by using

all cases with nonmissing values for this pair of variables. This makes better use of

the available data, but the resulting correlation matrix may not be positive deﬁnite.

Another strategy for handling missing data is simple imputation, which substitutes a

value for each missing value. Standard statistical procedures for complete data anal-

ysis can then be used with the ﬁlled-in data set. For example, each missing value

can be imputed with the variable mean of the complete cases, or it can be imputed

with the mean conditional on observed values of other variables. This approach treats

missing values as if they were known in the complete-data analysis. However, sin-

gle imputation does not reﬂect the uncertainty about the predictions of the unknown

missing values, and the resulting estimated variances of the parameter estimates will

be biased toward zero (Rubin 1987, p. 13).

Instead of ﬁlling in a single value for each missing value, multiple imputation (Rubin

1976; 1987) replaces each missing value with a set of plausible values that represent

the uncertainty about the right value to impute. The multiply imputed data sets are

then analyzed by using standard procedures for complete data and combining the

results from these analyses. No matter which complete-data analysis is used, the

process of combining results from different data sets is essentially the same.

Multiple imputation does not attempt to estimate each missing value through sim-

ulated values but rather to represent a random sample of the missing values. This

process results in valid statistical inferences that properly reﬂect the uncertainty due

to missing values; for example, conﬁdence intervals with the correct probability cov-

erage.

132



Chapter 9. The MI Procedure

Multiple imputation inference involves three distinct phases:

1. The missing data are ﬁlled in m times to generate m complete data sets.

2. The m complete data sets are analyzed using standard statistical analyses.

3. The results from the m complete data sets are combined to produce inferential

results.

The new MI procedure creates multiply imputed data sets for incomplete multivariate

data. It uses methods that incorporate appropriate variability across the m imputa-

tions. The method of choice depends on the patterns of missingness. A data set with

variables

, ...,

(in that order) is said to have a monotone missing pattern

when the event that a variable

is missing for a particular individual implies that all

subsequent variables

k>j

, are missing for that individual.

For data sets with monotone missing patterns, either a parametric regression method

(Rubin 1987) that assumes multivariate normality or a nonparametric method that

uses propensity scores (Rubin 1987; Lavori, Dawson, and Shera 1995) is appro-

priate. For data sets with arbitrary missing patterns, a Markov Chain Monte Carlo

(MCMC) method (Schafer 1997) that assumes multivariate normality is used to im-

pute all missing values or just enough missing values to make the imputed data sets

have monotone missing patterns.

Once the m complete data sets are analyzed using standard SAS procedures, the new

MIANALYZE procedure can be used to generate valid statistical inferences about

these parameters by combining results from the m analyses. These two procedures

are available in experimental form in Release 8.2 of the SAS System.

Often, as few as three to ﬁve imputations are adequate in multiple imputation (Rubin

1996, p. 480). The relative efﬁciency of the small

imputation estimator is high for

cases with little missing information (Rubin 1987, p. 114). Also see the “Multiple

Imputation Efﬁciency” section on page 174.

Multiple imputation inference assumes that the model (variables) you used to analyze

the multiply imputed data (the analyst’s model) is the same as the model used to im-

pute missing values in multiple imputation (the imputer’s model). But in practice, the

two models may not be the same. The consequence for different scenarios (Schafer

1997, pp. 139–143) is discussed in the “Imputer’s Model Versus Analyst’s Model”

section on page 174.

In addition to the multiple imputation method, a simulation-based method of pa-

rameter simulation can also be used to analyze the data for many incomplete-data

problems. Although the MI procedure does not offer a simulation-based method of

parameter simulation, the choice between the two methods (Schafer 1997, pp. 89–90,

135–136) is examined in the “Parameter Simulation Versus Multiple Imputation” sec-

tion on page 175.

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