Multiple Imputation for Missing Data: Concepts and New …

1 Multiple Imputation for Missing data : Concepts and NewDevelopment (Version )Yang C. Yuan, SAS Institute Inc., Rockville, MDAbstractMultiple Imputation provides a useful strategy for dealingwith data sets with Missing values. Instead of filling in asingle value for each Missing value, Rubin s (1987) multipleimputation procedure replaces each Missing value with aset of plausible values that represent the uncertainty aboutthe right value to impute. These multiply imputed data setsare then analyzed by using standard procedures for com-plete data and combining the results from these matter which complete- data analysis is used, the pro-cess of combining results from different imputed data setsis essentially the same.

2 This results in valid statistical in-ferences that properly reflect the uncertainty due to paper reviews methods for analyzing Missing data , in-cluding basic Concepts and applications of Multiple impu-tation techniques. The paper presents SAS procedures,PROC MI and PROC MIANALYZE, for creating Multiple im-putations for incomplete multivariate data and for analyzingresults from multiply imputed data MI and MIANALYZE procedures, which were intro-duced as experimental software in Releases and ,are production software in Version The syntax andexamples in this paper apply to Version The follow-ing enhancements have been made to the MI procedure inVersion.

3 A new REGPMM option in the MONOTONE state-ment and a new PMM option in the MCMC statementrequest the predicted mean matching method for im-putation. This method imputes an observed valuewhich is closest to the predicted value from the simu-lated regression model for each Missing value. A flexible model specification in the MONOTONE statement allows a different set of covariates to bespecified for each imputed following changes and enhancements have been madeto the MIANALYZE procedure in Version : A new MODELEFFECTS statement allows you tospecify the effects in the data set to be statement replaces the VAR statement, whichwas used in Releases and A new STDERR statement provides standard er-rors associated with effects in the MODELEFFECTS statement.

4 The statement can be used for univari-ate inference when the input data = data set con-tains both parameter estimates and standard errorsas variables. A new TEST statement tests linear hypotheses aboutthe paper also describes new experimental features inVersion for specification of classification variables in theMI and MIANALYZE SAS statistical procedures exclude observations withany Missing variable values from the analysis. These obser-vations are called incomplete cases. While using only com-plete cases has its simplicity, you lose information in theincomplete cases.

5 This approach also ignores the possi-ble systematic difference between the complete cases andincomplete cases, and the resulting inference may not beapplicable to the population of all cases, especially with asmaller number of complete SAS procedures use all the available cases in ananalysis, that is, cases with available information. For ex-ample, PROC CORR estimates a variable mean by usingall cases with nonmissing values on this variable, ignor-ing the possible Missing values in other variables. PROCCORR also estimates a correlation by using all cases withnonmissing values for this pair of variables.

6 This may makebetter use of the available data , but the resulting correlationmatrix may not be positive strategy is single Imputation , in which you substi-tute a value for each Missing value. Standard statistical pro-cedures for complete data analysis can then be used withthe filled-in data set. For example, each Missing value canbe imputed from the variable mean of the complete approach treats Missing values as if they were knownin the complete- data analyses. Single Imputation does notreflect the uncertainty about the predictions of the unknownmissing values, and the resulting estimated variances of theparameter estimates will be biased toward of filling in a single value for each Missing value, amultiple Imputation procedure (Rubin 1987) replaces eachmissing value with a set of plausible values that representthe uncertainty about the right value to impute.

7 The multiplyimputed data sets are then analyzed by using standard pro-cedures for complete data and combining the results fromthese analyses. No matter which complete- data analysis isused, the process of combining results from different datasets is essentially the Imputation does not attempt to estimate each miss-ing value through simulated values but rather to represent arandom sample of the Missing values. This process resultsin valid statistical inferences that properly reflect the uncer-tainty due to Missing values; for example, valid confidenceintervals for Imputation inference involves three distinct phases: The Missing data are filled inmtimes to generatemcomplete data sets.

8 Themcomplete data sets are analyzed by usingstandard procedures. The results from themcomplete data sets are com-bined for the MI procedure in the SAS/STAT Software is a multi-ple Imputation procedure that creates multiply imputed datasets for incompletep-dimensional multivariate data . It usesmethods that incorporate appropriate variability across themimputations. Once themcomplete data sets are ana-lyzed by using standard procedures, the MIANALYZE pro-cedure can be used to generate valid statistical inferencesabout these parameters by combining results from themcomplete data Missing - data MechanismLetYbe then pmatrix of complete data , which is notfully observed, and denote the observed part ofYbyYobsand the Missing part byYmis.

9 The SAS Multiple imputationprocedures assume that the Missing data are Missing atrandom (MAR), that is, the probability that an observation ismissing may depend onYobs, but not onYmis(Rubin 1976;1987, p. 53).For example, consider a trivariate data set with variablesY1andY2fully observed, and a variableY3that has missingvalues. MAR assumes that the probability thatY3is missingfor an individual may be related to the individual s valuesof variablesY1andY2, but not to its value ofY3. On theother hand, if a complete case and an incomplete case forY3with exactly the same values for variablesY1andY2havesystematically different values, then there exists a responsebias forY3and it is not MAR assumption is not the same as Missing com-pletely at random (MCAR), which is a special case of MCAR, the Missing data values are a simple randomsample of all data values.

10 The missingness does not dependon the values of any variables in the data , these SAS procedures also assume that theparameters of the data model and the parameters of themissing data indicators are distinct. That is, knowing the val-ues of does not provide any additional information about , and vice versa. If both MAR and distinctness assump-tions are satisfied, the Missing - data mechanism is said tobe MechanismsThis section describes three methods that are available inthe MI procedure. The method of choice depends on thetype of Missing data pattern. For monotone Missing datapatterns, either a parametric regression method that as-sumes multivariate normality or a nonparametric methodthat uses propensity scores is appropriate.