The SURVEYIMPUTE Procedure

Suppose you want to impute P items jointly (by using the IMPJOINT statement in PROC SURVEYIMPUTE). Let be the true response for the P items for unit i. is completely known if all P items are observed for unit i. However, the true response might not be known for some units because of item nonresponse. Let be categorical and have J levels for item j. Denote as the observed part and as the missing part of . Let be the population proportion that falls in category . Assume that it is possible to estimate the population proportion from the observed sample. That is, for example, the conditional probability, , in the observed data is the same as the conditional probability in the data where are missing. The conditional probabilities are estimated by

where

is the estimated joint probability, is an indicator function, and is the observation weight for unit i.

The FEFI method uses an EM-by-weighting algorithm similar to that of Ibrahim (1990). The detailed algorithm is described in Kim and Fuller (2013). The following steps describe the imputation technique. If you do not specify imputation cells by using the CELLS statement, PROC SURVEYIMPUTE uses the entire data set as one imputation cell. If you specify imputation cells, then all the probabilities in these steps are computed by using observations from the same imputation cell as the recipient unit. To simplify notation, subscripts are not used for imputation cells in the following description.

For given i, let and be the observed part and the missing part, respectively, of unit i. Let be the index set for the complete respondents. Suppose you want to impute the missing part of , . The index set contains the indexes for the all possible donor units for . Let be all the observed combinations of . The set of all observed combinations for unit i defines the donor cells (all possible realizations) for unit i. Let be the lth imputed value of . You must assume that at least one imputed value is available; otherwise the observation is not imputed.

The replicate weights are created by computing a replicated version of , , and by repeating the EM-by-weighting algorithm as described earlier. For the kth replicate sample, is computed by

The small data set shown in Figure 8 is used to illustrate the imputation technique. The data set contains nine observation units, and each unit has two items (X and Y). The variable Unit contains the observation identification. In this example, X is missing for units 5 and 9, and Y is missing for units 2 and 9.

The following SAS statements request joint imputation of X and Y by using the FEFI method. These statements also request imputation-adjusted replicate weights for the jackknife replication method. The CLASS statement specifies that both X and Y are CLASS variables. The OUTPUT statement stores the imputed values in the data set Imputed and stores the jackknife coefficients in the data set Ojkc. The FRACTIONALWEIGHTS= option in the OUTPUT statement saves the fractional weights in the Imputed data set.

proc surveyimpute data=test varmethod=jackknife;
   class x y;
   var x y;
   id Unit;
   output out=Imputed fractionalweights=FracWgt outjkcoefs=Ojkc;
run;

The initial fractional weights, FracWgt, after the initialization step are displayed in Figure 9.

The resulting data set contains 14 rows. There are six rows for fully observed units (ImpIndex = 0), two rows for unit 2, two rows for unit 5, and four rows for unit 9. The sum of initial fractional weights is 1 for all units.

The EM algorithm repeats the computation of the joint probabilities and the fractional weights until convergence. The fractional weights, FracWgt, after the EM step and the imputation-adjusted replicate weights (ImpRepWt_1, …, ImpRepWt_9) are displayed in Figure 10.