Propensity Score Method for Monotone Missing Data

The MI Procedure

Propensity Score Method for Monotone Missing Data

A propensity score is generally defined as the conditional probability of assignment to a particular treatment given a vector of observed covariates (Rosenbaum and Rubin 1983). In the propensity score method, for each variable with missing values, a propensity score is generated for each observation to estimate the probability that the observation is missing. The observations are then grouped based on these propensity scores, and an approximate Bayesian bootstrap imputation (Rubin 1987, p. 124) is applied to each group (Lavori, Dawson, and Shera 1995).

A data set with variables Y₁, Y₂, ..., Y_p (in that order) is said to have a monotone missing pattern when the event that a variable Y_j is observed for a particular individual implies that all previous variables Y_k, k < j, are also observed for that individual. The propensity score method uses the following steps to impute values for each variable Y_j with missing values:

1. Create an indicator variable R_j with the value 0 for observations with missing Y_j and 1 otherwise.

2. Fit a logistic regression model

${\rm logit} (p_{j})={\beta}_{0} + {\beta}_{1} \, Y_{1} + {\beta}_{2} \, Y_{2} + ... + {\beta}_{j-1} \, Y_{j-1}$

where p_j = Pr( R_j=0 | Y₁, Y₂, ... , Y_j-1 ) and logit (p) = log ( p / (1-p) ).

3. Create a propensity score for each observation to estimate the probability that it is missing.

4. Divide the observations into a fixed number of groups (typically assumed to be five) based on these propensity scores.

5. Apply an approximate Bayesian bootstrap imputation to each group. In group k, suppose that Y_obs denotes the n₁ observations with nonmissing Y_j values and Y_mis denotes the n₀ observations with missing Y_j. The approximate Bayesian bootstrap imputation first draws n₁ observations randomly with replacement from Y_obs to create a new data set Y_obs^*. This is a nonparametric analogue of drawing parameters from the posterior predictive distribution of the parameters. The process then draws the n₀ values for Y_mis randomly with replacement from Y_obs^*.

Steps 1 through 5 are repeated sequentially for each variable with missing values.

Note that the propensity score method was originally designed for a randomized experiment with repeated measures on the response variables. The goal was to impute the missing values on the response variables. The method uses only the covariate information that is associated with whether the imputed variable values are missing. It does not use correlations among variables. It is effective for inferences about the distributions of individual imputed variables, such as an univariate analysis, but it is not appropriate for analyses involving relationship among variables, such as a regression analysis. It can also produce badly biased estimates of regression coefficients when data on predictor variables are missing (Allison 2000).

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