Regression Method for Monotone Missing Data

The MI Procedure

Regression Method for Monotone Missing Data

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.

In the regression method, a regression model is fitted for each variable with missing values, with the previous variables as covariates. Based on the fitted regression coefficients, 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). The process is repeated sequentially for variables with missing values. That is, for a variable Y_j with missing values, a model

$Y_{j}={\beta}_{0} + {\beta}_{1} \, Y_{1} + {\beta}_{2} \, Y_{2} + ... + {\beta}_{j-1} \, Y_{j-1}$

is fitted using observations with observed values for variables Y₁, Y₂, ..., Y_j.

The fitted model includes the regression parameter estimates $\hat{\beta}=(\hat{\beta}_{0}, \hat{\beta}_{1}, ... , \hat{\beta}_{j-1})$ and the associated covariance matrix $\hat{\sigma}_{j}^2 V_{j}$ ,where V_j is the usual X'X inverse matrix derived from the intercept and variables Y₁, Y₂, ... , Y_j-1.

For each imputation, new parameters ${\beta}_{*}=({\beta}_{*0}, {\beta}_{*1}, ... , {\beta}_{*(j-1)})$ and ${\sigma}_{*j}^2$ are drawn from the posterior predictive distribution of the parameters. That is, they are simulated from $(\hat{\beta}_{0}, \hat{\beta}_{1}, ... , \hat{\beta}_{j-1})$ , ${\sigma}_{j}^2$ , and V_j. The variance is drawn as

${\sigma}_{*j}^2=\hat{\sigma}_{j}^2 (n_{j}-j) / g$

where g is a ${\chi}_{n_{j}-j}^2$ random variate and n_j is the number of nonmissing observations for Y_j. The regression coefficients are drawn as

${\beta}_{*}=\hat{\beta} + {\sigma}_{*j} V_{hj}' Z$

where V_hj' is the upper triangular matrix in the Cholesky decomposition, V_j = V_hj' V_hj, and Z is a vector of j independent random normal variates.

The missing values are then replaced by

${\beta}_{*0} + {\beta}_{*1} \, y_{1} + {\beta}_{*2} \, y_{2} + ... + {\beta}_{*(j-1)} \, y_{j-1} + z_{i} \, {\sigma}_{*j}$

where y₁, y₂, ... , y_j-1 are the covariate values of the first j-1 variables and z_i is a simulated normal deviate.

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