Combining Inferences from Multiply Imputed Data Sets

The MI Procedure

Combining Inferences from Multiply Imputed Data Sets

With m imputations, m different sets of the point and variance estimates for a parameter Q can be computed. Suppose $\hat{Q_i}$ and $\hat{U_i}$ are the point and variance estimates from the ith imputed data set, i=1, 2, ..., m. Then the combined point estimate for Q from multiple imputation is the average of the m complete-data estimates:

${\overline Q}=\frac{1}m \sum_{i=1}^m \hat{Q_i}$

Suppose ${\overline U}$ is the within-imputation variance, which is the average of the m complete-data estimates:

${\overline U}=\frac{1}m \sum_{i=1}^m \hat{U_i}$

and B is the between-imputation variance

$B=\frac{1}{m-1} \sum_{i=1}^m (\hat{Q_i}-{\overline Q})^2$

Then the variance estimate associated with ${\overline Q}$ is the total variance (Rubin 1987)

$T={\overline U} + (1+\frac{1}m)B$

The statistic $(Q-{\overline Q}) T^{-(1/2)}$ is approximately distributed as t with v_m degrees of freedom (Rubin 1987), where

$v_{m}=(m-1) [1 + \frac{{\overline U}}{(1+m^{-1})B} ]^2$

When the complete-data degrees of freedom v₀ is small, and there is only a modest proportion of missing data, the computed degrees of freedom, v_m, can be much larger than v₀, which is inappropriate. Barnard and Rubin (1999) recommend the use of an adjusted degrees of freedom

$v_{m}^{*}=\, [ \frac{1}{v_{m}} + \frac{1}{\hat{v}_{obs}} ] ^{-1}$

where $\hat{v}_{obs}=(1 - \gamma) \, v_{0} (v_{0}+1) / (v_{0}+3)$ and $\gamma=(1+m^{-1}) B / T$ .

Note that the MI procedure uses the adjusted degrees of freedom, v_m^*, for inference.

The degrees of freedom v_m depends on m and the ratio

$r=\frac{(1+m^{-1})B}{\overline U}$

The ratio r is called the relative increase in variance due to nonresponse (Rubin 1987). When there is no missing information about Q, the values of r and B are both zero. With a large value of m or a small value of r, the degrees of freedom v will be large and the distribution of $(Q-{\overline Q}) T^{-(1/2)}$ will be approximately normal.

Another useful statistic is the fraction of missing information about Q:

$\hat{\lambda}=\frac{r+2/(v+3)}{r+1}$

Both statistics r and $\lambda$ are helpful diagnostics for assessing how the missing data contribute to the uncertainty about Q.

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