Original Rubin degrees of freedom approximation

Compute the degrees of freedom according to the original Rubin (1987) approximation.

Usage

rubin_orig_df(v_com, var_b, var_t, M)

Arguments

v_com

Ignored in this function.

var_b

Between-imputation sample variance of the point estimates across multiply imputed datasets.

var_t

Estimate (according to Rubin's rules) of the variance of the point estimates.

M

Number of imputations (integer larger than 1).

Value

Degrees of freedom according to the original Rubin approximation. If the between-imputation variance is zero, returns Inf.

Details

Let \(V_W = V_T - (1 + 1 / M) V_B\) be the within-imputation variance and \(r = (1 + 1 / M) V_B / V_W\) be the relative increase in variance due to nonresponse. The returned degrees of freedom are \((M - 1) (1 + 1 / r)^2\). Note that the internal computation is algebraically simplified to \((M - 1) ((M V_T) / ((M + 1) V_B))^2\).

References

Rubin, D.B. (1987). Multiple Imputation for Nonresponse in Surveys. John Wiley & Sons, New York. [Section 3.3]