Table 2 Summary of significance tests for combining different estimates from m imputed datasets after MI
Estimate | F | Test statistic | Degrees of freedom (df) | Relative increase in variance ( r ) |
|---|---|---|---|---|
A) Scalar
| F 1, v |
| v = (m - 1)(1 + r -1)2 |
|
B) Multivariate
|
|
H0:Q = Q 0, k = number of parameters |
where a = k(m - 1) |
|
C) χ 2 statistics w 1,..., w m |
|
k = df associated with χ 2 tests |
|
|
D) Likelihood Ratio χ 2 statistics w L1,..., w Lm |
|
k = number of parameters in fitted model |
|
|
, H0: Q = Q
0
where a = k(m - 1)
- KEY: F = value from the F-distribution, which the test statistic is compared to.
-
= average of the m imputed data estimates. -
= within imputation variance. - B = between imputation variance.
- T = total variance for the combined MI estimate.
-
w
j
, j = 1,..., m = χ
2 statistics associated with testing the null hypothesis H
o
: Q = Q
o
on each imputed dataset, such that the significance level for the j
thimputed dataset is P{
> w
j
}, where is the χ
2 value with k degrees of freedom (Rubin 1987). -
= average of the repeated χ
2 statistics. -
= average of the m likelihood ratio statistics, w
L1,..., w
Lm
, evaluated using the average MI parameter estimates and the average of the estimates from a model fitted subject to the null hypothesis.
= average of the m imputed data estimates.
= within imputation variance.
> w
j
}, where 
= average of the repeated χ
2 statistics.
= average of the m likelihood ratio statistics, w
L1,..., w
Lm
, evaluated using the average MI parameter estimates and the average of the estimates from a model fitted subject to the null hypothesis.