Table 2

Summary of significance tests for combining different estimates from m imputed datasets after MI



Test statistic

Degrees of freedom (df)

Relative increase in variance (r)

A) Scalar

F1, v

, H0: Q = Q0

v = (m - 1)(1 + r-1)2

B) Multivariate

H0:Q = Q0,

k = number of parameters

where a = k(m - 1)


χ2 statistics

w1,..., wm

k = df associated with χ2 tests

D) Likelihood Ratio χ2 statistics

wL1,..., wLm

k = number of parameters in fitted model

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.

wj, j = 1,..., m = χ2 statistics associated with testing the null hypothesis Ho : Q = Qo on each imputed dataset, such that the significance level for the jth imputed dataset is P{ > wj}, where is the χ2 value with k degrees of freedom (Rubin 1987).

= average of the repeated χ2 statistics.

= average of the m likelihood ratio statistics, wL1,..., wLm, evaluated using the average MI parameter estimates and the average of the estimates from a model fitted subject to the null hypothesis.

Marshall et al. BMC Medical Research Methodology 2009 9:57   doi:10.1186/1471-2288-9-57

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