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Table 2 Summary of significance tests for combining different estimates from m imputed datasets after MI

From: Combining estimates of interest in prognostic modelling studies after multiple imputation: current practice and guidelines

Estimate F Test statistic Degrees of freedom (df) Relative increase in variance ( r )
A) Scalar F 1, v , H0: Q = Q 0 v = (m - 1)(1 + r -1)2
B) Multivariate
H0:Q = Q 0,
k = number of parameters

where a = k(m - 1)
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
where a = k(m - 1)
  1. KEY: F = value from the F-distribution, which the test statistic is compared to.
  2. = average of the m imputed data estimates.
  3. = within imputation variance.
  4. B = between imputation variance.
  5. T = total variance for the combined MI estimate.
  6. 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).
  7. = average of the repeated χ 2 statistics.
  8. = 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.