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In many applications we obtain test statistics by combining estimates from different experiments or studies. The usual combined estimator of the overall effect in independent studies leads to systematic overestimates of the significance level, see Li, Shi, and Roth (1994). This results in a great number of unjustified significant evidences. By examination of the convexity of composed functions involved and application of higher and inverse moments of the χ2 distribution we propose corrections for the estimated standard deviation of the overall effect estimator. Analytical results and simulations show that we improve the estimated significance level in such models.
Böckenhoff et al. (1998) studied this question.