Limitations of Independent Assumption
Dependence among statistical tests is generally positive, which means that the p-value of X2 is too small (anti-conservative) if the dependency is not taken into account. Thus, if Fisher's method for independent tests is applied in a dependent setting, and the p-value is not small enough to reject the null hypothesis, then that conclusion will continue to hold even if the dependence is not properly accounted for. However, if positive dependence is not accounted for, and the meta-analysis p-value is found to be small, the evidence for the alternative hypothesis is generally overstated. The approximate false discovery rate, reduced for positively correlated tests, may suffice to adjust alpha for useful comparison to an over-small p-value from Fisher's X2.
Read more about this topic: Fisher's Method
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