How can I combine fully correlated effect sizes and their standard errors?

Question

Let's say that I, for a meta-analysis, have two effect sizes, $ES_1$ and $ES_2$ (begin standardized mean differences) with standard errors, $SE_1$ and $SE_2$, and that I want to combine these into a single measure (they might for example represent two different outcomes on the same test) with a single standard error, $ES_{combined}$ and $SE_{combined}$. Further, I assume that they are fully correlated, meaning that the number of effect sizes that I combine should not affect how precise the final effect size is (that is, how narrow its standard error is).

I've found some sources claiming that you simply can compute the averages, that is: $ES_{combined}=\frac{ES_1+ES_2}{2}$ and $SE_{combined}=\frac{SE_1+SE_2}{2}$, but they have not supplied a rationale for this reasoning.

How can I do this correctly?

Answer 1

Keep in mind that $ES_i$ are random variables, in your case $ES_{combine}$ is just a linear combination of random variables. $$ ES_{combine}=\sum_i^n w_i ES_i $$ Where $n=2,w=1/2$. So,

$$ \begin{align} mean(ES_{combine})=\sum_i^n w_i \ mean(ES_i) \\ sd(ES_{combine})=\sqrt{\sum_i^n w_i^2 \ var(ES_i) + \sum_{ij\in n} 2 w_i w_j cov(ES_i ES_j)} \end{align} $$

Also, its interesting to point out that by taking average you are assuming the effective sizes play the same importance. But in reality some information source might be more reliable than others. There are some truth finding models that can help you calculate the reliabilities of the information sources.