# Theoretical foundations of meta-analysis

I am hoping someone can give me some guidance (or references) relating to the mean and variance of the distribution of the mean of sample means.

I am thinking about a large number of populations, each with $\mu_{i}$ and $\sigma^2_{i}$. I will refer to the mean of the $\mu_{i}$ as $\mu_{Meta}$ and the variance of the $\mu_{i}$ as $\sigma^2_{Meta}$.

Let's say random samples of size $n_{j}$ are drawn from a randomly chosen $k$ of these populations; $j=1:k$. The means ($m_{j}$) of each of these $k$ samples are computed. Then these $k$ means are averaged as a weighted sum to produce $M$, the mean of sample means. This is the statistic of interest.

If population variances are equal (i.e., $\sigma^2=\sigma^2_{1}=\sigma^2_{2},...,=\sigma^2_{N}$), and sample sizes are equal (i.e., $n=n_{1}=n_{2},...,=n_{k}$), then I assume that $M$ is an unbiased estimate of $\mu_{Meta}$, so $\mu_{M}$ = $\mu_{Meta}$. I also assume that the distribution of $M$ has variance $\sigma^2_{M}$ = ($\sigma^2_{Meta}$ + $\sigma^2$)/$n$.

My question is about what happens when sample sizes ($n_{j}$) and population variances ($\sigma^2_{i}$) are unequal. If we assume that $n_{1}+n_{2},...,+n_{k} = K$, my intuition (from Matlab simulations) is that $\sigma^2_{M}$ = ($\sigma^2_{Meta}$ + $m_{\sigma^2} )/m_{n}$, where $m_{\sigma^2}$ is the mean of all $\sigma^2_{i}$ and $m_{n}$ is the mean sample size, i.e., $K/k$. Is this true? If so it would seem (for those on this site) to be pretty elementary and references should exist in standard texts.

Rick

### Some terminology, definitions:

It sounds like you're describing a random effects model, a type of hierarchical model. Let $i$ index a group $j$ index an individual. Your model is:

$$y_{i,j} = \mu + u_i + w_{i,j}$$

$\mu$ is the population mean. $u_i$ is a random, mean zero group effect. $w_{i,j}$ is white noise.

If the variance of the error term $w_{i,j}$ differs by group, we have heteroskedasticity. Let us define $\sigma^2_i$ as the variance conditional on the group. $$\sigma^2_i = \mathrm{Var}\left(w_{i,j} \mid i \right) = \mathrm{Var}\left(y_{i,j} \mid i \right)$$

From here you should be able to lookup or the derive the properties of various estimators.

### Efficient estimation:

Something to be aware of is the concept of inverse variance weighting, of providing greater weight to observations that are observed more precisely. This leads into generalized least squares, a typical way to estimate the parameters of random effects models.

• Thanks Matthew, Your terminology does express what I was describing. And, yes, I was asking about RE meta-analysis with heteroscedastic populations. If I knew how to derive the properties of these distributions I would have. Since I don't, I thought someone at CV might know whether my assumption about the variance of the distribution of the mean of means is correct. If you know the answer it would help me a lot. Thanks again, Rick – Rick Sep 30 '16 at 19:16