# testing for difference in means across N pairs of samples

I have N different populations.

Within each population, I have an equally-sized pair of sub-populations: $(t_j, c_j)$ for $j=1,N$

For each pair in each population, I have the measurement of a mean.

$(\bar{x}^{\space t}_j, \bar{x}^{\space c}_j)$ for $j=1,N$ and the corresponding variance.

all $t$ were subjected to a treatment, the same one across populations.

How can I test whether the mean of the groups with treatment is significantly different from the control groups?

Note that:

• Each population has a different size.
• The variance is not the same across populations.
• Within each population, $t$ and $c$ were randomly allocated.

I would have thought this needs some sort of weighted pair t-test where the size and standard deviation of each $\bar{x}$ is taken into account.

Happy to hear suggestions.

• Probably a good bet is some kind of mixed hierarchical model where treatment and control groups are nested within populations. Check out lme4 and other packages in R. – NatWH Aug 31 '18 at 20:22