Combining multiple confidence intervals for comparison against another Suppose I have three sets of means and standard deviations or errors from separate studies that ask the same research question.
M1 +/- SD1
M2 +/- SD2
M3 +/- SD3

How can I combine these separate values to produce one "average" confidence interval, like a meta-analysis would do, to produce an M_new +/- SD_new? Furthermore, how could I statistically compare that "average" confidence interval to a new M4 +/- SD4 like a t-test would do?
 A: First a broad outline of what is involved in doing this via meta-analysis. You have estimates from each of three studies $y_i, i = 1,3$ and their sampling variances which you either get by squaring the standard errors or from the standard deviation and sample sizes. Call these $v_i$. You now work out the weights you need $w_i = \frac{1}{v_i}$. You then form the weighted average of the $y_i$ which is your overall average $\frac{\Sigma w_i y_i}{\Sigma w_i}$ with variance $\frac{1}{\Sigma w_1}$. This gives you the so-called fixed effects estimator.
At this point it is convenient to abandon explaining exactly what happens in algebra because realistically you need to use software for the next steps. To examine whether your three studies differ from another group (in your example a single study) you need the concept of meta-regression. You create a new dummy variable with value 0 for the first three and 1 for the fourth study. You then use this as a moderator in the meta-regression and its coefficient and the confidence interval for that tell you whether the means of the two group differ and by how much.
If instead you wanted to use the results from your meta-analysis to set up a prediction interval for the next observed study then you can do that too. Again this should be available in software. Prediction intervals are discussed in this paper by Riley and colleagues entitled "Interpretation of random effects meta--analyses".
