How to use the Scheffe correction I have 4 conditional correlation matrices $C_1, \ldots, C_4$. I want to test each entry in the $k$th matrix, say $\hat{r}_{ij}^{(k)}$, against a reference value $r_{ij}^{0}$ value in a matrix $C_0$. 
I do this by checking whether $$-n_{k}\log\left[1 - (\hat{r}_{ij}^{(k)} - r_{ij}^{0})^2\right] > 3.841$$
which is the critical value of a chi-square distribution with 1 degree of freedom at the 0.05 significance level, under the null hypothesis that $\hat{r}_{ij}^{(k)} = r_{ij}^{0}$.
I know I am prone to false positives because of multiple comparisons and I have heard that Scheffe's method can alleviate this problem. This method would also be appropriate since each conditional correlation matrix $C_k$ is estimated based on different sample sizes $n_k$. I understand how this method works in very broad strokes, but I am not sure how I can apply in this particular situation. What are the groups in this case: what contrasts are appropriate?
 A: In the context of anova, the Scheffe method is based on testing all possible contrasts, of which there are infinitely many, and using a critical value based on the distribution of the maximum $t$ value among all of them. 
But as I understand your question, this is not an anova situation and it'd be tough to figure out that worst-case distribution. Moreover, as I understand it, there are only 4 contrasts you want to test. So applying the Scheffe idea would be quite conservative, even if it is tractable. 
In the anova context again, the Bonferroni method is typically less conservative than Scheffe when there are just a handful of tests involved. 
I suggest applying a simple Bonferroni correction -- that is, for 4 tests, divide your alpha by 4 so that you use the $100 - 5/4 = 98.75$th percentile of the $\chi^2_1$ distribution for an overall type I error probability of at most 5%. The Bonferroni method is somewhat conservative but is based on a probability inequality that always holds, regardless of the model, sampling distribution, or intercorrelations among the statistics. 
