I have a group of correlation coefficients (more than two). They are all dependent on one variable A in the form of r_A1, r_A2, r_A3....r_Ak, where 1, 2 ...k denote other variables; they all have the same sample size.
My question is: what statistics should I use when I want to know whether any one of the correlation coefficients is any different to any of the others? I know if there are only two dependent correlation coefficients, this can be easily compared using most statistical tools, but this is a multiple correlations test. I know that a Chi-square test can be used to compare equality of several correlation coefficients (see http://home.ubalt.edu/ntsbarsh/business-stat/otherapplets/MultiCorr.htm for an example), but to my knowledge this approach is for testing the difference between INDEPENDENT correlation coefficients. So I am wondering whether there is any approach that is equivalent to Fisher's least significant difference that can be used to make comparisons among several dependent correlation coefficients?
EDIT: thanks @russ-lenth for your answer. In general, I found that CIs computed from lsmeans are larger than those computed using Fisher's Z method. Here's an example of the CIs that I get through the lsmeans function:
rep.meas lsmean SE df lower.CL upper.CL M1 0.76914236 0.13325688 23 0.4934795 1.0448052 M2 0.82346705 0.11830361 23 0.5787374 1.0681967 M3 0.89294217 0.09386717 23 0.6987631 1.0871212 M4 -0.09985512 0.20747224 23 -0.5290441 0.3293339 M5 0.56183690 0.17249315 23 0.2050076 0.9186662 M6 0.79086279 0.12760947 23 0.5268825 1.0548431 M7 0.14667681 0.20625924 23 -0.2800029 0.5733566
Take M1 whose r = 0.769 as an example: the width of the CI from lsmeans is (1.0448-0.4935) 0.5513. The width of the CI computed from Fisher’s Z is (0.8948-0.5302) 0.3646, which is much smaller than the former. Is the difference between the widths of the two confidence intervals too large?