How to combine many Goodman-Kruskal gamma measures?

Question

I have 5 samples (from the same population). I want to calculate the Goodman-Kruskal gamma ($\gamma$) for the population, based on the gamma of the 5 samples ($G_i$).

Each data has 2 ordinal variables (X and Y), but the X's and Y's of different samples are not comparable, so I cannot put all the samples in the same bag and calculate the gamma for this new set. I can calculate the $G_i$ for each sample, and the standard error for the $G_i$ and I need to combine these $G_i$.

Furthermore, I would like to test the hypothesis $ \gamma \ne 0$ and thus I need the standard error for the combined G.

Just to add information, I am working in R, and I use the function GKgamma from the vcdExtra package to calculate the gamma itself and the standard error.

Answer 1

This is how I end up solving that problem.

I indeed used the theories of meta-analysis on how to combine results of experiments, in this case regression values.

The two best sources of how to combine regression values (mainly Pearson and Spearman) I found were:

a) Fixed- and random-effects models in meta-analysis. Hedges, Larry V.; Vevea, Jack L. Psychological Methods, Vol 3(4), Dec 1998, 486-504. doi: 10.1037/1082-989X.3.4.486 (link changed to an open access version)

b) Meta-analysis of correlation coefficients: A Monte Carlo comparison of fixed-and random-effects methods. AP Field - Psychological methods, 2001 (link corrected also to an open access version)

I decided to use the fixed effect model. The procedure is then:

1) convert the correlations to z-values (Pearson r-to-z procedure).

$z_i = {{1}\over{2}} ln({{1-r_1}\over{1+r_i}})$

2) combine the $z_i$

$z= {{\sum w_i z_i}\over{\sum w_i}}$ where $w_i = n_i-3$, $n_i$ the number of data points used in calculating the $r_i$ correlation.

3) the standard error of $z$ is $SE(z) = {1\over{\sqrt(\sum(n_i-3))}}$

4) convert z back to r

$r = {{e^{2z}-1}\over{e^{2z}+1}}$

This procedure is implemented in R in the package meta The function metacor implements both the fixed effects (described above) and the more complex random effects model.

PROBLEM: both the fixed and the random effects models were defined for Pearson r correlation, and I believe, also used for Spearman rho correlation. But there is a problem with Kruskal gamma in the procedure above.

I don't know if the r-to-z transformation really transform the Kruskal gamma into a normally distributed variable (which is the whole foundation of the procedure). An indication that this may not be true is that it is NOT uncommon to get a gamma=1, but that would give us a z=$\infty$ ! I decided to cap the gamma in +-0.95, that is I would substitute a gamma>0.95 for gamma=.095 and the same for -0.95.