The SPSS output for the CMH procedure reports both the Cochran Mantel-Haenszel test statistic.
What considerations should go into choosing one or the other test statistic? (In my data set, one is statistically significant and the other is not. See output of the Crosstabs CHM command below.)
In section 6.3.2 of the second edition of Categorical Data Analysis, Agresti explains the distinction between the earlier Cochran and the later Mantel-Haenszel tests for 2 x 2 x K tables:
Mantel and Haenszel (1959)...treated response (column) marginal totals as fixed. Thus, in each partial table $k$ of cell counts $(n_{ijk})$, their analysis conditions on both the predictor totals $(n_{1\space+\space k} , n_{2\space+\space k})$ and the response outcome totals $(n_{+\space1\space k} , n_{+\space2\space k})$... Cochran (1954)...treated the rows in each 2x2 table as two independent binomials rather than a hypergeometric.
He nevertheless calls the Mantel-Haenszel formula the "the Cochran-Mantel-Haenszel (CMH) statistic" and continues:
The Mantel and Haenszel approach using the hypergeometric is more general in that it also applies to some cases in which the rows are not independent binomial samples from two populations. Examples are retrospective studies and randomized clinical trials with the available subjects randomly allocated to two treatments.
So the answer depends on whether Cochran's assumption of independent binomials in each partial table holds for your data.
That said, do you really want to base a claim of "significance" on whether one asymptotic test of independence says $p = 0.04$, another says $p = 0.06$, and the asymptotic test on the odds-ratio provides $p = 0.049$? Perhaps it's best to report the odds ratio and confidence limits along with both tests of independence, then discuss the implications of the result for your field of study.
