## Info

chi-squared statistics (or, taking square roots, z statistics) testing whether j8D = 0 using the likelihood-ratio, Wald, or score approaches, in the same way as just discussed for two-way tables.

To illustrate, applying the simpler model with a linear dose effect and dose scores (1, 2, 3, 4) to Table V, we get 0 = 0 205 (ASE = 0 058). The Wald chi-squared statistic equals 12-5, and the likelihood-ratio statistic comparing this model to the simpler one without the dose effect equals the difference in — 2-log-likelihood values for the two models, which is also 12-5 (z = 3-53). The P-value is less than 0 001.

More generally, one could extend model (9) or the simpler one with the linear effect by permitting dose-by-stratum interaction. The model simply then adds cross-product terms of the dose and strata variables (or dummy variables). One can test the hypothesis of no interaction by comparing the —2 log-likelihood values for this model and the corresponding model without interaction. When the degree of interaction seems substantively important, one can estimate and test the effect separately in each stratum using the dose effect estimates pertaining to that stratum, or one could simply fit the original model (for example, 6) separately to each stratum to obtain the separate effects. (This approach is not equivalent, because it estimates intercept parameters separately with each fit.) On the other hand, when the dose effects do not vary much among the strata, the overall test based on a lack of interaction tends to be much more powerful, and the overall estimate tends to be more efficient, since they summarize information across the strata.

In fact, there is some evidence of interaction in Table V. For the models with linear dose effect, the likelihood-ratio statistic comparing the model with separate slopes to the model with a single slope equals 3-85 with d.f. = 1. The model with separate slopes has estimates 0 = 0099 (ASE = 0-082) for the mild trauma group and 0 = 0-327 (ASE = 0-082) for the moderate/severe trauma group. Hence, there is a strong evidence of a dose effect only for the latter group. There are other approaches one could use both to check for interaction and to describe the separate effects, but we do not discuss them here because of space limitations.

4.2. Non-model-based approaches for the stratified case

The CMH approach generalizes naturally to combining information from several strata; in fact, the original statistic presented by Mantel and Haenszel46 was designed specifically for the stratified case with two groups and a binary response. For the case of several doses and an ordinal response, the correlation statistic (Mantel17) provides a large-sample chi-squared statistic with d.f. = 1 for detecting a linear trend in the effect. One can, as usual, treat the signed square root as a standard normal statistic. The CMH approach, like model (9), works well when the dose effects are similar in each stratum. It is available with the CMH1 option in PROC FREQ in SAS. For Table V, this approach used with equally-spaced scores for doses and response outcomes yields a chi-squared statistic of 16-2 and normal statistic of 4-0, for which the P-value is less than 0001.

Similarly, one could consider stratified versions of tests discussed in Section 2 that are special cases of a generalized CMH test, such as Tarone's test.42 In principle, this type of construction could also be used with other sorts of statistics, such as the Jonckheere-Terpstra statistic.

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