The methods for matched sets presented in Chapter 2 are appropriate in the case of balanced allocation. Example 2.3.2 does involve the use of weighting estimates across type of matched sets, although the sample sizes within the matched sets are the same in this case. A further direction of research for this methodology may involve the extension to situations of unbalanced allocation when the matched sets have variable size. The extension of the NPANCOVA methodology to randomized studies withM:N matching (where M and N are both ≥ 2 but ≤ 4) is straightforward. The difference in mean responses would be formed for the two treatment groups for each matched
set, and formation of f¯and Vf¯ would proceed as previously outlined. For the odds
ratio, the methodology presented in Section 2.2.3 could be extended via a Mantel- Haenszel estimate of the odds ratio or a conditional maximum likelihood estimate as in (Miettinen, 1970), but this is beyond the scope of this research. In practice, M and N are rarely greater than 3, and so the methodology here is most applicable for matched pairs and matched sets of size≤6. For larger matched sets, the methodology presented in Chapter 3 may be more appropriate.
For the NPANCOVA methodology as applied to matched sets in Chapter 2, several extensions of the NPANCOVA methdology are possible. While only dichotomous out- comes are considered, these methods may be easily extended to continuous outcomes through a covariate-adjusted difference in means. Repeated outcome measures through multiple assessments (e.g. clinic visits) could be managed by including entries in the difference vector for each pair corresponding to the differences that the respective as- sessments have for the responses for the two treatments. In the case of 1:1:1 randomized allocation to treatments A,B, or C (matched triples), the difference vector for stratum h could contain the A-C difference in response, the B-C difference in response, and such differences for the covariates for adjustment. Formation of the appropriate mean vector and its covariance matrix would follow per Section 2.2.1. Subsequent assessment of ordinality of the treatment effects could be accomplished via hypothesis testing of contrasts for the vector of covariate-adjusted estimates.
The methods in Chapter 3 and 4 could be combined in the case of a randomized study with stratification and multiple treatment groups. Extension to ordinal outcomes and counts is a possibility, although it has not yet been determined whether the cross- products of the DFBETA residuals from a treatment-only proportional odds model or Poisson regression model can be used to obtain a valid robust estimate of the covariance matrix for the log odds ratios or log rate ratios. In the case of repeated measures, there
is the possible extension of the Saville and Koch methodology using the DFBETA residuals from generalized estimating equations (GEE). Preisser and Qaqish (1996) have developed cluster-level DFBETA residuals (i.e. DFBETA residuals based on deleting an entire cluster’s observations) and observation-level DFBETA residuals (i.e. based on deleting one observation within a cluster) which are available in standard software, and the cross-products of the cluster-level DFBETA residuals have been shown to exactly equal a bias-corrected covariance estimator in the GEE setting (Mancl and Derouen, 2001). Accommodating baseline covariates in this setting is straightforward, and it would be possible to use two or more of the repeated measures (e.g. a pre-baseline measure and its baseline counterpart) as covariates. However, it is currently unclear how to easily manage adjustment of similar measures on different scales (e.g. a dichotomized baseline value included as an outcome measure and its continuous counterpart included as a baseline covariate).
While the research presented here offers a theoretical road map for the NPANCOVA methods in these settings, a future direction of the research could be to assess the per- formance of the NPANCOVA methods in this paper as they relate to the parametric modeling methods. Simulations would be a necessary next step to ensure that the NPANCOVA methods do not result in any Type I error increase beyond the nomi- nal level as well as to ensure that the methods have adequate power for testing the hypotheses of interest. From the examples presented here, one can observe that the power to test coefficients relating to the treatment effects is greater when there is co- variance adjustment than when there is no covariance adjustment (when the standard errors show a decrease after adjustment). However, regarding power, it is unclear how the NPANCOVA methods perform compared to the parametric ANCOVA methods. Jiang et. al. have conducted a simulation study comparing the NPANCOVA methods of Tangen and Koch (1999) for logrank and Wilcoxon scores to the Cox proportional
hazards model. They showed that the NPANCOVA methods did preserve the nominal Type I error and that there was a small loss in efficiency of the NPANCOVA methods when the proportional hazards assumption was correct (Jiang et al., 2008). However, in planning a study, the cost of having a greater sample size for NPANCOVA may be out- weighed by the benefit of having minimal statistical assumptions when pre-specifying the primary analysis method. Accordingly, the sample size needed for NPANCOVA may end up being less than what would be necessary in a nonparametric analysis that did not adjust for covariables (e.g. logrank test).