A novel approach to assess simultaneous trends in a two-way ANOVA: application to cholesterol data

Two-way ANOVA models are usually employed to see the homogeneity of row (column) effects. In various medical and psychological studies, prior information that these effects follow monotone orderings may be available; for example, the effects of the factors ‘Age’ and ‘Gender’ on low-density lipoprotein are observed to be monotonic with respect to both factors. Integrating this information yields computationally intensive yet powerful tests. Here we develop powerful procedures for testing simultaneous trends and constructing simultaneous confidence intervals for ordered effects in a two-way heteroscedastic additive ANOVA model. The likelihood ratio test and two union-intersection type tests are developed. For comparing various treatments, the reporting of confidence intervals for successive differences in effects that lead to the rejection of the null hypothesis is of interest. The proposed tests control type-I error rates, achieve high power, and remain robust under departures from normality. They also allow the construction of simultaneous confidence intervals. The test procedures developed here are implemented on cholesterol data of the patients classified according to age and gender and are seen to detect even small increments in effects of the factors. An ‘R’ software package has been developed to implement the proposed tests.