Simultaneous Confidence Intervals for Product-Type Interaction Contrasts

Abstract

Summary The set of product‐type interaction contrasts, which contains the subsets of interaction residuals, tetrad contrasts, double‐dichotomy contrasts and pooled‐tetrad contrasts, is discussed. In the Normal case with equal and known variances, simultaneous confidence intervals for a given set of such contrasts can be constructed in several ways. Five such methods (the Scheffé method, the Tukey method, the Dunn method, a method based on Roy's maximum‐root statistic and a modification of Tukey's method) are presented and compared in terms of the widths of the resulting 95 and 99 per cent confidence intervals. The Dunn method is found to yield the shortest intervals for interaction residuals, for tetrad contrasts and for double‐dichotomy contrasts. On the other hand, for the set of pooled‐tetrad contrasts, and therefore also for any larger set of product‐type interaction contrasts, the maximum‐root method yields the shortest intervals. A table of the half‐widths of the intervals given by the recommended methods is provided.

Morphological and biochemical responses of Balanites aegyptiaca to drought stress and recovery are provenance‐dependent

“…Data were split according to the levels of VWC, and all pair‐wise comparisons among provenances were tested (Tukey test) on each VWC level separately. Further, interaction contrasts (Gabriel, Putter, & Wax, ) were used to compare the magnitude of differences between VWC levels among the provenances. All computations were performed in R‐3.1.0 (R Core Team ) using the package multcomp (Hothorn, Bretz, & Westfall, ) for the Tukey test and compact letter display, the packages nlme (Pinheiro & Bates, ) and lme4 (Bates, Maechler, Bolker, & Walker, ) for GLS and random effect models.…”

Type IV Errors and Analysis of Simple Effects

“…By properly planning a factorial experiment, an experimenter could test hypotheses concerned with interaction, main effects and simple effects at levels close to the conventional alpha values with reasonable power by carefully considering the relationship involving the number of levels of his factors, cell sample size and choice of an overall hypothesis with appropriate experimentwise alpha level. (Marascuilo & Levin, 1970) or one might consider even wider sets of subhypotheses of interaction (Bradu and Gabriel, 1974;Gabriel, Putter and Wax, 1973). 8), which bounds the experimentwise error rate by the sum of the alphas for all hypotheses under consideration.…”

Analysis of Statistical Interactions in Factorial Experiments

“…Gabriel et al. () presented this general approach to construct comparisons of cell means, which they denoted as (Kronecker) product‐type interaction contrasts, to build simultaneous confidence intervals for interactions. All tetradic contrasts in Eq.…”