Covariance Analyses with Heterogeneity of Slopes in Fixed Models

Abstract

Techniques that describe the use of covariance when heterogeneity of slopes exists are severely limited. Although a few procedures for model selection have been recommended, none, except the hierarchical approach, is straightforward and usable with present computer programs. The hierarchical subset selection procedure presented in this paper is based on the proposition that heterogeneity may be present only for certain terms in the model. After hierarchical selection, those terms which do not involve heterogeneity are interpreted as in the usual analysis for covariance. The interpretations of those terms which do involve heterogeneity are modified with respect to significance tests performed at various values of the covariate. The hierarchical subset selection method allows one to investigate heterogeneity of slopes in covariance models as functions of the classification variables present in the design.

Ejaculate expenditure by malebush crickets decreases with sperm competition intensity

“…We used a one-factor analysis of covariance to examine the influence of prior encounters with females and female body size on ejaculate characteristics, following the procedure of Hendrix et al (1982), with an α 0.1 to avoid type II errors in the detection of interactions. For residual spermatophylax weight, there was no significant interaction between treatment and female body size so that the interaction term was removed from the model.…”

“…We thus examined treatment effects at the mean and at ±1 s.d. from the mean value for the covariate (see Hendrix et al 1982). When mating with small females, males increased spermatophylax weight under a female-biased OSR; the adjusted mean residual spermatophylax weight 1 s.d.…”

Cue reliability, risk sensitivity and inducible morphological defense in a marine snail

“…Because ANOVA/ANCOVA are robust to violations of normality, I used a parametric general linear model that included predator cue (presence/absence) and damaged prey cue (heterospeciWc, conspeciWc, nonspeciWc neutral cue) as Wxed factors and their interaction, replicate aquarium as a random factor nested in the predator cue £ prey cue interaction, shell length as a size covariate, and the covariate-by-factor interactions (to test for homogeneity of slopes). Shell length-by-factor interaction terms were dropped from the Wnal model if they were non-signiWcant (Hendrix et al 1982). I also implemented a two-way nonparametric nested ANCOVA (Conover and Iman 1982).…”

Allometric Analysis beyond Heterogeneous Regression Slopes: Use of the Johnson‐Neyman Technique in Comparative Biology

“…This allows systematic, though time-consuming, examination of the nature of intergroup differences. Hendrix et al (1982) describes a similar procedure for multiple comparisons. Groups are first compared pairwise, and those that are not found to have significantly different slopes are compared using ANCOVA.…”