A new approach for modeling generalization gradients: a case for hierarchical models

Abstract

A case is made for the use of hierarchical models in the analysis of generalization gradients. Hierarchical models overcome several restrictions that are imposed by repeated measures analysis-of-variance (rANOVA), the default statistical method in current generalization research. More specifically, hierarchical models allow to include continuous independent variables and overcomes problematic assumptions such as sphericity. We focus on how generalization research can benefit from this added flexibility. In a simulation study we demonstrate the dominance of hierarchical models over rANOVA. In addition, we show the lack of efficiency of the Mauchly's sphericity test in sample sizes typical for generalization research, and confirm how violations of sphericity increase the probability of type I errors. A worked example of a hierarchical model is provided, with a specific emphasis on the interpretation of parameters relevant for generalization research.

Training discrimination diminishes maladaptive avoidance of innocuous stimuli in a fear conditioning paradigm

“…To deal with these dependencies in the data we used a hierarchical modeling approach [27]. Use of hierarchical models is a more appropriate way of analyzing generalization data than using repeated measures ANOVA, as the latter treats the generalization stimuli as a categorical dimension whereas a dimensional approach seems more appropriate.…”

“…Moreover, the assumption of sphericity in a repeated measures ANOVA, that is equality of variances within stimuli and correlations between stimuli, is not realistic when working with generalization data, resulting in increased probability of type I error. For a full discussion of the benefits of the used approach over a repeated measures ANOVA, see [27]. All models were fitted in R [28] by means of the lme4 package [29].…”

Stimulus generalization as a mechanism for learning to trust

“…Accordingly, we investigated two key questions: ( i ) Is perceptual similarity of the morphs implicitly used to guide novel decisions to trust unfamiliar others, and ( ii ) do these putative generalization gradients evoke structurally similar behavioral tuning profiles (e.g., adaptively refraining from or choosing to trust at the same rate)? To answer these questions, we ran a hierarchal logistic regression ( 14 ), where both trustworthiness type (whether faces were morphed with the original trustworthy, untrustworthy, or neutral player) and perceptual similarity (increasing similarity to the original players) were entered as predictors of choosing to play with the morph. We found that as perceptual resemblance to the original trustworthy player increased subjects were significantly more likely to choose the morph as a partner for a future trust game (trust type × perceptual similarity: P < 0.001; Fig.…”

Rule-based generalization and peak shift in the presence of simple relational rules

“…The training and test data were analyzed in hierarchical mixed effects regressions using the ‘lme4’ package ([ 37 ]) in R ([ 38 ]). In the context of our experiments, hierarchical linear regression is more appropriate and powerful than traditional analysis of variance (ANOVA) (see [ 39 ]). Firstly, both our training and test data consist of multiple trial-level observations for each level of each factor.…”

“…Hierarchical linear models utilize the raw trial-level responses (e.g., correct/incorrect), minimizing information loss ([ 40 ]). Mixed effects models are especially useful in analyzing categorization test data consisting of binary (correct/incorrect) responses, which otherwise would need to be averaged to form continuous data to analyze in a repeated-measures ANOVA (see [ 39 ], for other arguments against using repeated-measures ANOVA to analyze generalization gradients).…”