Application of hierarchical linear modelling to the study of trajectories of behavioural development

Abstract

Students of behavioural development often collect andanalyse longitudinal data on a sample of individuals(Bateson 1981; Sackett et al. 1981; Chalmers 1987). Forexample, primate studies of behavioural and relationshipdevelopment have typically provided data on rates of,or percentages of time engaged in, the performance ofvarious behaviours across the individuals’ early ontogeny(e.g. Fairbanks 1996; Deputte 2000). The longitudinalrecords for each individual, collected on multiple occa-sions, are then pooled per age intervals of varying length.Commonly, such longitudinal data are analysed usingeither a mixed design analysis of variance (ANOVA) or aconventional (simple or multiple) regression analysis (Zar1999; David 2002). In this commentary, we demonstratea number of serious problems in using these methods,and describe an alternative that avoids them.Researchers applying mixed design ANOVA to thestudy of behavioural development typically use age of theindividual (i.e. time blocks) as the within-subject factor,and variables such as sex or species as the between-subjects factors (e.g. Maestripieri 1994; Suomi et al. 1996).Although ANOVA models can be used to analyse longitu-dinal data they have two major limitations. First, theyrequire a balanced data collection design, with the samenumber of individuals per measurement occasion and thesame interval between consecutive measurements. Whenthese requirements are not fulfilled, which is often thecase in observational studies of spontaneously occurringsocial interactions, a number of complications arise(e.g. Hox & Kreft 1994). Second, ANOVA models assumehomogeneous (co)variances at level 1, that is, a constantwithin-subject covariance structure (Zar 1999; David2002). This assumption is often not met by developmen-tal data, resulting in a Type I error greater than thespecified .Animal behaviour researchers using traditional regres-sion analysis have used two major approaches. The firsttreats all predictor or independent variables, such as ageof infant, sex, cohort, maternal experience or dominancerank, as if they pertained to a single level (e.g. Bramblett& Coelho 1985; Altmann & Samuels 1992) and theircontributions to the observed variance in the rate ofbehaviour during the whole study period are thenassessed. The second regression modelling approachis known as the intercepts-and-slopes-as-outcomesapproach (e.g. Bryk & Raudenbush 1992; Hox & Kreft1994). In this case, two separate steps are taken. First,linear (or curvilinear) regressions are fitted to thedevelopmental functions for each subject. Next, theparameters from these regression equations are usedas dependent variables in subsequent analyses to bepredicted by one or a set of independent variables(e.g. Wasser & Wasser 1995).The application of simple or multiple, single-levellinear regression to the analysis of longitudinal data isinadequate for several reasons (Bryk & Raudenbush 1992;Hox & Kreft 1994; Goldstein et al. 1998; van der Leeden1998; Snijders & Bosker 1999). First, this approach isconceptually unsound because it ignores the hierarchicalnature of longitudinal data in which observations withinindividuals are dependent. Second, conventional single-level methods disaggregate all higher level explanatoryvariables to the lowest level, followed by an ordinary leastsquares (OLS) regression analysis. Therefore, they do notprovide methods to test for the effect of independentvariables on the variation observed in the parameters ofchange (initial status, rate of development, and soon), which is one of the major theoretical concerns ofthose interested in the study of behavioural develop-ment. When OLS is applied to longitudinal data, theassumption of independence of residual error terms isviolated and this leads to inefficient estimates and a TypeI error rate that is much higher than the nominal level(Hox & Kreft 1994; van der Leeden 1998).In the intercepts-and-slopes-as-outcomes approach,separate fixed-effects regression models are first fittedwithin each individual, using level 1 explanatory vari-ables as predictors. Next, the within-individual growthcoefficients are, in turn, used as dependent variables to bepredicted by level 2 explanatory variables (Wasser &