Longitudinal Data Analysis Examples With Random Coefficient Models

Abstract

Key words: longitudinal data analysis, hierarchical linear models Longitudinal panel data examples are used to illustrate estimation methods for individual growth curve models. These examples constitute one of the basic multilevel analysis settings, and they are used to illustrate issues and concerns in the application of hierarchical modeling estimation methods, specifically, the widely advertised HLMprocedures ofBryk and Raudenbush. One main expository purpose is to demystify these analyses by showing equivalences with simpler approaches. Perhaps more importantly, these equivalences indicate useful data analytic checks and diagnostics to supplement the multilevel estimation procedures. In addition, we recommend the general use of standardized canonical examples for the checking and exposition of the various multilevel procedures; as part of this effort, methods for the construction of longitudinal data examples with known structure are described.This article attempts to give a thorough treatment of one small, but prominent, example in multilevel analysis: individual growth curve analyses of longitudinal panel data. The history of this article began with a presentation on longitudinal data analysis (Rogosa, 1989) at the October, 1989 conference "Best Methods for Analyzing Change" at the University of Southern California; one section of that presentation (prepared with Hilary Saner) compared results from simpler longitudinal methods with those from the HLM program

Salivary Cortisol Profiles of Children with Hearing Loss

“…For the purposes of this study, longitudinal refers to awakening, morning, afternoon, and evening cortisol levels as a repeated measure. The statistical approach was longitudinal—that is, the raw data were used to generate individual growth curves that were compared across groups (Rogosa & Saner, 1995). The general statistical approach was Hierarchical Linear Modeling (HLM) within the context of a mixed model statistical design between groups (CHL and CNL) and within groups (cortisol level across time; Hruschka, Kohrt, & Worthman, 2005).…”

“…The longitudinal analysis of cortisol was completed by creating an individual growth curve (Rogosa & Saner, 1995; Singer & Willett, 2003) from each participant’s cortisol samples each day. Growth in this context refers to any change, positive or negative.…”

Relationship Among Signal Fidelity, Hearing Loss, and Working Memory for Digital Noise Suppression

“…In the literature, it is also referred to as hierarchical linear modeling, mixed models, random coefficient model, and longitudinal modeling (Fitzmaurice et al 2004; Littell et al 2006; Rogosa & Saner 1995; Singer & Willett 2003; Snijders & Bosker 1999). For the current data, the model was specified hierarchically using two levels (Level-1 and Level-2), and can be expressed symbolically as the predicted intelligibility/quality score ( Y ij ) of listener i at condition j as a function of envelope fidelity ( F ij ) and as a function of hearing loss ( HFPTA i ), working memory ( RST i ), and age ( Age i ) with the following equation: $$Y_{\mathit{\text{ij}}}=P_{\mathrm{italic0}i}+P_{\mathrm{italic1}i}*false(F_{\mathit{\text{ij}}}false)+e_{\mathit{\text{ij}}},$$ where $$P_{\mathrm{italic0}i}=B_{\mathit{00}}+B_{\mathit{01}}*false({\mathit{\text{HFPTA}}}_{i}false)+B_{\mathit{02}}*false({\mathit{\text{RST}}}_{i}false)+B_{\mathit{03}}*false({\mathit{\text{Age}}}_{i}false)+r_{\mathrm{italic0}i}$$ $$P_{\mathrm{italic1}i}=B_{\mathit{10}}+B_{\mathit{11}}*false({\mathit{\text{HFPTA}}}_{i}false)+B_{\mathrm{italic12}}*false({\mathit{\text{RST}}}_{i}false)+B_{\mathit{13}}*false({\mathit{\text{Age}}}_{i}false)+r_{\mathrm{italic1}i}$$ Equation 1 represents the Level-1 model where intercept, P 0 j , corresponds to the expected value of intelligibility/quality when F ij equals zero, and the slope, P 1i , represents the rate of change in intelligibility/quality as fidelity increases.…”

Influence of gestational age and postnatal age on speech sound processing in NICU infants

“…Key et al Main analytic model. A random coefficients model for repeated measurements (Rogosa & Saner, 1995), also known as random regression (Hedeker & Gibbons, 2006), hierarchical linear modeling (Raudenbush & Bryk, 2002), multilevel modeling (Snijders & Bosker, 1999), or linear mixed effects modeling (Pinheiro & Bates, 2000) was used to address the research questions. This analysis can be thought of as a modern analysis of variance (ANOVA) for repeated measures when there are correlated errors that violate ANOVA's assumption of independent errors.…”