Data Analysis Using Regression and Multilevel/Hierarchical Models

Abstract

Data Analysis Using Regression and Multilevel/Hierarchical Models, first published in 2007, is a comprehensive manual for the applied researcher who wants to perform data analysis using linear and nonlinear regression and multilevel models. The book introduces a wide variety of models, whilst at the same time instructing the reader in how to fit these models using available software packages. The book illustrates the concepts by working through scores of real data examples that have arisen from the authors' own applied research, with programming codes provided for each one. Topics covered include causal inference, including regression, poststratification, matching, regression discontinuity, and instrumental variables, as well as multilevel logistic regression and missing-data imputation. Practical tips regarding building, fitting, and understanding are provided throughout.

Functional traits associated with plant colonizing and competitive ability influence species abundance during secondary succession: Evidence from subalpine meadows of the Qinghai–Tibetan Plateau

“…As our study design involved two replicate meadow sites for each successional age, with species measurements undertaken within meadows of all ages, a hierarchical regression model was warranted (Gelman & Hill, 2006). The trait–abundance intercepts and slopes were subject to random effects, and our key objective here was to model intercepts and slopes of the trait–abundance relationship as a function of successional age, while allowing for possible differences in these relationships between the two chronosequences.…”

Improving estimates of environmental change using multilevel regression models of Ellenberg indicator values

“…Slope and intercept parameters are constrained to come from bivariate normal distribution (MVN) with mean vector $({\mu }_{\alpha },{\mu }_{\beta })$ to account for correlation between them (Gelman & Hill, 2006). The covariance matrix is defined by the variance in plot intercepts $({\sigma }_{\alpha })$ and slopes $({\sigma }_{\beta })$, and the covariance between the two sets of parameters $(\rho {\sigma }_{\alpha }^2{\sigma }_{\beta }^2)$, where ρ is the correlation coefficient.…”

“…Environmental heterogeneity is an important factor in plant ecology studies generally (e.g., Maslov, 1989), and by failing to account for different levels of variation within a system, traditional methods discard much information, which may result in over‐ or underestimation of the extent of change over time (Gelman & Hill, 2006). Figure 1 depicts three distinct levels of variation that can be identified within a typical ecological study estimating environmental change using EIVs: (a) variation among EIV scores of species recorded within sampled plots ( σ species ); (b) variation between plots in mean EIV scores ( σ α ); and (c) variation in between time‐period differences in plot mean EIVs as environmental conditions change differentially across a landscape over time ( σ β ).…”

“…Instead of using point estimates of mean EIVs, data enter the model as species‐specific EIV scores, and inferred plot means—with all of their associated uncertainty—are estimated and used at a higher level within the model to infer differences between groups of plots in space and time (McElreath, 2016). In this way, information is shared between plots, with mean EIV estimates augmented through partial pooling—that is, plot‐level estimates being pulled towards the overall mean to an extent dependant on the number of species a mean estimate is composed of, and the variability of estimates between plots (Gelman & Hill, 2006). More fully accounting for uncertainty in this way should lead to more reliable estimates of individual plot mean values, and of differences between groups of plots in space and time (Gelman & Hill, 2006).…”

“…In this way, information is shared between plots, with mean EIV estimates augmented through partial pooling—that is, plot‐level estimates being pulled towards the overall mean to an extent dependant on the number of species a mean estimate is composed of, and the variability of estimates between plots (Gelman & Hill, 2006). More fully accounting for uncertainty in this way should lead to more reliable estimates of individual plot mean values, and of differences between groups of plots in space and time (Gelman & Hill, 2006). Furthermore, because estimates are pooled according to shared information content, differences between any pair or combination of individual plots or habitats in the system can be inferred without having to contend with the issue of multiple comparisons, which should provide more power to detect change over time in conventional null hypothesis testing frameworks (Gelman, Hill, & Yajima, 2012).…”