Multilevel statistical models and the analysis of experimental data

Abstract

Abstract. Data sets from ecological experiments can be difficult to analyze, due to lack of independence of experimental units and complex variance structures. In addition, information of interest may lie in complicated contrasts among treatments, rather than direct output from statistical tests. Here, we present a statistical framework for analyzing data sets containing non-independent experimental units and differences in variance among treatments (heteroscedasticity) and apply this framework to experimental data on interspecific competition among three tadpole species. Our framework involves three steps: (1) use a multilevel regression model to calculate coefficients of treatment effects on response variables; (2) combine coefficients to quantify the strength of competition (the target information of our experiment); and (3) use parametric bootstrapping to calculate significance of competition strengths. We repeated this framework using three multilevel regression models to analyze data at the level of individual tadpoles, at the replicate level, and at the replicate level accounting for heteroscedasticity. Comparing results shows the need to correctly specify the statistical model, with the model that accurately accounts for heteroscedasticity leading to different conclusions from the other two models. This approach gives a single, comprehensive analysis of experimental data that can be used to extract informative biological parameters in a statistically rigorous way.

Order of arrival structures arbuscular mycorrhizal colonization of plants

“…We first analysed the qPCR root abundance data in the simultaneous experiment with linear mixed models (LMMs) in the R package lme4 (Bates et al, 2013), using the logarithm of copy number mg À1 root mass as the dependent variable, AMF species, harvest date and the interaction between these two factors as fixed effects, and plant as a random effect. This takes into account nonindependence of measurements of different AMF species taken from the same root material (Behm et al, 2013;Engelmoer et al, 2014). For colonization data, we used a linear model with the harvest date as the explanatory variable to test for differences in colonization percentages between the three plant growth periods.…”

Intense competition between arbuscular mycorrhizal mutualists in an in vitro root microbiome negatively affects total fungal abundance

“…We used separate linear mixed effects models (lmer function from lme4 package (Bates et al 2001)) to investigate differences in intraradical species-specific abundances, extraradical species-specific abundances, and the investment ratio with species, competition treatment and P concentration as fixed factors and plate nested within species as a random factor. The latter term was used to take into account the nonindependence of the responses of the two species from the same experimental replicate (Behm et al 2013). The mixed treatments contained half the spore density of each species compared with the monoculture treatments.…”

“…equal investment; 0 because data were ln-transformed) using independent contrasts. We followed the mixed model analysis of the species-specific intraradical and extraradical abundances with planned contrasts of treatment means to explore the nature of competition among the species (Behm et al 2013). The first set of contrasts compares the abundance of R. irregularis to G. aggregatum in the mixed treatment and then in the monoculture.…”

“…The latter term was used to take into account the nonindependence of the responses of the two species from the same experimental replicate (Behm et al . ). The mixed treatments contained half the spore density of each species compared with the monoculture treatments.…”

Analysis and Prediction of Pedestrians’ Violation Behavior at the Intersection Based on a Markov Chain

“…where f o is the actual observation frequency; f e is the expected frequency. For two-dimensional interaction tables or univariate logistic regression models, χ 2 = 0.05 is usually used as the significant level for screening candidate variables; that is, when the χ 2 significance value is less than 0.05, the independent variables are valuable for predicting the results [14].…”