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40 CitationsR2 Statistics for Mixed Models
Abstract
The R 2 statistic, when used in a regression or ANOVA context, is appealing because it summarizes how well the model explains the data in an easy-to-understand way. R 2 statistics are also useful to gauge the effect of changing a model. Generalizing R 2 to mixed models is not obvious when there are correlated errors, as might occur if data are georeferenced or result from a designed experiment with blocking. Such an R 2 statistic might refer only to the explanation associated with the independent variables, or might capture the explanatory power of the whole model. In the latter case, one might develop an R 2 statistic from Wald or likelihood ratio statistics, but these can yield different numeric results. Example formulas for these generalizations of R 2 are given. Two simulated data sets, one based on a randomized complete block design and the other with spatially correlated observations, demonstrate increases in R 2 as model complexity increases, the result of modeling the covariance structure of the residuals.
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- The causal model is acceptable if the p value associated with the C statistic is >0.05, as this means that the data do not depart significantly from that predicted by the relationships defined in the causal graph (Shipley 2009). We calculated the R 2 of each endogenous variable in the path models using the likelihood ratio R 2 test (Kramer 2005) to give some indication of explanatory power.
[Show abstract] [Hide abstract] ABSTRACT: Vertebrate consumers can be important drivers of the structure and functioning of ecosystems, including the soil and litter invertebrate communities that drive many ecosystem processes. Burrowing seabirds, as prevalent vertebrate consumers, have the potential to impact consumptive effects via adding marine nutrients to soil (i.e. resource subsidies) and non-consumptive effects via soil disturbance associated with excavating burrows (i.e. ecosystem engineering). However, the exact mechanisms by which they influence invertebrates are poorly understood. We examined how soil chemistry and plant and invertebrate communities changed across a gradient of seabird burrow density on two islands in northern New Zealand. Increasing seabird burrow density was associated with increased soil nutrient availability and changes in plant community structure and the abundance of nearly all the measured invertebrate groups. Increasing seabird densities had a negative effect on invertebrates that were strongly influenced by soil-surface litter, a positive effect on fungal-feeding invertebrates, and variable effects on invertebrate groups with diverse feeding strategies. Gastropoda and Araneae species richness and composition were also influenced by seabird activity. Generalized multilevel path analysis revealed that invertebrate responses were strongly driven by seabird engineering effects, via increased soil disturbance, reduced soil-surface litter, and changes in trophic interactions. Almost no significant effects of resource subsidies were detected. Our results show that seabirds, and in particular their non-consumptive effects, were significant drivers of invertebrate food web structure. Reductions in seabird populations, due to predation and human activity, may therefore have far-reaching consequences for the functioning of these ecosystems.- Author's personal copy on a likelihood ratio (LR) test between the candidate model (e.g. GLMM or GAMM) and an intercept only (null) model (Kramer 2005; Magee 1990 ). Smaller values of DIC indicate a better-fitting model, while 0 R 2 LR 1, with R 2 LR ¼ 1 corresponding to a perfect fit, and R 2 LR ! 0 for any reasonable model specification.
[Show abstract] [Hide abstract] ABSTRACT: Dengue is the world’s most important vector-borne viral disease. The dengue mosquito and virus are sensitive to climate variability and change. Temperature, humidity and precipitation influence mosquito biology, abundance and habitat, and the virus replication speed. In this study, we develop a modelling procedure to quantify the added value of including climate information in a dengue model for the 76 provinces of Thailand, from 1982–2013. We first developed a seasonal-spatial model, to account for dependency structures from 1 month to the next and between provinces. We then tested precipitation and temperature variables at varying time lags, using linear and nonlinear functional forms, to determine an optimum combination of time lags to describe dengue relative risk. Model parameters were estimated using integrated nested Laplace approximation. This approach provides a novel opportunity to perform model selection in a Bayesian framework, while accounting for underlying spatial and temporal dependency structures and linear or nonlinear functional forms. We quantified the additional variation explained by interannual climate variations, above that provided by the seasonal-spatial model. Overall, an additional 8 % of the variance in dengue relative risk can be explained by accounting for interannual variations in precipitation and temperature in the previous month. The inclusion of nonlinear functions of climate in the model framework improved the model for 79 % of the provinces. Therefore, climate forecast information could significantly contribute to a national dengue early warning system in Thailand.- In addition, the harvesting of beans, potato, pumpkin, ground nuts, and a combination of maize and millet 0–8 months prior were considered as potential explanatory variables. The best model for each response variable was selected using Akaike's Information Criterion (AIC) and an R 2 statistic was calculated from likelihood ratio statistics and only models with an R 2 > 0.30 were retained [40]. The relationship between flea abundance per host and environmental conditions was examined using the same methods.
[Show abstract] [Hide abstract] ABSTRACT: Background The distribution of human plague risk is strongly associated with rainfall in the tropical plague foci of East Africa, but little is known about how the plague bacterium is maintained during periods between outbreaks or whether environmental drivers trigger these outbreaks. We collected small mammals and fleas over a two year period in the West Nile region of Uganda to examine how the ecological community varies seasonally in a region with areas of both high and low risk of human plague cases.Methods Seasonal changes in the small mammal and flea communities were examined along an elevation gradient to determine whether small mammal and flea populations exhibit differences in their response to seasonal fluctuations in precipitation, temperature, and crop harvests in areas within (above 1300 m) and outside (below 1300 m) of a model-defined plague focus.ResultsThe abundance of two potential enzootic host species (Arvicanthis niloticus and Crocidura spp.) increased during the plague season within the plague focus, but did not show the same increase at lower elevations outside this focus. In contrast, the abundance of the domestic rat population (Rattus rattus) did not show significant seasonal fluctuations regardless of locality. Arvicanthis niloticus abundance was negatively associated with monthly precipitation at a six month lag and positively associated with current monthly temperatures, and Crocidura spp. abundance was positively associated with precipitation at a three month lag and negatively associated with current monthly temperatures. The abundance of A. niloticus and Crocidura spp. were both positively correlated with the harvest of millet and maize.Conclusions The association between the abundance of several small mammal species and rainfall is consistent with previous models of the timing of human plague cases in relation to precipitation in the West Nile region. The seasonal increase in the abundance of key potential host species within the plague focus, but not outside of this area, suggests that changes in small mammal abundance may create favorable conditions for epizootic transmission of Y. pestis which ultimately may increase risk of human cases in this region.- We tested for differences in movement rates related to PTT type (tag) through a separate set of models that used the Argos location data only and included the other covariates used in the activity and GPS movement rate analyses (Table 1). We calculated likelihood-ratio R 2 values (R 2 L R ; Magee 1990; Kramer 2005) in order to assess the variance explained by the best supported models in comparison to an intercept only model.
[Show abstract] [Hide abstract] ABSTRACT: Context. The potential for research methods to affect wildlife is an increasing concern among both scientists and the public. This topic has a particular urgency for polar bears because additional research is needed to monitor and understand population responses to rapid loss of sea ice habitat. Aims. This study used data collected from polar bears sampled in the Alaska portion of the southern Beaufort Sea to investigate the potential for capture to adversely affect behaviour and vital rates. We evaluated the extent to which capture, collaring and handling may influence activity and movement days to weeks post-capture, and body mass, body condition, reproduction and survival over 6 months or more. Methods. We compared post-capture activity and movement rates, and relationships between prior capture history and body mass, body condition and reproductive success. We also summarised data on capture-related mortality. Key results. Individual-based estimates of activity and movement rates reached near-normal levels within 2–3 days and fully normal levels within 5 days post-capture. Models of activity and movement rates among all bears had poor fit, but suggested potential for prolonged, lower-level rate reductions. Repeated captures was not related to negative effects on body condition, reproduction or cub growth or survival. Capture-related mortality was substantially reduced after 1986, when immobilisation drugs were changed, with only 3 mortalities in 2517 captures from 1987–2013. Conclusions. Polar bears in the southern Beaufort Sea exhibited the greatest reductions in activity and movement rates 3.5 days post-capture. These shorter-term, post-capture effects do not appear to have translated into any long-term effects on body condition, reproduction, or cub survival. Additionally, collaring had no effect on polar bear recovery rates, body condition, reproduction or cub survival. Implications. This study provides empirical evidence that current capture-based research methods do not have long-term implications, and are not contributing to observed changes in body condition, reproduction or survival in the southern Beaufort Sea. Continued refinement of capture protocols, such as the use of low-impact dart rifles and reversible drug combinations, might improve polar bear response to capture and abate short-term reductions in activity and movement post-capture.- Therefore, it is important to test whether there is a remaining contribution of age while entering both factors in describing variance in these performance measures. Class 1 only consists of members from the children group (n 5), whereas children are not part of Class 4. Because of this unbalanced design, we used mixed models (McCulloch & Searle, 2001, p. 358) in SPSS instead of standard ANOVAs for the following analyses of performance and their corresponding effect sizes: R LR 2 and by R LR 2 (repeated measures), which were based on likelihood ratio (see Kramer, 2005). In all of the following statistical analyses, the number of different feedback trials was modeled with a mixed model with feedback (5) as a repeated effect and age group (3) and class (4) as the fixed effects.
[Show abstract] [Hide abstract] ABSTRACT: Developmental differences in dorsolateral prefrontal cortex (DLPFC), anterior cingulate cortex (ACC), and superior parietal cortex (SPC) activation are associated with differences in how children, adolescents, and adults learn from performance feedback in rule-learning tasks (Crone, Zanolie, Leijenhorst, Westenberg, & Rombouts, 2008). Both maturational differences and performance differences can potentially explain variance in functional brain activation. To disentangle those effects, we established strategy differences in the performance of participants on the task of Crone et al. (2008) by the application of latent mixture models (McLachlan & Peel, 2000). We found 4 categorically different strategies, which were divided across age groups. Both adults and adolescents were distributed among all strategy groups except for the worst performing one, whereas children were distributed among all strategy groups except for the best performing one. Strategy use was a mediator and largely explained the relation between age and variance in activation patterns in the DLPFC and the SPC but not in the ACC. These findings are interpreted vis-à-vis age versus performance predictors of brain development. (PsycINFO Database Record (c) 2014 APA, all rights reserved).- The GA-MM methodology makes use of the Simple Genetic Algorithm (Table 1), completely analogous to GA-OLS, producing a ranking of variables by their frequency in a set S of GA solutions. However, there is no single commonly used definition for the R 2 statistic as is the case for OLS [16,17]. Several definitions have been suggested that all have different interpretations in the presence of correlated errors.
[Show abstract] [Hide abstract] ABSTRACT: Different high-dimensional regression methodologies exist for the selection of variables to predict a continuous variable. To improve the variable selection in case clustered observations are present in the training data, an extension towards mixed-effects modeling (MM) is requested, but may not always be straightforward to implement.In this article, we developed such a MM extension (GA-MM-MMI) for the automated variable selection by a linear regression based genetic algorithm (GA) using multi-model inference (MMI). We exemplify our approach by training a linear regression model for prediction of resistance to the integrase inhibitor Raltegravir (RAL) on a genotype-phenotype database, with many integrase mutations as candidate covariates. The genotype-phenotype pairs in this database were derived from a limited number of subjects, with presence of multiple data points from the same subject, and with an intra-class correlation of 0.92. In generation of the RAL model, we took computational efficiency into account by optimizing the GA parameters one by one, and by using tournament selection. To derive the main GA parameters we used 3 times 5-fold cross-validation. The number of integrase mutations to be used as covariates in the mixed effects models was 25 (chrom.size). A GA solution was found when R2MM > 0.95 (goal.fitness). We tested three different MMI approaches to combine the results of 100 GA solutions into one GA-MM-MMI model. When evaluating the GA-MM-MMI performance on two unseen data sets, a more parsimonious and interpretable model was found (GA-MM-MMI TOP18: mixed-effects model containing the 18 most prevalent mutations in the GA solutions, refitted on the training data) with better predictive accuracy (R2) in comparison to GA-ordinary least squares (GA-OLS) and Least Absolute Shrinkage and Selection Operator (LASSO). We have demonstrated improved performance when using GA-MM-MMI for selection of mutations on a genotype-phenotype data set. As we largely automated setting the GA parameters, the method should be applicable on similar datasets with clustered observations.
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