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Population pharmacokinetic—pharmacodynamic modelling of the relationship between testosterone and prostate specific antigen in patients with prostate cancer during treatment with leuprorelin. Pharmacokinetics and pharmacodynamics of voxelotor GBT in healthy adults and patients with sickle cell disease. Activity and mRNA expression levels of selected cytochromes P in various sections of the human small intestine.
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The role of smooth muscle cells in plaque stability: Therapeutic targeting potential Jennifer L. Harman Helle F. Sloeserwij A. Hazen D. Zwart A. Leendertse J. Examples of collinear variables include climatic data such as temperature and rainfall, and morphometric data such as body length and mass.
One problem with these methods though is that they rely on a user-selected choice of threshold of either the correlation coefficient or the VIF, and use of more stringent lower is probably sensible. Some argue that one should always prefer inspection of VIF values over correlation coefficients of raw predictors because strong multicollinearity can be hard to detect with the latter. Both approaches will only be applicable if it is possible to group explanatory variables by common features, thereby effectively creating broader, but still meaningful explanatory categories.
For more elaborate versions of these simulations, see Freckleton Two common transformations for continuous predictors are i predictor centering, the mean of predictor x is subtracted from every value in x, giving a variable with mean 0 and SD on the original scale of x ; and ii predictor standardising, where x is centred and then divided by the SD of x , giving a variable with mean 0 and SD 1.
Rescaling the mean of predictors containing large values e. Both approaches also remove the correlation between main effects and their interactions, making main effects more easily interpretable when models also contain interactions Schielzeth, Note that this collinearity among coefficients is distinct from collinearity between two separate predictors see above.
Centring and standardising by the mean of a variable changes the interpretation of the model intercept to the value of the outcome expected when x is at its mean value. Standardising further adjusts the interpretation of the coefficient slope for x in the model to the change in the outcome variable for a 1 SD change in the value of x.
Scaling is therefore a useful tool to improve the stability of models and likelihood of model convergence, and the accuracy of parameter estimates if variables in a model are on large e. When using scaling, care must be taken in the interpretation and graphical representation of outcomes. Further reading: Schielzeth provides an excellent reference to the advantages of centring and standardising predictors.
Gelman provides strong arguments for standardising continuous variables by 2 SDs when binary predictors are in the model. Once a global model is specified, it is vital to quantify model fit and report these metrics in the manuscript. Information criteria scores should not be used as a proxy for model fit, because a large difference in AIC between the top and null models is not evidence of a good fit. AIC tells us nothing about whether the basic distributional and structural assumptions of the model have been violated. Similarly, a high R 2 value is in itself only a test of the magnitude of model fit and not an adequate surrogate for proper model checks.
Just because a model has a high R 2 value does not mean it will pass checks for assumptions such as homogeneity of variance. We strongly encourage researchers to view model fit and model adequacy as two separate but equally important traits that must be assessed and reported. Here we discuss some key metrics of fit and adequacy that should be considered.
In addition, there are further model checks specific to mixed models.
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First, inspect residuals versus fitted values for each grouping level of a random intercept factor Zuur et al. Another feature of fit that is very rarely tested for in G LMMs is the assumption of normality of deviations of the conditional means of the random effects from the global intercept. Zuur et al. Models with a Gaussian Normal error structure do not require adjustment for overdispersion, as Gaussian models do not assume a specific mean-variance relationship. For generalized mixed models GLMMs , however e.
Poisson, Binomial , the variance of the data can be greater than predicted by the error structure of the model Hilbe, Overdispersion can be caused by several processes influencing data, including zero-inflation, aggregation non-independence among counts, or both Zuur et al. The presence of overdispersion in a model suggests it is a bad fit, and standard errors of estimates will likely be biased unless overdispersion is accounted for Harrison, The use of canonical binomial and Poisson error structures, when residuals are overdispersed, tends to result in Type I errors because standard errors are underestimated.
Adding an observation-level random effect OLRE to overdispersed Poisson or Binomial models can model the overdispersion and give more accurate estimates of standard errors Harrison, , Researchers very rarely report the overdispersion statistic but see Elston et al. Further reading: Crawley , pp. In a linear modelling context, R 2 gives a measure of the proportion of explained variance in the model, and is an intuitive metric for assessing model fit. Unfortunately, the issue of calculating R 2 for G LMMs is particularly contentious; whereas residual variance can easily be estimated for a simple linear model with no random effects and a Normal error structure, this is not the case for G LMMS.
See Harrison for a cautionary tale of how the GLMM R 2 functions are artificially inflated for overdispersed models. When models are too complex relative to the amount of data available, GLMM variance estimates can collapse to zero they cannot be negative, not to be confused with co variance estimates which can be negative. However, when comparing two models with the same random structure but different fixed effects, ML estimation cannot easily be avoided. If the model is a good fit, after a sufficiently large number of iterations e.
Simulating 10, datasets from our model reveals that the proportion of zeroes in our real data is comparable to simulated expectation Fig. Conversely, simulating 1, datasets and refitting our model to each dataset, we see that the sum of the squared Pearson residuals for the real data is far larger than simulated expectation Fig. The dispersion statistic for our model is 3. Thus, simulations have allowed us to conclude that our model is overdispersed, but that this overdispersion is not due to zero-inflation. Vertical red line shows the proportion of zeroes in our real dataset.
There is no strong evidence of zero-inflation for these data. B Histogram of the sum of squared Pearson residuals for 1, parametric bootstraps where the Poisson GLMM has been re-fitted to the data at each step. Vertical red line shows the test statistic for the original model, which lies well outside the simulated frequency distribution.
Simulating from models provides a simple yet powerful set of tools for assessing model fit and robustness. Rykiel discusses the need for validation of models in ecology. When collecting ecological data it is often not possible to measure all of the predictors of interest for every measurement of the dependant variable.
Incomplete rows of data in dataframes i. We discuss the relative merits of each approach briefly here, before expanding on the use of information-theory and multi-model inference in ecology. We note that these discussions are not meant to be exhaustive comparisons, and we encourage the reader to delve into the references provided for a comprehensive picture of the arguments for and against each approach. Evaluating whether a term should be dropped or not can be done using NHST to arrive at a model containing only significant predictors see Crawley, , or using IT to yield a model containing only terms that cause large increases in information criterion score when removed.
Stepwise selection using NHST is by far the most common variant of this approach, and so we focus on this method here. Stepwise deletion procedures have come under heavy criticism; they can overestimate the effect size of significant predictors Whittingham et al. It is common to present the MAM as if it arose from a single a priori hypothesis, when in fact arriving at the MAM required multiple significance tests Whittingham et al.
Perhaps most importantly, LRT can be unreliable for fixed effects in GLMMs unless both total sample size and replication of the random effect terms is high see Bolker et al. Global model reporting should not replace other model selection methods, but provides a robust measure of how likely significant effects are to arise by sampling variation alone. Stephens et al.
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Halsey et al. Unlike NHST, which leads to a focus on a single best model, model selection using IT approaches allows the degree of support in the data for several competing models to be ranked using metrics such as AIC. Information criteria attempt to quantify the Kullback—Leibler distance KLD , a measure of the relative amount of information lost when a given model approximates the true data-generating process.
Thus, relative difference among models in AIC should be representative in relative differences in KLD, and the model with the lowest AIC should lose the least information and be the best model in that it optimises the trade-off between fit and complexity Richards, We do not expand on the specific details of the difference between NHST and IT here, but point the reader to some excellent references on the topic.
Instead, we use this section to highlight recent empirical developments in the best practice methods for the application of IT in ecology and evolution. Further reading: Grueber et al. All-Subsets selection is the act of fitting a global model, often containing every possible interaction, and then fitting every possible nested model.
If adopting an all subsets approach, it is worth noting the number of models to consider increases exponentially with the number of predictors, where five predictors require 2 5 32 models to be fitted, whilst 10 predictors requires 1, models, both without including any interactions but including the null model. Global models should not contain huge numbers of variables and interactions without prior thought about what the models represent for a study system.
Therefore, best practice is to consider only a handful of hypotheses and then build a single statistical model to reflect each hypothesis.
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However, we argue all subsets selection may be sensible in a limited number of circumstances when testing causal relationships between explanatory variables and the response variable. For example, if the most complex model contains two main effects and their interaction, performing all subsets selection on that model is identical to building the five competing models including the null model nested in the global model, all of which may be considered likely to be supported by the data.
A small number of models built to reflect well-reasoned hypotheses are only valid if the predictors therein are not collinear see Collinearity section above. Several information criteria are available to rank competing models, but their calculations differ subtly. Note QAIC is not required if the overdispersion in the dataset has been modelled using zero-inflated models, OLREs, or compound probability distributions. Bolker et al.
Therefore, the choice between the two metrics is not straightforward, and may depend on the goal of the study i. Using high cut-offs is not encouraged, to avoid overly complex model sets containing uninformative predictors Richards, ; Grueber et al. Doing so greatly reduces the number of models to be used for inference, and improves parameter accuracy Arnold, ; Richards, Recent work has demonstrated that this approach is flawed because Akaike weights are interpreted as relative model probabilities, and give no information about the importance of individual predictors in a model Cade, , and fail to distinguish between variables with weak or strong effects Galipaud et al.
A better measure of variable importance would be to compare standardised effect sizes Schielzeth, ; Cade, The aim of model averaging is to incorporate the uncertainty in the size and presence of effects among a set of candidate models with similar support in the data. Model averaging using Akaike weights proceeds on the assumption that predictors are on common scales across models and are therefore comparable. Unfortunately, the nature of multiple regression means that the scale and sign of coefficients will change across models depending on the presence or absence of other variables in a focal model Cade, Cade recommends standardising model parameters based on partial standard deviations to ensure predictors are on common scales across models prior to model averaging details in Cade, We hope this article will act as both a guide, and as a gateway to further reading, for both new researchers and those wishing to update their portfolio of analytic techniques.
Here we distil our message into a bulleted list. Rigorous testing of both model fit R 2 and model adequacy violation of assumptions like homogeneity of variance must be carried out. We must recognise that satisfactory fit does not guarantee we have not violated the assumptions of LMM, and vice versa. Collinearity among predictors is difficult to deal with and can severely impair model accuracy. Be especially vigilant if data are from field surveys rather than controlled experiments, as collinearity is likely to be present.
When including a large number of predictors is necessary, backwards selection and NHST should be avoided, and ranking via AIC of all competing models is preferred. A critical question that remains to be addressed is whether model selection based on IT is superior to NHST even in cases of balanced experimental designs with few predictors.
Data simulation is a powerful but underused tool. If the analyst harbours any uncertainty regarding the fit or adequacy of the model structure, then the analysis of data simulated to recreate the perceived structure of the favoured model can provide reassurance, or justify doubt. Wherever possible, provide diagnostic assessment of model adequacy, and metrics of model fit, even if in the supplemental information.
This paper is the result of a University of Exeter workshop on best practice for the application of mixed effects models and model selection in ecological studies. Xavier A. Harrison was funded by an Institute of Zoology Research Fellowship. Beth S. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests Xavier A. Harrison is an Academic Editor for PeerJ. The authors declare no further competing interests. Author Contributions Xavier A. Lynda Donaldson conceived and designed the experiments, authored or reviewed drafts of the paper, approved the final draft.
Maria Eugenia Correa-Cano conceived and designed the experiments, authored or reviewed drafts of the paper, approved the final draft. Julian Evans conceived and designed the experiments, analysed the data, authored or reviewed drafts of the paper, approved the final draft. David N. Fisher conceived and designed the experiments, authored or reviewed drafts of the paper, approved the final draft. Cecily E. Goodwin conceived and designed the experiments, authored or reviewed drafts of the paper, approved the final draft. Robinson conceived and designed the experiments, authored or reviewed drafts of the paper, approved the final draft.
David J. Hodgson conceived and designed the experiments, authored or reviewed drafts of the paper, approved the final draft. Richard Inger conceived and designed the experiments, authored or reviewed drafts of the paper, approved the final draft. Data Availability The following information was supplied regarding data availability:. National Center for Biotechnology Information , U. Journal List PeerJ v. Published online May Fisher , 4, 6 Cecily E. Goodwin , 2 Beth S. Robinson , 2, 7 David J. Hodgson , 4 and Richard Inger 2, 4. Author information Article notes Copyright and License information Disclaimer.
Corresponding author. Harrison: ku. Received Jul 26; Accepted Apr This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. This article has been cited by other articles in PMC. Abstract The use of linear mixed effects models LMMs is increasingly common in the analysis of biological data. Understanding Fixed and Random Effects A key decision of the modelling process is specifying model predictors as fixed or random effects.
Open in a separate window. Figure 1. Controlling for non-independence among data points This is one of the most common uses of a random effect.
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Improving the accuracy of parameter estimation Random effect models use data from all the groups to estimate the mean and variance of the global distribution of group means. Estimating variance components In some cases, the variation among groups will be of interest to ecologists. Making predictions for unmeasured groups Fixed effect estimates prevent us from making predictions for new groups because the model estimates are only relevant to groups in our dataset Zuur et al. Considerations when Fitting Random Effects Random effect models have several desirable properties see above , but their use comes with some caveats.
Choosing random effects I: crossed or nested? Choosing random effects II: random slopes Fitting random slope models in ecology is not very common. Choosing fixed effect predictors and interactions One of the most important decisions during the modelling process is deciding which predictors and interactions to include in models. How complex should my global model be? Assessing predictor collinearity With the desired set of predictors identified, it is wise to check for collinearity among predictor variables. Figure 2. The effect of collinearity on model parameter estimates.
Quantifying GLMM fit and performance Once a global model is specified, it is vital to quantify model fit and report these metrics in the manuscript. Overdispersion Models with a Gaussian Normal error structure do not require adjustment for overdispersion, as Gaussian models do not assume a specific mean-variance relationship. R 2 In a linear modelling context, R 2 gives a measure of the proportion of explained variance in the model, and is an intuitive metric for assessing model fit.
Stability of variance components and testing significance of random effects When models are too complex relative to the amount of data available, GLMM variance estimates can collapse to zero they cannot be negative, not to be confused with co variance estimates which can be negative. Figure 3. Dealing with missing data When collecting ecological data it is often not possible to measure all of the predictors of interest for every measurement of the dependant variable.
Information-theory and multi-model inference Unlike NHST, which leads to a focus on a single best model, model selection using IT approaches allows the degree of support in the data for several competing models to be ranked using metrics such as AIC. Practical Issues with Applying Information Theory to Biological Data Using all-subsets selection All-Subsets selection is the act of fitting a global model, often containing every possible interaction, and then fitting every possible nested model.
Deciding which information criterion to use Several information criteria are available to rank competing models, but their calculations differ subtly. Model averaging when predictors are collinear The aim of model averaging is to incorporate the uncertainty in the size and presence of effects among a set of candidate models with similar support in the data.
Conclusion We hope this article will act as both a guide, and as a gateway to further reading, for both new researchers and those wishing to update their portfolio of analytic techniques. Acknowledgments This paper is the result of a University of Exeter workshop on best practice for the application of mixed effects models and model selection in ecological studies. Funding Statement Xavier A. References Aarts et al. Multilevel analysis quantifies variation in the experimental effect while optimizing power and preventing false positives. BMC Neuroscience. Allegue et al. Statistical Quantification of Individual Differences SQuID : an educational and statistical tool for understanding multilevel phenotypic data in linear mixed models.
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