Diagnostic tools for hierarchical linear models

Loy, Adam; Hofmann, Heike

doi:10.1002/wics.1238

Cited by 25 publications

(17 citation statements)

References 32 publications

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“…However, unlike standard linear models, the distributional assumptions in mixed-effects models need to be checked at multiple levels, including the distribution of random effect coefficients (Snijders & Bosker, 2011). Data points with high leverage on model estimates can also be an issue, but such leverage can be assessed with influence diagnosis tools (Demidenko & Stukel, 2005;Loy & Hofmann, 2013;Santos Nobre & Singer, 2007;Zare & Rasekh, 2011).…”

Section: Introductionmentioning

confidence: 99%

Robustness of linear mixed‐effects models to violations of distributional assumptions

et al. 2020

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Linear mixed‐effects models are powerful tools for analysing complex datasets with repeated or clustered observations, a common data structure in ecology and evolution. Mixed‐effects models involve complex fitting procedures and make several assumptions, in particular about the distribution of residual and random effects. Violations of these assumptions are common in real datasets, yet it is not always clear how much these violations matter to accurate and unbiased estimation. Here we address the consequences of violations in distributional assumptions and the impact of missing random effect components on model estimates. In particular, we evaluate the effects of skewed, bimodal and heteroscedastic random effect and residual variances, of missing random effect terms and of correlated fixed effect predictors. We focus on bias and prediction error on estimates of fixed and random effects. Model estimates were usually robust to violations of assumptions, with the exception of slight upward biases in estimates of random effect variance if the generating distribution was bimodal but was modelled by Gaussian error distributions. Further, estimates for (random effect) components that violated distributional assumptions became less precise but remained unbiased. However, this particular problem did not affect other parameters of the model. The same pattern was found for strongly correlated fixed effects, which led to imprecise, but unbiased estimates, with uncertainty estimates reflecting imprecision. Unmodelled sources of random effect variance had predictable effects on variance component estimates. The pattern is best viewed as a cascade of hierarchical grouping factors. Variances trickle down the hierarchy such that missing higher‐level random effect variances pool at lower levels and missing lower‐level and crossed random effect variances manifest as residual variance. Overall, our results show remarkable robustness of mixed‐effects models that should allow researchers to use mixed‐effects models even if the distributional assumptions are objectively violated. However, this does not free researchers from careful evaluation of the model. Estimates that are based on data that show clear violations of key assumptions should be treated with caution because individual datasets might give highly imprecise estimates, even if they will be unbiased on average across datasets.

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Section: Introductionmentioning

confidence: 99%

Robustness of linear mixed‐effects models to violations of distributional assumptions

et al. 2020

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“…Partial residuals are widely useful in detecting many types of problems, although several authors have pointed out that they are not without limitations (Mallows, 1986;Cook, 1993). Various extensions and modifications of partial residuals have been proposed, and there is an extensive literature on regression diagnostics (Belsley et al, 1980;Cook and Weisberg, 1982); indeed, many diagnostics are specific to the type of model (e.g., Pregibon, 1981;Grambsch and Therneau, 1994;Loy and Hofmann, 2013). Partial residuals are a useful, easily generalized idea that can applied to virtually any type of model although it is certainly worth being aware of other types of diagnostics that are specific to the modeling framework in question.…”

Section: Introductionmentioning

confidence: 99%

Visualization of Regression Models Using visreg

Breheny¹,

Burchett²

2017

The R Journal

937

600

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Regression models allow one to isolate the relationship between the outcome and an explanatory variable while the other variables are held constant. Here, we introduce an R package, visreg, for the convenient visualization of this relationship via short, simple function calls. In addition to estimates of this relationship, the package also provides pointwise confidence bands and partial residuals to allow assessment of variability as well as outliers and other deviations from modeling assumptions. The package provides several options for visualizing models with interactions, including lattice plots, contour plots, and both static and interactive perspective plots. The implementation of the package is designed to be fully object-oriented and interface seamlessly with R's rich collection of model classes, allowing a consistent interface for visualizing not only linear models, but generalized linear models, proportional hazards models, generalized additive models, robust regression models, and many more.

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“…The variance–covariance matrix of estimated contemporary group effects is unchanged when moving from the reduced model, (only contemporary group is fitted) compared to a full model where other fixed effects are fitted. To measure how close the model considered could come to such a state, the covariance ratio [ 26 ] was considered. The covariance ratio is the ratio of determinants for between a full and reduced model.…”

Section: Discussionmentioning

confidence: 99%

Estimation of genetic connectedness diagnostics based on prediction errors without the prediction error variance–covariance matrix

2017

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BackgroundAn important issue in genetic evaluation is the comparability of random effects (breeding values), particularly between pairs of animals in different contemporary groups. This is usually referred to as genetic connectedness. While various measures of connectedness have been proposed in the literature, there is general agreement that the most appropriate measure is some function of the prediction error variance–covariance matrix. However, obtaining the prediction error variance–covariance matrix is computationally demanding for large-scale genetic evaluations. Many alternative statistics have been proposed that avoid the computational cost of obtaining the prediction error variance–covariance matrix, such as counts of genetic links between contemporary groups, gene flow matrices, and functions of the variance–covariance matrix of estimated contemporary group fixed effects.ResultsIn this paper, we show that a correction to the variance–covariance matrix of estimated contemporary group fixed effects will produce the exact prediction error variance–covariance matrix averaged by contemporary group for univariate models in the presence of single or multiple fixed effects and one random effect. We demonstrate the correction for a series of models and show that approximations to the prediction error matrix based solely on the variance–covariance matrix of estimated contemporary group fixed effects are inappropriate in certain circumstances.ConclusionsOur method allows for the calculation of a connectedness measure based on the prediction error variance–covariance matrix by calculating only the variance–covariance matrix of estimated fixed effects. Since the number of fixed effects in genetic evaluation is usually orders of magnitudes smaller than the number of random effect levels, the computational requirements for our method should be reduced.Electronic supplementary materialThe online version of this article (doi:10.1186/s12711-017-0302-9) contains supplementary material, which is available to authorized users.

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Diagnostic tools for hierarchical linear models

Cited by 25 publications

References 32 publications

Robustness of linear mixed‐effects models to violations of distributional assumptions

Robustness of linear mixed‐effects models to violations of distributional assumptions

Visualization of Regression Models Using visreg

Estimation of genetic connectedness diagnostics based on prediction errors without the prediction error variance–covariance matrix

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