Group-level variance estimates of zero often arise when fitting multilevel or hierarchical linear models, especially when the number of groups is small. For situations where zero variances are implausible a priori, we propose a maximum penalized likelihood approach to avoid such boundary estimates. This approach is equivalent to estimating variance parameters by their posterior mode, given a weakly informative prior distribution. By choosing the penalty from the log-gamma family with shape parameter greater than 1, we ensure that the estimated variance will be positive. We suggest a default log-gamma(2,λ) penalty with λ→0, which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero, thus remaining consistent with the data while being nondegenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior.

Our default method provides better estimates of model parameters and standard errors than the maximum likelihood or the restricted maximum likelihood estimators. The log-gamma family can also be used to convey substantive prior information. In either case—pure penalization or prior information—our recommended procedure gives nondegenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.

Repeated measures designs in psychology have traditionally been analyzed by the univariate mixed model approach, in which the repeated measures effect is tested against an error term based on the subject by treatment interaction. This paper considers the extension of this analysis to designs in which the individual repeated measures are multivariate. Sufficient conditions for a valid multivariate mixed model analysis are given, and a test is described to determine whether or not given data satisfy these conditions.

When using linear models for cluster-correlated or longitudinal data, a common modeling practice is to begin by fitting a relatively simple model and then to increase the model complexity in steps. New predictors might be added to the model, or a more complex covariance structure might be specified for the observations. When fitting models for binary or ordered-categorical outcomes, however, comparisons between such models are impeded by the implicit rescaling of the model estimates that takes place with the inclusion of new predictors and/or random effects. This paper presents an approach for putting the estimates on a common scale to facilitate relative comparisons between models fit to binary or ordinal outcomes. The approach is developed for both population-average and unit-specific models.

Circular data originates in a wide range of scientific fields and can be analyzed on the basis of directional statistics and special distributions wrapped around the circumference. However, both propensity to transform non-linear to linear data and complexity of directional statistics limited the generalization of the circular paradigm in the animal breeding framework, among others. Here, we generalized a circular mixed (CM) model within the context of Bayesian inference. Three different parametrizations with different hierarchical structures were developed on basis of the von Mises distribution; moreover, both goodness of fit and predictive ability from each parametrization were compared through the analyses of 110 116 lambing distribution records collected from Ripollesa sheep herds between 1976 and 2017. The naive circular (NC) model only accounted for population mean and homogeneous circular variance, and reached the lowest goodness-of-fit and predictive ability. The CM model assumed a hierarchical structure for the population mean by accounting for systematic (ewe age and lambing interval) and permanent environmental sources of variation (flock-year-season and ewe). This improved goodness of fit by reducing both the deviance information criterion (DIC; −2520 units) and the mean square error (MSE; −12.4%) between simulated and predicted lambing data when compared against the NC model. Finally, the last parametrization expanded CM model by also assuming a hierarchical structure with systematic and permanent environmental factors for the variance parameter of the von Mises distribution (i.e. circular canalization (CC) model). This last model reached the best goodness of fit to lambing distribution data with a DIC estimate 5425 units lower than the one for NC model (MSE reduced 13.2%). The same pattern revealed when models were compared in terms of predictive ability. The superiority revealed by CC model emphasized the relevance of heteroskedasticity for the analysis of lambing distribution in the Ripollesa breed, and suggested potential applications for the sheep industry, even genetic selection for canalization. The development of CM models on the basis of the von Mises distribution has allowed to integrate flexible hierarchical structures accounting for different sources of variation and affecting both mean and dispersion terms. This must be viewed as a useful statistical tool with multiple applications in a wide range of research fields, as well as the livestock industry. The next mandatory step should be the inclusion of genetic terms in the hierarchical structure of the models in order to evaluate their potential contribution to current selection programs.

The current study investigated the influence of selection for body weight (BW) on growth curve parameters in two lines of Japanese quail through a mixed model approach. Live BWs of 1400 Japanese quail were recorded at 3-day intervals from hatching to 42 days of age. Birds were distributed equally across lines (selected and control) and sexes (male and female). The asymptotic weight parameter (β0) values were always higher in Gompertz than Richards models in both lines. The values of β0 were higher in the selected than control lines and in females than males across models. Differences were found in the inflection point for age and weight across lines and sexes. Values of the growth rate parameter (β2) ranged from 0.06 to 0.10 in both models, favouring males over females in both lines. Lower weights at the inflection point of both models were observed in the control line. Determination coefficient (R2) of both models in different genetic groups and sexes was similar. Mean square error (MSE) values of the Gompertz model were lower for females in selected v. control lines. In contrast, MSE values of the Richards model were lower for selected males v. control males. According to the criteria of choice (highest R2 and lowest MSE, Akaike information criterion and Bayesian information criterion), the Richards model was considered the best fitting model for the growth data of males, while the Gompertz model was the best for growth data of female quails in both lines.

The ability to properly assess and accurately phenotype true differences in feed efficiency among dairy cows is key to the development of breeding programs for improving feed efficiency. The variability among individuals in feed efficiency is commonly characterised by the residual intake approach. Residual feed intake is represented by the residuals of a linear regression of intake on the corresponding quantities of the biological functions that consume (or release) energy. However, the residuals include both, model fitting and measurement errors as well as any variability in cow efficiency. The objective of this study was to isolate the individual animal variability in feed efficiency from the residual component. Two separate models were fitted, in one the standard residual energy intake (REI) was calculated as the residual of a multiple linear regression of lactation average net energy intake (NEI) on lactation average milk energy output, average metabolic BW, as well as lactation loss and gain of body condition score. In the other, a linear mixed model was used to simultaneously fit fixed linear regressions and random cow levels on the biological traits and intercept using fortnight repeated measures for the variables. This method split the predicted NEI in two parts: one quantifying the population mean intercept and coefficients, and one quantifying cow-specific deviations in the intercept and coefficients. The cow-specific part of predicted NEI was assumed to isolate true differences in feed efficiency among cows. NEI and associated energy expenditure phenotypes were available for the first 17 fortnights of lactation from 119 Holstein cows; all fed a constant energy-rich diet. Mixed models fitting cow-specific intercept and coefficients to different combinations of the aforementioned energy expenditure traits, calculated on a fortnightly basis, were compared. The variance of REI estimated with the lactation average model represented only 8% of the variance of measured NEI. Among all compared mixed models, the variance of the cow-specific part of predicted NEI represented between 53% and 59% of the variance of REI estimated from the lactation average model or between 4% and 5% of the variance of measured NEI. The remaining 41% to 47% of the variance of REI estimated with the lactation average model may therefore reflect model fitting errors or measurement errors. In conclusion, the use of a mixed model framework with cow-specific random regressions seems to be a promising method to isolate the cow-specific component of REI in dairy cows.

In recent years germination experiments have become more and more complex. Typically, they are replicated in time as independent runs and at each time point they involve hierarchical, often factorial experimental designs, which are now commonly analysed by means of linear mixed models. However, in order to characterize germination in response to time elapsed, specific event-time models are needed and mixed model extensions of these models are not readily available, neither in theory nor in practice. As a practical workaround we propose a two-step approach that combines and weighs together results from event-time models fitted separately to data from each germination test by means of meta-analytic random effects models. We show that this approach provides a more appropriate appreciation of the sources of variation in hierarchically structured germination experiments as both between- and within-experiment variation may be recovered from the data.

The replication of experiments over multiple environments such as locations and years is a common practice in field research. A major reason for the practice is to estimate the effects of treatments over a variety of environments. Environments are frequently classed as random effects in the model for statistical analysis, while treatments are almost always classed as fixed effects. Where environments are random and treatments are fixed, it is not always necessary to include all possible interactions between treatments and environments as random effects in the model. The rationale for decisions about the inclusion or exclusion of fixed by random effects in a mixed model is presented. Where the effects of treatments over broad populations of environments are to be estimated, it is often most appropriate to include only those fixed by random effects that reference experimental units.

There has been ongoing interest in studying wine judges' performance in evaluating wines. Most of the studies have reached a similar conclusion: a significant lack of consensus exists in wine quality ratings. However, a few studies, to the author's knowledge, have provided direct quantification of how much consensus (as opposed to randomness) exists in wine ratings. In this paper, a permutation-based mixed model is proposed to quantify randomness versus consensus in wine ratings. Specifically, wine ratings under the condition of randomness are generated with a permutation method, and wine ratings under the condition of consensus can be produced by sorting the ratings for each judge. Then the observed wine ratings are modeled as a mixture of ratings under randomness and ratings under consensus. This study shows that the model can provide excellent model fit, which indicates that wine ratings, indeed, consist of a mixture of randomness and consensus. A direct measure is easily computed to quantify randomness versus consensus in wine ratings. The method is demonstrated with data analysis from a major wine competition and a simulation study. (JEL Classifications: C10, C13, C15)

The mean or average value of a measure or indicator variable used to monitor a fish population can be defined in various ways, each with a correspondingly different statistical estimator for use in the context of a trawl survey. When, as is typical for many species, fish are heterogeneously clustered over space, then scooped in clusters from restricted localities using a trawl, these different estimators can produce different sample mean values with contrasting variations over time, possibly leading to different inferences about the fish population. Two mean parameters and their intuitive estimators, the mean “over fish”, and the mean “over stations”, are discussed and found to present contrasting statistical properties. A third estimator based on fitting a mixed model is proposed which has intermediate properties based on the within-haul correlation. The three estimators are applied illustratively to length data for cod caught in the North Sea by the English groundfish survey from 1992 to 2007. The time series of the mean over stations was smoothest, that for the mean over fish much more variable, and the mixed mean fell between in all years. Variance estimators derived from the fitted mixed model are also put forward. Estimates made from the example suggested that the mixed mean is most efficient. The type of estimator used for the mean should always be considered carefully and mentioned when reporting indicator studies.