Explore the vast range of titles available.

Understanding how both the mean (location) and variance (scale) of traits differ among species and lineages is fundamental to unveiling macroevolutionary patterns. Yet, traditional phylogenetic comparative methods primarily focus on modelling mean trait values, often overlooking variability and heteroscedasticity that can provide critical insights into evolutionary dynamics. Here, we introduce phylogenetic location–scale models (PLSMs), a novel framework that jointly analyses the evolution of trait means and variances. This dual approach captures heteroscedasticity and evolutionary changes in trait variability, allowing for the detection of clades with differing variances and revealing patterns of adaptation, diversification, and evolutionary constraints. Extending PLSMs to a multivariate context enables simultaneous analysis of multiple traits and their covariances, facilitating the testing of hypotheses about evolutionary trade‐offs, pleiotropy and phenotypic integration. By modelling covariances between phylogenetic effects in both the location and scale parts, we can discern whether changes in one trait's mean or variance are associated with changes in another's, thereby offering deeper insights into the mechanisms driving trait co‐evolution and co‐divergence or ‘contra‐divergence’. We also describe how an extended version of PLSMs incorporating within‐species variability can enhance our understanding of trait convergence and divergence arising from ecological and environmental factors. Our framework provides a powerful tool for exploring macroevolutionary patterns and can be used to reassess previously published comparative data, offering new insights into the mechanisms driving the diversity of life.

Across species, prenatal maternal stress has been shown to create heterogeneity in behavioural phenotypes. Research has recently highlighted that individuals vary in how predictable they are in their behavioural responses. This within-individual variation in behaviour is likely to be of biological importance, since individuals interact with the world not only through their mean behavioural phenotype, but also through their full range of behavioural variation. Yet, the underlying mechanisms that create and constrain between-individual variation in behavioural predictability remain largely unexplored. Here, we estimate whether experimental elevation of maternal corticosterone during egg laying (to model prenatal maternal stress) can cause variation in behavioural predictability in a population of chickens. Offspring’s behavioural predictability was quantified by testing them repeatedly (16 times) in a standard anxiety test (open-field test). Elevated maternal corticosterone resulted in less anxious and more predictable offspring compared to control offspring. These findings provide the first evidence that maternal corticosterone levels, via prenatal pathways, may influence multi-hierarchical behavioural plasticity by affecting both the magnitude and the predictability of behavioural responses. These results not only expand our current knowledge about the ways maternal stress can affect offspring’s behavioural phenotypes but also suggest a possible proximate mechanism underlying within-population variation in individual behavioural predictability.

Background There is growing attention to individuality in movement, its causes and consequences. Similarly to other well-established personality traits (e.g., boldness or sociability), conspecifics also differ repeatedly in their spatial behaviors, forming behavioral types (“spatial-BTs”). These spatial-BTs are typically described as the difference in the mean-level among individuals, and the intra-individual variation (IIV, i.e., predictability) is only rarely considered. Furthermore, the factors determining predictability or its ecological consequences for broader space-use patterns are largely unknown, in part because predictability was mostly tested in captivity (e.g., with repeated boldness assays). Here we test if (i) individuals differ in their movement and specifically in their predictability. We then investigate (ii) the consequences of this variation for home-range size and survival estimates, and (iii) the factors that affect individual predictability. Methods We tracked 92 barn owls ( Tyto alba ) with an ATLAS system and monitored their survival. From these high-resolution (every few seconds) and extensive trajectories (115.2 ± 112.1 nights; X̅ ± SD) we calculated movement and space-use indices (e.g., max-displacement and home-range size, respectively). We then used double-hierarchical and generalized linear mix-models to assess spatial-BTs, individual predictability in nightly max-displacement, and its consistency across time. Finally, we explored if predictability levels were associated with home-range size and survival, as well as the seasonal, geographical, and demographic factors affecting it (e.g., age, sex, and owls’ density). Results Our dataset (with 74 individuals after filtering) revealed clear patterns of individualism in owls’ movement. Individuals differed consistently both in their mean movement (e.g., max-displacement) and their IIV around it (i.e., predictability). More predictable individuals had smaller home-ranges and lower survival rates, on top and beyond the expected effects of their spatial-BT (max-displacement), sex, age and ecological environments. Juveniles were less predictable than adults, but the sexes did not differ in their predictability. Conclusion These results demonstrate that individual predictability may act as an overlooked axis of spatial-BT with potential implications for relevant ecological processes at the population level and individual fitness. Considering how individuals differ in their IIV of movement beyond the mean-effect can facilitate understanding the intraspecific diversity, predicting their responses to changing ecological conditions and their population management.

Increasing uniformity of traits is an important objective in livestock production. This study focused on the BWcomparison of a double hierarchical GLM (DHGLM) with the conventional analysis of uniformity, using within-litter variation in birth weight (BW0) in pigs as a case. In pigs, within-litter variation of BW0 is a trait in which uniformity is important in breeding practice. Traditionally, uniformity has been studied by analysis of SD or variances. In DHGLM, differences between animals are studied by analyzing the residual variance of the trait and estimating its variance components. Here we used data on BW0, recorded in 2 sow lines (Large White and Landrace), to compare the estimation of genetic parameters and breeding values for uniformity from DHGLM and traditional analysis of the variance. Comparison of DHGLM with the conventional analysis using the logarithm-transformed variance of BW0 was possible because both methods were on the same scale and the models contained the same random effects. In addition, the genetic CV at the residual SD level (GCV) was proposed as a measure expressing the potential response to selection. Three-fold cross-validation was performed to study predictive ability of both methods. The estimated GCV was highly similar using both methods. Results indicate that the SD of BW0 can be decreased by up to approximately 10% after 1 generation of selection, indicating good prospects for response to selection. The correlation between EBV (0.88 in both sow lines) obtained from both methods indicated high similarity between conventional analysis and DHGLM. Comparison of accuracies of EBV showed that the methods were comparable, with moderate accuracies achieved with approximately 100 piglets per maternal grandsire. Cross-validation also indicated very similar predictive ability in estimating EBV for BW0 variation for both methods. Therefore, it was concluded that conventional analysis and DHGLM produced highly comparable results. Still, the DHGLM potentially has a broader application than conventional analysis to study uniformity of traits, because it also can be used for traits with single observations per animal.

There is evidence for genetic variability in residual variance of livestock traits, which offers the potential for selection for increased uniformity of production. Different statistical approaches have been employed to study this topic; however, little is known about the concordance between them. The aim of our study was to investigate the genetic heterogeneity of residual variance on yearling weight (YW; 291.15 ± 46.67) in a Nellore beef cattle population; to compare the results of the statistical approaches, the two-step approach and the double hierarchical generalized linear model (DHGLM); and to evaluate the effectiveness of power transformation to accommodate scale differences. The comparison was based on genetic parameters, accuracy of EBV for residual variance, and cross-validation to assess predictive performance of both approaches. A total of 194,628 yearling weight records from 625 sires were used in the analysis. The results supported the hypothesis of genetic heterogeneity of residual variance on YW in Nellore beef cattle and the opportunity of selection, measured through the genetic coefficient of variation of residual variance (0.10 to 0.12 for the two-step approach and 0.17 for DHGLM, using an untransformed data set). However, low estimates of genetic variance associated with positive genetic correlations between mean and residual variance (about 0.20 for two-step and 0.76 for DHGLM for an untransformed data set) limit the genetic response to selection for uniformity of production while simultaneously increasing YW itself. Moreover, large sire families are needed to obtain accurate estimates of genetic merit for residual variance, as indicated by the low heritability estimates (<0.007). Box-Cox transformation was able to decrease the dependence of the variance on the mean and decreased the estimates of genetic parameters for residual variance. The transformation reduced but did not eliminate all the genetic heterogeneity of residual variance, highlighting its presence beyond the scale effect. The DHGLM showed higher predictive ability of EBV for residual variance and therefore should be preferred over the two-step approach.

The objective of this study was to estimate the genetic variance for within-litter variation of birth weight (BW0) using genomic (GRM) or pedigree relationship matrices (PRM) and to compare the accuracy of estimated breeding values (EBV) for within-litter variation of BW0 using GRM and PRM. The BW0 and residual variance of BW0 were modeled by the double hierarchical generalized linear model using GRM or PRM. Data came from 2 dam lines: Landrace and Large White. After editing, the data set in Landrace consisted of 748 sows with 1,938 litters and 29,430 piglets and in Large White of 989 sows with 3,320 litters and 51,818 piglets. To construct GRM, 46,466 (Landrace) and 44,826 (Large White) single nucleotide polymorphisms were used, whereas to construct PRM, 5 generations of pedigree were used. The accuracy of EBV with GRM was estimated with 8-fold cross-validation and compared to PRM. Estimated variance components were highly similar for GRM and PRM. The maternal genetic variance in residual variance of BW0 in Landrace was 0.05 with GRM and 0.06 with PRM. In Large White these were 0.04 with GRM and 0.05 with PRM. The genetic coefficient of variation (GCV SDe) was about 0.10 in both dam lines. This indicates a change of 10% in residual SD of BW0 when achieving a genetic response of 1 genetic standard deviation. The genetic correlation between birth weight and its residual variance was about 0.6 in both dam lines. The accuracies of selection for within-litter variation of birth weight were 0.35 with GRM and 0.23 with PRM in Landrace and 0.29 with GRM and 0.34 with PRM in Large White. In this case, using GRM did not significantly increase accuracies of selection. Results, however, show good opportunities to select for reduced within-litter variation of BW0. Genomic selection can increase accuracy of selection when reference populations contain at least 2,000 sows.

Random-effect models are becoming increasingly popular in the analysis of data. Lee and Nelder (2006) introduced double hierarchical generalized linear models (DHGLMs) in which not only the mean but also the residual variance (overdispersion) can be further modelled as random-effect models. In this article, we introduce DHGLMs that allow random-effect models for both the variances of random effects and the residual variance. We show how to use this general model class for the analysis of data and discuss how to select the best fitting model using the likelihood and various model-checking plots.

Background Genome-wide dense markers have been used to detect genes and estimate relative genetic values. Among many methods, Bayesian techniques have been widely used and shown to be powerful in genome-wide breeding value estimation and association studies. However, computation is known to be intensive under the Bayesian framework, and specifying a prior distribution for each parameter is always required for Bayesian computation. We propose the use of hierarchical likelihood to solve such problems. Results Using double hierarchical generalized linear models, we analyzed the simulated dataset provided by the QTLMAS 2010 workshop. Marker-specific variances estimated by double hierarchical generalized linear models identified the QTL with large effects for both the quantitative and binary traits. The QTL positions were detected with very high accuracy. For young individuals without phenotypic records, the true and estimated breeding values had Pearson correlation of 0.60 for the quantitative trait and 0.72 for the binary trait, where the quantitative trait had a more complicated genetic architecture involving imprinting and epistatic QTL. Conclusions Hierarchical likelihood enables estimation of marker-specific variances under the likelihoodist framework. Double hierarchical generalized linear models are powerful in localizing major QTL and computationally fast.

The variational properties of biological systems are an increasing focus of current research, and statistical methods are required for drawing inferences about the processes that determine them. Double‐hierarchical generalised linear models (DHGLM) are ideally suited for studying variational properties since they provide a direct way of modelling the distribution of variances. Although DHGLM have mainly been used to model heterogeneous residual variances over groups, models have been proposed that also allow heterogeneous random effect variances. However, these multi‐way DHGLM make the assumption that the residual variance of a group is independent of its random‐effect variance. Here, using a Bayesian approach, we extend multi‐way DHGLMs so that the correlation between residual‐ and random‐effect variances can be estimated. Using simulated data, the performance of the model is compared with the non‐DHGLM models that have traditionally been used to estimate such correlations. The proposed model is shown to perform well at estimating all model parameters, and in particular performs better than alternative models at estimating the correlation among variance components. Numerical analyses are complemented with theoretical work showing the expected bias when using non‐DHGLM models. In some cases, commonly used non‐DHGLM models are even expected to get the sign of the correlation wrong.

Conventional linear regression models assume homoscedastic error terms. This assumption often is violated in empirical applications. Various methods for evaluating the extent of such violations and for adjusting the estimated model parameters if necessary are generally available in books on regression methodology. Recent developments in statistics have taken a different approach by examining the data to ascertain whether the estimated heteroscedastic residuals (from a first-stage regression model of the conditional mean of an outcome variable as a function of a set of explanatory variables or covariates) are themselves systematically related to a set of explanatory variables in a second-stage regression. These extensions of the conventional models have been given various names but, most generally, are heteroscedastic regression models (HRMs). Instead of treating heteroscedasticity as a nuisance to be adjusted out of existence to reduce or eliminate its impact on regression model parameter estimates, the basic idea of HRMs is to model the heteroscedasticity itself. This chapter systematically reviews the specification of HRMs in both linear and generalized linear model forms, describes methods of estimation of such models, and reports empirical applications of the models to data on changes over recent decades in the US income distribution and in self-reported health/health disparities. A concluding section points to similarities and complementarities of the goals of the counterfactual approach to causal inference and heteroscedastic regression models.