Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.     

Semiparametric bayesian inference for multilevel repeated measurement data

We discuss inference for data with repeated measurements at multiple levels. The motivating example is data with blood counts from cancer patients undergoing multiple cycles of chemotherapy, with days nested within cycles. Some inference questions relate to repeated measurements over days within cycle, while other questions are concerned with the dependence across cycles. When the desired inference relates to both levels of repetition, it becomes important to reflect the data structure in the model. We develop a semiparametric Bayesian modeling approach, restricting attention to two levels of repeated measurements. For the top-level longitudinal sampling model we use random effects to introduce the desired dependence across repeated measurements. We use a nonparametric prior for the random effects distribution. Inference about dependence across second-level repetition is implemented by the clustering implied in the nonparametric random effects model. Practical use of the model requires that the posterior distribution on the latent random effects be reasonably precise.     

Long-term behavioral repeatability in wild adult and captive juvenile turtles (Terrapene carolina): Implications for personality development

Animal personality can be defined as conspecific individuals consistently differing in behavioral tendencies. Personality is typically identified by behavioral repeatability, which occurs when within-individual variance is low relative to among-individual variance in the population. Intraspecific comparisons of behavioral repeatability in juveniles and adults within and across years are rare, but would be useful for testing hypotheses related to origins of animal personality and whether individuals exhibit stable or diverging behavior with ontogeny. To examine within- and across-year behavioral repeatability for eastern box turtles (Terrapene carolina), we assessed boldness (movement latency after brief confinement) of captive-born juveniles twice within three days when eight months old. We then repeated these tests for the same individuals one year later. Juveniles exhibited repeatable boldness within and across years. Although increasing body temperature was slightly associated with decreased movement latency, test year (1 or 2), or housing experience (being raised in an enriched or unenriched condition) had no effects on boldness. We also assessed across-year repeatability of boldness (head emergence from the shell after brief confinement) for wild adults at 1–3 year intervals. Adults also exhibited repeatable across-year boldness that was of similar magnitude to juveniles. We found no indication that sex class or whether adults had been radio-tracked influenced boldness. Our results suggest eastern box turtles demonstrate consistent individuality in boldness from an early age that is largely unaffected by temporal or environmental variation, and these behavioral differences can be maintained for multiple years in captivity and the wild, contrasting with theoretical expectations for personality development. These findings add to recent accumulating evidence demonstrating juvenile and adult box turtles exhibit multiple repeatable behaviors over the short- and long-term. We suggest this species is quickly gaining traction as a model organism for studying the proximate and ultimate causes of personality development within long-lived animals.     

Genetic parameters for first lactation test-day milk flow in Holstein cows

Genetic parameters for test-day milk flow (TDMF) of 2175 first lactations of Holstein cows were estimated using multiple-trait and repeatability models. The models included the direct additive genetic effect as a random effect and contemporary group (defined as the year and month of test) and age of cow at calving (linear and quadratic effect) as fixed effects. For the repeatability model, in addition to the effects cited, the permanent environmental effect of the animal was also included as a random effect. Variance components were estimated using the restricted maximum likelihood method in single- and multiple-trait and repeatability analyses. The heritability estimates for TDMF ranged from 0.23 (TDMF 6) to 0.32 (TDMF 2 and TDMF 4) in single-trait analysis and from 0.28 (TDMF 7 and TDMF 10) to 0.37 (TDMF 4) in multiple-trait analysis. In general, higher heritabilities were observed at the beginning of lactation until the fourth month. Heritability estimated with the repeatability model was 0.27 and the coefficient of repeatability for first lactation TDMF was 0.66. The genetic correlations were positive and ranged from 0.72 (TDMF 1 and 10) to 0.97 (TDMF 4 and 5). The results indicate that milk flow should respond satisfactorily to selection, promoting rapid genetic gains because the estimated heritabilities were moderate to high. Higher genetic gains might be obtained if selection was performed in the TDMF 4. Both the repeatability model and the multiple-trait model are adequate for the genetic evaluation of animals in terms of milk flow, but the latter provides more accurate estimates of breeding values.     

Value of repeated blood pressure measurements in children--the Brompton study.

Systolic blood pressure were measured in 1797 infants aged 4 days and then repeated at 6 weeks, 6 months, 1 year and then yearly until 4 years of age. The mean pressure rose from 76 mm Hg at 4 days to 96 mm Hg at 6 weeks but did not vary appreciably between subsequent measurements. Serial correlation coefficients of blood pressure adjusted for weight and degree of consciousness were calculated, comparing measurements at each age. At ages under 1 year the correlation coefficients were relatively weak, though most were significant (r < 0.2). As the children grew older these serial correlations became stronger, so that the correlation coefficient in blood pressure between ages 3 and 4 years was 0.47. These observations suggest that "tracking" for blood pressure starts at about 1 year and is much stronger by 4 years. Taken in conjunction with the findings of other long-term follow-up studies in older children, this suggests that children develop blood pressures indicative of their adult values between 1 and 4 years.     

Statistical analysis of test-day milk yields using random regression models for the comparison of feeding groups during the lactation period

Random regression models are widely used in the field of animal breeding for the genetic evaluation of daily milk yields from different test days. These models are capable of handling different environmental effects on the respective test day, and they describe the characteristics of the course of the lactation period by using suitable covariates with fixed and random regression coefficients. As the numerically expensive estimation of parameters is already part of advanced computer software, modifications of random regression models will considerably grow in importance for statistical evaluations of nutrition and behaviour experiments with animals. Random regression models belong to the large class of linear mixed models. Thus, when choosing a model, or more precisely, when selecting a suitable covariance structure of the random effects, the information criteria of Akaike and Schwarz can be used. In this study, the fitting of random regression models for a statistical analysis of a feeding experiment with dairy cows is illustrated under application of the program package SAS. For each of the feeding groups, lactation curves modelled by covariates with fixed regression coefficients are estimated simultaneously. With the help of the fixed regression coefficients, differences between the groups are estimated and then tested for significance. The covariance structure of the random and subject-specific effects and the serial correlation matrix are selected by using information criteria and by estimating correlations between repeated measurements. For the verification of the selected model and the alternative models, mean values and standard deviations estimated with ordinary least square residuals are used.     

Multivariate t linear mixed models for irregularly observed multiple repeated measures with missing outcomes

Missing outcomes or irregularly timed multivariate longitudinal data frequently occur in clinical trials or biomedical studies. The multivariate t linear mixed model (MtLMM) has been shown to be a robust approach to modeling multioutcome continuous repeated measures in the presence of outliers or heavy‐tailed noises. This paper presents a framework for fitting the MtLMM with an arbitrary missing data pattern embodied within multiple outcome variables recorded at irregular occasions. To address the serial correlation among the within‐subject errors, a damped exponential correlation structure is considered in the model. Under the missing at random mechanism, an efficient alternating expectation‐conditional maximization (AECM) algorithm is used to carry out estimation of parameters and imputation of missing values. The techniques for the estimation of random effects and the prediction of future responses are also investigated. Applications to an HIV‐AIDS study and a pregnancy study involving analysis of multivariate longitudinal data with missing outcomes as well as a simulation study have highlighted the superiority of MtLMMs on the provision of more adequate estimation, imputation and prediction performances.     

On the analysis of phylogenetically paired designs

As phylogenetically controlled experimental designs become increasingly common in ecology, the need arises for a standardized statistical treatment of these datasets. Phylogenetically paired designs circumvent the need for resolved phylogenies and have been used to compare species groups, particularly in the areas of invasion biology and adaptation. Despite the widespread use of this approach, the statistical analysis of paired designs has not been critically evaluated. We propose a mixed model approach that includes random effects for pair and species. These random effects introduce a “two-layer” compound symmetry variance structure that captures both the correlations between observations on related species within a pair as well as the correlations between the repeated measurements within species. We conducted a simulation study to assess the effect of model misspecification on Type I and II error rates. We also provide an illustrative example with data containing taxonomically similar species and several outcome variables of interest. We found that a mixed model with species and pair as random effects performed better in these phylogenetically explicit simulations than two commonly used reference models (no or single random effect) by optimizing Type I error rates and power. The proposed mixed model produces acceptable Type I and II error rates despite the absence of a phylogenetic tree. This design can be generalized to a variety of datasets to analyze repeated measurements in clusters of related subjects/species.     

Validation and utility of novel volume reduction technique for determination of parallel conductance

The parallel conductance volume, created by the conductivity of structures surrounding the ventricular blood pool, can be estimated by using a saline dilution technique. This paper examines the use of a novel volume reduction method, during a standard vena caval preload reduction maneuver, as an alternative to the routinely used saline dilution method to calibrate conductance catheter measurements in the left (LV) and right ventricle (RV) of animals and humans. The serial reproducibility of both methods was examined by measurement of percent difference, and by assessing the coefficient of repeatability 1) between two measurements within the same subject, 2) between the two techniques, and 3) interobserver variability. The effect of ventricular size and contractile state on the volume reduction technique was also observed. It was essential to ensure the technique was not affected by inotropic state. The volume reduction technique and saline dilution method were repeated at three different loading states (baseline, 5, and 10 microg x kg(-1) x min(-1) of dobutamine). The coefficient of repeatability between serial measurements was similar for both the volume reduction and saline dilution methods, and good interobserver variability was demonstrated. The volume reduction technique was compared with the saline dilution technique over a large range of ventricular sizes. No significant difference was observed in the RV or LV of adult humans or in the LV of neonatal pigs and children. There was no significant effect on either the saline dilution or the volume reduction technique as the inotropic state increased. In conclusion, the volume reduction technique is neither affected by ventricular size nor contractile state, is repeatable between different observers, and can be used to substitute the saline dilution method when preload reduction of the ventricle is being employed.     

An approach to the analysis of repeated measurements

A linear model for repeated measurements is proposed in which the correlation structure within each time sequence of measurements includes parameters for measurement error, variation between experimental units, and serial correlation within units. An approach to data analysis is presented which involves preliminary analysis by ordinary least squares, use of the empirical semi-variogram of residuals to suggest a suitable correlation structure, and formal inference using likelihood-based methods. Applications to two biological data sets are described.     

Weighted normality-based estimator in correcting correlation coefficient estimation between incomplete nutrient measurements

Consider the problem of estimating the correlation between two nutrient measurements, such as the percent energy from fat obtained from a food frequency questionnaire (FFQ) and that from repeated food records or 24-hour recalls. Under a classical additive model for repeated food records, it is known that there is an attenuation effect on the correlation estimation if the sample average of repeated food records for each subject is used to estimate the underlying long-term average. This paper considers the case in which the selection probability of a subject for participation in the calibration study, in which repeated food records are measured, depends on the corresponding FFQ value, and the repeated longitudinal measurement errors have an autoregressive structure. This paper investigates a normality-based estimator and compares it with a simple method of moments. Both methods are consistent if the first two moments of nutrient measurements exist. Furthermore, joint estimating equations are applied to estimate the correlation coefficient and related nuisance parameters simultaneously. This approach provides a simple sandwich formula for the covariance estimation of the estimator. Finite sample performance is examined via a simulation study, and the proposed weighted normality-based estimator performs well under various distributional assumptions. The methods are applied to real data from a dietary assessment study.     

Estimating genetic covariance functions assuming a parametric correlation structure for environmental effects

A random regression model for the analysis of "repeated" records in animal breeding is described which combines a random regression approach for additive genetic and other random effects with the assumption of a parametric correlation structure for within animal covariances. Both stationary and non-stationary correlation models involving a small number of parameters are considered. Heterogeneity in within animal variances is modelled through polynomial variance functions. Estimation of parameters describing the dispersion structure of such model by restricted maximum likelihood via an "average information" algorithm is outlined. An application to mature weight records of beef cow is given, and results are contrasted to those from analyses fitting sets of random regression coefficients for permanent environmental effects.     

Multivariate bernoulli mixture models with application to postmortem tissue studies in schizophrenia

A novel mixture model is presented for repeated measurements in which correlation among repeated observations on the same subject is induced via correlated unobservable component indicators. The mixture components in our model are linear regressions, and the mixing proportions are logits with random effects. Inference is facilitated by sampling from the posterior distribution of the parameters via Markov chain Monte Carlo methods. The model is applied to a neuronal postmortem brain tissue study to examine the differences in neuron volumes between schizophrenic and control subjects.     

Repeatability of nest predation in passerines depends on predator species and time scale

It has been proposed that some specific locations of bird's nests have higher intrinsic chances of being depredated than other locations. This predicts that fates of consecutive nesting attempts at the same site should be repeatable. We used 20 pairs of old thrush nests to simulate repeated nesting attempts at the same sites, both within and between breeding seasons (n=40  sites×2  trials×2  years=160). Each nest was monitored by a camera to record multiple predation events and to identify predators. Predation by all predator species was repeatable during a 15-day trial. Predation by principal predators (jay Garrulus glandarius , marten Martes martes / foina ) and total predation (all species combined) was not correlated within pairs of simultaneously exposed nests or within samples of nests from particular study plot, and not repeatable for individual nests between-trials or between-years. These findings suggest short-term effect of predator memory causing revisitation of previously depredated nests during a current nesting trial (all predators); do not support an effect of nest site features on multiple nest discoveries and/or an effect of nest location on repeated random encounters with the same nest (principal predators). Long-term repeatability and correlation within pairs of simultaneously exposed nests was detectable only in occasional predators (great spotted woodpecker Dendrocopos major , possibly also squirrel Sciurus vulgaris ), which suggests effect of nest location combined with site fidelity and individual foraging specialization of these predators. We conclude that repeatability of nest predation depends on the time scale considered and the local predator community. We caution against spurious findings of repeatable nest predation resulting simply from statistical properties of correlation in binary data (nest fates).     

Maximin D-optimal designs for longitudinal mixed effects models

In this article, the optimal selection and allocation of time points in repeated measures experiments is considered. D-optimal cohort designs are computed numerically for the first- and second-degree polynomial models with random intercept, random slope, and first-order autoregressive serial correlations. Because the optimal designs are locally optimal, it is proposed to use a maximin criterion. It is shown that, for a large class of symmetric designs, the smallest relative efficiency over the model parameter space is substantial.     

Environmental and genetic variation factors of artificial insemination success in French dairy sheep

Artificial inseminations (n = 678 168) recorded during 5 years in five French artificial insemination (AI) centres (2 'Lacaune', 1 'Manech tête rousse', 1 'Manech tête noire' and 1 'Basco béarnaise') were analysed to determine environmental and genetic factors affecting the insemination results. Analyses within centre-breed were performed using a linear model, which jointly estimates male and female fertility. This model combined four categories of data: the environmental effects related to the female, those related to the male, the non-sex-specific effects and finally the pedigree data of these males and females. After selection, the environmental female effects considered were age, synchronisation (0/1) on the previous year, total number of synchronisations during the female reproductive life, time interval between previous lambing and insemination, already dry or still lactating (0/1) when inseminated, and milk quantity produced during the previous year expressed as quartiles intra herd * year. The environmental male effects were motility and concentration of the semen. The non-sex-specific effects were the inseminator, the interaction herd * year nested within the inseminator, considered as random effects and the interaction year * season considered as a fixed effect. The main variation factors of AI success were relative to non-sex-specific effects and to female effects. Heritability estimates varied from 0.001 to 0.005 for male fertility and from 0.040 to 0.078 for female fertility. Repeatability estimates varied from 0.007 to 0.015 for male fertility and from 0.104 to 0.136 for female fertility. These parameters indicate that genetic improvement of AI results through a classical polygenic selection would be difficult. Moreover, in spite of the large quantity of variation factors fitted by the joint model, a very large residual variance remained unexplained.     

Statistical analysis of test-day milk yields using random regression models for the comparison of feeding groups during the lactation period

Abstract

Random regression models are widely used in the field of animal breeding for the genetic evaluation of daily milk yields from different test days. These models are capable of handling different environmental effects on the respective test day, and they describe the characteristics of the course of the lactation period by using suitable covariates with fixed and random regression coefficients. As the numerically expensive estimation of parameters is already part of advanced computer software, modifications of random regression models will considerably grow in importance for statistical evaluations of nutrition and behaviour experiments with animals. Random regression models belong to the large class of linear mixed models. Thus, when choosing a model, or more precisely, when selecting a suitable covariance structure of the random effects, the information criteria of Akaike and Schwarz can be used. In this study, the fitting of random regression models for a statistical analysis of a feeding experiment with dairy cows is illustrated under application of the program package SAS. For each of the feeding groups, lactation curves modelled by covariates with fixed regression coefficients are estimated simultaneously. With the help of the fixed regression coefficients, differences between the groups are estimated and then tested for significance. The covariance structure of the random and subject-specific effects and the serial correlation matrix are selected by using information criteria and by estimating correlations between repeated measurements. For the verification of the selected model and the alternative models, mean values and standard deviations estimated with ordinary least square residuals are used.     

Use of quasi-least squares to adjust for two levels of correlation

This article considers data with two levels of association. Our motivating example is an international intervention trial with repeated observations on subjects who reside within geographically defined clusters. To account for the potential correlation within clusters and within the repeated measurements that pertain to each subject, we apply a method based on generalized estimating equations for a correlation structure proposed by Lefkopoulou, Moore, and Ryan (1989, Journal of the American Statistical Association 84, 810-815).     

Cross-validation analysis for genetic evaluation models for ranking in endurance horses

Ranking trait was used as a selection criterion for competition horses to estimate racing performance. In the literature the most common approaches to estimate breeding values are the linear or threshold statistical models. However, recent studies have shown that a Thurstonian approach was able to fix the race effect (competitive level of the horses that participate in the same race), thus suggesting a better prediction accuracy of breeding values for ranking trait. The aim of this study was to compare the predictability of linear, threshold and Thurstonian approaches for genetic evaluation of ranking in endurance horses. For this purpose, eight genetic models were used for each approach with different combinations of random effects: rider, rider–horse interaction and environmental permanent effect. All genetic models included gender, age and race as systematic effects. The database that was used contained 4065 ranking records from 966 horses and that for the pedigree contained 8733 animals (47% Arabian horses), with an estimated heritability around 0.10 for the ranking trait. The prediction ability of the models for racing performance was evaluated using a cross-validation approach. The average correlation between real and predicted performances across genetic models was around 0.25 for threshold, 0.58 for linear and 0.60 for Thurstonian approaches. Although no significant differences were found between models within approaches, the best genetic model included: the rider and rider–horse random effects for threshold, only rider and environmental permanent effects for linear approach and all random effects for Thurstonian approach. The absolute correlations of predicted breeding values among models were higher between threshold and Thurstonian: 0.90, 0.91 and 0.88 for all animals, top 20% and top 5% best animals. For rank correlations these figures were 0.85, 0.84 and 0.86. The lower values were those between linear and threshold approaches (0.65, 0.62 and 0.51). In conclusion, the Thurstonian approach is recommended for the routine genetic evaluations for ranking in endurance horses.     

Genotype by environment interaction for Holstein cattle populations using autoregressive and within- and across-country multi-trait reaction norms test-day models

The progenies of international bulls in diverse climatic conditions and management levels may lead to different expressions of their genetic potential resulting in a re-ranking of these bulls. Therefore, evaluate the presence of genotype by environment interaction (G × E) within and across countries is important to guide the decision-making on alternative selection strategies. Thus, a two-step reaction norm (RN) approach was used to investigate the presence of G × E in Portuguese and Brazilian Holstein cattle. In step 1, we performed a within-country genetic evaluation using an autoregressive model to obtain precorrected phenotypes and environmental gradients (herd test-day solutions, HTD levels). In step 2, the precorrected phenotypes were considered as two distinct traits in a bi-trait RN model to estimate variance components across HTD levels, genetic correlation between HTD levels in Portugal and Brazil, and RN of the estimated breeding values. Additionally, the genetic correlation between countries using a bi-trait random regression (RR) sire model was obtained. In step 1, genetic additive variance for milk yield (MY) in Portugal was 14.1% higher than in Brazil. For somatic cell score (SCS), the genetic additive variance in Portugal was 12.7% lower than in Brazil. Although similar heritability estimates for SCS were observed in both countries, MY heritabilities were 0.31 for Portugal and 0.23 for Brazil. Genetic correlations (SD) between both countries obtained using RR sire model were 0.78 (0.051) for MY and 0.75 (0.062) for SCS. In step 2, MY genetic correlations among HTD levels within countries were higher than 0.94 for Portugal and 0.98 for Brazil. Somatic cell score genetic correlations among HTD levels ranged from 0.70 to 0.99 for Portugal and from 0.84 to 0.99 for Brazil. The average (SD) of genetic correlation estimates between Portuguese and Brazilian HTD levels were 0.74 (0.009) for MY and 0.57 (0.060) for SCS. These results suggest the presence of G × E for MY and SCS of Holstein cattle between both countries. Although there was no indication of G × E between Brazilian herd environments, the low genetic correlation for SCS indicates potential re-ranking of bulls between extreme environmental gradient in Portugal. Overall, the results of this study evidence the importance of national and international genetic evaluation systems to assist dairy farmers in the selection of the best genotypes to obtain the expected returns from investments in imported semen and to realize genetic progress in dairy populations under local environmental conditions.