Bayesian Threshold Model Research Articles

A threshold model for respondent heterogeneity

Continuous models of respondent heterogeneity assume the existence of a response function where variables of interest are continuously related to explanatory variables. In many situations this assumption may not be true. In this paper we propose an approach of modeling respondent heterogeneity that identifies abrupt changes in the distribution of response coefficients around a threshold specification. Our model differs from traditional threshold models by introducing the threshold effect to describe across-unit behavior as opposed to within-unit behavior. We illustrate our proposed Bayesian threshold model for survey data from a large national retail bank that examines the effects of service wait times on customer satisfaction. We find evidence of a threshold effect where long in-process wait times are associated with bank branches characterized by weak associations between service quality drivers and overall perceptions of service quality. Branches with wait times below the threshold are found to have much stronger associations.

A Threshold Model for Respondent Heterogeneity

Continuous models of respondent heterogeneity assume the existence of a response function where variables of interest are continuously related to explanatory variables. In many situations this assumption may not be true. In this paper we propose an approach of modeling respondent heterogeneity that identifies abrupt changes in the distribution of response coefficients around a threshold specification. Our model differs from traditional threshold models by introducing the threshold effect to describe across-unit behavior as opposed to within unit behavior. We illustrate our proposed Bayesian threshold model for survey data from a large national retail bank that examines the effects of service wait times on customer satisfaction. We find evidence of a threshold effect where long in-process wait times are associated with bank branches characterized by weak associations between service quality drivers and overall perceptions of service quality. Branches with wait times below the threshold are found to have much stronger associations.

The relevance of purebred information for predicting genetic merit of survival at birth of crossbred piglets1

The objective of this study was to infer (co)variance components for piglet survival at birth in purebred and crossbred pigs. Data were from 13,643 (1,213 litters) crossbred and 30,919 (3,162 litters) purebred pigs, produced by mating the same 168 purebred boars to 460 Large White-derived crossbred females and 1,413 purebred sows, respectively. The outcome variable was piglet survival at birth as a binary trait. A Bayesian bivariate threshold model was implemented via Gibbs sampling. Flat priors were assigned to the effects of sex, parity of the dam, litter size, and year-month of birth. Gaussian priors were assigned to litter, dam, and sire effects. Marginal posterior means (SD) of the sire and dam variances for liability of piglet survival in purebred were 0.018 (0.008) and 0.077 (0.020), respectively. For crossbred, sire and dam variance estimates were 0.030 (0.018) and 0.120 (0.034), respectively. The posterior means (SD) of the heritability of liability of survival in purebred and crossbred and of the genetic correlation between these traits were 0.049 (0.023), 0.091 (0.054), and 0.248 (0.336), respectively. The greatest 95% confidence region (-0.406, 0.821) for the genetic correlation between purebred and crossbred liabilities of piglet survival included zero. Results suggest that the expected genetic progress for piglet survival in crossbreds when selection is based on purebred information may be nil.

Bayesian threshold analysis of direct and maternal genetic parameters for piglet mortality at farrowing in Large White, Landrace, and Pietrain populations1

A Bayesian threshold model was fitted to analyze the genetic parameters for farrowing mortality at the piglet level in Large White, Landrace, and Pietrain populations. Field data were collected between 1999 and 2006. They were provided by 3 pig selection nucleus farms of a commercial breeding company registered in the Spanish Pig Data Bank (BDporc). Analyses were performed on 3 data sets of Large White (60,535 piglets born from 4,551 litters), Landrace (57,987 piglets from 5,008 litters), and Pietrain (42,707 piglets from 4,328 litters) populations. In the analysis, farrowing mortality was considered as a binary trait at the piglet level and scored as 1 (alive piglet) or 0 (dead piglet) at farrowing or within the first 12 h of life. Each breed was analyzed separately, and operational models included systematic effects (year-season, sex, litter size, and order of parity), direct and maternal additive genetic effects, and common litter effects. Analyses were performed by Bayesian methods using Gibbs sampling. The posterior means of direct heritability were 0.02, 0.06, and 0.10, and the posterior means of maternal heritability were 0.05, 0.13, and 0.06 for Large White, Landrace, and Pietrain populations, respectively. The posterior means of genetic correlation between the direct and maternal genetic effects for Landrace and Pietrain populations were -0.56 and -0.53, and the highest posterior intervals at 95% did not include zero. In contrast, the posterior mean of the genetic correlation between direct and maternal effects was 0.15 in the Large White population, with the null correlation included in the highest posterior interval at 95%. These results suggest that the genetic model of evaluation for the Landrace and Pietrain populations should include direct and maternal genetic effects, whereas farrowing mortality could be considered as a sow trait in the Large White population.

Variance Component and Breeding Value Estimation for Reproductive Traits in Laying Hens Using a Bayesian Threshold Model

Variance components and breeding values were estimated for 3 reproductive traits in a pure line of White Leghorn laying hens. The traits were proportion of fertile eggs of eggs set, proportion of first quality chicks of eggs set, and proportion of first-quality chicks of fertile eggs. A total of 3,020 hens were tested up to 3 times over a period of 7 d. For the definition of the traits, each egg was scored for each trait either as 0 or 1. To account for the binomial distribution of the data, a Bayesian animal threshold model implemented in a Gibbs sampler was applied that considered the record of each egg set as a repeated observation of the hen. The estimated heritability was 0.067, 0.126, and 0.136 for the traits proportion of fertile eggs of eggs set, proportion of first quality chicks of eggs set, and proportion of first-quality chicks of fertile eggs, respectively, and the SE were small. It was shown that the heritability estimates were substantially higher from their expected values based on linear models. This results in a higher genetic progress and consequently favors the applied Bayesian threshold model for a genetic evaluation of binomial distributed reproductive traits.

Complex Segregation Analysis of Categorical Traits in Farm Animals: Comparison of Linear and Threshold Models

Main objectives of this study were to investigate accuracy, bias and power of linear and threshold model segregation analysis methods for detection of major genes in categorical traits in farm animals. Maximum Likelihood Linear Model (MLLM), Bayesian Linear Model (BALM) and Bayesian Threshold Model (BATM) were applied to simulated data on normal, categorical and binary scales as well as to disease data in pigs. Simulated data on the underlying normally distributed liability (NDL) were used to create categorical and binary data. MLLM method was applied to data on all scales (Normal, categorical and binary) and BATM method was developed and applied only to binary data. The MLLM analyses underestimated parameters for binary as well as categorical traits compared to normal traits; with the bias being very severe for binary traits. The accuracy of major gene and polygene parameter estimates was also very low for binary data compared with those for categorical data; the later gave results similar to normal data. When disease incidence (on binary scale) is close to 50%, segregation analysis has more accuracy and lesser bias, compared to diseases with rare incidences. NDL data were always better than categorical data. Under the MLLM method, the test statistics for categorical and binary data were consistently unusually very high (while the opposite is expected due to loss of information in categorical data), indicating high false discovery rates of major genes if linear models are applied to categorical traits. With Bayesian segregation analysis, 95% highest probability density regions of major gene variances were checked if they included the value of zero (boundary parameter); by nature of this difference between likelihood and Bayesian approaches, the Bayesian methods are likely to be more reliable for categorical data. The BATM segregation analysis of binary data also showed a significant advantage over MLLM in terms of higher accuracy. Based on the results, threshold models are recommended when the trait distributions are discontinuous. Further, segregation analysis could be used in an initial scan of the data for evidence of major genes before embarking on molecular genome mapping.

순서범주형자료 분석을 위한 베이지안 분계점 모형

A Bayesian threshold model is considered to analyze binary or ordered categorical traits. Gibbs sampler for making full Bayesian inferences about the category probability as well as the regression coefficients is described. The model can be regarded as an alternative to the ordered logit regression model. Numerical examples are shown to demonstrate the efficiency of the model.

Effects of trait definition on genetic parameter estimates and sire evaluation for clinical mastitis with threshold models

AbstractClinical mastitis records on 36 178 first-lactation Norwegian dairy cattle (NRF) cows, daughters of 245 sires from 5286 herds, were analysed to study the impact of trait definition on estimates of genetic parameters and sire evaluations for clinical mastitis. The opportunity interval for infection, going from 30 days pre-calving to 300 days post partum, was divided into either 11 periods (each 30 days long); four periods ((-30, 0), (1, 30), (31, 120), (121, 300)); a single period (-30, 300) or defined as the interval currently used for sire evaluation in Norway (-15,120). Within each period, clinical mastitis was scored as 1 if it occurred at least once and 0 otherwise. Analysis was with Bayesian threshold models, assuming that mastitis (presence v. absence) was a different trait in each period. By use of multivariate or univariate normal link functions, unobserved liabilities to disease were modelled as a linear function of year of calving, age-season of calving, herd, sire of cow and residual effects. Estimates of heritability of liability to clinical mastitis ranged from 0-06 to 0-14, depending on the model and stage of lactation. In multi-period models, estimates of genetic correlations between periods were positive and ranged from 0-13 to 0-55. This suggests that clinical mastitis resistance is not the same trait in different periods of the first lactation, which is not captured by the single-interval models. The single-interval (-30, 300) model gave slightly smaller sire-specific posterior probabilities of clinical mastitis during the first lactation than the multi-period models. Furthermore, the interval used in current Norwegian sire evaluation understated the posterior probabilities of clinical mastitis, relative to the multi-period specifications. This led to some differences in sire rankings between the four models, although there was agreement between the four- and 11-period models. In conclusion, the multi-period models captured more genetic variation than the single-interval models, but the four-period model gave sire rankings that differed little from those obtained with an 11-period definition of clinical mastitis.

Genetic parameters for stillbirth in Danish Holstein cows using a Bayesian threshold model.

The objective of this study was to make an inference about the direct and maternal genetic variation of stillbirth for first-calving Holstein cows and to estimate the effect of breed and heterosis for original Danish black and white and Holstein-Friesian. A Bayesian threshold model, which included correlated genetic effects of sires and maternal grandsires was used. Marginal posterior distributions of effects were obtained using Gibbs sampling. Point estimates were compared with results from a linear model using REML. Data with and without twins were analyzed and models with and without effects of breed and heterosis were fitted, but estimates of genetic parameters were almost identical. In all the analyses with threshold models, the marginal posterior mean (and standard deviation) was 0.10 (0.014) for the direct heritability, 0.13 (0.015) for the maternal heritability, and 0.05 (0.10) for the genetic correlation between direct and maternal effects. The stillbirth rate tended to increase with a higher proportion of Holstein-Friesian in the calf and in the dam, but no effects of breed and heterosis were significant. Joint sampling of all location parameters was found superior to univariate sampling in terms of much better mixing properties of the fixed effects. Based on the results showing genetic variation for stillbirth at first calving, both the direct and the maternal effect could be included in the breeding program.

Longitudinal analysis of clinical mastitis at different stages of lactation in Norwegian Cattle

Genetic variation and covariation of liability to clinical mastitis in the course of first lactation in Norwegian Cattle (NRF) were investigated. The data consisted of 36,178 first-lactation cows with 354,506 clinical mastitis (absence=0 vs. presence=1) monthly records. A longitudinal binary data analysis was carried out using Bayesian threshold models and Markov chain Monte Carlo (MCMC) procedures. Liability was related to stage of lactation using random regression functions: the Ali–Schaeffer function (AS), the Wilmink function (W) and Legendre Polynomials of order 2, 3 or 4 (L2, L3, L4). Models were compared using a pseudo Bayes factor and an analysis of residuals. The MCMC scheme for the AS function did not converge after 20,000 iterations, and was therefore excluded from further analysis. The pseudo Bayes factor strongly favored the L4 model. Most posterior means of the residuals fell in the range from −0.2 to 0 when cows were healthy (a residual is negative when mastitis is absent and positive otherwise). The L4 model tended to have smaller residuals than the other three models when cows had mastitis. The posterior means of the herd variance and of the cow-specific variance were 0.0645 and 0.1084, respectively, for the fourth order Legendre polynomial. Heritability of liability to clinical mastitis was from 7% to 13% before calving, and ranged between 3% and 11% from calving to 260 days after calving. Most genetic correlations of liability to clinical mastitis between different days of first-lactation ranged from 0.4 to 0.7.

Genetic Analysis of Longitudinal Trajectory of Clinical Mastitis in First-Lactation Norwegian Cattle

Clinical mastitis records for 36,178 first-lactation daughters of 245 Norwegian Cattle (NRF) sires were analyzed with a Bayesian longitudinal threshold model. For each cow, the period going from 30 d before calving to 300 d after calving was divided into 11 intervals of 30 d length each. Absence or presence of clinical mastitis within each interval was scored as “0” or “1”, respectively. A Bayesian threshold model consisting of a set of explanatory variables plus Legendre polynomials on time of order four was used to describe the trajectory of liability to clinical mastitis. Heritability ranged between 0.07 and 0.13 before calving, from 0.04 to 0.15 during the first 270 d after calving, and increased sharply thereafter, as a consequence of the form of the polynomial. Genetic correlations between adjacent days were close to 1, and decreased when days were further apart. Most genetic correlations were moderate to high. A measure of probability of future daughters contracting clinical mastitis during lactation was computed for each sire. A typical curve had a peak near calving followed by a decrease thereafter. The best sires had a low peak around calving and a low expected probability of mastitis among daughters throughout lactation. Expected fraction of days without mastitis was derived from the probability curves and used for ranking of sires. Rank correlations with genetic evaluations of sires obtained from cross-sectional models were high. However, sire selection was affected markedly, especially at high selection intensity. An advantage of the longitudinal model for clinical mastitis is its ability to take multiple treatments and time aspects into account.

Genetic analysis of mastitis in dairy cattle with a Bayesian threshold model

Genetic parameters of mastitis are required in genetic selection for mastitis resistance. Excluding cows that are culled at an early stage of lactation due to mastitis from genetic parameter estimation may introduce culling bias. The use of linear models, suitable for continuous traits, is inappropriate for analysis of mastitis records because it is recorded as an all or none (binary) trait. Here, a Bayesian-threshold model with Markov chain Monte Carlo (MCMC) techniques was used to analyze mastitis data. The objective was to estimate genetic parameters of mastitis in first and all-lactation cows using complete or incomplete records on disease.

Bayesian analysis of twinning and ovulation rates using a multiple-trait threshold model and Gibbs sampling.

The Multiple-Trait Gibbs Sampler for Animal Models programs were extended to allow analysis of ordered categorical data using a Bayesian threshold model. The algorithm is based on data augmentation, where a value on the unobserved underlying normally distributed variable (liability) is generated in each round of iteration for each categorical observation. The programs allow analysis of several continuous and ordered categorical traits. Categorical traits can have any number of response levels. Models can be different for each trait. The programs were used to analyze twinning and ovulation rates from a herd of cattle selected for twinning rate at the U.S. Meat Animal Research Center. Data included number of calves born at each parturition for the lifetime of a cow and number of eggs ovulated for several estrous cycles before first breeding as heifers. A total of 6,411 calvings was recorded for 2,087 cows with 83.2% single and 16.8% multiple births. A total of 19,849 ovulations was recorded for 2,332 heifers with 85.2% single and 14.8% multiple ovulations. Mean posterior estimates of heritability and fraction of variance accounted for by permanent environmental effects (PE) were .128 and .103 for twinning rate and .168 and .079 for ovulation rate. Mean posterior estimate of genetic correlation was .808, and correlation of PE effects was .517. Use of a threshold model could allow for more rapid genetic improvement of the twinning herd through improved identification and selection of genetically superior animals because of higher heritability on the underlying scale.