TG5 cattle studies
Section of Hierarchical Bayes, part of Bayesian statistics
TG5 cattle studies
| Type | Section |
| Taught in | Hierarchical Bayes |
| Part | Bayesian statistics |
Genetics is among the most statistically sophisticated applied fields, but that sophistication is often hidden behind software. A scientist may be using a recently developed Monte Carlo method because it has appeared in a software package, without knowing how the method works. Someone nevertheless has to establish that the computation and its assumptions are sound. That is a fairly concrete use for statistical expertise.
The practical difficulty begins even earlier. A proper meta-analysis tries to find every relevant study, including studies with small or null effects that were never published. In this case much of the material was unpublished, and the data were obtainable largely because a company already had commercial relationships with the breeders. The available records were study summaries rather than the complete farm-level records. Commercial questions and scientific questions were not always the same. Getting the data is not as easy as it sounds.
Example 1 (TG5 cattle hierarchy) A cattle-genetics meta-analysis illustrates hierarchical Bayes. The search was not exhaustive, but it recovered 14 independent studies, many of them unpublished, which produced 19 standardised estimates of a TG5 genotype effect on beef marbling. The studies covered four breed groups and four marbling measurements, so treating every estimate as an exchangeable replicate would erase scientifically meaningful structure.
The original study-level analyses compared the CC and TT genotypes with CT, then divided each contrast by its residual standard deviation. A schematic version of the hierarchy is \begin{align*} y_{ij}\mid\delta_i,w_{ij} &\sim\operatorname{N}(\delta_i,w_{ij}^{-1}),\\ \delta_i\mid\mu_{b(i)},\tau_{\mathrm{study}} &\sim\operatorname{N}(\mu_{b(i)},\tau_{\mathrm{study}}^{-1}),\\ \mu_b\mid\mu_0,\tau_{\mathrm{breed}} &\sim\operatorname{N}(\mu_0,\tau_{\mathrm{breed}}^{-1}),\\ \mu_0&\sim\operatorname{N}(0,\xi^2), \qquad \xi^2\text{ large}. \end{align*} Here w_{ij} records the within-study information, \delta_i is a study effect, and \mu_b is a breed-group effect. Precision parameters were given diffuse scaled inverse-\chi^2-type priors in the applied analysis. The diagrammatic flow is
\text{overall effect}\longrightarrow \text{breed effect}\longrightarrow \text{study effect}\longrightarrow \text{observed contrast}.
The hierarchy performs partial pooling at more than one level. The reported posterior for TT minus CC lay mostly above zero, with a smaller shift for one T copy, but the important methodological point is that every conclusion and every uncertainty interval came from one coherent multilevel fit. The example connects Bayesian random parameters to mixed models. The analysis was fitted in WinBUGS using a Gibbs sampler; the “GS” in the software name stands for Gibbs sampling.
Used in
See also
Hierarchical Bayesian model · Hyperparameter · Normal hierarchical shrinkage · Exchangeability · Empirical Bayes · Study-rate full conditional · Marginal hyperposterior · Uniformity depends on a domain