Rat-tumour data
Section of Hierarchical Bayes, part of Bayesian statistics
Rat-tumour data
| Type | Section |
| Taught in | Hierarchical Bayes |
| Part | Bayesian statistics |
Example 1 (Rat-tumour hierarchy) Tumour rates in female F344 rats supply a second hierarchical example. In a current control study, y=4 of n=14 rats developed one or more tumours over two years. A one-study model is
Y\mid\theta\sim\operatorname{Bin}(14,\theta), \qquad \theta\sim\operatorname{Beta}(\alpha,\beta),
giving
\theta\mid Y=4 \sim\operatorname{Beta}(\alpha+4,\beta+10).
The question is how to choose (\alpha,\beta) when 70 earlier control studies are available.
Let study j contribute y_j tumour-bearing rats from n_j animals. The hierarchical model is Y_j\mid\theta_j,n_j\sim\operatorname{Bin}(n_j,\theta_j), \qquad \theta_j\mid\alpha,\beta \stackrel{\mathrm{iid}}{\sim}\operatorname{Beta}(\alpha,\beta), \quad j=1,\ldots,J. Each study has its own rate, but all study rates are drawn from a common population distribution.
Good applied statistics begins before the algebra. Downloading a data set without understanding why the experiment was conducted, how an outcome was defined, or which sources of variation changed between studies is not enough. The hierarchy should represent the scientific design we actually have, not a generic stack of random effects added after the fact. The version of the rat example used here follows Gelman’s presentation of the Dempster, Selwyn, and Weeks analysis.
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