Fit both hierarchies to the historical data and compare their posterior predictive checks. Use discrepancies sensitive to between-individual tails, such as the maximum standardized personal mean, tail counts and the spread of individual sample means. Simulate new individual means as well as repeated observations; conditioning only on fitted personal means would conceal a poor population model.

Leave whole individuals out for cross-validation, rather than just measurements, when the aim is prediction for a new individual. Compare held-out predictive log densities or interval calibration, with uncertainty in the comparison. Check both fits' Markov chain Monte Carlo convergence diagnostics and sensitivity to scale priors. Better tail prediction and adequate calibration would support the heavier-tailed model; the data are required to decide, so no empirical winner can be asserted here.

The Student t-distribution has heavier tails than the normal distribution, so unusual personal means need not force the entire population's spread upward or be shrunk as strongly toward its center. The latent ${\lambda }_i$ allows a large local prior variance for an outlying individual. This normal scale mixture representation of Student t is a useful robust version of the hierarchical Bayesian model. It still assumes a symmetric, unimodal population; a genuinely multimodal population may need a different model.

Conditionally on ${\lambda }_i$, the proposed precision gives ${\theta }_i\mid {\lambda }_i\sim N(\mu ,4{\tau }^2/{\lambda }_i)$. Thus, with independent $Z\sim N(0,1)$ and ${\lambda }_i\sim {\chi }_4^2$,By the Student t-distribution construction, $\boxed{({\theta }_i-\mu )/\tau \sim t_4}$. Here $\tau$ is a scale rather than the standard deviation: the prior variance of ${\theta }_i$ is $2{\tau }^2$.

Add the new individual's latent mean and log variance to the same population hierarchy. Enter only that individual's first two measurements as observations; use the historical data and these two values to fit the model. The third measurement must be held out, or the predictive check reuses the value it is meant to test. For example, augment the unchanged population hyperpriors with this WinBUGS structure:

new.mean ~ dnorm(mu, invtau2)
new.logvar ~ dnorm(phi, invpsi2)
new.precision <- exp(-new.logvar)
for (k in 1:2) {
    new.Y[k] ~ dnorm(new.mean, new.precision)
}
future.Y ~ dnorm(new.mean, new.precision)
upper.indicator <- step(future.Y - held.out.third)

future.Y is unobserved. Its simulated values give the posterior predictive distribution, including uncertainty in the population parameters and personal variance. A central 99.9% prediction interval uses its $0.0005$ and $0.9995$ empirical quantiles. Flag the third value if it is outside that interval. Equivalently, estimate its predictive tail probability by the mean of upper.indicator and flag if that mean is below $0.0005$ or above $0.9995$. This is two-sided, unlike the earlier one-sided upper limit.

The hierarchical Bayesian model separates variability between individuals from variability between repeated measurements on an individual. In WinBUGS, dnorm takes a precision parameter, so inverse variance is the appropriate second argument.

Line (1) gives $\log {\sigma }_j^2\sim N(\varphi ,{\psi }^2)$, hence a positive log-normal distribution for each within-individual variance. Estimating the shared parameters supplies partial pooling of these variances without forcing them equal. Line (2) assigns a proper bounded flat prior to the between-individual standard deviation $\tau$, and line (3) does the same for the standard deviation $\psi$ on the log-variance scale. The broad bounded priors on $\mu$ and $\varphi$ are also proper.

This is a reasonable structure for heterogeneous repeated measurements, but flat does not mean uninformative on all transformed scales. The bounds must be wide enough for plausible values, while such enormous ranges can put substantial mass on implausible predictions. Use prior predictive checks and sensitivity analysis, and check the Markov chain Monte Carlo convergence diagnostics. Serial dependence or systematic trends would require an extension of the independent-measurement likelihood.

After $k$ measurements the normal-normal conjugacy formulas areThe upper monitoring limit is $m_{ik}+z_{0.999}\sqrt{v_{ik}+{\sigma }_i^2}$. Its center moves toward the athlete's own mean and the population's influence diminishes. Its predictive width decreases toward $z_{0.999}{\sigma }_i$, since uncertainty about the mean vanishes but the next measurement's error remains. The absolute upper limit need not decrease monotonically: its center can move upward or downward with the observations. This is sequential normal predictive monitoring.

Taking the upper $0.999$ quantile of the posterior predictive distribution givesThe same predictive false-alarm probability is retained after updating with the first measurement.

Model the count relative to exposure, for exampleHere $E_i$ is a measured exposure such as patient-days, $x_i$ contains hospital predictors, and $b_i$ is a random effect. This Poisson regression separates hospital size from rate variation and supplies partial pooling. Use proper, scale-appropriate priors on coefficients and variation; add regional or temporal effects if the data demand them. A gamma mixing law with an estimated shape is another way to allow more flexible overdispersion. Compare the fitted model's counts and dispersion through posterior predictive checks.

The model treats hospitals as exchangeable and assumes the same exponential mixing law for their total expected annual counts. Its shape is fixed: the prior standard deviation equals its mean. Real hospitals differ in size, patient mix, ascertainment and observation effort, so expected counts should not generally share one distribution without accounting for these predictors. Common regional conditions can also invalidate their assumed conditional independence. The resulting geometric distribution for observed counts is a restrictive distributional prediction that can be checked.

The gamma full conditional distribution has mean $(y_i+1)/(1+\psi )=(y_i+1)\varphi$. The tower property and the beta distribution mean giveWriting $\overline{y}=s/n$ and $w=s/(s+n)=\overline{y}/(1+\overline{y})$ yieldsThus partial pooling pulls every local mean toward the overall sample mean. The common weight arises from the fixed shape-one mixing distribution; it is the Poisson–exponential posterior shrinkage formula.

Multiplying the likelihood in part (d) by the Haldane prior gives $p(\varphi \mid y)\propto {\varphi }^{s-1}(1-\varphi {)}^{n-1}$. ConsequentlyFor at least one hospital, $n>0$, this posterior is proper exactly when $s>0$. If every count is zero, the kernel diverges at $\varphi =0$ and does not define a posterior distribution; the improper prior cannot then be used for the subsequent expectation.

A flat density in $\log \psi$ induces $p(\psi )\propto 1/\psi$. Since $\psi =(1-\varphi )/\varphi$, the absolute Jacobian determinant is $|d\psi /d\varphi |={\varphi }^{-2}$. ThereforeThis is the Haldane prior, an improper prior with formal beta parameters $(0,0)$.

Consistent independent replications should strengthen confidence; contradictory results can weaken it. The change depends on the actual evidence. If $k$ of $n$ studies are positive and all labels are available, the posterior odds areFor the normal model, retaining the full independent statistics instead givesEach study multiplies the existing odds by its own Bayes factor. Reading a complete synthesis is therefore different from learning only that somebody obtained a positive result. Publication bias, dependence between studies and changes in study design must be accounted for before applying the independent-product formula.

For $0<\alpha ,\beta <1$, both event probabilities tend to one, so the posterior odds tend to $\boxed{R}$. If $1-\beta >\alpha$, the event's Bayes factor decreases toward one. To see monotonicity, put $a=1-\alpha >\beta$ and writeThe last quotient is a weighted average of the decreasing values $(\beta /a{)}^k$, with weights $a^k$. Adding the next, smaller value decreases the average.

Thus the selection rule becomes less informative as more teams try: eventually a positive study is almost certain even under the null. This explains the hotter-field interpretation in Ioannidis's model. It describes evidence retained by this particular selection event, rather than proving that every individual study becomes less reliable or that the full accumulated evidence gets weaker.

Let $A_n$ denote the event that at least one study is positive. Conditional independence gives $\mathbb{P}(A_n\mid T)=1-{\beta }^n$ and $\mathbb{P}(A_n\mid F)=1-(1-\alpha {)}^n$. Updating the prior odds by this event's Bayes factor givesThese are the odds under at-least-one-positive selection, not the odds using every study's actual result.

Conditional on a false relationship, the independent studies are each negative with probability $1-\alpha$. The complement of at least one positive is that all are negative. Hence the familywise error rate is $\boxed{1-(1-\alpha {)}^n}$.

Yes: replacing the observed statistic by a significance label loses information. The binary positive event has Bayes factor $(1-\beta )/\alpha \simeq 16$, but the observed-statistic Bayes factor varies continuously with $y$. A result just at the threshold has $B_{10}(1.65)\simeq 2.74$, while $B_{10}(3)\simeq 79.0$. Both would receive the same positive label.

The binary calculation is correct for that deliberately coarsened observation; its problem is treating it as the whole evidence when the statistic is available. This is the point developed in Goodman and Greenland's response.

By balanced factorial orthogonality, omitting A adds $S{S}_A$ to the additive residual: ${\mathrm{RSS}}_{B\text{-only}}=2.6867+12.7400=15.4267$. This model has $2$ parameters and $10$ residual degrees of freedom.

By balanced factorial orthogonality, omitting B adds $S{S}_B$ to the full additive residual: ${\mathrm{RSS}}_{A\text{-only}}=2.6867+0.4033=3.0900$. This model has $3$ parameters and $9$ residual degrees of freedom.

The additive-model residual sum of squares is the reported residual entry: ${\mathrm{RSS}}_1=2.6867$, with $8$ residual degrees of freedom.