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Choose a parametric baseline hazard $h_0(t;\eta )$ and fitUnder independent right censoring, the full survival likelihood isEstimate $(\eta ,\beta )$ by maximum likelihood estimation and test $H_0:\beta =0$ with a likelihood-ratio test, Wald test, or score test. Equality of the two event-time distributions is exactly $\beta =0$ within this model.

Under a country-specific constant hazard survival model, let $D_k$ be the number of deaths and $Y_k$ the total person-time at risk in country $k$. The likelihood is proportional toso the maximum-likelihood estimator is ${\hat{\lambda }}_k=D_k/Y_k$. Test $H_0:{\lambda }_0={\lambda }_1$ by a likelihood-ratio test, score test, or Wald test; equivalently, give $D_k$ a Poisson distribution with mean ${\lambda }_k{Y}_k$ and test the country coefficient in a Poisson regression with offset $\log Y_k$.

The variance $\mu +{\mu }^2/\theta$ approaches the Poisson distribution variance $\mu$ when $\theta \to \infty$. Equivalently, with $\alpha =1/\theta \ge 0$, the Poisson model is the boundary value $\alpha =0$. The usual Wilks theorem for a likelihood-ratio test assumes that the null parameter is an interior point of a smooth parameter space, so comparing the statistic with an ordinary ${\chi }_1^2$ law is invalid here.

A valid parametric bootstrap proceeds as follows. First fit the null Poisson model and retain its fitted means ${\hat{\mu }}_i$. For each bootstrap repetition $b=1,\dots ,B$, independently simulateusing the original doses, refit both the Poisson and negative-binomial models to that simulated data, and calculateFor the observations calculate the analogous $T_{obs}$. The bootstrap $p$-valueuses the null distribution with its boundary and finite-sample fitting behaviour automatically reproduced. A large $B$ controls the Monte Carlo error of this estimate.

The Neyman–Pearson lemma says that among tests of size at most $\alpha$ for two simple hypotheses, a likelihood-ratio test rejecting where $p(x;{\theta }_1)/p(x;{\theta }_0)>k$ is most powerful, with boundary randomisation if required. If $\phi$ is that test and $\psi$ any competing test, choose $k$ so the sizes agree. Pointwise,Integration and the size inequality yield $E_1\phi \ge E_1\psi$.

Herewhich decreases with $|x|$. Thus reject for $|X|\le c$, where

The third fit is a negative binomial regression. The Poisson model is obtained at the boundary where the negative-binomial overdispersion tends to zero, so the likelihood-ratio test has the asymptotic null distribution $\frac{1}{2}{\chi }_0^2+\frac{1}{2}{\chi }_1^2$. Since model3 has one additional parameter,The supplied output gives $\mathbb{P}({\chi }_1^2\ge D)=0.04608555$, and hence the boundary-corrected p-value is

Under the two-rate alternative, maximize the piecewise-exponential survival model likelihood separately to obtain the estimates in part (i). Under $H_0:{\theta }_1={\theta }_2=\theta$, there are 15 deaths and 2200 days of exposure, so $\hat{\theta }=15/2200$. Twice the maximized log-likelihood difference isUnder the null this likelihood-ratio test statistic is asymptotically ${\chi }_1^2$.

Testwith a likelihood-ratio test. Model1 has three likelihood parameters and model2 has four, so their AIC values giveThe statistic isUsing a naive ${\chi }_1^2$ reference gives $p=0.0341$. Because the null variance lies on the boundary, the standard asymptotic reference is the mixture $\frac{1}{2}{\chi }_0^2+\frac{1}{2}{\chi }_1^2$, giving $p=0.0170$. Either calibration rejects at the five-percent level and supports a subject random effect.

The number of free dimensions remaining after specified constraints often determines a reference sampling distribution. In a full-rank normal linear model with $n$ observations and $p$ mean coefficients, the residual space has dimension $n-p$. A regular nested likelihood-ratio test instead uses the difference in free statistical parameter dimensions. Boundary constraints may invalidate this regular calibration.