= Solution
Let $X$ have columns $1,t$, and let $Z$ have one column per incubator, with entry $t_i$ in the column for observation $i$'s incubator and zeros elsewhere. Integrating the Gaussian <random effects> gives
$$
Y\sim N(X\beta,V),\qquad V=\sigma^2I+\tau^2ZZ^T.
$$
Consequently its marginal <log-likelihood> is
$$
\ell(\beta,\sigma^2,\tau^2)=-\frac12\{n\log(2\pi)+\log\det V+(Y-X\beta)^TV^{-1}(Y-X\beta)\}.
$$
The <likelihood-ratio test> of $H_0:\tau^2=0$ against $\tau^2>0$ uses
$$
\boxed{T=2\left[\sup_{\beta,\sigma^2>0,\tau^2\ge0}\ell-\sup_{\beta,\sigma^2>0}\ell(\beta,\sigma^2,0)\right].}
$$
For each <covariance>, the optimizing coefficients are the <generalized least squares> estimator $(X^TV^{-1}X)^{-1}X^TV^{-1}Y$. Both fits should use the same marginal likelihood convention, for example fitting the alternative with `update(fly.model, REML=FALSE)` and the null as a Gaussian linear model. The displayed alternative <REML> criterion alone cannot supply $T$.
The <variance> component is constrained to be nonnegative, and zero lies on the boundary. Thus the interior-parameter hypothesis of <Wilks theorem> fails; the usual $\chi_1^2$ approximation is inappropriate. A $\tfrac12\delta_0+\tfrac12\chi_1^2$ limit sometimes applies to a single <variance> component under additional asymptotic conditions, but four incubators do not give a persuasive large-number-of-groups approximation.
Use a <parametric bootstrap> under the null: estimate $\beta,\sigma^2$ there; simulate independent Gaussian errors with that fitted <variance> at exactly the observed times and grouping labels; refit both models by maximum likelihood for each simulated sample; and calculate $T^*$. The upper-tail proportion, conventionally $(1+\#\{T^*\ge T\})/(B+1)$, estimates the <p-value>. Boundary fits with estimated $\tau^2=0$ must remain in the simulation distribution. A restricted-likelihood ratio and a correspondingly calibrated bootstrap are another option when the fixed-effect design is identical, but one must not mix ordinary and restricted likelihoods in a single statistic.
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