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Let $RS{S}_0$ and $RS{S}_1$ be the residual sums of squares under the restricted model with $p_0$ coefficients and the full model with $p$ coefficients. Assuming $X$ has full column rank, the nested-model F-test usesUnder the null hypothesis this has the $F_{p-p_0,n-p}$ distribution.

Maximizing the Gaussian likelihood over $\beta$ and the unknown variance gives ${\hat{\sigma }}^2=RSS/n$. Thus the likelihood-ratio test statistic isSincewe obtainThe logarithm is strictly increasing, so the two test statistics give exactly the same rejection ordering.

The null hypothesis is that price and score are independent, equivalently that the independence log-linear model for a two-way contingency table is correct. Against the saturated alternative, the reported Poisson deviance is the likelihood-ratio test statisticUnder the null and the usual large-sample regularity assumptions, Wilks theorem givesThe observations must be independent, the cell probabilities must not lie on the boundary of the parameter space, and the expected counts must be large enough for the chi-squared asymptotic approximation. The fitted counts are either $31/3$ or $28/3$, so the customary expected-count check is comfortably satisfied. Since(equivalently, the p-value is about $0.329$), we do not reject the null at the $1\%$ significance level. The data provide no significant lack of fit for independence.

Model 2 is obtained from model 3 by setting ${\beta }_0=0$. Model 3 is contained in model 1 by choosing the symmetric block parametersHence

Assuming every college-pair cell is observed and the models are identifiable, model 1 has $C(C+1)/2$ parameters and model 3 has $C+1$. The likelihood-ratio test statistic for model 3 against model 1 isUnder model 3 and the regularity conditions of Wilks theorem,

The full parameter space has dimension two and the null space dimension one. The regular Wilks theorem therefore gives $W\stackrel{d}{\to }{\chi }_1^2$ under $H_0$ as $n\to \infty$. If $F_1$ denotes the cumulative distribution function of the chi-squared distribution with one degree of freedom, the approximate size-$\alpha$ rejection region isThe corresponding approximate p-value isThe one degree of freedom here describes the large-sample likelihood-ratio test statistic; it is different from the exact residual degrees of freedom $n-1$ in the next part.

Put $S={\sum }_i(x_i-\bar{x}{)}^2$ and $q=S/{\sigma }_0^2$. Under the null hypothesis, maximizing over the nuisance mean gives ${\hat{\mu }}_0=\bar{x}$. The unrestricted normal mean and variance maximum-likelihood estimators are $\hat{\mu }=\bar{x}$, $\widehat{{\sigma }^2}=S/n$. Thus, for $S>0$,The likelihood-ratio test statistic is consequentlyThe usual model assumes $n\ge 2$. The event $S=0$ has probability zero under a nondegenerate normal distribution; at that observation the unrestricted likelihood is unbounded as the variance tends to zero, and the ratio has limiting value zero. This is the likelihood-ratio test for a normal variance with unknown mean.

Testing a single nonnegative random-effect variance against zero puts the null on the boundary of the parameter space. In the usual regular increasing-independent-groups limit, the likelihood-ratio test statistic has limit $\frac{1}{2}{\delta }_0+\frac{1}{2}{\chi }_1^2$, rather than ${\chi }_1^2$. A finite-sample simulation calibrated for the actual design avoids relying on this asymptotic approximation.