15.3 The likelihood ratio test

$$t_{LR} = 2[l(\hat{\theta})-l(\theta_0)] \sim \chi^2_v$$

where v is the degree of freedom.

Compare the height of the log-likelihood of the sample estimate in relation to the height of log-likelihood of the hypothesized population parameter

Alternatively,

This test considers a ratio of two maximizations,

$$\begin{aligned} L_r &= \text{maximized value of the likelihood under $H_0$ (the reduced model)} \\ L_f &= \text{maximized value of the likelihood under $H_0 \cup H_a$ (the full model)} \end{aligned}$$

Then, the likelihood ratio is:

$$\Lambda = \frac{L_r}{L_f}$$

which can’t exceed 1 (since $L_f$ is always at least as large as $L-r$ because $L_r$ is the result of a maximization under a restricted set of the parameter values).

The likelihood ratio statistic is:

$$\begin{aligned} -2ln(\Lambda) &= -2ln(L_r/L_f) = -2(l_r - l_f) \\ \lim_{n \to \infty}(-2ln(\Lambda)) &\sim \chi^2_v \end{aligned}$$

where $v$ is the number of parameters in the full model minus the number of parameters in the reduced model.

If $L_r$ is much smaller than $L_f$ (the likelihood ratio exceeds $\chi_{\alpha,v}^2$), then we reject he reduced model and accept the full model at $\alpha \times 100 \%$ significance level