LikelihoodRatioTest¶

LikelihoodRatioTest(model0NbParameters, model0LogLikelihood, model1NbParameters, model1LogLikelihood, level=0.05)¶

Nested likelihood model selection.

This test helps selecting between two nested models $\mathcal{M}_0 \subset \mathcal{M}_1$ estimated by likelihood maximization.

Suppose that $\mathcal{M}_1$ is a model with $d$ -dimensional parameter $\vect{\theta} = (\vect{\theta}^{(1)}, \vect{\theta}^{(2)})$ where $\vect{\theta}^{(1)}$ is a $k$ -dimensional subset of $\vect{\theta}$ . We suppose that $\mathcal{M}_0$ is obtained by constraining $\vect{\theta}^{(1)}$ to be equal to a fixed value denoted by $\vect{\theta}^{(1)}_0$ . Then $\mathcal{M}_0$ is a model with parameter $\vect{\theta}^{(2)}$ .

Let $\hat{\vect{\theta}}$ denote the maximum likelihood estimator of $\vect{\theta}$ for model $\mathcal{M}_1$ and $\ell(\hat{\vect{\theta}})$ its maximized log-likelihood.

The profile log-likelihood for $\vect{\theta}^{(1)}$ is defined as:

$l_p(\vect{\theta}^{(1)}) = \max_{\vect{\theta}^{(2)}} \ell(\vect{\theta}^{(1)}, \vect{\theta}^{(2)})$

We define the profile deviance statistic depending on $\vect{\theta}^{(1)}$ as the maximized log-likelihood with respect to $\vect{\theta}^{(2)}$ when $\vect{\theta}^{(1)}$ is frozen:

$\mathcal{D}_p (\theta^{(1)}) = 2(\ell(\hat{\vect{\theta}}) - l_p(\vect{\theta}^{(1)}))$

Under suitable regularity conditions, for large $n$ , $\mathcal{D}_p (\theta^{(1)})$ follows a $\chi^2_k$ distribution.

We use the profile deviance statistic to define the $(1-\alpha)$ confidence region for the true value of parameter $\vect{\theta}^{(1)}$ :

$\mathcal{C}_{\alpha} = \left\{\vect{\theta}^{(1)} : \mathcal{D}_p (\theta^{(1)}) \leq c_{\alpha} \right\}$

where $c_{\alpha}$ is the $(1-\alpha)$ quantile of the $\chi^2_k$ distribution. The level $\alpha$ is the Type 1 error of the test, which means the mistaken rejection of model $\mathcal{M}_0$ .

In order to test the validity of model $\mathcal{M}_0$ with parameter $(\vect{\theta}^{(1)}_0, \vect{\theta}^{(2)})$ relative to $\mathcal{M}_1$ with parameter $(\vect{\theta}^{(1)}, \vect{\theta}^{(2)})$ at the $\alpha$ -level of significance, we check the value $\mathcal{D}_p (\theta^{(1)}_0)$ :

if $\mathcal{D}_p (\theta^{(1)}_0) \leq c_{\alpha}$ , we accept $\mathcal{M}_0$ rather than $\mathcal{M}_1$ ,
if $\mathcal{D}_p (\theta^{(1)}_0) \geq c_{\alpha}$ , we reject $\mathcal{M}_0$ in favor of $\mathcal{M}_1$ .

Parameters:

model0NbParametersint, $\geq 1$ ,: Number of parameters of $\mathcal{M}_0$
m0llhfloat: $\mathcal{M}_0$ log-likelihood
model1NbParametersint, $\geq 1$ ,: Number of parameters of $\mathcal{M}_1$
m1llhfloat: $\mathcal{M}_1$ log-likelihood
levelfloat, $0 \leq \alpha \leq 1$ , optional: Risk of wrongly rejecting $\mathcal{M}_0$ Default value is 0.05

Returns: