Is the likelihood ratio test ($-2 \log L$) basically like the partial F-test in that you are using it for logistic regression instead of linear regression?
Solved – Likelihood ratio test
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I think that you might be confusing an extra-sum-of-squares F-test with a likelihood ratio test. Although, both are used to compare two models.
A likelihood ratio statistic, denoted by $\Lambda$, is given by
$$\Lambda = \frac{L\text{(reduced model})}{L(\text{full model})}$$
Taking $-2\log\Lambda$ produces a statistic that has $\chi^2_{d.f(\text{reduced model})-d.f(\text{full model})}$ distribution. That is to say that taking $-2\log$ of the $\Lambda$ gives you a $\chi^2$ distribution.
I have not used SAS so I cannot comment on the output, but I hope that I have been able to answer your question.
Note: that $\Lambda$ is equivalent to your L
Janne: For linear regression you could use either the likelihood ratio test or the extra-sum-squares F-test and you should end up with the same p-value. Despite, this they are not the same thing.
As has been mentioned above the likelihood ratio test produces a statistic that has $\chi^2_{d.f(\text{reduced model})-d.f(\text{full model})}$ distribution. Where as an extra-sum-of-squares F-test, given by
$$F = \frac{(SSR_{\text{reduced model}}-SSR_{\text{full model}})/d.f_{\text{reduced model}} - d.f_{\text{full model}}}{\hat{\sigma}^2_\text{full model}}$$
producing a statistic that has $F_{d.f(\text{reduced model})-d.f(\text{full model}),d.f(\text{full model})}$ distribution. Where SSR is the sum of squared residuals and $\hat{\sigma}^2$ is our standard estimate.
- In general, a higher value of the likelihood (or log-likelihood) is preferred. The relationship to the value of 0 is irrelevant. However, there are a number of caveats: if you add more parameters (predictors) to the model, the likelihood can only go up, so selecting the maximum value would always give the model with the most variables. There are multiple ways around it via various information criteria, that penalize the model by its number of parameters. Look up the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC).
- I don't know exactly what the "likelihood ratio chi-squared" is, but it could be the value of the likelihood-ratio test statistic compared to the null model (not the other competing models). It is unlikely to be the Wald or the score test statistic. However, if the competing models are nested (one is a special case of the other) you can use the log-likelihood values to compute the likelihood-ratio test statistics of one model versus the other. $$ \mathrm{LRT} = -2 *( \log L(M_1) - \log L(M_2)) \sim \chi^2_{p_2-p_1}$$ under $H_0$, where $p_1$ and $p_2$ are the number of parameters
- There are no "typical" values for the value of the log-likelihood, just as there are no typical values for the sum of all the
yvalues in your data. They are only meaningful when comparing different fits to the same set of values for the outcome.
Best Answer
For logistic regression you use the asymptotic distribution of the log of likelihood ratio test statistic for variable selection (testing hypotheses or model selection). In the case of linear regression, due to the assumed normality for the error distribution, there is no need to use asymptotics, and the likelihood ratio test static trivially reduces to a ratio of chi-squares, that is, F-distribution.