# Solved – Likelihood ratio tests

Is likelihood ratio test (\$F\$-test) of significance of difference of two linear models the same as chi-square test of difference of \$-2log L\$?

SAS `PROC GLM` produces \$F\$-statistics and `PROC MIXED` \$-2log L\$.

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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{Ltext{(reduced model})}{L(text{full model})}\$\$

Taking \$-2logLambda\$ produces a statistic that has \$chi^2_{d.f(text{reduced model})-d.f(text{full model})}\$ distribution. That is to say that taking \$-2log\$ 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.

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