Score test

A score test is a statistical test of a simple null hypothesis that a parameter of interest $heta$ isequal to some particular value $heta\_0$. It is the most powerful test when the true value of $heta$ is close to $heta\_0$.

ingle parameter test

The statistic

Let $L$ be the likelihood function which depends on a univariate parameter $heta$ and let $x$ be the data. The score is $U(\; heta)$ where:$U(\; heta)=frac\{partial\; log\; L(\; heta\; |\; x)\}\{partial\; heta\}.$

The observed Fisher information is,

:$I(\; heta)\; =\; -frac\{partial^2\; log\; L(\; heta\; |\; x)\}\{partial\; heta^2\}.$

The statistic to test $H\_0:\; heta=\; heta\_0$ is:$frac\{U(\; heta\_0)^2\}\{I(\; heta\_0)\}$

which takes a $chi^2\_1$ distribution asymptotically when $H\_0$ is true.

Justification

The case of a likelihood with nuisance parameters

As most powerful test for small deviations

:$left(frac\{partial\; log\; L(\; heta\; |\; x)\}\{partial\; heta\}\; ight)\_\{\; heta=\; heta\_0\}\; geq\; C$Where $L$ is the likelihood function,$heta\_0$ is the value of the parameter of interest under the null hypothesis, and $C$ is a constant set depending onthe size of the test desired (i.e. the probability of rejecting $H\_0$ if$H\_0$ is true; see Type I error).

The score test is the most powerful test for small deviations from $H\_0$.To see this, consider testing $heta=\; heta\_0$ versus $heta=\; heta\_0+h$. By the Neyman-Pearson lemma, the most powerful test has the form

:$frac\{L(\; heta\_0+h|x)\}\{L(\; heta\_0|x)\}\; geq\; K;$

Taking the log of both sides yields

:$log\; L(\; heta\_0\; +\; h\; |\; x\; )\; -\; log\; L(\; heta\_0|x)\; geq\; log\; K.$

The score test follows making the substitution

: $log\; L(\; heta\_0+h|x)\; approx\; log\; L(\; heta\_0|x)\; +\; h\; imes\; left(frac\{partial\; log\; L(\; heta\; |\; x)\}\{partial\; heta\}\; ight)\_\{\; heta=\; heta\_0\}$

and identifying the $C$ above with $log(K)$.

Relationship with Wald test

Multiple parameters

A more general score test can be derived when there is more than one parameter. Suppose that $hat\{\; heta\}\_0$ is the Maximum Likelihood estimate of $heta$ under the null hypothesis $H\_0$. Then

:$U^T(hat\{\; heta\}\_0)\; I^\{-1\}(hat\{\; heta\}\_0)\; U(hat\{\; heta\}\_0)\; sim\; chi^2\_k$

asymptotically under $H\_0$, where $k$ is the number of constraints imposed by the null hypothesis and

:$U(hat\{\; heta\}\_0)\; =\; frac\{partial\; log\; L(hat\{\; heta\}\_0\; |\; x)\}\{partial\; heta\}$

and

:$I(hat\{\; heta\}\_0)\; =\; -frac\{partial^2\; log\; L(hat\{\; heta\}\_0\; |\; x)\}\{partial\; heta\; partial\; heta\text{'}\}.$

This can be used to test $H\_0$.

ee also

*Fisher information
*Uniformly most powerful test
*Wald test
*Likelihood Ratio Test
*Score (statistics)