Score test

The score test is a statistical test of a simple null hypothesis (that the parameter of interest $\theta$ is equal to some particular value $\theta _{0}$ ):

\left({\frac {\partial \log L(\theta |x)}{\partial \theta }}\right)_{\theta =\theta _{0}}\geq C

Where $L$ is the likelihood function, $\theta _{0}$ is the value of the parameter of interest under the null hypothesis, and $C$ is a constant set depending on the 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 $\theta =\theta _{0}$ versus $\theta =\theta _{0}+h$ . By the Neyman-Pearson lemma, the most powerful test has the form

{\frac {L(\theta _{0}+h|x)}{L(\theta _{0}|x)}}\geq K

Taking the log of both sides yeilds

\log L(\theta _{0}+h|x)-\log L(\theta _{0}|x)\geq \log K

The score test follows making the substitution

\log L(\theta _{0}+h|x)\approx \log L(\theta _{0}|x)+h\times \left({\frac {\partial \log L(\theta |x)}{\partial \theta }}\right)_{\theta =\theta _{0}}

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

See Also