---
tag: 8085SD9EYY
kind: definition
name: Score test
msc: [62F03, 62F05]
defines: Y5HMW0BDTP
uses: [CP97GRVTQ0, G1Z3WSTE8Q, 087P03NEND, 0RFMMBHFTV]
sources:
  - url: https://en.wikipedia.org/wiki/Score_test
---

## Definition

Let $X$ be data with log-likelihood $\ell(\theta) = \log L(\theta \mid X)$, differentiable in $\theta$ in an open set $\Theta \subseteq \R^k$, and let $\theta_0 \in \Theta$.
Let $U(\theta) = \nabla \ell(\theta)$ be the [[score|CP97GRVTQ0]] and let $I(\theta)$ be the [[Fisher information|G1Z3WSTE8Q]] of $X$, assumed invertible at $\theta_0$.
The *score statistic* for $H_0 \colon \theta = \theta_0$ is
$$S = U(\theta_0)^{T} I(\theta_0)^{-1} U(\theta_0).$$
The *score test* at level $\alpha$ rejects $H_0$ when $S$ exceeds the $1 - \alpha$ quantile of the [[chi-squared distribution|0RFMMBHFTV]] with $k$ degrees of freedom.

## Remarks

Under $H_0$ and the usual regularity conditions for i.i.d. samples, $S$ converges in distribution to $\chi^2_k$, so the test has asymptotic level $\alpha$.
For a composite $H_0$ with nuisance parameters, one evaluates the score and the information at the maximum likelihood estimate under $H_0$.
Example: for $X$ binomial with parameters $n$ and $p$, one has $U(p) = (X - np)/(p(1 - p))$ and $I(p) = n/(p(1 - p))$.
So $S = (X - np_0)^2/(np_0(1 - p_0))$, the square of the usual z-statistic for a proportion.
Non-example: the statistic $U(\theta_0)^T U(\theta_0)$, without the factor $I(\theta_0)^{-1}$, does not have a $\chi^2_k$ limit in general, so it is not the score statistic.