Instatistics, thescore test assessesconstraints onstatistical parameters based on thegradient of thelikelihood function—known as thescore—evaluated at the hypothesized parameter value under thenull hypothesis. Intuitively, if the restricted estimator is near themaximum of the likelihood function, the score should not differ from zero by more thansampling error. While thefinite sample distributions of score tests are generally unknown, they have an asymptoticχ²-distribution under the null hypothesis as first proved byC. R. Rao in 1948,^[1] a fact that can be used to determinestatistical significance.

Since function maximization subject to equality constraints is most conveniently done using a Lagrangean expression of the problem, the score test can be equivalently understood as a test of themagnitude of theLagrange multipliers associated with the constraints where, again, if the constraints are non-binding at the maximum likelihood, the vector of Lagrange multipliers should not differ from zero by more than sampling error. The equivalence of these two approaches was first shown byS. D. Silvey in 1959,^[2] which led to the nameLagrange Multiplier (LM) test that has become more commonly used, particularly in econometrics, sinceBreusch andPagan's much-cited 1980 paper.^[3]

The main advantage of the score test over theWald test andlikelihood-ratio test is that the score test only requires the computation of the restricted estimator.^[4] This makes testing feasible when the unconstrained maximum likelihood estimate is aboundary point in theparameter space.^{[citation needed]} Further, because the score test only requires the estimation of the likelihood function under the null hypothesis, it is less specific than the likelihood ratio test about the alternative hypothesis.^[5]

Let${\displaystyle L}$ be thelikelihood function which depends on a univariate parameter${\displaystyle \theta }$ and let${\displaystyle x}$ be the data. The score${\displaystyle U(\theta )}$ is defined as

${\displaystyle U(\theta )=\frac{\mathrm{\partial }\log L(\theta \mid x)}{\mathrm{\partial }\theta }.}$

TheFisher information is^[6]

${\displaystyle I(\theta )=-\mathrm{E}\left[\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^2}\log f(X;\theta )|\theta \right],}$

where ƒ is the probability density.

The statistic to test${\displaystyle {\mathcal{H}}_0:\theta ={\theta }_0}$ is${\displaystyle S({\theta }_0)=\frac{U({\theta }_0{)}^2}{I({\theta }_0)}}$

which has anasymptotic distribution of${\displaystyle {\chi }_1^2}$, when${\displaystyle {\mathcal{H}}_0}$ is true. While asymptotically identical, calculating the LM statistic using theouter-gradient-product estimator of the Fisher information matrix can lead to bias in small samples.^[7]

Note that some texts use an alternative notation, in which the statistic${\displaystyle S^{\ast }(\theta )=\sqrt{S(\theta )}}$ is tested against a normal distribution. This approach is equivalent and gives identical results.

${\displaystyle {\left(\frac{\mathrm{\partial }\log L(\theta \mid x)}{\mathrm{\partial }\theta }\right)}_{\theta ={\theta }_0}\ge C}$

where${\displaystyle L}$ is thelikelihood function,${\displaystyle {\theta }_0}$ is the value of the parameter of interest under the null hypothesis, and${\displaystyle C}$ is a constant set depending on the size of the test desired (i.e. the probability of rejecting${\displaystyle H_0}$ if${\displaystyle H_0}$ is true; seeType I error).

The score test is the most powerful test for small deviations from${\displaystyle H_0}$. To see this, consider testing${\displaystyle \theta ={\theta }_0}$ versus${\displaystyle \theta ={\theta }_0+h}$. By theNeyman–Pearson lemma, the most powerful test has the form

${\displaystyle \frac{L({\theta }_0+h\mid x)}{L({\theta }_0\mid x)}\ge K;}$

Taking the log of both sides yields

${\displaystyle \log L({\theta }_0+h\mid x)-\log L({\theta }_0\mid x)\ge \log K.}$

The score test follows making the substitution (byTaylor series expansion)

${\displaystyle \log L({\theta }_0+h\mid x)\approx \log L({\theta }_0\mid x)+h\times {\left(\frac{\mathrm{\partial }\log L(\theta \mid x)}{\mathrm{\partial }\theta }\right)}_{\theta ={\theta }_0}}$

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

If the null hypothesis is true, thelikelihood ratio test, theWald test, and the score test are asymptotically equivalent tests of hypotheses.^[8][9] When testingnested models, the statistics for each test then converge to a Chi-squared distribution with degrees of freedom equal to the difference in degrees of freedom in the two models. If the null hypothesis is not true, however, the statistics converge to a noncentral chi-squared distribution with possibly different noncentrality parameters.

A more general score test can be derived when there is more than one parameter. Suppose that${\displaystyle {\hat{\theta }}_0}$ is themaximum likelihood estimate of${\displaystyle \theta }$ under the null hypothesis${\displaystyle H_0}$ while${\displaystyle U}$ and${\displaystyle I}$ are respectively, the score vector and the Fisher information matrix. Then

${\displaystyle U^T({\hat{\theta }}_0)I^{-1}({\hat{\theta }}_0)U({\hat{\theta }}_0)\sim {\chi }_k^2}$

asymptotically under${\displaystyle H_0}$, where${\displaystyle k}$ is the number of constraints imposed by the null hypothesis and

${\displaystyle U({\hat{\theta }}_0)=\frac{\mathrm{\partial }\log L({\hat{\theta }}_0\mid x)}{\mathrm{\partial }\theta }}$

and

${\displaystyle I({\hat{\theta }}_0)=-\mathrm{E}\left(\frac{{\mathrm{\partial }}^2\log L({\hat{\theta }}_0\mid x)}{\mathrm{\partial }\theta \mathrm{\partial }{\theta }^{\prime }}\right).}$

This can be used to test${\displaystyle H_0}$.

The actual formula for the test statistic depends on which estimator of the Fisher information matrix is being used.^[10]

In many situations, the score statistic reduces to another commonly used statistic.^[11]

Inlinear regression, the Lagrange multiplier test can be expressed as a function of theF-test.^[12]

When the data follows a normal distribution, the score statistic is the same as thet statistic.^{[clarification needed]}

When the data consists of binary observations, the score statistic is the same as the chi-squared statistic in thePearson's chi-squared test.

• Fisher information
• Uniformly most powerful test
• Score (statistics)
• Sup-LM test

• Buse, A. (1982). "The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note".The American Statistician.36 (3a):153–157.doi:10.1080/00031305.1982.10482817.
• Godfrey, L. G. (1988). "The Lagrange Multiplier Test and Testing for Misspecification : An Extended Analysis".Misspecification Tests in Econometrics. New York: Cambridge University Press. pp. 69–99.ISBN 0-521-26616-5.
• Ma, Jun; Nelson, Charles R. (2016). "The superiority of the LM test in a class of econometric models where the Wald test performs poorly".Unobserved Components and Time Series Econometrics. Oxford University Press. pp. 310–330.doi:10.1093/acprof:oso/9780199683666.003.0014.ISBN 978-0-19-968366-6.
• Rao, C. R. (2005). "Score Test: Historical Review and Recent Developments".Advances in Ranking and Selection, Multiple Comparisons, and Reliability. Boston: Birkhäuser. pp. 3–20.ISBN 978-0-8176-3232-8.

• Statistical tests

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