Score test

Rao's score test, or the score test (often known as the Lagrange multiplier test in econometrics^[1]) is a statistical test of a simple null hypothesis that a parameter of interest $\theta$ is equal to some particular value $\theta_0$ . It is the most powerful test when the true value of $\theta$ is close to $\theta_0$ . The main advantage of the Score-test is that it does not require an estimate of the information under the alternative hypothesis or unconstrained maximum likelihood. This constitutes a potential advantage in comparison to other tests, such as the Wald test and the generalized likelihood ratio test (GLRT). This makes testing feasible when the unconstrained maximum likelihood estimate is a boundary point in the parameter space.

1 Single parameter test
2 Multiple parameters
3 Special cases
4 See also
5 References

Single parameter test

The statistic

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

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

The Fisher information is^[2]

\mathcal{I}(\theta) = - \operatorname{E} \left[\left. \frac{\partial^2}{\partial\theta^2} \log L(X;\theta)\right|\theta \right]\,.

The statistic to test $\mathcal{H}_0:\theta=\theta_0$ is $S(\theta_0) = \frac{U(\theta_0)^2}{I(\theta_0)}$

which has an asymptotic distribution of $\chi^2_1$ , when $\mathcal{H}_0$ is true.

Note on notation

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

Justification

The case of a likelihood with nuisance parameters

As most powerful test for small deviations

\left(\frac{\partial \log L(\theta \mid 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\mid x)}{L(\theta_0\mid x)} \geq K;

Taking the log of both sides yields

\log L(\theta_0 + h \mid x ) - \log L(\theta_0\mid x) \geq \log K.

The score test follows making the substitution (by Taylor series expansion)

\log L(\theta_0+h\mid x) \approx \log L(\theta_0\mid x) + h\times\left(\frac{\partial \log L(\theta \mid x)}{\partial \theta}\right)_{\theta=\theta_0}

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

Relationship with other hypothesis tests

The likelihood ratio test, the Wald test, and the Score test are asymptotically equivalent tests of hypotheses.^[3] When testing nested models, the statistics for each test converge to a Chi-squared distribution with degrees of freedom equal to the difference in degrees of freedom in the two models.

Multiple parameters

A more general score test can be derived when there is more than one parameter. Suppose that $\hat{\theta}_0$ is the maximum likelihood estimate of $\theta$ under the null hypothesis $H_0$ . Then

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

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

U(\hat{\theta}_0) = \frac{\partial \log L(\hat{\theta}_0 | x)}{\partial \theta}

and

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

This can be used to test $H_0$ .

Special cases

In many situations, the score statistic reduces to another commonly used statistic.^[4]

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

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

When the data consists of failure time data in two groups, the score statistic for the Cox partial likelihood is the same as the log-rank statistic in the log-rank test. Hence the log-rank test for difference in survival between two groups is most powerful when the proportional hazards assumption holds.

References

^ Bera, Anil K.; Bilias, Yannis (2001). "Rao's score, Neyman's C(α) and Silvey's LM tests: An essay on historical developments and some new results". Journal of Statistical Planning and Inference 97: 9–44. doi:10.1016/S0378-3758(00)00343-8. Engle, Robert F (1984) . Wald, Likelihood Ratio and Lagrange Multiplier tests in Econometrics. in Handbook of Econometrics, Volume II, Edited by Z. Griliches and M.D. Intriligator. Elsevier Science Publishers BV.
^ Lehmann and Casella, eq. (2.5.16).
^ Engle, Robert F. (1983). "Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics". In Intriligator, M. D.; and Griliches, Z. Handbook of Econometrics II. Elsevier. pp. 796–801. ISBN 978-0-444-86185-6.
^ Cook, T. D.; DeMets, D. L., eds. (2007). Introduction to Statistical Methods for Clinical Trials. Chapman and Hall. pp. 296–297. ISBN 1-58488-027-9.

fonte: https://web.archive.org/web/20151223003752/https://en.wikipedia.org/wiki/Score_test

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