Score test

In statistics, the score test assesses constraints on statistical parameters based on the gradient of the likelihood function—known as the score—evaluated at the hypothesized parameter value under the null hypothesis. Intuitively, if the restricted estimator is near the maximum of the likelihood function, the score should not differ from zero by more than sampling error. While the finite sample distributions of score tests are generally unknown, it has an asymptotic χ2-distribution under the null hypothesis as first proved by C. R. Rao in 1948, a fact that can be used to determine statistical 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 the magnitude of the Lagrange 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 by S. D. Silvey in 1959, which led to the name Lagrange multiplier test that has become more commonly used, particularly in econometrics, since Breusch and Pagan's much-cited 1980 paper.
The main advantage of the score test over the Wald test and likelihood-ratio test is that the score test only requires the computation of the restricted estimator. This makes testing feasible when the unconstrained maximum likelihood estimate is a boundary point in the parameter space. Further, because the score test only requires the estimation of the likelihood function under the null hypothesis, it is less specific than the other two tests about the precise nature of the alternative hypothesis.
Single parameter test.
The statistic.
Let formula_1 be the likelihood function which depends on a univariate parameter formula_2 and let formula_3 be the data. The score formula_4 is defined as
The Fisher information is
The statistic to test formula_7 is
formula_8
which has an asymptotic distribution of formula_9, when formula_10 is true. While asymptotically identical, calculating the LM statistic using the outer-gradient-product estimator of the Fisher information matrix can lead to bias in small samples.
Note on notation.
Note that some texts use an alternative notation, in which the statistic formula_11 is tested against a normal distribution. This approach is equivalent and gives identical results.
As most powerful test for small deviations.
where formula_1 is the likelihood function, formula_14 is the value of the parameter of interest under the null hypothesis, and formula_15 is a constant set depending on the size of the test desired (i.e. the probability of rejecting formula_16 if formula_16 is true; see Type I error).
The score test is the most powerful test for small deviations from formula_16. To see this, consider testing formula_19 versus formula_20. By the Neyman–Pearson lemma, the most powerful test has the form
Taking the log of both sides yields
The score test follows making the substitution (by Taylor series expansion)
and identifying the formula_15 above with formula_25.
Relationship with other hypothesis tests.
If the null hypothesis is true, the likelihood ratio test, the Wald test, and the Score test are asymptotically equivalent tests of hypotheses. When testing nested 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.
Multiple parameters.
A more general score test can be derived when there is more than one parameter. Suppose that formula_26 is the maximum likelihood estimate of formula_2 under the null hypothesis formula_16 while formula_29 and formula_30 are respectively, the score and the Fisher information matrices under the alternative hypothesis. Then
asymptotically under formula_16, where formula_33 is the number of constraints imposed by the null hypothesis and
and
This can be used to test formula_16.
Special cases.
In many situations, the score statistic reduces to another commonly used statistic.
In linear regression, the Lagrange multiplier test can be expressed as a function of the "F"-test.
When the data follows a normal distribution, the score statistic is the same as the t statistic.
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.