Rao's score test, or the score test (often known as the Lagrange multiplier test in econometrics) is a statistical test of a simple null hypothesis that a parameter of interest
Contents
The statistic
Let
The Fisher information is
The statistic to test
which has an asymptotic distribution of
Note on notation
Note that some texts use an alternative notation, in which the statistic
As most powerful test for small deviations
Where
The score test is the most powerful test for small deviations from
Taking the log of both sides yields
The score test follows making the substitution (by Taylor series expansion)
and identifying the
Relationship with other hypothesis tests
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 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
asymptotically under
and
This can be used to test
Special cases
In many situations, the score statistic reduces to another commonly used statistic.
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.