Score test
Encyclopedia
A score test is a statistical test of a simple null hypothesis
that a parameter of interest
is
equal to some particular value
. It is the most powerful
test when the true value of
is close to 
. 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 makes testing feasible when the unconstrained maximum likelihood estimate is a boundary point in the parameter space.
be the likelihood function
which depends on a univariate parameter
and let 
be the data. The score
is
where
The observed Fisher information is,
The statistic to test
is
which takes a
distribution asymptotically when 
is true.
is tested against a normal distribution. This approach is equivalent and gives identical results.
Where
is the likelihood function
,
is the value of the parameter of interest under the
null hypothesis, and
is a constant set depending on
the size of the test desired (i.e. the probability of rejecting
if
is true; see Type I error).
The score test is the most powerful test for small deviations from
.
To see this, consider testing
versus
. 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
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
that a parameter of interest
isequal to some particular value
. It is the most powerfulStatistical power
The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...
test when the true value of
is close to . 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 makes testing feasible when the unconstrained maximum likelihood estimate is a boundary point in the parameter space.The statistic
Let be the likelihood functionLikelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...
which depends on a univariate parameter
and let be the data. The scoreScore (statistics)
In statistics, the score, score function, efficient score or informant plays an important role in several aspects of inference...
is
whereThe observed Fisher information is,
The statistic to test
iswhich takes a
distribution asymptotically when is true.Note on notation
Note that some texts use an alternative notation, in which the statistic is tested against a normal distribution. This approach is equivalent and gives identical results.As most powerful test for small deviations
Where
is the likelihood functionLikelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...
,
is the value of the parameter of interest under thenull hypothesis, and
is a constant set depending onthe size of the test desired (i.e. the probability of rejecting
if is true; see Type I error).The score test is the most powerful test for small deviations from
.To see this, consider testing
versus. By the Neyman-Pearson lemma, the most powerful test has the formTaking the log of both sides yields
The score test follows making the substitution
-
and identifying the
above with 
.
Relationship with other hypothesis tests
The likelihood ratio testLikelihood-ratio testIn statistics, a likelihood ratio test is a statistical test used to compare the fit of two models, one of which is a special case of the other . The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other...
, the Wald testWald testThe Wald test is a parametric statistical test named after Abraham Wald with a great variety of uses. Whenever a relationship within or between data items can be expressed as a statistical model with parameters to be estimated from a sample, the Wald test can be used to test the true value of the...
, 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
is the maximum likelihoodMaximum likelihoodIn statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....
estimate of
under the null hypothesis 
. Then
asymptotically under
, where 
is the number of constraints imposed by the null hypothesis and
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 testPearson's chi-squared testPearson's chi-squared test is the best-known of several chi-squared tests – statistical procedures whose results are evaluated by reference to the chi-squared distribution. Its properties were first investigated by Karl Pearson in 1900...
.
When the data consists of failure time data in two groups, the score statistic for the Cox partial likelihoodProportional hazards modelsProportional hazards models are a class of survival models in statistics. Survival models relate the time that passes before some event occurs to one or more covariates that may be associated with that quantity. In a proportional hazards model, the unique effect of a unit increase in a covariate...
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.
See also
- Fisher informationFisher informationIn mathematical statistics and information theory, the Fisher information is the variance of the score. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior...
- Uniformly most powerful testUniformly most powerful testIn statistical hypothesis testing, a uniformly most powerful test is a hypothesis test which has the greatest power 1 − β among all possible tests of a given size α...
- Score (statistics)Score (statistics)In statistics, the score, score function, efficient score or informant plays an important role in several aspects of inference...
- Fisher information