The likelihood ratio, often denoted by (the capital Greek letter lambda), is the ratio of the maximum probability of a result under two different hypotheses. A likelihood-ratio test is a statistical test for making a decision between two hypotheses based on the value of this ratio.
A statistical model is often a parametrized family of probability density functions or probability mass functions . A simple-vs-simple hypotheses test hypothesises single values of under both the null and alternative hypotheses:
If reject .
If accept (or don't reject) .
The critical value c is usually chosen to obtain a specified significance level , through the relation: (if x is discrete, some randomization on the boundary may be needed). The Neyman-Pearson lemma states that this likelihood ratio test is the most powerful among all level- tests for this problem.
Being a function of the data , the LR is therefore a statistic. The likelihood-ratio test rejects the null hypothesis if the value of this statistic is too small. How small is too small depends on the significance level of the test, i.e., on what probability of Type I error is considered tolerable ("Type I" errors consist of the rejection of a null hypothesis that is true).
The numerator corresponds to the maximum probability of an observed result under the null hypothesis. The denominator corresponds to the maximum probability of an observed result under the alternative hypothesis. Under certain regularity conditions, the numerator of this ratio is less than the denominator. The likelihood ratio under those conditions is between 0 and 1. Lower values of the likelihood ratio mean that the observed result was less likely to occur under the null hypothesis. Higher values mean that the observed result was more likely to occur under the null hypothesis.
One example of a likelihood ratio would be the likelihood that a given test result would be expected in a patient with a certain disorder compared to the likelihood that same result would occur in a patient without the target disorder.
As another example, one can imagine that one is trying to figure out whether one is in line for tickets to a football game or to the opera (assuming that one cannot ask people which line one is in, that one does not see any signs, etcetera). The only thing that one is allowed to do is ask other people in line whether or not they like football. One estimates that 90% of people in the line for a football game like football, while 10% of people in the line for the opera like football. Then the likelihood ratio is computed as:
(Probability of liking football given that someone is in line for football game)/(Probability of liking football given that someone's in line for the opera) = .9/.1 = 9
The larger one's likelihood ratio, the higher the chance that one will be able to correctly infer whether one is at the football game or at the opera given the people's responses. In other words, if one's LR is large, one can be more confident in one's decision as to whether one in line for football tickets or not given that one only asked a limited number of people whether or not they liked football. For an infinite likelihood ratio, one would be 100% sure that one is in line for the football game after only asking one person, who said "yes."
Writing for the best values for under the hypothesis , maximum likelihood is achieved with
The hypothesis and null hypothesis can be rewritten slightly so that they satisfy the constraints for the logarithm of the likelihood ratio to have the desired nice distribution. Since the constraint causes the two-dimensional to be reduced to the one-dimensional , the asymptotic distribution for the test will be , the distribution with one degree of freedom.
For the general contingency table, we can write the log-likelihood ratio statistic as
Instead they put forward methods such as Bayes factors, which explicitly take uncertainty about the parameters into account, and which are based on the evidence that did occur.
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