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In mathematics, the ratio test is a test (or "criterion") for the convergence of a series =, where each term is a real or complex number and a n is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.
The likelihood-ratio test, also known as Wilks test, [2] is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test. [3] In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent.
In fact, post-test probability, as estimated from the likelihood ratio and pre-test probability, is generally more accurate than if estimated from the positive predictive value of the test, if the tested individual has a different pre-test probability than what is the prevalence of that condition in the population.
To be clear: These limitations on Wilks’ theorem do not negate any power properties of a particular likelihood ratio test. [3] The only issue is that a χ 2 {\displaystyle \chi ^{2}} distribution is sometimes a poor choice for estimating the statistical significance of the result.
The sequential probability ratio test (SPRT) is a specific sequential hypothesis test, developed by Abraham Wald [1] and later proven to be optimal by Wald and Jacob Wolfowitz. [2] Neyman and Pearson's 1933 result inspired Wald to reformulate it as a sequential analysis problem.
The ratio estimator is a statistical estimator for the ratio of means of two random variables. Ratio estimates are biased and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymmetrical and symmetrical tests such as the t test should not be used to generate confidence intervals.
There is nothing magical about a sample size of 1 000, it's just a nice round number that is well within the range where an exact test, chi-square test, and G–test will give almost identical p values. Spreadsheets, web-page calculators, and SAS shouldn't have any problem doing an exact test on a sample size of 1 000 . — John H. McDonald [2]
There exist several alternatives to the Wald test, namely the likelihood-ratio test and the Lagrange multiplier test (also known as the score test). Robert F. Engle showed that these three tests, the Wald test, the likelihood-ratio test and the Lagrange multiplier test are asymptotically equivalent. [17]