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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 that event, the likelihood test is still a sensible test statistic and even possess some asymptotic optimality properties, but the significance (the p-value) can not be reliably estimated using the chi-squared distribution with the number of degrees of freedom prescribed by Wilks. In some cases, the asymptotic null-hypothesis distribution of ...
Production of a small p-value by multiple testing. 30 samples of 10 dots of random color (blue or red) are observed. On each sample, a two-tailed binomial test of the null hypothesis that blue and red are equally probable is performed. The first row shows the possible p-values as a function of the number of blue and red dots in the sample.
The interpretation of a p-value is dependent upon stopping rule and definition of multiple comparison. The former often changes during the course of a study and the latter is unavoidably ambiguous. (i.e. "p values depend on both the (data) observed and on the other possible (data) that might have been observed but weren't"). [68]
Rather than the Wald method, the recommended method [21] to calculate the p-value for logistic regression is the likelihood-ratio test (LRT), which for these data give (see § Deviance and likelihood ratio tests below).
The p-value was first formally introduced by Karl Pearson, in his Pearson's chi-squared test, [39] using the chi-squared distribution and notated as capital P. [39] The p-values for the chi-squared distribution (for various values of χ 2 and degrees of freedom), now notated as P, were calculated in (Elderton 1902), collected in (Pearson 1914 ...
In statistics, Wilks' lambda distribution (named for Samuel S. Wilks), is a probability distribution used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test and multivariate analysis of variance (MANOVA).
The likelihood ratio is central to likelihoodist statistics: the law of likelihood states that the degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio. In frequentist inference, the likelihood ratio is the basis for a test statistic, the so-called likelihood-ratio test.