Wilks' theorem
In statistics Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produce confidence intervals for maximum-likelihood estimates or as a test statistic for performing the likelihood-ratio test. Statistical tests (such as hypothesis testing) generally require knowledge of the probability distribution of the test statistic. This is often a problem for likelihood ratios, where the probability distribution can be very difficult to determine. A convenient result by Samuel S. Wilks says that as the sample size approaches formula_1, the distribution of the test statistic formula_2 asymptotically approaches the chi -squared (formula_3) distribution under the null hypothesis formula_4. Here, formula_5 denotes the likelihood ratio, and the formula_3 distribution has degrees of freedom equal to the difference in dimensionality of formula_7 and formula_8, where formula_7 is the full parameter space and formula_8 is the subset of the parameter space associated with formula_4. This result means that for large samples and a great variety of hypotheses, a practitioner can compute the likelihood ratio formula_5 for the data and compare formula_13 to the formula_3 value corresponding to a desired statistical significance as an approximate statistical test. The theorem no longer applies when any  one of the estimated parameters is at its upper or lower limit: Wilks’ theorem assumes that the ‘true’ but unknown values of the estimated parameters lie within the interior of the supported parameter space. The likelihood maximum may no longer have the assumed ellipsoidal shape if the maximum value for the population likelihood function occurs at some boundary-value of one of the parameters, i.e. on an edge of the parameter space. In that event, the likelihood test will still be valid and optimal as guaranteed by the Neyman-Pearson lemma, but the significance (the -value) can not be reliably  estimated using the chi-squared distribution with the number of degrees of freedom prescribed by Wilks. Use. Each of the two competing models, the null model and the alternative model, is separately fitted to the data and the log-likelihood recorded. The test statistic (often denoted by ) is twice the log of the likelihoods ratio, "i.e.", it is twice the difference in the log-likelihoods: The model with more parameters (here "alternative") will always fit at least as well — i.e., have the same or greater log-likelihood — than the model with fewer parameters (here "null"). Whether the fit is significantly  better and should thus be preferred is determined by deriving how likely (-value) it is to observe such a difference  by "chance alone", if the model with fewer parameters were true. Where the null hypothesis represents a special case of the alternative hypothesis, the probability distribution of the test statistic is approximately a chi-squared distribution with degrees of freedom equal to formula_16, respectively the number of free parameters of models "alternative" and "null". For example: If the null model has 1 parameter and a log-likelihood of −8024 and the alternative model has 3 parameters and a log-likelihood  of −8012, then the probability of this difference is that of chi-squared value of formula_17 with formula_18 degrees of freedom, and is equal to formula_19. Certain assumptions must be met for the statistic to follow a chi-squared distribution, but empirical -values may also be computed if those conditions are not met. Examples. Coin tossing. An example of Pearson's test is a comparison of two coins to determine whether they have the same probability of coming up heads. The observations can be put into a contingency table with rows corresponding to the coin and columns corresponding to heads  or tails. The elements of the contingency table will be the number of times each coin came up heads or tails. The contents of this table are our observations . Here consists of the possible combinations of values of the parameters formula_21, formula_22, formula_23, and formula_24, which are the probability that coins 1 and 2 come up heads or tails. In what follows, formula_25 and formula_26. The hypothesis space is constrained by the usual constraints on a probability distribution, formula_27, and formula_28. The space of the null hypothesis formula_4 is the subspace where formula_30. The dimensionality of the full parameter space  is 2 (either of the formula_31 and either of the formula_32 may be treated as free parameters under the hypothesis formula_33), and the dimensionality of formula_34 is 1 (only one of the formula_35 may be considered a free parameter under the null hypothesis formula_36). Writing formula_37 for the best estimates of formula_38 under the hypothesis , the maximum likelihood estimate is given by Similarly, the maximum likelihood estimates of formula_38 under the null hypothesis formula_4 are given by which does not depend on the coin . 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 distribution. Since the constraint causes the two-dimensional to be reduced to the one-dimensional formula_4, the asymptotic distribution for the test will be formula_44, the formula_3 distribution with one degree of freedom. For the general contingency table, we can write the log-likelihood ratio statistic as Invalidity for random or mixed effects models. Wilks’ theorem assumes that the true but unknown values of the estimated parameters are in the interior of the parameter space. This is commonly violated in random or mixed effects models, for example, when one of  the variance components is negligible relative to the others. In some such cases, one variance component can be effectively zero, relative to the others, or in other cases the models can be improperly nested. To be clear: These limitations on Wilks’ theorem do "not" negate any power properties of a particular likelihood ratio test. The only issue is that a formula_3 distribution is sometimes a poor choice for estimating the statistical significance of the result. Bad examples. Pinheiro and Bates (2000) showed that the true distribution of this likelihood ratio chi-square statistic could be substantially different from the naïve  formula_3 – often dramatically so. The naïve assumptions could give significance probabilities (-values) that are, on average, far too large in some cases and far too small in others. In general, to test random effects, they recommend using Restricted maximum likelihood (REML). For fixed-effects testing, they say, “a likelihood ratio test for REML fits is not feasible”, because changing the fixed effects specification changes the meaning of the mixed effects, and the restricted model is therefore not nested within the larger model. As a demonstration, they set either one or two random effects variances to zero in simulated tests. In  those particular examples, the simulated -values with restrictions most closely matched a 50–50 mixture of formula_49 and formula_50. (With , formula_51 is 0 with probability 1. This means that a good approximation was formula_52) Pinheiro and Bates also simulated tests of different fixed effects. In one test of a factor with 4 levels (degrees of freedom = 3), they found that a 50–50 mixture of formula_53 and formula_54 was a good match for actual -values obtained by simulation – and the error in using the naïve formula_53 “may not be too alarming.” However, in another test of a factor with 15 levels , they found a reasonable match to formula_56 – 4 more degrees of freedom than the 14 that one would get from a naïve (inappropriate) application of Wilks’ theorem, "and" the simulated -value was several times the naïve formula_57. They conclude that for testing fixed effects, “it's wise to use simulation.”