Likelihood-ratio test

Not to be confused with the use of likelihood ratios in diagnostic testing.

In statistics, a likelihood ratio test is a statistical test used to compare the fit of two models, one of which (the null model) is a special case of the other (the alternative model). The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other. This likelihood ratio, or equivalently its logarithm, can then be used to compute a p-value, or compared to a critical value to decide whether to reject the null model in favour of the alternative model. When the logarithm of the likelihood ratio is used, the statistic is known as a log-likelihood ratio statistic, and the probability distribution of this test statistic, assuming that the null model is true, can be approximated using Wilks' theorem.

In the case of distinguishing between two models, each of which has no unknown parameters, use of the likelihood ratio test can be justified by the Neyman–Pearson lemma, which demonstrates that such a test has the highest power among all competitors.

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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 (usually denoted D) is twice the difference in these log-likelihoods:

\begin{align} D & = -2\ln\left( \frac{\text{likelihood for null model}}{\text{likelihood for alternative model}} \right) \\ &= -2\ln(\text{likelihood for null model}) %2B 2\ln(\text{likelihood for alternative model})] \\ \end{align}

The model with more parameters will always fit at least as well (have a greater log-likelihood). Whether it fits significantly better and should thus be preferred is determined by deriving the probability or p-value of the difference D. In many cases, the probability distribution of the test statistic is approximately a chi-squared distribution with degrees of freedom equal to df2 − df1 , if the nested model with fewer parameters is correct. Symbols df1 and df2 represent the number of free parameters of models 1 and 2, the null model and the alternative model, respectively. The test requires nested models, that is: models in which the more complex one can be transformed into the simpler model by imposing a set of constraints on the parameters.need examples or references to describe

For example: if model 1 has 1 free parameter and a log-likelihood of −8024 and the alternative model has 3 degrees of freedom and a LL of −8012, then the probability of this difference is that of chi-squared value of +2·(8024 − 8012) = 24 with 3 − 1 = 2 degrees of freedom. Certain assumptions must be met for the statistic to follow a chi-squared distribution and often empirical p-values are computed.

Background

The likelihood ratio, often denoted by \Lambda (the capital Greek letter lambda), is the ratio of the likelihood function varying the parameters over two different sets in the numerator and denominator. A likelihood-ratio test is a statistical test for making a decision between two hypotheses based on the value of this ratio.

It is central to the Neyman–Pearson approach to statistical hypothesis testing, and, like statistical hypothesis testing generally, is both widely used and much criticized; see Criticism, below.

Simple-versus-simple hypotheses

A statistical model is often a parametrized family of probability density functions or probability mass functions f(x|\theta). A simple-vs-simple hypotheses test has completely specified models under both the null and alternative hypotheses, which for convenience are written in terms of fixed values of a notional parameter \theta:

\begin{align} H_0 &:& \theta=\theta_0 ,\\ H_1 &:& \theta=\theta_1 . \end{align}

Note that under either hypothesis, the distribution of the data is fully specified; there are no unknown parameters to estimate. The likelihood ratio test statistic can be written as[1]:

\Lambda(x) = \frac{ L(\theta_0|x) }{ L(\theta_1|x) } = \frac{ f(x|\theta_0) }{ f(x|\theta_1) }

or

\Lambda(x)=\frac{L(\theta_0\mid x)}{\sup\{\,L(\theta\mid x):\theta\in\{\theta_0,\theta_1\}\}},

where L(\theta|x) is the likelihood function. Note that some references may use the reciprocal as the definition.[2] In the form stated here, the likelihood ratio is small if the alternative model is better than the null model and the likelihood ratio test provides the decision rule as:

If \Lambda > c , do not reject H_0;
If \Lambda < c , reject H_0;
Reject with probability q if \Lambda = c .

The values c, \; q are usually chosen to obtain a specified significance level \alpha, through the relation: q\cdot P(\Lambda=c \;|\; H_0) %2B P(\Lambda < c \; | \; H_0) = \alpha . The Neyman-Pearson lemma states that this likelihood ratio test is the most powerful among all level-\alpha tests for this problem.

Definition (likelihood ratio test for composite hypotheses)

A null hypothesis is often stated by saying the parameter \theta is in a specified subset \Theta_0 of the parameter space \Theta.

\begin{align} H_0 &:& \theta \in \Theta_0\\ H_1 &:& \theta \in \Theta_0^{\complement} \end{align}

The likelihood function is L(\theta|x) = f(x|\theta) (with f(x|\theta) being the pdf or pmf) is a function of the parameter \theta with x held fixed at the value that was actually observed, i.e., the data. The likelihood ratio test statistic is [3]

\Lambda(x)=\frac{\sup\{\,L(\theta\mid x):\theta\in\Theta_0\,\}}{\sup\{\,L(\theta\mid x):\theta\in\Theta\,\}}.

Here, the \sup notation refers to the Supremum function.

A likelihood ratio test is any test with critical region (or rejection region) of the form \{x|\Lambda \le c\} where c is any number satisfying 0\le c\le 1. Many common test statistics such as the Z-test, the F-test, Pearson's chi-squared test and the G-test are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof.

Interpretation

Being a function of the data x, 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 likelihood of an observed outcome under the null hypothesis. The denominator corresponds to the maximum likelihood of an observed outcome varying parameters over the whole parameter space. The numerator of this ratio is less than the denominator. The likelihood ratio hence is between 0 and 1. Lower values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared to the alternative. Higher values of the statistic mean that the observed outcome was more than or equally likely or nearly as likely to occur under the null hypothesis as compared to the alternative, and the null hypothesis cannot be rejected.

Distribution: Wilks' theorem

If the distribution of the likelihood ratio corresponding to a particular null and alternative hypothesis can be explicitly determined then it can directly be used to form decision regions (to accept/reject the null hypothesis). In most cases, however, the exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine. A convenient result, attributed to Samuel S. Wilks, says that as the sample size n approaches \infty, the test statistic -2 \log(\Lambda) for a nested model will be asymptotically \chi^2 distributed with degrees of freedom equal to the difference in dimensionality of \Theta and \Theta_0.[4] This means that for a great variety of hypotheses, a practitioner can compute the likelihood ratio \Lambda for the data and compare -2\log(\Lambda) to the chi squared value corresponding to a desired statistical significance as an approximate statistical test.

Examples

Coin tossing

An example, in the case of Pearson's test, we might try to compare two coins to determine whether they have the same probability of coming up heads. Our observation 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 the coin for that row came up heads or tails. The contents of this table are our observation X.

Heads Tails
Coin 1 k_{1H} k_{1T}
Coin 2 k_{2H} k_{2T}

Here \Theta consists of the parameters p_{1H}, p_{1T}, p_{2H}, and p_{2T}, which are the probability that coins 1 and 2 come up heads or tails. The hypothesis space H is defined by the usual constraints on a distribution, 0 \le p_{ij} \le 1, and p_{iH} %2B p_{iT} = 1 . The null hypothesis H_0 is the sub-space where p_{1j} = p_{2j}. In all of these constraints, i = 1,2 and j = H,T.

Writing n_{ij} for the best values for p_{ij} under the hypothesis H, maximum likelihood is achieved with

n_{ij} = \frac{k_{ij}}{k_{iH}%2Bk_{iT}}.

Writing m_{ij} for the best values for p_{ij} under the null hypothesis H_0, maximum likelihood is achieved with

m_{ij} = \frac{k_{1j}%2Bk_{2j}}{k_{1H}%2Bk_{2H}%2Bk_{1T}%2Bk_{2T}},

which does not depend on the coin i.

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 H to be reduced to the one-dimensional H_0, the asymptotic distribution for the test will be \chi^2(1), the \chi^2 distribution with one degree of freedom.

For the general contingency table, we can write the log-likelihood ratio statistic as

-2 \log \Lambda = 2\sum_{i, j} k_{ij} \log \frac{n_{ij}}{m_{ij}}.

Criticism

Bayesian criticisms of classical likelihood ratio tests focus on two issues:

  1. the supremum function in the calculation of the likelihood ratio, saying that this takes no account of the uncertainty about θ and that using maximum likelihood estimates in this way can promote complicated alternative hypotheses with an excessive number of free parameters;
  2. testing the probability that the sample would produce a result as extreme or more extreme under the null hypothesis, saying that this bases the test on the probability of extreme events that did not happen.

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. From a frequentist approach, uncertainty about the parameters is taken into account in the probability distribution of the test statistic.

A frequentist reply to this critique is that likelihood ratio tests provide a practicable approach to statistical inference – they can easily be computed, by contrast to Bayesian posterior probabilities, which are more computationally intensive. The Bayesian reply to the latter is that computers obviate any such advantage.

A point in favour of using likelihood-ratio tests is that these satisfy the likelihood principle, which expresses the requirement put forward by many statisticians that statistical inference should follow procedures based on the likelihood function. Bayesian inference embodies the likelihood principle (ie that all the information from the sample data is expressed through the likelihood function), which using Bayes theorem it combines in a natural way with prior knowledge available prior to taking the sample data. In some fields, such as medical diagnosis and epidemiology, such information can be very important, reflecting for example the background prevalence of a condition or disease.

References

  1. ^ Mood, Duane C. Boes; Introduction to theory of Statistics (page 410)
  2. ^ Cox, D. R. and Hinkley, D. V Theoretical Statistics, Chapman and Hall, 1974. (page 92)
  3. ^ George Casella, Roger L. Berger Statistical Inference, Second edition (page 375)
  4. ^ Wilks, S. S. (1938). "The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses". The Annals of Mathematical Statistics 9: 60–62. doi:10.1214/aoms/1177732360.  edit

External links
 
Statistical theory
Bayesian inference
Frequentist inference
Specific tests
General estimation