Likelihood-ratio test: Definition from Answers.com

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In statistics, a likelihood ratio test is used to compare the fit of two models one of which is nested within the other.

Both models are fitted to the data and their log-likelihood recorded. The test statistic (usually denoted D) is twice the difference in these log-likelihoods:

$\begin{align} D & = -2(\ln(\text{likelihood for first model}) - \ln(\text{likelihood for second model})) \\ & = -2\ln\left( \frac{\text{likelihood for first model}}{\text{likelihood for second model}} \right). \end{align}$

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 can be determined by deriving the probability or p-value of the obtained difference D. In many cases, the probability distribution of the test statistic can be approximated by a chi-squared distribution with (df1 − df2) degrees of freedom, where df1 and df2 are the degrees of freedom of models 1 and 2 respectively.

The test requires nested models, that is, models in which the more complex one can be transformed into the simpler by fixing one or more parameters.

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

1 Background
2 Simple-versus-simple hypotheses
3 Definition (likelihood ratio test for composite hypotheses)
- 3.1 Interpretation
- 3.2 Approximation
4 Examples
- 4.1 Medical
- 4.2 Coin tossing
5 Criticism
- 5.1 Theoretical
- 5.2 Practical
6 References
7 See also
- 7.1 Context
8 External links

Background

The likelihood ratio, often denoted by $Λ$ (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 $f (x | θ)$ . 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 $θ$ :

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

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]:

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

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

where $L (θ | x)$ 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:

Λ > c

, do not reject

H 0

;

Λ < c

, reject

H 0

;

Reject with probability

q

Λ = c .

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

Definition (likelihood ratio test for composite hypotheses)

A null hypothesis is often stated by saying the parameter $θ$ is in a specified subset $Θ 0$ of the parameter space $Θ$ .

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

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

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

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-square 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 probability of an observed outcome under the null hypothesis. The denominator corresponds to the maximum probability 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 alternate. 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 alternate, and the null hypothesis cannot be rejected.

Approximation

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, though, says that as the sample size $n$ approaches $\infty$ , the test statistic $- 2log(Λ)$ for a nested model will be asymptotically $χ 2$ distributed with degrees of freedom equal to the difference in dimensionality of $Θ$ and $Θ 0$ . This means that for a great variety of hypotheses, a practitioner can compute the likelihood ratio $Λ$ for the data and compare $- 2log(Λ)$ to the chi squared value corresponding to a desired statistical significance as an approximate statistical test.

Examples

Medical

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.

Some sources distinguish between LR+ and LR−.^[4] A worked example is shown below.

Relationships among terms

		Condition (as determined by "Gold standard")
		Positive	Negative
Test outcome	Positive	True Positive	False Positive (Type I error, P-value)	→ Positive predictive value
Test outcome	Negative	False Negative (Type II error)	True Negative	→ Negative predictive value
		↓ Sensitivity	↓ Specificity

A worked example: the fecal occult blood (FOB) screen test is used in 203 people to look for bowel cancer:

		Patients with bowel cancer (as confirmed on endoscopy)
		Positive	Negative	?
FOB test	Positive	TP = 2	FP = 18	= TP / (TP + FP) = 2 / (2 + 18) = 2 / 20 ≡ 10%
FOB test	Negative	FN = 1	TN = 182	= TN / (TN + FN) 182 / (1 + 182) = 182 / 183 ≡ 99.5%
		↓ = TP / (TP + FN) = 2 / (2 + 1) = 2 / 3 ≡ 66.67%	↓ = TN / (FP + TN) = 182 / (18 + 182) = 182 / 200 ≡ 91%

Related calculations

False positive rate (α) = FP / (FP + TN) = 18 / (18 + 182) = 9% = 1 − specificity
False negative rate (β) = FN / (TP + FN) = 1 / (2 + 1) = 33% = 1 − sensitivity
Power = sensitivity = 1 − β
Likelihood-ratio positive = sensitivity / (1 − specificity) = 66.67% / (1 − 91%) = 7.4
Likelihood-ratio negative = (1 − sensitivity) / specificity = (1 − 66.67%) / 91% = 0.37

Hence with large numbers of false positives and few false negatives, a positive FOB screen test is in itself poor at confirming cancer (PPV = 10%) and further investigations must be undertaken, it will, however, pick up 66.7% of all cancers (the sensitivity). However as a screening test, a negative result is very good at reassuring that a patient does not have cancer (NPV = 99.5%) and at this initial screen correctly identifies 91% of those who do not have cancer (the specificity).

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	$k 1 H$	$k 1 T$
Coin 2	$k 2 H$	$k 2 T$

Here $Θ$ consists of the parameters $p 1 H$ , $p 1 T$ , $p 2 H$ , and $p 2 T$ , which are the probability that coin 1 (2) comes up heads (tails). The hypothesis space $H$ is defined by the usual constraints on a distribution, $0 \le p_{ij} \le 1$ , and $p i H + p i T = 1$ . The null hypothesis $H 0$ is the sub-space where $p 1 j = p 2 j$ . In all of these constraints, $i = 1,2$ and $j = H, T$ .

Writing $n i j$ for the best values for $p i j$ under the hypothesis $H$ , maximum likelihood is achieved with

$n_{ij} = \frac{k_{ij}}{k_{iH}+k_{iT}}.$

Writing $m i j$ for the best values for $p i j$ under the null hypothesis $H 0$ , maximum likelihood is achieved with

$m_{ij} = \frac{k_{1j}+k_{2j}}{k_{1H}+k_{2H}+k_{1T}+k_{2T}},$

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 $χ 2 (1)$ , the $χ 2$ 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

Theoretical

Bayesian criticisms of classical likelihood ratio tests focus on two issues:^{[citation needed]}

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;
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.

There are two frequentist replies to this critique.^{[citation needed]} The first is that in practice, likelihood ratio tests are used as the basis of confidence intervals, which do reflect the uncertainty about θ – though in turn there is Bayesian criticism of confidence intervals as themselves ill-conceived, and that credible intervals are a better alternative. The second 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.

Practical

It has been suggested that the results of diagnostic tests could be more accurately interpreted if presented in terms of likelihood ratios ^[5]. A large likelihood ratio, for example 10 or more, suggests the disease is present, while small ratios (<0.1), help rule out disease^[6].

In practice, however, physicians rarely make these calculations^[7] and when they do, they often make errors.^[8] A randomized controlled trial compared how well physicians interpreted diagnostic tests that were presented as either sensitivity and specificity, a likelihood ratio, or an inexact graphic of the likelihood ratio, found no difference between the three modes in interpretation of test results.^[9]

References

^ Mood, Duane C. Boes; Introduction to theory of Statistics (page 410)
^ Cox, D. R. and Hinkley, D. V Theoretical Statistics, Chapman and Hall, 1974. (page 92)
^ George Casella, Roger L. Berger Statistical Inference, Second edition (page 375)
^ "Likelihood ratios". http://www.poems.msu.edu/InfoMastery/Diagnosis/likelihood_ratios.htm. Retrieved 2009-04-04.
^ Jaeschke R, Guyatt GH, Sackett DL (1994). "Users’ guides to the medical literature. III. How to use an article about a diagnostic test. B. What are the results and will they help me in caring for my patients? The Evidence-Based Medicine Working Group". JAMA 271 (9): 703–7. doi:10.1001/jama.271.9.703. PMID 8309035.
^ McGee S (2002). "Simplifying likelihood ratios". Journal of general internal medicine : official journal of the Society for Research and Education in Primary Care Internal Medicine 17 (8): 646–9. PMID 12213147.
^ Reid MC, Lane DA, Feinstein AR (1998). "Academic calculations versus clinical judgments: practicing physicians’ use of quantitative measures of test accuracy". Am. J. Med. 104 (4): 374–80. doi:10.1016/S0002-9343(98)00054-0. PMID 9576412.
^ Steurer J, Fischer JE, Bachmann LM, Koller M, ter Riet G (2002). "Communicating accuracy of tests to general practitioners: a controlled study". BMJ 324 (7341): 824–6. doi:10.1136/bmj.324.7341.824. PMID 11934776.
^ Puhan MA, Steurer J, Bachmann LM, ter Riet G (2005). "A randomized trial of ways to describe test accuracy: the effect on physicians' post-test probability estimates". Ann. Intern. Med. 143 (3): 184–9. PMID 16061916.

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