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| {{Distinguish2|the use of [[likelihood ratios in diagnostic testing]]}}
| | == Vibram 5 Fingers Cheap They do regional file hosting == |
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| {{Citations missing|date=September 2009}}
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| {{Expert-subject|Statistics|date=November 2008}}
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| {{More footnotes|date=November 2010}}
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| In [[statistics]], a '''likelihood ratio test''' is a [[statistical test]] used to compare the fit of two models, one of which (the ''[[null hypothesis|null]] model'') is a special case of the other (the ''[[alternative hypothesis|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#Statistics|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'''.
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| | | 相关的主题文章： |
| In the case of distinguishing between two models, each of which has no unknown [[statistical parameters|parameters]], use of the likelihood ratio test can be justified by the [[Neyman–Pearson lemma]], which demonstrates that such a test has the highest [[statistical power|power]] among all competitors.<ref name=NP1>{{cite doi|10.1098/rsta.1933.0009}}</ref>
| | <ul> |
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| ==Use==
| | <li>[http://verdamilio.net/tonio/spip.php?article1544/ http://verdamilio.net/tonio/spip.php?article1544/]</li> |
| Each of the two competing models, the null model and the alternative model, is separately fitted to the data and the log-[[likelihood function|likelihood]] recorded. The test statistic (often denoted by ''D'') is twice the difference in these log-likelihoods:
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| | | <li>[http://www.diyrom.com/forum.php?mod=viewthread&tid=451780&fromuid=88498 http://www.diyrom.com/forum.php?mod=viewthread&tid=451780&fromuid=88498]</li> |
| : <math>
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| \begin{align}
| | <li>[http://classifieds.mpsoftechnology.com/index.php?page=item&id=449888 http://classifieds.mpsoftechnology.com/index.php?page=item&id=449888]</li> |
| D & = -2\ln\left( \frac{\text{likelihood for null model}}{\text{likelihood for alternative model}} \right) \\
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| &= -2\ln(\text{likelihood for null model}) + 2\ln(\text{likelihood for alternative model}) \\
| | <li>[http://ks35439.kimsufi.com/spip.php?article453/ http://ks35439.kimsufi.com/spip.php?article453/]</li> |
| \end{align}
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| </math> | | <li>[http://web.zaiwww.com/news/html/?186853.html http://web.zaiwww.com/news/html/?186853.html]</li> |
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| The model with more parameters will always fit at least as well (have an equal or 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''. 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 (statistics)|degrees of freedom]] equal to ''df''2 − ''df''1 .<ref name="Huelsenbeck1997">{{cite jstor|2952500}}</ref> Symbols ''df''1 and ''df''2 represent the number of free parameters of models 1 and 2, the null model and the alternative model, respectively.
| | </ul> |
| 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.<ref>An example using phylogenetic analyses is described at {{cite doi|10.1093/sysbio/45.4.546 }}</ref>
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| For example: if the null model has 1 free 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 +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.
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| ==Background==
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| {{ref improve section|date=July 2012}}
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| The '''likelihood ratio,''' often denoted by <math>\Lambda</math> (the capital [[Greek alphabet|Greek letter]] [[lambda]]), is the ratio of the [[likelihood function]] varying the parameters over two different sets in the numerator and denominator.
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| A '''likelihood ratio test''' is a statistical test for making a decision between two hypotheses based on the value of this ratio.
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| It is central to the [[Jerzy Neyman|Neyman]]–[[Egon Pearson|Pearson]] approach to statistical hypothesis testing, and, like statistical hypothesis testing in general, is both widely used and criticized.{{Citation needed|date=July 2012}}
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| ==Simple-vs-simple hypotheses==
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| {{Main|Neyman–Pearson lemma}}
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| A statistical model is often a [[parametrized family]] of [[probability density function]]s or [[probability mass function]]s <math>f(x|\theta)</math>. A simple-vs-simple hypotheses test has completely specified models under both the [[Null hypothesis|null]] and [[Alternative hypothesis|alternative]] hypotheses, which for convenience are written in terms of fixed values of a notional parameter <math>\theta</math>:
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| :<math>
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| \begin{align}
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| H_0 &:& \theta=\theta_0 ,\\
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| H_1 &:& \theta=\theta_1 .
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| \end{align}
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| </math>
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| 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:<ref>Mood, A.M.; Graybill, F.A. (1963) ''Introduction to the Theory of Statistics'', 2nd edition. McGraw-Hill ISBN 978-0070428638 (page 286)</ref><ref>Kendall, M.G., Stuart, A. (1973) ''The Advanced Theory of Statistics, Volume 2'', Griffin. ISBN 0852642156 (page 234)</ref>
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| :<math> | |
| \Lambda(x) = \frac{ L(\theta_0|x) }{ L(\theta_1|x) } = \frac{ f(x|\theta_0) }{ f(x|\theta_1) }
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| </math>
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| or
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| :<math>\Lambda(x)=\frac{L(\theta_0\mid x)}{\sup\{\,L(\theta\mid x):\theta\in\{\theta_0,\theta_1\}\}},</math>
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| where <math>L(\theta|x)</math> is the [[likelihood function]]. Note that some references may use the reciprocal as the definition.<ref>Cox, D. R. and Hinkley, D. V ''Theoretical Statistics'', Chapman and Hall, 1974. (page 92)</ref> 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: | |
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| :If <math>\Lambda > c </math>, do not reject <math>H_0</math>;
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| :If <math>\Lambda < c </math>, reject <math>H_0</math>;
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| :Reject with probability <math>q</math> if <math>\Lambda = c .</math>
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| The values <math>c, \; q</math> are usually chosen to obtain a specified [[significance level]] <math>\alpha</math>, through the relation: <math>q\cdot P(\Lambda=c \;|\; H_0) + P(\Lambda < c \; | \; H_0) = \alpha </math>.{{cn|date=June 2012}} The [[Neyman-Pearson lemma]] states that this likelihood ratio test is the [[Statistical power|most powerful]] among all level <math>\alpha</math> tests for this problem.<ref name=NP1/> | |
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| == Definition (likelihood ratio test for composite hypotheses) ==
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| A null hypothesis is often stated by saying the parameter <math>\theta</math> is in a specified subset <math>\Theta_0</math> of the parameter space <math>\Theta</math>.
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| :<math>
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| \begin{align}
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| H_0 &:& \theta \in \Theta_0\\
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| H_1 &:& \theta \in \Theta_0^{\complement}
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| \end{align}
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| </math>
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| The [[likelihood function]] is <math>L(\theta|x) = f(x|\theta)</math> (with <math>f(x|\theta)</math> being the pdf or pmf) is a function of the parameter <math>\theta</math> with <math>x</math> held fixed at the value that was actually observed, ''i.e.'', the data. The '''likelihood ratio test statistic''' is <ref>Casella, George; Berger, Roger L. (2001) ''Statistical Inference'', Second edition. ISBN 978-0534243128 (page 375)</ref>
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| : <math>\Lambda(x)=\frac{\sup\{\,L(\theta\mid x):\theta\in\Theta_0\,\}}{\sup\{\,L(\theta\mid x):\theta\in\Theta\,\}}.</math>
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| Here, the <math>\sup</math> notation refers to the [[Supremum]] function.
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| A '''likelihood ratio test''' is any test with critical region (or rejection region) of the form <math>\{x|\Lambda \le c\}</math> where <math>c</math> is any number satisfying <math>0\le c\le 1</math>. 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.
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| ===Interpretation===
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| Being a function of the data <math>x</math>, 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).
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| 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. Low values of the likelihood ratio mean that the observed result was less likely to occur under the null hypothesis as compared to the alternative. High values of the statistic mean that the observed outcome was nearly as likely to occur under the null hypothesis as compared to the alternative, and the null hypothesis cannot be rejected.
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| ==={{anchor|Wilks' theorem}} Distribution: Wilks' theorem===
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| 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 <math>n</math> approaches [[Infinity|<math>\infty</math>]], the test statistic <math>-2 \log(\Lambda)</math> for a nested model will be asymptotically [[chi-squared distribution|<math>\chi^2</math> distributed]] with [[degrees of freedom (statistics)|degrees of freedom]] equal to the difference in dimensionality of <math>\Theta</math> and <math>\Theta_0</math>.<ref>{{cite doi|10.1214/aoms/1177732360}}</ref> This means that for a great variety of hypotheses, a practitioner can compute the likelihood ratio <math>\Lambda</math> for the data and compare <math>-2\log(\Lambda)</math> to the chi squared value corresponding to a desired [[statistical significance]] as an approximate statistical test.
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| == Examples == | |
| === Coin tossing ===
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| 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 <math>X</math>.
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| <table align=center>
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| <tr> | |
| <td></td> | |
| <td> '''Heads''' </td>
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| <td> '''Tails'''</td>
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| </tr><tr>
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| <td> '''Coin 1''' </td>
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| <td align=center> <math>k_{1H}</math> </td>
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| <td align=center> <math>k_{1T}</math> </td>
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| </tr><tr> | |
| <td> '''Coin 2''' </td> | |
| <td align=center> <math>k_{2H}</math> </td>
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| <td align=center> <math>k_{2T}</math> </td>
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| </tr> | |
| </table>
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| Here <math>\Theta</math> consists of the parameters <math>p_{1H}</math>, <math>p_{1T}</math>, <math>p_{2H}</math>, and <math>p_{2T}</math>, which are the probability that coins 1 and 2 come up heads or tails. The hypothesis space <math>H</math> is defined by the usual constraints on a distribution, <math>0 \le p_{ij} \le 1</math>, and <math> p_{iH} + p_{iT} = 1 </math>. The null hypothesis <math>H_0</math> is the subspace where <math> p_{1j} = p_{2j}</math>. In all of these constraints, <math>i = 1,2</math> and <math>j = H,T</math>.
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| Writing <math>n_{ij}</math> for the best values for <math>p_{ij}</math> under the hypothesis <math>H</math>, maximum likelihood is achieved with
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| :<math>n_{ij} = \frac{k_{ij}}{k_{iH}+k_{iT}}.</math>
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| Writing <math>m_{ij}</math> for the best values for <math>p_{ij}</math> under the null hypothesis <math>H_0</math>, maximum likelihood is achieved with
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| :<math>m_{ij} = \frac{k_{1j}+k_{2j}}{k_{1H}+k_{2H}+k_{1T}+k_{2T}},</math> | |
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| which does not depend on the coin <math>i</math>.
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| 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 <math>H</math> to be reduced to the one-dimensional <math>H_0</math>, the asymptotic distribution for the test will be <math>\chi^2(1)</math>, the <math>\chi^2</math> distribution with one degree of freedom.
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| For the general contingency table, we can write the log-likelihood ratio statistic as
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| :<math>-2 \log \Lambda = 2\sum_{i, j} k_{ij} \log \frac{n_{ij}}{m_{ij}}.</math>
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| ==References==
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| {{Reflist|2}}
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| ==External links==
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| * [http://www.itl.nist.gov/div898/handbook/apr/section2/apr233.htm Practical application of likelihood ratio test described]
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| * [http://faculty.vassar.edu/lowry/clin2.html Richard Lowry's Predictive Values and Likelihood Ratios] Online Clinical Calculator
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| {{Statistics|inference}}
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| {{DEFAULTSORT:Likelihood-Ratio Test}}
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| [[Category:Statistical ratios]]
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| [[Category:Statistical tests]]
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| [[Category:Statistical theory]]
| |
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