Talk:Likelihood-ratio test

Added General Audience Introduction and Created Examples Contents

The instructions for creating less technical articles suggest starting with a simplier explanation upfront and then get into the technical details later. With a table of contents, the instructions indicate that provides for having something accessible for those that haven't extensively studied this topic; while at the same time, leaving a meaty article for those interested in something more sophisticated. I dont know if I pulled it off perfectly, but I think it improves the article in a way in which who ever put the "to technical" banner would approve.

At the same time I moved the example into its own contents tab to seperate it from the theory portion.

Jeremiahrounds 18:59, 20 June 2007 (UTC)

difficult take on the likelihood viewpoint?

I believe this essentially obscures the idea here:

$\Lambda (x)={\frac {\sup\{\,L(\theta \mid x):\theta \in \Theta _{0}\,\}}{\sup\{\,L(\theta \mid x):\theta \in \Theta \,\}}}.$

The likelihood ratio test is the ratio of the probability of the result GIVEN the maximum likelihood estimator in the domain of the null and alternative hypothesis.

The supremums in that equation sort of combine the maximum likelihood method into the theory of likelihood ratios.

I am not making this up. For example, the text Hoel, Introduction of Statistical Theory uses L(x| theta0) / L(x | theta) where each theta is the maximum likelihood estimate applicable to each hypothesis.

You can more simply state it as Hoel does and just note that the thetas are produced by maximum likelihood estimates. So the supremum doesnt need to appear in the theory of likelihood ratios. Then you get a ratio of probabilities that is easier to read and even think about.

I actually initially called the offered equation an error. But that is a bridge to far I think. Putting the supremums in the context where you appear to be maximizing something after the data is taken isnt very useful for understanding the actual method though.

Jeremiahrounds 12:11, 20 June 2007 (UTC)

I don't think there is any maximum involved in the Likelihood-ratio test, you just have to make the ratio of the likelihood under hypothesis H0 and H1. I'm not an expert in statistics but I think this equation introduces a confusion between Likelihood-ratio test and maximum likelihood estimation. I have never seen it presented this way anyway... Sylenius 14:45, 27 June 2007 (UTC)

I think Jeremiahrounds is mistaken. In case the MLEs actually exist, the likelihood-ratio test statistic is in fact equal to what Hoel's book says it is, and also it is equal to the expression in TeX above, which appears in this article. But the likelihood-ratio test statistic can exist even in cases where MLEs don't exist, simply because the sup exists and the max does not, i.e. the sup is not actually attained. Moreover, the problem of non-unique MLEs doesn't matter, since it is only the value of the sup rather than the value of θ where the sup occurs that matters. Michael Hardy 19:05, 27 June 2007 (UTC)

Untitled

Can someone please replace the awful ascii-art in this article with TeX, please?

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I may get to that if someone doesn't beat me to it. Hundreds of articles here are in need of TeX to replace what was used here before 2003. Michael Hardy 22:57 Feb 2, 2003 (UTC)

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The article uses λ in some places, and Λ in others -- is this intentional, or should they all be one or the other?

This article needs thorough checking and copyediting.

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(Capital) Λ is the most frequently used notation for the test statistic. Michael Hardy 20:12 Feb 4, 2003 (UTC)

Can the Likelihoor ratio test be used in place of the F-test for a fixed effects models. Any diffrences from the F-test in this case? What about using LRT for testing fixed effects in mixed model?

The F-test is the likelihood ratio test in such models. Michael Hardy 22:30, 3 September 2005 (UTC)

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Hi. I may be misguided or mistaken here, I'm hardly expert. But I think the definition of the test statistic given is inconsistent with the test statistic given. The unrestricted numerator will be larger than the restricted denominator, so the ratio will be greater than 1, and its log will be positive, so -2 log Λ will be negative and can hardly be chi-square distributed. I think that either the ratio should be inverted, or the test statistic multiplied by negative 1, to keep things consistent. (My apologies again if I'm making a basic mistake, a possibility of which the likelihood is high.) Stevewaldman (talk) 00:58, 20 January 2008 (UTC)

"asymptotically"

"If the null hypothesis is true, then −2 log Λ will be asymptotically χ2 distributed" The validity conditions of this theorem should be given. "asymptotically" when what tends to what value ?

I have now answered this question in the article. Michael Hardy 02:36, 28 October 2005 (UTC)

There's really no further restriction on the random variables ("n independent identically distributed random variables")? Dchudz 15:22, 13 July 2007 (UTC)

References

This article lacks references. For a instance, who proved that the likelihood ratio has density function is $\chi ^{2}+O_{p}(n^{-1})$?

I believe the critical paper is WILKS, SS (1938): "The Large Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses," Annals of Mathematical Statistics, 9, 60-62.

Freely available online at http://projecteuclid.org/euclid.aoms/1177732360 —Preceding unsigned comment added by 61.18.170.102 (talk) 18:15, 6 April 2008 (UTC)

Coins

Hi, I think your example of the coins is fine but needs elaborating.

• You haven't defined m_ij which I assume is the probability of event j when the two coins have the same probability of event j. It might be better calling it m_j then m_ij.
• I think you should put in the equation for the likelihood ratio lambda, then follow it with the -2 log lambda (-2LL) equation
• I'm not sure your -2LL equation is right, though I may be wrong. It looks to me as if your -2LL equation converts to lambda squared equals the ratio of the max likelihood of the data for the two hypotheses.

Desmond D.Campbell@iop.kcl.ac.uk 89.241.126.245 01:36, 24 March 2007 (UTC)

This page is FUNDAMENTALLY WRONG.

Where to begin? A likelihood ratio test is for simple-vs-simple hypothesis. The test statistic given is a generalized, or maximum, likelihood ratio statistic. It may be commonly referred to in conversation as an LRT, but no competent mathematical statistics text will refer to it as such.

The distinction is critical. For example, the Neyman-Pearson lemma, mentioned in the article, is only directly applicable to the simple-vs-simple test. It may be extended to some composite alternatives (UMP test) through eg. Monotone likelihood ratios. For most practical composite hypotheses, the best results are generally more restrictive, eg. UMPU.

As for the flag about "too technical for a general audience". Blah. No choice, but one has to understand some mathematical statistics to have a chance of understanding LRT's. Conversely, a "General Audience" will have little concern over LRT's.

Anyway, another page for the "expert needed" flag... --Zaqrfv (talk) 09:37, 25 August 2008 (UTC)

Blah? I wholeheartedly disagree that this article can't be directed at a more general audience (e.g. physicians wanting to interpret the diagnostic validity of a test). The more complex stuff is fine towards the end of the article, but let's put the accessible stuff up front. Currently, the Wikipedia article is one of the least accessible articles on LRs on the web. After having read this article, I still have no idea what they are.164.111.16.221 (talk) 13:50, 5 November 2008 (UTC)

What is f(.)

Are we talking cdf or pdf? The probability that x is observed exactly as is? or that x or something more extreme than x was observed cancan101 (talk) 03:02, 18 February 2009 (UTC)