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Reading this answer about p-values https://stats.stackexchange.com/a/130772/137921

"Likelihood ratios" are referred to, but there's no definition of what this is there and searching for these gives lots of results relating to medicine and such, but I can't seem to find a clear plain english explanation of what is meant by this term with respect to the context that it's used in.

Relevant part of the post and the graph used for it:

you now realize that outcomes with large likelihood ratios are evidence for the alternative and outcomes with small likelihood ratios are evidence against the alternative

enter image description here

baxx
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1 Answers1

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A likelihood is a measure of how well a particular model (probability distribution and set of parameters) fits a given set of data.

If we want to compare how 2 different models fit the same data then we can compare the likelihoods. The most common way to compare the 2 likelihoods is to look at their ratio (because of the mathematical and theoretical properties), this is the likelihood ratio.

That is the short answer. The reason you saw so many confusing things in the search is that there are a lot of details that depend on which models you want to compare, what assumptions you are making, how well some things approximate other things, how big of a ratio is meaningful, etc.

Edit

Here is a simple example:

Say I have a friend that claims he can read my mind, it is not perfect but he thinks he can do better than guessing. We decide to do an experiment where I flip a coin and then concentrate on whether the coin shows heads or tails (while hiding the coin from my friend), he then tries to read my mind or guess and we do this a total of 10 times. My skeptical model is that he will be right about half the time (pure guess or always choosing heads would give this result). My friend's model is that he will be right about 70% of the time. We both agree that the binomial distribution is appropriate. So if the number of correct guesses is $x$ then the likelihood for my model is ${10\choose x} \cdot (0.5)^x \cdot (1-0.5)^{10-x}$ and for my friends model is ${10\choose x} \cdot(0.7)^x \cdot (1-0.7)^{10-x}$. So simplifying a little the likelihood ratio would be $ \frac{0.7^x \cdot 0.3^{10-x}}{0.5^{10}}$.

The more common significance testing approach would keep my $0.5$ value as the null hypothesis, but then replace the $0.7$ with the actual proportion observed (this guarantees that the ratio will be at least 1) then use an approximation on the distribution under the null to compute a p-value. Algebra can convert this into the usual normal approximation test for proportions.

Greg Snow
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  • Thanks - so If I have two differnet models (in this case they're $H_0$ and $H_1$) then the likelihood here is ... ? I assume that I've divided the one by the other, but I'm not clear on this from the post there and I'm not too sure how to go about it. Perhaps I divide $H_1$(s) by $H_0$, then for values of `Value` (from the graph) close to zero the likelihood ratio would be such that $H_1 > 1$ , and for `values` increasingly far from zero the quotient would decrease, so at `Value = 3` we would have $H_1/H_0 < 1$ Is this correct? – baxx Apr 27 '17 at 15:42
  • @baxx, see my added example above. – Greg Snow Apr 27 '17 at 16:05
  • Thanks - so I'm familiar with the binomial (familiar enough to write it out and read it at least), you've expressed this as the "likelihood for my model is ...", where as this is just the binomial model for $\sim Bin(n, p)$, so you're saying that using model and likelihood here are interchangable terms? We're finding $H_1 / H_0$ here, what's the reasoning for this? Why not $H_0 / H_1$? `this guarantees the ratio will be at least 1`, at least 1 where? – baxx Apr 27 '17 at 16:37
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    @baxx, Simple answer: If n and p are fixed and you are generating x then the binomial is a probability distribution. If x and n are fixed and you are estimating p then the binomial is a likelihood. Same formula, different interpretation. More complex answer is that for cases beyond the binomial, the likelihood is often the product of the formula for each data point. Convention is that we prefer ordering the hypotheses so that the ratio is >1 which means the log likelihood ratio is positive. Try plugging the values 0 through 10 into the ratio above with the 0.7 and with the observed p. – Greg Snow Apr 27 '17 at 17:42