How to Calculate a Likelihood Ratio
The likelihood ratio is a method of statistically comparing two models of data, one of which is more complex than the other. A more complex model will always fit the data at least as well as a simpler one, and nearly always it will fit better. But most people do not want overly complex models. The likelihood ratio is one way of seeing if the improvement in fit is worth the added complexity.
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Instructions
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1
Compute the likelihood of the data under the simpler model. The likelihood of the data set is the product of the likelihoods of each data point. In all but the very simplest cases, this will be computed by a statistics program. In very simple cases, it can be computed with a calculator.
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2
Compute the likelihood of the data under the more complex model. This is done in the same way as the likelihood of the simpler model.
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3
Compute the ratio of the likelihood of the simpler model divided by the likelihood of the more complex model.
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4
Compute the natural (base e) logarithm of the ratio. Multiply this by -2.
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5
Find the number of parameters in the simpler model. Find the number of parameters in the more complex model. Compute the difference in number of parameters.
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6
Compare the value found in Step 4 to a chi-squared distribution with degrees of freedom equal to the difference in parameters found in Step 5. This can be done by the statistics program or by looking in a table in a statistics book.
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Tips & Warnings
In addition to the likelihood ratio test, consider using other methods of model fit, such as the Akaike information criterion (and modifications of it) or the Bayesian information criterion. These will be part of the output from your statistics program.
Only use the likelihood ratio test for models that are nested. Models A and B are nested if model B contains every parameter in model A, plus some others, but model A contains no parameter not in model B.
Do not rely exclusively on likelihood ratio tests or any statistical test. Also look at effect sizes and scientific meaning.
References
- "The Cambridge Dictionary of Statistics"; Brian Everitt; 1998
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