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## What is the -2LL or the Log-likelihood Ratio?

If you have ever read the literature on pharmacokinetic modeling and simulation, you are likely to have run across the phrase  “-2LL” or “log-likelihood ratio”. These are statistical terms that are used when comparing two possible models. In this post, I hope to explain with the log-likelihood ratio is, how to use it, and what it means. At the end of this post, you should feel comfortable interpreting this information as you read about or perform modeling and simulation.

A pharmacokinetic model is a mathematical model that describes the concentration-time profile of a specific drug. When choosing the best model, one must compare a group of related models to find the one that fits the data the best. For example, when performing linear regression, the “best” model is chosen when the statistic called sum of squares is at a minimum. This gives the best-fit line for the observed data. Thus, linear regression can be performed by minimizing the sum of squares values using iterative mathematics. Pharmacokinetic models are non-linear, thus the statistics used to compare models are a bit more complex; however conceptually, they are identical to the linear regression.

When a pharmacokinetic model is fit, a value called the objective function value is calculated. This value is analogous to the sum of squares statistic. The best-fit model will contain the minimum objective function value, just like linear regression.

However, perhaps we want to compare two different models. For example, consider a model that includes body weight as a factor and a model that does not include body weight. Both models will be minimized and provide objective function values. How do we decide which model (with or without body weight) is “better”?

We use a statistical test called the log-likelihood ratio test. This test takes the following form: $D=-2*ln(\frac{\text{likelihood for null model}}{\text{likelihood for alternative model}})$

The likelihood is the objective function value, and D is the test statistic. For pharmacokinetic model comparison, D is part of a chi2 distribution, thus the statistical significance between two models can be tested based on the difference D, the significance level, and the number of parameters different between the two models. If D is greater than a critical value, then the difference in the models is statistically significant. However, if the D is less than the critical value, then the difference in the models is not statistically significant.  A table of critical values is shown at the end of this post for informational purposes.

So when you read log-likelihood ratio test or -2LL, you will know that the authors are simply using a statistical test to compare two competing pharmacokinetic models. And reductions in -2LL are considered better models as long as they exceed the critical values shown in the table below.