# Model Comparison Power Analysis

POWERComputes the power of between-model likelihood ratio tests, where power is defined as the probability of rejecting the null hypothesis when it is false. A description of an analytical-numerical approximation approach can be found in:

• Burnham, K.P., D.R. Anderson, G.C. White, C. Brownie and K.H. Pollock. 1987. Design and Analysis Methods for Fish Survival Experiments Based on Release-Recapture. Am. Fish. Soc. Monogr. 5, pages 216, 289-295.

### Run POWER online

This web-page (JavaScript program) computes the power of between-model likelihood ratio tests, where power is defined as the probability of rejecting the null hypothesis when it is false. A description of an analytical-numerical approximation approach can be found in:

Burnham, K.P., D.R. Anderson, G.C. White, C. Brownie and K.H. Pollock. 1987. Design and Analysis Methods for Fish Survival Experiments Based on Release-Recapture. Am. Fish. Soc. Monogr. 5, pages 216, 289-295.

The basic idea: Given the alternative model, Ha, is true, what is the probability that the likelihood-ratio test will reject the null model, Ho? If Ho is true, the power of the test should equal the significance level α.

Data should be generated by computing expected values under the alternative model for specified sample sizes, compute estimates under the alternative and null models, then compute a likelihood-ratio test between models. The likelihood-ratio test will give you the χ² value and degrees of freedom needed for the power analysis.

Specifically, the χ² value approximates the noncentrality parameter of a noncentral χ² distribution, from which power is directly obtained.

The formula used by the program is:

POWER = 1 - CHINC( a, DF ,χ² )

where CHINC is the non-central χ² cdf and a is the critical value for a chi-square distribution at level, α.

### R function to compute power:

```powr <- function(x2,df,alflvl) 1-pchisq(qchisq(1-alflvl,df),df,x2)
```

Enter Likelihood-Ratio χ² value, degrees of freedom and significance level for the test between the null model and the alternative model.
```   χ²                   :
degrees-of-freedom   :
significance level(α):
```