## Relationships between distributions

t^{2 } = approx. (for large df) z^{2 } = chi-square on 1 degree of freedom

chi-squared = (numerator degrees of freedom) * F.

For instance, if you tell me that you have an F(2,71) = 2.05, the corresponding chi-squared is 2 * 2.05 = 4.1 and, by the way, the tail probabilities are virtually the same:

F(2,71) = 2.05 p = .1363 chi2(2) = 4.1 p = .1287

This can be useful in checking the output from mixed models in R using lmer().

For example anova(modelA, modelA + extra predictor) gives a chi-square value on 1 df for the extra predictor (assuming the predictor is continuous). The t ratio for the regression coefficient in the lmer output when squared should approximately give the chi-square value outputted so that the p-value for regression coefficient of the fixed effect which is not outputted by lmer() is obtainable using anova().