Extension F: Why is the deviance expressed as the log-likelihood multiplied by -2?
It is not necessary to read this extension in order to understand and interpret logistic regression. It is included only for the stat-curious among you!
The maximum likelihood estimation procedure generates a likelihood function which will range from 0 up to a maximum of 1 for a perfect model (a very unlikely scenario, but theoretically one where every case where the event occurs (Y=1) has a predicted value of 1 and every case where the event does not occur (Y=0) has a predicted value of 0). The log of this likelihood function (log likelihood) will range from an infinitely low negative number for very poorly fitting models up to a maximum of 0 for a perfect model (0 because the log of 1 is 0). However to make the interpretation of the LL more in line with interpretations in linear regression, we multiply the LL by -2 to convert negative numbers to positive numbers so that a ‘perfect’ model will have a -2LL of 0 and increasing positive values indicate increasingly poorer model fit. This make the interpretation of the deviance (-2LL) similar to the interpretation of the sum of square in linear regression. In linear regression better explanatory models will minimise the sum of squares of the difference between the predicted and observed values. Similarly in logistic regression better explanatory models will minimise the deviance. The change in the -2LL statistic can be tested using a X2 (chi-square) distribution so that the significance of the change can be calculated.