Deviance And Model Fit

Deviance is a key concept in logistic regression.

• it measure the deviance of the fitted logistic model w.r.t. a perfect model for \(\mathbb{P}[Y=1|X_1=x_1,...,X_k=x_k]\) perfect model is known as a saturated model

  • an abstract model that fits perfectly the sample

    deviance is defined as the difference of likelihoods between the fitted model and saturated model

    \(D=-2\log lik(\hat{\beta})+2 \log lik(saturated model)\) since the the likelihood of the saturated model is exactly one

    • then the deviance is simply another expression of the likelihood… \(D=-2\log lik(\hat{\beta})\) deviance is always larger or equal than zero

      evaluating the magnitude of deviance is the null deviance

      \(D_0 = -2 \log like(\hat\beta)\)

      deviance of the worst model, the one fitted without any predictor to the perfect model

      \(Y|(X_1=x_1,...,X_k=x_k)\sim Ber(logistic(\beta_0))\)

      in this case, \(\hat\beta_0 = logit(\frac{m}{n}) = \log \frac{\frac{m}{n}}{1-\frac{m}{n}}\) where \(m\) is the number of \(1\) s in \(Y_1,...,Y_n\)

      null deviance serves for comparing how much the model has improved

      • by adding the predictors and can be done by means of the \(R^2\) statistic
        • which is a generalization of the determination coefficient

          • mulitple linear regression

            \(R^2 = 1 - \frac{D}{D_0} = 1 - \frac{deviance(fitted logistic, saturated model)}{deviance(null model, saturated model)}\)

in logistic regression, \(R^2\) does have the same interpretation

• it is a ratio of indicating how close is the fit

  • to being perfect or the worst

    The connexion between and the determination coefficient is given by the expressions of the deviance and null of the deviance for the linear model:

    \(D=SSE (or D = RSS) and D_0 = SST\)