Deviance residual
Topic of Paired comparisons, and how far a fit is from the data, part of Generalised linear models
Deviance residual
| Type | Topic |
| Taught in | Paired comparisons, and how far a fit is from the data |
| Part | Generalised linear models |
r_D=\operatorname{sign}(y_i-\hat\mu_i)\sqrt{d_i}, where d_i is observation i’s contribution to the deviance, so the squared deviance residuals add up to the deviance. It measures misfit on the likelihood’s own scale.
In Paired comparisons, and how far a fit is from the data
The deviance residual takes one observation’s contribution d_i to D and attaches the sign of y_i-\hat\mu_i: r_D=\mathrm{sign}(y_i-\hat\mu_i)\sqrt{d_i}. At \hat p=0.7 the binary deviance residuals are +0.844 and -1.55. At \hat p=0.5 they are \pm 1.177, not \pm 1. For the Poisson, seeing 7 when expecting 2 gives \sqrt{2\bigl\{7\log(7/2)-(7-2)\bigr\}}=2.75, and seeing 45 when expecting 50 gives -0.72. The two deviations of five are no longer equal in magnitude once the likelihood, rather than the variance alone, is scoring them. That is the reason to prefer r_D when the distribution is skewed.
Used in
See also
Deviance · Residual deviance · Null deviance · Scaled deviance · Saturated model · Likelihood function · Bradley–Terry model · Three links, one cholesterol slope · Analysis of deviance