Pearson residual
Topic of Paired comparisons, and how far a fit is from the data, part of Generalised linear models · Also in Probability-integral residuals
Pearson residual
| Type | Topic |
| Taught in | Paired comparisons, and how far a fit is from the data · Probability-integral residuals |
| Part | Generalised linear models |
| Questions using it | 1 |
r_P=(y-\hat\mu)/\sqrt{\hat\phi V(\hat\mu)}, the raw residual divided by its estimated standard deviation under the model. For binary or count data a plot of Pearson residuals is striped, because the response takes only a few values at each fitted mean; probability-integral residuals remove the stripes.
In Paired comparisons, and how far a fit is from the data
The Pearson residual divides by the estimated standard deviation: r_P=\frac{y-\hat\mu}{\sqrt{\hat\phi\, V(\hat\mu)}}. For a binary observation with \hat p=0.7, \frac{1-0.7}{\sqrt{0.7\times 0.3}}=0.655, \qquad \frac{0-0.7}{\sqrt{0.7\times 0.3}}=-1.53. Landing tails is the larger Pearson residual. At \hat p=0.5 the two residuals are +1 and -1. For Poisson counts, with \phi=1, expecting 2 and seeing 7 gives (7-2)/\sqrt{2}=3.54. Expecting 50 and seeing 45 gives (45-50)/\sqrt{50}=-0.71. The unit is fairer. The sign is still symmetric: 55 instead of 45 would flip the sign and keep the magnitude, and a Poisson distribution is not symmetric.
In Probability-integral residuals
A Pearson plot of a binary or a count fit is striped. The stripes are the values the response is allowed to take, not a missed term. This is the last diagnostic that still belongs with generalised linear models. It replaces that plot with a transform that does not care what the distribution was.
Used in
- Taught in Paired comparisons, and how far a fit is from the data · Probability-integral residuals
- Also used in Underdispersion, positive responses, and quasi-likelihood · Probability-integral residuals
- Questions that use it (1) — C30 (answer)
- All 4 questions on Paired comparisons, and how far a fit is from the data
See also
Count responses · Bradley–Terry model · Deviance · Three links, one cholesterol slope · Analysis of deviance · Residuals in a GLM · Saturated model · Residual deviance