Residuals: Residuals from a binomial regression model
Description
Function to extract residuals from a binomial regression modelUsage
Residuals(object, type = c("approx.deletion", "exact.deletion",
"standard.deviance", "standard.pearson", "deviance",
"pearson", "working", "response", "partial"))
Arguments
object
An object of class glm
with a binomial family
type
The type of residuals to be returned. Default is
approx.deletion
residuals
Details
A considerable terminology inconsistency regarding residuals is found in
the litterature, especially concerning the adjectives
standardized and studentized. Here, we use the term
standardized about residuals divided by $\sqrt(1-h_i)$ and
avoid the term studentized in favour of deletion to
avoid confusion. See Hardin and Hilbe (2007) p. 52 for a short
discussion of this topic.
The objective of Residuals
is to enhance transparency of
residuals of binomial regression models in Rand to uniformise the
terminology. With the exception of exact.deletion
all residuals
are extracted with a call to rstudent
,
rstandard
and residuals
from
the stats
package (see the description of the individual
residuals below).
response
: response residuals$$y_i - \hat{y}_i$$The response residuals are also called raw residuals
The residuals are extracted with a call toresiduals
.
pearson
: Pearson residuals
$$X_i =
\frac{y_i - n_i \hat{p}_i}{\sqrt{n_i\hat{p}_i(1-\hat{p}_i)}}$$
The residuals are extracted with a call to residuals
.
standard.pearson
: standardized Pearson residuals
$$r_{P,i} = \frac{X_i}{\sqrt{1-h_i}} =
\frac{y_i+n_i\hat{p}_i}{\sqrt{n_i\hat{p}_i(1-\hat{p}_i)(1-h_i)}}$$
where $X_i$ are the Pearson residuals and $h_i$ are the
hatvalues obtainable with hatvalues
.
The standardized Pearson residuals have many names including
studentized Pearson residuals, standardized residuals, studentized
residuals, internally studentized residuals.
The residuals are extracted with a call to rstandard
.
deviance
: deviance residual
The deviance residuals are the signed square roots of the individual
observations to the overall deviance
$$d_i = sgn(y_i-\hat{y}_i)
\sqrt{2 y_i \log\left( \frac{y_i}{\hat{y}_i}\right) + 2(n_i-y_i)
\log\left( \frac{n_i-y_i}{n_i-\hat{y}_i}\right)}$$
The residuals are extracted with a call to residuals
.
standard.deviance
: standardized deviance residuals
$$r_{D,i} = \frac{d_i}{\sqrt{1-h_i}}$$
where $d_i$ are the deviance residuals and $h_i$ are the
hatvalues that can be obtained with hatvalues
.
The standardized deviance residuals are also called studentized
deviance residuals.
The residuals are extracted with a call to rstandard
.
approx.deletion
: approximate deletion residuals
$$sgn(y_i-\hat{y}_i)\sqrt{h_i r^2_{P,i}+(1-h_i)r^2_{D,i}}$$
where $r_{P,i}$ are the standardized Pearson residuals,
$r_{D,i}$ are the standardized deviance residuals and $h_i$
are the hatvalues that is obtained with hatvalues
The approximate deletion residuals are approximations to the exact
deletion residuals (see below) as suggested by Williams (1987).
The approximate deletion residuals are called many different names in the
litterature including likelihood residuals, studentized residuals,
externally studentized residuals, deleted studentized residuals and
jack-knife residuals.
The residuals are extracted with a call to rstudent
.
exact.deletion
: exact deletion residuals
The $i$th deletion residual is calculated subtracting the
deviances when fitting a linear logistic model to the full set of
$n$ observations and fitting the same model to a set of $n-1$
observations excluding the $i$th observation, for $i =
1,...,n$. This gives rise to $n+1$ fitting processes and may be
computationally heavy for large data sets.
working
: working residuals
The difference between the working response and the linear predictor
at convergence
$$r_{W,i} = (y_i -
\hat{y}_i)\frac{\partial\hat{\eta}_i}{\partial\hat{\mu}_i}$$
The residuals are extracted with a call to residuals
.
partial
: partial residuals
$$r_{W,i} + x_{ij} \hat{\beta}_j$$
where $j = 1,...,p$ and $p$ is the number of
predictors. $x_{ij}$ is the $i$th observation of the $j$th
predictor and $\hat{\beta}_j$ is the $j$th fitted coefficient.
The residuals are useful for making partial residuals plots. They
are extracted with a call to residuals
References
Collett, D. (2003) Modelling binary data. Second
edition. Chapman & Hall/CRC.
Fox, J. (2002) An R and S-Plus Companion to Applied
Regression. Sage Publ.
Hardin, J.W., Hilbe, J.M. (2007). Generalized Linear Models and
Extensions. Second edition. Stata Press.
Williams, D. A. (1987) Generalized linear model diagnostics using the
deviance and single case deletions. Applied Statistics
36, 181-191.Examples
data(serum)
serum.glm <- glm(cbind(y, n-y) ~ log(dose), family = binomial, data = serum)
Residuals(serum.glm, type='standard.deviance')