```
Residuals(object, type = c("approx.deletion", "exact.deletion",
"standard.deviance", "standard.pearson", "deviance",
"pearson", "working", "response", "partial"))
```

object

An object of class

`glm`

with a binomial familytype

The type of residuals to be returned. Default is

`approx.deletion`

residuals- A vector of residuals

`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 to`residuals`

.

`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`

data(serum) serum.glm <- glm(cbind(y, n-y) ~ log(dose), family = binomial, data = serum) Residuals(serum.glm, type='standard.deviance')