Residuals: Residuals from a binomial regression model

In binomTools: Performing diagnostics on binomial regression models

Description

Function to extract residuals from a binomial regression model

Usage

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 √(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 R and 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 = y_i-n_i \hat{p}_i/√{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} = X_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)) √{2 y_i log(y_i/ \hat{y}_i) + 2(n_i-y_i) log((n_i-y_i)/(n_i-\hat{y}_i))}

  The residuals are extracted with a call to residuals.

• standard.deviance: standardized deviance residuals

  r_{D,i} = d_i/√(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) √{h_i r_{P,i}^2 + (1-h_i) r_{D,i}^2}

  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 ith 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 ith 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)d\hat{eta}_i/d\hat{mu}_i

  The residuals are extracted with a call to residuals.

• partial: partial residuals

  r_{W,i} + x_{ij} \hat{β}_j

  where j = 1,...,p and p is the number of predictors. x_{ij} is the ith observation of the jth predictor and \hat{β}_j is the jth fitted coefficient.

  The residuals are useful for making partial residuals plots. They are extracted with a call to residuals

Value

A vector of residuals

Author(s)

Merete K Hansen

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')

Example output

         1          2          3          4          5 
 1.2831240 -0.8082727 -0.8980337  1.3897974 -0.1749601 

binomTools documentation built on May 29, 2017, 10:12 p.m.