sigma Extract Residual Standard Deviation 'Sigma'

Description

Extract the estimated standard deviation of the errors, the “residual standard deviation” (misnamed also “residual standard error”, e.g., in summary.lm()'s output, from a fitted model).

Many classical statistical models have a scale parameter, typically the standard deviation of a zero-mean normal (or Gaussian) random variable which is denoted as σ. sigma(.) extracts the estimated parameter from a fitted model, i.e., sigma^.

Usage

sigma(object, ...)

## Default S3 method:
sigma(object, use.fallback = TRUE, ...)

Arguments

object	an R object, typically resulting from a model fitting function such as lm.
use.fallback	logical, passed to nobs.
...	potentially further arguments passed to and from methods. Passed to deviance(*, ...) for the default method.

Details

The stats package provides the S3 generic and a default method. The latter is correct typically for (asymptotically / approximately) generalized gaussian (“least squares”) problems, since it is defined as

   sigma.default <- function (object, use.fallback = TRUE, ...)
       sqrt( deviance(object, ...) / (NN - PP) )
 

where NN <- nobs(object, use.fallback = use.fallback) and PP <- sum(!is.na(coef(object))) – where in older R versions this was length(coef(object)) which is too large in case of undetermined coefficients, e.g., for rank deficient model fits.

Value

typically a number, the estimated standard deviation of the errors (“residual standard deviation”) for Gaussian models, and—less interpretably—the square root of the residual deviance per degree of freedom in more general models. In some generalized linear modelling (glm) contexts, sigma^2 (sigma(.)^2) is called “dispersion (parameter)”. Consequently, for well-fitting binomial or Poisson GLMs, sigma is around 1.

Very strictly speaking, σ^ (“σ hat”) is actually √(hat(σ^2)).

For multivariate linear models (class "mlm"), a vector of sigmas is returned, each corresponding to one column of Y.

Note

The misnomer “Residual standard error” has been part of too many R (and S) outputs to be easily changed there.

See Also

deviance, nobs, vcov.

Examples

## -- lm() ------------------------------
lm1 <- lm(Fertility ~ . , data = swiss)
sigma(lm1) # ~= 7.165  = "Residual standard error"  printed from summary(lm1)
stopifnot(all.equal(sigma(lm1), summary(lm1)$sigma, tolerance=1e-15))

## -- nls() -----------------------------
DNase1 <- subset(DNase, Run == 1)
fm.DN1 <- nls(density ~ SSlogis(log(conc), Asym, xmid, scal), DNase1)
sigma(fm.DN1) # ~= 0.01919  as from summary(..)
stopifnot(all.equal(sigma(fm.DN1), summary(fm.DN1)$sigma, tolerance=1e-15))


## -- glm() -----------------------------
## -- a) Binomial -- Example from MASS
ldose <- rep(0:5, 2)
numdead <- c(1, 4, 9, 13, 18, 20, 0, 2, 6, 10, 12, 16)
sex <- factor(rep(c("M", "F"), c(6, 6)))
SF <- cbind(numdead, numalive = 20-numdead)
sigma(budworm.lg <- glm(SF ~ sex*ldose, family = binomial))

## -- b) Poisson -- from ?glm :
## Dobson (1990) Page 93: Randomized Controlled Trial :
counts <- c(18,17,15,20,10,20,25,13,12)
outcome <- gl(3,1,9)
treatment <- gl(3,3)
sigma(glm.D93 <- glm(counts ~ outcome + treatment, family = poisson()))
## (currently) *differs* from
summary(glm.D93)$dispersion # == 1
## and the *Quasi*poisson's dispersion
sigma(glm.qD93 <- update(glm.D93, family = quasipoisson()))
sigma (glm.qD93)^2 # 1.282285 is close, but not the same
summary(glm.qD93)$dispersion # == 1.2933

## -- Multivariate lm() "mlm" -----------
utils::example("SSD", echo=FALSE)
sigma(mlmfit) # is the same as {but more efficient than}
sqrt(diag(estVar(mlmfit)))