sigma {stats}R Documentation

Extract Residual Standard Deviation 'Sigma'


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 \sigma. sigma(.) extracts the estimated parameter from a fitted model, i.e., \hat\sigma.


sigma(object, ...)

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



an R object, typically resulting from a model fitting function such as lm.


logical, passed to nobs.


potentially further arguments passed to and from methods. Passed to deviance(*, ...) for the default method.


The stats package provides the S3 generic, a default method, and a method for objects of class "glm". The default method 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(! – 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.


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.

Very strictly speaking, \hat{\sigma} (“\sigma hat”) is actually \sqrt{\widehat{\sigma^2}}.

For generalized linear models (class "glm"), the sigma.glm method returns the square root of the dispersion parameter (See summary.glm). For families with free dispersion parameter, sigma is estimated from the root mean square of the Pearson residuals. For families with fixed dispersion, sigma is not estimated from the residuals but extracted directly from the family of the fitted model. Consequently, for binomial or Poisson GLMs, sigma is exactly 1.

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


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, summary.glm.


## -- 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()))
## equal to
sqrt(summary(glm.D93)$dispersion) # == 1
## and the *Quasi*poisson's dispersion
sigma(glm.qD93 <- update(glm.D93, family = quasipoisson()))
sigma (glm.qD93)^2 # 1.2933 equal to
summary(glm.qD93)$dispersion # == 1.2933

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

[Package stats version 4.4.0 Index]