sigma

0th

Percentile

Extract Residual Standard Deviation 'Sigma'

Extract the estimated standard deviation of the errors, the residual standard deviation (misnomed 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., $\hat\sigma$.

Keywords
models
Usage
sigma(object, ...)

## S3 method for class 'default': sigma(object, use.fallback = TRUE, ...)

Arguments
object
an Robject, 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 <- length(coef(object)).

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, $\hat{\sigma}$ ($\sigma$ hat) is actually $\sqrt{\widehat{\sigma^2}}$.

    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.

Aliases
  • sigma
  • sigma.default
  • sigma.mlm
Examples
library(stats) ## -- lm() ------------------------------ lm1 <- lm(Fertility ~ . , data = swiss) sigma(lm1) # ~= 7.165 = "Residual standard error" printed from summary(lm1) stopifnot(all.equal(sigma(lm1), summary(lm1)$sigma, tol=1e-15)) ## -- nls() ----------------------------- DNase1 <- subset(DNase, Run == 1) fm.DN1 <- nls(density ~ SSlogis(log(conc), Asym, xmid, scal), DNase1) sigma(fm.DN1) # ~= 0.01919 as from summary(..) stopifnot(all.equal(sigma(fm.DN1), summary(fm.DN1)$sigma, tol=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))) stopifnot(all.equal(sigma(mlmfit), sqrt(diag(estVar(mlmfit)))))
Documentation reproduced from package stats, version 3.3, License: Part of R 3.3

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