# [R] Where is the SD in output of glm with Gaussian distribution

Fox, John j|ox @end|ng |rom mcm@@ter@c@
Mon Dec 9 16:32:29 CET 2019

```Dear Marc,

For your simple model, the standard deviation of y is the square-root of the estimated dispersion parameter:

> set.seed(123)
> y <- rnorm(100)
> gnul <- glm(y ~ 1)
> summary(gnul)

Call:
glm(formula = y ~ 1)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-2.39957  -0.58426  -0.02865   0.60141   2.09693

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.09041    0.09128    0.99    0.324

(Dispersion parameter for gaussian family taken to be 0.8332328)

Null deviance: 82.49  on 99  degrees of freedom
Residual deviance: 82.49  on 99  degrees of freedom
AIC: 268.54

Number of Fisher Scoring iterations: 2

> sqrt(0.8332328)
 0.9128159
> mean(y)
 0.09040591
> sd(y)
 0.9128159

I hope this helps,
John

-----------------------------
John Fox, Professor Emeritus
McMaster University
Web: http::/socserv.mcmaster.ca/jfox

> On Dec 9, 2019, at 10:16 AM, Marc Girondot via R-help <r-help using r-project.org> wrote:
>
> Let do a simple glm:
>
> > y=rnorm(100)
> > gnul <- glm(y ~ 1)
> > gnul\$coefficients
> (Intercept)
>   0.1399966
>
> The logLik shows the fit of two parameters (DF=2) (intercept) and sd
>
> > logLik(gnul)
> 'log Lik.' -138.7902 (df=2)
>
> But where is the sd term in the glm object?
>
> If I do the same with optim, I can have its value
>
> > dnormx <- function(x, data) {1E9*-sum(dnorm(data, mean=x["mean"], sd=x["sd"], log = TRUE))}
> > parg <- c(mean=0, sd=1)
> > o0 <- optim(par = parg, fn=dnormx, data=y, method="BFGS")
> > o0\$value/1E9
>  138.7902
> > o0\$par
>      mean        sd
>
> 0.1399966 0.9694405
>
> But I would like have the value in the glm.
>
> (and in the meantime, I don't understand why gnul\$df.residual returned 99... for me it should be 98=100 - number of observations) -1 (for mean) - 1 (for sd); but it is statistical question... I have asked it in crossvalidated [no answer still] !)
>
> Thanks
>
> Marc
>
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