Question

There is the 2-parameter linear model where e is the error vector of length n given by e ~ N(0, 2² I_n) where I_n is the n by n identity matrix.

I want to use glm(y ~ x, family = gaussian(), data = ...) in R. Please advise how to specify the given error vector.

Answer 1

# The required function is

n=30
## independent variable
x=rnorm(n, 4, 2)

## Error with mean zero and variance 22
e=rnorm(n, 0, sd=sqrt(22))

## Dependent variable with variance 22
y=x+e

data=cbind(y, x)
data=data.frame(data)

## Required model

model=glm(y~x, family = gaussian, data)
summary(model)

## For check the error variance

model=lm(y~x)
summary(model)

#########################

### Run the model

> n=30
> ## independent variable
> x=rnorm(n, 4, 2)
>
> ## Error with mean zero and variance 22
> e=rnorm(n, 0, sd=sqrt(22))
>
> ## Dependent variable with variance 22
> y=x+e
>
> data=cbind(y, x)
> data=data.frame(data)
>
> ## Required model
>
> model=glm(y~x, family = gaussian, data)
> summary(model)

Call:
glm(formula = y ~ x, family = gaussian, data = data)

Deviance Residuals:
Min 1Q Median 3Q Max
-9.0524 -1.4839 -0.0806 2.1880 11.1185

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.03522 1.89826 0.019 0.9853
x 1.06081 0.45707 2.321 0.0278 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 17.304)

Null deviance: 577.72 on 29 degrees of freedom
Residual deviance: 484.51 on 28 degrees of freedom
AIC: 174.59

Number of Fisher Scoring iterations: 2

>
> model=lm(y~x)
> summary(model)

Call:
lm(formula = y ~ x)

Residuals:
Min 1Q Median 3Q Max
-9.0524 -1.4839 -0.0806 2.1880 11.1185

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.03522 1.89826 0.019 0.9853
x 1.06081 0.45707 2.321 0.0278 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.16 on 28 degrees of freedom
Multiple R-squared: 0.1613, Adjusted R-squared: 0.1314
F-statistic: 5.387 on 1 and 28 DF, p-value: 0.0278