The Simple Linear Regression Model
When conducting a regression analysis on an explanatory variable and a response
variable, the following population model is assumed
Y = β 0 β x1+ ε
β0is the population y-intercept
β1is the population slope
ε are random errors, where ε ~ N (0, σ )
Without ε, all observed points would fall on the regression line
My = β 0 β x 1
Mean of response variable
Assumptions in the model
The distribution of ε at any particular X value has mean 0.
The distribution of ε at any particular X value is normal.
The variance of ε, denoted σ , is the same for any particular X value.
ε is homoscedastic
The random observations ε , ε ,1… 2 assocnated with different observations are
independent of each other.
We can sum up these assumptions as
εi N (0, σ ), i = 1, …., n
iid = independently and identically distributed In order to perform a regression analysis, we need to avoid any serious violations of the
assumptions. We check these assumptions graphically.
Recall that e = i - i i are residuals/errors for our fitted model.
We can standardize our residuals
di= esimateds.d.of e