Ch. 8. Multiple Regression
I. Situation
A. Have more than one X variable.
B. Want to establish a linear function to predict Y using
the linear combination of all X variables of interest.
II. Model
A. In the population
yi = β0 + β1xi1 + β2xi2 + . . . + βpxip + εi
where,
β0 - βp : regression coefficients,
xi1 - xip : predictor variables, and
εi : error associated with the prediction.
y
ˆ
i = μY|X1,X2,...,Xp = β0 + β1xi + β2xi2 + . . . + βpxip
where
y
ˆ
i : predicted value of y.
Thus,
yi = β0 + β1xi + β2xi2 + . . . + βpxip + εi
=
y
ˆ
i + εi
and,
εi = yi -
y
ˆ
i
B. Sample Regression Model
yi = b0 + b1xi1 + b2xi2 + . . . + bpxip + ei
y
ˆ
i = b0 + b1xi1 + b2xi2 + . . . + bpxip
ei = yi -
y
ˆ
i
III. Assumptions
A. Assumption 1: Existence
For a given value of
y
ˆ
, y has a certain (usually
normal) distribution with a finite mean (μY|X1,X2,...,Xp)
and variance (σ²Y|X1,X2,...,Xp).
B. Assumption 2: Independence
For a given value of
y
ˆ
, y is independently
distributed.
C. Assumption 3: Linearity
The mean values of y for a given
y
ˆ
make a straight-
line function with the
y
ˆ
values.
D. Assumption 4: Homoscedasticity
The variance of y-score distribution is the same
Docsity.com
