Outline
• Multiple Regression– Data and notation–Model – Inference
• Recall notes from Topic 3 for simple linear regression
Data for Multiple Regression
• Yi is the response variable
• Xi1, Xi2, … , Xi,p-1 are p-1 explanatory (or predictor) variables
• Cases denoted by i = 1 to n
Multiple Regression Model
• Yi is the value of the response variable for the ith case
• β0 is the intercept• β1, β2, … , βp-1 are the regression
coefficients for the explanatory variables
0 1 1 2 2 1 , 1...i i i p i p iY X X X
Multiple Regression Model
• Xi,k is the value of the kth explanatory variable
for the ith case
• ei are independent Normally distributed
random errors with mean 0 and variance σ2
0 1 1 2 2 1 , 1...i i i p i p iY X X X
Multiple Regression Parameters
• β0 is the intercept
• β1, β2, … , βp-1 are the regression coefficients for the explanatory variables
• σ2 the variance of the error term
Interesting special cases
• Yi = β0 + β1Xi + β2Xi2
+…+ βp-1Xip-1+ ei
(polynomial of order p-1)
• X’s can be indicator or dummy variables taking the values 0 and 1 (or any other two distinct numbers)
• Interactions between explanatory variables (represented as the product of explanatory variables)
Interesting special cases
• Consider the model
Yi= β0 + β1Xi1+ β2Xi2+β3X i1Xi2+ ei
• If X2 a dummy variable
–Yi = β0 + β1Xi + ei (when X2=0)
–Yi = β0 + β1Xi1+β2+β3Xi1+ ei (when X2=1)
= (β0+β2) + (β1+β3)Xi1+ ei
–Modeling two different regression lines at same time
Covariance Matrix of residuals
• Cov(e)=σ2(I-H)(I-H)΄= σ2(I-H)
• Var(ei)= σ2(1-hii)
• hii= X΄i(X΄X)-1Xi
• X΄i =(1,Xi1,…,Xi,p-1)
• Residuals are usually correlated
• Cov(ei,ej)= -σ2hij
Distribution of b
• b = (X΄X)-1X΄Y
• Since Y~N(Xβ, σ2I)
• E(b)=((X΄X)-1X΄)Xβ=β
• Cov(b)=σ2 ((X΄X)-1X΄)((X΄X)-1X΄)΄
=σ2(X΄X)-1
• σ2 (X΄X)-1 is estimated by s2 (X΄X)-1
ANOVA Table• Sources of variation are
–Model (SAS) or Regression (KNNL)
–Error (SAS, KNNL) or Residual
–Total
• SS and df add as before
–SSM + SSE =SSTO
–dfM + dfE = dfTotal
ANOVA F test
• H0: β1 = β2 = … = βp-1 = 0
• Ha: βk ≠ 0, for at least one k=1,., p-1
• Under H0, F ~ F(p-1,n-p)
• Reject H0 if F is large, use P-value
P-value of F test• The P-value for the F significance test
tells us one of the following:– there is no evidence to conclude that
any of our explanatory variables can help us to model the response variable using this kind of model (P ≥ .05)–one or more of the explanatory
variables in our model is potentially useful for predicting the response variable in a linear model (P ≤ .05)
R2
• The squared multiple regression
correlation (R2) gives the proportion of
variation in the response variable
explained by all the explanatory variables
• It is usually expressed as a percent
• It is sometimes called the coefficient of
multiple determination (KNNL p 226)
R2
• R2 = SSM/SST
– the proportion of variation explained
• R2 = 1 – (SSE/SST)
– 1 – the proportion not explained
• Can express F test is terms of R2
F = [ (R2)/(p-1) ] / [ (1- R2)/(n-p) ]