ENG-256: Linear Models
ENG-256: Linear Models
Checking normality
Normality is a key assumption in order to obtain the right
properties of the estimates and justify the tests of hypotheses based
on t a
ENG-256: Linear Models
ENG-256: Linear Models
Estimation of 2
The distribution of s2 , given 2 , is
In order to obtain an estimate of 2 recall that
SSE = |Y Y |2 =
n
s2
(yi yi )2
i=1
2 2
nk1
nk1
2
s
ENG-256: Linear Models
ENG-256: Linear Models
Model Building
Automatic methods of model building like stepwise regression have
to be taken with care. The aim of model building is to obtain a
model tha
ENG-256: Linear Models
ENG-256: Linear Models
Coecient of Correlation
The estimator of the slope of a regression line, 1 , provides a
measure of the linear relationship between two variables. But it
d
ENG-256: Linear Models
ENG-256: Linear Models
Matrix multiplication
Some linear algebra
Let A be an r c matrix and x a c-dimensional vector. Then Ax
corresponds to a linear combination of the columns
ENG-256: Linear Models
ENG-256: Linear Models
Multiple regression
Example 1
The percent survival of a certain type of animal semen after storage
(y) was measured at various combinations of concentrati
ENG-256: Linear Models
ENG-256: Linear Models
To consider the previous problem in a hypothesis testing framework
we assume that the group of stayers has a population mean equal
to 1 and the group of l
ENG-256: Linear Models
ENG-256: Linear Models
Simple linear regression
0 is the intercept
1 is the slope
3
2
Sales Revenue
3
1
y is the response or dependent variable
Sales Revenue
x is the independen
ENG-256: Linear Models
ENG-256: Linear Models
Observational data
Example
The data that correspond to the regressors can be obtained by
designing a specic experiment and setting the values of the
covar
ENG-256: Linear Models
ENG-256: Linear Models
Prediction
An estimate of the mean response at a particular value of the
regressors x01 , . . . , x0k is given by
Condence intervals
y = 0 + 1 x01 + . . .
ENG-256: Linear Models
ENG-256: Linear Models
LSE
Matrix notation
For a set of n observations the linear regression can be written as
yi = 0 + 1 xi1 + . . . + k xik + i i = 1, . . . , n
This can be wr
ENG-256: Linear Models
ENG-256: Linear Models
Examples:
Introduction
Statistics: Collect, organize, classify and summarize DATA in
order to perform analyses and interpretation, produce forecasts and
p
ENG-256: Linear Models
ENG-256: Linear Models
Introduction to R
Mathematical expressions
How to get help
The usual mathematical expressions are available in R
To get online help you can type help(fun)
ENG-256: Linear Models
ENG-256: Linear Models
Maximum Likelihood
Notice that the likelihood depends on the data only through and
SSE. These are the sucient statistics for the linear model.
By consider
ENG-256: Linear Models
ENG-256: Linear Models
Piecewise linear regression
dening
1
2
strength
3
4
Nonlinear relationships between the response and the explanatory
variables can be sometimes successful
ENG-256: Linear Models
ENG-256: Linear Models
F test
Analysis of variance
To compare the between sample variation with the within levels
variation we can compute the estimates
We consider the problem
ENG-256: Linear Models
ENG-256: Linear Models
Estimating the mean
Given a sample y1 , . . . , yn from a normal population we want to
obtain an estimator of the population mean .
Central Limit Theorem