Lecture 2

Last: Sections 1.1-1.3 (Read these)

Today: Quick Review of sections 1.4, 1.6, 1.7 and 1.9 with examples

Will not cover section 1.5

Next Day: 2.1-2.3

Review of Regression and Analysis of Variance (ANOVA)… Saturday at 12:00 in Frieze B166

Experiment

In an experiment, the experimenter adjusts the settings of input factors to observe the impact on the system

Better understanding of how the factors impact the system allows the experimenter predict future values or optimize the process

Should select experiment treatments so that data is easy (as possible) to analyze

Linear Regression Model

Have N observations (y1 , y2 ,…, yN)

Have p covariates (regressors or explanatory variables)

Model:

Where the error terms are independent, identically distributed (iid) normal random variables

iippii xxy ...110

),0(~ 2 Ni

Linear Regression Model

In matrix notation,

Where,

Xy

NpN

p

p

xx

xx

xx

X

...1

.

.

.

...1

...1

1

221

111

Tp ),...,,( 10

TN ),...,( 1

The experiment is run in order to estimate the model

Least squares estimator of the regression coefficients:

Variance of least squares estimators:

yXXX TT 1ˆ

21)ˆ( XXV T

Notice, estimator and variance are functions of X

Does this indicate a strategy for choosing the design points or treatments?

How would one test to see if a particular explanatory variable is statistically significant?

What does this imply about our choice of treatments?

Residuals

Can verify some model assumptions by looking at residuals from model fit

Do this using plots

One-Way ANOVA

Example - Comparing battery lifetimes

Is there a difference in battery life by brand?

Four brands of batteries when used in a one of those 'radio controlled' cars for kids (Schwarz, 1995).

A selection of brands was bought, and used in random order. The total time the car was able to be used was recorded to the nearest 1/2 hour.

Here is the raw data:

Which battery brand would you buy? Why?

A B C D5.5 7.0 4.0 4.55.0 7.5 3.5 4.05.3 7.8 3.8 4.25.8 7.3 3.7 4.8

BRAND

TIM

E

8

7

6

5

4

3

Model:

In matrix form:

Interpretation:

Estimation of model parameters:

Constraints:

Assumptions:

This experiment is an example of a completely randomized experiment

ANOVA Table

ANOVA

TIME

30.637 3 10.212 104.074 .000

1.177 12 9.812E-02

31.814 15

Between Groups

Within Groups

Total

Sum ofSquares df

MeanSquare F Sig.

Hypothesis

Want to test:

Test statistic:

Multiple Comparisons

Which treatments are different?

Will use Tukey method:

General Procedure:

Design experiment (collect data) Plot data to gain intuition and check assumptions Fit model Residuals Test hypothesis Multiple comparisons (if necessary)