Unsorted UnsortedTreatments Random Numbers 1 0.533 1 0.683 2 0.702 2 0.379 3 0.411 3 0.962 3 0.139

Sorted Sorted ExperimentalTreatments Random Units Numbers 3 0.139 1 2 0.379 2 3 0.411 3 1 0.533 4 1 0.683 5 2 0.702 6 3 0.962 7

Randomization

Bread Rise Experiment

1. Mix The Dough

2. Divide the dough into 12 small loaves of the same size.

3. Randomly assign 4 loaves to rise 35 minutes, 4 to rise 40 minutes, etc.

4. After allowing each loaf to rise the specified time, measure the height of the loaf.

Model for CRD Design

ijiijY

tirj

N

i

ijij

,,1 ,,1

tindependenmutually s' ),,0(~ 2

Cell Means Model

Alternate Model

iiijiitY

tirj

N

i

ijij

,,1 ,,1

tindependenmutually s' ),,0(~ 2

Effects Model

),(~ 2 iit NY

Notation

Sample means

ir

jij

ii Yr

Y1

1

t

ti

r

ii

t

i

r

jit rnyy

ny

ny

n

ii

111 1

where,111

Grand Mean

Least Squares Estimates Cell Means

021 1

t

i

r

jiij

i

i

yssE

t

i

r

jiij

i

yssE1 1

2Choose estimates to minimize

ii y

3 ,4321 rrr

Matrix Notation for Alternate Model

34

33

32

31

24

23

22

21

14

13

12

11

3

2

1

34

33

32

31

24

23

22

21

14

13

12

11

1001

1001

1001

1001

0101

0101

0101

0101

0011

0011

0011

0011

εβXy

y

y

y

y

y

y

y

y

y

y

y

y

εXβy

LS Estimatorsare solutionto

yXβXX ˆ

Problem is singular

XX

yXβXX ˆ

21XXX

1001

1001

1001

1001

0101

0101

0101

0101

0011

0011

0011

0011

SAS proc glm

112212

2111 XXXXXX

XXXXXX

Non-singular

00

0XXXX

111 is a generalized inverse for XX

yXXXβ )(ˆ Biased Estimates

0

ˆˆ

ˆˆ

ˆˆ

ˆ 32

31

3

β

3

2

1

β

Bread Rise Experiment

0

ˆˆ

ˆˆ

ˆˆ

ˆ 32

31

3

β

=^

:

Matrix Notation for Estimable Functions

βL ˆ is an unbiased estimator for

when the rows of L are linear combination of the rows of for example

Lβ

X

1001

1001

1001

1001

0101

0101

0101

0101

0011

0011

0011

0011

X

1010

0110L

3

2

1

β

is a linear

))(,(~ˆ 2 LXXLLMVNL

ssE must represent varation in experimental units not subsamples, repeated measures or duplicates

Teaching Example (illustration of problems)

Classes randomized to different teaching methods experimental unit=class

No replicate classes no way to compute ssE

Teaching method confounded with difference in classes

Use of student to student variability (i.e. subsamples) to calculate ssECould be totally misleading

● Independence of error terms εij

● Equality of variance across levels of treatment factor

● Normal distribution of εij

Check equal variance assumption 1. plot data vs treatment factor level 2. plot residuals vs predicted values or cell means

Check normality with normal plot of residuals

ods graphics on;proc glm data=bread plots=diagnostics; class time; model height=time/solution;run;ods graphics off;

λ = 1 -1.294869

Solutions►Analysis►Design of Experiments

Two-Level FactorialResponse Surface

MixtureMixed-Level Factorial

Optimal Design

Split-Plot Design

GeneralFactorial

Define Variables►Add> ►Add qualitative factorial variable

Customize…►Replicate Runs

Edit Responses… Design►Randomize Design …

Fit …

Model ► Fit Details…Model►Check Assumptions►Perform Residual AnalysisModel►Check Transformation►Box-Cox Plot

Teaching Experiment

Objective: Compare student satisfaction between 3 different teaching methods

Experimental unit: class

Two replicate classes for each teaching method.

Response: rating given by each student, summarized over class as multinomial vector of counts

),,,,,(~ 54321

5

4

3

2

1

iiiiii

i

i

i

i

i

pppppnMN

y

y

y

y

y

y

iiiii yi

yi

yi

yi

yi

iiiii

i pppppyyyyy

nP 44321

5432154321

)(

y

power, 1-β

practicalsignificance

power

Size of a practical difference

3. H0: μ3 = μ4

Does a mix of artificial fertilizers enhance yield?

Is there a difference in plowed and broadcast?

Does Timing of Application change Yield?

^ ^

Option 1 Option 2

Option 1

Option 2

Review Important Concepts

• Experimental Unit

• Randomization

• Replication

• Practical Difference

• Determining the number of replicates by calculating the power