2

Design of an Experimenti. Define experimental unit

ii. Define treatment/factor

iii. Define levels of treatment/factor• This will be cost associated during

the implementation

iv. Define response variable• The effects that you expect to see

v. Allocate treatment levels to experimental units based on some selection probability• Completely Randomized Design

(Single-factor / 2-factor Experiment)

A. Experimental unit • An experiment unit is an individual or plot or area

that receives a treatment randomly assigned to it• Measurement of effects can be done with the

experimental unit as a whole or some portion of the experimental unit

B. Treatment • Applied to experimental units such as fertilization• “Factorial experiment” = “Treatment experiment”

❑ Two factors = two treatments = two groupsC. Treatment Level

• Specify how to implement a treatment on experimental units

• One treatment may have two or more ways of applying it on experimental units with each “way” being a specific amount or quantity

3

1. Goal/Objectives:• Infer causation (effects) between treatments and

response

• Will be able to do controlled (manipulative) experiment to some extent

2. Procedures i. We start with a uniform population

ii. Randomly divide the population into subpopulations

iii. Apply a treatment for each subpopulation that we expect to influence the subpopulations’ means

iv. We measure effect by examining variation within each treatment to variation between each treatment

v. If treatment:• No Effect: Treatment means are the same with the

population and the between treatment variation will equal 0 (Treatment variation ‹ Population variation)

• Have Effect: Big differences between treatment means (Treatment variation › Population variation)

vi. We will test differences in means by assessing proportions of variation

4

• With just one treatment of two or more levels

(Thinning spacing with five levels)

i. Objective: Effect of spacing on tree height

ii. Define experimental unit:20 plots

iii.Define treatment/factor: Thinning spacing

iv.Define levels of treatment/factor:No thinning / 1m / 2m / 3m / 4m

v. Define response variableMean height growth for each plot

5

Ho: There is no difference in means between treatments

Ha: There is difference(at least one) in means between treatments

• Analysis of Variance (ANOVA) F‐test is a common way to tests the differences between means by comparing the amounts of variability explained by different sources

• In ANOVA, the hypothesis set up

Treatment S0 S1 S2 S3 S4

Reduced (Equal means) Model µ µ µ µ µ

Full (Separate means) Model µ0 µ1 µ2 µ3 µ4

6

Source DF Sum of Squares Mean Squares F-statistic

Between

treatmentt-1

Within

residualt(r-1)

Total tr-1

2

. ..

1

( )t

i

i

SST r =

= −

2

.

1 1

( )t r

ij i

i j

SSR = =

= −

2

..

1 1

( )t r

ij

i j

SSTOT = =

= −

1

SSTMST

t=

−

( 1)

SSRMSR

t r=

−

MSTF

MSR=

t= the number of treatment levels

r= the number of replicates

7

•Model is :( )ij i j iT = + +

µ= fixed constant

ε𝑗(𝑖)~𝑁𝐼𝐷(0, σε2)

Source DFExpected Mean Squares

Fixed Random

𝑇𝑖 t-1

𝜀𝑗(𝑖) t(r-1)

2

Tr +

2

2

2 2

Tr +

2 0i TT = =

2

10;

t

i

i

Ti i

r T

SST rt

T = = =−

− =

8

SourceFixed Random

Expected Mean Squares

𝑇𝑖

𝜀𝑗(𝑖)

Step 1: Write the variable terms in the model as row headings, include subscripts and bracketed subscripts.

9

SourceFixed Random

Expected Mean Squaresi = 5 j = 4

𝑇𝑖

𝜀𝑗(𝑖)

Step 2: Write the subscripts in the model as column headings and the number of observations.

10

SourceFixed Random

Expected Mean Squaresi = 5 j = 4

𝑇𝑖 4

𝜀𝑗(𝑖)

Step 3: For each row, copy the number of observations under each subscript, providing the subscript does not appear in the row heading.

11

SourceFixed Random

Expected Mean Squaresi = 5 j = 4

𝑇𝑖 4

𝜀𝑗(𝑖) 1

Step 4: For any bracketed subscripts, place a “1”under those subscripts that are in the brackets.

12

SourceFixed Random

Expected Mean Squaresi = 5 j = 4

𝑇𝑖 0 4

𝜀𝑗(𝑖) 1 1

Step 5: Fill the remaining cells with “0” or “1”, depending upon whether the factor is Fixed (0) or Random (1).

13

SourceFixed Random

Expected Mean Squaresi = 5 j = 4

𝑇𝑖 0 4

𝜀𝑗(𝑖) 1 1

Step 6: Cover the column(s) that contain non-bracketed subscript letters; multiply the remaining numbers in each row, these products are the coefficients for the factor contribution to expected mean squares.

24 T +

14

SourceFixed Random

Expected Mean Squaresi = 5 j = 4

𝑇𝑖 0 4

𝜀𝑗(𝑖) 1 1

Step 6: Cover the column(s) that contain non-bracketed subscript letters; multiply the remaining numbers in each row, these products are the coefficients for the factor contribution to expected mean squares.

24 T +

2

15

i. Objective: Effects of spacing and fertilization on tree height

ii. Define experimental unit:20 plots

iii.Define treatment/factor: Thinning spacing / fertilization

iv.Define levels of treatment/factor:No thinning / 1m / 2m / 3m / 4m

No fertilization / With fertilization

v. Define response variableMean height growth for each plot

• With two or more treatments of two or more levels

• Thinning spacing with 5 levels;

• Fertilization with 2 levels

16

•Model is : abr a b ab abrA B AB = + + + +

Source DFExpected Mean Squares

Fixed Random

Aa a-1

Bb b-1

ABab (a-1)(b-1)

εr(ab) ab(r-1)

Total abr-1

2

Arb +

2

2

22 2

A ABr rb + +2

Bra +2

ABr +

22 2

B ABr ra + +22

ABr +

17

Step 1: List factors.

SourceFixed Fixed Random

Expected Mean Squaresa=5 b=2 r=2

𝐴𝑎

𝐵𝑏

𝐴𝐵𝑎𝑏

𝜀𝑟(𝑎𝑏)

18

SourceFixed Fixed Random

Expected Mean Squaresa=5 b=2 r=2

𝐴𝑎 2 2

𝐵𝑏 5 2

𝐴𝐵𝑎𝑏 2

𝜀𝑟(𝑎𝑏)

Step 2:If the subscript does not appear in the row heading, copy the number of observations under each subscript.

19

SourceFixed Fixed Random

Expected Mean Squaresa=5 b=2 r=2

𝐴𝑎 2 2

𝐵𝑏 5 2

𝐴𝐵𝑎𝑏 2

𝜀𝑟(𝑎𝑏) 1 1

Step 3:For any bracketed subscripts in the model, place a “1” under those subscripts.

20

SourceFixed Fixed Random

Expected Mean Squaresa=5 b=2 r=2

𝐴𝑎 0 2 2

𝐵𝑏 5 0 2

𝐴𝐵𝑎𝑏 0 0 2

𝜀𝑟(𝑎𝑏) 1 1 1

Step 4:Fill the remaining cells with “0” or “1”, depending upon whether the factor is Fixed (0) or Random (1).

21

SourceFixed Fixed Random

Expected Mean Squaresa=5 b=2 r=2

𝐴𝑎 0 2 2

𝐵𝑏 5 0 2

𝐴𝐵𝑎𝑏 0 0 2

𝜀𝑟(𝑎𝑏) 1 1 1

Step 5:Cover the column(s) that contain non-bracketed subscript letters; multiply the remaining numbers in each row.

24 A +

22

SourceFixed Fixed Random

Expected Mean Squaresa=5 b=2 r=2

𝐴𝑎 0 2 2

𝐵𝑏 5 0 2

𝐴𝐵𝑎𝑏 0 0 2

𝜀𝑟(𝑎𝑏) 1 1 1

Step 5:Cover the column(s) that contain non-bracketed subscript letters; multiply the remaining numbers in each row.

24 A +

210 B +

23

SourceFixed Fixed Random

Expected Mean Squaresa=5 b=2 r=2

𝐴𝑎 0 2 2

𝐵𝑏 5 0 2

𝐴𝐵𝑎𝑏 0 0 2

𝜀𝑟(𝑎𝑏) 1 1 1

Step 5:Cover the column(s) that contain non-bracketed subscript letters; multiply the remaining numbers in each row.

24 A +

210 B +

22 AB +

24

SourceFixed Fixed Random

Expected Mean Squaresa=5 b=2 r=2

𝐴𝑎 0 2 2

𝐵𝑏 5 0 2

𝐴𝐵𝑎𝑏 0 0 2

𝜀𝑟(𝑎𝑏) 1 1 1

Step 5:Cover the column(s) that contain non-bracketed subscript letters; multiply the remaining numbers in each row.

24 A +

210 B +

22 AB +

2

25

SourceExpected Mean Squares for Mixed Factor Models

A Fixed B Random A Random B Fixed

Aa

Bb

ABab

εj(ab)2

22

ABr +

22

ABArb r + +22

ABra + 22

ABBra r + +

22

ABrb +

22

ABr +2

a=Number of treatment A levels = 5

b=Number of treatment B levels = 2

r=Number of replicates = 2

26

•What happened to the F-statistics if we use fixed and random, respectively?

• The use of random models result in losing many degrees of freedom

Treatment Fixed Random

A F(0.05,5-1,20-9)=3.4 F(0.05,5-1,(5-1)*(2-1))=6.4

B F(0.05,2-1,20-9) F(0.05,2-1,(5-1)*(2-1))

Only determined by number of treatment levels

27

• F-value > F-critical

• P-value < α• Treatment effect is significant

• Reject the null hypothesis

• F-value < F-critical

• P-value > α• Treatment effect is

not significant

• Fail to reject the null

hypothesis

Design of Your Experimenti. What is your experimental

unit?

ii. What treatment you want to test?

iii. How many levels of your treatment are?

iv. What observed responses

you need to collect?