1

Love does not come by demanding from others, but it is a self initiation.

2

Two Factor DesignsConsider studying the impact of two factors on the yield (response):

Here we have R = 3 rows (levels of the Row factor), C = 4 (levels of the column factor), and n = 2 replicates per cell

[nij for (i,j)th cell if not all equal]

NOTE: The “1”, “2”,etc...mean Level 1, Level 2,etc..., NOT metric values

1 2 3 417.9, 18.1 17.8, 17.8 18.1, 18.2 17.8, 17.9

18.0, 18.2 18.0, 18.3 18.4, 18.1 18.1, 18.5

18.0, 17.8 17.8, 18.0 18.1, 18.3 18.1, 17.9

BRAND

123

DEVICE

3

MODEL:

i = 1, ..., Rj = 1, ..., Ck= 1, ..., n

In general, n observations per cell, R • C cells.

Yijk = ijijijk

4

the grand meanithe difference between the ith

row mean and the grand meanj the difference between the jth

column mean and the grand meanij the interaction associated with

the i-th row and the j-th columnijij

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Where Y••• = Grand mean

Yi•• = Mean of row iY•j• = Mean of column jYij• = Mean of cell (i,j)

[All the terms are somewhat “intuitive”, except for (Yij• -Yi•• - Y•j• + Y•••)]

Yijk = Y•••+ (Yi•• - Y•••) + (Y•j• - Y•••)+ (Yij• - Yi•• - Y•j• + Y•••)+ (Yijk - Yij•)

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The term (Yij• -Yi•• - Y•j• + Y•••) is more intuitively written as:

how a cellmean differs

from grand mean

adjustmentfor “row

membership”

adjustmentfor “column

membership”

We can, without loss of generality, assume (for a moment) that there is no error (random part); why then might the above be non-zero?

(Yij• - Y•••) (Yi•• - Y•••) (Y•j• - Y•••)

7

ANSWER:

Two basic ways to look at interaction:

BL BH

AL 5 8AH 10 ?

If AHBH = 13, no interactionIf AHBH > 13, + interactionIf AHBH < 13, - interaction

- When B goes from BLBH, yield goes up by 3 (58).- When A goes from AL AH, yield goes up by 5 (510).- When both changes of level occur, does yield go up by the sum, 3 + 5 = 8?

Interaction = degree of difference from sum of separate effects

1)

“INTERACTION”

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2)

- Holding BL, what happens as A goes from AL AH? +5

- Holding BH, what happens as A goes from AL AH? +9If the effect of one factor (i.e., the impact of changing its level) is DIFFERENT for different levels of another factor, then INTERACTION exists between the two factors.

BL BH

AL 5 8AH 10 17

NOTE:- Holding AL, BL BH has impact + 3- Holding AH, BL BH has impact + 7

(AB) = (BA) or (9-5) = (7-3).

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(Yijk - Y•••) = (Yi•• - Y•••) + (Y•j• - Y•••)

+ [(Yij• - Yi••) - (Y•j• - Y•••)]

+ (Yijk - Yij•)

Going back to the (model) equation on page 4, and bringing Y... to the other side of the equation, we get

If we then square both sides, triple sum both sides over i, j, and k, we get, (after noting that all cross-product terms cancel):

Effect of column j at row i. Effect of column j

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TSS = SSBRows + SSBCols + SSIR,C+ SSWError

and, in terms of degrees of freedom,

R.C.n-1 = (R-1) + (C-1) + (R-1)(C-1) + R.C.(n-1); DF of Interaction = (RC-1)-(R-1)-(C-1) = (R-1)(C-1).

OR,

(Yijk - Y•••)n.C.Yi•• - Y•••

i j k i

+ n.R.Y•j• - Y•••)2

+ n.Yij• - Yi•• - Y•j• +Y•••

i j

(Yijk - Yij•

i j k

j

11

17.9, 18.1 17.8, 17.8 18.1, 18.2 17.8,17.9

18.1 17.8 18.15 17.85

18.2, 18.0 18.0, 18.3 18.4, 18.1 18.1, 18.5

18.1 18.15 18.25 18.3

18.0, 17.8 17.8, 18.0 18.1, 18.3 18.1, 17.9

17.9 17.9 18.2 18.0

1 2 3 4

18.00 17.95 18.20 18.05

1

2

3

In our example:

DEV ICE

17.95

18.20

18.00

18.05

BRAND

12

SSBrows =2 4[(17.95-18.05) 2 + (18.20-18.05) 2 + (18.0-18.05) 2]

= 8 (.01 + .0225 + .0025) = .28

SSBcol =2•3[(18-18.05) 2+(17.95-18.05) 2+(18.2-18.05) 2+( 18.05-18.05) 2]

= 6 (.0025 + .001 + .0225 + 0) = .21

SSIR,C = 2(18-17.95-18+18.05)2 + (17.8-17.95-17.95+18.05)2 ....… + (18-18-18.05+18.05)2

[]

= 2 [.055] = .11

SSW = (17.9-18.0) 2 + (18.1-18.0) 2 + (17.8-17.8) 2 + (17.8-17.8) 2 + …....... (18.1-18.0) 2 + (17.9-18.0) 2

= .30

TSS = .28 + .21+ .11 + .30 = .90

•

•

13

FTV (2, 12) = 3.89 Reject Ho

FTV (3, 12) = 3.49 Accept Ho

FTV (6, 12) = 3.00 Accept Ho

1) Ho: All Row Means EqualH1: Not all Row Means Equal

2) Ho: All Col. Means EqualH1: Not All Col. Means Equal

3) Ho: No Int’n between factorsH1: There is int’n between factors

ANOVA

.05

SOURCE SSQ df M.S. FcalcRows .28 2 .14 5.6COL .21 3 .07 2.8Int’n .11 6 .0183 .73Error .30 12 .025

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Minitab: Stat >> Anova >> General Linear Model

General Linear Model: time versus brand, device

Factor Type Levels Valuesbrand fixed 4 1, 2, 3, 4device fixed 3 1, 2, 3

Analysis of Variance for time, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F Pbrand 3 0.21000 0.21000 0.07000 2.80 0.085device 2 0.28000 0.28000 0.14000 5.60 0.019brand*device 6 0.11000 0.11000 0.01833 0.73 0.633Error 12 0.30000 0.30000 0.02500Total 23 0.90000

S = 0.158114 R-Sq = 66.67% R-Sq(adj) = 36.11%

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Assumption:

ijk follows N(0, 2) for all i, j, k, and they are independent.

17

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Test for Normality

1. Restore residuals when doing Anova.2. Stat >> Basic Statistics >> Normality Test

Mean -4.44089E-16StDev 0.1142N 24AD 0.815P-Value 0.030

Not really normal but not too far from normal.

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Test for Equal Variances

Minitab: Stat >> Anova >> Test for Equal Variances

Test for Equal Variances: time versus device, brand

95% Bonferroni confidence intervals for standard deviations

device brand N Lower StDev Upper 1 1 2 0.0463363 0.141421 49.6486 1 2 2 * 0.000000 * … 3 4 2 0.0463363 0.141421 49.6486

Bartlett's Test (normal distribution)Test statistic = 2.33, p-value = 0.993

* NOTE * Levene's test cannot be computed for these data.

Fixed Effect Model

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Random Effect ModelAdditional assumptions:i follows N(0, 2

) for all i, and they are independent.j follows N(0, 2

) for all j, and they are independent.ij follows N(0, 2

) for all I, j, and they are independent.•All these random components i j , ij ijk are (mutually) independent.

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Mixed Effect Model (fixed rows and random columns)

22

23

Fixed Random Mixed MSRows

cnn

cn

ncn

MSColRnnRnn

Rn

MSRC

nnn

MSError

Another issue:Table 17.17 (O/L 6th ed., p. 1057)

MEANSQUARE EXPECTATIONS

col = randomrow= fixed

Reference: Design and Analysis of Experiments by D.C. Montgomery, 4th edition, Chapter 11.

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Fixed: Specific levels chosen by the experimenterRandom: Levels chosen randomly from a large

number of possibilities

Fixed: All Levels about which inferences are to be made are included in the experiment

Random: Levels are some of a large number possible

Fixed: A definite number of qualitatively distinguishable levels, and we plan to study them all, or a continuous set of quantitative settings, but we choose a suitable, definite subset in a limited region and confine inferences to that subset

Random: Levels are a random sample from an infinite ( or large) population

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“In a great number of cases the investigator may argue either way, depending on his mood and his handling of the subject matter. In other words, it is more a matter of assumption than of reality.”

Some authors say that if in doubt, assume fixed model. Others say things like “I think in most experimental situations the random model is applicable.” [The latter quote is from a person whose experiments are in the field of biology].

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My own feeling is that in most areas of management, a majority of experiments involve the fixed model [e.g., specific promotional campaigns, two specific ways of handling an issue on an income statement, etc.] . Many cases involve neither a “pure” fixed nor a “pure” random situation [e.g., selecting 3 prices from 6 “practical” possibilities].

Note that the issue sometimes becomes irrelevant in a practical sense when (certain) interactions are not present. Also note that each assumption may yield you the same “answer” in terms of practical application, in which case the distinction may not be an important one.

How to Fit these Models in Minitab

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• “Balanced ANOVA” can fit restricted and unrestricted version. By default, it shows unrestricted model.

• “General Linear Model” can only fit unrestricted model.

• There are no difference between restricted or unrestricted versions for fixed effect and random effect model. It only matters for the mixed effect model.

More on Minitab

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• The notation in EMS output under restricted model matches with ours but it under unrestricted model is different to ours.

• General suggestion: use “General Linear Model” to fit the models BUT use “Balanced ANOVA, option of restricted model” to find the EMS for fixed and random effect models.

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Two-Way ANOVA in Minitab

Stat>>Anova>>General Linear Model:

Model device brand device*brand

Random factors

Results

Factor plots

Graphs

device

Tick “Display expected mean squares and variance components”

Main effects plots & Interactions plots

Use standardized residuals for plots

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General Linear Model: time versus device, brand

Factor Type Levels Values device random 3 1 2 3brand fixed 4 1 2 3 4

Analysis of Variance for time, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F Pdevice 2 0.28000 0.28000 0.14000 7.64 0.022brand 3 0.21000 0.21000 0.07000 3.82 0.076device*brand 6 0.11000 0.11000 0.01833 0.73 0.633Error 12 0.30000 0.30000 0.02500Total 23 0.90000

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Exercise: Lifetime of a Special-purpose Battery

It is important in battery testing to consider different temperatures and modes of use; a battery that is superior at one temperature and mode of use is not necessarily superior at other treatment combination. The batteries were being tested at 4 different temperatures for three modes of use (I for intermittent, C for continuous, S for sporadic). Analyze the data.

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Battery Lifetime (2 replicates)

Mode of use 1 2 3 4

I 12, 16 15, 19 31, 39 53, 55

C 15, 19 17, 17 30, 34 51, 49

S 11, 17 24, 22 33, 37 61, 67

Temperature

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M FInteresting Example:*

Frontiersman

April

50 peopleper cell

Mean Scores “Frontiersman” “April” “Frontiersman” “April”Dependent males males females femalesVariables (n=50) (n=50) (n=50) (n=50)

Intent-to-purchase 4.44 3.50 2.04 4.52

(*) Decision Sciences”, Vol. 9, p. 470, 1978

Brand Name Appeal for Men & Women:

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1 2

1 2

2

3

4

gender

brandM

ean

Interaction Plot - Data Means for y

12Y

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ANOVA Results

Dependent Source d.f. MS FVariable

Intent-to- Sex (A) 1 23.80 5.61* purchase Brand name (B) 1 29.64 6.99**(7 pt. scale) A x B 1 146.21 34.48***

Error 196 4.24

*p<.05**p<.01

***p<.001

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Two Factors with No Replication,

When there’s no replication, there is no “pure” way to estimate ERROR.Error is measured by considering more than one observation (i.e., replication) at the same “treatment combination” (i.e., experimental conditions).

1 2 31 7 3 42 10 6 83 6 2 54 9 5 7

A

B

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Our model for analysis is “technically”:

Yij = i j + Iij

i = 1, ..., R

j = 1, ..., C

We can write:

Yij = Y•• + (Yi• - Y••) + (Y•j - Y••)

+ (Yij - Yi• - Y•j+ Y••)

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After bringing Y•• to the other side of the equation, squaring both sides, and double summing over i and j,

We Find:Yij - Y••)2 = C • Yi•-Y••)2

+ R • Y•j - Y••)2

+ (Yij - Yi• - Y•j +

Y••)2

R

i = 1

C

j=1

R

i=1

C

j=1

R

i=1

C

j=1

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TSS = SSBROWS + SSBCol + SSIR, C

R•C - 1 = (R - 1) + (C - 1) + (R - 1) (C - 1)Degrees of Freedom :

We Know, E(MSInt.) = VInt.

If we assume VInt. = 0, E(MSInt.) = 2,

and we can call SSIR,C SSW

MSInt MSW

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And, our model may be rewritten:

Yij = + i + j + ij,

and the “labels” would become:

TSS = SSBROWS+ SSBCol + SSWError

In our problem: SSBrows = 28.67

SSBcol = 32

SSW = 1.33

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Source SSQ df MSQ Fcalc

rows

col

Error

28.67

32.00

1.33

9.55

16.00

00.22

3

2

6

43

72

TSS = 62 11

at = .01,

FTV (3,6)

= 9.78

FTV(2,6)

= 10.93

ANOVAand:

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What if we’re wrong about there being no interaction?

If we “think” our ratio is,

in Expectation, 2 + VROWS , (Say, for ROWS) 2

and it really is (because there’s interaction)

2 + VROWS,2 + Vint’n

being wrong can lead only to giving us an underestimated Fcalc.

43

Thus, if we’ve REJECTED Ho, we can feel confident of our conclusion, even if there’s

interaction

If we’ve ACCEPTED Ho, only then could the no interaction assumption be CRITICAL.