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Experimental StatisticsExperimental Statistics Spring 2006 Spring 2006

- week 6 - week 6

Experimental StatisticsExperimental Statistics Spring 2006 Spring 2006

- week 6 - week 6

Chapter 15: Factorial Models (15.5)

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STIMULUS EXAMPLE:

Personal computer presents stimulus, and person responds.

Study of how RESPONSE TIME is effected by a WARNING given prior to the stimulus:

2-factors of interest:

Warning Type --- auditory or visual

Time between warning and stimulus -- 5 sec, 10 sec, or 15 sec. 

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.204 .257

.170 .279

.190 .269

.167 .282

.182 .255

.187 .274

.192 .256

.200 .281

.216 .258

Auditory Visual

5 sec

10 sec

15 sec

WarningTime

Note: “Sort of like RCB” -- what is the difference?

Question: How would you randomize? - 18 subjects - 1 subject

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Observed data

ijky

Level of Factor A

Level of Factor B

Replication

(warning type) (time)

(response time)

Stimulus Data

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FactorA

Factor B

2-Factor ANOVA Data

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.

..

. .

...

ij

i

j

y

y

y

y

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A Possible Model for STIMULUS Data

ij i j

ijk i j ijky

Note:

so according to this model

1 2 difference between types 1 and 2 at time

j j j

Note: The model assumes that the difference between types is the same for all times

i = type, j = time

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Auditory

Visual

5 10 15

Hypothetical Cell MeansHypothetical Cell Means

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ij ij j j

Similarly

i.e. the model says the difference between times j and j' is the same for all types

We may not want to make these assumptions!!

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Auditory

Visual

5 10 15

Hypothetical Cell MeansHypothetical Cell Means

Auditory

Visual

5 10 15

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Model for 2-factor Design

ijk i j ij ijky

1 1 1 1

0a b a b

i j ij iji j i j

where

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2 2... .. ...

1 1 1 1

2. . ...

1

2. .. . . ...

1 1

2...

1

( ) ( )

( )

( )

( )

a b n a

ijk ii j k i

b

jj

a b

ij i ji j

ijkk

y y bn y y

an y y

n y y y y

y y

1 1

a b n

i j

Sum-of-Squares Breakdown

(2-factor ANOVA)

SSA

SSB

SSAB

SSE

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2-Factor ANOVA Table(2-Factor Completely Randomized Design)

 

 

Source SS df MS F 

Main Effects

A SSA a 1  

B SSB b1

Interaction

AB SSAB (a 1)(b1)

Error SSE ab(n 1)   Total TSS abn  

/( 1)MSB SSB b

/ ( 1)MSE SSE ab n

/MSA MSE

See page 900

/( 1)( 1)MSAB SSAB a b

/MSB MSE

/( 1)MSA SSA a

/MSAB MSE

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0

( 1, ( 1))

H

MSBF F b ab n

MSE

Reject at level if

0 1 2: 0

: 0

a

a i

H

H

at least one

Hypotheses:

Main Effects:

0 1 2: 0

: 0

b

a j

H

H

at least one

0

( 1, ( 1))

H

MSAF F a ab n

MSE

Reject at level if

0

(( 1)( 1), ( 1))

H

MSABF F a b ab n

MSE

Reject at level if

Interactions:

0 11 12: 0

: 0

ab

a ij

H

H

at least one

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data stimulus;input type$ time response;datalines;A 5 .204A 5 .170A 5 .190A 10 .167A 10 .182A 10 .187A 15 .192A 15 .200A 15 .216V 5 .257V 5 .279V 5 .269V 10 .282V 10 .255V 10 .274V 15 .256V 15 .281V 15 .258;PROC GLM; CLASSES type time; MODEL response=type time type*time; means type/lsd; means time/lsd; TITLE ‘Stimulus Data';run;

Stimulus Data -- SAS

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The GLM Procedure

Dependent Variable: response

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 5 0.02837783 0.00567557 32.24 <.0001

Error 12 0.00211267 0.00017606

Corrected Total 17 0.03049050

R-Square Coeff Var Root MSE response Mean

0.930711 5.798365 0.013269 0.228833

Source DF Type I SS Mean Square F Value Pr > F

type 1 0.02745606 0.02745606 155.95 <.0001

time 2 0.00026533 0.00013267 0.75 0.4917

type*time 2 0.00065644 0.00032822 1.86 0.1972

GLM Output

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Testing ProcedureTesting Procedure2 factor CRD Design

Step 1. Test for interaction.

Step 2.(a) IF there IS NOT a significant interaction - test the main effects

(b) IF there IS a significant interaction - compare cell means

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Stimulus Example

1.86MSAB

FMSE

Test for Interaction:

.1972P

Therefore we DO NOT reject the null hypothesis of no interaction.

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Stimulus Data

t ype A V

r esponse

0. 17

0. 18

0. 19

0. 20

0. 21

0. 22

0. 23

0. 24

0. 25

0. 26

0. 27

0. 28

t i me

5 6 7 8 9 10 11 12 13 14 15

t i me 5 10 15

r esponse

0. 17

0. 18

0. 19

0. 20

0. 21

0. 22

0. 23

0. 24

0. 25

0. 26

0. 27

0. 28

t ype

A V

B

B

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Stimulus Example

1.86MSAB

FMSE

Test for Interaction:

.1972P

Therefore we DO NOT reject the null hypothesis of no interaction.

Thus - based on the testing procedure, we next test for main effects.

1 2 22

( )α/MSE

y y tN

| |

Testing Main Effects:Testing Main Effects:

For each main effect (i.e. A and B)

0H- test

0H- if is rejected, compare marginal means

using an appropriate procedure (eg. LSD or BON)

Note: I’ll use LSD from this point on unless otherwise noted.

1 2(y y2 marginal means and ) are declared

to be significantly different (using LSD) ifIn General:

where N denotes the # of observations involved in the computation of a marginal mean.

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Auditory Visual

5 sec

10 sec

15 sec

WarningTime

1.. .190y 2.. .268y

.1. .228y

.2. .225y

.3. .234y

.204 .257

.170 .279

.190 .269

.167 .282

.182 .255

.187 .274

.192 .256

.200 .281

.216 .258

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Stimulus Example

155.95MSA

FMSE

Test for Main Effects:

Thus, there is a significant effect due to type but not time

A (type): .0001P

B (time): 0.75MSB

FMSE

.4917P

- i.e. we can use LSD to compare marginal means for type

- we will do this here for illustration although MC not needed when there are only 2 groups

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The GLM Procedure

t Tests (LSD) for response

NOTE: This test controls the Type I comparisonwise error rate, not

the experimentwise error rate.

Alpha 0.05

Error Degrees of Freedom 12

Error Mean Square 0.000176

Critical Value of t 2.17881

Least Significant Difference 0.0136

Means with the same letter are not significantly different.

t Grouping Mean N type

A 0.267889 9 V

B 0.189778 9 A

GLM Output -- Comparing “Types”

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The GLM Procedure t Tests (LSD) for response

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.

Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 0.000176 Critical Value of t 2.17881 Least Significant Difference 0.0167

Means with the same letter are not significantly different.

t Grouping Mean N time A 0.233833 6 15 A A 0.228167 6 5 A A 0.224500 6 10

GLM Output -- Comparing “Times”

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A5 A10 A15 V5 V10 V15

0. 150

0. 175

0. 200

0. 225

0. 250

0. 275

0. 300

response

cel l i d

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- 0. 016 - 0. 008 0 0. 008 0. 016

0

5

10

15

20

25

30

Percent

r es i d

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Pilot Plant Data Variable = Chemical Yield Factors: A – Temperature (160, 180)

B – Catalyst (C1 , C2)  160 C1 59160 C1 61160 C1 50 160 C1 58180 C1 74180 C1 70180 C1 69180 C1 67160 C2 50160 C2 54160 C2 46160 C2 44180 C2 81180 C2 85180 C2 79180 C2 81

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Pilot Plant Data 

Variable = Chemical Yield 

Factors: A – Temperature (160, 180) B – Catalyst (C1 , C2) 

59 74 61 70 50 69 58 67

50 81 54 85 46 79 44 81

o o160 180

C1

C2

Catalyst

Temperature

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- 6 - 3 0 3 6

0

5

10

15

20

25

30

35

40

Percent

r es i d

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Pilot Plant -- Probability Plot of Residuals

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DATA one;INPUT temp catalyst$ yield;datalines;160 C1 59160 C1 61 . . .180 C2 79180 C2 81;PROC GLM; class temp catalyst; MODEL yield=temp catalyst temp*catalyst; Title 'Pilot Plant Example -- 2-way ANOVA'; MEANS temp catalyst/LSD; RUN;PROC SORT;BY temp catalyst;PROC MEANS; BY temp catalyst; OUTPUT OUT=cells MEAN=yield;RUN;

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Pilot Plant Example -- 2-way ANOVA  General Linear Models Procedure Dependent Variable: YIELD Sum of MeanSource DF Squares Square F Value Pr > F Model 3 2525.0000000 841.6666667 58.05 0.0001 Error 12 174.0000000 14.5000000 Corrected Total 15 2699.0000000  R-Square C.V. Root MSE YIELD Mean  0.935532 5.926672 3.8078866 64.250000  Source DF Type I SS Mean Square F Value Pr > F TEMP 1 2116.0000000 2116.0000000 145.93 0.0001CATALYST 1 9.0000000 9.0000000 0.62 0.4461TEMP*CATALYST 1 400.0000000 400.0000000 27.59 0.0002 

Pilot Plant -- GLM Output

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RECALL: RECALL: Testing ProcedureTesting Procedure 2 factor CRD Design

Step 1. Test for interaction.

Step 2.(a) IF there IS NOT a significant interaction - test the main effects

(b) IF there IS a significant interaction - compare cell means

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Pilot Plant Example

27.59MSAB

FMSE

Test for Interaction:

.0002P

Therefore we reject the null hypothesis of no interaction - and conclude that there is an interaction between temperature and catalyst.

Thus, we DO NOT test main effects

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38

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Since there is a significant interaction, we do not test for main effects!

- instead compare “Cell Means”

- NOTE: interaction plot is a plot of the cell means

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Pilot Plant Data 

Variable = Chemical Yield 

Factors: A – Temperature (160, 180) B – Catalyst (C1 , C2) 

59 74 61 70 50 69 58 67

50 81 54 85 46 79 44 81

o o160 180

C1

C2

Catalyst

Temperature

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Pilot Plant Data -- cell means 

57.0 70.0 48.5 81.5

o o160 180

C1

C2

Catalyst

Temperature

1 2 22

( )α/MSE

y y tN

| |

Comparing Cell Means:Comparing Cell Means:

If there is significant interaction, then we compare the a x b cell means using the criteria below.

1 2(y y2 cell means and ) are declared

to be significantly different (using LSD) if

Procedure similar to that for comparing marginal means:

where N denotes the # of observations involved in the computation of a cell mean.

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The GLM Procedure

t Tests (LSD) for yield NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.   Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 14.5 Critical Value of t 2.17881 Least Significant Difference 4.1483  Means with the same letter are not significantly different.   t Grouping Mean N temp  A 75.750 8 180  B 52.750 8 160

GLM Output -- Comparing “Temps”

- disregard

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The GLM Procedure  t Tests (LSD) for yield

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.   Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 14.5 Critical Value of t 2.17881 Least Significant Difference 4.1483  Means with the same letter are not significantly different.   t Grouping Mean N catalyst  A 65.000 8 C2 A A 63.500 8 C1

GLM Output -- Comparing “Catalysts”

- disregard

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Note:

- SAS does not provide a comparison of cell means

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Pilot Plant Data -- cell means 

57.0 70.0 48.5 81.5

o o160 180

C1

C2

Catalyst

Temperature

1 2 22

( )α/MSE

y y tN

| | LSD:

.025 2.1788t

MSE =

N =

LSD =

C2/160 C1/160 C1/180 C2/180 48.5 57.0 70.0 81.5

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Testing Procedure RevistedTesting Procedure Revisted2 factor CRD Design

Step 1. Test for interaction.

Step 2.(a) IF there IS NOT a significant interaction - test the main effects

(b) IF there IS a significant interaction - compare a x b cell means (by hand)

Main Idea:

We are trying to determine whether the factors effect the response either individually or collectively.