1

Experimental StatisticsExperimental Statistics - week 6 - week 6Experimental StatisticsExperimental Statistics - week 6 - week 6

Chapter 15: Randomized Complete Block Design (15.3)

Factorial Models (15.5)

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Caution: Chapter 15 introduces some new notation - i.e. changes notation already defined

3

2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y

Recall: Sum-of-Squares Identity 1-Factor ANOVA

TSS SSB SSW Notation:

In words:

Total SS = SS between samples + within sample SS

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2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y

Recall: Sum-of-Squares Identity 1-Factor ANOVA

TSS SSB SSW Notation:

- new notation for Chapter 15

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2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y

Recall: Sum-of-Squares Identity 1-Factor ANOVA

TSS S SSWST Notation:

- new notation for Chapter 15

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2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y

Recall: Sum-of-Squares Identity 1-Factor ANOVA

TSS SST SSE Notation:

- new notation for Chapter 15

In words:

Total SS = SS for “treatments” + SS for “error”

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Revised ANOVA Table for 1-Factor ANOVA(Ch. 15 terminology - p.857) 

 

Source SS df MS F 

Treatments SST t 1  

Error SSE N t  Total TSS N  

/( 1)MST SST t

/( )MSE SSE N t

/MST MSE

1 2 t

N

nt

n n n

total # of observations

(if equal # obs.)

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Recall CRD Model for Gasoline Data

yij = iij

yij = iij

or

unexplained partmean for ith gasoline

observed octane

-- car-to-car differences-- temperature-- etc.

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Gasoline Data

Question:

What if car differences are obscuring gasoline differences?

Similar to diet t-test example: Recall: person-to-person differences obscured effect of diet

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Possible Alternative Design:

Test all 5 gasolines on the same car

- in essence we test the gasoline effect directly and remove effect of car-to-car variation

Question:How would you randomize an experiment with 4 cars?

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Blocking an Experiment

- dividing the observations into groups (called blocks) where the observations in each block are collected under relatively similar conditions

- comparisons can many times be made more precisely this way

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Terminology is based on Agricultural Experiments

Consider the problem of testingfertilizers on a crop - t fertilizers - n observations on each

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Completely Randomized DesignCompletely Randomized DesignCompletely Randomized DesignCompletely Randomized Design

A

A

BB

C

C

B

A

C

C

B

A

A

B

C

t = 3 fertilizersn = 5 replications

- randomly select 15 plots- randomly assign fertilizers to the 15 plots

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Randomized Complete Block Randomized Complete Block StrategyStrategy

Randomized Complete Block Randomized Complete Block StrategyStrategy

B | A | C

A | C | B

C | A | B

A | B | C C | B | A

t = 3 fertilizers

- select 5 “blocks”- randomly assign the 3 treatments to each block

Note: The 3 “plots” within each block are similar - similar soil type, sun, water, etc

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Randomized Complete Block Design Randomly assign each treatment once to every block

Car Example Car 1: randomly assign each gas to this car

Car 2: ....

etc.

Agricultural Example Randomly assign each fertilizer to one of the 3 plots within each block

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yij = ijij

Model For Randomized Complete Block (RCB) Design

effect of ith treatment

effect of jth block

unexplained error

(car)(gasoline)

1 1

0t b

i ji j

As before:

-- temperature-- etc.

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Back to CAR data:

Suppose that instead of 20 cars, there were only 4 cars, and we tested each gasoline on each car.

“Restructured” CAR Data

A 91.7 91.2 90.9 90.6B 91.7 91.9 90.9 90.9C 92.4 91.2 91.6 91.0D 91.8 92.2 92.0 91.4E 93.1 92.9 92.4 92.4

Old Data Format1 2 3 4

Car

Gas

A 91.7 91.2 90.9 90.6B 91.7 91.9 90.9 90.9C 92.4 91.2 91.6 91.0D 91.8 92.2 92.0 91.4E 93.1 92.9 92.4 92.4

Gas

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Back to CAR data:

Suppose that instead of 20 cars, there were only 4 cars, and we tested each gasoline on each car.

“Restructured” CAR Data

A 91.7 91.2 90.9 90.6B 91.7 91.9 90.9 90.9C 92.4 91.2 91.6 91.0D 91.8 92.2 92.0 91.4E 93.1 92.9 92.4 92.4

Old Data Format1 2 3 4

Car

Gas

A 91.7 91.2 90.9 90.6B 91.7 91.9 90.9 90.9C 92.4 91.2 91.6 91.0D 91.8 92.2 92.0 91.4E 93.1 92.9 92.4 92.4

Gas

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2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y

Recall: Sum-of-Squares Identity 1-Factor ANOVA

TSS SST SSE Notation:

- new notation for Chapter 15

In words:

Total SS = SS for “treatments” + SS for “error”

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2 2 2 2.. . .. . .. . . ..

1 1 1 1 1 1

( ) ( ) ( ) ( )t b t b t b

ij i j ij i ji j i j i j

y y b y y t y y y y y y

A New Sum-of-Squares Identity

TSS SST SSB SSE Not atio n:

In words:

Total SS = SS for treatments + SS for blocks + SS for error

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Hypotheses:

To test for treatment effects - i.e. gas differenceswe test

0 1 2: tH

To test for block effects - i.e. car differences (not usually the research hypothesis)we test

0 1 2: bH

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Randomized Complete Block Design ANOVA Table  

 

Source SS df MS F 

Treatments SST t 1  

Blocks SSB

Error SSE  Total TSS bt  

/( 1)MST SST t

/( 1)( 1)MSE SSE b t

/MST MSE

See page 866

( 1)( 1)b t

1b /( 1)MSB SSB b /MSB MSE

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0

( 1,( 1)( 1))

H

MSTF F t b t

MSE

We reject at significance level if

0 1 2:

: 0t

a i

H

H

at least one

Test for Treatment Effects

Note:2MSE estimates

2 2

1

1

1

t

ii

MSTt

estimates

1F - if no treatment effects, we expect ; 1F - if treatment effects, we expect

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0

( 1,( 1)( 1))

H

MSBF F b b t

MSE

We reject at significance level if

Test for Block Effects

0 1 2:

: 0b

a j

H

H

at least one

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The first variable (A - E) indicates gas as it did with the CompletelyRandomized Design. The second variable (B1 - B4) indicates car.

A B1 91.7A B2 91.2A B3 90.9A B4 90.6B B1 91.7B B2 91.9B B3 90.9B B4 90.9C B1 92.4C B2 91.2C B3 91.6C B4 91.0D B1 91.8D B2 92.2D B3 92.0D B4 91.4E B1 93.1E B2 92.9E B3 92.4E B4 92.4

“Restructured” CAR Data - SAS Format

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SAS file - Randomized Complete Block Design for CAR Data 

INPUT gas$ block$ octane;PROC GLM; CLASS gas block; MODEL octane=gas block; TITLE 'Gasoline Example -Randomized Complete Block Design'; MEANS gas/LSD;RUN;

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CRD ANOVA Table Output - car data 

 

Source SS df MS F p-value 

Gas 6.108 4 1.527 6.80 0.0025 (treatments)

Error 3.370 15 0.225 

Totals 9.478 19 

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RCB ANOVA Table Output - car data 

 

Source SS df MS F p-value 

Gas 6.108 4 1.527 15.58 0.0001 (treatments)

Cars 2.194 3 0.731 7.46 0.0044 (blocks)

Error 1.176 12 0.098 

Totals 9.478 19 

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Dependent Variable: OCTANE Sum of MeanSource DF Squares Square F Value Pr > F Model 7 8.30200000 1.18600000 12.10 0.0001 Error 12 1.17600000 0.09800000 Corrected Total 19 9.47800000  R-Square C.V. Root MSE OCTANE Mean  0.875923 0.341347 0.3130495 91.710000  Source DF Anova SS Mean Square F Value Pr > F GAS 4 6.10800000 1.52700000 15.58 0.0001BLOCK 3 2.19400000 0.73133333 7.46 0.0044

SAS Output -- RCB CAR Data

1 2

1 2

y y

y y

and are significantly different if

| |

Multiple Comparisons in RCB AnalysisMultiple Comparisons in RCB Analysis

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( )α/MSE

tb

(LSD)

(2 )2

( )α/ mMSE

tb

(Bonferroni)

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  t Grouping Mean N gas  A 92.7000 4 E  B 91.8500 4 D B C B 91.5500 4 C C B C B 91.3500 4 B C C 91.1000 4 A

t Grouping Mean N gas  A 92.7000 4 E  B 91.8500 4 D B C B 91.5500 4 C C C 91.3500 4 B C C 91.1000 4 A 

CAR Data -- LSD Results

CRD Analysis

RCB Analysis

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  Bon Grouping Mean N gas  A 92.7000 4 E A B A 91.8500 4 D B B 91.5500 4 C B B 91.3500 4 B B B 91.1000 4 A

CAR Data -- Bonferroni Results

CRD Analysis

RCB Analysis

Bon Grouping Mean N gas

A 92.7000 4 E

B 91.8500 4 D B B 91.5500 4 C B B 91.3500 4 B B B 91.1000 4 A

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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

.181 .269

.167 .283

.182 .235

.187 .260

.202 .256

.198 .281

.236 .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 2j j j

difference between types 1 and 2 at time

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

j jthe difference between times and 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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****************************************************** Two-Way ANOVA using PROC GLM ** showing Interaction Plots ******************************************************;data stimulus;input type$ time response;datalines;A 5 .204A 5 .170A 5 .181A 10 .167A 10 .182A 10 .187A 15 .202A 15 .198A 15 .236V 5 .257V 5 .279V 5 .269V 10 .283V 10 .235V 10 .260V 15 .256V 15 .281V 15 .258;PROC GLM; CLASSES type time; MODEL response=type time type*time;TITLE ‘Stimulus Data';run;PROC SORT;BY type time;PROC MEANS; BY type time; OUTPUT OUT=cells MEAN=response;RUN;* OUTPUT MEAN INTERACTION PLOTS;PROC GPLOT;PLOT response*type=time;SYMBOL1 V=CIRCLE I=JOIN C=BLACK;SYMBOL2 V=DOT I=JOIN C=BLACK;symbol3 V=BOX I=JOIN C=BLACK;RUN;PROC GPLOT;PLOT response*time=type;SYMBOL1 V=CIRCLE I=JOIN C=BLACK;SYMBOL2 V=DOT I=JOIN C=BLACK;RUN;PROC PRINT;RUN;

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****************************************************** Two-Way ANOVA using PROC GLM * showing Interaction Plots *****************************************************;data stimulus;input type$ time response;datalines;A 5 .204A 5 .170A 5 .181A 10 .167A 10 .182A 10 .187A 15 .202A 15 .198A 15 .236V 5 .257V 5 .279V 5 .269V 10 .283V 10 .235V 10 .260V 15 .256V 15 .281V 15 .258;PROC GLM; CLASSES type time; MODEL response=type time type*time;TITLE ‘Stimulus Data';run;

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Stimulus Data The GLM Procedure Dependent Variable: response  Sum of Source DF Squares Mean Square F Value Pr > F  Model 5 0.02554894 0.00510979 17.66 <.0001  Error 12 0.00347200 0.00028933  Corrected Total 17 0.02902094   R-Square Coeff Var Root MSE response Mean  0.880362 7.458622 0.017010 0.228056   Source DF Type I SS Mean Square F Value Pr > F  type 1 0.02354450 0.02354450 81.38 <.0001 time 2 0.00115811 0.00057906 2.00 0.1778

type*time 2 0.00084633 0.00042317 1.46 0.2701

GLM Output

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PROC SORT;BY type time;PROC MEANS; BY type time; OUTPUT OUT=cells MEAN=response;RUN;

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---------------------------------------- type=A time=5 ----------------------------------  The MEANS Procedure  Analysis Variable : response  N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 3 0.1850000 0.0173494 0.1700000 0.2040000 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ---------------------------------------- type=A time=10 ---------------------------------  Analysis Variable : response  N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 3 0.1786667 0.0104083 0.1670000 0.1870000 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ---------------------------------------- type=A time=15 ---------------------------------  Analysis Variable : response  N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 3 0.2120000 0.0208806 0.1980000 0.2360000 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ---------------------------------------- type=V time=5 ----------------------------------  Analysis Variable : response  N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 3 0.2683333 0.0110151 0.2570000 0.2790000 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ---------------------------------------- type=V time=10 ---------------------------------  The MEANS Procedure  Analysis Variable : response  N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 3 0.2593333 0.0240069 0.2350000 0.2830000 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ---------------------------------------- type=V time=15 ---------------------------------  Analysis Variable : response  N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 3 0.2650000 0.0138924 0.2560000 0.2810000 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

PROC MEANS Output

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  Stimulus Data Obs type time _TYPE_ _FREQ_ response  1 A 5 0 3 0.18500 2 A 10 0 3 0.17867 3 A 15 0 3 0.21200 4 V 5 0 3 0.26833 5 V 10 0 3 0.25933

6 V 15 0 3 0.26500

PROC PRINT Output (dataset CELLS)

PROC SORT;BY type time;PROC MEANS; BY type time; OUTPUT OUT=cells MEAN=response;RUN;PROC PRINT;RUN;

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Stimulus Data

Obs type time response  1 A 5 0.204 2 A 5 0.170 3 A 5 0.181 4 A 10 0.167 5 A 10 0.182 6 A 10 0.187 7 A 15 0.202 8 A 15 0.198 9 A 15 0.236 10 V 5 0.257 11 V 5 0.279 12 V 5 0.269 13 V 10 0.283 14 V 10 0.235 15 V 10 0.260 16 V 15 0.256 17 V 15 0.281 18 V 15 0.258

PROC PRINT Output (dataset STIMULUS)

PROC PRINT data=stimulus;RUN;

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Stimulus Data

* OUTPUT MEAN INTERACTION PLOTS;PROC GPLOT DATA=cells;PLOT response*type=time;SYMBOL1 V=CIRCLE I=JOIN C=BLACK;SYMBOL2 V=DOT I=JOIN C=BLACK;SYMBOL3 V=BOX I=JOIN C=BLACK;RUN;

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Stimulus Data

* OUTPUT MEAN INTERACTION PLOTS;PROC GPLOT DATA=cells;PLOT response*time=type;SYMBOL1 V=CIRCLE I=JOIN C=BLACK;SYMBOL2 V=DOT I=JOIN C=BLACK;RUN;

56

Stimulus Data

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Lab Assignment

Run a 2-factor ANOVA on the Pilot Plant Data

a. Print ANOVA Table

b. Graph Interaction Plots

c. Plot Histogram and Probability Plot of Residuals

i. using standard GCHART and UNIVARIATE procedures

ii. using Analyst

http://support.sas.com/techsup/sample/sample_graph.html

http://support.sas.com/documentation/onlinedoc/sas9doc.html

SAS Online Resources

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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