Completely Randomized Design. 1. Experimental Units (Subjects) Are Assigned Randomly to Treatments Subjects are Assumed HomogeneousSubjects are Assumed

Completely Completely Randomized DesignRandomized Design

Completely Randomized DesignCompletely Randomized Design

1.1. Experimental Units (Subjects) Are Experimental Units (Subjects) Are Assigned Randomly to TreatmentsAssigned Randomly to Treatments• Subjects are Assumed HomogeneousSubjects are Assumed Homogeneous

2.2. One Factor or Independent VariableOne Factor or Independent Variable• 2 or More Treatment Levels or 2 or More Treatment Levels or

ClassificationsClassifications

3.3. Analyzed by One-Way ANOVAAnalyzed by One-Way ANOVA

Factor (Training Method)Factor levels(Treatments)

Level 1 Level 2 Level 3

Experimentalunits

Dependent 21 hrs. 17 hrs. 31 hrs.

variable 27 hrs. 25 hrs. 28 hrs.

(Response) 29 hrs. 20 hrs. 22 hrs.

Factor (Training Method)Factor levels(Treatments)

Level 1 Level 2 Level 3

Experimentalunits

Dependent 21 hrs. 17 hrs. 31 hrs.

variable 27 hrs. 25 hrs. 28 hrs.

(Response) 29 hrs. 20 hrs. 22 hrs.

Randomized Design ExampleRandomized Design Example

The Linier Model The Linier Model

ijiijy

i = 1,2,…, t j = 1,2,…, r

yij = the observation in ith treatment and the jth replication

= overall meani = the effect of the ith treatmentij = random error

One-Way ANOVA F-One-Way ANOVA F-TestTest

One-Way ANOVA F-TestOne-Way ANOVA F-Test

1.1. Tests the Equality of 2 or More (Tests the Equality of 2 or More (pp) ) Population MeansPopulation Means

2.2. VariablesVariables• One Nominal Scaled Independent VariableOne Nominal Scaled Independent Variable

2 or More (2 or More (pp) Treatment Levels or ) Treatment Levels or ClassificationsClassifications

• One Interval or Ratio Scaled Dependent VariableOne Interval or Ratio Scaled Dependent Variable

3.3. Used to Analyze Completely Randomized Used to Analyze Completely Randomized Experimental DesignsExperimental Designs

One-Way ANOVA F-Test One-Way ANOVA F-Test AssumptionsAssumptions

1.1. Randomness & Independence of ErrorsRandomness & Independence of Errors• Independent Random Samples are Drawn for Independent Random Samples are Drawn for

each conditioneach condition

2.2. NormalityNormality• Populations (for each condition) are Normally Populations (for each condition) are Normally

DistributedDistributed

3.3. Homogeneity of VarianceHomogeneity of Variance• Populations (for each condition) have Equal Populations (for each condition) have Equal

VariancesVariances

One-Way ANOVA F-Test One-Way ANOVA F-Test HypothesesHypotheses

HH00: : 11 = = 22 = = 33 = ... = = ... = tt

• All Population Means are EqualAll Population Means are Equal• No Treatment EffectNo Treatment Effect

HHaa: Not All : Not All ii Are Equal Are Equal• At Least 1 Pop. Mean is DifferentAt Least 1 Pop. Mean is Different• Treatment EffectTreatment Effect 11 22 ... ... tt

One-Way ANOVA F-Test One-Way ANOVA F-Test HypothesesHypotheses

HH00: : 11 = = 22 = = 33 = ... = = ... = tt

• All Population Means All Population Means are Equalare Equal

• No Treatment EffectNo Treatment Effect

HHaa: Not All : Not All ii Are Equal Are Equal

• At Least 1 Pop. Mean At Least 1 Pop. Mean is Differentis Different

• Treatment EffectTreatment Effect11 22 ... ... tt

XX

f(X)f(X)

11 = = 22 = = 33

XX

f(X)f(X)

11 = = 22 33

Why Variances?Why Variances? Observe one sample from each treatment Observe one sample from each treatment

groupgroup Their means may be slightly differentTheir means may be slightly different How different is enough to conclude How different is enough to conclude

population means are different?population means are different? Depends on variability within each Depends on variability within each

populationpopulation• Higher variance in population Higher variance in population higher higher

variance in meansvariance in means• Statistical tests are conducted by comparing Statistical tests are conducted by comparing

variability between means to variability within variability between means to variability within each sampleeach sample

Two PossibleTwo PossibleExperiment OutcomesExperiment Outcomes

Same treatment variationSame treatment variationDifferent random variationDifferent random variation

AA

Can’t reject equality of means!Can’t reject equality of means!

Reject equality of means!Reject equality of means!

Two More PossibleTwo More PossibleExperiment OutcomesExperiment Outcomes

Pop 1 Pop 2 Pop 3

Pop 4 Pop 6

Pop 1 Pop 2 Pop 3

Pop 4 Pop 6Pop 5 Pop 5

Same treatment variationSame treatment variationDifferent random variationDifferent random variation

AA BB

Different treatment Different treatment variationvariation

Same random variationSame random variation

Can’t reject equality of means!Can’t reject equality of means!

RejectRejectRejectReject

1.1. Compares 2 Types of Variation to Compares 2 Types of Variation to Test Equality of MeansTest Equality of Means

2.2. Comparison Basis Is Ratio of Comparison Basis Is Ratio of Variances Variances

3.3. If Treatment Variation Is Significantly If Treatment Variation Is Significantly Greater Than Random Variation then Greater Than Random Variation then Means Are Means Are NotNot Equal Equal

4.4. Variation Measures Are Obtained by Variation Measures Are Obtained by ‘Partitioning’ Total Variation‘Partitioning’ Total Variation

One-Way ANOVA One-Way ANOVA Basic IdeaBasic Idea

One-Way ANOVA One-Way ANOVA Partitions Total VariationPartitions Total Variation


Total variationTotal variation


Variation due to treatment



One-Way ANOVAOne-Way ANOVA Partitions Total VariationPartitions Total Variation



Variation due to random samplingVariation due to

random sampling






random sampling


Sum of Squares AmongSum of Squares Among Sum of Squares BetweenSum of Squares Between Sum of Squares TreatmentSum of Squares Treatment Among Groups VariationAmong Groups Variation





random sampling


Sum of Squares WithinSum of Squares Within Sum of Squares Sum of Squares

Error (SSE)Error (SSE) Within Groups Within Groups

VariationVariation

Sum of Squares AmongSum of Squares Among Sum of Squares BetweenSum of Squares Between Sum of Squares Sum of Squares

Treatment (SST)Treatment (SST) Among Groups VariationAmong Groups Variation

Total VariationTotal Variation

XX

Group 1Group 1 Group 2Group 2 Group 3Group 3

Response, XResponse, X

22

21

2

11 XXXXXXTotalSS ij 22

21

2

11 XXXXXXTotalSS ij

Treatment VariationTreatment Variation

XX

XX33

XX22XX11



2222

211

XtXtnXXnXXnSST 22

222

11X

tXtnXXnXXnSST

Random (Error) VariationRandom (Error) Variation

XX22XX11

XX33



22

121

2

111 ttj XXXXXXSSE 22

121

2

111 ttj XXXXXXSSE

SS=SSE+SSTSS=SSE+SST

1

1

1 1

2....

1 1..

n

i

n

jiiij

n

i

n

jij

i

i

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...1 1

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2.

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2 XXXX

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i

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0

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Thus, SS=SSE+SSTThus, SS=SSE+SST

11

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XXnXX

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One-Way ANOVA F-Test One-Way ANOVA F-Test Test StatisticTest Statistic

1.1. Test StatisticTest Statistic• FF = = MSTMST / / MSEMSE

MSTMST Is Mean Square for Treatment Is Mean Square for Treatment MSEMSE Is Mean Square for Error Is Mean Square for Error

2.2. Degrees of FreedomDegrees of Freedom 11 = = tt -1 -1 22 = tr - = tr - tt

tt = # Populations, Groups, or Levels = # Populations, Groups, or Levels trtr = Total Sample Size = Total Sample Size

One-Way ANOVA One-Way ANOVA Summary TableSummary Table

Source ofSource ofVariationVariation

DegreesDegreesofof

FreedomFreedom

Sum ofSum ofSquaresSquares

MeanMeanSquareSquare

(Variance)(Variance)

FF

TreatmentTreatment t - 1t - 1 SSTSST MST =MST =SST/(t - 1)SST/(t - 1)

MSTMSTMSEMSE

ErrorError tr - ttr - t SSESSE MSE =MSE =SSE/(tr - t)SSE/(tr - t)

TotalTotal tr - 1tr - 1 SS(Total) =SS(Total) =SST+SSESST+SSE

ANOVA Table for aANOVA Table for aCompletely Randomized DesignCompletely Randomized Design

Source of Sum of Degrees of MeanSource of Sum of Degrees of Mean

Variation Squares Freedom Squares FVariation Squares Freedom Squares F

TreatmentsTreatments SSTSST tt - 1 - 1 SST/t-1 SST/t-1 MST/MSE MST/MSE

ErrorError SSE trSSE tr - - tt SSE/tr-t SSE/tr-t

TotalTotal SSTotSSTot tr - 1tr - 1

The F distributionThe F distribution

Two parametersTwo parameters• increasing either one decreases F-alpha increasing either one decreases F-alpha

(except for(except for v2<3) v2<3)

• I.e., the distribution gets smashed to the leftI.e., the distribution gets smashed to the left

FF v1v1 v2v2(( ,, ))00 FF

One-Way ANOVA F-Test Critical One-Way ANOVA F-Test Critical ValueValue

If means are equal, If means are equal, FF = = MSTMST / / MSEMSE 1. 1. Only reject large Only reject large FF!!

Always One-Tail!Always One-Tail!

FF tt trtr -t)-t)(( ,, 1100

Reject HReject H00

Do NotDo NotReject HReject H00

FF

© 1984-1994 T/Maker Co.© 1984-1994 T/Maker Co.

Example: Home Products, Inc.Example: Home Products, Inc.

Completely Randomized DesignCompletely Randomized Design

Home Products, Inc. is considering marketing a long-Home Products, Inc. is considering marketing a long-lasting car wax. Three different waxes (Type 1, Type 2, lasting car wax. Three different waxes (Type 1, Type 2, and Type 3) have been developed.and Type 3) have been developed.

In order to test the durability of these waxes, 5 new cars In order to test the durability of these waxes, 5 new cars were waxed with Type 1, 5 with Type 2, and 5 with Type were waxed with Type 1, 5 with Type 2, and 5 with Type 3. Each car was then repeatedly run through an 3. Each car was then repeatedly run through an automatic carwash until the wax coating showed signs automatic carwash until the wax coating showed signs of deterioration. The number of times each car went of deterioration. The number of times each car went through the carwash is shown on the next slide.through the carwash is shown on the next slide.

Home Products, Inc. must decide which wax to market. Home Products, Inc. must decide which wax to market.

Are the three waxes equally effective?Are the three waxes equally effective?

Example: Home Products, Inc.Example: Home Products, Inc. WaxWax Wax Wax Wax Wax

ObservationObservation Type 1 Type 2 Type 1 Type 2 Type 3Type 3

11 27 27 33 33 29 29 22 30 30 28 28 28 28 33 29 29 31 31 30 30 44 28 28 30 30 32 32 55 31 31 30 30 31 31

Sample MeanSample Mean 29.0 29.0 30.4 30.4 30.0 30.0

Sample VarianceSample Variance 2.52.5 3.3 3.3 2.5 2.5

HypothesesHypotheses

HH00: : 11==22==33

HHaa: Not all the means are equal: Not all the means are equal

where: where:

1 1 = mean number of washes for Type 1 wax= mean number of washes for Type 1 wax




Mean Square Between TreatmentsMean Square Between Treatments

Since the sample sizes are all Since the sample sizes are all equal:equal:

μμ= (x= (x11 + x + x22 + x + x33)/3 = (29 + 30.4 + 30)/3 = 29.8)/3 = (29 + 30.4 + 30)/3 = 29.8

SSTR= 5(29–29.8)SSTR= 5(29–29.8)22+ 5(30.4–29.8)+ 5(30.4–29.8)22+ 5(30–29.8)+ 5(30–29.8)22= 5.2= 5.2

MSTR = 5.2/(3 - 1) = 2.6MSTR = 5.2/(3 - 1) = 2.6 Mean Square ErrorMean Square Error

SSE = 4(2.5) + 4(3.3) + 4(2.5) = 33.2SSE = 4(2.5) + 4(3.3) + 4(2.5) = 33.2

MSE = 33.2/(15 - 3) = 2.77MSE = 33.2/(15 - 3) = 2.77

== ______


Rejection RuleRejection RuleUsing test statistic:Using test statistic: Reject Reject HH00 if if FF > 3.89 > 3.89

Using Using pp-value:-value: Reject Reject HH00 if if pp-value < .05-value < .05

where where FF.05.05 = 3.89 is based on an = 3.89 is based on an FF distribution distribution with 2 with 2 numerator degrees of freedom and 12 numerator degrees of freedom and 12 denominator denominator degrees of freedomdegrees of freedom



Test StatisticTest Statistic F F = MST/MSE = 2.6/2.77 = .939 = MST/MSE = 2.6/2.77 = .939

ConclusionConclusionSince Since FF = .939 < = .939 < FF.05.05 = 3.89, we cannot = 3.89, we cannot reject reject HH00. There is insufficient evidence . There is insufficient evidence to conclude that the mean number of to conclude that the mean number of washes for the three wax types are not washes for the three wax types are not all the same.all the same.

ANOVA TableANOVA Table


Variation Squares Freedom Squares Variation Squares Freedom Squares FF

TreatmentsTreatments 5.25.2 2 2 2.60 .93982.60 .9398

Error Error 33.233.2 12 12 2.77 2.77

TotalTotal 38.4 38.4 1414


Value Worksheet (top portion)Value Worksheet (top portion)

Using Excel’s ANOVA: Single Factor Using Excel’s ANOVA: Single Factor ToolTool

A B C D E

1 ObservationWax

Type 1Wax

Type 2Wax

Type 32 1 27 33 29 3 2 30 28 284 3 29 31 305 4 28 30 32 6 5 31 30 31 7

Value Worksheet (bottom portion)Value Worksheet (bottom portion)

Using Excel’s ANOVA:Using Excel’s ANOVA: Single Factor Tool Single Factor Tool

A B C D E F G8 Anova: Single Factor9

10 SUMMARY11 Groups Count Sum Average Variance12 Wax Type 1 5 145 29 2.513 Wax Type 2 5 152 30.4 3.314 Wax Type 3 5 150 30 2.51516 ANOVA17 Source of Variation SS df MS F P-value F crit18 Between Groups 5.2 2 2.6 0.939759 0.41768 3.8852919 Within Groups 33.2 12 2.766672021 Total 38.4 14

Conclusion Using the Conclusion Using the pp-Value-Value• The value worksheet shows a The value worksheet shows a pp-value of .418-value of .418

• The rejection rule is “The rejection rule is “Reject Reject HH00 if if pp-value < .05”-value < .05”

• Because .418 > .05, we cannot reject Because .418 > .05, we cannot reject HH00. . There is insufficient evidence to conclude that There is insufficient evidence to conclude that the mean number of washes for the three wax the mean number of washes for the three wax types are not all the same.types are not all the same.

Using Excel’s ANOVA: Single Factor Using Excel’s ANOVA: Single Factor ToolTool

RCBDRCBD((Randomized Complete Block DesignRandomized Complete Block Design))

Randomized Complete Block DesignRandomized Complete Block Design

An experimental design in which there is one An experimental design in which there is one independent variable, and a second variable known independent variable, and a second variable known as a blocking variable, that is used to control for as a blocking variable, that is used to control for confounding or concomitant variables.confounding or concomitant variables.

It is used when the experimental unit or material are It is used when the experimental unit or material are heterogeneousheterogeneous

There is a way to block the experimental units or There is a way to block the experimental units or materials to keep the variability among within a materials to keep the variability among within a block as small as possible and to maximize block as small as possible and to maximize differences among blockdifferences among block

The block (group) should consists units or materials The block (group) should consists units or materials which are as uniform as possiblewhich are as uniform as possible

ConfoundingConfounding or or concomitantconcomitant variable are not variable are not being controlled by the analyst but can have an being controlled by the analyst but can have an effect on the outcome of the treatment being effect on the outcome of the treatment being studiedstudied

Blocking variableBlocking variable is a variable that the analyst is a variable that the analyst wants to control but is not the treatment variable wants to control but is not the treatment variable of interest.of interest.

Repeated measures design Repeated measures design is a randomized block is a randomized block design in which each block level is an individual design in which each block level is an individual item or person, and that person or item is item or person, and that person or item is measured across all treatments.measured across all treatments.

Randomized Complete Block DesignRandomized Complete Block Design

The Blocking PrincipleThe Blocking Principle BlockingBlocking is a technique for dealing with is a technique for dealing with nuisancenuisance

factorsfactors A A nuisance nuisance factor is a factor that probably has factor is a factor that probably has

some effect on the response, but it is of no interest some effect on the response, but it is of no interest to the experimenter…however, the variability it to the experimenter…however, the variability it transmits to the response needs to be minimizedtransmits to the response needs to be minimized

Typical nuisance factors include batches of raw Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units(shifts, days, etc.), different experimental units

ManyMany industrial experiments involve blocking (or industrial experiments involve blocking (or should)should)

Failure to block is a common flaw in designing an Failure to block is a common flaw in designing an experiment (consequences?)experiment (consequences?)

The Blocking PrincipleThe Blocking Principle If the nuisance variable is If the nuisance variable is knownknown and and controllablecontrollable, ,

we use we use blockingblocking If the nuisance factor is If the nuisance factor is knownknown and and uncontrollableuncontrollable, ,

sometimes we can use the sometimes we can use the analysis of covarianceanalysis of covariance (see Chapter 14) to statistically remove the effect of (see Chapter 14) to statistically remove the effect of the nuisance factor from the analysisthe nuisance factor from the analysis

If the nuisance factor is If the nuisance factor is unknownunknown and and uncontrollableuncontrollable (a “lurking” variable (a “lurking” variable)), we hope that , we hope that randomizationrandomization balances out its impact across the balances out its impact across the experimentexperiment

Sometimes several sources of variability are Sometimes several sources of variability are combinedcombined in a block, so the block becomes an in a block, so the block becomes an aggregate variableaggregate variable

Partitioning the Total Sum of Squares Partitioning the Total Sum of Squares in the Randomized Block Designin the Randomized Block Design

Partitioning the Total Sum of Squares Partitioning the Total Sum of Squares in the Randomized Block Designin the Randomized Block Design

SStotal(total sum of squares)

SST(treatment

sum of squares)

SSE(error sum of squares)

SSB(sum of squares

blocks)

SSE’(sum of squares

error)

A Randomized Block DesignA Randomized Block DesignA Randomized Block DesignA Randomized Block Design

Individualobservations

.

.

.

.

.

.

.

.

.

.

.

.

Single Independent Variable

BlockingVariable

.

.

.

.

.

MSE

MST

The Linier ModelThe Linier Model

ijε

jρ

iτμ

ijy

i = 1,2,…, t j = 1,2,…,r

yij = the observation in ith treatment in the jth block

= overall meani = the effect of the ith treatmentj = the effect of the jth block

ij = random error

No interaction between blocks and treatments

Extension of the ANOVA to the RCBDExtension of the ANOVA to the RCBD

ANOVA partitioning of total ANOVA partitioning of total variability:variability:

t

1i

r

1j

2...ji.ij

r

1j

2...j

t

1i

2..i.

t

1i

r

1j

2...ji.ij...j..i.

t

1i

r

1j

2..ij

)yyy(y)yy(t)yy(r

)yyy(y)yy()yy()y(y

EBlocksTreatmentsT SSSSSSSS

Extension of the ANOVA to the RCBDExtension of the ANOVA to the RCBD

The degrees of freedom for the sums of squares inThe degrees of freedom for the sums of squares in

are as followsare as follows::

Ratios of sums of squares to their degrees of Ratios of sums of squares to their degrees of freedom result in mean squares, and freedom result in mean squares, and

The ratio of the mean square for treatments to The ratio of the mean square for treatments to the error mean square is an the error mean square is an FF statistic statistic used to used to test the hypothesis of equal treatment meanstest the hypothesis of equal treatment means

EBlocksTreatmentsT SSSSSSSS

)]1)(1[( )1( )1( 1 rtrttr

ANOVA ProcedureANOVA Procedure The ANOVA procedure for the randomized block design The ANOVA procedure for the randomized block design

requires us to partition the sum of squares total (SST) requires us to partition the sum of squares total (SST) into three groups: sum of squares due to treatments, into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due sum of squares due to blocks, and sum of squares due to error.to error.

The formula for this partitioning isThe formula for this partitioning is

SSTot = SST + SSB + SSESSTot = SST + SSB + SSE

The total degrees of freedom, The total degrees of freedom, nnTT - 1, are partitioned - 1, are partitioned such that such that kk - 1 degrees of freedom go to treatments, - 1 degrees of freedom go to treatments,

bb - 1 go to blocks, and ( - 1 go to blocks, and (kk - 1)( - 1)(bb - 1) go to the error term. - 1) go to the error term.

ANOVA Table for aANOVA Table for aRandomized Block DesignRandomized Block Design


Variation Squares Freedom Squares FVariation Squares Freedom Squares F

TreatmentsTreatments SSTSST t t – 1– 1 SST/t-1 MST/MSE SST/t-1 MST/MSE

BlocksBlocks SSBSSB rr - 1 - 1

ErrorError SSE (SSE (t t - 1)(- 1)(rr - 1) SSE/(t-1)(r-1) - 1) SSE/(t-1)(r-1)

TotalTotal SSTotSSTottrtr - 1 - 1

Example: Eastern Oil Co.Example: Eastern Oil Co.

Randomized Block DesignRandomized Block Design

Eastern Oil has developed three new blends of Eastern Oil has developed three new blends of gasoline and must decide which blend or gasoline and must decide which blend or blends to produce and distribute. A study of blends to produce and distribute. A study of the miles per gallon ratings of the three blends the miles per gallon ratings of the three blends is being conducted to determine if the mean is being conducted to determine if the mean ratings are the same for the three blends.ratings are the same for the three blends.

Five automobiles have been tested using each Five automobiles have been tested using each of the three gasoline blends and the miles per of the three gasoline blends and the miles per gallon ratings are shown on the next slide.gallon ratings are shown on the next slide.


Automobile Type of Gasoline (Treatment)Automobile Type of Gasoline (Treatment) Blocks Blocks

(Block)(Block) Blend XBlend X Blend YBlend Y Blend Z Blend Z MeansMeans

11 3131 30 30 3030 30.333 30.333

22 3030 29 29 2929 29.333 29.333

33 2929 29 29 2828 28.667 28.667

44 3333 31 31 2929 31.000 31.000

55 2626 25 25 2626 25.667 25.667

TreatmentTreatment MeansMeans 29.8 29.8 28.8 28.8 28.4 28.4

Example: Eastern Oil Co.Example: Eastern Oil Co. Mean Square Due to TreatmentsMean Square Due to Treatments

The overall sample mean is 29. Thus,The overall sample mean is 29. Thus,SST= 5[(29.8 - 29)SST= 5[(29.8 - 29)22+ (28.8 - 29)+ (28.8 - 29)22+ (28.4 - 29)+ (28.4 - 29)22]= 5.2]= 5.2

MST = 5.2/(3 - 1) = 2.6MST = 5.2/(3 - 1) = 2.6 Mean Square Due to BlocksMean Square Due to Blocks

SSB = 3[(30.333 - 29)SSB = 3[(30.333 - 29)22 + . . . + (25.667 - 29) + . . . + (25.667 - 29)22] = 51.33] = 51.33

MSB = 51.33/(5 - 1) = 12.8 MSB = 51.33/(5 - 1) = 12.8 Mean Square Due to ErrorMean Square Due to Error

SSE = 62 - 5.2 - 51.33 = 5.47SSE = 62 - 5.2 - 51.33 = 5.47

MSE = 5.47/[(3 - 1)(5 - 1)] = .68MSE = 5.47/[(3 - 1)(5 - 1)] = .68

Rejection RuleRejection Rule

Using test statistic:Using test statistic: Reject Reject HH00 if if FF > 4.46 > 4.46

Using Using pp-value:-value: Reject Reject HH00 if if pp-value < .05-value < .05

Assuming Assuming = .05, = .05, FF.05.05 = 4.46 (2 d.f. = 4.46 (2 d.f. numerator numerator and 8 d.f. denominator)and 8 d.f. denominator)



Test StatisticTest Statistic

FF = MST/MSE = 2.6/.68 = 3.82 = MST/MSE = 2.6/.68 = 3.82 ConclusionConclusion

Since 3.82 < 4.46, we cannot reject Since 3.82 < 4.46, we cannot reject HH00. There is not . There is not sufficient evidence to conclude that the miles per gallon sufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends.ratings differ for the three gasoline blends.

Using Excel’s Anova:Using Excel’s Anova:Two-Factor Without Replication ToolTwo-Factor Without Replication Tool

Step 1Step 1 Select the Select the ToolsTools pull-down menu pull-down menu Step 2Step 2 Choose the Choose the Data AnalysisData Analysis option option Step 3Step 3 Choose Choose Anova: Two Factor Without Anova: Two Factor Without

Replication Replication from the list of Analysis Toolsfrom the list of Analysis Tools

… … continuedcontinued

Step 4Step 4 When the Anova: Two Factor Without When the Anova: Two Factor Without

Replication dialog box appears:Replication dialog box appears:

Enter A1:D6 in the Enter A1:D6 in the Input RangeInput Range box box

Select Select LabelsLabels

Enter .05 in the Enter .05 in the AlphaAlpha box box

Select Select Output RangeOutput Range

Enter A8 (your choice) in the Enter A8 (your choice) in the Output RangeOutput Range box box

Click Click OKOK


Value Worksheet (top portion)Value Worksheet (top portion)


A B C D E F G1 Automobile Blend X Blend Y Blend Z2 1 31 30 30 3 2 30 29 294 3 29 29 28 5 4 33 31 29 6 5 26 25 26

Value Worksheet (middle portion)Value Worksheet (middle portion)


A B C D E F G8 Anova: Two-Factor Without Replication910 SUMMARY Count Sum Average Variance11 1 3 91 30.3333 0.3333312 2 3 88 29.3333 0.3333313 3 3 86 28.6667 0.3333314 4 3 93 31 415 5 3 77 25.6667 0.333331617 Blend X 5 149 29.8 6.718 Blend Y 5 144 28.8 5.219 Blend Z 5 142 28.4 2.3

Value Worksheet (bottom portion)Value Worksheet (bottom portion)


A B C D E F G8 ANOVA9 Variation SS df MS F P-value F crit10 Rows 51.3333 4 12.8333 18.7805 0.0004 3.8378511 Columns 5.2 2 2.6 3.80488 0.06899 4.4589712 Error 5.46667 8 0.683331314 Total 62 14

Conclusion Using the p-ValueConclusion Using the p-Value• The value worksheet shows that the The value worksheet shows that the pp-value -value

is .06899is .06899

• The rejection rule is “The rejection rule is “Reject Reject HH00 if if pp-value < .05”-value < .05”

• Thus, we cannot reject Thus, we cannot reject HH00 because the because the pp-value -value = .06899 > = .06899 > = .05 = .05

• There is not sufficient evidence to conclude There is not sufficient evidence to conclude that the miles per gallon ratings differ for the that the miles per gallon ratings differ for the three gasoline blendsthree gasoline blends


Similarities and differences between Similarities and differences between CRD and RCBD: ProceduresCRD and RCBD: Procedures

RCBD: Every level of “treatment” encountered RCBD: Every level of “treatment” encountered by each experimental unit; CRD: Just one level by each experimental unit; CRD: Just one level eacheach

Descriptive statistics and graphical display: the Descriptive statistics and graphical display: the same as CRDsame as CRD

Model adequacy checking procedure: the same Model adequacy checking procedure: the same except: specifically, except: specifically, NO Block x Treatment NO Block x Treatment InteractionInteraction

ANOVA: Inclusion of the Block effect; ANOVA: Inclusion of the Block effect; dfdferrorerror change from change from tt((rr – 1) to ( – 1) to (tt – 1)( – 1)(rr – 1) – 1)

Latin Square DesignLatin Square Design

DefinitionDefinition A Latin square is a square array of objects A Latin square is a square array of objects

(letters A, B, C, …) such that each object (letters A, B, C, …) such that each object appears once and only once in each row appears once and only once in each row and each column. and each column.

Example - 4 x 4 Latin Square.Example - 4 x 4 Latin Square.

A B C DA B C DB C D AB C D AC D A BC D A BD A B CD A B C

The Latin Square DesignThe Latin Square Design This design is used to simultaneously control (or eliminate) This design is used to simultaneously control (or eliminate)

two sources of nuisance variabilitytwo sources of nuisance variability It is called “Latin” because we usually specify the treatment It is called “Latin” because we usually specify the treatment

by the Latin lettersby the Latin letters ““Square” because it always has the same number of levels Square” because it always has the same number of levels

((tt) for the row and column nuisance factors) for the row and column nuisance factors A significant assumption is that the three factors A significant assumption is that the three factors

(treatments and two nuisance factors) (treatments and two nuisance factors) do not interactdo not interact More restrictive than the RCBDMore restrictive than the RCBD Each treatment appears once and only once in each row and Each treatment appears once and only once in each row and

columncolumn If you can block on two (perpendicular) sources of variation If you can block on two (perpendicular) sources of variation

(rows x columns) you can reduce experimental error when (rows x columns) you can reduce experimental error when compared to the RCBDcompared to the RCBD

A

B C D

A

B C D A

BC D

A

B CD

Advantages and DisadvantagesAdvantages and Disadvantages

Advantage:Advantage:• Allows the experimenter to control two sources Allows the experimenter to control two sources

of variationof variation

Disadvantages:Disadvantages:• Error degree of freedom (df) is small if there Error degree of freedom (df) is small if there

are only a few treatmentsare only a few treatments• The experiment becomes very large if the The experiment becomes very large if the

number of treatments is largenumber of treatments is large• The statistical analysis is complicated by The statistical analysis is complicated by

missing plots and mis-assigned treatmentsmissing plots and mis-assigned treatments

Selected Latin SquaresSelected Latin Squares3 x 33 x 3 4 x 44 x 4A B CA B CA B C DA B C D A B C DA B C D A B C DA B C D A B C DA B C DB C AB C AB A D CB A D C B C D AB C D A B D A CB D A C B A D CB A D CC A BC A BC D B AC D B A C D A BC D A B C A D BC A D B C D A BC D A B

D C A BD C A B D A B CD A B C D C B AD C B A D C B AD C B A

5 x 55 x 5 6 x 66 x 6A B C D EA B C D E A B C D E FA B C D E FB A E C DB A E C D B F D C A EB F D C A EC D A E BC D A E B C D E F B AC D E F B AD E B A CD E B A C D A F E C BD A F E C BE C D B AE C D B A F E B A D CF E B A D C

Latin Square DesignsLatin Square Designs

In a Latin square You have three factors:In a Latin square You have three factors: Treatments (Treatments (tt) (letters A, B, C, …)) (letters A, B, C, …) Rows (Rows (tt) ) Columns (Columns (tt) )

The number of treatments = the number of rows = the number of columns = t.

The row-column treatments are represented by cells in a t x t array.

The treatments are assigned to row-column combinations using a Latin-square arrangement

ExampleExample

A courier company is interested in deciding A courier company is interested in deciding between five brands (D,P,F,C and R) of car between five brands (D,P,F,C and R) of car for its next purchase of fleet cars. for its next purchase of fleet cars.

The brands are all comparable in purchase price.

The company wants to carry out a study that will enable them to compare the brands with respect to operating costs.

For this purpose they select five drivers (Rows).

In addition the study will be carried out over a five week period (Columns = weeks).

Each week a driver is assigned to a car using randomization and a Latin Square Design.

The average cost per mile is recorded at the end of each week and is tabulated below: Week

1 2 3 4 5 1 5.83 6.22 7.67 9.43 6.57 D P F C R 2 4.80 7.56 10.34 5.82 9.86 P D C R F

Drivers 3 7.43 11.29 7.01 10.48 9.27 F C R D P 4 6.60 9.54 11.11 10.84 15.05 R F D P C 5 11.24 6.34 11.30 12.58 16.04 C R P F D

The Linier ModelThe Linier Model

kijjikkijy

i = 1,2,…, t j = 1,2,…, t

yij(k) = the observation in ith row and the jth column receiving the kth treatment

= overall meank = the effect of the ith treatmenti = the effect of the ith row

ij(k) = random error

k = 1,2,…, t

j = the effect of the jth column

No interaction between rows, columns and treatments

A Latin Square experiment is assumed to be a three-factor experiment.

The factors are rows, columns and treatments.

It is assumed that there is no interaction between rows, columns and treatments.

The degrees of freedom for the interactions is used to estimate error.

The Anova Table for a Latin Square ExperimentThe Anova Table for a Latin Square Experiment

SourcSourcee

S.S.S.S. d.f.d.f. M.S.M.S. FF p-p-valuevalue

TreatTreat SSSSTT t-1t-1 MSMSTT MSMST T /MS/MSEE

RowsRows SSSSRowRow t-1t-1 MSMSRowRow MSMSRow Row

/MS/MSEE

ColsCols SSSSColCol t-1t-1 MSMSColCol MSMSCol Col /MS/MSEE

ErrorError SSSSEE(t-1)(t-2)(t-1)(t-2) MSMSEE

TotalTotal SSSSTT tt22 - 1 - 1

The Anova Table for ExampleThe Anova Table for Example

SourceSource S.S.S.S. d.f.d.f. M.S.M.S. FF p-valuep-value

WeekWeek 51.1788751.17887 44 12.7947212.79472 16.0616.06 0.00010.0001

DriverDriver 69.4466369.44663 44 17.3616617.36166 21.7921.79 0.00000.0000

CarCar 70.9040270.90402 44 17.7260117.72601 22.2422.24 0.00000.0000

ErrorError 9.563159.56315 1212 0.796930.79693

TotalTotal 201.09267201.09267 2424

ExampleExample

In this Experiment theIn this Experiment the we are again interested we are again interested in how weight gain (Y) in rats is affected by in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low). by Level of Protein (High or Low).

There are a total of t = 3 X 2 = 6 treatment combinations of the two factors.

Beef -High Protein Cereal-High Protein Pork-High Protein Beef -Low Protein Cereal-Low Protein and Pork-Low Protein

In this example we will consider using a In this example we will consider using a Latin SquareLatin Square design design

Six Initial Weight categories are identified for the Six Initial Weight categories are identified for the test animals in addition to Six Appetite categories. test animals in addition to Six Appetite categories.

A test animal is then selected from each of the 6 X 6 = 36 combinations of Initial Weight and Appetite categories.

A Latin square is then used to assign the 6 diets to the 36 test animals in the study.

In the latin square the letter In the latin square the letter

A represents the high protein-cereal diet

B represents the high protein-pork diet C represents the low protein-beef Diet D represents the low protein-cereal

diet E represents the low protein-pork diet

and F represents the high protein-beef diet.

The weight gain after a fixed period is measured for The weight gain after a fixed period is measured for each of the test animals and is tabulated below:each of the test animals and is tabulated below:

Appetite Category 1 2 3 4 5 6 1 62.1 84.3 61.5 66.3 73.0 104.7 A B C D E F 2 86.2 91.9 69.2 64.5 80.8 83.9 B F D C A E

Initial 3 63.9 71.1 69.6 90.4 100.7 93.2 Weight C D E F B A

Category 4 68.9 77.2 97.3 72.1 81.7 114.7 D A F E C B 5 73.8 73.3 78.6 101.9 111.5 95.3 E C A B F D 6 101.8 83.8 110.6 87.9 93.5 103.8 F E B A D C

The Anova Table for ExampleThe Anova Table for Example


InwtInwt 1767.08361767.0836 55 353.41673353.41673 111.1111.1 0.00000.0000

AppApp 2195.43312195.4331 55 439.08662439.08662 138.03138.03 0.00000.0000

DietDiet 4183.91324183.9132 55 836.78263836.78263 263.06263.06 0.00000.0000

ErrorError 63.6199963.61999 2020 3.1813.181

TotalTotal 8210.04998210.0499 3535

Diet SS partioned into main effects for Source and Level of Protein


InwtInwt 1767.08361767.0836 55 353.41673353.41673 111.1111.1 0.00000.0000

AppApp 2195.43312195.4331 55 439.08662439.08662 138.03138.03 0.00000.0000

SourceSource 631.22173631.22173 22 315.61087315.61087 99.2299.22 0.00000.0000

LevelLevel 2611.20972611.2097 11 2611.20972611.2097 820.88820.88 0.00000.0000

SLSL 941.48172941.48172 22 470.74086470.74086 147.99147.99 0.00000.0000

ErrorError 63.6199963.61999 2020 3.1813.181

TotalTotal 8210.04998210.0499 3535

Graeco-Latin Graeco-Latin Square DesignsSquare Designs

Mutually orthogonal Squares

DefinitionDefinitionA A Greaco-LatinGreaco-Latin square consists of two latin square consists of two latin squares (one using the letters A, B, C, … the squares (one using the letters A, B, C, … the other using greek letters a, b, c, …) such that other using greek letters a, b, c, …) such that when the two latin square are supper imposed on when the two latin square are supper imposed on each other the letters of one square appear once each other the letters of one square appear once and only once with the letters of the other and only once with the letters of the other square. The two Latin squares are called square. The two Latin squares are called mutually orthogonalmutually orthogonal..Example: Example: a 7 x 7 Greaco-Latin Squarea 7 x 7 Greaco-Latin Square

AA BB CC DD EE FF GGBB CC DD EE FF GG AA

CC DD EE FF GG AA BBDD EE FF GG AA BB CCEE FF GG AA BB CC DDFF GG AA BB CC DD EEGG AA BB CC DD EE FF

The Graeco-Latin Square DesignThe Graeco-Latin Square Design

√ This design is used to simultaneously control (or This design is used to simultaneously control (or eliminate) eliminate) three sources of nuisance three sources of nuisance variabilityvariability

√ It is called “Graeco-Latin” because we usually It is called “Graeco-Latin” because we usually specify the third nuisance factor, represented by specify the third nuisance factor, represented by the Greek letters, orthogonal to the Latin lettersthe Greek letters, orthogonal to the Latin letters

√ A significant assumption is that the four factors A significant assumption is that the four factors (treatments, nuisance factors) (treatments, nuisance factors) do not interactdo not interact

√ If this assumption is violated, as with the Latin If this assumption is violated, as with the Latin square design, it will not produce valid resultssquare design, it will not produce valid results

√ Graeco-Latin squares exist for all Graeco-Latin squares exist for all tt ≥ 3 except ≥ 3 except tt = 6= 6

Note:Note:

At most (At most (tt –1) –1) tt x x tt Latin squares Latin squares LL11, , LL22, …, , …, LLt-1t-1 such that any pair are such that any pair are mutually mutually orthogonalorthogonal..

It is possible that there exists a set of six 7 x 7 mutually orthogonal Latin squares L1, L2, L3, L4, L5, L6 .

The Greaco-Latin Square DesignThe Greaco-Latin Square Design - An Example - An Example

A researcher is interested in determining the A researcher is interested in determining the effect of two factorseffect of two factors

the percentage of Lysine in the diet and percentage of Protein in the diet

have on Milk Production in cows.

Previous similar experiments suggest that interaction between the two factors is

negligible.

For this reason it is decided to use a For this reason it is decided to use a Greaco-Latin square design to Greaco-Latin square design to

experimentally determine the two effects experimentally determine the two effects of the two factors (of the two factors (LysineLysine and and ProteinProtein). ).

Seven levels of each factor is selected• 0.0(A), 0.1(B), 0.2(C), 0.3(D), 0.4(E),

0.5(F), and 0.6(G)% for Lysine and • 2(), 4(), 6(), 8(), 10(), 12() and 14()

% for Protein. • Seven animals (cows) are selected at

random for the experiment which is to be carried out over seven three-month periods.

A Greaco-Latin Square is the used to assign the 7 X 7 A Greaco-Latin Square is the used to assign the 7 X 7 combinations of levels of the two factors (combinations of levels of the two factors (LysineLysine and and ProteinProtein) ) to a period and a cow. The data is tabulated on below:to a period and a cow. The data is tabulated on below:

P e r i o d

1 2 3 4 5 6 7

1 3 0 4 4 3 6 3 5 0 5 0 4 4 1 7 5 1 9 4 3 2

( A ( B ( C ( D ( E ( F ( G 2 3 8 1 5 0 5 4 2 5 5 6 4 4 9 4 3 5 0 4 1 3 B ( C ( D ( E ( F ( G ( A 3 4 3 2 5 6 6 4 7 9 3 5 7 4 6 1 3 4 0 5 0 2 ( C ( D ( E ( F ( G ( A ( B C o w s 4 4 4 2 3 7 2 5 3 6 3 6 6 4 9 5 4 2 5 5 0 7 ( D ( E ( F ( G ( A ( B ( C 5 4 9 6 4 4 9 4 9 3 3 4 5 5 0 9 4 8 1 3 8 0 ( E ( F ( G ( A ( B ( C ( D 6 5 3 4 4 2 1 4 5 2 4 2 7 3 4 6 4 7 8 3 9 7 ( F ( G ( A ( B ( C ( D ( E 7 5 4 3 3 8 6 4 3 5 4 8 5 4 0 6 5 5 4 4 1 0 ( G ( A ( B ( C ( D ( E ( F

The Linear ModelThe Linear Model

klijjilkklijy

i = 1,2,…, t j = 1,2,…, t

yij(kl) = the observation in ith row and the jth column receiving the kth Latin treatment and the lth

Greek treatment

k = 1,2,…, t l = 1,2,…, t

= overall mean

k = the effect of the kth Latin treatment

i = the effect of the ith row

ij(k) = random error

j = the effect of the jth column

No interaction between rows, columns, Latin treatments and Greek treatments

l = the effect of the lth Greek treatment

A Greaco-Latin Square experiment is assumed to be a four-factor experiment.

The factors are rows, columns, Latin treatments and Greek treatments.

It is assumed that there is no interaction between rows, columns, Latin treatments and Greek treatments.

The degrees of freedom for the interactions is used to estimate error.

Documents

Completely Randomized Design. 1. Experimental Units (Subjects) Are Assigned Randomly to Treatments Subjects are Assumed HomogeneousSubjects are Assumed