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### Graeco-Latin Square Designs

Completely Randomized Design

1. Experimental Units (Subjects) Are Assigned Randomly to Treatments

- Subjects are Assumed Homogeneous

2. One Factor or Independent Variable

- 2 or More Treatment Levels or Classifications

3. Analyzed by One-Way ANOVA

The Linier Model

i = 1,2,…, t

j = 1,2,…, r

Xij = the observation inith treatment and thejth replication

m = overall mean

t i= the effect of theithtreatment

eij = random error

One-Way ANOVA F-Test

1. Tests the Equality of 2 or More (t) Population Means

2. Variables

- One Nominal Scaled Independent Variable
- 2 or More (t) Treatment Levels or Classifications
- One Interval or Ratio Scaled Dependent Variable

3. Used to Analyze Completely Randomized Experimental Designs

Assumptions

1. Randomness & Independence of Errors

- Independent Random Samples are Drawn for each condition

2. Normality

- Populations (for each condition) are Normally Distributed

3. Homogeneity of Variance

- Populations (for each condition) have Equal Variances

Hypotheses

- H0: 1 = 2 = 3 = ... = t
- All Population Means are Equal
- No Treatment Effect
- Ha: Not All i Are Equal
- At Least 1 Pop. Mean is Different
- Treatment Effect

NOT 12 ... t

Hypotheses

- H0: 1 = 2 = 3 = ... = t
- All Population Means are Equal
- No Treatment Effect
- Ha: Not All i Are Equal
- At Least 1 Pop. Mean is Different
- Treatment Effect

NOT 12 ... t

f(X)

X

=

=

1

2

3

f(X)

X

=

1

2

3

One-Way ANOVA Basic Idea

1. Compares 2 Types of Variation to Test Equality of Means

2. Comparison Basis Is Ratio of Variances

3. If Treatment Variation Is Significantly Greater Than Random Variation then Means Are Not Equal

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

One-Way ANOVA Partitions Total Variation

Total variation

One-Way ANOVAPartitions Total Variation

Total variation

Variation due to treatment

Variation due to random sampling

Sum of Squares Among

Sum of Squares Between

Sum of Squares Treatment

Among Groups Variation

One-Way ANOVAPartitions Total VariationTotal variation

Variation due to treatment

Variation due to random sampling

Sum of Squares Within

Sum of Squares Error (SSE)

Within Groups Variation

Sum of Squares Among

Sum of Squares Between

Sum of Squares Treatment (SST)

Among Groups Variation

One-Way ANOVAPartitions Total VariationTotal variation

Variation due to treatment

Variation due to random sampling

One-Way ANOVA F-Test Test Statistic

1. Test Statistic

- F = MST / MSE
- MST Is Mean Square for Treatment
- MSE Is Mean Square for Error

2. Degrees of Freedom

- 1 = t -1
- 2 = tr - t
- t = # Populations, Groups, or Levels
- tr = Total Sample Size

One-Way ANOVA Summary Table

Source of

Degrees

Sum of

Mean

F

Variation

of

Squares

Square

Freedom

(Variance)

Treatment

t - 1

SST

MST =

MST

SST/(t - 1)

MSE

Error

tr - t

SSE

MSE =

SSE/(tr - t)

Total

tr - 1

SS(Total) =

SST+SSE

ANOVA Table for aCompletely Randomized Design

Source of Sum of Degrees of Mean

Variation Squares Freedom Squares F

TreatmentsSST t - 1 SST/t-1 MST/MSE

ErrorSSE tr- t SSE/tr-t

TotalSSTot tr - 1

The F distribution

- Two parameters
- increasing either one decreases F-alpha (except for v2<3)
- I.e., the distribution gets smashed to the left

F

0

F

(

v1

,

v2

)

One-Way ANOVA F-Test Critical Value

If means are equal, F = MST / MSE1. Only reject large F!

Reject H

0

Do Not

Reject H

0

F

0

F

a

(

t

1

,

tr

-t)

Always One-Tail!

© 1984-1994 T/Maker Co.

Example: Home Products, Inc.

- Completely Randomized Design

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

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 3. Each car was then repeatedly run through an automatic carwash until the wax coating showed signs of deterioration. The number of times each car went through the carwash is shown on the next slide.

Home Products, Inc. must decide which wax to market. Are the three waxes equally effective?

Example: Home Products, Inc.

Wax Wax Wax

Observation Type 1 Type 2 Type 3

1 27 33 29

2 30 28 28

3 29 31 30

4 28 30 32

5 31 30 31

Sample Mean 29.0 30.4 30.0

Sample Variance2.5 3.3 2.5

Example: Home Products, Inc.

- Hypotheses

H0: 1=2=3

Ha: Not all the means are equal

where:

1 = mean number of washes for Type 1 wax

2 = mean number of washes for Type 2 wax

3 = mean number of washes for Type 3 wax

Example: Home Products, Inc.

- Mean Square Between Treatments

Since the sample sizes are all equal:

μ= (x1 + x2 + x3)/3 = (29 + 30.4 + 30)/3 = 29.8

SSTR= 5(29–29.8)2+ 5(30.4–29.8)2+ 5(30–29.8)2= 5.2

MSTR = 5.2/(3 - 1) = 2.6

- Mean Square Error

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

MSE = 33.2/(15 - 3) = 2.77

_

_

_

=

Example: Home Products, Inc.

- Rejection Rule

Using test statistic: Reject H0 if F > 3.89

Using p-value: Reject H0 if p-value < .05

where F.05 = 3.89 is based on an F distribution with 2 numerator degrees of freedom and 12 denominator degrees of freedom

Example: Home Products, Inc.

- Test Statistic

F = MST/MSE = 2.6/2.77 = .939

- Conclusion

Since F = .939 < F.05 = 3.89, we cannot reject H0. There is insufficient evidence to conclude that the mean number of washes for the three wax types are not all the same.

Example: Home Products, Inc.

- ANOVA Table

Source of Sum of Degrees of Mean

Variation Squares Freedom Squares F

Treatments5.2 2 2.60 .9398

Error 33.2 12 2.77

Total38.4 14

Using Excel’s ANOVA: Single Factor Tool

- Value Worksheet (top portion)

Using Excel’s ANOVA: Single Factor Tool

- Value Worksheet (bottom portion)

Using Excel’s ANOVA: Single Factor Tool

- Conclusion Using the p-Value
- The value worksheet shows a p-value of .418
- The rejection rule is “Reject H0 if p-value < .05”
- Because .418 > .05, we cannot reject H0. There is insufficient evidence to conclude that the mean number of washes for the three wax types are not all the same.

RCBD

(Randomized Complete Block Design)

Randomized Complete Block Design

- An experimental design in which there is one independent variable, and a second variable known as a blocking variable, that is used to control for confounding or concomitant variables.
- It is used when the experimental unit or material are heterogeneous
- There is a way to block the experimental units or materials to keep the variability among within a block as small as possible and to maximize differences among block
- The block (group) should consists units or materials which are as uniform as possible

Randomized Complete Block Design

- Confounding or concomitant variable are not being controlled by the analyst but can have an effect on the outcome of the treatment being studied
- Blocking variable is a variable that the analyst wants to control but is not the treatment variable of interest.
- Repeated measures designis a randomized block design in which each block level is an individual item or person, and that person or item is measured across all treatments.

The Blocking Principle

- Blocking is a technique for dealing with nuisancefactors
- A nuisance factor is a factor that probably has some effect on the response, but it is of no interest to the experimenter…however, the variability it transmits to the response needs to be minimized
- Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units
- Many industrial experiments involve blocking (or should)
- Failure to block is a common flaw in designing an experiment

The Blocking Principle

- If the nuisance variable is known and controllable, we use blocking
- If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance to statistically remove the effect of the nuisance factor from the analysis
- If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experiment
- Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable

Partitioning the Total Sum of Squares in the Randomized Block Design

SStotal

(total sum of squares)

SSE

(error sum of squares)

SST

(treatment

sum of squares)

SSB

(sum of squares

blocks)

SSE’

(sum of squares

error)

The Linier Model

i = 1,2,…, t

j = 1,2,…,r

yij = the observation inith treatment in thejth block

m = overall mean

ti = the effect of theithtreatment

No interaction between blocks and treatments

rj = the effect of the jth block

eij = random error

Extension of the ANOVA to the RCBD

ANOVA partitioning of total variability:

Extension of the ANOVA to the RCBD

The degrees of freedom for the sums of squares in

are as follows:

- Ratios of sums of squares to their degrees of freedom result in mean squares, and
- The ratio of the mean square for treatments to the error mean square is an F statistic used to test the hypothesis of equal treatment means

ANOVA Procedure

- The ANOVA procedure for the randomized block design requires us to partition the sum of squares total (SST) into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error.
- The formula for this partitioning is

SSTot = SSTreatment + SSBlock + SSE

- The total degrees of freedom, tr - 1, are partitioned such that t - 1 degrees of freedom go to treatments, r - 1 go to blocks, and (t - 1)(r - 1) go to the error term.

ANOVA Table for aRandomized Block Design

Source of Sum of Degrees of Mean

Variation Squares Freedom Squares F

TreatmentsSST t – 1 SST/t-1 MST/MSE

BlocksSSB r - 1

ErrorSSE (t - 1)(r - 1) SSE/(t-1)(r-1)

TotalSSTot tr - 1

Example: Eastern Oil Co.

Randomized Block Design

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

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

Example: Eastern Oil Co.

Automobile Type of Gasoline (Treatment) Blocks

(Block)Blend XBlend YBlend Z Means

1 32 31 30 31

2 29 28 27 28

3 30 30 27 29

4 32 31 30 31

5 27 25 26 26

Treatment

Means 30 29 28 29

Example: Eastern Oil Co.

- Mean Square Due to Treatments

The overall sample mean is 29. Thus,

SSTreatment = 5[(30-29)2+ (29-29)2+ (28-29)2]= 10.00

MSTreatment = 10/(3 - 1) = 5

- Mean Square Due to Blocks

SSBlock = 3[(31-29)2 + . . . + (26-29)2] = 54

MSBlock= 54/(5 - 1) = 13.5

- Mean Square Due to Error

CF = (435)2 /(3x5) =12615

SStotal = (32)2+ …+ (26)2 – 12615 = 62

SSE = 68 - 10 – 54 = 4

MSE = 4/[(3 - 1)(5 - 1)] = .5

Example: Eastern Oil Co.

- Rejection Rule

Using test statistic:Reject H0 if F > 4.46

Assumingα = .05, F.05 = 4.46 (2 d.f. numerator and 8 d.f. denominator)

Example: Eastern Oil Co.

- Test Statistic

F = MSTreatment/MSE = 5/.5 = 10

- Conclusion

Since 10.00 > 4.46, we reject H0. There is sufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends.

Using Excel’s Anova

- Step 1Select the Tools pull-down menu
- Step 2Choose the Data Analysis option
- Step 3 Choose Anova: Two Factor Without Replication from the list of Analysis Tools

Using Excel’s Anova

- Step 4When the Anova: Two Factor Without

Replication dialog box appears:

Enter A1:D6 in the Input Range box

Select Labels

Enter .05 in the Alpha box

Select Output Range

Enter A8 (your choice) in the Output Range box

Click OK

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

- Value Worksheet (top portion)

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

- Value Worksheet (middle portion)

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

- Conclusion Using F table
- The value worksheet shows that F table for column is 4.459
- The rejection rule is “Reject H0 if F calculated > F Table”
- Thus, we reject H0 because F calculated > F Table for a = .05
- There is sufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends

Similarities and differences between CRD and RCBD: Procedures

- RCBD: Every level of “treatment” encountered by each experimental unit; CRD: Just one level each
- Descriptive statistics and graphical display: the same as CRD
- Model adequacy checking procedure: the same except: specifically, NO Block x Treatment Interaction
- ANOVA: Inclusion of the Block effect; dferror change from t(r – 1) to (t – 1)(r – 1)

Definition

- A Latin square is a square array of objects (letters A, B, C, …) such that each object appears once and only once in each row and each column.
- Example - 4 x 4 Latin Square.

A B C D

B C D A

C D A B

D A B C

B

C

D

D

B

C

A

C

D

A

B

D

A

B

C

The Latin Square Design- This design is used to simultaneously control (or eliminate) two sources of nuisance variability (confounding variables)
- It is called “Latin” because we usually specify the treatment by the Latin letters
- “Square” because it always has the same number of levels (t) for the row and column nuisance factors
- A significant assumption is that the three factors (treatments and two nuisance factors) do not interact
- More restrictive than the RCBD
- Each treatment appears once and only once in each row and column
- If you can block on two (perpendicular) sources of variation (rows x columns) you can reduce experimental error when compared to the RCBD

Useful in Animal Nutrition Studies

- Suppose you had four feeds you wanted to test on dairy cows. The feeds would be tested over time during the lactation period
- This experiment would require 4 animals (think of these as the rows)
- There would be 4 feeding periods at even intervals during the lactation period beginning early in lactation (these would be the columns)
- The treatments would be the four feeds. Each animal receives each treatment one time only.

1

2

3

4

1 2 3 4

A

B

C

D

B

C

D

A

C

D

A

B

D

A

B

C

‘Column’

Uses in Field Experiments- When two sources of variation must be controlled
- Slope and fertility
- Furrow irrigation and shading
- If you must plant your plots perpendicular to a linear gradient

- Practically speaking, use only when you have more than four but fewer than ten treatments
- a minimum of 12 df for error

Randomization

First row in alphabetical order

A B C D E

Subsequent rows - shift letters one position

4 3 5 1 2

A B C D E 2 C D E A B A B D C E

B C D E A 4 A B C D E D E B A C

C D E A B 1 D E A B C B C E D A

D E A B C 3 B C D E A E A C B D

E A B C D 5 E A B C D C D A E B

Randomize the order of the rows: 2 4 1 3 5

Finally, randomize the order of the columns: 4 3 5 1 2

Analysis

- Set up a two-way table and compute the row and column totals
- Compute a table of treatment totals and means
- Set up an ANOVA table divided into sources of variation
- Rows
- Columns
- Treatments
- Error
- Significance tests
- FT tests difference among treatment means
- FR and FC test if row and column groupings are effective

Advantages and Disadvantages

- Advantage:
- Allows the experimenter to control two sources of variation
- Disadvantages:
- Error degree of freedom (df) is small if there are only a few treatments
- The experiment becomes very large if the number of treatments is large
- The statistical analysis is complicated by missing plots and mis-assigned treatments

Selected Latin Squares

3 x 34 x 4

A B C A B C D A B C D A B C D A B C D

B C A B A D C B C D A B D A C B A D C

C A B C D B A C D A B C A D B C D A B

D C A B D A B C D C B A D C B A

5 x 56 x 6

A B C D E A B C D E F

B A E C D B F D C A E

C D A E B C D E F B A

D E B A C D A F E C B

E C D B A F E B A D C

In a Latin square, there are three factors:

- Treatments (t) (letters A, B, C, …)
- Rows (t)
- Columns (t)

- 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

The Linier Model

i = 1,2,…, t

j = 1,2,…, t

k = 1,2,…, t

yij(k) = the observation inithrow and thejthcolumnreceiving thekthtreatment

m = overall mean

tk = the effect of thekthtreatment

No interaction between rows, columns and treatments

ri = the effect of theithrow

gj = the effect of thejthcolumn

eij(k) = random error

- 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.

Example

1.

A courier company is interested in deciding between five brands (D,P,F,C and R) of car 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:

Example

2.

we are again interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and 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
- Pork-Low Protein

In this example we will consider using a Latin Square design

Six Initial Weight categories are identified for the 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

- 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
- F represents the high protein-beef diet.

The weight gain after a fixed period is measured for each of the test animals

Mutually orthogonal Squares

Definition

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

Example: a 7 x 7 Greaco-Latin Square

Aa Be Cb Df Ec Fg Gd

Bb Cf Dc Eg Fd Ga Ae

Cc Dg Ed Fa Ge Ab Bf

Dd Ea Fe Gb Af Bc Cg

Ee Fb Gf Ac Bg Cd Da

Ff Gc Ag Bd Ca De Eb

Gg Ad Ba Ce Db Ef Fc

The Graeco-Latin Square Design

- This design is used to simultaneously control (or eliminate) three sources of nuisance variability
- It is called “Graeco-Latin” because we usually specify the third nuisance factor, represented by the Greek letters, orthogonal to the Latin letters
- A significant assumption is that the four factors (treatments, nuisance factors) do not interact
- If this assumption is violated, as with the Latin square design, it will not produce valid results
- Graeco-Latin squares exist for all t ≥ 3 except t = 6

Note:

At most (t –1) t x t Latin squares L1, L2, …, Lt-1 such that any pair are mutually orthogonal.

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 Design - An Example

A researcher is interested in determining the effect 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 Greaco-Latin square design to experimentally determine the two effects of the two factors (Lysine and Protein).

- 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(a), 4(b),6(c), 8(d),10(e), 12(f) and14(g)% 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 combinations of levels of the two factors (Lysine and Protein) to a period and a cow. The data is tabulated on below:

The Linear Model

j = 1,2,…, t

i = 1,2,…, t

k = 1,2,…, t

l = 1,2,…, t

yij(kl) = the observation in ithrow and the jth column receivingthe kth Latin treatment and the lth Greek treatment

tk = the effect of the kth Latin treatment

ll = the effect of thelthGreek treatment

ri = the effect of theithrow

gj = the effect of thejthcolumn

eij(k) = random error

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

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.

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