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Chapter 16

Other Analysis of Variance Designs

I Some Basic Experimental Design Concepts

A. Definition of Experimental Design

1. A randomization plan for assigning participants

to experimental conditions and the associated

statistical analysis.

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B. Procedures for Controlling Nuisance Variables

1. Hold the nuisance variables constant.

2. Assign participants randomly to the treatment

levels.

3. Include the nuisance variable as one of the factors

in the experiment. This procedure is referred

to as blocking.

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C. Blocking Variable

1. Any variable that is positively correlated with

the dependent variable is a candidate for blocking.

D. Procedures for Forming Blocks of Dependent Samples

1. Obtain repeated measures on each participant 2. Match subjects on a relevant variable

3. Use participants who are genetically similar

4. Use participants who are matched by mutual

selection

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II Randomized Block Design (RB-p Design)

A. Characteristics of the RB-p Design1. Design has one treatment, treatment A, with j = 1,

. . . , p levels and i = 1, . . . , n blocks.

2. A block contains p dependent participants or a

participant who is observed p times.

3. The p participants in each block are randomly

assigned to the treatment levels. Alternatively,

the order in which the levels are presented to

a participant is randomized for each block.

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B. Comparison of Layouts for a t-Test Design for Dependent Samples and an RB-3 Design

Bock1 a1 a2

Bock2 a1 a2

Bockn a1 a2

Bock1 a1 a2 a3

Bock2 a1 a2 a3

Bockn a1 a2 a3

Treat. level

Treat. level

Treat. level

Treat. level

Treat. level

X.1 X.2

X1.

X2.

Xn.

X.1 X.2 X.2

X1.

X2.

Xn.

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C. Sample Model Equation for a Score in Blocki and Treatment Level j

X ij X. . ( X. j X. .) ( X i. X. .)

Score Grand Treatment Block

Mean Effect Effect

( X ij X i. X. j X. .)

Error Effect

(Residual)

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D. Partition of the Total Sum of Squares (SSTO)

1. The total variability among scores

is a composite that can be decomposed into

treatment A sum of squares (SSA)

block sum of squares (SSBL)

SSTO ( X ij X. .

i1

n

j1

p )2

SSAn ( X. j X..

j1

p )2

SSBLp ( X i. X..

i1

n )2

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error, residual, sum of squares (SSRES)

SSRES ( X ij X i. X. j X. .

i1

n

j1

p )2

E. Degrees of Freedom for SSTO, SSA, SSBL, and SSRES

1. dfTO = np – 1

2. dfA = p – 1

3. dfBL = n – 1

4. dfRES = (n – 1)(p – 1)

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F. Mean Squares (MS)

1. SSTO/(np – 1) = MSTO

2. SSA/(p – 1) = MSA

3. SSBL/(n – 1) = MSBL

4. SSRES/(n – 1)(p – 1) = MSRES

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G. Hypotheses and F Statistics

1. Treatment A

F = MSA/MSRES

2. Blocks

F = MSBL/MSRES

H1 : . j . j for some j and j

H0 : block population means are all equal

H1 : block population means are not all equal

pH 210 :

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Table 1. Computational Procedures for RB-3 Design (Diet Data)

a1 a2 a3

Block1 7 10 12 34

Block2 9 13 11 28

Block3 8 9 15 32

Block10 6 7 14 22 X ij 80 90 120

X ij

000.29014897 ijX

000.302614897 22222 ASX ij

12

667.28883

223

34 22

1

2

1

S

n

Xn

i

p

jij

000.289010

1201090

1080 222

1

2

1

An

Xn

j

p

iij

333.2803103290)( 22 XnpX ij

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H. Sum of Squares Formulas for RB-3 Design

SSTO [AS] [X ]3026.000 2803.333222.667

SSA[A] [X ]2890.000 2803.33386.667

SSBL [S] [X ]2888.667 2803.33385.333

SSRES [AS] [A] [S] [X ]3026.000 2890.000

2888.667 2803.33350.667

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Table 2. ANOVA Table for Weight-Loss Data

Source SS df MS F

1. Treatment 86.667 p – 1 = 2 43.334 15.39**A (three diets)

2. Blocks 85.333 n – 1 = 9 9.481 3.37*(initial wt.)

3. Residual 50.667 (n – 1)(p – 1) = 18 2.815

4. Total 222.667 np – 1 = 29

*p < .02 *p < .0002

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SS TOTAL = 222.667

df = 29

SSBG = 86.667

df = 2

SSWG = 136.000

df = 27

SSA = 86.667

df = 2

SS BLOCKS = 85.333

df = 9

SS RESIDUAL = 50.667

df = 18

CR-3 Design

RB-3 Design

Figure 1. Partition of the total sum of squares and degrees of freedom for a CR-3 design and an RB-3 design.

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I. Assumptions for RB-p Design

1. The model equation reflects all of the sources of

variation that affect Xij.

2. The blocks are a random sample from a population

of blocks, each block population is normally

distributed, and the variances of the block

populations are homogeneous.

3. The population variances of differences for all

pairs of treatment levels are homogeneous.

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4. The population error effects are normally

distributed, the variances are homogeneous, and

the error effects are independent and independent

of other effects in the model equation.

III Multiple Comparisons

A. Fisher-Hayter Test Statistic

qFH X. j X. j

MSRESn

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1. Critical value for the Fisher-Hayter statistic is

B. Scheffé Test Statistic

1. Critical value for the Scheffé statistic is

q; p 1, .

( p 1)F ;1 ,2

.

p

p

pp

nc

nc

ncMSREG

XcXcXcFS

2

2

22

1

21

22211

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C. Scheffé Two-Sided Confidence Interval

i (p 1)F;1,2MSRES

c j2

nj1

p i

i (p 1)F;1,2MSRES

c j2

nj1

p

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IV Practical Significance

A. Partial Omega Squared

1. Treatment A, ignoring blocks

2. Computation for the weight-loss data

X |A.BL2

(p 1)(FA 1)(p 1)(FA 1) np

X |A.BL2

(3 1)(15.394 1)(3 1)(15.394 1) (10)(3)

.49

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B. Hedges’s g Statistic

1. g is used to assess the effect size of contrasts

g X. j X. j

Pooled

Pooled SSBL SSRES

p(n 1)

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2. Computational example for the weight-loss data

g |8 12|2.244

1.78

g |9 12|2.244

1.34

g |8 9|2.244

0.45

Pooled SSBL SSRES

p(n 1)

85.333 50.6673(10 1)

2.244

g X. j X. j

Pooled

211ˆ XX

312ˆ XX

323ˆ XX