Chapter 16 Other Analysis of Variance Designs ISome Basic Experimental Design Concepts
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Transcript of Chapter 16 Other Analysis of Variance Designs ISome Basic Experimental Design Concepts
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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
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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