1. Other Analysis of Variance Designs
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.

2. 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.

3. 1. Any variable that is positively correlated with
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

4. II Randomized Block Design (RB-p Design)
A. Characteristics of the RB-p Design
1. 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.

5. Dependent Samples and an RB-3 Design
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

6. C. Sample Model Equation for a Score in Block
i and Treatment Level j

7. 1. The total variability among scores
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)

8. and SSRES  error, residual, sum of squares (SSRES)
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)

9. 1. SSTO/(np – 1) = MSTO F. Mean Squares (MS) 2. SSA/(p – 1) = MSA
3. SSBL/(n – 1) = MSBL
4. SSRES/(n – 1)(p – 1) = MSRES

10. G. Hypotheses and F Statistics
1. Treatment A
 F = MSA/MSRES
2. Blocks
 F = MSBL/MSRES

11. Table 1. Computational Procedures for RB-3 Design (Diet Data)
a1 a2 a3
Block
Block
Block
Block

13. H. Sum of Squares Formulas for RB-3 Design

14. Table 2. ANOVA Table for Weight-Loss Data
Source SS df MS F
1. Treatment p – 1 = ** A (three diets)
2. Blocks n – 1 = *
(initial wt.)
3. Residual (n – 1)(p – 1) =
4. Total np – 1 = 29
*p < *p < .0002

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

16. 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.

17. III Multiple Comparisons
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

18. B. Scheffé Test Statistic
1. Critical value for the Fisher-Hayter statistic is
B. Scheffé Test Statistic
1. Critical value for the Scheffé statistic is

19. C. Scheffé Two-Sided Confidence Interval

20. IV Practical Significance
A. Partial Omega Squared
1. Treatment A, ignoring blocks
2. Computation for the weight-loss data

21. B. Hedges’s g Statistic
1. g is used to assess the effect size of contrasts

22. 2. Computational example for the weight-loss data