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.

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.

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

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.

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

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

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)

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)

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

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

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

H. Sum of Squares Formulas for RB-3 Design

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

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.

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.

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

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

C. Scheffé Two-Sided Confidence Interval

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

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

2. Computational example for the weight-loss data