Two-way Repeated Measures ANOVA

This presentation explains the procedure involved in two-way repeated measures ANOVA(within-within design). An illustration has been discussed by using the functionality of SPSS.

1. Two-way Repeated Measures Design Presented by Dr.J.P.Verma MSc ( Statistics), PhD, MA(Psychology), Masters(Computer Application ) Professor(Statistics) Lakshmibai National Institute of Physical Education, Gwalior, India (Deemed University) Email: [email protected]

2. Two-Way Repeated Measures Design Where the effect of two within-subjects factor on a dependent variable needs to be investigated simultaneously Where individual variations of the subjects cannot be controlled Recruiting large sample in the study is difficult within-within design, two-way repeated measures design(RMD) or two-way ANOVA with repeated measures. Also known as When to Use 2

3. Features of Two-way RMD All subjects are tested in each level of both the factors. Mean differences between groups, split on two within-subjects factors are compared. Structure Highlights If Factor A has two levels A1 and A2 and Factor B has three levels B1, B2 and B3 Then there will be six treatment conditions A 1 B 1 , A 1 B 2 , A 1 B 3 A 2 B 1 , A 2 B 2 , A 2 B 3   A randomly drawn sample is then tested in all the six treatment conditions 3

4. What research questions we investigate? Whether the factor A affects the dependent variable? Whether the factor B affects the dependent variable? Investigated through main effects Investigated through simple effects Whether interaction between the factor A and B is significant? 4

5. 5 This Presentation is based on Chapter 5 of the book Repeated Measures Design for Empirical Researchers Published by Wiley , USA Complete Presentation can be accessed on Companion Website of the Book

6. Main and Simple Effect Objective: To compare the effect of teaching methods on learning < 20 years( b 1 ) 21 - 40 years (b 2 ) 18 19 21 35 32 29 24 28 34 18 19 22 Traditional( a 1 ) Audio-visual( a 2 ) Factor A : Teaching Methods Factor B : Age Main Effect of A : Effect of Teaching methods on learning across all levels of factor B (Age) Simple Effect of A Effect of Factor A on learning in each level of factor B >4o years (b 3 ) 20 22 29 20 21 17 Main Effect of B : Effect of Age on learning across all the levels of factor A (Teaching methods) Simple Effect of B Effect of Factor B on learning in each level of factor A Investigated only when Interaction is significant 6

7. Understanding Interaction < 20 ( b 1 ) 21 – 40 (b 2 ) 18 19 21 35 32 29 24 28 34 18 19 22 Traditional( a 1 ) Audio-visual( a 2 ) Factor A: Teaching Methods Factor B : Age >4o (b 3 ) 20 22 29 20 21 17 Interaction Joint effect of Teaching method and Age (A×B) on learning If Interaction (A×B) is significant Association exists between teaching method and age Pattern of learning response differs in each teaching methods b1 b2 b3 a 1 a 2 b1 b2 b3 a 1 a 2 No Interaction There is Interaction 7

8. Characteristics of Two-way RMD If factor levels are large, subjects get tired/bored resulting inaccurate observations Design becomes less efficient if variability among subjects becomes insignificant Advantage Disadvantage  Requires limited number of subjects Study can be completed quickly Increased efficiency in comparison to independent measures ANOVA Can be used for the longitudinal studies 8

9. Testing protocol Factor 1: Caffeine Factor 2: Environmental S1 S2 S5 S6 S3 S4 Evening First phase testing S3 S4 S1 S2 S5 S6 S5 S6 S3 S4 S1 S2 Second phase testing Third phase testing Afternoon Morning S3 S4 S1 S2 S5 S6 S1 S2 S5 S6 S3 S4 S5 S6 S3 S4 S1 S2 Coffee Placebo Subjects First phase testing Second phase testing Third phase testing Case I: Levels of the within-subjects variable are different treatment conditions Example: Investigate the effect of caffeine (coffee and placebo) and time of testing on the mathematical ability on six subjects. Layout Procedure Within-subjects factors 1. Caffeine 2. Time Divide subjects into 3(number of levels) groups Allocate treatments randomly on these groups   Like (1,1,1), 2,1,3 and 3,1,2 as shown in figure (1,1,1): Group will undergo the first treatment condition thereafter second and then the third When to use Two-way RMD Used in Two Types of Situations Figure 5.1 Layout design 9

10. 3 weeks S1 S2 S3 S4 S5 S6 S1 S2 S3 S4 S5 S6 S1 S2 S3 S4 S5 S6 6 weeks 9 weeks Factor 2: Time Initial S1 S2 S3 S4 S5 S6 Coffee Placebo Subjects Testing protocol Factor 1: Caffeine S1 S2 S3 S4 S5 S6 S1 S2 S3 S4 S5 S6 S1 S2 S3 S4 S5 S6 S1 S2 S3 S4 S5 S6 Case II: levels of the within-subjects variable are different time periods When to use Two-way RMD Used in Two Types of Situations Example: To see the effect of caffeine on mathematical ability in four different time duration i.e. before experiment, after 3 weeks, 6 weeks and 9 weeks. Let us have the sample of size six. Figure 5.2 Layout design 10

11. Application of Two-Way RMD To study the effect of caffeine(coffee and placebo) on memory retention over a period of time(0, 1 and 2 weeks) To see the impact of fat consumption(no fat, medium fat and high fat) and time(morning afternoon and evening) of the day on the performance in a comprehension test   A market researcher may like to investigate the effect of time and season on the sale in grocery outlets of a company 11

12. Steps in Two-way RMD Test normality assumption in all treatment conditions Describe design layout Write research questions Write different H to be tested Decide family wise error rates (α) Use SPSS to generate outputs Descriptive statistics Mauchly's test of sphericity F table for within-subjects effect Pair-wise comparison of means for IVs if found significant Different Means plots Marginal Means for each cell and IV Cont ….. 12

13. Steps in Two-way RMD Generate following results using SPSS Descriptive statistics Mauchly's test of sphericity F table for within-subjects effect Pair-wise comparison of means for IVs if found significant F table for within-subjects effect Interaction Significant No Test Main Effect if Significant Do pair-wise comparison of means Yes Test Simple Effect of each IV 13

14. Steps in Two-way RMD Check sphericity assumption while testing main or simple effect p<.05 Test F ratio by assuming sphericity N Y Check <.75 Test F by using Huynh- Feldt correction N Test F by using Greenhouse- Geisser correction Y If F is significant apply t tests for comparison of means using Bonferroni correction. Report findings 14

15. Table 5.1 Number of match box prepared per hour in a day Environment Hot Humid Cold _____________________________________________ 20 16 27 18 17 24 No music 22 16 26 16 19 17 18 20 26 20 22 23 22 21 23 20 25 21 Jazz 24 27 22 19 21 20 22 27 25 20 26 25 24 26 21 26 22 20 Instrumental 25 22 18 26 21 24 24 19 18 25 22 21 _______________________________________________ Music Solving Two-way RMD with SPSS To investigate the effect of environment and music on the performance of six employees in a cottage industry of packaging. Objective Environment : hot, humid and cold Types of music : Instrumental, Classical Jazz and no music 15

16. Testing protocol Factor 1: Music Factor 2: Environment S1 S2 S3 S4 S5 S6 Cold First testing Second testing Third testing Humid Hot No Music Subjects S5 S6 S1 S2 S3 S4 S3 S4 S5 S6 S1 S2 S5 S6 S1 S2 S3 S4 First testing Second testing Third testing Jazz S3 S4 S5 S6 S1 S2 S1 S2 S3 S4 S5 S6 S3 S4 S5 S6 S1 S2 First testing Second testing Third testing Instrumental S1 S2 S3 S4 S5 S6 S5 S6 S1 S2 S3 S4 Two-Way RMD with SPSS Figure 5.3 Layout of the repeated measures design with two factor 16

17. Distribution of SS in Two-way RMD   Total SS = SS Subjects + SS Withing Subjects   = SS Subjects + ( SS Music + SS Error_Music ) + ( SS Envir + SS Error_Envir ) + ( SS Music×Envir + SS Error_Music×Envir ) Hot Humid Cold 20 16 27 18 17 24 No music 22 16 26 16 19 17 18 20 26 20 22 23 22 21 23 20 25 21 Jazz 24 27 22 19 21 20 22 27 25 20 26 25 24 26 21 26 22 20 Instru 25 22 18 26 21 24 24 19 18 25 22 21 Music Environment 17

18. SS Between_Subjects n-1 Total SS df = nrc-1 SS Within_Subjects n(rc-1) 53 5 48 SS Error_Music (r-1)(n-1) 10 SS Music r-1 SS Error_Music×Envir (r-1)(c-1)(n-1) SS Music×Envir (r-1)(c-1) SS Error_Envir (c-1)(n-1) 10 SS Envir c-1 20 4 2 2 Distribution of SS and df in Two-way RMD Figure 5.4 Scheme of distributing total SS and df in two-way repeated measures design 18

19.  Whether back ground music affects the performance of workers. Whether performance of workers is affected by the environment. Whether interaction between background music and type of environment affects the worker’s performance. Research Issues and Hypothesis Construction  against H 1 : At least one group mean differs  Research Questions Hypotheses Construction Main effect of Music  against H 1 : At least one group mean differs  Main effect of Environment Interaction Effect (Music × Environment)   H : There is no interaction between Music and Environment against H 1 : The interaction between Music and Environment is significant 19

20. Level of Significance   Bonferroni correction shall be applied for correcting the level of significance Family wise error rate(α) shall be taken as .05 20

21. Two-way RMD with SPSS NOM_Hot NOM_Humid NOM_Cold Jz_Hot Jz_Humid Jz_Cold Inst_Hot Inst_Humid Inst_Cold How to Prepare Data File in SPSS? In Variable View define the following nine treatment combinations as variables 21

22. Figure 5.5 Data format in the repeated measures design with two factors Figure 5.5 Data format in the repeated measures design with two factors   Analyze General Linear Model Repeated Measures   Data File for Two-way RMD in SPSS While being in Data View click on the following command sequence 22

23. 23 To buy the book Repeated Measures Design for Empirical Researchers and all associated presentations Click Here Complete presentation is available on companion website of the book