Module 5 - Two-Way Anova

 

MODULE 5 - TWO-WAY ANALYSIS OF VARIANCE (2-WAY ANOVA)

 A two-way (or two factor) factor) analysis of variance (ANOVA) is a procedure procedure that designates designates a single dependent variable (always categorical) to gain an understanding of how the independent variables influence the dependent variable. A thorough grasp of one-way ANOVA is necessary before two-way ANOVA can be understood. Importantly, this also paves the way for three-way  ANOVA and and the influence influence of of covariates.  Although the two-way two-way ANOVA can be regarded regarded as an efficient efficient design design insofar as it allows allows two different independent variables to be incorporated into the study, its ability to identify interactions may be more important. An interaction is simply a situation in which the combined effect of two variables is greater than the sum of the effects of each of the two variables acting separately (Goodwin, 2008). When to use two-way ANOVA Two-way ANOVA, like other forms of ANOVA, are ideally suited to randomized experimental studies. ANOVA is very closely related to multiple regression which may be preferred for survey-type data. Data requirements for two-way ANOVA The independent (grouping) variables should consist of relatively small number of categories otherwise the data analysis will be exceedingly cumbersome.  As usual, itit is better if the scores scores approximate approximate an equal equal interval scale scale of measuremen measurement. t. There does not have to be the same number of participants in each condition of the study though it is generally better if there are when the requirements of ANOVA are violated, such as when the variances of scores in each condition are different (Howitt & Cramer, 2008).

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2-Way ANOVA  2 x 2 Independent Groups  A 2-way ANOVA ANOVA is used used for a factorial factorial design design with two two independent independent variables. variables. If both variables variables are testing within subjects, an analysis similar to the 1-way ANOVA for repeated measures is completed. Finally, if the design is a mixed one, involving within-subjects and between-subjects factors, the analysis combines elements of independent groups ANOVA and repeatedmeasures ANOVA. Statistical Power Power is simply the probability of finding a significant difference in a sample, given a difference (between groups) of a particular size, and a specific sample size. Often power is calculated before conducting a study. If you know the expected magnitude of a difference between  

Dr. Felicidad T. Villavicencio

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groups, you can (for example) calculate how large a sample you need to be 80% sure of finding a significant difference between groups. Statistics—Power Analysis—Power calculation (or sample size calculation)—Several means,  ANOVA, 2Way--Ok 2Way--Ok Steps: 1. In a 2-way ANOVA, three separate F  ratios  ratios are calculated, one for each of the two main effects and one for the interaction. 2. Determine if the calculated F values are significant. 3. Perform subsequent analyses. As described previously in 1-way ANOVA, subsequent testing in factorial ANOVA can take two forms. If there is a subsequent main effect for a factor with more than two levels, then pairwise comparisons of the overall means would be done; perhaps using Tukey’s HSD test again. If a significant significa nt interaction occurred, occurred, simple effects testing could be done. Problem In an experiment, high and low self-esteem subjects did a concept formation task either alone or with an audience. Use the data below to determine the effects of level o off self-esteem and the absence/presence of an audience on the number of errors committed in the concept formation task. Data Entry:

 

Self-Esteem High High High High High High High

Audience Without Without Without Without Without Without With

No. of Errors 3 6 2 2 4 7 9

High High High High High Low Low Low Low Low Low Low Low Low Low Low Low

With With With With With Without Without Without Without Without Without With With With With With With

4 5 8 4 6 7 7 2 6 8 6 10 14 11 15 11 11

Dr. Felicidad T. Villavicencio

PSYSTA2

Module 4

2012

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STATISTICA Output Statistics/Advanced Linear/General Linear Models/Factorial ANOVA/Ok/Variables/Ok/All effects (graph)

…………………………………/All effects/graphs effects /graphs

(means)

…………………………/click spreadsheet (unclick graph) 

Univariate Tests of Significance for No. of Errors (Data two-way AN Sigma-restricted parameterization Effective hypothesis decomposition SS Effect 1 1 7 6 .0 0 Intercept 9 6 .0 0 Self-Esteem  Audi  Au di en ce 9 6 .0 0 Self-Esteem*Audience 2 4 .0 0 Error  8 6 .0 0

De g r. o f   MS F Freedom 1 1 1 7 6 .0 0 2 7 3 .4 8 8 1 9 6 .0 0 2 2 .3 2 5 1 9 6 .0 0 2 2 .3 2 5 1 2 4 .0 0 5 .5 8 1 20 4 .3 0 0

p 0 .0 0 0 0 0 0 .0 0 0 1 3 0 .0 0 0 1 3 0 .0 2 8 3 9  

 Aud i en ce; ce ; L S M ea ns (Sprea (Sp readshee dshee t1 t1)) Current effect: F(1, 20)=22.326, p=.00013 Effective hypothes hypothesis is decompo sitio n  Audi  Au di en ce Errors Errors Errors Errors N Cell No. Mean Std.Err. -95.00% +95.00% 1 Wi th o u 5 .0 0 0 0 0 0 .5 9 8 6 0 3 .7 5 1 3 2 6 .2 4 8 6 1 2 2 Wi th th 9 .0 0 0 0 0 0 .5 9 8 6 0 7 .7 5 1 3 2 1 0 .2 4 8 6 1 2

   Audience; LS Means Means Current effect: F(1, 20)=22.326, p=.00013 Effecti Effectiv ve e hypothesis decomposition Vertical Vertical bars denote 0.95 confidence intervals

Self-Esteem; LS Means Current effect: F(1, 20)=22.326, p=.00013 Effective hypothesis decomposition

11

Vertical bars denote 0. 95 confidence intervals 11

10 10

9 9

8 8 s r o

7 E  r   r    o r    s 

r r E . o N

5

6 5

4

4

3 W ith o u t

3

With  Audience

 

7 f o

6

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High

 

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Low Self-Esteem

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2012

 

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Self-Esteem*Audience; LS Means (Data two-way ANOVA) Current effect: F(1, 20)=5.5814, p=.02840 Effective hypo thesis thesis decompo sitio n Se l ff-Este e m A u d i e en n ce Cell No. 1 2 3 4

Hi g h Hi g h Lo Lo

Wi th o u Wi th Wi th o u Wi th

No . o f   Errors 4 .0 0 0 0 6 .0 0 0 0 6 .0 0 0 0 1 2 .0 0 0 0

No. of  Errors 0 .8 4 6 5 6 0 .8 4 6 5 6 0 .8 4 6 5 6 0 .8 4 6 5 6

No. of  Errors 2 .2 3 4 1 4 .2 3 4 1 4 .2 3 4 1 1 0 .2 3 4 1

No. of  N Errors 5 .7 6 5 9 6 7 .7 6 5 9 6 7 .7 6 5 9 6 1 3 .7 6 5 9 6  

Self-Esteem*Audience; LS Means Current effect: F(1, 20)=5.5814, p=.02840 Effective hypothesis decomposition Vertical bars denote 0.95 confidenc e intervals intervals 16 14 12 10 8 E  r   r    o r    s 

6 4 2  Audience  Without  Audience  With

0 Hi gh

Low Self-Esteem

 

Dr. Felicidad T. Villavicencio

PSYSTA2

 

Module 4

2012

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Self-Esteem*Audienc e; LS M Means eans Current effect: F(1, 20)=5.5814, p=.02840 Effective hypothesis decomposition Vertical bars denote 0.95 confidence intervals 16 14 12 10 8 E  r   r    o r    s 

6 4 2  Self-Esteem  High  Self-Esteem  Low

0 W i thout

W i th  Audi ence

 

Bon ferr ferroni oni tes test; t; variab le No. of Errors Errors (D (Data ata twotwo-way way ANO Homogenou s Groups, Groups, alpha = .05000 Error: Err or: Betw Between een M S = 4.3000 , df = 20 .000 Se l ff-Est e e m Au d i e en n ce Cell No. 1 3 2 4

Hi g h Lo Hi g h Lo

Wi th o u Wi th o u Wi th Wi th

No . o f   1 2 Errors 4 .0 0 0 0 **** 6 .0 0 0 0 **** 6 .0 0 0 0 **** 1 2 .0 0 0 0 ****

 

Tukey HSD test; variable No. of Errors (Data two-way ANOVA)  App roxima roxi ma te P roba bi li ti ties es for Post Hoc Tests T ests Error: Err or: Betw Between een M S = 4.300 0, df = 20 .000 S e l ff-Este e m Au d i e en n ce Cell No. 1 2 3 4

 

Hi g h Hi g h Lo Lo

Wi th o u Wi th Wi th o u Wi th

Dr. Felicidad T. Villavicencio

{1 } 4.0000

{2} {3} {4} 6.0000 6.0000 12.000 0 .3 6 4 4 8 0 .3 6 4 4 8 0.00018 0 0..3 6 4 4 8 1 .0 0 0 0 0 0.00050 0 .3 6 4 4 8 1 .0 0 0 0 0 0.00050 0 ..0 0 00 00 18 18 0 .0 .0 00 00 50 50 0 .0 .0 00 00 50 50  

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LSD test; variabl e No. of E rrors rrors (Data (Data two-w two-way ay ANOVA) Probabilities for Post Hoc Tests Error: Err or: Betw Between een M S = 4.300 0, df = 20 .000 S e l ff-Este e m Au d i e en n ce Cell No. 1 2 3 4

Hi g h Hi g h Lo Lo

Wi th o u Wi th Wi th o u Wi th

{1 } 4.0000

{2} {3} {4} 6.0000 6.0000 12.000 0 .1 1 0 3 8 0 .1 1 0 3 8 0.00000 0 0..1 1 0 3 8 1 .0 0 0 0 0 0.00006 0 .1 1 0 3 8 1 .0 0 0 0 0 0.00006 0 ..0 0 00 00 00 00 0 .0 .0 00 00 06 06 0 .0 .0 00 00 06 06  

Scheffe tes test; t; variab le No. of Errors Errors (Data (Data two-w two-way ay ANOVA) Probabilities for Post Hoc Tests Error: Err or: Betw Between een M S = 4.300 0, df = 20 .000 S e l ff-Este e m Au d i e en n ce Cell No. 1 2 3 4

Hi g h Hi g h Lo Lo

Wi th o u Wi th Wi th o u Wi th

{1 } 4.0000

{2} {3} {4} 6.0000 6.0000 12.000 0 .4 4 4 3 7 0 .4 4 4 3 7 0.00002 0 0..4 4 4 3 7 1 .0 0 0 0 0 0.00083 0 .4 4 4 3 7 1 .0 0 0 0 0 0.00083 0 ..0 0 00 00 02 02 0 .0 .0 00 00 83 83 0 .0 .0 00 00 83 83  

Questions regarding results of analyses:

1 2 3 4 5

6 7 8

9 10 11

 

What is the independent variable? What is the dependent variable? What are the null and alternative hypotheses for testing the main effect of self-esteem? What are the F, df’s, MS, MS , and p-values for testing the above null hypothesis? Is there a significant main effect of self-esteem self-est eem on number of errors? Explain the effect of self-esteem on the number of errors in terms that a person who has not had a course in statistics could understand. What are the null and alternative hypotheses for testing the main effect of audience? What are the F, df’s, MS, MS , and p-values for testing the above null hypothesis? Is there a significant main effect of audience on number of errors? Explain the effect of audience on the number of errors in terms that a person who has not had a course in statistics could understand. What are the null and alternative hypotheses for testing the Self-Esteem X Audience interaction effect? What are the F, df’s, MS, MS , and p-values for testing the above null hypothesis? Is there a significant interaction effect? Explain the presence or absence of an interaction effect in terms that a person who has not had a course in statistics could understand. Dr. Felicidad T. Villavicencio

PSYSTA2

Module 4

2012

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