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Factorial Designs & Managing Violated Statistical Assumptions Dr James Betts Developing Study Skills and Research Methods (HL20107)

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Lecture Outline: Factorial Research Designs Revisited Mixed Model 2-way ANOVA Fully independent/repeated measures 2-way ANOVA Statistical Assumptions of ANOVA.

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Last Week Recap In last weeks lecture we saw two worked examples of 1-way analyses of variance However, many experimental designs have more than one independent variable (i.e. factorial design)

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Factorial Designs: Technical Terms Factor Levels Main Effect Interaction Effect

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Factorial Designs: Multiple IVs Hypothesis: –The HR response to exercise is mediated by gender We now have three questions to answer: 1) 2) 3)

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Factorial Designs: Interpretation 210 180 150 120 90 60 30 0 Heart Rate (beats min -1 ) Resting Exercise Main Effect of Exercise Not significant Main Effect of Gender Not significant Exercise*Gender Interaction Not significant

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Factorial Designs: Interpretation 210 180 150 120 90 60 30 0 Heart Rate (beats min -1 ) Resting Exercise Main Effect of Exercise Significant Main Effect of Gender Not significant Exercise*Gender Interaction Not significant

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Factorial Designs: Interpretation 210 180 150 120 90 60 30 0 Heart Rate (beats min -1 ) Resting Exercise Main Effect of Exercise Not significant Main Effect of Gender Significant Exercise*Gender Interaction Not significant

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Factorial Designs: Interpretation 210 180 150 120 90 60 30 0 Heart Rate (beats min -1 ) Resting Exercise Main Effect of Exercise Not significant Main Effect of Gender Not Significant Exercise*Gender Interaction Significant

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Factorial Designs: Interpretation 210 180 150 120 90 60 30 0 Heart Rate (beats min -1 ) Resting Exercise ?

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Factorial Designs: Interpretation 210 180 150 120 90 60 30 0 Heart Rate (beats min -1 ) Resting Exercise ?

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Systematic Variance (resting vs exercise) Error Variance (between subjects) Systematic Variance (male vs female) Systematic Variance (Interaction) 2-way mixed model ANOVA: Partitioning = variance between means due to exercise = variance between means due to gender = variance between means due IV interaction = uncontrolled factors and within group differences for males vs females. Error Variance (within subjects) = uncontrolled factors plus random changes within individuals for rest vs exercise

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Procedure for computing 2-way mixed model ANOVA Step 1: Complete the table i.e. -square each raw score -total the raw scores for each subject (e.g. X T ) -square the total score for each subject (e.g. ( X T ) 2 ) -Total both columns for each group -Total all raw scores and squared scores (e.g. X & X 2 ).

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Step 2: Calculate the Grand Total correction factor GT = ( X) 2 N Procedure for computing 2-way mixed model ANOVA

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Step 3: Compute total Sum of Squares SS total = X 2 - GT Procedure for computing 2-way mixed model ANOVA

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Step 4: Compute Exercise Effect Sum of Squares SS ex = - GT = + - GT ( X ex ) 2 n ex ( X Rmale + X Rfemale ) 2 n Rmale+fem Procedure for computing 2-way mixed model ANOVA ( X Emale + X Efemale ) 2 n Emale+fem

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Step 5: Compute Gender Effect Sum of Squares SS gen = - GT = + - GT ( X gen ) 2 n gen ( X Rmale + X Emale ) 2 n maleR+E Procedure for computing 2-way mixed model ANOVA ( X Rfem + X Efem ) 2 n femR+E

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Step 6: Compute Interaction Effect Sum of Squares SS int = - GT = + + + - (SS ex +SS gen ) - GT ( X ex+gen ) 2 n ex+gen n Rmale Procedure for computing 2-way mixed model ANOVA n Rfem n Emale n Efem ( X Rmale ) 2 ( X Rfem ) 2 ( X Emale ) 2 ( X Efem ) 2

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Step 7: Compute between subjects Sum of Squares SS bet = -SS gen - GT = -SS gen - GT ( X S ) 2 nk Procedure for computing 2-way mixed model ANOVA ( X T ) 2 +( X D ) 2 +( X H ) 2 +( X J ) 2 +( X K ) 2 +( X A ) 2 +( X S ) 2 +( X L ) 2 nk

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Step 8: Compute within subjects Sum of Squares SS wit = SS total - (SS ex +SS gen +SS int +SS bet ) Procedure for computing 2-way mixed model ANOVA

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Step 9: Determine the d.f. for each sum of squares df total = (N - 1) df ex = (k - 1) df gen = (r - 1) df int = (k - 1)(r - 1) df bet = r(n - 1) df wit = r(n - 1)(k - 1) Procedure for computing 2-way mixed model ANOVA

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Step 10: Estimate the Variances Procedure for computing 2-way mixed model ANOVA Systematic Variance (resting vs exercise) Error Variance (between subjects) Systematic Variance (male vs female) Systematic Variance (Interaction) Error Variance (within subjects) SS ex df ex SS gen df gen SS int df int SS wit df wit SS bet df bet = = = = =

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Step 11: Compute F values Procedure for computing 2-way mixed model ANOVA Systematic Variance (resting vs exercise) Error Variance (between subjects) Systematic Variance (male vs female) Systematic Variance (Interaction) Error Variance (within subjects) SS ex df ex SS gen df gen SS int df int SS wit df wit SS bet df bet = = = = =

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Step 12: Consult F distribution table as before Exercise Gender Procedure for computing 2-way mixed model ANOVA Interaction

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2-way mixed model ANOVA: SPSS Output Calculated F ex SS ex df ex Systematic Variance ex SS wit df wit Error Variance wit

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2-way mixed model ANOVA: SPSS Output Calculated F gen SS gen SS bet df gen df bet Error Variance bet GT Systematic Variance gen

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Systematic Variance (resting vs exercise) Error Variance (between subjects) Systematic Variance (male vs female) Systematic Variance (Interaction) Error Variance (within subjects) 2-way mixed model ANOVA The previous calculation and associated partitioning is an example of a 2-way mixed model ANOVA –i.e. exercise = repeated measures gender= independent measures

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Systematic Variance (resting vs exercise) Systematic Variance (male vs female) Systematic Variance (Interaction) Error Variance (within subjects) 2-way Independent Measures ANOVA So for a fully unpaired design –e.g. males vs females & rest group vs exercise group Error Variance (between subjects)

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Systematic Variance (resting vs exercise) Systematic Variance (male vs female) Systematic Variance (Interaction) Error Variance (within subjects) 2-way Independent Measures ANOVA Error Variance (between subjects)

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Systematic Variance (resting vs exercise) Systematic Variance (am vs pm) Systematic Variance (Interaction) Error Variance (within subjects exercise ) 2-way Repeated Measures ANOVA …but for a fully paired design –e.g. morning vs evening & rest vs exercise Error Variance (within subjects time ) Error Variance (within subjects interact )

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Systematic Variance (resting vs exercise) Systematic Variance (am vs pm) Systematic Variance (Interaction) Error Variance (within subjects exercise ) 2-way Repeated Measures ANOVA Error Variance (within subjects time ) Error Variance (within subjects interact )

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Summary: 2-way ANOVA 2-way (factorial) ANOVA may be appropriate whenever there are multiple IVs to compare We have worked through a mixed model but you should familiarise yourself with paired/unpaired procedures You should also ensure you are aware what these effects actually look like graphically.

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Statistical Assumptions As with other parametric tests, ANOVA is associated with a number of statistical assumptions When these assumptions are violated we often find that an inferential test performs poorly We therefore need to determine not only whether an assumption has been violated but also whether that violation is sufficient to produce statistical errors.

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16 17 18 19 20 Sustained Isometric Torque (seconds) e.g. ND assumption from last year

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Assumptions of ANOVA N acquired through random sampling Data must be of at least the interval LOM (continuous) Independence of observations Homogeneity of variance All data is normally distributed

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ANOVA is generally robust to violations of the normality assumption, in that even when the data are non-normal, the actual Type I error rate is usually close to the nominal (i.e., desired) value. Maxwell & Delaney (1990) Designing Experiments & Analyzing Data: A Model Comparison Perspective, p. 109 If the data analysis produces a statistically significant finding when no test of sphericity is conducted…you should disregard the inferential claims made by the researcher. Huck & Cormier (1996) Reading Statistics & Research, p. 432

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Group B Group C Group A Supplement 1 Supplement 2 Placebo

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Plac.Supp. 1Supp. 2Plac.-Supp. 1Supp. 1-Supp. 2Plac.-Supp. 2 Tom 2.43.03.3-0.6-0.3-0.9 Dick 2.22.52.4-0.30.1-0.2 Harry 1.81.92.2-0.1-0.3-0.4 James 1.61.11.20.5-0.10.4 Mean 2.02.12.3-0.1-0.2-0.3 SD 2 0.10.7 0.20.040.3

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Many texts recommend Mauchleys Test of Sphericity –A χ 2 test which, if significant, indicates a violation to sphericity However, this is not advisable on four counts: 1.) 2.) 3.) 4.)

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How should we analyse aspherical data? Option 1 Option 2 Option 3 Managing Violations to Sphericity

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Paired 1-way MANOVA: SPSS Output

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Paired 1-way ANOVA: SPSS Output

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Summary ANOVA and Sphericity ANOVA is a generally robust inferential test Unpaired data are susceptible to heterogeneity of variance only if group sizes are unequal Paired data are susceptible to asphericity only if multiple comparisons are made Suggested solutions for the latter include either MANOVA or epsilon corrected df depending on sample size relative to number of levels.

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