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**Factorial Designs & Managing Violated Statistical Assumptions**

Developing Study Skills and Research Methods (HL20107) Dr James Betts

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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 week’s 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**

Levels Main Effect Interaction Effect

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**Factorial Designs: Multiple IV’s**

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

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**Main Effect of Exercise Exercise*Gender Interaction**

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

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**Main Effect of Exercise Exercise*Gender Interaction**

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

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**Main Effect of Exercise Exercise*Gender Interaction**

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

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**Main Effect of Exercise Exercise*Gender Interaction**

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

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**Factorial Designs: Interpretation**

210 180 150 ? Heart Rate (beatsmin-1) 120 90 60 30 Resting Exercise

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**Factorial Designs: Interpretation**

210 180 150 Heart Rate (beatsmin-1) 120 ? 90 60 30 Resting Exercise

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**2-way mixed model ANOVA: Partitioning**

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

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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. XT) -square the total score for each subject (e.g. (XT)2) -Total both columns for each group -Total all raw scores and squared scores (e.g. X & X2).

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**Procedure for computing 2-way mixed model ANOVA**

Step 2: Calculate the Grand Total correction factor GT = (X)2 N

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**Procedure for computing 2-way mixed model ANOVA**

Step 3: Compute total Sum of Squares SStotal= X2 - GT

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**Procedure for computing 2-way mixed model ANOVA**

Step 4: Compute Exercise Effect Sum of Squares SSex= GT = GT (Xex)2 nex (XRmale+XRfemale)2 (XEmale+XEfemale)2 nRmale+fem nEmale+fem

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**Procedure for computing 2-way mixed model ANOVA**

Step 5: Compute Gender Effect Sum of Squares SSgen= GT = GT (Xgen)2 ngen (XRmale+XEmale)2 (XRfem+XEfem)2 nmaleR+E nfemR+E

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**Procedure for computing 2-way mixed model ANOVA**

Step 6: Compute Interaction Effect Sum of Squares SSint= GT = (SSex+SSgen) - GT (Xex+gen)2 nex+gen (XRmale)2 (XRfem)2 (XEmale)2 (XEfem)2 nRmale nRfem nEmale nEfem

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**Procedure for computing 2-way mixed model ANOVA**

Step 7: Compute between subjects Sum of Squares SSbet= SSgen- GT = SSgen- GT (XS)2 nk (XT)2+(XD)2+(XH)2+(XJ)2+(XK)2+(XA)2+(XS)2+(XL)2 nk

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**Procedure for computing 2-way mixed model ANOVA**

Step 8: Compute within subjects Sum of Squares SSwit= SStotal - (SSex+SSgen+SSint+SSbet)

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**Procedure for computing 2-way mixed model ANOVA**

Step 9: Determine the d.f. for each sum of squares dftotal = (N - 1) dfex = (k - 1) dfgen = (r - 1) dfint = (k - 1)(r - 1) dfbet = r(n - 1) dfwit = r(n - 1)(k - 1)

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**Procedure for computing 2-way mixed model ANOVA**

Systematic Variance (resting vs exercise) Step 10: Estimate the Variances SSex = dfex Systematic Variance (male vs female) SSgen = dfgen Systematic Variance (Interaction) SSint = dfint Error Variance (within subjects) SSwit = dfwit Error Variance (between subjects) SSbet = dfbet

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**Procedure for computing 2-way mixed model ANOVA**

Systematic Variance (resting vs exercise) Step 11: Compute F values SSex = dfex Systematic Variance (male vs female) SSgen = dfgen Systematic Variance (Interaction) SSint = dfint Error Variance (within subjects) SSwit = dfwit Error Variance (between subjects) SSbet = dfbet

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**Procedure for computing 2-way mixed model ANOVA**

Step 12: Consult F distribution table as before Exercise Gender Interaction

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**2-way mixed model ANOVA: SPSS Output**

Systematic Varianceex SSex Calculated Fex dfex SSwit Error Variancewit dfwit

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**2-way mixed model ANOVA: SPSS Output**

GT Calculated Fgen SSgen Systematic Variancegen dfgen dfbet SSbet Error Variancebet

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

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**2-way Independent Measures ANOVA**

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

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**2-way Independent Measures ANOVA**

Systematic Variance (resting vs exercise) Systematic Variance (male vs female) Systematic Variance (Interaction) Error Variance (within subjects) Error Variance (between subjects)

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**2-way Repeated Measures ANOVA**

Systematic Variance (resting vs exercise) …but for a fully paired design e.g. morning vs evening & rest vs exercise Systematic Variance (am vs pm) Systematic Variance (Interaction) Error Variance (within subjectsexercise) Error Variance (within subjectstime) Error Variance (within subjectsinteract)

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**2-way Repeated Measures ANOVA**

Systematic Variance (resting vs exercise) Systematic Variance (am vs pm) Systematic Variance (Interaction) Error Variance (within subjectsexercise) Error Variance (within subjectstime) Error Variance (within subjectsinteract)

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Summary: 2-way ANOVA 2-way (factorial) ANOVA may be appropriate whenever there are multiple IV’s 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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**e.g. ND assumption from last year**

Sustained Isometric Torque (seconds)

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**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 A Placebo Supplement 1 Group B Supplement 2 Group C

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Plac. Supp. 1 Supp. 2 Plac.-Supp. 1 Supp. 1-Supp. 2 Plac.-Supp. 2 Tom 2.4 3.0 3.3 -0.6 -0.3 -0.9 Dick 2.2 2.5 0.1 -0.2 Harry 1.8 1.9 -0.1 -0.4 James 1.6 1.1 1.2 0.5 0.4 Mean 2.0 2.1 2.3 SD2 0.7 0.2 0.04 0.3

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**Many texts recommend ‘Mauchley’s Test of Sphericity’**

A χ2 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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**Managing Violations to Sphericity**

How should we analyse aspherical data? Option 1 Option 2 Option 3

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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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