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Analysis of Variance (ANOVA) Developing Study Skills and Research Methods (HL20107) Dr James Betts

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Lecture Outline: Multiple Comparisons and Type I Errors 1-way ANOVA for Unpaired data 1-way ANOVA for Paired Data Factorial Research Designs.

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Males Females

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Placebo Lucozade

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Placebo Lucozade Gatorade Powerade Placebo Lucozade Gatorade Powerade

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Placebo Lucozade Gatorade Powerade Placebo Lucozade Gatorade Powerade

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What is Analysis of Variance? ANOVA is an inferential test designed for use with 3 or more data sets t-tests are just a form of ANOVA for 2 groups ANOVA only interested in establishing the existence of a statistical differences, not their direction (last slide) Based upon an F value (R. A. Fisher) which reflects the ratio between systematic and random/error variance…

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Total Variance between means Systematic Variance Error Variance Dependent Variable Extraneous/ Confounding (Error) Variables Independent Variable

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Group A Group B Group C

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Group B Group C Group A

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Procedure for computing 1-way ANOVA for independent samples Step 1: Complete the table i.e. -square each raw score -total the raw scores for each group -total the squared scores for each group.

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Procedure for computing 1-way ANOVA for independent samples Step 2: Calculate the Grand Total correction factor GT = = ( X) 2 N ( X A + X B + X C ) 2 N

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Procedure for computing 1-way ANOVA for independent samples Step 3: Compute total Sum of Squares SS total = X 2 - GT = ( X A 2 + X B 2 + X C 2 ) - GT

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Procedure for computing 1-way ANOVA for independent samples Step 4: Compute between groups Sum of Squares SS bet = - GT = GT ( X) 2 n ( X A ) 2 nAnA ( X B ) 2 nBnB ( X C ) 2 nCnC

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Procedure for computing 1-way ANOVA for independent samples Step 5: Compute within groups Sum of Squares SS wit = SS total - SS bet

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Procedure for computing 1-way ANOVA for independent samples Step 6: Determine the d.f. for each sum of squares df total = (N - 1) df bet = (k - 1) df wit = (N - k)

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Systematic Variance (between means) Error Variance (within means) Procedure for computing 1-way ANOVA for independent samples Step 7/8: Estimate the Variances & Compute F = SS bet df bet SS wit df wit

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Procedure for computing 1-way ANOVA for independent samples Step 9: Consult F distribution table -d 1 is your df for the numerator (i.e. systematic variance) -d 2 is your df for the denominator (i.e. error variance)

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Independent 1-way ANOVA: SPSS Output

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Group B Group C Group A Trial 2 Trial 3 Trial 1

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Procedure for computing 1-way ANOVA for paired samples Step 1: Complete the table i.e. -square each raw score -total the raw scores for each trial & subject -total the squared scores for each trial & subject.

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Procedure for computing 1-way ANOVA for paired samples Step 2: Calculate the Grand Total correction factor GT = = = = 54.6 ( X) 2 N ( X 1 + X 2 + X 3 ) 2 N ( ) 2 12 …so GT just as with unpaired data

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Procedure for computing 1-way ANOVA for paired samples Step 3: Compute total Sum of Squares SS total = X 2 - GT = ( X X X 3 2 ) - GT

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Procedure for computing 1-way ANOVA for paired samples Step 4: Compute between trials Sum of Squares SS betT = - GT = GT ( X T ) 2 nTnT ( X 1 ) 2 n1n1 ( X 2 ) 2 n2n2 ( X 3 ) 2 n3n3

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Procedure for computing 1-way ANOVA for paired samples Step 5: Compute between subjects Sum of Squares SS betS = - GT = GT ( X S ) 2 nTnT ( X T ) 2 nTnT ( X D ) 2 nDnD ( X H ) 2 nHnH ( X J ) 2 nJnJ

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Procedure for computing 1-way ANOVA for paired samples Step 6: Compute interaction Sum of Squares SS int = SS total - (SS betT + SS betS )

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Procedure for computing 1-way ANOVA for paired samples Step 7: Determine the d.f. for each sum of squares df total = (N - 1) df betT = (k - 1) df betS = (r - 1) df int = (r-1)(k-1) = df betT x df betS

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Step 8/9: Estimate the Variances & Compute F values = Systematic Variance (between trials IV) Error Variance Procedure for computing 1-way ANOVA for paired samples SS betT df betT SS int df int SS betS df betS Systematic Variance (between subjects)

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Procedure for computing 1-way ANOVA for paired samples Step 10: Consult F distribution table as before

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Paired 1-way ANOVA: SPSS Output

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Next week we will continue to work through some examples of 2-way ANOVA (i.e. factorial designs) However, you will come across 2-way ANOVA in this weeks lab class so there are a few terms & concepts that you should be aware of in advance... Introduction to 2-way ANOVA

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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 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 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 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 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 Heart Rate (beats min -1 ) Resting Exercise Main Effect of Exercise Main Effect of Gender Exercise*Gender Interaction ?

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Factorial Designs: Interpretation Heart Rate (beats min -1 ) Resting Exercise Main Effect of Exercise Main Effect of Gender Exercise*Gender Interaction ?

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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 = 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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Systematic Variance (resting vs exercise) Systematic Variance (male vs female) Systematic Variance (Interaction) Error Variance (within subjects) 2-way mixed model ANOVA So for a fully unpaired design –e.g. males vs females & rest group vs exercise group …between subject variance (i.e. SD) has a negative impact upon all contrasts 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 mixed model ANOVA …but for a fully paired design –e.g. morning vs evening & rest vs exercise …between subject variance (i.e. SD) can be removed from all contrasts. Error Variance (within subjects time ) Error Variance (within subjects interact ) Refer back to this partitioning in your lab class

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