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Between Groups & Within-Groups ANOVA

Between Groups & Within-Groups ANOVA. BG & WG ANOVA Partitioning Variation “making” F “making” effect sizes Things that “influence” F Confounding Inflated within-condition variability Integrating “stats” & “methods”. ANOVA  AN alysis O f VA riance Variance means “variation”

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Between Groups & Within-Groups ANOVA

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  1. Between Groups & Within-Groups ANOVA • BG & WG ANOVA • Partitioning Variation • “making” F • “making” effect sizes • Things that “influence” F • Confounding • Inflated within-condition variability • Integrating “stats” & “methods”

  2. ANOVA ANalysis Of VAriance • Variance means “variation” • Sum of Squares (SS) is the most common variation index • SS stands for, “Sum of squared deviations between each of a set of values and the mean of those values” • SS = ∑ (value – mean)2 • So, Analysis Of Variance translates to “partitioning of SS” • In order to understand something about “how ANOVA works” we need to understand how BG and WG ANOVAs partition the SS differently and how F is constructed by each.

  3. Variance partitioning for a BG design Called “error” because we can’t account for why the folks in a condition -- who were all treated the same – have different scores. Tx C 20 30 10 30 10 20 20 20 Mean 15 25 Variation among all the participants – represents variation due to “treatment effects” and “individual differences” Variation among participants within each condition – represents “individual differences” Variation between the conditions – represents variation due to “treatment effects” SSTotal= SSEffect + SSError

  4. How a BG F is constructed Mean Square is the SS converted to a “mean”  dividing it by “the number of things” SSTotal= SSEffect + SSError dfeffect = k - 1 represents # conditions in design MSeffect SSeffect / dfeffect F = = SSerror / dferror MSerror dferror = ∑n - k represents # participants in study

  5. How a BG r is constructed r2 = effect / (effect + error)  conceptual formula = SSeffect / ( SSeffect + SSerror)  definitional formula = F / (F + dferror)  computational forumla MSeffect SSeffect / dfeffect F = = SSerror / dferror MSerror

  6. An Example … SStotal = SSeffect + SSerror 1757.574 = 605.574 + 1152.000 r2 = SSeffect / ( SSeffect + SSerror ) = 605.574 / ( 605.574 + 1152.000 ) = .34 r2 = F / (F + dferror) = 9.462 / ( 9.462 + 18) = .34

  7. Variance partitioning for a WG design Called “error” because we can’t account for why folks who were in the same two conditions -- who were all treated the same two ways – have different difference scores. SumDif Tx C 20 30 10 30 10 20 20 20 50 40 30 40 10 20 10 0 Mean 15 25 Variation among participant’s difference scores – represents “individual differences” Variation among participants – estimable because “S” is a composite score (sum) SSTotal= SSEffect + SSSubj + SSError

  8. How a WG F is constructed Mean Square is the SS converted to a “mean”  dividing it by “the number of things” SSTotal= SSEffect + SSSubj + SSError dfeffect = k - 1 represents # conditions in design MSeffect SSeffect / dfeffect F = = SSerror / dferror MSerror dferror = (k-1)*(n-1) represents # data points in study

  9. How a WG r is constructed r2 = effect / (effect + error)  conceptual formula = SSeffect / ( SSeffect + SSerror)  definitional formula = F / (F + dferror)  computational forumla MSeffect SSeffect / dfeffect F = = SSerror / dferror MSerror

  10. An Example … Don’t ever do this with real data !!!!!! SStotal = SSeffect + SSsubj + SSerror 1757.574 = 605.574 + 281.676 + 870.325 Professional statistician on a closed course. Do not try at home! r2 = SSeffect / ( SSeffect + SSerror ) = 605.574 / ( 605.574 + 281.676 ) = .68 r2 = F / (F + dferror) = 19.349 / ( 19.349 + 9) = .68

  11. What happened?????Same data. Same means & Std. Same total variance. Different F ??? BG ANOVA SSTotal= SSEffect + SSError WG ANOVA SSTotal= SSEffect + SSSubj + SSError The variation that is called “error” for the BG ANOVA is divided between “subject” and “error” variation in the WG ANOVA. Thus, the WG F is based on a smaller error term than the BG F  and so, the WG F is generally larger than the BG F.

  12. What happened?????Same data. Same means & Std. Same total variance. Different r ??? r2 = effect / (effect + error)  conceptual formula = SSeffect / ( SSeffect + SSerror)  definitional formula = F / (F + dferror)  computational forumla The variation that is called “error” for the BG ANOVA is divided between “subject” and “error” variation in the WG ANOVA. Thus, the WG r is based on a smaller error term than the BG r  and so, the WG r is generally larger than the BG r.

  13. Both of these models assume there are no confounds, and that the individual differences are the only source of within-condition variability SSeffect / dfeffect BGSSTotal= SSEffect + SSError F = WG SSTotal= SSEffect+SSSubj+SSError SSerror / dferror A “more realistic” model of F IndDif  individual differences BGSSTotal= SSEffect + SSconfound + SSIndDif + SSwcvar WG SSTotal= SSEffect + SSconfound +SSSubj + SSIndDif + SSwcvar SSconfound between conditionvariability caused by anything(s) other than the IV (confounds) SSwcvar  inflated within condition variability caused by anything(s) other than “natural population individual differences”

  14. Imagine an educational study that compares the effects of two types of math instruction (IV)uponperformance (% - DV) Participants were randomly assigned to conditons, treated, then allowed to practice (Prac) as many problems as they wanted to before taking the DV-producing test Control GrpExper. Grp PracDV PracDV S15 75 S210 82 S35 74 S4 1084 S51078 S615 88 S71079S8 1589 • IV • compare Ss 5&2 - 7&4 • Confounding due to Prac • mean prac dif between cond • WG variability inflated by Prac • wg corrrelation or prac & DV • Individual differences • compare Ss 1&3, 5&7, 2&4, or 6&8

  15. The problem is that the F-formula will … • Ignore the confounding caused by differential practice between the groups and attribute all BG variation to the type of instruction (IV)  overestimating the effect • Ignore the inflation in within-condition variation caused by differential practice within the groups and attribute all WG variation to individual differences  overestimating the error • As a result, the F & r values won’t properly reflect the relationship between type of math instruction and performance  we will make a statistical conclusion error ! • Our inability to procedurally control variables like this will lead us to statistical models that can “statistically control” them SSeffect / dfeffect F = r = F / (F + dferror) SSerror / dferror

  16. How research design impacts F  integrating stats & methods! SSeffect / dfeffect SSTotal =SSEffect+SSconfound+SSIndDif+SSwcvar F = SSerror / dferror SSEffect “bigger” manipulations produce larger mean difference between the conditions  larger F • SSconfound between group differences – other than the IV -- change mean difference  changing F • if the confound “augments” the IV  F will be inflated • if the confound “counters” the IV  F will be underestimated SSIndDif  more heterogeneous populations have larger within- condition differences  smaller F • SSwcvar within-group differences – other than natural individual differences  smaller F • could be “procedural”  differential treatment within-conditions • could be “sampling”  obtain a sample that is “more heterogeneous than the target population”

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