Factorial BG ANOVAPsy 420AinsworthTopics in Factorial Designs• Factorial?• Crossing and Nesting• Assumptions• Analysis• Traditional and Regression Approaches• Main Effects of IVs• Interactions among IVs• Higher order designs• “Dangling control group” factorial designs• Specific Comparisons• Main Effects• Simple Effects• Interaction Contrasts• Effect Size estimates• Power and Sample SizeFactorial?• Factorial – means that all levels of one IV are completely crossed with all level of the other IV(s).• Crossed – all levels of one variable occur in combination with all levels of the other variable(s)• Nested – levels of one variable appear at different levels of the other variable(s)Factorial?• Crossing example• Every level of teaching method is found together with every level of book• You would have a different randomly selected and randomly assigned group of subjects in each cell• Technically this means that subjects are nested within cells Teaching Method Lecture Media Lecture/MediaTabachnick and Fidell L & TF M & TF LM & TF Text book Keppel and Wickens L & KW M & KW LM & KWFactorial?• Crossing Example 2 – repeated measures• In repeated measures designs subjects cross the levels of the IV Pre - test Mid - test Post - tests1 s1 s1 s2 s2 s2 s3 s3 s3 s4 s4 s4 Subjects s5 s5 s5Factorial?• Nesting Example• This example shows testing of classes that are pre-existing; no random selection or assignment • In this case classes are nested within each cell which means that the interaction is confounded with class Teaching Method Lecture Media Lecture/Media T and F L & TF/ Class 1 M & TF/ Class 3 LM & TF/ Class 5Text book K and W L & KW/ Class 2 M & KW/ Class 4 LM & KW/ Class 6Assumptions• Normality of Sampling distribution of means• Applies to the individual cells• 20+ DFs for error and assumption met• Homogeneity of Variance• Same assumption as one-way; applies to cells• In order to use ANOVA you need to assume that all cells are from the same populationAssumptions• Independence of errors• Thinking in terms of regression; an error associated with one score is independent of other scores, etc.• Absence of outliers• Relates back to normality and assuming a common populationEquations• Extension of the GLM to two IVs• α = deviation of a score, Y, around the grand mean, µ, caused by IV A (Main effect of A)• β = deviation of scores caused by IV B (Main effect of B)• αβ = deviation of scores caused by the interaction of A and B (Interaction of AB), above and beyond the main effectsYµαβαβε=+++ +Equations• Performing a factorial analysis essentially does the job of three analyses in one• Two one-way ANOVAs, one for each main effect• And a test of the interaction• Interaction – the effect of one IV depends on the level of another IV• e.g. The T and F book works better with a combo of media and lecture, while the K and W book works better with just lectureEquations• The between groups sums of squares from previous is further broken down;• Before SSbg= SSeffect• Now SSbg= SSA+ SSB+ SSAB• In a two IV factorial design A, B and AxB all differentiate between groups, therefore they all add to the SSbgEquations• Total variability = (variability of A around GM) + (variability of B around GM) + (variability of each group mean {AxB} around GM) + (variability of each person’s score around their group mean)• SSTotal= SSA+ SSB+ SSAB + SSS/AB2222222()()()()()()()iab a a b biabab ab a a b babiab abiabYGM n YGM n YGMn Y GM n Y GM n Y GMYY−= −+ −+−−−−−+−∑∑∑ ∑ ∑∑∑ ∑ ∑∑∑∑Equations• Degrees of Freedom• dfeffect= #groupseffect–1• dfAB= (a – 1)(b – 1)• dfs/AB= ab(s – 1) = abs – ab = abn – ab= N – ab• dftotal= N – 1 = a – 1 + b – 1 + (a – 1)(b – 1) + N – abEquations• Breakdown of sums of squaresSSbgSSASSBSSABSStotalSSwgSSs/abEquations• Breakdown of degrees of freedomab - 1a - 1 b - 1 (a - 1)(b - 1)N - 1N - abN - abEquations• Mean square• The mean squares are calculated the same• SS/df = MS• You just have more of them, MSA, MSB, MSAB, and MSS/AB• This expands when you have more IVs• One for each main effect, one for each interaction (two-way, three-way, etc.)Equations• F-test• Each effect and interaction is a separate F-test• Calculated the same way: MSeffect/MSS/ABsince MSS/ABis our variance estimate• You look up a separate Fcritfor each test using the dfeffect, dfS/AB and tabled valuesSample data B: Vacation Length A: Profession b1: 1 week b2: 2 weeks b3: 3 weeks 0 4 5 1 7 8 a1: Administrators0 6 6 5 5 9 7 6 8 a2: Belly Dancers 6 7 8 5 9 3 6 9 3 a3: Politicians 8 9 2 222 20 1 2 1046Y =+++=∑"Sample data• Sample info• So we have 3 subjects per cell• A has 3 levels, B has 3 levels• So this is a 3 x 3 designAnalysis – Computational• Marginal Totals – we look in the margins of a data set when computing main effects• Cell totals – we look at the cell totals when computing interactions• In order to use the computational formulas we need to compute both marginal and cell totalsAnalysis – Computational• Sample data reconfigured into cell and marginal totals B: Vacation Length A: Profession b1: 1 week b2: 2 weeks b3: 3 weeks Marginal Sums for Aa1: Administrators 1 17 19 a1 = 37 a2: Belly Dancers 18 18 25 a2 = 61 a3: Politicians 19 27 8 a3 = 54 Marginal Sums for B b1 = 38 b2 = 62 b3 = 52 T = 152Analysis – Computational• Formulas for SS()()()()()()2222222222/22jAkBjk jkABjkSABTaTSSbn abnbTSSan abnab abTSSn bn an abnabSS YnTSS Yabn=−=−=−−+=−=−∑∑∑∑∑∑ ∑∑∑∑∑∑∑∑Analysis – Computational• Example222 2222 222222 22 22222 2 2 2 237 61 54 152889.55 855.7 33.853(3) 3(3)(3)38 62 52 152888 855.7 32.303(3) 3(3)(3)1171918182519 278337 61 54 38 62 52 1523(3) 3(3) 3(3)(3)1026 889.55 888 855ABABSSSSSS++=−=−=++=−=−=+++++++ +=++ ++−−+=− −+.7 104.15=Analysis – Computational• Example22222 22 22/2117191818 2519 278104631046 1026 201521046 1046 855.7 190.303(3)(3)SABTSSSS+++++++ +=−=−==− =− =Analysis – Computational• Example/13121312( 1)( 1) (3 1)(3 1) 2(2) 427 9 18127126ABABSABtotaldf adf bdf a bdf abn abdf abn=−=−==−=−==− −=− −= ==−=−==−=−=Analysis – Computational• ExampleSource SS df MS F Profession 33.85 2 16.93 15.25Length 32.3 2