Factorial BG ANOVATopics in Factorial DesignsFactorial?Slide 4Slide 5Slide 6AssumptionsSlide 8EquationsSlide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Sample dataSlide 19Analysis – ComputationalSlide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Factorial 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 cellsFactorial?•Crossing Example 2 – repeated measures•In repeated measures designs subjects cross the levels of the IVFactorial?•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 classAssumptions•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 = deviation of a score, Y, around the grand mean, , caused by IV A (Main effect of A) = deviation of scores caused by IV B (Main effect of B) = deviation of scores caused by the interaction of A and B (Interaction of AB), above and beyond the main effectsY m a b ab e= + + + +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/AB2 2 22 2 22( ) ( ) ( )( ) ( ) ( )( )iab a a b bi a bab ab a a b ba biab abi a bY GM n Y GM n Y GMn Y GM n Y GM n Y GMY Y- = - + -� �+ - - - - -� �� �+ -��� � ��� � ����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/AB since MSS/AB is our variance estimate•You look up a separate Fcrit for 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: Administrators 0 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 2 2 2 20 1 2 1046Y     LSample 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 A a1: 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( )( )( ) ( )( )( )22222 22222/22jAkBjk jkABjkS ABTaTSSbn abnbTSSan abnab abTSSn bn an abnabSS YnTSS Yabn= -= -= - - += -= -������ ��������Analysis – Computational•Example2 2 2 22 2 2 22 2 2 2 2 2 2 2 22 2 2 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)1 17 19 18 18 25 19 27 8337 61 54 38 62 52 1523(3) 3(3) 3(3)(3)1026 889.55 888 855ABABSSSSSS+ += - = - =+ += - = - =+ + + + + + + +=+ + + +- - += - - + .7 104.15=Analysis – Computational•Example2 2 2 2 2 2 2 2 2/21 17 19 18 18 25 19 27 8104631046 1026 201521046 1046 855.7 190.303(3)(3)S ABTSSSS+ + + + + + + += -= - == - = - =Analysis – Computational•Example/1 3 1 21 3 1 2( 1)( 1) (3 1)(3 1) 2(2) 427 9 181 27 1 26ABABS ABtotaldf adf bdf a bdf abn abdf abn= - = - == - = - == - - = - - = == - = - == - = - =Analysis – Computational•ExampleSource SS df MS F Profession 33.85 2 16.93 15.25 Length 32.3 2 16.15 14.55 Profession x Length 104.15 4 26.04 23.46 Subjects/Profession x Length