University:
California State University, Northridge
Course:
PSY 534/S | Latent Variable Analysis and Seminar
Academic year:

2024
Views:

143

Pages:

27
Author:

hesmyonlyonehcg

Factorial BG ANOVA Topics 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 Size Factorial? • 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 Text book Tabachnick and Fidell Keppel and Wickens Teaching Method Lecture Media Lecture/Media M & TF L & TF LM & TF L & KW M & KW LM & KW • 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 Factorial? • Crossing Example 2 – repeated measures Pre - test Mid - test Post - test s1 s1 s1 s2 s2 s2 Subjects s3 s3 s3 s4 s4 s4 s5 s5 s5 • In repeated measures designs subjects cross the levels of the IV Factorial? • Nesting Example Teaching Method Lecture Media Lecture/Media T and F L & TF/ Class 1 M & TF/ Class 3 LM & TF/ Class 5 Text book K and W L & KW/ Class 2 M & KW/ Class 4 LM & KW/ Class 6 • This example shows testing of classes that are preexisting; no random selection or assignment • In this case classes are nested within each cell which means that the interaction is confounded with class Assumptions • 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 population Assumptions • 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 population Equations • Extension of the GLM to two IVs Y = µ + α + β + αβ + ε • 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 effects 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 lecture Equations • 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 SSbg Equations • 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/AB 2 2 2 − = − + − Y GM n Y GM n Y GM ( ) ( ) ( ) ∑∑∑ iab a∑ a b∑ b i a b  2 2 2 +  nab ∑∑ (Yab − GM ) − na ∑ (Ya − GM ) − nb ∑ (Yb − GM )  a b   + ∑∑∑ (Yiab − Yab ) 2 i a b 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 – ab Equations • Breakdown of sums of squares SStotal SSbg SSA SSB SSwg SSAB SSs/ab Equations • Breakdown of degrees of freedom N-1 ab - 1 a-1 b-1 N - ab (a - 1)(b - 1) N - ab Equations • 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 Ftest • 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 values Sample data A: Profession a1: Administrators a2: Belly Dancers a3: Politicians B: Vacation Length b1: 1 week b2: 2 weeks b3: 3 weeks 0 4 5 1 7 8 0 6 6 5 5 9 7 6 8 6 7 8 5 9 3 6 9 3 8 9 2 2 2 2 2 Y = 0 + 1 +  + 2 = 1046 ∑ 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 design Analysis – 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 totals Analysis – Computational • Sample data reconfigured into cell and marginal totals A: Profession a1: Administrators a2: Belly Dancers a3: Politicians Marginal Sums for B B: Vacation Length b1: 1 week b2: 2 weeks b3: 3 weeks 1 17 19 18 18 25 19 27 8 b1 = 38 b2 = 62 b3 = 52 Marginal Sums for A a1 = 37 a2 = 61 a3 = 54 T = 152 Analysis – Computational • Formulas for SS 2 aj ) ( T2 ∑ ∑ = − SS A = SS B bn ∑ (∑ b ) abn 2 k an T2 − abn ∑ ( ∑ ab ) − ∑ ( ∑ a ) − ∑ ( ∑ b ) 2 SS AB = SS= S / AB 2 jk n ∑Y 2 j bn ab ) ( ∑ ∑ − 2 T = SST ∑ Y 2 − abn jk n 2 k an 2 T2 + abn Analysis – Computational • Example 37 2 + 612 + 542 1522 − = 889.55 − 855.7 = 33.85 SS A = 3(3) 3(3)(3) 382 + 622 + 522 1522 − = SS B = 888 − 855.7 = 32.30 3(3) 3(3)(3) 12 + 17 2 + 192 + 182 + 182 + 252 + 192 + 27 2 + 82 SS AB = 3 37 2 + 612 + 542 382 + 622 + 522 1522 − − + 3(3) 3(3) 3(3)(3) = 1026 − 889.55 − 888 + 855.7 = 104.15 Analysis – Computational • Example 12 + 17 2 + 192 + 182 + 182 + 252 + 192 + 27 2 + 82 1046 − SS S= / AB 3 = 1046 − 1026 = 20 1522 SST =1046 − =1046 − 855.7 =190.30 3(3)(3) Analysis – Computational • Example df A = a − 1 = 3 − 1 = 2 df B = b − 1 = 3 − 1 = 2 df AB = (a − 1)(b − 1) = (3 − 1)(3 − 1) = 2(2) = 4 df S / AB = abn − ab = 27 − 9 = 18 dftotal = abn − 1= 27 − 1= 26 Analysis – Computational • Example Source SS df Profession 33.85 2 Length 32.3 2 Profession x Length 104.15 4 Subjects/Profession x Length 20 18 Total 190.3 26 MS F 16.93 15.25 16.15 14.55 26.04 23.46 1.11 Analysis – Computational • Fcrit(2,18)=3.55 • Fcrit(4,18)=2.93 • Since 15.25 > 3.55, the effect for profession is significant • Since 14.55 > 3.55, the effect for length is significant • Since 23.46 > 2.93, the effect for profession * length is significant

Factorial BG ANOVA

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