Two-way ANOVA: Factorial Design with Replications and Balanced Fixed Effects - Prof. B. Ha, Study notes of Data Analysis & Statistical Methods

Download Two-way ANOVA: Factorial Design with Replications and Balanced Fixed Effects - Prof. B. Ha and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! Overview of the Two-way ANOVA Factorial, With Replications, Balanced, Fixed Effect Say there are two factors Factor A has levels numbered i=1, ... A Factor C has levels numbered j=1, ...C and each combination has k=1, ... n replicatons. (Note that the names we give to the factors don’t matter! On page 343 of the text they use A and B... on 348 they use G and R. ) The data could be laid out as follows: Factor A ▼ j=1 Factor C j=2 ... j=C Means for Factor A ▼ i=1 nx x x 11 112 111
11x nx x x 12 122 121
12x ... Cn C C x x x 1 21 11
Cx1 1A i=2 nx x x 21 212 211
21x nx x x 22 222 221
22x ... Cn C C x x x 2 22 12
Cx2 2A





i=A nA A A x x x 1 12 11
1Ax nA A A x x x 2 22 21
2Ax ... ACn AC AC x x x
2 1 ACx AA Means for ► Factor C 1C 2C ... CC x The model equation for this two way ANOVA could be written as: xijk=µbaseline+αi+γj+(αγ)ij+εijk for i=1,...A, j=1,...C, and k=1,...n where the xijk are the observations µbaseline is the baseline α1, α2, ... αA are the main effects for the levels of factor A γ1, γ2, ... γC are the main effects for the levels of factor C (αγ)11, (αγ)12,... (αγ)21,... (αγ)AC are the interactions for the combinations of A and C and the εijk are the errors that satisfy the conditions of mean equal to 0, equal variances, normality, and independence The basic ANOVA table could then be written as: Source SS df MS F Between ∑ ∑ ∑ −= = = = A i C j n k ijbet xxSS 1 1 1 2)( AC-1 1− = AC betSS betMS witMS betMSF = ►Factor A ∑ ∑ ∑ −= = = = A i C j n k iA xASS 1 1 1 2)( A-1 1− = A ASSMSA witMS AMSF = ►Factor C ∑ ∑ ∑ −= = = = A i C j n k jC xCSS 1 1 1 2)( C-1 1− = C SS MS CC witMS CMSF = ►AC Interaction CAbetAC SSSSSSSS −−= (AC-1)-(A-1)-(C-1) =(A-1)(C-1) )1)(1( −− = CA SS MS ACAC witMS ACMSF = Within ∑ ∑ ∑ −= = = = A i C j n k ijijkwit xxSS 1 1 1 2)( ACn-AC ACACn betSS betMS − = Total ∑ ∑ ∑ −= = = = A i C j n k ijktot xxSS 1 1 1 2)( ACn-1