Question

. Discuss when N-way ANOVA is appropriate to use

Answer 1

Introduction to N-Way ANOVA

Use N-way ANOVA to determine if the means in a set of data differ with respect to groups (levels) of multiple factors. By default, anovan treats all grouping variables as fixed effects. For an example of ANOVA with random effects, see ANOVA with Random Effects. For repeated measures,

N-way ANOVA is a generalization of two-way ANOVA. For three factors, for example, the model can be written as

yijkr=μ+αi+βj+γk+(αβ)ij+(αγ)ik+(βγ)jk+(αβγ)ijk+εijkr,

where

• yijkr is an observation of the response variable. i represents group i of factor A, i = 1, 2, ..., I, j represents group j of factor B, j = 1, 2, ..., J, k represents group k of factor C, and r represents the replication number, r = 1, 2, ..., R. For constant R, there are a total of N = I*J*K*R observations, but the number of observations does not have to be the same for each combination of groups of factors.

• μ is the overall mean.

• αi are the deviations of groups of factor A from the overall mean μ due to factor A. The values of αi sum to 0.

  αi=0.

• βj are the deviations of groups in factor B from the overall mean μ due to factor B. The values of βj sum to 0.

  =1βj=0.

• γk are the deviations of groups in factor C from the overall mean μ due to factor C. The values of γk sum to 0.

  1γk=0.

• (αβ)ij is the interaction term between factors A and B. (αβ)ij sum to 0 over either index.

  (αβ)ij=Jj=1(αβ)ij=0.

• (αγ)ik is the interaction term between factors A and C. The values of (αγ)ik sum to 0 over either index.

  (αγ)ik=Kk=1(αγ)ik=0.

• (βγ)jk is the interaction term between factors B and C. The values of (βγ)jk sum to 0 over either index.

  (βγ)jk=Kk=1(βγ)jk=0.

• (αβγ)ijk is the three-way interaction term between factors A, B, and C. The values of (αβγ)ijk sum to 0 over any index.

  (αβγ)ijk=Jj=1(αβγ)ijk=Kk=1(αβγ)ijk=0.

• εijkr are the random disturbances. They are assumed to be independent, normally distributed, and have constant variance.

Three way ANOVA:

Three-way ANOVA tests hypotheses about the effects of factors A, B, C, and their interactions on the response variable y. The hypotheses about the equality of the mean responses for groups of factor A are

H0:α1=α2⋯=αIH1: at least one αi is different, i=1, 2, ..., I.

The hypotheses about the equality of the mean response for groups of factor B are

H0:β1=β2=⋯=βJH1: at least one βj is different,  j=1, 2, ..., J.

The hypotheses about the equality of the mean response for groups of factor C are

H0:γ1=γ2=⋯=γKH1: at least one γk is different, k=1, 2, ..., K.

The hypotheses about the interaction of the factors are

H0:(αβ)ij=0H1:at least one (αβ)ij≠0

H0:(αγ)ik=0

H1:at least one (αγ)ik≠0

H0:(βγ)jk=0

H1:at least one (βγ)jk≠0

H0:(αβγ)ijk=0

H1:at least one (αβγ)ijk≠0