Suppose you have (a) different groups of (m) similar subjects each. You give them all ( i.e the m xa subjects) the same treatment repeatedly for (b) days.
You take daily measurements of the effect of the treatment on all groups and you want to decide if the treatment has made any difference after all these days.
How do you analyze this ANOVA and how do you split up the total Sum of Squares into smaller sums and how the df are assigned ?
Hello Elias,
As Ajit kumar Roy
suggests, the observations across days will be considered the repeated measures dimension, with B levels.Let N = m1 + m2 + ... + ma (the total number of cases across A groups.
The basic design is a one-between (groups; having A levels), one-within (days/observations, having B levels). Sometimes called a split plot design.
Total data points = N * B
Total df = N * B - 1
which can be partitioned into:
Between-subjects df = N - 1
Within-subjects df = N * (B - 1)
Between-subjects df can be partitioned into:
Groups differences (df = A - 1)
Subjects within groups ("error"; df = A*(m - 1), assuming all groups have same size)
F(groups) = MS(groups) / MS(subjects within groups)
Within-subjects df can be partitioned into:
Days/Observations (df = B - 1)
Group X Day interaction (df = (A - 1) * (B - 1) )
Subjects within groups X day ("error"; df = A * (m - 1) * (B - 1) )
F(Days) = MS(days) / MS(subjects within groups X day)
F(Group X Day) = MS(group X day) / MS(subjects within groups X day)
There are many good texts that can explain the particulars of between-within designs. Here are a couple:
Field, A. Discovering statistics using IBM SPSS statistics. SAGE.
Howell, D. Statistical methods for psychology. Wadsworth/Cengage.
Good luck with your work.
To me, this seems like requiring a common repeated measures ANOVA. Andy Field names the design involved a Mixed (Between x Within) design. Any statistical software offers ANOVA procedures. One only needs to take into account the assumptions (o.a. normality of distribution) underlying such an analysis before running it.
See: https://sites.ualberta.ca/~lkgray/uploads/7/3/6/2/7362679/slides_-_anova_assumptions.pdf
There are 'nonparametric' alternatives in the case of violations of these assumptions, e.g. the 'art3.anova' (a nonparametric analysis of variance using ART (Aligned Rank Transform) for Mixed designs (split plot or repeated measures designs) by Haiko Lüpsen: http://www.uni-koeln.de/~a0032/R/
Thank you all so much for your prompt responses !!
My question concerns :
Apparently this ANOVA resembles to Two Factor with Replications, however it is different due to the fact that we have the same subjects (m xa ) repeatedly tested over (b) days. In vue of this
1) I am interested on what model we must choose for the Measurement Random variable Xi,j,k ( i for (a), j for (b) , k for (m)), or in what parts you split the SStotal.
2) In the evaluation of the degrees of freedom, as in particular for the error term these are :
df = a(b-1)(m-1) ; Jos Feys why b-1 and not b ?
3) Where can I find some simple explanatory literature on the theory of it ? Preferably internet sites if possible.
You say you have different groups, but the same subjects are tested repeatedly. This is a 'common' 2-way Mixed design (Between (different) subjects x Within (the same) subjects, but repeatedly). Just trust the available repeated measures factorail ANOVA in any statistical package. You only need to check the distributions of your dependent values (repeated measures), e.g. with histograms.
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