# Degrees of Freedom in Full Factorial

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• #238393

noambsn
Participant

Hello,
if I have a design of 2^k*n when k=4 factors and n=3 replications,
how can I calculate for the whole model the df_total,df_model, and df_error?
thank you

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#238438

Robert Butler
Participant

Give two levels for each factor you have the following:
Factor A, df = (a-1)
Factor B, df = (b-1)
Factor C, df = (c-1)
Factor D, df = (d-1)

in each case a = b = c = d = 2

For each interaction
AxB, df = (a-1)*(b-1)
AxC, df = (a-1)*(c-1)
etc.
AxBxC, df = (a-1)*(b-1)*(c-1)
etc.
and
AxBxCxD = (a-1)*(b-1)*(c-1)*(d-1)
Error, df = a*b*c*d*(n-1) where n = number of replicates
Total df = a*b*c*d*n -1

As an aside – 3 full replicates is gross overkill. The whole point of experimental design is minimum effort for maximum information. Unless you have a very unusual process it is unlikely that 3 and 4 way interactions will mean anything to you physically. Under those circumstances just take the df associated with the three and the 4 way interactions and build a model using only mains and two ways – this dumps all of the df for the higher terms into the error and instead of running 48 experiments you will have run 16.

The other option, assuming you actually have a process where 3 way and 4 way interactions are physically meaningful and the variables under consideration can all be treated as continuous, is to replicate just one or perhaps two of the design points – this will give you df for error and will also allow you to check the higher order interactions. Given that they can be treated as continuous a better bet would be to just add 2 center design points to your 2^4 design. This will give you a run of 18 experiments, an estimate of all mains, 2,3, and 4 way interactions, an estimate of error, and a check on the possible curvilinear behavior.

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