If both factors are repeated factors:

Suppose factor1 has i levels and factor2 has j levels and you have n subjects tested

df for factor1 = i-1

df for factor2 = j-1

df for interaction factor1 x factor2 = (i-1)*(j-1)

df for error(factor1) = (i-1)*(n-1)

df for error(factor2) = (j-1)*(n-1)

df for error(factor1xfactor2) = (i-1)*(j-1)*(n-1)

F for factor1 = MeanSquare of factor1 divided by MeanSquare of error of factor1

F for factor2 = MeanSquare of factor2 divided by MeanSquare of error of factor2

Usually sphericity is tested for repeated measured effects. If sphericity assumption is not violated you don't have to correct the degrees of freedom. If sphericity assumption is viloated (you get a significant Chi-Squared value in the Sphericity test or the Huynh-Feldt Epsilon is lower than 1 you should correct the degrees of freedom for the F-tests by multiply them by the Huynh-Feldt Epsilon (which corrects optimal according to the error variance covariance matrix). The multiplication will not change the F-value, it will change only the origin of the F-value since it will be from another distribution with the corrected degrees of freedom...

Source                   SumOfSq        df            Mean Squares                  F

factor1                    SS1                  df1         1) SS1 / df1                1) / 2)   

err(f1)                     SS(err(f1))       dfe1       2) SS(err(f1)) / dfe1            

factor2                    SS2                  df2          3) SS2 / df2               3) / 4)

err(f2)                     SS(err(f2))       dfe2        4) SS(err(f2)) / dfe2

factor1 x factor2    SS1x2              df1x2       5) SS1x2 / df1x2         5) / 6)

err(f1x2)                 SS(err(f1x2))   dfe1xe2   6) SS(err(f1xf2)) / dfe1x2

The term "two-way" could mean two things - "2 factors + time as an additional factor" or "1 factor + time as another factor". Assuming the later, your RM ANOVA will look like [assuming you have used a Randomized Complete Block Design - say with r replications (=blocks)]

Source                      DF

Block                         r-1

Treat                         t-1  (t treatments)                                     (1)

Residual (a)           (r-1)(t-1)

Time                         T-1 (repeated measurements made at T different times) (2)

Treat x Time            (t-1) (T-1)                                                 (3)

Residual (b)            T(r-1)(t-1)

For (1), the DF for F = [t-1, (r-1)(t-1)]

For (2), the DF for F = [T-1, T(r-1)(t-1)]                                 (4)

For (3), the DF for F = [(t-1)(T-1), T(r-1)(t-1)]                       (5)

Note that in (4) and (5), the denominator (ie residual) DF will be adjusted for any dependence in observations over time (eg by a measure called 'epsilon' that GenStat software implements).

If both factors are repeated factors:

Suppose factor1 has i levels and factor2 has j levels and you have n subjects tested

df for factor1 = i-1

df for factor2 = j-1

df for interaction factor1 x factor2 = (i-1)*(j-1)

df for error(factor1) = (i-1)*(n-1)

df for error(factor2) = (j-1)*(n-1)

df for error(factor1xfactor2) = (i-1)*(j-1)*(n-1)

F for factor1 = MeanSquare of factor1 divided by MeanSquare of error of factor1

F for factor2 = MeanSquare of factor2 divided by MeanSquare of error of factor2

Usually sphericity is tested for repeated measured effects. If sphericity assumption is not violated you don't have to correct the degrees of freedom. If sphericity assumption is viloated (you get a significant Chi-Squared value in the Sphericity test or the Huynh-Feldt Epsilon is lower than 1 you should correct the degrees of freedom for the F-tests by multiply them by the Huynh-Feldt Epsilon (which corrects optimal according to the error variance covariance matrix). The multiplication will not change the F-value, it will change only the origin of the F-value since it will be from another distribution with the corrected degrees of freedom...

Source                   SumOfSq        df            Mean Squares                  F

factor1                    SS1                  df1         1) SS1 / df1                1) / 2)   

err(f1)                     SS(err(f1))       dfe1       2) SS(err(f1)) / dfe1            

factor2                    SS2                  df2          3) SS2 / df2               3) / 4)

err(f2)                     SS(err(f2))       dfe2        4) SS(err(f2)) / dfe2

factor1 x factor2    SS1x2              df1x2       5) SS1x2 / df1x2         5) / 6)

err(f1x2)                 SS(err(f1x2))   dfe1xe2   6) SS(err(f1xf2)) / dfe1x2

Really, the question is not very clear, because if you have blocks as another factor or direction, then you have to use factorial RCBD and the block df should be added to the calculation of F value, but if you just have two factors without any other directions or internal factor, you 'll use Factorial CRD without having df for blocks.

( number of columns - 1 ) * ( number of rows - 1 ) 

( number of columns - 1 ) * ( number of rows - 1 )

Hi, I have similar problem. I have 6 subjects and 3 levels of treatment repeated for 4 days for all subjects i.e., 72 observations (no random block design). I want to see day and treatment interaction. How should I calculate the df2 here for main effects and interaction effects. Sigmplot13 gives residual df as 27 and total df as 68 which does not match the calculations given here. Could anybody help? Thanks.

Hello Sandeep,

You have a two-within subjects factor design (e.g., both day and treatment are repeated measures factors). Assuming no missing data, the test of the day x treatment interaction should be an F based on 6 and 30 df.

day x treatment df = (4 - 1) * (3 - 1) = 6

subject x day x treatment df = (6 - 1) * (4 - 1) * (3 - 1) = 30

Note that there will be three different error terms for the test of day (3, 15 df), treatment (2, 10 df), or day x treatment (6, 30 df).

Good luck with your work.

My question is : I have (a) no of groups, (m) subjects per group . All of them (m x a) are tested repeatedly for (b) no of days with the same kind of treatment and the effect on each subject is recorded. The question is to decide whether the treatment was effective.

Can someone, briefly, explain the logic of spliting the total variation to various parts ? I.e what is the underlying model (equation) for the random variable Xi,j,k (i for group, j for day, m for subjct)

What are the df for each ?

In particular I can't explain why the df for the SSerror (days x subjects w/i groups) is a(b-1)(m-1) ; why b-1 and not b ?