Degrees of Freedom

#1
I hope you are all well. This question is about degrees of freedom for G*power ANOVA A priori and extends to calculating degrees of freedom for replicates of replicates (Replicated dishes).

Scenario: 3 treatments (Solutions 1, 2 and 3) with 4 levels each (Red, green, orange and blue). This creates 12 Petri dishes, each with a different treatment. Numerator degrees of freedom (3-1)*(4-1) = 6. In each dish we are measuring how much dye is absorbed in wooden pellets. There are 50 small pellets in each Petri dish. Denominator degrees of freedom 50-1 = 49.

As it is, we may say we have numerator df = 6, denominator df = 49. Sample size n = 50. Total sample size (50 x 12) N = 600.

But if we were calculating for sample size, we may give significance 0.05, power 0.8, numerator df 6, denominator degrees of freedom 49. (and effect size if needed).

We want to replicate each combination 4 times. In detail similar to above, how would you treat the replicated plates in the numerator degrees of freedom?

Cheers G
 

katxt

Active Member
#2
As I see it, your experimental unit it the petri dish and value is the average of the 50 pellets. You don't need to worry about the df of the pellets.
 
#3
As I see it, your experimental unit it the petri dish and value is the average of the 50 pellets. You don't need to worry about the df of the pellets.
Thank you katxt yes good point, either the pellets (if individually measured) or Petri dishes could be the experimental unit, that is the denominator degrees of freedom. In the first case how many degrees of freedom would you give the numerator?
 
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katxt

Active Member
#4
Perhaps we are talking at cross purposes, because I'm not familiar with G*power ANOVA. If this is any help, as I see it the final anova table would have solutions 2df, colors 3df, solutionXcolour 6df, these are the numerators. Dishes error36df. This is the denominator for the f tests for the main effects. The df for the pellets would not be used for testing the main effects.
 
#5
I see that, it's interesting isn't it. Now imagine the variation between the individual pellets is wanted, parametric pellet SD, not n=4 plate SD, but we'll perhaps model plates as a random effect
 

katxt

Active Member
#6
You can include the pellets and include the extra df to see if the dish to dish variation is greater than can be explained by pellet to pellet variation but because pellets are nested in dishes, the main effects will still be calculated from the dishes.
 
#7
If this was done without Petri dish replicates it would have treatment df2, colour df3 and pellets df49. So for a 2-way ANOVA df(3)*(4)*(50-1) on the denominator df for the F test according to Sokal&Rohlf. Then it is frustrating to lose so many df (i.e. so much 1-beta) by replicating 4 Petri dishes if denominator df becomes (3)*(4)*(4-1). Then it comes down to resources allocation: estimate the minimum pellets to be parametrically representative on the average (for each dish) and up the Petri dishes to meet minimum numbers for the desired power.
 
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katxt

Active Member
#8
There is a difference between repeats and replicates. Replicates means doing the experiment several times, mixing up the various solutions each time (rather than making the solution once and pouring it into four dishes.) This means that possible dish to dish variation can and will be allowed for. Repeats are making several measurements under identical dish conditions. It is tempting to combine all the pellets into one big sample but this won't work because the pellets aren't independent because of dish to dish differences.
 
#9
In 'repeats' as you define it the dye is mixed once and poured into four dishes. 'replicates' as you define it are a new batch of dye made for each Petri dish. With both ways a test could be done to see if there is more variation between the plates than is expected than if the plates were from the same population, assuming the parametric variation is normally distributed or otherwise known. The first instance if nested would test for dish effects, the second instance if 'significant' may infer added variation between the dyes or between the dishes