Dear Allstat members,
I hope this list is the right place to post this kind of question. I Apoligize if it is not the case.
I might need your help on a problem of sample size calculation.
In brief : does a formula to calculate sample size of a three way ANOVA (or rather a two-way ANOVA with repeated samples) exist, given the power wanted, the alpha level, the expected means and variances of each group ?
With more details :
The idea is to evaluate the effect of a breed on the resistance to ticksBreed 1 = sensitive
Breed 2 = resistant
We want to compare F1 (father breed 2 * mother breed 1) vs pure breed (mother and father = breed 1)
We expect F1 to be more resistant ( = have less ticks) than pure breed
The experimental design is the following :
Unit = animal
Dependant variable = nb of ticks on each animal (quantitative)
Factors :
- Breed (F1 vs pure), Fixed. Factor of interest
-Farm (the maximum number of farms that we can sample is 4 = 4 blocks, correlation intra-farm = probably +++). Random
- Time : 4 samples will be taken on each animal at different times of the year (the same animals will be followed for all 4 counts). Random
Therefore, to evaluate the effect "breed" on the number of ticks, I thought I would perform an ANOVA with two-factors and repeated measures
There will be n F1 and n pure animals randomly sampled in each of the 4 farms.
They want to be able to see a difference of at least 1 to 3 in the number of ticks (if 100 ticks on pure breeds, they want to be able to detect a difference if there is less than 30 ticks on F1)
I am asked to calculate n so that they will be sure to see that difference.
I expect to find no significant interaction between the "breed" factor and the others (anymay, if in the end there are some interactions, the answer to the problem will be straight forward)
I found some formula for sample size calculation for two-way Anova, but not for two way Anova with repeated measures.
Do these formula exist ?
Or is my only option to simulate a LOT of samples to answer the question ?
(I have to state here that I do not have acces to Zar book on this matter, and therfore don't have acces to the power tables for different tests....)
I found several software to perform sample size calculation, but find them hard to use.
And would much rather understand what I'm doing exactly, what formula is used, and what it refers to exactly, the hypothesis underneath, etc.
For example, I tried to find this formula for a simpler design, with only one farm and one sample in time (see below)
I'd like to build the same kind of formula for a two-way ANOVA with repeated sampling
Can I treat this problem as a three-way Anova, with the thrid factor being time ?
Should I keep on looking for the right formula, or should I start simulate data straight away ?
Thank you in advance for your precious help
Yours sincerely
Magali Teurlai
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EXAMPLE
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I calculated the formula if I had only 1 farm, and no repetition (simple t-test), to have an idea, based on the following thoughts
let m1 be the mean number of ticks in group pure breed, an estimation of true mu1
let m2 be the mean number of ticks in group F1, an estimation of true mu2
let A be the alpha level for the one-sided t.test (0.025)
let 1- B be the power wanted
let s1 be the expected variance in each sample (supposed equal for both breeds)
CI [1 - alpha] (mu1) = [m1 - Phi(1-A) * s1/sqrt(n) ; m1 + Phi(1-A) * s1/sqrt(n) ]
Phi being the normal distribution
cutting value for the t.test : m1 + Phi(1-A) * s1/sqrt(n)
==> to be able to detect the minimum difference between m1 and m2, with A alpha level and 1-B power :
m2 - Phi (1-B) * s/sqrt(n) = m1 + Phi(1-A) * s1/sqrt(n)
let be dm the minimum difference between mu1 and mu2 that they want to be able to detect
the formula becomes :
m1 - dm - Phi (1-B) * s1/sqrt(n) = m1 + Phi(1-A) * s1/sqrt(n)
and thus I find :
n = s1^2 * ( Phi(1-B) + Phi (1-A) )^2 / dm^2
Here is the R-code
# parameters
s1 <- 22 # expected variance of each group
m1 <- 50 # expected mean number of ticks in pure breeed group
B <- 0.1 # error type 2 accepted (1-power)
A <- 0.025 # alpha level
ratio <- 1/3 # difference between m1 and m2 that they want to see
### One farm, One sample in time, F1 vs pure
m2 <- m1 * ratio # expected mean nb of ticks for the F1 group
dm <- m1 - m2 # mean difference that they want to see
nmin <- s1^2 * (qnorm(1-A) + qnorm(1-B))^2 / dm^2
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#### Then I can make a table of the sample size needed for different power, different alpha level, different expected values for m1 and s1....
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