Replication is the number of random independent replicates from which an ANOVA model calculates the unmeasured variation that is used to calibrate the significance of effects. A balanced ANOVA model has the same number of random independent observations in each sample. This sample size defines the replication of at least the most nested term in a hierarchical model, and the highest order interaction term in a cross-factored model. Main effects have more independent replication than interactions (except in randomized-block designs with Model-1 analysis), which generally gives them higher power.
The independent replication available for any effect in a balanced ANOVA is given by the product of all the variables contributing to the denominator degrees of freedom, q, that do not also contribute to the numerator degrees of freedom, p. For power analysis, the effective samples size, n, is given by the product of all of the variables contributing to the total degrees of freedom of the model that do not also contribute to the p of the term, or equally, the total number of observations in the design divided by the number of samples for the term. The table below shows some examples of designs with treatment factors A, B, C, each with a, b, c levels, and s random independent replicate sampling units S΄ (e.g., subjects or plots or blocks).
|
Design |
Terms |
Effect |
Error |
p |
q |
Total d.f. |
Independent replication |
Effective sample size, n |
|
One-factor fully randomized 1 |
S΄(A) |
A |
S΄(A) |
a - 1 |
(s - 1)a |
sa - 1 |
s |
s |
|
Three-factor nested 2 |
S΄(C΄(B΄(A))) |
A |
B΄(A) |
a - 1 |
(b - 1)a |
scba - 1 |
b |
scb |
|
Three-factor fully randomized 3 |
S΄(C|B|A) |
C*B*A |
S΄(C*B*A) |
(c - 1)(b - 1)(a - 1) |
(s - 1)cba |
scba - 1 |
s |
s |
|
Three-factor fully randomized |
S΄(C|B|A) |
B*A |
S΄(C*B*A) |
(b - 1)(a - 1) |
(s - 1)cba |
scba - 1 |
sc |
sc |
|
Three-factor fully randomized |
S΄(C|B|A) |
A |
S΄(C*B*A) |
a - 1 |
(s - 1)cba |
scba - 1 |
scb |
scb |
|
Two-factor randomized block 4 |
S΄|B|A |
B*A |
S΄*B*A |
(b - 1)(a - 1) |
(s - 1)(b - 1)(a - 1) |
sba - 1 |
s |
s |
|
Two-factor randomized block 5 |
S΄|B|A |
A |
S΄*A |
a - 1 |
(s - 1)(a - 1) |
sba - 1 |
s |
sb |
|
Two-factor randomized block 5 |
S΄|B|A |
B |
S΄*A |
a - 1 |
(s - 1)(a - 1) |
sba - 1 |
s |
sa |
|
Two-factor split plot 6,7 |
B|S΄(A) |
B*A |
B*S΄(A) |
(b - 1)(a - 1) |
(b - 1)(s - 1)a |
sba - 1 |
s |
s |
|
Two-factor split plot |
B|S΄(A) |
A |
S΄(A) |
a - 1 |
(s - 1)a |
sba - 1 |
s |
sb |
|
Two-factor split plot |
B|S΄(A) |
B |
B*S΄(A) |
b - 1 |
(b - 1)(s - 1)a |
sba - 1 |
sa |
sa |
1 For example, s = 6 subjects nested in each of a = 3 levels of factor A, requiring a total of sa = 18 observations and giving 6 independent replicates for testing the A main effect.
2 For example, s = 24 subjects nested in each of c = 12 towns in each of b = 6 counties in each of a = 3 countries, requiring a total of scba = 5184 observations and giving 6 independent replicates for testing the effect A of country.
3 For example, s = 2 plots nested in each of cba = 27 combinations of three-level factors C, B, A, requiring a total of scba = 54 observations and giving 2, 6, 18 independent replicates for testing respectively the three-way interaction, two-way interactions, and main effects.
4 For example, s = 6 blocks, each with ba = 9 combinations of three-level factors B, A, requiring a total of sba = 54 observations and giving 6 independent replicates for testing both the interaction and main effects.
5 Model-1 analysis of main effects; Model-2 analysis has a single variance component defining the pooled error, and effective sample size n = s.
6 For example, s = 6 blocks nested in each of a = 2 levels of factor A, and each block split into b = 3 levels of factor B, requiring a total of sba = 36 observations and giving 6, 6, 12 independent replicates for testing respectively the interaction, A main effect, and B main effect.
7 For example, s = 6 subjects nested in each of a = 2 levels of intervention factor A: treatment and control, and repeated measures on each subject at b = 2 levels of time factor B: before and after intervention, requiring a total of sba = 24 observations and giving 6 independent replicates for testing the treatment effect in the B*A interaction.
The program Power.exe will estimate the power of any balanced ANOVA design to detect effects from an effective sample size n. For any of the models described on these web pages, the program CritiF.exe will list each estimable effect and its threshold effect size for a power of 0.8, given specified sample sizes and levels of treatment factors.
Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press.
http://southampton.likn.co/~cpd/anovas/datasets/