Dear useRs,
I've just encountered a serious problem involving the F-test being carried
out in aov() and anova(). In the provided example, aov() is not making the
correct F-test for an hypothesis involving the expected mean square (EMS) of
a factor divided by the EMS of another factor (i.e., instead of the error
EMS).
Here is the example:
Expected Mean Square df
Mi σ2+18σ2M 1
Ij σ2+6σ2MI+12Ф(I) 2
MIij σ2+6σ2MI 2
ε(ijk)l σ2 30
The clear test for Ij is EMS(I) / EMS(MI) - F(2,2)
However, observe the following example carried out in R,
M <- rep(c("M1", "M2"), each = 18)
I <- as.ordered(rep(rep(c(5,10,15), each = 6), 2))
y <-
c(44,39,48,40,43,41,27,20,25,21,28,22,35,30,29,34,31,38,12,7,6,11,7,12,15,10,12,17,11,13,22,15,27,22,21,19)
dat <- data.frame(M, I, y)
summary(aov(y~M*I, data = dat))
Df Sum Sq Mean Sq F value
Pr(>F)
m 1 3136.0 3136.0 295.85 <
2e-16 ***
i 2 513.7 256.9 24.23
5.45e-07 ***
m:i 2 969.5 484.7 45.73
7.77e-10 ***
Residuals 30 318.0 10.6
---
In this example aov has taken the F-ratio of MS(I) / MS(ε) - F(2,30) =
24.23 with F-crit = qf(0.95,2,3) = 9.55 -- significant
However, as stated above, the correct F-ratio is MS(I) / MS(MI) - F(2,2) =
0.53 with F-crit = qf(0.95,2,2) = 19 -- non-significant
Why is aov() miscalculating the F-ratio, and is there a way to fix this
without prior knowledge of the appropriate test (e.g., EMS(I)/EMS(MI)?
Thanks for your help,
Patrick
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