Violations of Assumptions (11.9)
1. random samples, interval or ratio scores
You can violate the random samples assumption. If you replicate the study, you are
more likely to be able to generalize the findings. The data must come from
independent samples (there are other methods for dependent samples) and must be
an interval or ratio variable. Cannot use ANOVA for nominal or ordinal data.
2. normal distribution
ANOVA is robust to violations of this assumption. It is important that each sample is
from the same shape distribution but they can all be skewed or rectangular, etc.
3. homogeneity of variance
ANOVA is robust to this assumption but only if the sample sizes are the same. Also,
the largest variance should be no more than 4 times the smallest.
Levene’s test of Homogeneity of Variance
Levene’s test seen in Chapter 7 (and covered in lab) takes the deviations from the
group mean and testing to see if those two groups of deviations are significantly
different. This test can be extended to more than 2 groups and the same procedure is
Alternatives to Levene
Box (1954) indicates that the degrees of freedom can be adjusted to deal with
violations. The most conservative test would be to compare Fobt to Fcrit(1, n-1). If you
still have a significant result than violations of assumptions are irrelevant. However, this
is very conservative.
The Welch Procedure
The Welch procedure has the advantage of power (lost with Box) and protection against
type I error. it should be used whenever a Levene test indicates heterogeneity of
variance and especially if you have unequal sample sizes.
An alternative way to deal with violations of assumptions is to transform data.
Transform data to a form that yields homogeneous variance.
Useful when standard deviation is proportional to the mean and when the data are
positively skewed. XNew = logX or XNew = lnX
The logarithmic transformation makes smaller numbers transform less and larger
numbers more (positive skewed data affected more by the transformation). If SD is
proportional (smaller means have smaller SD) then the transformation will reduce the
SD of the larger means more than the smaller means (making them more equal). You
can use log base 10 or log e. It doesn’t matter. If you have data points that are
negative or near zero you can add a constant to make them positive before doing the
Square Root Transformation
When the mean is proportional to the variance and no the SD, you can use a square
root transformation to stabilize the variance. X New X
When you have very large outliers in the positive tail a reciprocal transformation can
reduce the influence of these. XNew = 1/X
To decrease the effect of large tails in a distribution (rectangular), a sample may be
trimmed by removing 5% of the extreme scores in each tail.
Magnitude of Experimental Effect (11.12)
Eta Squared 2
Eta Squared is related to r2
Eta Squared is subject to sample bias.
Omega Squared 2 is an alternative (less biased)
SStreat (k 1)MSerror
That’s it. I am not going to cover Power for ANOVA. In your future research
careers, if you need to calculate power for ANOVA, hire a statistical consultant!