PYB210 Lecture Notes – Week 9
Chapter 8 - Power and Effect Size
Overview of Lecture
o Eta squared (R )
o Omega squared
o Sample size
A significant F simply tells us that there is a difference between means. It does not tell us
how big this difference is or how important this effect is.
An F that is significant at 0.01 does not necessarily imply a bigger or more important effect
size than an F significant at 0.05. This is because the significance of F is dependent on the
sample size and the number of conditions.
Therefore, we need a statistics which summarizes the strength of the treatment effect:
o Eta squared (η ) = R
o Omega squared (ω ) 2
Both of these indicate the proportion of the total variability in the data accounted for by the
effect of the IV.
Eta squared – or η – treats our ANOVA data as though it came from a correlation st2dy. η is
simply the squared correlation coefficient from a regression on this data – a.k.a. it is R .
There is a sum for calculating eta squared which essentially the sum of squares for the
treatment variability divided by the sum of squares for the total variability (treatment +
These sums of squares totals come from the following table (example below).
The example sum provided above tells us that 65% of the variability comes from the IV.
Problems with Eta Squared
The eta squared relies on numerically ordered data to provided a regression from which the
η is calculated. In other words, it wrongly assumes that the levels of the IV can be treated as
quantitative. It also wrongly assumes that the regression line will pass through the groups’ means.
Thus, in instances where our independent variable is categorical, there is no logical way to
order our levels of the IV into a correlation.
However, SPSS still uses this as its go-to measure of effect size and in most cases it is
This on the other hand is a better estimate of effect size as it does not have the two
problems that eta squared d2es (highlighted previously).
There are two versions of ω . One is for fixed model ANOVAs and the other is for random
A fixed model ANOVA is where the particular levels of the IV have been deliberately chosen
by the experimenter (e.g. deliberately choosing 4, 8 and 12 hrs sleep deprivation).
A random model ANOVA is where the particular levels of the IV in the experiment have been
selected at random from a number of possible levels of the IV.
In PYB210 we always assume that we are using a fixed model ANOVA.
The conceptual formula for ω is as follows:
There is also a computational formula which is much larger but easy and could possibly be
on the exam! An example of this formula is given below:
Where a = number of levels of IV; F = F ratio for ANOVA; n = number of scores per group.
The above example tells us that the IV accounted for 54.3% of the variability in the data (this
is a large effect).
Interpreting Effect Size
Once you have obtained either you eta or omega squared you need to interpret it.
Remember that they are both a proportion and can range from 0 (IV had no effect) to 1
(100% variability accounted for by IV).
If the F is < 1 than the proportion will be expressed as a negative number.
Cohen (1988) proposed the following scale for effect size:
o 0.01 = small effect
o 0.06 = medium effect
o > 0.15 = l