PYB210 Lecture Notes – Week 8
Effects of Breaches of Assumptions
Breaches of the assumptions can result in either an inflated or deflated estimate of the true
BG or WG variability.
As F = BG/WG, this will subsequently either inflate or deflate the obtained F value.
This in turn will either raise or lower the Type I error rate.
o For example, if a breach causes an overestimate of BG variability relative to WG
variability then we will get a bigger F value than we should and thus will conclude
that there is a significant difference when there is actually not.
Linear Structural Model for One-Way Between Groups ANOVA
The total variability associated with any score can be partitioned into WG and BG
components. This is called the linear structural model.
Another way of looking at this is score = pop. Mean + treatment effect + error.
The linear structural model is breached if an extraneous variable confounds the IV (that is,
the EV has a systematic effect which varies between the levels of the IV).
As a result, the BG variability will either be raised or lowered depending upon whether the
EV adds to or subtracts from the effects of the IV.
Thus the F-ratio will be either inflated or deflated leading to an increase in either Type I or
Type II error rate respectively (F = BG/WG).
Therefore you need to control all EVs or include them as extra IVs.
Random Sampling Assumptions
The independence assumption states that it is not possible to predict any one score in the
data from any other score. In other words, this means that the score one participant has
does not necessarily impact any other participant’s scores.
The identical distribution (within groups) assumption states that the error associated with
each score within groups is derived from a single distribution. In other words, this means
that we do not know any more about one score than about any other score.
There are no specific tests we can conduct to control these assumptions. They are expected
to be controlled within a between groups design by a good experimental design.
o This includes the random allocation of participants to groups within the IV.
o Also, the random selection of participants from the population(s) of interest (this is
particularly important to an individual differences IV where random allocation is not
o Each participant must contribute only one score to the analysis (this may be the
average of many observations).
Note that this is routinely breached in a repeated measures design but is not
a problem there.
o Each participant should be tested alone. The Distributional Assumptions
The normality assumption states that the samples are drawn from normally distributed
populations and that the error component is normally distributed within each treatment
group (level of the IV).
It has been shown (Hsu and Feldt 1969) that ANOVA is capable of dealing with breaches of
this assumption provided that the following are met:
o An equal number of participants are included in each condition
o There are at least 12 participants included in each condition
o The distributions are symmetrical and not skewed
o If either skewness or kurtosis is present, the effects are small provided that the
departures from normality are similar in each condition.
In order to determine if this assumption has been breached we can either inspect frequency
histograms for each experimental condition or use PASW/SPSS EXPLORE command to get
skewness and kurtosis statistics.
Homogeneity of Variance
The homogeneity of variance states that the variances of each treatment condition are
equal. Breaches of this assumption are more serious than breaches of independence or
normality and cannot be as easily controlled by a good design.