1. Introduction
The assumptions required for an analysis of variance (ANOVA) are not
always perfectly met by real-world data. However, researchers who choose this
procedure need to be assured that, even if they do not fully meet the necessary
assumptions (normality of residuals and homogeneity of variances), their data
will still be suitable.
It is known that minor deviations from normality do not seriously
compromise the validity of the ANOVA, especially when group sizes are equal or
similar. Similarly, minor violations of homogeneity of variances have little
practical relevance - except in two critical situations:
1) when there is
asymmetry in the residuals;
(2) when there is
positive kurtosis in the residuals.
In any case, the F-test remains the most powerful of the available tests
provided that its assumptions are met. Otherwise, researchers should consider
using non-parametric tests or resorting to data transformation. Transformations
are particularly useful for stabilising the variance, but also generally help
to approximate the distribution to normality.
2. What does it mean to
transform data?
Transforming data involves applying a mathematical operation to each
observation and conducting statistical analyses using the resulting values. The
best-known transformations are listed below.
2.1.
Square Root
In general, variables obtained by counting do not have a constant
variance or a normal distribution. For count data (e.g. the number of insects
or bacterial colonies, or the prevalence of lesions), it is recommended that
the square root is applied to each observation before proceeding with ANOVA. This
usually results in a more constant variance.
Practical note: If the observed
values are small (less than 10) or there are many zeros, it is recommended, to
avoid problems with the square root of zero, use
the Anscombe transformation:
or a simplified, older correction that is also
effective:
before conducting the analysis.
2.2. Logarithm
Many biological variables (such as tree height, body weight and survival
time) follow a lognormal distribution. In these cases, taking the logarithm
(decimal or natural) of the variable helps stabilize the variance and
approximate the distribution to normality. One classic indication that this
transformation is needed is when the variance of the groups increases
proportionally with the mean.
2.3. Arc sine of the square
root of the proportion
If the variable is a proportion or percentage (e.g. the percentage of
seeds that germinate), ANOVA can only be applied directly if the proportions
strictly vary between 0.3 and 0.7. If many values fall outside this range, it
is recommended that the transformation is applied.
Y
= arcsin(√p).
3. Final considerations
For those unfamiliar with statistics, transforming data may seem like
suspicious 'manipulation'. It is not. It is a legitimate and widely accepted
technique that is often necessary when alternatives are unavailable.
Although modern software offers alternative methods, such as Welch's
test for one-way analysis of variance, transforming the original variable
may be the only feasible and robust approach to satisfy the model assumptions
for more complex analysis of variance models, such as split-plot designs or
hierarchical models.
Researchers must always be able to justify their chosen transformation
and, ideally, use the most common transformation in their field of study.
Important: even if the statistical analysis was performed using transformed data,
the descriptive results (means, standard errors, graphs, etc.) must be
presented on the original scale of the variable. To achieve this, the
transformation must be 'undone' (back-transformed) using the inverse function
of the original transformation.
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