As all statistical models, ANOVAs have a number of assumptions that should hold for valid inferences. These assumptions are:
The most important assumption generally is the i.i.d. assumption (i.e., if it does not hold, the inferences are likely invalid), specifically the independent part. This assumption cannot be tested empirically but needs to hold on conceptual or logical grounds. For example, in an ideal completely between-subjects design each observation comes from a different participant that is randomly sampled from a population so we know that all observations are independent. Often, we collect multiple observations from the same participant in a within-subject or repeated-measures design. To ensure the i.i.d. assumption holds in this case, we need to specify an ANOVA with within-subject factors. However, if we have a data set with multiple sources of non-independence – such as participants and items – ANOVA models cannot be used but we have to use a mixed model.
The other assumptions can be tested empirically, either graphically
or using statistical assumption tests. However, there are different
opinions on how useful statistical assumptions tests are when done in an
automatic manner for each ANOVA. Whereas this is the position taken in
some statistics books, this runs the risk of reducing the
statistical analysis to a “cookbook” or “flowchart”. Real life data
analysis is often more complex than such simple rules. Therefore, it is
often more productive to explore ones data using both descriptive
statistics and graphical displays. This data exploration should allow
one to judge whether the other ANOVA assumptions hold to a sufficient
degree. For example, plotting ones ANOVA results using
afex_plot
and including a reasonable display of the
individual data points often allows one to judge both the homogeneity of
variance and the normality of the residuals assumption.
Let us take a look at all three empirically testable assumptions in detail. ANOVAs are often robust to light violations to the homogeneity of variances assumption. If this assumption is clearly violated, we have learned something important about the data, namely variance heterogeneity, that requires further study. Some further statistical solutions are discussed below.
If the main goal of an ANOVA is to see whether or not certain effects are significant, then the assumption of normality of the residuals is only required for small samples, thanks to the central limit theorem. As shown by Lumley et al. (2002), with sample sizes of a few hundred participants even extreme violations of the normality assumptions are unproblematic. So mild violations of this assumptions are usually no problem with sample sizes exceeding 30.
Finally, the default afex
behaviour is to correct for
violations of sphericity using the Greenhouse-Geisser correction.
Whereas this default may in some situation produce a small loss in
statistical power, this seems preferable to a situation in which
violations of sphericity are overlooked and tests become
anti-conservative (i.e., more false positive results).
Thus, my position as the afex
developer is that an
appropriate exploratory data analysis is often better than just blindly
applying statistical assumption tests. Nevertheless, assumption tests
are of course an important tool in the statistical toolbox and can be
helpful in many situations. Thus, I am thankful to Mattan S. Ben-Shachar
who has provided them for ANOVAs in afex
. The following
text provides his introduction to the assumption tests based on the performance
and see
packages.
afex
comes with a set of built-in functions to help in
the testing of the assumptions of ANOVA design. Generally speaking, the
testable assumptions of ANOVA are^{1}:
performance::check_homogeneity()
.performance::check_sphericity()
.performance::check_normality()
.What follows is a brief review of these assumptions and their tests.
This assumption, for between subject-designs, states that the within group errors all share a common variance around the group’s mean. This can be tested with Levene’s test:
data(obk.long, package = "afex")
o1 <- aov_ez("id", "value", obk.long,
between = c("treatment", "gender"))
## Warning: More than one observation per design cell, aggregating data using `fun_aggregate = mean`.
## To turn off this warning, pass `fun_aggregate = mean` explicitly.
## OK: There is not clear evidence for different variances across groups (Levene's Test, p = 0.350).
These results indicate that homogeneity is not significantly violated.
ANOVAs are generally robust to “light” heteroscedasticity, but there
are various other methods (not available in afex
) for
getting robust error estimates.
Another alternative is to ditch this assumption altogether and use
permutation tests (e.g. with permuco
)
or bootstrapped estimates (e.g. with boot
).
data("fhch2010", package = "afex")
a1 <- aov_ez("id", "log_rt", fhch2010,
between = "task",
within = c("density", "frequency", "length", "stimulus"))
## Warning: More than one observation per design cell, aggregating data using `fun_aggregate = mean`.
## To turn off this warning, pass `fun_aggregate = mean` explicitly.
We can use check_sphericity()
to run Mauchly’s test of
sphericity:
## Warning in summary.Anova.mlm(object$Anova, multivariate = FALSE): HF eps > 1 treated as 1
## Warning: Sphericity violated for:
## - length:stimulus (p = 0.021)
## - task:length:stimulus (p = 0.021).
We can see that both the error terms of the
length:stimulus
and task:length:stimulus
interactions significantly violate the assumption of sphericity at
p = 0.021. Note that as task
is a between-subjects
factor, both these interaction terms share the same error term!
afex
offers both the Greenhouse-Geisser (which is used by
default) and the Hyunh-Feldt corrections.emmeans
, a multivariate
model can be used, which does not assume sphericity (this is used by
default since afex
1.0).Both can be set globally with:
The normalicy of residuals assumption is concerned with the errors that make up the various error terms in the ANOVA. Although the Shapiro-Wilk test can be used to test for deviation from a normal distribution, this test tends to have high type-I error rates. Instead, one can visually inspect the residuals using quantile-quantile plots (AKA qq-plots). For example:
data("stroop", package = "afex")
stroop1 <- subset(stroop, study == 1)
stroop1 <- na.omit(stroop1)
s1 <- aov_ez("pno", "rt", stroop1,
within = c("condition", "congruency"))
## Warning: More than one observation per design cell, aggregating data using `fun_aggregate = mean`.
## To turn off this warning, pass `fun_aggregate = mean` explicitly.
If the residuals were normally distributed, we would see them falling close to the diagonal line, inside the 95% confidence bands around the qq-line.
We can further de-trend the plot, and show not the expected quantile, but the deviation from the expected quantile, which may help reducing visual bias.
Wow! The deviation from normalicy is now visually much more pronounced!
As with the assumption of homogeneity of variances, we can resort to using permutation tests for ANOVA tables and bootstrap estimates / contrasts.
Another popular solution is to apply a monotonic transformation to the dependent variable. This should not be done lightly, as it changes the interpretability of the results (from the observed scale to the transformed scale). Luckily for us, it is common to log transform reaction times, which we can easily do^{2}:
## Warning: More than one observation per design cell, aggregating data using `fun_aggregate = mean`.
## To turn off this warning, pass `fun_aggregate = mean` explicitly.
Success - after the transformation, the residuals (on the log scale) do not deviate more than expected from errors sampled from a normal distribution (are mostly contained in the 95%CI bands)!
There is also the assumptions that (a) the model is correctly specified and that (b) errors are independent, but there is no “hard” test for these assumptions.↩︎
But note ANOVA no longer tests if any differences between the means is significantly different from 0, but if any ratio between the means is significantly different from 1.↩︎