The unequal variance t-test is an underused alternative to Student's t-test and the Mann–Whitney U test
The unequal variance t-test is an underused alternative to Student's t-test and the Mann-Whitney U test
Graeme D. Ruxton 0
0 Division of Environmental and Evolutionary Biology, Institute of Biomedical and Life Sciences , Graham Kerr Building , University of Glasgow , Glasgow G12 8QQ , United Kingdom
Often in the study of behavioral ecology, and more widely in science, we require to statistically test whether the central tendencies (mean or median) of 2 groups are different from each other on the basis of samples of the 2 groups. In surveying recent issues of Behavioral Ecology (Volume 16, issues 1-5), I found that, of the 130 papers, 33 (25%) used at least one statistical comparison of this sort. Three different tests were used to make this comparison: Student's t-test (67 occasions; 26 papers), Mann-Whitney U test (43 occasions; 21 papers), and the t-test for unequal variances (9 occasions; 4 papers). My aim in this forum article is to argue for the greater use of the last of these tests. The numbers just related suggest that this test is not commonly used. In my survey, I was able to identify tests described simply as ''t-tests'' with confidence as either a Student's t-test or an unequal variance t-test because the calculation of degrees of freedom from the 2 sample sizes is different for the 2 tests (see below). Hence, the neglect of the unequal variance t-test illustrated above is a real phenomenon and can be explained in several (nonexclusive ways) ways: 1. Authors are unaware that Student's t-test is unreliable when variances differ between underlying populations. 2. Authors are aware of this but consider their samples to have similar variances. 3. Authors believe than the Mann-Whitney U test can effectively substitute for Student's t-test when variances are unequal. 4. Because the t distribution tends to the normal distribution for large sample sizes, authors may consider that their sample sizes are sufficiently large for concerns about unequal variance and nonnormality of the samples to be ignored. Argument 4 relies on the central limit theorem and would require a combined sample size of at least 30 (Sokal and Rohlf 1987, p. 107); however, in my survey, the majority (47 out of 61) of tests for which sample sizes were provided had a combined sample size below 30. The fallacy of argument 3 has been demonstrated previously on several occasions (e.g., Kasuya 2001; Neuhauser 2002). To explore argument 1 further, imagine that we have 2 sample groups (labeled ''1'' and ''2,'' with means [l1 and l2], variances [s12 and s22], and sample sizes [N1 and N2]). For the unpaired Student's t-test, the t statistic is calculated as The Author 2006. Published by Oxford University Press on behalf of the International Society for Behavioral Ecology. All rights reserved. For permissions, please e-mail:
where the pooled variance sp2 is given by
The variances of the 2 samples are pooled in order to achieve
the best estimate of the (assumed equal) variances of the 2
populations. Hence, we can see the need for the underlying
assumption of equal population variances in this test. The
Student’s t-test performs badly when these variances are
actually unequal, both in terms of Type I and Type II errors (Zar
1996). Figure 1 suggests that unequal sample variances are
common in behavioral ecology. Although it is true that
unequal variances are less problematic if sample sizes are similar,
in practice, we often have quite unequal sample sizes (Figure 2).
Hence, I suggest that the Student’s t-test is frequently used in
behavioral ecology when one of its important underlying
assumptions is violated, and consequently, its performance is
The unequal variance t-test does not make the assumption
of equal variances. Coombs et al. (1996) presented measured
Type I errors obtained by simulated sampling from normal
distributions for the Student’s t-test and the unequal variance
t-test (their result are summarized in Table 1). In the
examples in Table 1, we see that the Type I error rate of the unequal
variance t-test never deviates far from the nominal 5% value,
whereas the Type I error rate for the Student’s t-test was over 3
times the nominal rate when the higher variance was
associated with the smaller sample size and less than a quarter the
nominal rate when the higher variance was associated with the
higher sample size. These results concur qualitatively with
other studies of these 2 tests (e.g., Zimmerman and Zumbo
1993). Notice that even when the variances are identical, the
unequal variance t-test performs just as effectively as the
Student’s t-test in terms of Type I error. The power of the unequal
variance t-test is similar to that of the Student’s t-test even
when the population variances are equal (e.g., Moser et al.
1989; Moser and Stevens 1992; Coombs et al. 1996). Hence,
I suggest that the unequal variance t-test performs as well as,
or better than, the Student’s t-test in terms of control of both
Type I and Type II error rates whenever the underlying
distributions are normal.
Let us now consider convenience of calculation: the
unequal variance t-test involves calculation of a t statistic that is
compared with the appropriate value in standard t tables. The
test statistic for the unequal variance t-test (t#) is actually
slightly simpler than that of the Student’s t-test:
However, the calculation of the degrees of freedom (v) is
more involved but not prohibitively so. For the Student’s t-test,
v ¼ N1 1 N2 2; for the unequal variance t-test, it is given
(e.g., Moser and Stevens 1992) by
Ruxton • Unequal variance t-test
In general, v calculated from Equation 4 will take a noninteger
value; it is conventional to round down to the nearest integer
before consulting standard t tables. Hence, the calculation of
the unequal variance t-test is straightforward. Further, the test
is available in several commonly used statistics packages: for
example, Excel, Minitab, SPSS, SAS, and SYSTAT. Hence, ease
of calculation is not a valid reason for choosing a Student’s
t-test over an unequal variance t-test.
The unequal variance t-test has no performance benefits
over the Student’s t-test when the underlying population
variances are equal. Hence, you might consider that an effective
way to conduct your analysis would be to perform an initial
test for homogeneity of variance and then perform either
a Student’s t-test when the variances are equal or an unequal
variance t-test when they are not. The problem with this
flexible approach is that the combination of this preliminary test
plus whichever of the subsequent tests is ultimately used
controls Type I error rates less well than simply always performing
an unequal variance t-test on every occasion (Gans 1992;
Moser and Stevens 1992), this is one reason why it is generally
unwise to decide whether to perform one statistical test on the
basis of the outcome of another (Zimmerman 2004 and
references therein). There are further reasons for not
recommending preliminary tests of variances (e.g., Markowski CA and
Markowski EP 1990; Quinn and Keough 2002, p. 42). Hence,
I suggest avoiding preliminary tests and adopting the unequal
variance t-test unless an argument based on logical, physical,
or biological grounds can be made as to why the variances
are very likely to be identical for the 2 populations under
It is important to remember that although the unequal
variance t-test is more reliable than the Student’s t-test in
Histogram of the ratio of the highest over the lowest sample size for
61 Student’s t-tests and Mann–Whitney U tests in my survey from
Behavioral Ecology for which sample sizes were provided. For ease of
presentation, the following four ratios were not plotted: 3.1, 3.3, 4.2,
terms of violation of the assumption of homogeneity of
variances, it is not necessarily any more reliable than the
Student’s t-test if the assumption of normality of the
underlying populations is violated. However, Zimmerman and Zumbo
(1993) argue that the unequal variance t-test performed on
ranked data performs just as well as the Mann–Whitney U test
(in terms of control of Type I errors) when variances are equal
and considerably better than the U test when variances are
unequal (see Table 2 for an example). This behavior was
found when tested with populations coming from 8 different
types of nonnormal distribution. Thus, Zimmerman and
Zumbo (1993) suggest that the unequal variance t-test can
effectively replace the Mann–Whitney U test if the data are
first ranked before the test is applied. There are alternatives
to the unequal variance t-test that perform even better, in
particular, being more robust to nonnormality in the
underling populations (e.g., Coombs et al. 1996; Keselman et al.
2004). However, I recommend the unequal variance t-test as
having the best combination of performance and ease of use.
I have used the name unequal variance t-test as this is its
most common name in the literature, you may also find in
referred to as the Welch test deriving from Welch (1938,
Calculated Type I error rate for the t-test and unequal variance t-test
with a nominal a value of 0.05 (adapted from Coombs et al. 1996)
Unequal 0.051 0.054 0.051
Calculated Type I error rate for the Mann–Whitney U test and the
unequal variance t-test performed on the ranked data from normal
distributions with a nominal a value of 0.05 (adapted from
Zimmerman and Zumbo 1993)
Unequal 0.049 0.052 0.051
1947). Welch actually proposed several ways to evaluate the
degrees of freedom, and the method I describe in Equations 4
and 5 is sometimes referred to as the Welch Approximate
Degrees of Freedom (APDF) test. Note that statistical
packages may use other methods for calculating the degrees of
freedom. You may also encounter the unequal variances t-test
called simply the unpooled variances t-test or Satterwaite’s test
or the Welch–Satterthwaite test, after Satterwaite (1946). You
may also find it called as the Smith/Welch/Satterwaite test,
acknowledging the work in Smith (1936).
The importance of considering whether or not to pool
variances extends beyond the simple case of comparing 2 groups.
Julious (2005) argues against the standard practice of using
the pooled variance across all groups when performing a
comparison between 2 groups from several used in an analysis of
variance. Indeed, Julious (2005) argues that using a pooled
variance across more than 2 groups can be even more serious
than the issues covered in this paper. No matter the number
of groups, the decision as to whether to pool or not also needs
careful consideration in the construction of randomization
tests as well as the analytic tests considered here.
IN CONCLUSION: A STEP-BY-STEP SUMMARY
If you want to compare the central tendency of 2 populations
based on samples of unrelated data, then the unequal
variance t-test should always be used in preference to the
Student’s t-test or Mann–Whitney U test. To use this test, first
examine the distributions of the 2 samples graphically. If
there is evidence of nonnormality in either or both
distributions, then rank the data. Take the ranked or unranked data
and perform an unequal variance t-test. Draw your
conclusions on the basis of this test. Note that some packages (e.g.,
SPSS) perform a Student’s t-test and unequal variances t-test
simultaneously and provide output for both. The
experimenter ought to have decided which test they consider most
appropriate beforehand and thus look at the output for that
test alone, ignoring the other.
In presenting the outcome of the unequal variance t-test,
provide a suitable reference for the adoption of the test and
its exact formulation (e.g., Moser et al. 1989 or this paper) as
well as providing the mean, variance, and number of samples
in each group, the calculated t# value, the calculated degrees
of freedom (v), and finally the P value.
Thanks to Steven Julious and 2 anonymous referees for helpful
comments on a previous version.
Address correspondence to G.D. Ruxton. E-mail: g.ruxton@bio.
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