Levene's test for equality of variances is a test of whether the variances of
two samples or groups are approximately equal, or homogeneous. The independent
samples t-test assumes that the variances of the sample groups are
approximately equal, or that the samples have homogeneity of variance. Let's break
down that word "homogeneity". The word "homo" means same. "Geneity" refers to the genus,
or the nature, or the type, so homogeneity means of the same nature. Variance of
course refers to the variability in each group.
Therefore, homogeneity of variance means that the variance in the group's is of
the same nature, or that the two groups have the same kind of variability. The
variance does not have to be precisely equal, just close enough. So how do you
know if the variance is close enough to be called homogeneous or not? We use a
test called Levene's test for equality of variance. Levene's test for equality
of variance (which I will just call Levine's test) tests whether the
variances of two samples are approximately equal, so it tests our
assumption of homogeneity of variance. Like any hypothesis test, Levene's test
starts with a null hypothesis. In this case, the null hypothesis is that there
is no difference between the variance of the first group and the variance of the
second group. Unlike a t-test, in which we are testing whether the mean of the
experimental group is different than the mean of the control group after a
treatment, and we expect or perhaps even want those means to be different, with
Levene's test we want the variances to be the same. We would like Levene's test
to be non-significant. That is the assumption of the
independent samples t-test. The group variances are the same. They do
not differ. So we want Levene's tests to be non-significant because we do not
want the variances to be different. In SPSS, a Levene's test is conducted
automatically anytime you do an independent samples t-test. Levene's test
is going to be included in the output for that t-test. Levene's test is an F
test, the same kind of test as an ANOVA, so we could interpret the significance
value exactly the same as we would for any hypothesis test.
Check this box labeled "sig" if the significance is greater than .05,
i.e. non-significant, the variances are not significantly different, so equal
variances are assumed. Now in this case, the probability is .363. The assumption
of homogeneity of variance has been met. The variances are equal. They're not
different, or at least they're close enough. Since the assumption has been met,
we will interpret the t-test using the top line equal variances assumed. On the
other hand, if the significance value is less than .05, i.e., it is
significant, then the variances are statistically significantly different, so
equal variances are not assumed. In this case, the probability equals .042
the assumption of homogeneity of variance has been violated, therefore
we would interpret the t-test on the bottom line, equal variances not assumed.
So remember that you, as a researcher, can set the alpha level to the value that is
most important to your research. Therefore, many researchers choose to
interpret Levene's test using a cut-off significance criteria of .01 or even
.001 instead of .05. We want the
groups to be really screwy before we start making adjustments to the tests.
Using a more restrictive alpha level like .001 for testing
assumptions would be especially useful when you have a large sample size for
your t-test. Any hypothesis test is more likely to be statistically significant
when you have a large sample size, including the Levene's test. Now you
should remember that as long as your total sample size is greater than 30, and
the sample size in the first sample is approximately equal to the sample size
of the second sample, then the t-test is robust to violations of homogeneity of
variance. As I mentioned, a Levene's test will be automatically included in the
output anytime you do an independent samples t-test. I'm going to show you how
to do the t-test in the next video, but I want to show you that you can do a
Levene's test in SPSS anytime you like. I am using the job satisfaction data set that
I will use for correlation and regression. We are going to do a Levene's
test on job satisfaction between men and women. Go to analyze >
descriptive statistics > explore. Move "job satisfaction" into the dependents list,
and "gender" into the factor list. This will create two groups based on gender
for the job satisfaction variable. This time we're going to turn off some of the
default output. Click on plots > turn off the box plots > and uncheck stem-and-leaf.
Under spread versus level with Levene's test, choose untransformed. Click continue
and ok. Now most of this output we will not need. Scroll down until you find
Levene's test. We are interested in the test based on the mean, which we can see
is non significant. P equals .822 the variances are equal, or at least,
close enough. To make sure that this is working, let's do an independent samples
t-test. Go to analyze > compare means > independent samples t-test. Move "job
satisfaction" into the test variables box, and "gender" into grouping variables. Click
on define groups, and define the groups as one and two - for male and female. Click
continue and okay. In the output, notice that we have two
sections. Now remember, anytime we have a group of numbers we can calculate a mean
and a standard deviation, or variance. What SPSS has given us are two types of
hypothesis tests. These are the results for a test of equality of variances,
this is Levene's test, and we would like the variances to be the same or
homogeneous. These are the results for the test for equality of means. This is
the t-test, which typically we would prefer to be
different, indicating that our intervention worked. The Levene's test f
value was .051, with a significance or p-value of 0.822,
again non-significant, so the variances can be assumed to be equal. Now let's
check our previous output. Scroll up and you will see that the Levene statistic
based on the mean was .051 with a significance of .822,
exactly the same, although here we are given the degrees of freedom for the f
test (1, 198) now scroll back down. Although we are not given the degrees of
freedom with the t-test output, we can figure them out pretty
easily. The degrees of freedom between is always one, because the t-test only
compares two groups. The degrees of freedom within is the same as the
degrees of freedom for the t-test itself, 198.
We would write up the Levene statistic as F(1,198) = .051, p = .822.