In this video I want to take a look at the relationship between the independent
samples t-test and the one-way ANOVA. Now it may seem that these tests are quite
different, it may look like that on the surface, after all they're located in
different chapters in intro stats texts and so on, but we'll take a look today at
how these tests are actually very similar. So let's go and start with this
example here on your screen. Here we have a two group problem where we have volume,
where we have 1 and 2, where this corresponds to either a no music
condition, or no volume condition, and a high-volume condition. So in this
hypothetical study we had 20 people, 10 were randomly assigned to study for an
exam with no music, and then 10 were randomly assigned to study for an exam
under high volume. And then the dependent variable here is the exam scores. So our
independent variable is type of volume, where we have none or high, and then our
dependent is exam scores. So let's start by running an independent samples t test.
So do that we go to Analyze, Compare Means and then select Independent-
Samples T Test. Here we'll move our dependent variable, exam scores, into the
Test Variable(s) box. We'll move volume into the Grouping Variable box. And, as we can see
here, we have 1s and 2s, so under Define Groups, group 1 was assigned a 1 and
group 2 was assigned a 2. Click Continue and then OK. And then here's the
results for the independent samples t test. Before we take a look at that,
let's go back and now run the One-Way ANOVA. So we'll go to Analyze, Compare
Means and then One-Way ANOVA. This time we move exam scores into what's called
the Dependent List, and volume into the Factor box, factor standing for the
independent variable. Go ahead and click OK. And then now here's the ANOVA
results. Now the first thing here that I want you to notice is the p-value. Look
at the p-value for the Independent Samples T Test under Equal variances
assumed, we have a p-value of .014. Now here, in the ANOVA results, if you
look at this p-value,
notice what it is. It's the exact same, .014. And just to make sure, let's go ahead
and double-click on this table here,
double-click on the p-value and we can see it's .013996, and then we'll do the
same here for the ANOVA table, and this is also .013996. Well, if the p-values
are the exact same, that indicates that we're really running the same test here.
So the first thing we can see is that we get the exact same p-value running our
two groups under the independent t as we do when we run our two groups under the
one way ANOVA. But there's something else here. You might notice here you see
a t of 2.722 and an F of 7.407. Well, despite the p-values looking the same, we
can see here that t and F definitely are not equal.
So, in summary, we had an F of 7.407 and a t of 2.722. Well, while those are not equal,
if we square the t, we would actually get the value for F, within rounding error,
it's off by a couple one-thousandths of a place just because SPSS rounded. But this
relationship does hold: F is equal to t- squared, within rounding error. Or in
other words, when we have two groups, F is equal to t-squared.
So we saw that the p-values were the same in SPSS and now we can see that F is equal
to t-squared.
So therefore we could summarize the results as follows here,
the one-way ANOVA and t test are equivalent With two groups. They will
provide the same answer or decision in terms of the hypothesis test (as they
produce the exact same p value). So, in other words, if you reject the null with
the ANOVA, you will reject the null with the t test. However, this property only
applies with two groups. So, for example, if we had three groups and we ran the
ANOVA we got a p-value is let's say .003, for example. If we have three groups
and we run the ANOVA, that would require that we run three separate independent
samples t tests, group 1 vs. 2, group 1 vs. 3, and group 2 vs.
3. Now if our p-value for the ANOVA was .003, that doesn't mean in any way,
shape, or form, that the p-value for any or all of those t-tests will be .003. So
the relationship, once again, only holds when there's two groups. But with two
groups, we can run the independent samples t, or we can run the one-way
ANOVA, as they produce the exact same p value, meaning you'll draw the same
conclusion about the hypothesis test. And once again the relationship between F
and t is that F equals t-squared with two groups.
This concludes the video for looking at the relationship between the independent t
and the ANOVA with two groups.
Thanks for watching.