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This presentation is on the One-Way ANOVA. ANOVA stands for Analysis of Variance. It
is an analytical technique that's used a lot in experimental tests. In the social sciences,
psychologists use the ANOVA technique a lot. ANOVA is appropriate for testing whether or
not several sample mean values for groups that are drawn from a larger sample are statistically
different from one another. Thus, it goes beyond the independent samples t-test, which
tests whether or not two sample means differ significantly from one another. In other words,
ANOVA is even more flexible in its use than the independent samples t-test.
The logic of the One-Way ANOVA is to check to see how much the observations vary within
each group and compare that variation to how much the observations vary between the groups.
Essentially, if the mean values between the groups vary more than we expect based on the
variability of observations within the groups, then we conclude that the sample means, together,
are not equal. That is the alternative hypothesis. The null hypothesis is that the sample means,
together, are equal.
The test statistic is an F-test. This is a new test for us. The F-test is a distribution
of variances, and as you can see from the graph of its distribution, it has a positive
skew to it, because variances are skewed, whereas means are not, inherently, skewed.
Recall that if our sample size of means is large enough, then the distribution looks
normal.
Let's look at an example of the One-Way ANOVA in order to understand how it works. Our example
is the "Workweek in the United States in 1998". The data come from a large survey of U.S.
residents called the General Social Survey, from the University of Chicago. It's one of
the most important surveys in the U.S. on societal attitudes in the country.
As you can see from the example, we are asking the question of the number of hours that the
respondent worked in the past week. Further, we have our data broken down by five educational
groups, ranging from those respondents who have less than a high school education, to
those who have a high school education, to those who have a junior college degree, to
those with a bachelor's degree, and finally to those who have a graduate degree. Notice
the column headed by "Mean". That is the mean of the number of hours worked last week for
those respondents in each educational category. If we track from top to bottom in the column,
then we see that there's a gradual increase in the number of hours worked per week, all
the way up to 50.27 hours for those with graduate degrees.
If we look at the box-and-whiskers plot below the descriptive statistics box, we see that
the mean values are plotted out, as are their confidence intervals, which means that if
the sample were given an additional 100 times, then 95 times the mean value would fall within
the lower bound and upper bound range of values. The
key thing to look for is: do the whiskers overlap? If they do, then there is the possibility
that one of the group mean values could be the same as at least one other group mean
value. If they do not, then there is a very remote statistical possibility of the group
mean values overlapping. For the null hypothesis to be accepted, the group mean values' confidence
intervals must all overlap. For the alternative hypothesis to be accepted, at least one group
mean value's confidence interval must not overlap with at least one other group mean
value's confidence interval.
As we examine the box-and-whisker plots, we can see that the graduate degree whisker does
not overlap either the less than high school whisker or the high school degree whisker.
This means that in all likelihood, we will reject the null hypothesis and accept the
alternative hypothesis. But the definitive answer can only come from the One-Way ANOVA
test, to which we will now proceed.
You can see that the ANOVA box contains information about the "Mean Square", then the F-test,
then the "Sig." Let's talk about each of these values. The Mean Square refers to the average
sum of squared standard deviations. The Mean Square Within Groups refers to the average
sum of squared standard deviations from the mean value within each group (less than high
school education, high school education, junior college education, bachelor's degree, graduate
degree). The Mean Square Between Groups refers to the average sum of squared standard deviations
from the mean value between the groups (less than high school education, and so on). If
the mean square between the group mean values is really large, then that indicates there's
a big difference between the groups. If the mean square between the groups is much larger
in proportion to the mean square within the groups, then that means that each group is
highly distinct from one another. If the mean square between the groups is much smaller
in proportion to the mean square within the groups, then that means that each group is
not distinct at all from the other groups, with wide variation in values within each
group.
The F-test value is simply the division of the Between Groups Mean Square by the Within
Groups Mean Square. Try it yourself: 456.479 divided by 125.201 equals 3.646.
Now, I would ordinarily say that we should look to a table of critical values for the
F-test in order to make the correct decision on accepting or rejecting the null hypothesis.
However, we don't have such a table in the *** I book, so instead we will make use
of the "Sig." statistic, which is the significance level of the F-test statistic, which in our
example is .006. It can be interpreted in this way: given our sample size of five groups
and 741 cases we will incorrectly reject the null hypothesis six one-thousandths of a time
with a calculated F-test statistic of 3.646. (We can tell the N based on the "Total df"
value + 1.) Our standard for feeling very confident in rejecting the null hypothesis
is Sig. = .05, or that we would incorrectly reject the null hypothesis five times out
of 100. Any Sig. that is lower than .05 means that we will correctly reject the null hypothesis,
with 95 percent confidence. This also goes for significance = .05.
Thus, there are statistically significant differences between the group mean values
for the number of hours worked in the previous week, in 1998. Graduate degree recipients
report more hours worked. From personal experience, I've worked a lot of hours!
We thus conclude this presentation of the One-Way ANOVA.