Lecture
9
Analysis of Variance
Two Groups
Analysis of Variance, commonly
referred to as ANOVA (uhnoveuh), is the same as a between groups ttest when
used with two groups. It is a more general test, though, that allows one to
compare several groups at once, not just two. Instead of using the t statistic
as in the ttest, we use an F statistic. Why F? Because statisticians like to
use letters to to abbreviate so that everyone else will be confused about what
the heck they are talking about. That way they seem really smart. No, I'm mostly
kidding. The letter F is a tribute to the inventer of ANOVA, Sir Ronald Fisher, knighted for his accomplishments in
statistics. Fisher was a statistician who studied agricultural genetics among
other things. It turns out that the Ftest (or ANOVA) with two groups is
equivalent to the ttest. You'll get the same result with either. But the ANOVA
test is more general because it can be used in more complex studies that
compare more than two groups.
In an earlier lecture, I described
the ttest as a type of ratioa ratio between the group difference and
sampling variability (i.e., the standard error).
t
= difference between groups
sampling variability
And I said the standard error is
really based on the standard deviation, a measure of variability within the
sample. ANOVA is really based on the same idea, but Fisher conceptualized it
slightly differently. He thought of it as a ratio of two types of variances,
the variance between group means and overall variance in the sample.
F
= Variance Between Groups
Overall Variance
In the between groups ttest, we
examined the difference between two means (). Another way to think of the difference
between two means is as a type of variation among the means. Here we have just
two means, but a difference is the same thing as a variation. (Remember the
calculation of the variance of a group of numbers involves subtracting x from ?). If there are
several groups, their group means may differ or vary. The overall variance is
just the variation of scores in the sample. So, we could restate the Ftest
this way:
F
= Difference Among the Group Means
Difference Among the Scores
If the differences between the group
means is large relative to the amount of variability in the scores, the group
differences are probably significant. If, however, there is a lot of
variability in the scores, then the difference between (or among) the group
means will not seem so large, a nonsignificant difference.
Notation
The notation in the formulas begins to get a little more complex with ANOVA,
because we are planning on adding more groups. For some of you this is not a
problem. For others, the subscripts are a little overwhelming. If so, click here
for some detailed
explanation of the notation.
Steps and Formulas
for the TwoGroup ANOVA
The steps for conducting are really the same as with the other tests. You need
to know the hypotheses, then do the calculations to get the statistics, and
then compare to the tabled value to see if there is a significant difference.
Because the ANOVA is conceptualized
as a ratio of varianceshence the term Analysis of Variance, we need to
compute the variance of the scores and the variance of the means. The variance
of the scores is based on the Sum of Squares Within (SSW). The SSW is based on
how the scores vary around the mean in each of the groups. The Sum of Squares
Among the groups (SSA) is the squared deviations of the means from the total
meanhow much the means vary from each other. If you put these two components
togther, they add up to the Sum of Squares Total (SST). Here are the formulas:
Formula 
Name 
How To 
Concept 
Sum of Square Total 
Subtract each of the scores from the mean of the entire sample. Square each of those deviations. Add those up for each group, then add the two groups together. 
This is just like computing the variance. The purpose is to find the deviation of all the scores from the middle. We haven't yet divided by how many there are, but we will. 

Sum of Squares Among 
Each group mean is subtracted from the overall sample mean, squared, multiplied by how many are in that group, then those are summed up. For two groups, we just sum together two numbers. 
This is a measure of how much the means differ from one another. Its conceptualized a little differently, because it is thought of as the variation of each mean from the mean of the total sample. 

Sum of Squares Within 
Here's a shortcut. Just find the SST and the SSA and find the difference. What's left over is the SSW. 
If we calculated the SSW from scratch, it would be the deviation of the scores in each group from its group mean, then added together. Represents variation within groups. 
The next step is to turn each of
these sums into averages. Just as when the variance is computed, we want the
average deviation from the mean. So for these Sum of Squares, we want to divide
by approximately how many things we summed up. The degrees of freedom is based
on the number of groups or cases.
Formula 
Name 
How To 
Concept 
Mean Square Among Groups 
Divide the SSA by the number of groups minus 1. 
Average of the sum of squares among the groups. Represents the amount of difference between the groups. 

Mean Square Within 
Divide the SSW by the number of cases in the total sample minus the number of groups. 
Average of the sum of squares within the groups. Represents the amount of variation of the scores within the groups. 

There is no MST used 



The Ftest is just the ratio
of the MSA and MSW:
F
= MSA
MSW
It’s a ratio of the variance
between the group means relative to the amount of variation in the sample
(i.e., variation within each of the groups).
The Statistical
Test
Next we need to make our decision about whether the groups are signfircantly
different. Again, we calculate the Ftest for the sample, then compare it to a
table that tells us how extreme our sample F is relative to what we expect from
other randomly drawn samples. To do that, we need a new table, Table G in the
back of the book. To look up the value, we need to know the d.f. There are two,
based on the numbers we used to divide the SSA and SSW. So, the numerator
degrees of freedom is k  1 (referring to the top MSA and the top part of the F
ratio), and the denominator is N  k (referring to the MSW and the bottom part
of the F ratio).
Example
Let's redo the ttest we did in the the one of the earlier lectures (Between Groups ttest).
Remember the first born/second born example? Here's the data again.
Introversion score First Born 
Introversion score Second Born 



65 
5.4 
29.16 
61 
1.4 
1.96 
48 
11.6 
134.56 
42 
17.6 
309.76 
63 
3.4 
11.56 
66 
6.4 
40.96 
52 
7.6 
57.76 
52 
7.6 
57.76 
61 
1.4 
1.96 
47 
12.6 
158.76 
53 
6.6 
43.56 
58 
1.6 
2.56 
63 
3.4 
11.56 
65 
5.4 
29.16 
70 
10.4 
108.16 
62 
2.4 
5.76 
65 
5.4 
29.16 
64 
4.4 
19.36 
66 
6.4 
40.96 
69 
9.4 
88.36 


The mean of all the 20 participants
is .





Looking up in Table G under
Numberator DF = 1 and Denomator d.f. = 18, the critical value needed for
significance is . So, our test is not significant.
Relation between
ttest and Ftest
Before, I stated that the t and Ftests are equivalent. They will lead to the
same answer. The number you caluclate is slightly different. It turns out that (or ). So, we should
get the tvalue we obtained before if we take the square root of F. .
Oops, this is not the tvalue I
obtained, but it is the tvalue I should have obtained. Well, even your
instructor makes mistakes! In the ttest example I made an error when
calculating the pooled variance. I used the standard deviation instead of the
variance, where and were called for in the formula. The error has now been
corrected in that lecture. You should always be double checking my work, never trust your
instructor!!well maybe you shouldn't go that far.
Interpretation
The ratio of the amonggroups
variance to the withingroups variance was not large enough for significance.
In other words, there was a nonsignificant difference between the groups.
Thus, we assume, at least for the moment, that there is not a difference in the
introvertedness of first and second born children in the population. If the
Ftest was significant, we would conclude the birth order does have an effect.
Three or More
Groups
I'm not going to go into much detail about ANOVA with more than two groups ,
because the book gives an example of that. I do want to note a couple of
things. ANOVA is sometimes referred to as an "omnibus" test, meaning
that it is an overall test of differences among the means. When there are three
or more groups, and we find significance among the group means, we do not know
which groups are different from which. The overall, omnibus, test may be
significant if just one mean is different from the rest. Or all group means may
be significantly different from one another. A signficant ANOVA does not
distinguish between these possibilites, it merely indicates there are some
differences.
Alpha Inflation
So, often there are follow up tests conducted to see, specifically, which means
are different from which other means. One way to do this would be to conduct
some separate ttests after one obtains a significant ANOVA, comparing two
group means at a time. This is a big nono in statistics. There are problems
with this approach, although occaisionally one will see a published article that
does this. One reason it is a nono to conduct separate ttests after a
signficant ANOVA is because of something called alpha inflation. Alpha
inflation (sometimes also called familywise error, or experimentwise error)
occurs because multiple tests are conducted on the same data set. When too many
tests are conducted, the original alpha value for each test is actually higher
than expected (this is a probability thing). Instead of a=.05
it might actually be higher than that. This implies that the Type I error rate
is higher than expected, and our chance of finding a significant difference in
error is greater.
The remedy is to use a slightly
modified version of the ttest to correct for the problem. There are a lot of
these tests (often called posthoc tests) out there. Some examples are the
Bonferroni, the Scheffe, the NeumanKeuls, and the Tukey. The Tukey test, also
called the Honest Significant Difference test (HSD) is the best. The HSD
compares all possible pairs of means withouth inflating alpha. Daniel provides
a nice discussion of this test, so I'm not going to go over it here. So, if you
need to find out which group means are different from one another, you should
use the Tukey HSD test.
For an example SPSS output, click here.