Analysis of Variance (ANOVA)
W&W, Chapter 10
Introduction
Last time we learned about the chi square
test for independence, which is useful for
data that is measured at the nominal or
ordinal level of analysis.
If we have data measured at the interval
level, we can compare two or more
population groups in terms of their
population means using a technique called
analysis of variance, or ANOVA.
Completely randomized design
Population 1
Mean = 1
Variance=12
Population 2….. Population k
Mean = 2 …. Mean = k
Variance=22 … Variance = k2
We want to know something about how the
populations compare. Do they have the same
mean? We can collect random samples from each
population, which gives us the following data.
Completely randomized design
Mean = M1
Variance=s12
Mean = M2 ..…
Variance=s22 ….
N1 cases
N2 cases
….
Mean = Mk
Variance = sk2
Nk cases
Suppose we want to compare 3 college majors in a
business school by the average annual income
people make 2 years after graduation. We collect
the following data (in $1000s) based on random
surveys.
Completely randomized design
Accounting
27
22
33
25
38
29
Marketing
23
36
27
44
39
32
Finance
48
35
46
36
28
29
Completely randomized design
Can the dean conclude that there are
differences among the major’s incomes?
Ho: 1 = 2 = 3
HA: 1  2  3
In this problem we must take into account:
1) The variance between samples, or the actual
differences by major. This is called the sum of
squares for treatment (SST).
Completely randomized design
2) The variance within samples, or the
variance of incomes within a single major.
This is called the sum of squares for error
(SSE).
Recall that when we sample, there will always
be a chance of getting something different
than the population. We account for this
through #2, or the SSE.
F-Statistic
For this test, we will calculate a F statistic,
which is used to compare variances.
F = SST/(k-1)
SSE/(n-k)
SST=sum of squares for treatment
SSE=sum of squares for error
k = the number of populations
N = total sample size
F-statistic
Intuitively, the F statistic is:
F = explained variance
unexplained variance
Explained variance is the difference between
majors
Unexplained variance is the difference based
on random sampling for each group (see
Figure 10-1, page 327)
Calculating SST
SST = ni(Mi - )2
 = grand mean or =  Mi/k or the sum of
all values for all groups divided by total
sample size
Mi = mean for each sample
k= the number of populations
Calculating SST
By major
Accounting
M1=29, n1=6
Marketing
M2=33.5, n2=6
Finance
M3=37, n3=6
 = (29+33.5+37)/3 = 33.17
SST = (6)(29-33.17)2 + (6)(33.5-33.17)2 +
(6)(37-33.17)2 = 193
Calculating SST
Note that when M1 = M2 = M3, then SST=0
which would support the null hypothesis.
In this example, the samples are of equal size,
but we can also run this analysis with
samples of varying size also.
Calculating SSE
SSE = (Xit – Mi)2
In other words, it is just the variance for each sample
added together.
SSE = (X1t – M1)2 + (X2t – M2)2 +
(X3t – M3)2
SSE = [(27-29)2 + (22-29)2 +…+ (29-29)2]
+ [(23-33.5)2 + (36-33.5)2 +…]
+ [(48-37)2 + (35-37)2 +…+ (29-37)2]
SSE = 819.5
Statistical Output
When you estimate this information in a computer
program, it will typically be presented in a table as
follows:
Source of
Variation
Treatment
Error
Total
df
k-1
n-k
n-1
Sum of
squares
SST
SSE
SS=SST+SSE
Mean
squares
F-ratio
MST=SST/(k-1) F=MST
MSE=SSE/(n-k)
MSE
Calculating F for our example
F = 193/2
819.5/15
F = 1.77
Our calculated F is compared to the critical
value using the F-distribution with
F, k-1, n-k degrees of freedom
k-1 (numerator df)
n-k (denominator df)
The Results
For 95% confidence (=.05), our critical F is
3.68 (averaging across the values at 14 and
16
In this case, 1.77 < 3.68 so we must accept the
null hypothesis.
The dean is puzzled by these results because
just by eyeballing the data, it looks like
finance majors make more money.
The Results
Many other factors may determine the salary
level, such as GPA. The dean decides to
collect new data selecting one student
randomly from each major with the
following average grades.
New data
Average Accounting
A+
41
A
36
B+
27
B
32
C+
26
C
23
M(t)1=30.83
 = 33.72
Marketing
45
38
33
29
31
25
M(t)2=33.5
Finance M(b)
51
M(b1)=45.67
45
M(b2)=39.67
31
M(b3)=30.83
35
M(b4)=32
32
M(b5)=29.67
27
M(b6)=25
M(t)3=36.83
Randomized Block Design
Now the data in the 3 samples are not
independent, they are matched by GPA
levels. Just like before, matched samples
are superior to unmatched samples because
they provide more information. In this case,
we have added a factor that may account for
some of the SSE.
Two way ANOVA
Now SS(total) = SST + SSB + SSE
Where SSB = the variability among blocks,
where a block is a matched group of
observations from each of the populations
We can calculate a two-way ANOVA to test
our null hypothesis. We will talk about this
next week.