One Way ANOVA.pdf

One-Way Analysis of Variance
Note: Much of the math here is tedious but straightforward. We’ll skim over it in class but you
should be sure to ask questions if you don’t understand it.
I. Overview
A. We have previously compared two populations, testing hypotheses of the form
H0: µ1 = µ2
HA: µ1 ≠ µ2
But in many situations, we may be interested in more than two populations.
Examples:
T Compare the average income of blacks, whites, and others.
T Compare the educational attainment of Catholics, Protestants, Jews.
B. Q: Why not just compare pairwise - take each possible pairing, and see
which are significant?
A: Because by chance alone, some contrasts would be significant. For example,
suppose we had 7 groups. The number of pairwise combinations is 7C2 = 21. If α = .05, we
expect one of the differences to be significant.
Therefore, you want to simultaneously investigate differences between the means
of several populations.
C. To do this, you use ANOVA - Analysis of Variance. ANOVA is appropriate
when
T You have a dependent, interval level variable
T You have 2 or more populations, i.e. the independent variable is categorical. In
the 2 population case, ANOVA becomes equivalent to a 2-tailed T test (2 sample tests, Case II,
σ's unknown but assumed equal).
D. Thus, with ANOVA you test
H0: µ1 = µ2 = µ3 = ... = µJ
HA: The means are not all equal.
E. Simple 1-factor model: Suppose we want to compare the means of J different
populations. We have j samples of size Nj. Any individual score can be written as follows:
yij = µ + τj + εij, where j = 1, J (# groups) and i = 1, 2, ..., Nj
That is, an observation is the sum of three components:
1. The grand mean µ of the combined populations. For example, the overall
average income might be $15,000.
One-Way Analysis of Variance - Page 1

2. A treatment effect τj associated with the particular population from which
the observation is taken; put another way, τj is the deviation of the group mean from the overall
mean. For example, suppose the average White income is $20,000. Then τwhites = $5,000.
3. A random error term εij. This reflects variability within each population.
Not everyone in the group will have the same value. For example, the average white income
might be $20,000, but some whites will make more, some will make less. (For a white who
makes $18,000, εij = -2,000.)
F. An alternative way to write the model is
yij = µj + εij,
where µj = mean of the jth population = µ + τj.
G. We are interested in testing the hypothesis
H0: µ1 = µ2 = µ3 = ... = µJ
But if the J means are equal, this means that µj = µ, which means that there are no
treatment effects. That is, the above hypothesis is equivalent to
H0: τ1 = τ2 = τ3 = ... = τJ = 0
H. Estimating the treatment effects: As usual, we use sample information to
estimate the population parameters. It is pretty simple to estimate the treatment effects:
y
-
y
=
N
T
N
y
y
N
y
=
y
= j
j
j
j
-
=
,
=
=
=
, j
j
j
A
j
ij
N
1
=
i
j
j
ij
N
1
=
i
J
1
=
j
µ
µ
τ
µ
µ ˆ
ˆ
ˆ
ˆ
ˆ
∑
∑
∑
Example: A firm wishes to compare four programs for training workers to perform a certain
manual task. Twenty new employees are randomly assigned to the training programs, with 5 in
each program. At the end of the training period, a test is conducted to see how quickly trainees
can perform the task. The number of times the task is performed per minute is recorded for each
trainee, with the following results:

Observation Program 1 Program 2 Program 3 Program 4
1 9 10 12 9
2 12 6 14 8
3 14 9 11 11
4 11 9 13 7
5 13 10 11 8
TAj = Σyij 59 44 61 43
j
µ̂ = TAj/Nj
11.8 8.8 12.2 8.6
Estimate the treatment effects for the four programs.
Solution. Note that ΣΣyij = 207, so µ̂ = 207/20 = 10.35. Since µ
µ
τ ˆ
ˆ
ˆ −
= j
j , we get
1
ˆ
τ = 11.8 - 10.35 = 1.45,
2
ˆ
τ = 8.8 - 10.35 = -1.55,
3
ˆ
τ = 12.2 - 10.35 = 1.85,
4
ˆ
τ = 8.6 - 10.35 = -1.75
I. Computing the treatment effects is easy - but how do we test whether the
differences in effects are significant???
Note the following:
Total
MS
=
Total
DF
Total
SS
=
1
-
N
)
y
-
y
(
=
s
=
s
2
ij
2
2
total
∑
∑
where SS = Sum of squares (i.e. sum of the squared deviations from the mean), DF = degrees of
freedom, and MS = Mean square. Also,
Between
SS
+
Within
SS
=
Total
SS
Where
Residual
SS
=
Errors
SS
=
Within
SS
=
=
)
y
-
y
( 2
ij
2
j
ij ε
ˆ
∑
∑
∑
∑
Explained
SS
=
Between
SS
=
N
=
)
y
y
( 2
j
j
j
2
j
i
j
2
j
i
j
= τ
τ ˆ
ˆ ∑
∑
∑
∑
∑ −

SS Within captures variability within each group. If all group members had the same score, SS
Within would equal 0. It is also called SS Errors or SS Residual, because it reflects variability
that cannot be explained by group membership. Note that there are Nj degrees of freedom
associated with each individual sample, so the total number of degrees of freedom within = Σ(Nj
- 1) = N - J.
SS Between captures variability between each group. If all groups had the same mean, SS
Between would equal 0. The term SS Explained is also used because it reflects variability that is
“explained” by group membership. Note that there are J samples, one grand mean, hence DF
Between = J - 1.
We further define
Variance
Total
=
1
-
N
Total
SS
=
1
-
N
Between
SS
+
Within
SS
=
Total
MS
,
1
-
J
Between
SS
=
Between
DF
Between
SS
=
Between
MS
,
J
-
N
Within
SS
=
Within
DF
Within
SS
=
Within
MS
Proof (Optional): Note that
y
-
y
+
y
-
y
=
y
-
y
and
,
y
+
y
-
y
=
y
j
j
ij
ij
j
j
ij
ij
We simply add and subtract y j. Why do we do this? Note that j
ij y
y − = deviation of the
individual's score from the group score = ij
ε
ˆ ; and y
y j − = deviation of the group score from the
total score = j
τ
ˆ . Hence,
τ
ε
τ
ε
τ
ε ˆ
ˆ
ˆ
ˆ
ˆ
ˆ j
ij
2
j
2
ij
2
j
ij
2
j
j
ij
2
ij 2
+
+
=
)
+
(
=
)
y
y
+
y
y
(
=
)
y
y
(
=
Total
SS ∑
∑
∑
∑
∑
∑
∑
∑
−
−
∑
∑
−
∑
∑
Let us deal with each term in turn:
Residual
SS
=
Errors
SS
=
Within
SS
=
=
)
y
-
y
( 2
ij
2
j
ij ε
ˆ
∑
∑
∑
∑
SS Within captures variability within each group. If all group members had the same score, SS
Within would equal 0. It is also called SS Errors or SS Residual, because it reflects variability

that cannot be explained by group membership. Note that there are Nj degrees of freedom
associated with each individual sample, so the total number of degrees of freedom within = Σ(Nj
- 1) = N - J.
Explained
SS
=
Between
SS
=
N
=
)
y
y
( 2
j
j
j
2
j
i
j
2
j
i
j
= τ
τ ˆ
ˆ ∑
∑
∑
∑
∑ −
(The third equation is valid because all cases within a group have the same value for y j.) SS
Between captures variability between each group. If all groups had the same mean, SS Between
would equal 0. The term SS Explained is also used because it reflects variability that is
“explained” by group membership. Note that there are J samples, one grand mean, hence DF
Between = J - 1.
0
=
0
*
2
2
=
2
=
)
y
y
)(
y
y
( = j
j
ij
i
j
j
j
ij
i
j
j
j
ij
i
j
τ
ε
τ
τ
ε ˆ
ˆ
ˆ
ˆ
ˆ ∑
∑
∑
∑
∑
∑ −
−
∑
2
(The latter is true because the deviations from the mean must sum to 0). Hence,
Between
SS
+
Within
SS
=
Total
SS
J. Now that we have these, what do we do with them? For hypothesis testing,
e have to make certain assumptions. Recall that yij = µ + τj + εij. εij is referred to as a "random
error te or
or all samples,
pendent (Note that these assumptions basically
mean that the ε are iid, independent and identically distributed);
w
rm" "disturbance." If we assume:
(1) εij - N(0, σ2
),
(2) σ2
is the same f
(3) the random error terms are inde
's
Then, if H0 is true,
1
=
E(F)
and
J),
-
N
1,
-
(J
F
~
Within
MS
Between
MS
=
F
That is, if H0 is true, then the test statistic F has an F distribution with J - 1 and N - J degrees of
Freedom.
ix E, Table V (Hayes, pp. 935-941), for tables on the F distribution. See especially
See Append
tables 5-3 (Q = .05) and 5-5 (Q = .01).

K. Rationale:
T The basic idea is to determine whether all of the variation in a set of data is
attribut to chance) or whether some of the variation is attributable to chance
and some is att
is seen to
e composed of two parts: the numerator, which is a sum of squares, and the denominator, which
is the degrees o
m of squares can be partitioned into SS Between and SS Within,
nd the total degrees of freedom can be partitioned into d.f. between and d.f. Within.
nd MS
ithin are determined; these represent the sample variability between the different samples and
the sample var
be due to random error alone,
ccording to the assumptions of the one-factor model.
the other hand, may be attributable
oth to chance and to any differences in the J population means.
MS Within (as measured by
e F-test), then the null hypothesis of zero treatment effects must be rejected.
an 1.
ve.
e right-hand side of the tail.
give
e d.f. for MS Within (N - J).
5-3, column 1; compare with Table 3 for the T distribution, the
olumn labeled 2Q = .05. Note that F = T2
. A two sample test, case II, σ1 = σ2 = σ, with a 2-
tailed alternati
able random error (
ributable to differences in the means of the J populations of interest.
T First, the sample variance for the entire set of data is computed and
b
f freedom.
T The total su
a
T By dividing each sum of squares by the respective d.f., MS between a
w
iability within all the samples, respectively.
T But the variability within the samples must
a
T The variability between the samples, on
b
T Thus, if MS Between is significantly greater than
th
L. Comments on the F distribution:
T There are two sets of d.f., rather th
T F is not symmetric. All values are positi
T Like χ2
, we are only interested in values in th
T In the tables, columns give the d.f. for MS Between (J - 1), while the rows
th
T Look at Table
c
ve hypothesis, can also be tested using ANOVA.

M. Computational procedures for ANOVA. The above formulas are, in practice, a
little awkward to deal with. When doing computations by hand, the following procedure is
generally easier:
One Way Anova: Computational Procedures
Formula Explanation
y
=
T ij
N
i
A
j
j ∑
TAj = the sum of the scores in group Aj, where A1 = first
group, A2 = second group, etc. Add up the values for the
observations for group A1, then A2, etc. Also sometimes
called just Tj.
Y
N
=
N
)
y
(
=
(1) 2
2
ij
∑
∑
Sum all the observations. Square the result. Divide by the
total number of observations.
y
=
(2) 2
ij
∑
∑
Square each observation. Sum the squared observations.
N
T
=
(3)
A
A
2
j j
j
∑
Square TA1, and divide by NA1. Repeat for each of the J
groups, and add the results together.
SS Total = (2) - (1) Total Sum of Squares
SS Between = (3) - (1). Or, if treatment effects have been
computed, use
2
ˆ j
j
N
∑ τ
Between Sum of Squares. This is also sometimes called
SSA, SS Treatment, or SS Explained
SS Within = (2) - (3) Within sum of squares. Also called SS error, or SS
Residual
MS Total = SS Total / (N - 1) Mean square total. Same as s2
, the sample variance.
MS Between = SS Between / (J - 1) Mean square between. Also called MSA, MS Treatment, or
MS Explained
MS Within = SS Within / (N - J) Mean Square Within. Also called MS error or MS Residual
F = MS Between / MS Within Test statistic. d.f. = (J - 1, N - J)

N. The ANOVA Table. The results of an analysis of variance are often presented in
a table that looks something like the following (with the appropriate values filled in):
Source SS D.F. Mean Square F
A (or Treatment, or
Explained)
SS Between J - 1 SS Between/ (J - 1)
Error (or Residual) SS Within N - J SS Within / (N - J)
Total SS Total N - 1 SS Total / (N - 1)
MS Between
MS Within
O. Hypothesis testing using ANOVA. As usual, we determine the critical value of
the test statistic for a given value of α. If the test statistic is less than the critical value, we accept
H0, if it is greater than the critical value we reject H0.
EXAMPLES:
1. Again consider this problem: A firm wishes to compare four programs for
training workers to perform a certain manual task. Twenty new employees are randomly
assigned to the training programs, with 5 in each program. At the end of the training period, a
test is conducted to see how quickly trainees can perform the task. The number of times the task
is performed per minute is recorded for each trainee, with the following results:
Program 1: 9, 12, 14, 11, 13
Program 2: 10, 6, 9, 9, 10
Program 3: 12, 14, 11, 13, 11
Program 4: 9, 8, 11, 7, 8
(a) Construct the ANOVA table
(b) Using α = .05, determine whether the treatments differ in their effectiveness.
Solution.
(a) As we saw before, TA1 = 59, TA2 = 44, TA3 = 61, TA4 = 43. Also,
2142.45
=
20
207
=
N
)
y
(
=
(1)
2
2
ij
∑
∑
2239
=
8
+
...
+
12
+
10
+
9
=
y
=
(2) 2
2
2
2
2
ij
∑
∑
2197.4
=
5
43
+
5
61
+
5
44
+
5
59
=
N
T
=
(3)
2
2
2
2
A
A
2
j j
j
∑

SS Total = (2) - (1) = 2239 - 2142.45 = 96.55,
SS Between = (3) - (1) = 2197.4 - 2142.45 = 54.95; or,
SS Between = = 5 * 1.45
2
ˆ j
j
N
∑ τ 2
+ 5 * 1.552
+ 5 * 1.852
+ 5 * 1.752
= 54.95
SS Within = (2) - (3) = 2239 - 2197.4 = 41.6,
MS Total = SS Total/ (N - 1) = 96.55 / 19 = 5.08,
MS Between = SS Between/ (J - 1) = 54.95/3 = 18.32,
MS Within = SS Within/ (N - J) = 41.6/16 = 2.6,
F = MS Between / MS Within = 18.32 / 2.6 = 7.04
The ANOVA Table therefore looks like this:
A (or Treatment, or
Explained)
SS Between =
54.95
J - 1 =
3
SS Between/ (J - 1)
= 18.32
Error (or Residual) SS Within =
41.6
N - J =
16
SS Within / (N - J) =
2.6
Total SS Total =
96.55
N - 1 =
19
SS Total / (N - 1) =
5.08
MS Between =
MS Within
7.04
NOTE: Most computer programs would not be nice enough to spell out "SS Between =", etc.;
that is, you would have to know from the location of the number in the table whether it was SS
Between, MS Within, or whatever. See the SPSS examples below.
(b) For α = .05, the critical value for an F with d.f. (3, 16) is 3.24. Ergo, we reject the null
hypothesis. More formally,
Step 1:
H0: µ1 = µ2 = µ3 = µ4, i.e. treatments are equally effective
HA: The means are not all equal.
Step 2:An F statistic is appropriate, since the dependent variable is continuous and there are 2 or
more groups.
Step 3:Since α = .05 and d.f. = 3, 16, accept H0 if F3,16 # 3.24
Step 4:The computed value of the F statistic is 7.04
Step 5:Reject H0. The treatments are not equally effective.

There are several SPSS routines that can do one-way Anova. These include ANOVA (which,
alas, requires that you enter the syntax directly rather than use menus; but it will give you the
MCA table if you want it), MEANS, and ONEWAY. Which you use depends on any additional
information you might like as well as the format you happen to like best. I’ll use ONEWAY but
feel free to try the others. If using the SPSS pull-down menus, after entering the data select
ANALYZE/ COMPARE MEANS/ ONE WAY ANOVA.
* Problem 1. Employee training.
DATA LIST FREE / program score.
BEGIN DATA.
1 9
1 12
1 14
1 11
1 13
2 10
2 6
2 9
2 9
2 10
3 12
3 14
3 11
3 13
3 11
4 9
4 8
4 11
4 7
4 8
END DATA.
ONEWAY
score BY program
/STATISTICS DESCRIPTIVES
/MISSING ANALYSIS .
Descriptives
SCORE
5 11.8000 1.9235 .8602 9.4116 14.1884 9.00 14.00
5 8.8000 1.6432 .7348 6.7597 10.8403 6.00 10.00
5 12.2000 1.3038 .5831 10.5811 13.8189 11.00 14.00
5 8.6000 1.5166 .6782 6.7169 10.4831 7.00 11.00
20 10.3500 2.2542 .5041 9.2950 11.4050 6.00 14.00
1.00
2.00
3.00
4.00
Total
N Mean Std. Deviation Std. Error Lower Bound Upper Bound
95% Confidence Interval for
Mean
Minimum Maximum

ANOVA
SCORE
54.950 3 18.317 7.045 .003
41.600 16 2.600
96.550 19
Between Groups
Within Groups
Total
Sum of
Squares df Mean Square F Sig.
2. For each of the following, indicate whether H0 should be accepted or rejected.
a. A researcher has collected data from 21 Catholics, 21 Protestants, and 21 Jews.
She wants to see whether the groups significantly differ at the .05 level in their incomes. Her
computed F = 3.0.
Solution. Note that n = 63, j = 3. Hence, d.f. = 3 - 1, 63 - 3 = 2, 60. Looking at table V, we see
that for α = .05 we should accept H0 if F # 3.15. Since the researcher got an F of 3.0, she should
accept H0.
b. A manager wants to test (using α = .025) whether the mean delivery time of
components supplied by 5 outside contractors is the same. He draws a random sample of 5
delivery times for each of the 5 contractors. He computes the following:
SS Between = 4
SS Within = 50
Solution. Note that n = 25 (5 delivery times for each of 5 contractors) and J = 5 (5 contractors).
Hence
MS Between = SS Between/(J - 1) = 4/4 = 1
MS Within = SS Within/(N - J) = 50/20 = 2.5
F = MS Between/MS Within = 1/2.5 = .4
D.F. = (J - 1, N - J) = (4, 20)
For α = .025, accept H0 if F # 3.51.
Therefore, accept H0.

3. An economist wants to test whether mean housing prices are the same regardless of
which of 3 air-pollution levels typically prevails. A random sample of house purchases in 3
areas yields the price data below.
MEAN HOUSING PRICES (THOUSANDS OF DOLLARS):
Pollution Level
Observation Low Mod High
1 120 61 40
2 68 59 55
3 40 110 73
4 95 75 45
5 83 80 64
Σ 406 385 277
(a) Compute the treatment effects
(b) Construct the ANOVA Table
(c) At the .025 level of significance, test whether housing prices differ by level of
pollution.
Solution.
(a)
8
.
15
2
.
71
4
.
55
ˆ
8
.
5
2
.
71
0
.
77
ˆ
10
2
.
71
2
.
81
ˆ
2
.
71
ˆ
,
4
.
55
ˆ
,
77
ˆ
,
2
.
81
ˆ
3
2
1
3
2
1
−
=
−
=
=
−
=
=
−
=
=
=
=
=
τ
τ
τ
µ
µ
µ
µ

(b) TA1 = 406, TA2 = 385, TA3 = 277,
76041.6
=
15
1068
=
N
)
y
(
=
(1)
2
2
ij
∑
∑
83940
=
64
+
...
+
61
+
120
=
y
=
(2) 2
2
2
2
ij
∑
∑
77958
=
5
277
+
5
385
+
5
406
=
N
T
=
(3)
2
2
2
A
A
2
j j
j
∑
SS Total = (2) - (1) = 83940 - 76041.6 = 7898.4,
SS Between = (3) - (1) = 77958 - 76041.6 = 1916.4; or,
SS Between = = 5 * 10
2
ˆ j
j
N
∑ τ 2
+ 5 * 5.82
+ 5 * -15.82
= 1916.4,
SS Within = (2) - (3) = 83940 - 77958 = 5982,
MS Total = SS Total/ (N - 1) = 7898.4 / 14 = 564.2,
MS Between = SS Between/ (J - 1) = 1916.4 / 2 = 958.2,
MS Within = SS Within / (N - J) = 5982 / 12 = 498.5,
F = MS Between / MS Within = 958.2 / 498.5 = 1.92
A (or Treatment, or
Explained)
SS Between =
1916.4
J - 1 =
2
SS Between/ (J - 1)
= 958.2
Error (or Residual) SS Within =
5982.0
N - J =
12
SS Within / (N - J) =
498.5
Total SS Total =
7898.4
N - 1 =
14
SS Total / (N - 1) =
564.2
MS Between =
MS Within
1.92
(c) For α = .025 and df = 2, 12, accept H0 if the computed F is # 5.10. Since F = 1.92, do not
reject H0. More formally,
Step 1.
H0: The τ's all = 0 (i.e. prices are the same in each area)
HA: The τ's are not all equal (prices not all the same)
Step 2. Appropriate stat is
F = MS Between/ MS Within.
Since n = 15 and j = 3, d.f. = 2, 12.
Step 3. For α = .025, accept H0 if F # 5.10

Step 4.Compute test stat. As shown above, F = 1.92
Step 5.Do not reject H0 [NOTE: the SPSS solutions follows later]
Here is how you could solve this problem using SPSS. If using the SPSS pull-down menus, after
entering the data select ANALYZE/ COMPARE MEANS/ ONE WAY ANOVA.
* Problem 3. Housing Prices.
DATA LIST FREE / plevel price.
BEGIN DATA.
1 120
1 68
1 40
1 95
1 83
2 61
2 59
2 110
2 75
2 80
3 40
3 55
3 73
3 45
3 64
END DATA.
ONEWAY
price BY plevel
/MISSING ANALYSIS .
Oneway
Descriptives
PRICE
5 81.2000 29.8781 13.3619 44.1015 118.2985 40.00 120.00
5 77.0000 20.5061 9.1706 51.5383 102.4617 59.00 110.00
5 55.4000 13.5019 6.0382 38.6352 72.1648 40.00 73.00
15 71.2000 23.7523 6.1328 58.0464 84.3536 40.00 120.00
1.00
2.00
3.00
Total
Mean
Minimum Maximum
ANOVA
PRICE
1916.400 2 958.200 1.922 .189
5982.000 12 498.500
7898.400 14
Between Groups
Within Groups
Total
Sum of

Comment: Some Anova routines would also report that R2
= .243. Note that R2
= SS Between
/ SS Total = 1916.4/7898.4 = .243. That is, R2
= Explained Variance divided by total variance.
We will talk more about R2
later.
F Test versus T Test. Finally, for good measure, we will do an F-Test vs. T-Test comparison.
We will do a modified version of problem 1, combining treatments 1 and 3 (the most effective),
and 2 and 4 (the least effective). We’ll let SPSS do the work.
* F test versus T-test comparison.
DATA LIST FREE / program score.
BEGIN DATA.
1 9
1 12
1 14
1 11
1 13
2 10
2 6
2 9
2 9
2 10
3 12
3 14
3 11
3 13
3 11
4 9
4 8
4 11
4 7
4 8
END DATA.
RECODE PROGRAM (1, 3 = 1) (2, 4 = 2).
ONEWAY
score BY program
/MISSING ANALYSIS .
Oneway
Descriptives
SCORE
10 12.0000 1.5635 .4944 10.8816 13.1184 9.00 14.00
10 8.7000 1.4944 .4726 7.6309 9.7691 6.00 11.00
20 10.3500 2.2542 .5041 9.2950 11.4050 6.00 14.00
1.00
2.00
Total
Mean
Minimum Maximum

ANOVA
SCORE
54.450 1 54.450 23.280 .000
42.100 18 2.339
96.550 19
Between Groups
Within Groups
Total
Sum of
Note that the F value is 23.28.
T-TEST / GROUPS PROGRAM (1, 2) / VARIABLES SCORE.
T-Test
Group Statistics
10 12.0000 1.5635 .4944
10 8.7000 1.4944 .4726
PROGRAM
1.00
2.00
SCORE
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
.010 .921 4.825 18 .000 3.3000 .6839 1.8631 4.7369
4.825 17.963 .000 3.3000 .6839 1.8629 4.7371
Equal variances
assumed
Equal variances
not assumed
SCORE
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means
COMMENT: Note that 4.822
= 23.28 (approximately), i.e. t2
= F. When you only have two
groups, both the F test and the T-Test are testing
H0: µ1 = µ2
HA: µ1 ≠ µ2
Not surprisingly, then, both tests yield the same conclusion.

One Way ANOVA.pdf

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