Analysis of Variance ANOVA
ECON1320 LECTURE 2 (ANOVA)
Analysis of Variance (ANOVA)
•
•
•
compare the means of multiple populations using one-way ANOVA (the completely
randomised design)
use the Tukey-Kramer procedure to determine which means are significantly different
compare the means of multiple populations using two-way ANOVA (the randomised
block design)
independent variable (or factor)
I. The treatment variable is under the control of the analyst
II.
classification variable is an existing characteristic of the experimental
subjects which is outside the control of the analyst
ANOVA assumptions
1. Samples should be independently selected and randomly assigned to
the levels of the treatment factor.
Randomness and independence must be met – drawing a random sample
or assigning treatments randomly will ensure independence
2. The variable level of interest for each population has a normal
distribution.
Normality – various tests available: e.g., goodness-of-fit test, residual plot,
etc.
3. The variance associated with each variable level in the population is the same
(equal) => homogeneity of variance
Equal variances – F test
Partitioning total variation
The One-way ANOVA Table
ECON1320 LECTURE 2 (ANOVA)
Hypothesis test
(i) H0: m1 = m2 = … = mc
HA: the means are not all equal (at least one mean is different)
(ii) Decion rule: Reject H0 if Ftest > Fa(C-1, N-C)
SSC
MSC
C−1
(iii) Test statistic: Ftest =
=
MSE
SSE
N−C
(iv) Decision
(v) Conclusion
Multiple Comparison Tests- Tukey-Kramer
Steps:
1. Compute all possible pairs of differences
2. For each pair, compute the critical range:
C ritic a l ra n g e = q
α , C , N -C
M SE 1
1
+
2 n
n
r
s
where MSE = Mean Square Within
qa,C,N-C = Table A.10 (pp.603-604), with df = (C, N-C)
3. A given pair is significantly different at if the absolute difference,
sample means exceeds the critical range.
|X́ i − X́ j|
, in the
EXAMPLE
A management consulting company presents a 3-day seminar on project management
to various clients. The seminar is basically the same each time it is given. However,
sometimes it is presented to high-level managers, sometimes to mid-level managers
and sometimes to low-level managers. The seminar organizers believe evaluations of
the seminar may vary with the audience. Suppose the following data are random
scores for seminar attendees.
ECON1320 LECTURE 2 (ANOVA)
The Randomised Block Design - ANOVA table for twofactor design
Hypothesis Tests for the Randomised Block Design
1.
Treatment effects (due to factor B)
H0: 1. = 2. = 3. =… c. => no treatment effects
HA: not all means are equal => treatment effects
2.
Blocking effects (due to factor A)
H0: .1 = .2 = .3 =… R => no blocking effects
HA: not all means are equal => blocking effects
EXAMPLE
•
•
•
A randomised block design study was undertaken to ascertain whether the
perception of economic recovery in Australia differs according to political
affiliation. The sample had three levels of political affiliation – Australian Labor
Party (ALP), The Liberal-National Coalition, and the Greens. To control for
differences in socioeconomic class, a blocking variable that had five
socioeconomic categories was used.
The respondents were asked to give a score on a 25-point scale from 0 =
economy was definitely not in recovery to 25 = the economy was definitely in
complete recovery, and some value in between for more uncertain responses.
Given the results below, use a = .01 to determine whether there is a significant
difference in mean responses according to political affiliation.
ECON1320 LECTURE 2 (ANOVA)
(i)
Hypotheses
Treatment effects:
H0: m1 = m2 = m3
HA: at least one treatment mean is different
Blocking effects:
H0: m1 = m2 = m3 = m4 = m5
HA: at least one blocking mean is different
(ii)
Decision rules: a =.01; dfC= C-1 = 3-1 = 2
•
•
•
•
(iii)
dfR= n-1 = 5-1 = 4
dfE= (C-1)(n-1)=2(4)=8
Critical values: FC= F.01,2,8= 8.65; Reject H0 if Ftest > 8.65
FR= F.01,4,8 = 7.01 Reject H0 if Ftest > 7.01
Test Statistics:
•
Blocking effects (Columns): Ftest=
•
Treatment effects (Rows): Ftest=
SSC /df C
64.53 /2
=
= 15.36
SSE/df E
16.8 /8
SSR/ df C
137.6 /4
=
= 16.38
SSE /df E
16.8/ 8
(iv) Decision:
• Blocking: Reject H0 at the 1% level
• Treatment: Reject H0 at the 1% level
(v) Conclusion:
1. At least one of the population means of the treatment levels is different
from the others
i.e. there is a significant difference in the perception of economic recovery
among supporters of the different political parties
• 2. Blocking effects are significant.
• i.e. socioeconomic background significantly affects one’s perception of
economic recovery.
Therefore, the blocking has been advantageous in reducing the random error and
improving the accuracy of the test
•
ANOVA
Source of Variation
SS
df
MS
F
Rows
137.6
4
34.400
16.381
Columns
64.53
2
32.267
15.365
Error
16.80
8
2.1
Total
218.93
14
ECON1320 LECTURE 2 (ANOVA)
Analysis of Variance (ANOVA)
•
•
•
compare the means of multiple populations using one-way ANOVA (the completely
randomised design)
use the Tukey-Kramer procedure to determine which means are significantly different
compare the means of multiple populations using two-way ANOVA (the randomised
block design)
independent variable (or factor)
I. The treatment variable is under the control of the analyst
II.
classification variable is an existing characteristic of the experimental
subjects which is outside the control of the analyst
ANOVA assumptions
1. Samples should be independently selected and randomly assigned to
the levels of the treatment factor.
Randomness and independence must be met – drawing a random sample
or assigning treatments randomly will ensure independence
2. The variable level of interest for each population has a normal
distribution.
Normality – various tests available: e.g., goodness-of-fit test, residual plot,
etc.
3. The variance associated with each variable level in the population is the same
(equal) => homogeneity of variance
Equal variances – F test
Partitioning total variation
The One-way ANOVA Table
ECON1320 LECTURE 2 (ANOVA)
Hypothesis test
(i) H0: m1 = m2 = … = mc
HA: the means are not all equal (at least one mean is different)
(ii) Decion rule: Reject H0 if Ftest > Fa(C-1, N-C)
SSC
MSC
C−1
(iii) Test statistic: Ftest =
=
MSE
SSE
N−C
(iv) Decision
(v) Conclusion
Multiple Comparison Tests- Tukey-Kramer
Steps:
1. Compute all possible pairs of differences
2. For each pair, compute the critical range:
C ritic a l ra n g e = q
α , C , N -C
M SE 1
1
+
2 n
n
r
s
where MSE = Mean Square Within
qa,C,N-C = Table A.10 (pp.603-604), with df = (C, N-C)
3. A given pair is significantly different at if the absolute difference,
sample means exceeds the critical range.
|X́ i − X́ j|
, in the
EXAMPLE
A management consulting company presents a 3-day seminar on project management
to various clients. The seminar is basically the same each time it is given. However,
sometimes it is presented to high-level managers, sometimes to mid-level managers
and sometimes to low-level managers. The seminar organizers believe evaluations of
the seminar may vary with the audience. Suppose the following data are random
scores for seminar attendees.
ECON1320 LECTURE 2 (ANOVA)
The Randomised Block Design - ANOVA table for twofactor design
Hypothesis Tests for the Randomised Block Design
1.
Treatment effects (due to factor B)
H0: 1. = 2. = 3. =… c. => no treatment effects
HA: not all means are equal => treatment effects
2.
Blocking effects (due to factor A)
H0: .1 = .2 = .3 =… R => no blocking effects
HA: not all means are equal => blocking effects
EXAMPLE
•
•
•
A randomised block design study was undertaken to ascertain whether the
perception of economic recovery in Australia differs according to political
affiliation. The sample had three levels of political affiliation – Australian Labor
Party (ALP), The Liberal-National Coalition, and the Greens. To control for
differences in socioeconomic class, a blocking variable that had five
socioeconomic categories was used.
The respondents were asked to give a score on a 25-point scale from 0 =
economy was definitely not in recovery to 25 = the economy was definitely in
complete recovery, and some value in between for more uncertain responses.
Given the results below, use a = .01 to determine whether there is a significant
difference in mean responses according to political affiliation.
ECON1320 LECTURE 2 (ANOVA)
(i)
Hypotheses
Treatment effects:
H0: m1 = m2 = m3
HA: at least one treatment mean is different
Blocking effects:
H0: m1 = m2 = m3 = m4 = m5
HA: at least one blocking mean is different
(ii)
Decision rules: a =.01; dfC= C-1 = 3-1 = 2
•
•
•
•
(iii)
dfR= n-1 = 5-1 = 4
dfE= (C-1)(n-1)=2(4)=8
Critical values: FC= F.01,2,8= 8.65; Reject H0 if Ftest > 8.65
FR= F.01,4,8 = 7.01 Reject H0 if Ftest > 7.01
Test Statistics:
•
Blocking effects (Columns): Ftest=
•
Treatment effects (Rows): Ftest=
SSC /df C
64.53 /2
=
= 15.36
SSE/df E
16.8 /8
SSR/ df C
137.6 /4
=
= 16.38
SSE /df E
16.8/ 8
(iv) Decision:
• Blocking: Reject H0 at the 1% level
• Treatment: Reject H0 at the 1% level
(v) Conclusion:
1. At least one of the population means of the treatment levels is different
from the others
i.e. there is a significant difference in the perception of economic recovery
among supporters of the different political parties
• 2. Blocking effects are significant.
• i.e. socioeconomic background significantly affects one’s perception of
economic recovery.
Therefore, the blocking has been advantageous in reducing the random error and
improving the accuracy of the test
•
ANOVA
Source of Variation
SS
df
MS
F
Rows
137.6
4
34.400
16.381
Columns
64.53
2
32.267
15.365
Error
16.80
8
2.1
Total
218.93
14
ECON1320 LECTURE 2 (ANOVA)