Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences
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Can one conclude on the basis of these data that there is a difference in the reduction of headache
frequency between the control and treatment groups? Can one conclude that there is a difference
in the reduction of headache frequency between migraine and non-migraine sufferers? Can one
conclude that there is interaction between treatments and migraine status? Let for each
test.
8.5.4 The purpose of a study by Porcellini et al. (A-23) was to study the difference in CD4 cell response
in patients taking highly active antiretroviral therapy (HAART, treatment 1) and patients taking
HAART plus intermittent interleukin (IL-2, treatment 2). Another factor of interest was the HIV-
RNA plasma count at baseline of study. Subjects were classified as having fewer than 50 copies/ml
(plasma 1) or having 50 or more copies/ml (plasma 2). The outcome variable is the percent change
in CD4 T cell count from baseline to 12 months of treatment. Can one conclude that there is a dif-
ference in the percent change in CD4 T cell count between the two treatments? The results are shown
in the following table. Can one conclude that there is a difference in the percent change in CD4 T
cell count between those who have fewer than 50/ml plasma copies of HIV-RNA and those who do
not? Can one conclude that there is interaction between treatments and plasma levels? Let
for each test.
a = .05
a = .05
EXERCISES 367
Change in Change in
Frequency of Migraine Sufferer Frequency of Migraine Sufferer
Headaches ( ) Treatment Headaches ( ) Treatment
33 1 1 14 1 2
21 22
2153 12
2126 22
12 1 1 3 1 2
64 1 1 15 1 2
36 2 1 3 1 2
6214112
4211612
11 2 1 2 2
021 22
912
922
22
922
312
422
Source: A. Main, H. Abu-Saad, R. Salt, l. Vlachonikolis, and A. Dowson, “Management by Nurses of Primary
Headache: A Pilot Study,” Current Medical Research Opinion, 18 (2002), 471–478. Used with Permission.
-3
-6
-4
-12
-1
-11-7
1 No, 2 Yes1 No, 2 Yes
(Continued)
Percent Change in CD4 T Cell Treatment Plasma
11
21
28.10 2 1
-14.60
-12.60
8.5.5 A study by Górecka et al. (A-24) assessed the manner in which among middle-aged smokers the
diagnosis of airflow limitation (AL) combined with advice to stop smoking influences the smok-
ing cessation rate. Their concerns were whether having AL, whether the subject successfully quit
smoking, and whether interaction between AL and smoking status were significant factors in regard
to baseline variables and lung capacity variables at the end of the study. Some of the variables of
interest were previous years of smoking (pack years), age at which subject first began smoking,
forced expiratory volume in one second (FEV
1
), and forced vital capacity (FVC). There were 368
subjects in the study. What are the factors in this study? At how many levels does each occur?
Who are the subjects? What is (are) the response variable(s)? Can you think of any extraneous
variables whose effects are included in the error term?
8.5.6 A study by Meltzer et al. (A-25) examined the response to 5 mg desloratadine, an H1-receptor
antagonist, in patients with seasonal allergies. During the fall allergy season, 172 subjects were
randomly assigned to receive treatments of desloratadine and 172 were randomly assigned to
receive a placebo. Subjects took the medication for 2 weeks after which changes in the nasal symp-
tom score were calculated. A significant reduction was noticed in the treatment group compared
to the placebo group, but gender was not a significant factor. What are the factors in the study?
At how many levels does each occur? What is the response variable?
8.6 SUMMARY
The purpose of this chapter is to introduce the student to the basic ideas and techniques
of analysis of variance. Two experimental designs, the completely randomized and the
randomized complete block, are discussed in considerable detail. In addition, the con-
cept of repeated measures designs and a factorial experiment as used with the completely
randomized design are introduced. Individuals who wish to pursue further any aspect of
analysis of variance will find the methodology references at the end of the chapter most
helpful.
368
CHAPTER 8 ANALYSIS OF VARIANCE
Percent Change in CD4 T Cell Treatment Plasma
77.30 1 1
11
50.20 1 1
48.60 2 2
86.20 2 2
205.80 1 2
100.00 1 2
34.30 1 2
82.40 1 2
118.30 1 2
Source: Simona Porcellini, Guiliana Vallanti, Silvia Nozza, Guido Poli,
Adriano Lazzarin, Guiseppe Tabussi, and Antonio Grassia, “Improved
Thymopoietic Potential in Aviremic HIV Infected Individuals with HAART
by Intermittent IL-2 Administration,” AIDS, 17 (2003), 1621–1630.
Used with permission.
-0.44
SUMMARY OF FORMULAS FOR CHAPTER 8 369
SUMMARY OF FORMULAS FOR CHAPTER 8
Formula
Number Name Formula
8.2.1 One-way ANOVA
model
8.2.2 Total sum-of-squares
8.2.3 Within-group
sum-of-squares
8.2.4 Among-group
sum-of-squares
8.2.5 Within-group
variance
8.2.6 Among-group
variance I
8.2.7 Among-group
variance II
(equal sample sizes)
8.2.8 Among-group
variance III (unequal
sample sizes)
8.2.9 Tukey’s HSD
(equal sample sizes)
8.2.10 Tukey’s HSD
(unequal sample
sizes)
8.3.1 Two-way ANOVA
model
8.3.2 Sum-of-squares SST SSBl SSTr SSE
representation
X
ij
= m + b
i
+ t
i
+P
ij
HSD* = q
a,k,N-k
A
MSE
n
a
1
n
i
+
1
n
j
b
HSD = q
a,k,N-k
A
MSE
n
MSA =
a
k
j = 1
n
j
1x
#
j
- x
##
2
2
1k - 12
MSA =
n
a
k
j = 1
1x
#
j
- x
##
2
2
1k - 12
s
2
= ns
x
2
MSW =
a
k
j = 1
a
n
j
i = 1
1x
ij
- x
#
j
2
2
a
k
j = 1
1n
j
- 12
SSA =
a
k
j = 1
n
j
1x
#
j
- x
##
2
2
SSW =
a
k
j = 1
a
n
j
i = 1
1x
ij
- x
#
j
2
2
SST =
a
k
j = 1
a
n
j
i = 1
1x
ij
- x
##
2
2
X
ij
= m + t
i
+P
ij
(Continued )
370 CHAPTER 8 ANALYSIS OF VARIANCE
8.3.3 Sum-of-squares total
8.3.4 Sum-of-squares block
8.3.5 Sum-of-squares
treatments
8.3.6 Sum-of-squares error SSE SST SSBl SSTr
8.4.1 Fixed-effects, additive
single-factor,
repeated-measures
ANOVA model
8.5.1 Two-factor completely
randomized fixed-
effects factorial model
8.5.2 Probabilistic
representation of
8.5.3 Sum-of-squares
total I
8.5.4 Sum-of-squares SST SSTr SSE
total II
8.5.5 Sum-of-squares
treatment partition
Symbol Key • probability of type I error
• treatment A effect
• treatment B effect
• block effect
• interaction effect
• error term
• HSD honestly significant difference
• k number of treatments
P
ij
1ab2
ij
b
i
b
j
a
i
a
+
a
a
i = 1
a
b
j = 1
a
n
k = 1
1x
ij
#
- x
i
##
- x
#
j
#
+ x
###
2
2
+
a
a
i = 1
a
b
j = 1
a
n
k = 1
1x
#
j
#
- x
###
2
2
a
a
i = 1
a
b
j = 1
a
n
k = 1
1x
ij
#
- x
###
2
2
=
a
a
i = 1
a
b
j = 1
a
n
k = 1
1x
i
##
- x
###
2
2
+
a
a
i = 1
a
b
j = 1
a
n
k = 1
1x
ijk
- x
ij
#
2
a
a
i = 1
a
b
j = 1
a
n
k = 1
1x
ijk
- x
###
2=
a
a
i = 1
a
b
j = 1
a
n
k = 1
1x
ij
#
- x
###
2
a
a 6 1 - 11 - a¿211 - a–211 - a‡2
X
ijk
= m + a
i
+ b
j
+ 1ab2
ij
+P
ijk
x
ij
= m + b
i
+ t
i
+P
ij
SSTr =
a
k
j = 1
a
n
i = 1
1x
#
j
- x
##
2
2
SSBl =
a
k
j = 1
a
n
i = 1
1x
i
#
- x
##
2
2
SST =
a
k
j = 1
a
n
i = 1
1x
ij
- x
##
2
2
REVIEW QUESTIONS AND EXERCISES 371
REVIEW QUESTIONS AND EXERCISES
1. Define analysis of variance.
2. Describe the completely randomized design.
3. Describe the randomized block design.
4. Describe the repeated measures design.
5. Describe the factorial experiment as used in the completely randomized design.
6. What is the purpose of Tukey’s HSD test?
7. What is an experimental unit?
8. What is the objective of the randomized complete block design?
9. What is interaction?
10. What is a mean square?
11. What is an ANOVA table?
12. For each of the following designs describe a situation in your particular field of interest where the
design would be an appropriate experimental design. Use real or realistic data and do the appro-
priate analysis of variance for each one:
(a) Completely randomized design
(b) Randomized complete block design
(c) Completely randomized design with a factorial experiment
(d) Repeated measures designs
13. Werther et al. (A-26) examined the -leucocyte count in 51 subjects with colorectal can-
cer and 19 healthy controls. The cancer patients were also classified into Dukes’s classification (A,
B, C) for colorectal cancer that gives doctors a guide to the risk, following surgery, of the cancer
coming back or spreading to other parts of the body. An additional category (D) identified patients
with disease that had not been completely resected. The results are displayed in the following table.
Perform an analysis of these data in which you identify the sources of variability and specify the
degrees of freedom for each. Do these data provide sufficient evidence to indicate that, on the aver-
age, leucocyte counts differ among the five categories? Let and find the p value. Use
Tukey’s procedure to test for significant differences between individual pairs of sample means.
a = .01
1*10
9
>L2b
• mean of population (or the grand mean)
• n number of blocks
• n
x
sample size
• variance
• SSX sum-of-squares (where X: A among,
Bl block, T total, Tr treatment, W within)
• treatment effect
• X
xxx
measurement
t
i
s
2
m
372 CHAPTER 8 ANALYSIS OF VARIANCE
Healthy A B C D
6.0 7.7 10.4 8.0 9.5
6.3 7.8 5.6 6.7 7.8
5.1 6.1 7.0 9.3 5.7
6.2 9.6 8.2 6.6 8.0
10.4 5.5 9.0 9.3 9.6
4.4 5.8 8.4 7.2 13.7
7.4 4.0 8.1 5.2 6.3
7.0 5.4 8.0 9.8 7.3
5.6 6.5 6.2 6.2
5.3 9.1 10.1
2.6 11.0 9.3
6.3 10.9 9.4
6.1 10.6 6.5
5.3 5.2 5.4
5.4 7.9 7.6
5.2 7.6 9.2
4.3 5.8
4.9 7.0
7.3
4.9
6.9
4.3
5.6
5.1
Source: Kim Werther, M.D., Ph.D. Used with permission.
14. In Example 8.4.1, we examined data from a study by Licciardone et al. (A-15) on osteopathic manip-
ulation as a treatment for chronic back pain. At the beginning of that study, there were actually 91
subjects randomly assigned to one of three treatments: osteopathic manipulative treatment (OMT),
sham manipulation (SHAM), or non-intervention (CONTROL). One important outcome variable was
the rating of back pain at the beginning of the study. The researchers wanted to know if the treat-
ment had essentially the same mean pain level at the start of the trial. The results are displayed in
the following table. The researchers used a visual analog scale from 0 to 10 cm where 10 indi-
cated “worst pain possible.” Can we conclude, on the basis of these data, that, on the average, pain
levels differ in the three treatment groups? Let and find the p value. If warranted, use
Tukey’s procedure to test for differences between individual pairs of sample means.
a = .05
CONTROL SHAM OMT
2.6 5.8 7.8 3.5
5.6 1.3 4.1 3.4
3.3 2.4 1.7 1.1
4.6 1.0 3.3 0.5
8.4 3.2 4.3 5.1
0.0 0.4 6.5 1.9
2.5 5.4 5.4 2.0
5.0 4.5 4.0 2.8
(Continued)
REVIEW QUESTIONS AND EXERCISES 373
15. The goal of a study conducted by Meshack and Norman (A-27) was to evaluate the effects of weights
on postural hand tremor related to self-feeding in subjects with Parkinson’s disease (PD). Each of the
16 subjects had the tremor amplitude measured (in mm) under three conditions: holding a built-up
spoon (108 grams), holding a weighted spoon (248 grams), and holding the built-up spoon while wear-
ing a weighted wrist cuff (470 grams). The data are displayed in the following table.
CONTROL SHAM OMT
1.7 1.5 4.1 3.7
3.8 0.0 2.6 1.6
2.4 0.6 3.2 0.0
1.1 0.0 2.8 0.2
0.7 7.6 3.4 7.3
2.4 3.5 6.7 1.7
3.3 3.9 7.3 7.5
6.6 7.0 2.1 1.6
0.4 7.4 3.7 3.0
0.4 6.5 2.3 6.5
0.9 1.6 4.4 3.0
6.0 1.3 2.8 3.3
6.6 0.4 7.3
6.3 0.7 4.6
7.0 7.9 4.8
1.3 4.9
Source: J. C. Licciardone, D.O. Used with permission.
Tremor Amplitude (mm)
Subject Built-Up Spoon Weighted Spoon Built-Up Spoon Wrist Cuff
1 .77 1.63 1.02
2 .78 .88 1.11
3 .17 .14 .14
4 .30 .27 .26
5 .29 .27 .28
6 1.60 1.49 1.73
7 .38 .39 .37
8 .24 .24 .24
9 .17 .17 .16
10 .38 .29 .27
11 .93 1.21 .90
12 .63 .52 .66
13 .49 .73 .76
14 .42 .60 .29
15 .19 .21 .21
16 .19 .20 .16
Source: Rubia P. Meshack and Kathleen E. Norman, “A Randomized Controlled Trial of the Effects of
Weights on Amplitude and Frequency of Postural Hand Tremor in People with Parkinson’s Disease,”
Clinical Rehabilitation, 16 (2003), 481– 492.
374 CHAPTER 8 ANALYSIS OF VARIANCE
Can one conclude on the basis of these data that the three experimental conditions, on the aver-
age, have different effects on tremor amplitude? Let Determine the p value.
16. In a study of pulmonary effects on guinea pigs, Lacroix et al. (A-28) exposed 18 ovalbumin-
sensitized guinea pigs and 18 nonsensitized guinea pigs to regular air, benzaldehyde, and
acetaldehyde. At the end of exposure, the guinea pigs were anesthetized and allergic responses
were assessed in bronchoalveolar lavage (BAL). The following table shows the alveolar cell
count by treatment group for the ovalbumin-sensitized and nonsensitized guinea
pigs.
1* 10
6
2
a = .05.
Ovalbumin-Sensitized Treatment Alveolar Count
no acetaldehyde 49.90
no acetaldehyde 50.60
no acetaldehyde 50.35
no acetaldehyde 44.10
no acetaldehyde 36.30
no acetaldehyde 39.15
no air 24.15
no air 24.60
no air 22.55
no air 25.10
no air 22.65
no air 26.85
no benzaldehyde 31.10
no benzaldehyde 18.30
no benzaldehyde 19.35
no benzaldehyde 15.40
no benzaldehyde 27.10
no benzaldehyde 21.90
yes acetaldehyde 90.30
yes acetaldehyde 72.95
yes acetaldehyde 138.60
yes acetaldehyde 80.05
yes acetaldehyde 69.25
yes acetaldehyde 31.70
yes air 40.20
yes air 63.20
yes air 59.10
yes air 79.60
yes air 102.45
yes air 64.60
yes benzaldehyde 22.15
yes benzaldehyde 22.75
yes benzaldehyde 22.15
yes benzaldehyde 37.85
yes benzaldehyde 19.35
yes benzaldehyde 66.70
Source: G. Lacroix, Docteur en Toxicologie. Used with permission.
: 10
6
REVIEW QUESTIONS AND EXERCISES 375
Test for differences (a) between ovalbumin-sensitized and nonsensitized outcomes, (b) among the three
different exposures, and (c) interaction. Let for all tests.
17. Watanabe et al. (A-29) studied 52 healthy middle-aged male workers. The researchers used the
Masstricht Vital Exhaustion Questionnaire to assess vital exhaustion. Based on the resultant scores,
they assigned subjects into three groups: VE1, VE2, and VE3. VE1 indicates the fewest signs of
exhaustion, and VE3 indicates the most signs of exhaustion. The researchers also asked subjects
about their smoking habits. Smoking status was categorized as follows: SMOKE1 are nonsmok-
ers, SMOKE2 are light smokers (20 cigarettes or fewer per day), SMOKE3 are heavy smokers
(more than 20 cigarettes per day). One of the outcome variables of interest was the amplitude of
the high-frequency spectral analysis of heart rate variability observed during an annual health
checkup. This variable, HF-amplitude, was used as an index of parasympathetic nervous function.
The data are summarized in the following table:
a = .05
HF-Amplitude
Smoking Status
Vital Exhaustion
Group SMOKE1 SMOKE2 SMOKE3
VE1 23.33 13.37 16.14 16.83
31.82 9.76 20.80 29.40
10.61 22.24 15.44 6.50
42.59 8.77 13.73 10.18
23.15 20.28 13.86
17.29
VE2 20.69 11.67 44.92 27.91
16.21 30.17 36.89
28.49 29.20 16.80
25.67 8.73 17.08
15.29 9.08 18.77
7.51 22.53 18.33
22.03 17.19
10.27
VE3 9.44 17.59 5.57
19.16 18.90 13.51
14.46 17.37
10.63
13.83
Source: Takemasa Watanabe, M.D., Ph.D. Used with permission.
Perform an analysis of variance on these data and test the three possible hypotheses. Let
Determine the p values.a¿=a–=a‡=.05.
376 CHAPTER 8 ANALYSIS OF VARIANCE
19. Eleftherios Kellis (A-31) conducted an experiment on 18 pubertal males. He recorded the elec-
tromyographic (EMG) activity at nine different angular positions of the biceps femoris muscle.
The EMG values are expressed as a percent (0–100 percent) of the maximal effort exerted with
the muscle and represent an average in a range of flexion angles. The nine positions correspond
to testing knee flexion angles of
and The results are displayed in the following table. For subject 1, for exam-
ple, the value of 30.96 percent represents the average maximal percent of effort in angular posi-
tions from 1 to 10 degrees.
81– 90°.71– 80°,
61–70°,51– 60°,41–50°,31– 40°,21–30°,11–20°,1–10°,
Location 1 Location 2 Location 3
Length Width Height Length Width Height Length Width Height
7.20 6.10 4.45 7.25 6.25 4.65 5.95 4.75 3.20
7.50 5.90 4.65 7.23 5.99 4.20 7.60 6.45 4.56
6.89 5.45 4.00 6.85 5.61 4.01 6.15 5.05 3.50
6.95 5.76 4.02 7.07 5.91 4.31 7.00 5.80 4.30
6.73 5.36 3.90 6.55 5.30 3.95 6.81 5.61 4.22
7.25 5.84 4.40 7.43 6.10 4.60 7.10 5.75 4.10
7.20 5.83 4.19 7.30 5.95 4.29 6.85 5.55 3.89
6.85 5.75 3.95 6.90 5.80 4.33 6.68 5.50 3.90
7.52 6.27 4.60 7.10 5.81 4.26 5.51 4.52 2.70
7.01 5.65 4.20 6.95 5.65 4.31 6.85 5.53 4.00
6.65 5.55 4.10 7.39 6.04 4.50 7.10 5.80 4.45
7.55 6.25 4.72 6.54 5.89 3.65 6.81 5.45 3.51
7.14 5.65 4.26 6.39 5.00 3.72 7.30 6.00 4.31
7.45 6.05 4.85 6.08 4.80 3.51 7.05 6.25 4.71
7.24 5.73 4.29 6.30 5.05 3.69 6.75 5.65 4.00
7.75 6.35 4.85 6.35 5.10 3.73 6.75 5.57 4.06
6.85 6.05 4.50 7.34 6.45 4.55 7.35 6.21 4.29
6.50 5.30 3.73 6.70 5.51 3.89 6.22 5.11 3.35
6.64 5.36 3.99 7.08 5.81 4.34 6.80 5.81 4.50
7.19 5.85 4.05 7.09 5.95 4.39 6.29 4.95 3.69
7.15 6.30 4.55 7.40 6.25 4.85 7.55 5.93 4.55
7.21 6.12 4.37 6.00 4.75 3.37 7.45 6.19 4.70
7.15 6.20 4.36 6.94 5.63 4.09 6.70 5.55 4.00
7.30 6.15 4.65 7.51 6.20 4.74
6.35 5.25 3.75 6.95 5.69 4.29
7.50 6.20 4.65
Source: John Brooker, M.S. and the Wright State University Statistical Consulting Center. Used with
permission.
18. The effects of thermal pollution on Corbicula fluminea (Asiatic clams) at three different geograph-
ical locations were analyzed by John Brooker (A-30). Sample data on clam shell length, width,
and height are displayed in the following table. Determine if there is a significant difference in
mean length, height, or width (measured in mm) of the clam shell at the three different locations
by performing three analyses. What inferences can be made from your results? What are the
assumptions underlying your inferences? What are the target populations?