Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences
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REVIEW QUESTIONS AND EXERCISES 377
Can we conclude on the basis of these data that the average EMG values differ among the nine
angular locations? Let
20. In a study of Marfan syndrome, Pyeritz et al. (A-32) reported the following severity scores of
patients with no, mild, and marked dural ectasia. May we conclude, on the basis of these data,
that mean severity scores differ among the three populations represented in the study? Let
and find the p value. Use Tukey’s procedure to test for significant differences among individual
pairs of sample means.
No dural ectasia: 18, 18, 20, 21, 23, 23, 24, 26, 26, 27, 28, 29, 29, 29, 30, 30, 30,
30, 32, 34, 34, 38
Mild dural ectasia: 10, 16, 22, 22, 23, 26, 28, 28, 28, 29, 29, 30, 31, 32, 32, 33, 33,
38, 39, 40, 47
Marked dural ectasia: 17, 24, 26, 27, 29, 30, 30, 33, 34, 35, 35, 36, 39
Source: Reed E. Pyeritz, M.D., Ph.D. Used with permission.
21. The following table shows the arterial plasma epinephrine concentrations (nanograms per milli-
liter) found in 10 laboratory animals during three types of anesthesia:
Animal
Anesthesia 1 234567 8910
A .28 .50 .68 .27 .31 .99 .26 .35 .38 .34
B .20 .38 .50 .29 .38 .62 .42 .87 .37 .43
C 1.23 1.34 .55 1.06 .48 .68 1.12 1.52 .27 .35
Can we conclude from these data that the three types of anesthesia, on the average, have differ-
ent effects? Let a = .05.
a = .05
a = .05.
Subject 1–10° 11–20° 21–30° 31–40° 41–50° 51–60° 61–70° 71–80° 81–90°
1 30.96 11.32 4.34 5.99 8.43 10.50 4.49 10.93 33.26
2 3.61 1.47 3.50 10.25 3.30 3.62 10.14 11.05 8.78
3 8.46 2.94 1.83 5.80 11.59 15.17 13.04 10.57 8.22
4 0.69 1.06 1.39 1.08 0.96 2.52 2.90 3.27 5.52
5 4.40 3.02 3.74 3.83 3.73 10.16 9.31 12.70 11.45
6 4.59 9.80 10.71 11.64 9.78 6.91 8.53 8.30 11.75
7 3.31 3.31 4.12 12.56 4.60 1.88 2.42 2.46 2.19
8 1.98 6.49 2.61 3.28 10.29 7.56 16.68 14.52 13.49
9 10.43 4.96 12.37 24.32 17.16 34.71 35.30 37.03 45.65
10 20.91 20.72 12.70 15.06 12.03 11.31 28.47 26.81 25.08
11 5.59 3.13 2.83 4.31 6.37 13.95 13.48 11.15 30.97
12 8.67 4.32 2.29 6.20 13.01 19.30 9.33 12.30 12.20
13 2.11 1.59 2.40 2.56 2.83 2.55 5.84 5.23 8.84
14 3.82 5.04 6.81 10.74 10.10 13.14 19.39 13.31 12.02
15 39.51 62.34 70.46 20.48 17.38 54.04 25.76 50.32 46.84
16 3.31 4.95 12.49 9.18 14.00 16.17 25.75 11.82 13.17
17 11.42 7.53 4.65 4.70 7.57 9.86 5.30 4.47 3.99
18 2.97 2.18 2.36 4.61 7.83 17.49 42.55 61.84 39.70
Source: Eleftherios Kellis, Ph.D. Used with permission.
378 CHAPTER 8 ANALYSIS OF VARIANCE
22. The aim of a study by Hartman-Maeir et al. (A-33) was to evaluate the awareness of deficit profiles
among stroke patients undergoing rehabilitation. She studied 35 patients with a stroke lesion in the
right hemisphere and 19 patients with a lesion on the left hemisphere. She also grouped lesion size as
and
One of the outcome variables was a measure of each patient’s total unawareness of their own limi-
tations. Scores ranged from 8 to 24, with higher scores indicating more unawareness.
4 = 5 cm
or greater.3 = 3 – 5 cm,2 = 1–3cm,
Unawareness Score
Lesion Size Left
Group Hemisphere Right Hemisphere
2 11 10 8
13 11 10
10 13 9
11 10 9
913 9
10 10
910
89
10 8
3131110
810 11
10 10 12
10 14 11
10 8
4111011
13 13 9
14 10 19
13 10 10
14 15 9
8 10
Test for a difference in lesion size, hemisphere, and interaction. Let for all tests.
23. A random sample of the records of single births was selected from each of four populations. The
weights (grams) of the babies at birth were as follows:
a = .05
Sample
ABCD
2946 3186 2300 2286
2913 2857 2903 2938
2280 3099 2572 2952
3685 2761 2584 2348
2310 3290 2675 2691
Source: Adina Hartman-Maeir, Ph.D.,
O.T.R. Used with permission.
(Continued)
REVIEW QUESTIONS AND EXERCISES 379
Sample
ABCD
2582 2937 2571 2858
3002 3347 2414
2408 2008
2850
2762
Do these data provide sufficient evidence to indicate, at the .05 level of significance, that the four
populations differ with respect to mean birth weight? Test for a significant difference between all
possible pairs of means.
24. The following table shows the aggression scores of 30 laboratory animals reared under three dif-
ferent conditions. One animal from each of 10 litters was randomly assigned to each of the three
rearing conditions.
Rearing Condition
Extremely Moderately Not
Litter Crowded Crowded Crowded
130 20 10
230 10 20
330 20 10
425 15 10
535 25 20
630 20 10
720 20 10
830 30 10
925 25 10
10 30 20 20
Do these data provide sufficient evidence to indicate that level of crowding has an effect on aggres-
sion? Let
25. The following table shows the vital capacity measurements of 60 adult males classified by occu-
pation and age group:
a = .05.
Occupation
Age
Group A B C D
1 4.31 4.68 4.17 5.75
4.89 6.18 3.77 5.70
4.05 4.48 5.20 5.53
4.44 4.23 5.28 5.97
4.59 5.92 4.44 5.52
(Continued)
Test for differences among occupations, for differences among age groups, and for interaction. Let
for all tests.
26. Complete the following ANOVA table and state which design was used.
a = .05
380 CHAPTER 8 ANALYSIS OF VARIANCE
Occupation
Age
Group A B C D
2 4.13 3.41 3.89 4.58
4.61 3.64 3.64 5.21
3.91 3.32 4.18 5.50
4.52 3.51 4.48 5.18
4.43 3.75 4.27 4.15
3 3.79 4.63 5.81 6.89
4.17 4.59 5.20 6.18
4.47 4.90 5.34 6.21
4.35 5.31 5.94 7.56
3.59 4.81 5.56 6.73
Source SS d.f. MS V.R. p
Treatments 154.9199 4
Error
Total 200.4773 39
27. Complete the following ANOVA table and state which design was used.
Source SS d.f. MS V.R. p
Treatments 3
Blocks 183.5 3
Error 26.0
Total 709.0 15
28. Consider the following ANOVA table.
Source SS d.f. MS V.R. p
A 12.3152 2 6.15759 29.4021 .005
B 19.7844 3 6.59481 31.4898 .005
AB 8.94165 6 1.49027 7.11596 .005
Treatments 41.0413 11
Error 10.0525 48 0.209427
Total 51.0938 59
REVIEW QUESTIONS AND EXERCISES 381
(a) What sort of analysis was employed?
(b) What can one conclude from the analysis? Let
29. Consider the following ANOVA table.
a = .05.
(a) What design was employed?
(b) How many treatments were compared?
(c) How many observations were analyzed?
(d) At the .05 level of significance, can one conclude that there is a difference among treatments?
Why?
30. Consider the following ANOVA table.
Source SS d.f. MS V.R.
Treatments 5.05835 2 2.52917 1.0438
Error 65.42090 27 2.4230
Source SS d.f. MS V.R.
Treatments 231.5054 2 115.7527 2.824
Blocks 98.5000 7 14.0714
Error 573.7500 14 40.9821
(a) What design was employed?
(b) How many treatments were compared?
(c) How many observations were analyzed?
(d) At the .05 level of significance, can one conclude that the treatments have different effects?
Why?
31. In a study of the relationship between smoking and serum concentrations of high-density lipopro-
tein cholesterol (HDL-C), the following data (coded for ease of calculation) were collected from
samples of adult males who were nonsmokers, light smokers, moderate smokers, and heavy
smokers. We wish to know if these data provide sufficient evidence to indicate that the four pop-
ulations differ with respect to mean serum concentration of HDL-C. Let the probability of com-
mitting a type I error be .05. If an overall significant difference is found, determine which pairs
of individual sample means are significantly different.
Smoking Status
Nonsmokers Light Moderate Heavy
12 9 5 3
10 8 4 2
11 5 7 1
13 9 9 5
9954
91076
12 8 6 2
Subject No. Pre-op 3 Months 6 Months 12 Months
1 108.0 200.0 94.3 92.0
2 96.7 119.0 84.0 93.0
3 77.0 130.0 76.0 74.0
4 92.0 181.0 82.5 80.5
5 97.0 134.0 81.0 76.0
6 94.0 163.0 96.0 71.0
7 76.0 125.0 74.0 75.5
8 100.0 189.0 97.0 88.5
9 82.0 282.0 91.0 93.0
10 103.5 226.0 86.0 80.5
11 85.5 145.0 83.5 83.0
12 74.5 156.0 71.0 87.0
Source: Theodore K. Alexandrides, M.D. Used with permission.
382 CHAPTER 8 ANALYSIS OF VARIANCE
32. Polyzogopoulou et al. (A-34) report the effects of bariatric surgery on fasting glucose levels
(mmol/L) on 12 obese subjects with type 2 diabetes at four points in time: pre-operation, at 3
months, 6 months, and 12 months. Can we conclude, after eliminating subject effects, that fasting
glucose levels differ over time after surgery? Let a = .05.
33. Refer to Review Exercise 32. In addition to studying the 12 type 2 diabetes subjects (group 1),
Polyzogopoulou et al. (A-34) studied five subjects with impaired glucose tolerance (group 2), and
eight subjects with normal glucose tolerance (group 3). The following data are the 12-month post-
surgery fasting glucose levels for the three groups.
Group
1.0 92.0
1.0 93.0
1.0 74.0
1.0 80.5
1.0 76.0
1.0 71.0
1.0 75.5
1.0 88.5
1.0 93.0
1.0 80.5
1.0 83.0
1.0 87.0
2.0 79.0
2.0 78.0
2.0 100.0
2.0 76.5
2.0 68.0
3.0 81.5
3.0 75.0
3.0 76.5
(Continued)
REVIEW QUESTIONS AND EXERCISES 383
Group
3.0 70.5
3.0 69.0
3.0 73.8
3.0 74.0
3.0 80.0
Source: Theodore K. Alexandrides,
M.D. Used with permission.
Can we conclude that there is a difference among the means of the three groups? If so, which
pairs of means differ? Let for all tests.
For exercises 34 to 38 do the following:
(a) Indicate which technique studied in this chapter (the completely randomized design, the ran-
domized block design, the repeated measures design, or the factorial experiment) is appropriate.
(b) Identify the response variable and treatment variables.
(c) As appropriate, identify the factors and the number of levels of each, the blocking variables,
and the subjects.
(d) List any extraneous variables whose effects you think might be included in the error term.
(e) As appropriate, comment on carry-over and position effects.
(f) Construct an ANOVA table in which you indicate the sources of variability and the number
of degrees of freedom for each.
34. Johnston and Bowling (A-35) studied the ascorbic acid content (vitamin C) in several orange juice
products. One of the products examined was ready-to-drink juice packaged in a re-sealable, screw-
top container. One analysis analyzed the juice for reduced and oxidized vitamin C content at time
of purchase and reanalyzed three times weekly for 4 to 5 weeks.
35. A study by Pittini et al. (A-36) assessed the effectiveness of a simulator-based curriculum on 30
trainees learning the basic practice of amniocentesis. Pre- and post-training performance were eval-
uated with the same instrument. The outcome variable was the post-training scorepretraining
score. Trainees were grouped by years of postgraduate experience: PGY 0–2, PGY 3–5, Fellows,
and Faculty.
36. Anim-Nyame et al. (A-37) studied three sets of women in an effort to understand factors related
to pre-eclampsia. Enrolled in the study were 18 women with pre-eclampsia, 18 normal pregnant
women, and 18 nonpregnant female matched controls. Blood samples were obtained to measure
plasma levels of vascular endothelial growth factor, leptin, - plasma protein concentrations,
and full blood count.
37. In a study by lwamoto et al. (A-38) 26 women were randomly assigned to the medication alfacal-
cidol for treatment of lumbar bone mineral density (BMD). BMD of the lumbar spine was meas-
ured at baseline and every year for 5 years.
38. Inoue et al. (A-39) studied donor cell type and genotype on the efficiency of mouse somatic cell
cloning. They performed a factorial experiment with two donor cell types (Sertoli cells or cumu-
lus) and six genotypes. Outcome variables were the cleavage rate and the birth rate of pups in each
treatment combination.
aTNF
a = .05
384 CHAPTER 8 ANALYSIS OF VARIANCE
For the studies described in Exercises 39 through 66, do the following:
(a) Perform a statistical analysis of the data (including hypothesis testing and confidence inter-
val construction) that you think would yield useful information for the researchers.
(b) Determine p values for each computed test statistic.
(c) State all assumptions that are necessary to validate your analysis.
(d) Describe the population(s) about which you think inferences based on your analysis would be
applicable.
39. Shirakami et al. (A-40) investigated the clinical significance of endothelin (ET), natriuretic peptides,
and the renin-angiotensin-aldosterone system in pediatric liver transplantation. Subjects were chil-
dren ages 6 months to 12 years undergoing living-related liver transplantation due to congenital
biliary atresia and severe liver cirrhosis. Among the data collected were the following serum total
bilirubin (mg/dl) levels after transplantation (h–hours, d–days):
Time After Reperfusion of Donor Liver
Liver Anhepatic
Preoperative Transection Phase 1 h 2 h 4 h 8 h 1 d 2 d 3 d
6.2 1.2 0.9 0.8 1.1 1.5 2 1.4 1.6 1.3
17.6 11.9 9.3 3.5 3 6.1 9 6.3 6.4 6.2
13.2 10.2 7.9 5.3 4.9 3.3 3.6 2.8 1.9 1.9
3.9 3.3 3 2.9 2.3 1.4 1.2 0.8 0.8 0.9
20.8 19.4 * 9.4 8.4 6.8 7.1 3.7 3.8 3.2
1.8 1.8 1.6 1.4 1.4 1.1 1.9 0.7 0.8 0.7
8.6 6.5 4.8 3.1 2.1 1 1.3 1.5 1.6 3.2
13.4 12 10.1 5.8 5.6 4.5 4.1 3 3.1 3.6
16.8 13.9 8.3 3.7 3.7 2.2 2.1 1.9 3.1 4.1
20.4 17.8 17 10.8 9.3 8.9 7 2.8 3.8 4.8
25 21.5 13.8 7.6 7 5 11.5 12.3 10.1 11.4
9.2 6.3 6.8 5.3 4.8 0.2 4 4.2 3.7 3.5
8 6.5 6.4 4.1 3.8 3.8 3.5 3.1 2.9 2.8
2.9 3 4.1 3.4 3.4 3.7 4.2 3.3 2 1.9
21.3 17.3 13.6 9.2 7.9 7.9 9.8 8.6 4.7 5.5
25 25 24 20.1 19.3 18.6 23.6 25 14.4 20.6
23.3 23.7 15.7 13.2 11 9.6 9.3 7.2 6.3 6.3
17.5 16.2 14.4 12.6 12.7 11.5 10 7.8 5.5 4.9
*Missing observation.
Source: Dr. Gotaro Shirakami. Used with permission.
Note that there is a missing observation in the data set. You may handle this problem in at least
three ways.
(a) Omit the subject whose datum is missing, and analyze the data of the remaining 17 subjects.
(b) Use a computer package that automatically deals with missing data.
(c) Analyze the data using a missing data procedure. For such a procedure, see Jerome L. Myers
and Arnold D. Well, Research Design and Statistical Analysis, Erlbaum Associates, Hillsdale, NJ,
1995, pp. 256–258.
REVIEW QUESTIONS AND EXERCISES 385
40. The purpose of a study by Sakakibara and Hayano (A-41) was to examine the effect of volun-
tarily slowed respiration on the cardiac parasympathetic response to a threat (the anticipation
of an electric shock). Subjects were 30 healthy college students whose mean age was 23 years
with a standard deviation of 1.5 years. An equal number of subjects were randomly assigned
to slow (six males, four females), fast (seven males, three females), and nonpaced (five males,
five females) breathing groups. Subjects in the slow- and fast-paced breathing groups regulated
their breathing rate to 8 and 30 cpm, respectively. The nonpaced group breathed spontaneously.
The following are the subjects’ scores on the State Anxiety Score of State-Trait Anxiety Inven-
tory after baseline and period of threat:
Slow Paced Fast Paced Nonpaced
Baseline Threat Baseline Threat Baseline Threat
39 59 37 49 36 51
44 47 40 42 34 71
48 51 39 48 50 37
50 61 47 57 49 53
34 48 45 49 38 52
54 69 43 44 39 56
34 43 32 45 66 67
38 52 27 54 39 49
44 48 44 44 45 65
39 65 41 61 42 57
41. Takahashi et al. (A-42) investigated the correlation of magnetic resonance signal intensity with
spinal cord evoked potentials and spinal cord morphology after 5 hours of spinal cord compres-
sion in cats. Twenty-four adult cats were divided into four groups on the basis of a measure of
spinal cord function plus a control group that did not undergo spinal compression. Among the data
collected were the following compression ratio [(sagittal diameter/transverse diameter) ] val-
ues after 5 hours of compression:
* 100
Control 80.542986 Group III 36.923077
79.111111 31.304348
70.535714 53.333333
87.323944 55.276382
80.000000 40.725806
82.222222
Group IV 66.666667
Group I 83.928571 29.565217
84.183673 12.096774
48.181818 34.274194
98.461538 24.000000
Group II 30.263158
34.865900
43.775100
82.439024
Source: Dr. Toshiaki Takahashi.
Used with permission.
Source: Dr. Masahito
Sakakibara. Used
with permission.
386 CHAPTER 8 ANALYSIS OF VARIANCE
42. The objective of a study by Yamashita et al. (A-43) was to investigate whether pentoxifylline admin-
istered in the flush solution or during reperfusion would reduce ischemia-reperfusion lung injury in
preserved canine lung allografts. Three groups of animals were studied. Pentoxifylline was not admin-
istered to animals in group 1 (C), was administered only during the reperfusion period (P) to animals
in group 2, and was administered only in the flush solution to animals in group 3 (F). A total of 14
left lung allotransplantations were performed. The following are the aortic pressure readings for each
animal during the 6-hour assessment period:
43. In a study investigating the relative bioavailability of beta-carotene (BC) and alpha-carotene (AC)
from different sources of carrots, Zhou et al. (A-44) used ferrets as experimental animals. Among
the data collected were the following concentrations of BC, AC, and AC/BC molar ratios in the
sera of 24 ferrets provided with different sources of carotenoids for 3 days in their drinking water:
0 60 120 180 240 300 360
Group min min min min min min min
C 85.0 100.0 120.0 80.0 72.0 75.0 *
C 85.0 82.0 80.0 80.0 85.0 80.0 80.0
C 100.0 75.0 85.0 98.0 85.0 80.0 82.0
C 57.0 57.0 57.0 30.0 * * *
C 57.0 75.0 52.0 56.0 65.0 95.0 75.0
P 112.0 67.0 73.0 90.0 71.0 70.0 66.0
P 92.0 70.0 90.0 80.0 75.0 80.0 *
P 105.0 62.0 73.0 75.0 70.0 55.0 50.0
P 80.0 73.0 50.0 35.0 * * *
F 70.0 95.0 105.0 115.0 110.0 105.0 100.0
F 60.0 63.0 140.0 135.0 125.0 130.0 120.0
F 67.0 65.0 75.0 75.0 80.0 80.0 80.0
F 115.0 107.0 90.0 103.0 110.0 112.0 95.0
F 90.0 99.0 102.0 110.0 117.0 118.0 103.0
*Missing observation.
Source: Dr. Motohiro Yamashita. Used with permission.
BC AC AC/BC
Unheated Juice
0.637 0.506 0.795
0.354 0.297 0.840
0.287 0.249 0.869
0.533 0.433 0.813
0.228 0.190 0.833
0.632 0.484 0.767
Heated Juice
0.303 0.266 0.878
0.194 0.180 0.927
(mol>mol)(Mmol>g)(Mmol>g)
(Continued)