B I O S T A T I
S T I C S
CHAIR AND DEPARTMENT OF COMPUTER
SCIENCE AND STATISTICS
2
1. Pathology
The data in Table 2.10 are measurements from a group of 10 normal males and 11 males with
left heart disease taken at autopsy at a particular hospital. Measurements were made on
several variables at that time, and the table presents the measurements on total heart weight
(THW) and total body weight (BW). Assume that the diagnosis of left heart disease is made
independently of these variables. Please check, if really left heart disease increases total heart
weight.
Table 2.10 Autopsy data
Left heart disease males
Normal males
Observation
number
THW (g)
BW
(kg)
Observation
number
THW (g)
BW (kg)
1
450
54.6
1
245
40.8
2
760
73.5
2
350
67.4
3
325
50.3
3
340
53.3
4
495
44.6
4
300
62.2
5
285
58.1
5
310
65.5
6
450
61.3
6
270
47.5
7
460
75.3
7
300
51.2
8
375
41.1
8
360
74.9
9
310
51.5
9
405
59.0
10
615
41.7
10
290
40.5
11
425
59.7
3
2. Infectious Disease
The data in Table 2.11 are a sample from a larger dataset collected on persons discharged
from a selected Pennsylvania hospital as part of a retrospective chart review of antibiotic
usage in hospitals.
Please give the answer for following questions:
1.
2.
3.
4.
5.
6.
Is there the difference in age between males and females?
Does age influences first temperature after admission?
Is there the dependence between type of service and gender?
Is there the difference in WBC level between antibiotic treatment groups?
Is there the linear correlation between duration of stay and age?
Calculate the probability that the length of the hospital stay will be greater than 20
days.
Table 2.11 Hospital stay data
ID
no.
Duration
of
hospital
stay
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5
10
6
11
5
14
30
11
17
3
9
3
8
8
5
5
7
4
3
7
9
11
11
9
4
Age
Sex
(1 = M,
2 = F)
First
temp.
following
admission
First
WBC
(x 103)
following
admission
Received
antibiotic
(1 = yes.
2 = no)
Received
bacterial
culture
(1 = yes,
2 = no)
Service
(1 = med.,
2 = surg.)
30
73
40
47
25
82
60
56
43
50
59
4
22
33
20
32
36
69
47
22
11
19
67
43
41
2
2
2
2
2
1
1
2
2
1
2
1
2
2
2
1
1
1
1
1
1
1
2
2
2
99.0
98.0
99.0
98.2
98.5
96.8
99.5
98.6
98.0
98.0
97.6
97.8
99.5
98.4
98.4
99.0
99.2
98.0
97.0
98.2
98.2
98.6
97.6
98.6
98.0
8
5
12
4
11
6
8
7
7
12
7
3
11
14
11
9
6
6
5
6
10
14
4
5
5
2
2
2
2
2
1
1
2
2
2
2
2
1
1
2
2
1
2
1
2
2
1
2
2
2
2
1
2
2
2
2
1
2
2
1
1
2
2
1
1
2
2
2
2
2
2
2
2
2
2
1
1
2
2
2
2
1
1
1
2
1
2
2
2
2
2
2
2
1
2
2
2
1
2
1
4
3. Cerebrovascular Disease
One problem with using the angiogram to diagnose stroke is the slight risk of mortality
associated with this test (< 1%). Some investigators have attempted to use the PET scanner
(which measures blood flow in the brain) to detect stroke disease noninvasively as an
alternative to the angiogram. A comparison was made on the same patients between these two
methodologies for detecting stroke, with the results given in Table 3.3.
Table 3.3 Comparison of a noninvasive test for detecting stroke with an angiogram
-
Noninvasive
test
+
+
-
3
+
+
32
Angiogram
5
Frequency
21
8
4. Hypertension
Hypertensive patients are screened at a neighborhood health clinic and are given methyl dopa,
a strong antihypertensive medication for their condition. They are asked to come back 1 week
later and have their blood pressures measured again. Suppose the initial and follow-up
systolic blood pressures of the patients are given in Table 6.9.
To test the effectiveness of the drug, we want to answer the question if the difference
between initial and follow-up blood pressures is statistically significant.
Table 6.9 Initial and follow-up systolic bp (mm Hg) for hypertensive patients given methyl
dopa
Patient no.
1
2
3
4
5
6
7
8
9
10
Initial systolic bp
200.0
194.0
236.0
163.0
240.0
225.0
203.0
180.0
177.0
240.0
Follow-up systolic bp
188.0
212.0
186.0
150.0
200.0
222.0
190.0
154.0
180.0
225.0
6
5. Epidemiology
Height and weight are often used in epidemiological studies as possible predictors of disease
outcomes. If the persons in the study are assessed in a clinic, then heights and weights are
usually measured directly. However, if the persons are interviewed at home or by mail, then a
person's self-reported height and weight are often used instead. Suppose we conduct a study
on 10 people to test the comparability of these two methods. The data for weight are given in
Table 7.4.
Table 7.4 A comparison of self-reported and measured weight (lb) for 10 subjects
SelfPerson
Measured
reported
number
weight
weight
1
120
125
Difference
-5
2
120
118
+2
3
135
139
-4
4
118
120
-2
5
120
125
-5
6
190
198
-8
7
124
128
-4
8
175
176
-1
9
133
131
+2
10
125
125
0
7
6. Environmental Health, Pediatrics
A topic of recent interest is the short- and long-term health effects of exposure to various
agents in the environment. A study was performed by Landrigan et al. to look at the effect of
chronic exposure to low levels of lead in children who lived near a lead smelter in El Paso,
Texas. A lead absorption group and a control group of children were identified by the amount
of lead in blood samples, and various neurology, and psychological tests were performed to
compare the two groups. In particular, the WISC performance IQ test was performed on 34
children, 5 years of age and older, in the lead absorption group and 36 comparably aged
children in the control group. Assess the statistical significance of the IQ test results, which
are given in Table 8.3.
Table 8.3 Results of WISC performance IQ test for exposed and control children
Mean
Lead absorption 96.44
Control
103.29
8
sd
n
13.74
34
17.87
36
7. Hypertension
An investigator wishes to determine if sitting upright in a chair vs. lying down on a bed will
affect a person's blood pressure. The investigator decides to use each of 10 patients as his or
her own control and collects systolic blood pressure data in both the sitting and lying positions
as given in Table 8.9.
Table 8.9 Effect of position on level of bp (mm Hg)
Patient
Sitting upright
Lying down
1
142
154
2
100
106
3
112
110
4
92
100
5
104
112
6
100
100
7
108
120
8
94
90
9
104
104
10
98
114
9
8. Pharmacology
One method for assessing the effectiveness of a drug is to note its concentration in blood
and/or urine samples at certain periods of time after giving the drug. Suppose we wish to
compare the concentrations of two types of aspirin (types A and B) in urine specimens taken
from the same person, 1 hour after he or she has taken the drug. Hence we give a specific
dosage of either type A or type B aspirin at one time and measure the 1-hour urine
concentration. One week later, after the first aspirin has presumably been cleared from the
system we give the same dosage of the other aspirin to the same person and note the 1-hour
urine concentration. Since the order of giving the drugs may affect the results, we use a table
of random numbers to decide which of the two types of aspirin to give first. We perform this
experiment on 10 people; the results are given in Table 8.10
Table 8.10. Concentration of aspirin in urine samples
Person
Aspirin A
1-hour
concentration
(mg%)
Aspirin B
1-hour
concentration
(mg%)
1
15
13
2
26
20
3
13
10
4
28
21
5
17
17
6
20
22
7
7
5
8
36
30
9
12
7
10
18
11
Mean
19.20
15.60
sd
8.63
7.78
10
9. Hypertension
Blood pressure measurements taken on the left and right arms of a person are assumed to be
comparable. To test this assumption, 10 volunteers are obtained and systolic blood pressure
readings are taken simultaneously on both arms by two different observers, Mr. Jones for the
left arm and Mr. Smith for the right arm. The data are given in Table 8.11.
Table 8.11 Effect of arm on level of blood pressure (mm Hg)
Patient Left arm
Right arm
1
130
126
2
120
124
3
135
127
4
100
95
5
98
102
6
110
109
7
123
124
8
136
132
9
140
137
10
155
156
11
10. Nutrition
A hypothesis of current clinical interest is that vitamin C prevents the common cold. A study
is organized to test this hypothesis using 20 prisoners as participants. In the study 10 are
randomly allocated to receive vitamin C capsules and 10 are randomly allocated to receive
placebo capsules. The number of colds over a 12-month period for each participant is given in
Table 8.12. We wish to test the hypothesis that vitamin C prevents the common cold.
Table 8.12 Number of colds over a 12-month period for persons taking vitamin C and placebo
capsules
Vitamin C
i
Placebo
xi1
i
xi2
Di(xi1 – xi2)
1
4
1
7
-3
2
0
2
8
-8
3
3
3
4
-1
4
4
4
6
-2
5
4
5
6
-2
6
3
6
4
-1
7
4
7
6
-2
8
3
8
4
-1
9
2
9
6
-4
10
6
10
6
0
12
11. Hypertension
A recent study by the Lipid Research Clinics looked at the relationship between alcohol
consumption and level of systolic blood pressure in women not using oral contraceptives.
Alcohol consumption was categorized as follows: no alcohol use; <=10oz per week alcohol
consumption; >10oz per week alcohol consumption. The results for women 30-39 years of
age are given in Table 8.15. Do we observe statistically significant difference in systolic
blood pressure between analyzed groups?
Table 8.15 Relationship of systolic blood pressure and alcohol consumption in 30—39-yearold women not using oral contraceptives
A. No alcohol use
B. ≤ 10 oz per week
alcohol
consumption
C. > 10 oz per week
alcohol
consumption
Systolic blood
pressure (mm
Hg)
Mean sd
n
110.5 13.3 357
109.1 13.4 440
114.5
13
14.9
23
12. Hypertension
A study of the relationship between salt intake and blood pressure of infants is in the planning
stages. A pilot study is done, comparing 5 1-year-old infants on a high-salt diet with 5 1-yearold infants on a low-salt diet. The results are given in Table 8.20. Is there statistically
significant difference in systolic blood pressure between both groups of infants?
Table 8.20 Relationship between salt intake and level of blood pressure
Mean systolic bp
sd systolic bp
n
Highsalt
90.8
diet
10.3
5
14
Low-salt diet
87.2
9.2
5
13. Sports Medicine
Many tennis players develop acute lateral epicondylitis whereby they experience acute elbow
pain (tennis elbow). A variety of nonsurgical treatments are used for this condition, including
rest, heat, and anti-inflammatory agents. A clinical trial was set up to compare the effects of
the anti-inflammatory agent ibuprofen (Motrin) with those of a placebo in the treatment of
tennis elbow. Patients were given 400 mg of ibuprofen orally or an identical looking placebo
4 times per day and were evaluated after 3 weeks of treatment. Patients subjectively rated
their pain as (1) worse, (2) unchanged, (3) slightly improved, (4) moderately improved, (5)
mostly improved, and (6) completely improved. The results are given in Table 9.6.
Table 9.6 A comparison of pain during maximal activity after 3 weeks of therapy with Motrin
and placebo as compared with baseline
Treatment
group
Total
Worse
Unchanged
Slightly
improved
Moderately
Improved
Mostly
improved
Completely
improved
Motrin
43
0
14
7
10
10
2
Placebo
44
5
20
11
3
5
0
15
14. Psychiatry
Much attention has been given in recent years to the role of transcendental meditation in
improving health, particularly in lowering blood pressure. One hypothesis that emerges from
this work is that transcendental meditation might also be useful in treating psychiatric patients
with symptoms of anxiety. Suppose that a protocol of meditational therapy is administered
once per day to 20 patients with anxiety. The patients are given a psychiatric exam at baseline
and at a follow-up exam 2 months later. The degree of improvement is rated on a 10-point
scale, with 1 indicating the most improvement and 10 the least improvement. Similarly, 26
comparably affected patients with anxiety are given standard psychotherapy and are asked to
come back 2 months later for a follow-up exam. The results are given in Table 9.8.
Is meditation comparably effective in lowering blood pressure as psychotherapy is?
Table 9.8 Degree of improvement in patients with anxiety who are treated with transcendental
meditation or psychotherapy
Meditation Psychotherapy
d*
1
2
3
4
5
6
7
8
9
10
Ġ
d*
Ġ
3
1
0
4
2
2
7
3
5
3
4
3
2
5
8
1
6
4
0
7
2
0
8
1
0
9
1
0
10
0
* d = degree of
improvement
† ƒ = frequency
16
15. Psychiatry
Suppose we are conducting a study of the effectiveness of lithium therapy for manicdepressive patients. The study is carried out at two different centers, and we wish to determine
if the patient populations are comparable at baseline. We administer a self-rating
questionnaire to the prospective patients at the two different centers about their general
psychological well-being, in which the outcome measure is a four-category scale as follows:
(1) = feel good; (2) = usually feel good, once in a while feel nervous; (3) = feel nervous half
the time; (4) = usually feel nervous. Suppose the data at the two different centers are as
follows:
Center 1
3, 4, 1, 1, 3, 2, 3, 4, 4, 3, 2, 4, 4, 4
Center 2
1, 2, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3
17
16. Obstetrics
A subject of recent research in obstetrics is the medical management of women who go into
labor prematurely, which we will define as labor occurring between 20 and 36 weeks of
pregnancy. Premature labor is undesirable because infants born prematurely have a higher rate
of infant mortality and morbidity. Terbutaline is a drug that is effectively used to arrest labor
and prolong the duration of the pregnancy beyond 36 weeks. However, it sometimes has
undesirable cardiovascular and metabolic side effects, such as tremors and hyperglycemia,
which limit its use for some women. Another possibility is to give terbutaline in combination
with another drug (metoprolol), which would neutralize these side effects.
However, before using this combination therapy, we want to compare its effectiveness in
prolonging pregnancy with that of terbutaline alone. Suppose that we set up a clinical trial to
compare the effectiveness of the two therapies, where the outcome is whether the pregnancy
lasts beyond 36 weeks. We find that in 200 patients treated with terbutaline alone, 130 have a
pregnancy lasting more than 36 weeks, whereas in 200 patients treated with terbutaline and
metoprolol, 120 have a pregnancy lasting more than 36 weeks. Assess the significance of
these results. Is combined therapy similarly effective In arresting labor as terbutaline alone?
Duration of pregnancy
>36 weeks
≤36 weeks
130
70
120
80
250
150
Treatment group
Terbutaline
Terbutaline and metoprolol
Total
18
Total
200
200
400
17. Cardiovascular Disease - Nutrition
Suppose we do a retrospective study on the deaths of all men aged 50-54 in a specific county
over a 1-month period. We find that of 35 persons who died from CVD, 5 were on a high-salt
diet before they died, whereas of 25 persons who died from other causes, 2 were on such a
diet. These data, presented in Table 10.9, are in the form of a 2 x 2 contingency table, and
thus the methods of Section 10.2.2 may be applicable. We find, however, that the expected
values of this table are too small to validly use such methods.
Cause of
death
Non-CVD
CVD
Total
Type c
High salt
2
5
7
f diet
Low salt
23
30
53
19
Total
25
35
60
18. Hypertension
A recent phenomenon in the recording of blood pressure is the development of the automated
blood pressure machine, where for a small fee a person can sit in a booth and have his or her
blood pressure measured by a computer device. A study is conducted to compare the
computer device with standard methods of measuring blood pressure. Twenty patients are
recruited, and their hypertensive status is assessed by both the computer device and a trained
observer. Hypertensive status is defined as either hypertensive (+), if either systolic bp > 160
or diastolic bp > 95, or normotensive ( —) otherwise. The data are given in Table 10.16.
Assess the statistical significance of these data.
Hypertensive status
Hypertensive status
Person
Computer
device
Trained
observer
Person
Computer
device
Trained
observer
1
-
-
11
+
-
2
-
-
12
+
-
3
+
-
13
-
-
4
+
+
14
+
-
5
-
+
15
-
+
6
+
-
16
+
-
7
-
-
17
+
-
8
+
+
18
-
-
9
+
+
19
-
-
10
-
-
20
-
-
Computer Trained observer
device
+
-
+
-
3.
7
1
9
20
19. Cancer
Suppose we wish to compare the following two treatments for breast cancer: simple
mastectomy (S) and radical mastectomy (R). We form matched pairs of women who are
within the same decade of age and with the same clinical condition. They receive the two
treatments, and we monitor their subsequent 5-year survival. The results are given in Table
10.26. We wish to test for significant differences between the treatments.
Table 10.26 Comparison of simple and radical mastectomy in treating breast cancer
Pair
Treatment S Treatment
woman
R woman
1
L*
L
2
L
D
3
L
L
4
L
L
5
L
L
6
D†
L
7
L
L
8
L
D
9
L
D
10
L
L
11
D
D
12
L
D
13
L
L
14
L
L
15
L
D
16
L
L
17
L
D
18
L
D
19
L
L
20
L
D
* L = lived at least 5 years
† D = died within 5 years
21
20. Obstetrics
Greene and Touchstone conducted a study to relate birthweight to the estriol level of pregnant
women. The data from the study are listed in Table 11.1. We want to check , if there exists
linear relationship between estriol level and birthweight.
Estriol
Birthweight
(mg/24 hr) (g/100)
Estriol
(mg/24 hr)
Birthweight
(g/100)
i
xi
yi
i
xi
yi
1
7
25
17
17
32
2
9
25
18
25
32
3
9
25
19
27
34
4
12
27
20
15
34
5
14
27
21
15
34
6
16
27
22
15
35
7
16
24
23
16
35
8
14
30
24
19
34
9
16
30
25
18
35
10
16
31
26
17
36
11
17
30
27
18
37
12
19
31
28
20
38
13
21
30
29
22
40
14
24
28
30
25
39
15
15
32
31
24
43
16
16
32
22
21. Cardiology
The following data are given for 27 patients with acute dilated cardiomyopathy .
1. Test for the significance of the regression line
2. What is the expected LVEF for a 45-year-old patient with this condition?
Patient
number
Age
x
Left ventricular ejection
fraction (LVEF)
1
35
0.19
2
28
0.24
3
25
0.17
4
75
0.40
5
42
0.40
6
19
0.23
7
54
0.20
8
35
0.20
9
30
0.30
10
65
0.19
11
26
0.24
12
56
0.32
13
60
0.32
14
47
0.28
15
50
0.24
16
43
0.18
17
30
0.22
18
56
0.23
19
23
0.14
20
26
0.14
21
58
0.30
22
65
0.07
23
34
0.12
24
63
0.13
25
23
0.17
26
23
0.24
27
46
0.19
23
22. Pathology
Two pathologists independently review biopsy specimens from 40 persons with self-reported
malignant melanoma as ascertained by mail questionnaire. The results are given in Table
11.18. Are there any statistically significant differences between their diagnosis?
Pathologist 1 Pathologist 2 diagnosis
diagnosis
+
Total
+
24
8
32
4
4
8
Total
28
12
40
24
23. Environmental Health
Suppose we are interested in the linear relationship between carbon monoxide concentrations
and the density of cars in some geographical area. We measure the number of cars per hour to
the nearest 500 cars per hour and the concentration of carbon monoxide (CO) in parts per
million at a particular street corner and group the data by cars per hour. The data are given in
Table 11.20. Please check the statistical significance of the results.
Cars/hour
(x 103)
CO concentrations
Number of
samples
1.0
9.0
6.8
7.7
3
1.5
9.6
6.8 11.3
3
2.0
12.3 11.8
2
3.0
20.7 19.2 21.6 20.6
4
25
24. Cardiovascular Disease
Many studies have demonstrated that the level of HDL cholesterol (high-density lipoprotein
cholesterol) is positively related to alcohol consumption. This relationship has intrigued
researchers because the level of HDL cholesterol is inversely correlated with the incidence of
heart disease.
One possible mechanism was explored by Kuller et al. in an analysis of participants in the
MRFIT study relating the level of HDL cholesterol to the level of SGOT, a parameter
commonly used to assess liver function . The data in Table 11.29 were presented. Is there
statistically significant relationship between analyzed parameters?
SGOT
<9
10-12
13-14
15-16
17-18
19-20
≥21
Mean HDL
cholesterol
40.0
41.2
42.3
42.8
43.8
43.6
46.5
(Reprinted with permission of
the American Journal of
Epidemiology 117(4): 406418,1983.)
26
25. Hypertension
Adults show a strong relationship between blood pressure and age. Data were collected from
schoolchildren in Muscatine, Iowa, to see if this relationship continued to hold in the age
group 5-18. The data in Table 11.25 were obtained from boys in this age group. Do we
observe linear dependence between presented data?
Age, x
Mean systolic
blood pressure, y
(mm Hg)
5
6
7
8
9
10
11
12
13
14
15
16
17
18
94.4
97.7
101.9
104.5
106.3
109.3
112.6
113.8
117.7
121.6
122.3
123.6
124.9
131.0
∑ xi = 161 ∑ xi2 = 2079
∑ yi =
∑ yi2 = 180.271
1581.6
∑ xi yi
=18.787.3
27
26. Pulmonary Disease
A topic of recent public health interest is whether or not passive smoking (i.e., exposure to
cigarette smoke in the atmosphere among nonsmokers) has a measurable effect on pulmonary
health. White and Froeb studied this question by measuring pulmonary function in several
ways in the following six groups [1]:
1. Nonsmokers (NS) People who themselves did not smoke and were not exposed to cigarette
smoke either at home or on the job.
2. Passive Smokers (PS) People who themselves did not smoke and were not exposed to
cigarette smoke in the home, but were employed for 20 or more years in an enclosed working
area that routinely contained tobacco smoke.
3. Noninhaling Smokers (NI) People who smoked pipes, cigars, or cigarettes, but who did not
inhale.
4. Light Smokers (LS) People who smoked and inhaled 1-10 cigarettes per day for 20 or more
years. (Note: There are 20 cigarettes in a pack.)
5. Moderate Smokers (MS) People who smoked and inhaled 11 - 39 cigarettes per day for 20
or more years.
6. Heavy Smokers (HS) People who smoked and inhaled 40 or more cigarettes per da\ for 20
or more years.
A principal measure used by the authors to assess pulmonary function was forced midexpiratory flow (FEF). The authors were interested in comparing FEF in the six groups.
Group Group
number name
1
2
3
4
5
6
Mean FEF
(l/s)
sd FEF
(l/s)
N
3.78
3.30
3.32
3.23
2.73
2.59
0.79
0.77
0.86
0.78
0.81
0.82
200
200
50
200
200
200
NS
PS
NI
LS
MS
HS
28
27. Ophthalmology
Arachidonic acid is well known to have an effect on ocular metabolism.
In particular, topical application of arachidonic acid has caused lid closure, itching, and ocular
discharge, among other effects. A study was conducted to compare the anti-inflammatory
effects of four different drugs in Albino rabbits after administration of arachidonic acid [3].
For all groups the anti-inflammatory agent was administered to one eye and a saline solution
was administered to the other eye. Ten minutes later arachidonic acid (sodium arachidonate)
was delivered to both eyes. Both eyes were evaluated every 15 minutes thereafter for lid
closure.
At each assessment the lids of both eyes were examined and a lid closure score from 0-3 was
determined, where 0 = eye completely open, 3 = eye completely closed, and 1,2 = intermediate states. The measure of effectiveness (x) is the change in lid closure scores (from baseline
to
follow-up) in the treated eye minus the change in lid closure scores in the saline eye. A high
value for x is indicative of an effective drug. The data, after 15 minutes of follow-up, are presented in Table 12.6. Which drugs are effective in the treatment of inflammatory status?
Rabbit Indomethacin
numbe
Score* Rank
r
1
+2
13.5
Aspirin
Piroxicam
BW755C
Score
Rank
Score
Rank
Score
Rank
+1
9.0
+3
20.0
+1
9.0
2
+3
20.0
+3
20.0
+1
9.0
0
4.0
3
+3
20.0
+1
9.0
+2
13.5
0
4.0
4
+3
20.0
+2
13.5
+1
9.0
0
4.0
5
+3
20.0
+2
13.5
+3
20.0
0
4.0
6
0
4.0
+3
20.0
+3
20.0
+1
1.0
29
28. Nutrition, Arthritis
A comparison was made of the protein intake among three groups of premenopausal women:
(1) women eating a standard American diet (STD), (2) women eating a lacto-ovo-vegetarian
diet (LAC), and (3) women eating a strict vegetarian diet (VEG). The averaged results for
protein intake (mg) are presented in Table 12.11. Is there statistically significant difference in
protein intake between diet groups?
Group
Mean
sd
n
STD
75
9
10
LAC
57
13
10
VEG
47
17
6
30