Vidakovic B. Statistics for Bioengineering Sciences: With Matlab and WinBugs Support

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10.9 Exercises 399

10.11. A. pantherina and A. rubescens. Making spore prints is an enormous

help in identifying genera and species of mushrooms. To make a spore print,

mushroom fans take a fresh, mature cap and lay it on a clean piece of glass.

Left overnight or possibly longer the cap should give you a good print.

The Amanitas Family is one that has the most poisonous (A. phalloides,

A. verna, A. virosa, A. pantherina, etc.), as well as the most delicious (A. ce-

sarea, A. rubescens) species. Two independent samples of m

=12 and n =15

spores of A. pantherina (“Panther”) and A. rubescens (“Blusher”), respec-

tively, were analyzed. In both species of mushrooms the spores are smooth

and elliptical and the largest possible measurement was taken (great axis

of the ellipse). It was found that the means were

= 6.3 microns and

= 7.5 microns with standard deviations of s

= 2.12 µm and s

= 1.94

µm.

(a) A researcher is interested in testing the hypothesis that the population

mean sizes of spores for these two mushrooms,

and µ

, are the same,

versus the two-sided alternative. Use

α =0.05.

(b) What sample sizes are needed so that the researcher should be able to

reject the null hypothesis of no effect (no difference between the population

means) with power 1

−β =0.9 versus the “medium” standardized effect size

of d

=0.5? The signiﬁcance α is, as in (a), 0.05.

10.12. Blood Volume in Infants. The total blood volume of normal newborn in-

fants was estimated by Schücking (1879) who took into account the addition

of placental blood to the circulation of the newborn infant when clamping

of he umbilical cord is delayed. Demarsh et al. (1942) further studied the

importance of early and late clamping. For 16 babies in whom the cord was

clamped early the total blood (as a percentage of weight) on the third day

was

13.8 8.0 8.4 8.8 9.6 9.8 8.2 8.0

10.3 8.5 11.5 8.2 8.9 9.4 10.3 12.6

For 16 babies in whom the cord was not clamped until the placenta began

to descend, the corresponding ﬁgures were

13.8 8.0 8.4 8.8 9.6 9.8 8.2 8.0

10.3 8.5 11.5 8.2 8.9 9.4 10.3 12.6

(a) Do these two samples provide evidence of a signiﬁcant difference be-

tween the blood volumes? Perform the test at

α =0.05.

(b) Using WinBUGS ﬁnd the posterior probability of the hypothesis that

there is no difference in blood volumes. Use noninformative priors.

10.13. Biofeedback. In the past, many bodily functions were thought to be be-

yond conscious control. However, recent experimentation suggests that it

400 10 Two Samples

may be possible for a person to control certain bodily functions if that per-

son is trained in a program of biofeedback exercises. An experiment is con-

ducted to show that blood pressure levels can be consciously reduced in

people trained in this program. The blood pressure measurements (in mil-

limeters of mercury) listed in the table represent readings before and after

the biofeedback training of ﬁve subjects.

Subject Before After

1 137 130

2 201 180

3 167 150

4 150 153

5 173 162

(a) If we want to test whether the mean blood pressure decreases after the

training, what are the appropriate null and alternative hypotheses?

(b) Perform the test in (a) with

α =0.05.

10.14. Hypertension. Dernellis and Panaretou (2002) examined a small number

of subjects with hypertension and healthy control subjects. One of the vari-

ables of interest was the aortic stiffness index. Measures of this variable

were calculated from the aortic diameter evaluated by M-mode echocardio-

graphy and blood pressure measured by a sphygmomanometer. Generally,

physicians wish to reduce aortic stiffness. From n

= 15 patients with hy-

pertension (group 1), the mean aortic stiffness index was

= 16.16 with

a standard deviation of s

=4.29. In the n

=16 control subjects (group 2),

the mean aortic stiffness index was

= 10.53 with a standard deviation

of s

=3.33.

(a) Test the hypothesis that the population mean aortic stiffness indices for

the two groups differ (use the two-sided alternative). Perform the test by

both the rejection-region and p-value methods. Take

α = 0.05 and assume

that the population variances are the same.

(b) What is the (observed, retrospective)

power of this test versus the spe-

ciﬁc alternative

∆ =|µ

−µ

|=3. Keep α =0.05.

orate clinical trials of the same type. What group sample sizes are needed

to assure a 99% power versus the ﬁxed alternative

∆ =|µ

−µ

|=3?

10.15. Hemopexin in DMD Cases II. Refer to Exercises 2.16 and 9.16 and data

set

dmd.dat|mat|xls.

(a) Form a 95% conﬁdence interval for the difference in mean responses

of hemopexin

h in two populations: population 1 consisting of all women

Attributes observed and retrospective stand for the power analysis of a test that has al-

ready been performed. The procedure is the same as for the prospective analysis but elicited

σs are replaced by observed estimators s.

10.9 Exercises 401

who are not DMD carriers (carrier=0) and population 2 consisting of all

women who are DMD carriers (

carrier=1).

(b) It is believed that the mean level of hemopexin in the population of

women DMD carriers exceeds the mean level in women noncarriers by 10.

Test the one-sided alternative that the mean levels of hemopexin for the

two populations differ by more than 10, that is, H

: µ

−µ

= −10 versus

: µ

−µ

< −10. Use α = 5%. Report the p-value as well. Compare the

population variances prior to deciding how to test the difference of means.

Hint:

In testing the hypothesis H

: µ

−µ

= C, the associated t-statistic

has as its numerator

−X

−C instead of X

−X

10.16. Risk of Stroke. Abbott et al. (1986) evaluate the risk of stroke among

smokers and nonsmokers in a 12-year prospective Honolulu Hearth Pro-

gram study. The data are given in the table below.

Stroke yes Stroke no total

Smoker 171 3264 3435

Nonsmoker 117 4320 4437

Total 288 7584 7872

Estimate

(a) the risk difference and ﬁnd the 95% conﬁdence interval for the popula-

tion risk difference,

(b) the risk ratio and ﬁnd the 95% conﬁdence interval for the population

risk ratio, and

odds ratio.

10.17. Cell Counts. Refer to Example 1.2 and assume that data is approximately

normal.

(a) Do automated and manual counts signiﬁcantly differ for a ﬁxed conﬂu-

ency level? Use a paired t-test with a two-sided alternative at signiﬁcance

level

α =0.05.

(b) What are 95% conﬁdence intervals for the population differences?

error, are the errors comparable for the two conﬂuency levels? This is a

DoD (difference-of-differences) test and is equivalent to a two-sample t-test

when measurements are the differences. Use the two-sided alternative and

α =0.05.

10.18. Impulses from Crayﬁsh. The crayﬁsh is utilized in numerous neuro-

science labs in order to explore the role of the central pattern generator

in locomotion. In an experiment with crayﬁsh, you design glass electrodes

to record from the abdominal ganglia nerves that contribute to the return

and power strokes of the swimmerets. What this means is that the nerves

you are recording from contain the motor output signals that tell the swim-

merets to move water around their abdomen anterior or posterior. In the

402 10 Two Samples

experimental setup, glass extracellular electrodes are used to measure the

action potentials or voltage waveform. Because this is a CPG, you normally

expect bursting activity, but when you suction onto the nerve cord, you no-

tice what seems to be regular spiking. A spike detector was applied on the

voltage time trace and the interspike intervals recorded. The data were ob-

tained under two treatments. (1) Drug: Carbachol (also known as carbamyl-

choline), a drug that binds and activates the acetylcholine receptor, was

added to possibly induce faster spiking; and (2) control group. The data ﬁles

carbachol.dat and control.dat containing the respective interspike

times can be found at the book’s Web site. Also, you can load both ﬁles by

importing

frankdata.mat or frankdata.xls. All interspike time mea-

surements are given in seconds. (Thanks to Dr. Frank Lin for the data and

description.)

(a) Find fundamental descriptive statistics for both samples

carbachol

and control. Is Gaussianity a reasonable assumption for the two popu-

lations?

(b) Find the 95% conﬁdence interval for the population mean interspike

time in the control case.

of 5 Hz. Test this hypothesis by testing that the mean population interspike

time is 0.2, versus the proper one-sided alternative. Use both RR and p-

value approaches;

α =0.05.

(d) Test the hypothesis that the mean interspike time in the control and

carbachol populations are the same versus the one-sided alternative that

states that carbachol increases the frequency of spikes (decreases the inter-

spike times);

α =0.05.

(e) Do (b) and (c) in a Bayesian fashion building on

spikes.odc, which

contains control data and hints on how to set the priors. Compare and dis-

cuss the 95% conﬁdence interval from (b) and the 95% credible set from

BUGS. Compare tests, the p-value from (c), and the probability of H

from

WinBUGS.

10.19. Aerobic Capacity. The peak oxygen intake per unit of body weight, called

the aerobic capacity of an individual performing a strenuous activity, is a

measure of work capacity. For comparative study, measurements of aero-

bic capacities are recorded (Frisancho, 1975) for a group of 20 Peruvian

Highland natives and for a group of 10 Peruvian lowlanders acclimatized

as adults to high altitudes. The measurements are taken on a bicycle er-

gometer at high altitude (ml kg

−1

min

−1

Peruvian Peruvian Lowlanders

highland natives acclimatized as adults

Sample mean 46.3 38.0

Sample st. deviation 5.0 5.2

10.9 Exercises 403

(a) Test the hypothesis that the population mean aerobic capacities are the

same versus the one-sided alternative. Assume an equality of population

variances and take

α =0.05.

(b) If you were to repeat this experiment, what sample size (per group)

would give you a power of 90% to detect the difference between the means

of magnitude 4, if you assumed that the common population variance was

=25? The level of the test, α, is to be kept at 5%.

10.20. Cataract and Diabetes. Hiller and Kahn (1976) consider diabetes as a

risk factor for cataracts and provide the results of a case/control study.

Diabetes Cataract cases No cataract Total

Present 56 84 140

Absent 552 1927 2479

Total 608 2011 2619

Find 95% conﬁdence intervals for the risk difference, risk ratio, and odds

ratio.

10.21. Beginnings of Antiseptic Surgeries. Sir Joseph Lister (1827–1912),

Professor of Surgery at Glasgow University, inﬂuenced by Pasteur’s ideas,

found that a wound wrapped in bandages treated by Carbolic acid (phenol)

would often not become infected. Here are Lister’s data on treating open

fractures and amputations:

Period Carbolic acid Results

1864–1866 No Treated 34 patients, 19 recovered and 15 died

1867–1870 Yes Treated 40 patients, 34 recovered and 6 died

(a) Find and interpret the risk difference, risk ratio, and odds ratio.

(b) Find a 95% conﬁdence intervals for the parameters estimated in (a).

10.22. Reaction Times. Researchers are interested in reactions times to differ-

ent color light stimuli, speciﬁcally, green and red. A randomized design is

proposed – each subject is given a number of trials, such as GGRRGRRG,

etc. As a measurement of reaction time for a subject we report an average

speed of reaction to each color. The table below contains the measurements.

Subject X (red) Y (green)

1 18 22

2 16 20

3 23 29

4 30 35

5 32 27

6 30 29

7 31 33

8 25 29

9 27 31

10 21 24

404 10 Two Samples

Are the reaction times to red and green lights the same? Use a two-sided

alternative and

α =0.05.

10.23. Gamma Globulin and Aspirin. A clinical trial of gamma globulin in the

treatment of children with Kawasaki syndrome randomized approximately

half of the patients to receive gamma globulin. The standard treatment

for Kawasaki syndrome was a regimen of aspirin; nevertheless, about one

quarter of these patients developed coronary abnormalities, even under the

standard treatment. The outcome of interest was the development of coro-

nary abnormalities (CA) over a 7-week period. The following 2

×2 table

summarizes the results of the trial.

Treatment group

CA =1 C A =0 Total

GG =1, A =0 5 78 83

=0, A =1 21 63 84

Total 26 141 167

Find the 95% conﬁdence interval for the (a) risk difference, (b) risk ratio,

and (c) odds ratio.

10.24. High/Low Protein Diet in Rats. Armitage and Berry (1994, p. 111)

report data on the weight gain of 19 female rats between 28 and 84 days

after birth. The rats were placed on diets with high (12 animals) and low (7

animals) protein content.

High protein Low protein

134 70

146 118

104 101

119 85

124 107

161 132

107 94

113

129

123

We want to test the hypothesis on dietary effect. Did a low protein diet

result in signiﬁcantly lower weight gain?

(a) What test should be used?

(b) Perform the test at an

α =0.05 level.

=σ

=450 and we are inter-

ested in detecting the difference

∆ =|µ

−µ

|=20 as a signiﬁcant deviation

from H

95% of the time (power = 0.95). Use α =0.05.

10.25. Spider Monkey DNA. In the context of Exercise 9.25

(a) Test that proportion p

is signiﬁcantly smaller than p

10.9 Exercises 405

(b) Demonstrate that proportions p

and p

are not signiﬁcantly different.

Use

α =0.05.

Hint:

Since this is a multinomial model, the variance for difference

−

needed for Z statistic is Var (

−

) = p

(1 − p

)/n + p

(1 − p

)/n +

/n. Under H

from (a) both p

and p

are estimated by (

)/2

which leads to Z

−

10.26. PBSC versus BM for Unrelated Donor Allogeneic Transplants. We

are interested in determining whether bone marrow (BM) is equivalent to

peripheral blood stem cells (PBSC) in myeloablative unrelated donor trans-

plantation, using the data from Eapen et al. (2007). The greater graft-

versus-host disease burden of PBSC might make clinicians less likely to

use PBSC in this context.

By using equivalence margins of

±10%, and proportions of relapse within

6 months, test whether PBSC and BM are equivalent at level

α =0.05.

Method Number of patients Relapsed after 6 months

BM 583 93 (16%)

PBSC 328 58 (18%)

Hint:

By mimicking Example 10.15 devise a TOST using a normal approx-

imation to the binomial:

−

−(−δ)

(1−

)

(1−

)

, p

=1 −Φ(z

);

−

−δ

(1−

)

(1−

)

, p

=Φ(z

);

−value = max{p

, p

10.27. Hydrogels. It has been demonstrated that poly(ethylene glycol) diacrylate

(PEG-DA)-based hydrogels could serve as direct cell carriers for engineer-

ing soft orthopedic tissues. Recent studies have also shown that PEG-DA

can be rendered degradable by introducing an enzyme-sensitive peptide se-

quence into the polymer chains.

The data from Dr. Temenoff’s lab at Georgia Tech represent proportions of

surviving cells after gel degradation by the enzyme, a bacterial collagenase,

in durations of 30 and 60 minutes.

It is postulated that the proportions of live cells after 30 minutes and

60 minutes of exposure to the enzyme are equivalent

The data collected in eight independent experiments represent the number

of cells alive among the total number of cells recovered, as in the table

below.

406 10 Two Samples

30 minutes 60 minutes

Alive Total Alive Total

20250 44250 40500 69750

51000 126000 42750 76500

77250 100500 78750 155250

39000 58500 42750 67500

Form ratios alive/total for the two durations. The ratios could be as-

sumed as approximately normal given the large number of cells recov-

ered. Show that H

: µ

= µ

is not rejected against one- or two-sided al-

ternatives. This, however, is not an evidence of equivalence. Test the hy-

pothesis of equivalence with equivalence margins

= −θ

= 0.1, that is,

: µ

−µ

<−0.1 or µ

−µ

>0.1 versus H

: −0.1 <µ

−µ

<0.1.

Use a Westlake interval and

α =0.05.

MATLAB AND WINBUGS FILES AND DATA SETS USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch10.Two/

argon.m, dnatest2.m, equivalence1.m, hemoc.m, hemopexin2.m,

leadexposure.m, miammt.m, microdamage.m, neanderthal.m, nounfirst.m,

piaget.m, plazmaacid.m, ratiopoissons.m, risk.m, schizo.m,

simulrisks.m, temenoff.m

argon.odc, braintissue.odc, cancerprop.odc, equivalence.odc,

microdamage.odc, plasma1.odc, plasma2.odc, psoriasis.odc, spikes.odc,

stressacids.odc,

carbachol.dat, lice.xls, pasi.dat|mat|xls

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