Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences

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8. The subjects of a study by J. Jose and S. R. Ell (A-13) were 303 healthy volunteers who self-

assessed their own nasal flow status by indicating whether their nasal airway was (1) totally

clear, (2) not very clear, (3) very blocked, or (4) totally blocked. Following the self-assessment,

an In-Check meter was used to measure peak inspiratory nasal flow rate (PINFR, L/min). Data

on 175 subjects in three of the self-assessment categories are displayed in the following table.

The authors performed a Kruskal–Wallis test to determine if these data provide sufficient evi-

dence to indicate a difference in population centers of PINFR among these three response groups.

Let What is the test statistic value for this test?

Peak Inspiratory Nasal Flow Rate (L/min)

Totally Clear Not Very Clear Partially Blocked

180 105 150 120 160 190 130 100

150 150 110 95 200 95 110 100

200 240 130 140 70 130 110 100

130 120 100 135 75 240 130 105

200 90 170 100 150 180 125 95

120 135 80 130 80 140 100 85

150 110 125 180 130 150 230 50

150 155 115 155 160 130 110 105

160 105 140 130 180 90 270 200

150 140 140 140 90 115 180

110 200 95 120 180 130 130

190 170 110 290 140 210 125

150 150 160 170 230 190 90

120 120 90 280 220 135 210

180 170 135 150 130 130 140

140 200 110 185 180 210 125

130 160 130 150 140 90 210

230 180 170 150 140 125 120

200 170 130 170 120 140 115

140 160 115 210 140 160 100

150 150 145 140 150 230 130

170 100 130 140 190 100 130

180 100 170 160 210 120 110

160 180 160 120 130 120 150

200 130 90 230 190 150 110

90 200 110 100 220 110 90

130 120 130 190 160 150 120

140 145 130 90 105 130 115

200 130 120 100 120 150 140

220 100 130 125 140 130 130

200 130 180 180 130 145 160

120 160 140 200 115 160 110

310 125 175 160 115 120 165

160 100 185 170 100 220 120

115 140 190 85 150 145 150

a = .01.

REVIEW QUESTIONS AND EXERCISES 747

(Continued)

Peak Inspiratory Nasal Flow Rate (L/min)

Totally Clear Not Very Clear Partially Blocked

170 185 130 150 130 150 170

130 180 160 280 130 120 110

220 115 160 140 170 155 120

250 260 130 100 130 100 85

160 160 135 140 145 140

130 170 130 90

130 115 120 190

150 150 190 130

160 130 170

Source: J. Jose, MS, FRCS. Used with permission.

9. Ten subjects with bronchial asthma participated in an experiment to evaluate the relative effective-

ness of three drugs. The following table shows the change in (forced expired volume in 1

second) values (expressed as liters) 2 hours after drug administration:

Drug Drug

Subject A B C Subject A B C

1 .00 .13 .26 6 .03 .18 .25

2 .04 .17 .23 7 .05 .21 .32

3 .02 .20 .21 8 .02 .23 .38

4 .02 .27 .19 9 .00 .24 .30

5 .04 .11 .36 10 .12 .08 .30

Are these data sufﬁcient to indicate a difference in drug effectiveness? Let What is the

p value for this test?

10. One facet of the nursing curriculum at Wright State University requires that students use mathe-

matics to perform appropriate dosage calculations. In a study by Wendy Gantt (A-14), undergrad-

uate nursing students were given a standardized mathematics test to determine their mathematical

aptitude (scale: 0–100). The students were divided into two groups: traditional college age (18–24

years, 26 observations) and nontraditional ( eight observations). Scores on the mathematics

test appear in the following table:

Traditional Students’ Scores Nontraditional Students’ Scores

70688 77

57 79 68 72

85 14 88 54

55 82 92 87

87 45 85 85

25+,

a = .05.

FEV

748 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS

(Continued)

Traditional Students’ Scores Nontraditional Students’ Scores

84 57 56 62

56 91 31 77

68 76 80 86

94 60

Source: Wendy Gantt and the Wright State University Statistical Consulting Center. Used with permission.

Do these data provide sufﬁcient evidence to indicate a difference in population medians? Let

What is the p value for this test? Use both the median test and the Mann–Whitney test and compare

the results.

11. The following are the PaCO

(mm Hg) values in 16 patients with bronchopulmonary disease:

39, 40, 45, 48, 49, 56, 60, 75, 42, 48, 32, 37, 32, 33, 33, 36

Use the Kolmogorov–Smirnov test to test the null hypothesis that PaCO

values in the sampled

population are normally distributed with and

12. The following table shows the caloric intake (cal/day/kg) and oxygen consumption VO

(ml/min/kg)

in 10 infants:

Calorie Calorie

Intake (X)VO

(Y) Intake (X)VO

(Y)

50 7.0 100 10.8

70 8.0 150 12.0

90 10.5 110 10.0

120 11.0 75 9.5

40 9.0 160 11.9

Test the null hypothesis that the two variables are mutually independent against the alternative that

they are directly related. Let What is the p value for this test?

13. Mary White (A-15) surveyed physicians to measure their opinions regarding the importance of

ethics in medical practice. The measurement tool utilized a scale from 1 to 5 in which a higher

value indicated higher opinion of the importance of ethics. The ages and scores of the study sub-

jects are shown in the following table. Can one conclude on the basis of these results that age and

ethics score are directly related? Let the probability of committing a type I error be .05. What is

the p value?

Age Ethics Age Ethics Age Ethics

25 4.00 26 4.50 26 4.50

34 4.00 29 4.75 27 5.00

30 4.25 30 4.25 22 3.75

31 3.50 26 4.50 22 4.25

25 4.75 30 4.25 24 4.50

a = .05.

s = 12.m = 44

a = .05.

REVIEW QUESTIONS AND EXERCISES 749

(Continued)

Age Ethics Age Ethics Age Ethics

25 3.75 25 3.75 22 4.25

25 4.75 24 4.75 24 3.75

29 4.50 24 4.00 38 4.50

29 4.50 25 4.50 22 4.50

26 3.75 25 4.00 22 4.50

25 3.25 26 4.75 25 4.00

29 4.50 34 3.25 23 3.75

27 3.75 23 4.50 22 4.25

29 4.25 26 3.25 23 4.00

25 3.75 23 5.00 22 4.25

25 4.50 24 4.25 25 3.50

25 4.00 45 3.25 26 4.25

26 4.25 23 3.75 25 4.25

26 4.00 25 3.75 27 4.75

24 4.00 25 3.75 23 3.75

25 4.00 23 3.75 22 4.00

22 3.75 23 4.75 26 4.75

26 4.50 26 4.00 22 4.25

23 4.00

14. Dominic Sprott (A-16) conducted an experiment with rabbits in which the outcome variable was

the fatty inﬁltration in the shoulder mass (PFI, measured as a percent). At baseline, 15 rabbits had

a randomly chosen shoulder muscle detached. The shoulder was then reattached. Six weeks later,

ﬁve randomly chosen rabbits were sacriﬁced and the differences in the PFI between the reattached

shoulder and the nondetached shoulder were recorded (group A). Six months later, the 10 remain-

ing rabbits were sacriﬁced and again the differences in the PFI between the reattached shoulder

and the nondetached shoulder were recorded (group B).

Percent Fatty Inﬁltration Difference

(Nondetached–Reattached)

Group A Group B

2.55 1.04 1.38

0.9 3.29 0.75

0.2 0.99 0.36

0.29 1.79 0.74

1.11 0.85 0.3

Can we conclude, at the .05 level of signiﬁcance, that the treatments have a differential effect on

PFI between the two shoulder muscles? What is the p value for the test?

In each of the Exercises 15 through 29, do one or more of the following that you think are

appropriate:

(a) Apply one or more of the techniques discussed in this chapter.

(b) Apply one or more of the techniques discussed in previous chapters.

750 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS

Source: Mary White, Ph.D. and Wright

State University Statistical Consulting

Center. Used with permission.

Source: Dominic Sprott, M.D. and the Wright

State University Statistical Consulting Center.

Used with permission.

(d) State the statistical decisions and clinical conclusions that the results of your hypothesis tests

justify.

(e) Describe the population(s) to which you think your inferences are applicable.

(f) State the assumptions necessary for the validity of your analyses.

15. The purpose of a study by Damm et al. (A-17) was to investigate insulin sensitivity and insulin secre-

tion in women with previous gestational diabetes (GDM). Subjects were 12 normal-weight glucose-

tolerant women (mean age, 36.6 years; standard deviation, 4.16) with previous gestational diabetes

and 11 controls (mean age, 35 years; standard deviation, 3.3). Among the data collected were the

following fasting plasma insulin values (mmol/L). Use the Mann–Whitney test to determine if you

can conclude on the basis of these data that the two populations represented differ with respect to

average fasting plasma insulin level.

Controls Previous GDM Controls Previous GDM

46.25 30.00 40.00 31.25

40.00 41.25 30.00 56.25

31.25 56.25 51.25 61.25

38.75 45.00 32.50 50.00

41.25 46.25 43.75 53.75

38.75 46.25 62.50

Source: Dr. Peter Damm. Used with permission.

16. Gutin et al. (A-18) compared three measures of body composition, including dual-energy x-ray

absorptiometry (DXA). Subjects were apparently healthy children (21 boys and 22 girls) between

the ages of 9 and 11 years. Among the data collected were the following measurements of body-

composition compartments by DXA. The investigators were interested in the correlation between

all possible pairs of these variables.

Bone Fat-Free

Fat-Free Mineral Soft

Percent Fat Fat Mass Mass Content Tissue

11.35 3.8314 29.9440 1.19745 28.7465

22.90 6.4398 21.6805 0.79250 20.8880

12.70 4.0072 27.6290 0.95620 26.6728

42.20 24.0329 32.9164 1.45740 31.4590

24.85 9.4303 28.5009 1.32505 27.1758

26.25 9.4292 26.4344 1.17412 25.2603

23.80 8.4171 26.9938 1.11230 25.8815

37.40 20.2313 33.8573 1.40790 32.4494

14.00 3.9892 24.4939 0.95505 23.5388

19.35 7.2981 30.3707 1.45545 28.9153

29.35 11.1863 26.8933 1.17775 25.7156

REVIEW QUESTIONS AND EXERCISES 751

(Continued)

Bone Fat-Free

Fat-Free Mineral Soft

Percent Fat Fat Mass Mass Content Tissue

18.05 5.8449 26.5341 1.13820 25.3959

13.95 4.6777 28.9144 1.23730 27.6771

32.85 13.2474 27.0849 1.17515 25.9097

11.40 3.7912 29.5245 1.42780 28.0967

9.60 3.2831 30.8228 1.14840 29.6744

20.90 7.2277 27.3302 1.24890 26.0813

44.70 25.7246 31.8461 1.51800 30.3281

17.10 5.1219 24.8233 0.84985 23.9734

16.50 5.0749 25.7040 1.09240 24.6116

14.35 5.0341 30.0228 1.40080 28.6220

15.45 4.8695 26.6403 1.07285 25.5674

28.15 10.6715 27.2746 1.24320 26.0314

18.35 5.3847 23.9875 0.94965 23.0379

15.10 5.6724 31.9637 1.32300 30.6407

37.75 25.8342 42.6004 1.88340 40.7170

39.05 19.6950 30.7579 1.50540 29.2525

22.25 7.2755 25.4560 0.88025 24.5757

15.50 4.4964 24.4888 0.96500 23.5238

14.10 4.3088 26.2401 1.17000 25.0701

26.65 11.3263 31.2088 1.48685 29.7219

20.25 8.0265 31.5657 1.50715 30.0586

23.55 10.1197 32.8385 1.34090 31.4976

46.65 24.7954 28.3651 1.22575 27.1394

30.55 10.0462 22.8647 1.01055 21.8541

26.80 9.5499 26.0645 1.05615 25.0083

28.10 9.4096 24.1042 0.97540 23.1288

24.55 14.5113 44.6181 2.17690 42.4412

17.85 6.6987 30.8043 1.23525 29.5690

20.90 6.5967 24.9693 0.97875 23.9905

33.00 12.3689 25.1049 0.96725 24.1377

44.00 26.1997 33.3471 1.42985 31.9172

19.00 5.0785 21.6926 0.78090 20.9117

Source: Dr. Mark Litaker. Used with permission.

17. The concern of a study by Crim et al. (A-19) was the potential role of ﬂow cytometric analysis of

bronchoalveolar lavage ﬂuid (BALF) in diagnosing acute lung rejection. The investigators note that

previous studies suggested an association of acute lung rejection with increases in CD8 lympho-

cytes, and increased expression of human lymphocyte antigen (HLA)-DR antigen and interleukin-2

receptor (IL-2R). Subjects consisted of lung transplant (LT) recipients who had no histologic evi-

dence of rejection or infection, normal human volunteers (NORM), healthy heart transplant (HT)

recipient volunteers, and lung transplant recipients who were experiencing acute lung rejection (AR).

Among the data collected were the following percentages of BALF CD8 lymphocytes that also

express IL-2R observed in the four groups of subjects.

752 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS

Norm HT LT AR

001612

20060

15589

040167

060242

20556

3 0 18 3 14

0422210

082103

18800

080

731

241

540

118 0

18. Ichinose et al. (A-20) studied the involvement of endogenous tachykinins in exercise-induced air-

way narrowing in patients with asthma by means of a selective neurokinin 1-receptor antagonist,

FK-888. Nine subjects (eight male, one female) ages 18 to 43 years with at least a 40 percent fall

in the speciﬁc airway conductance participated in the study. The following are the oxygen con-

sumption (ml/min) data for the subjects at rest and during exercise while under treatment with a

placebo and FK-888:

Placebo FK-888

At Rest Exercise At Rest Exercise

303 2578 255 2406

288 2452 348 2214

285 2768 383 3134

280 2356 328 2536

295 2112 321 1942

270 2716 234 2652

274 2614 387 2824

185 1524 198 1448

364 2538 312 2454

19. Transforming growth factor (TGF ), according to Tomiya and Fujiwara (A-21), is alleged to

play a role in malignant progression as well as normal cell growth in an autocrine manner, and its

serum levels have been reported to increase during this progression. The present investigators have

developed an enzyme-linked immunosorbent assay (ELISA) for measuring serum TGF levels in

the diagnosis of hepatocellular carcinoma (HCC) complicating cirrhosis. In a study in which they

evaluated the significance of serum TGF levels for diagnostic purposes, they collected thea

REVIEW QUESTIONS AND EXERCISES 753

Source: Dr. Courtney Crim.

Used with permission.

Source: Dr. Kunio Shirato.

Used with permission.

following measurements on the liver function tests, TGF (pg/ml), and serum -fetoprotein (AFP)

(ng/ml) from HCC patients:

TGF AFP TGF AFP TGF AFP TGF AFP

32.0 12866 44.0 23077 100.0 479 15.0 921

65.9 9 75.0 371 12.0 47 34.0 118

25.0 124.3 36.0 291 32.0 177 100.0 6.2

30.0 9 65.0 700 98.0 9 26.0 19

22.0 610 44.0 40 20.0 1063 53.0 594

40.0 238 56.0 9538 20.0 21 140.0 10

52.0 153 34.0 19 9.0 206 24.0 292

28.0 23 300.0 11 58.0 32 20.0 11

11.0 28 39.0 42246 39.0 628 35.0 37

45.0 240 82.0 12571 52.0 35

29.0 66 85.0 20 50.0 742

45.0 83 24.0 29 95.0 10

21.0 4 40.0 310 18.0 291

38.0 214 9.0 19

Source: Dr. Kenji Fujiwara. Used with permission.

20. The objective of a study by Sakhaee et al. (A-22) was to ascertain body content of aluminum (A1)

noninvasively using the increment in serum and urinary Al following the intravenous administration

of deferoxamine (DFO) in patients with kidney stones and osteoporotic women undergoing long-term

treatment with potassium citrate or tricalcium dicitrate respectively. Subjects con-

sisted of 10 patients with calcium nephrolithiasis and ﬁve patients with osteoporosis who were main-

tained on potassium citrate or calcium citrate for 2–8 years, respectively, plus 16 normal volunteers

without a history of regular aluminum-containing antacid use. Among the data collected were the fol-

lowing 24-hour urinary aluminum excretion measurements ( g/day) before (PRE) and after (POST)

2-hour infusion of DFO.

Group PRE POST Group PRE POST

Control 41.04 135.00 Control 9.39 12.32

Control 70.00 95.20 Control 10.72 13.42

Control 42.60 74.00 Control 16.48 17.40

Control 15.48 42.24 Control 10.20 14.20

Control 26.90 104.30 Control 11.40 20.32

Control 16.32 66.90 Control 8.16 12.80

Control 12.80 10.68 Control 14.80 62.00

Control 68.88 46.48 Patient 15.20 27.15

Control 25.50 73.80 Patient 8.70 38.72

Patient 0.00 14.16 Patient 5.52 7.84

Patient 2.00 20.72 Patient 13.28 31.70

Patient 4.89 15.72 Patient 3.26 17.04

Patient 25.90 52.40 Patient 29.92 151.36

Patient 19.35 35.70 Patient 15.00 61.38

1Ca

Cit

2,1K

Cit2

AAAA

754 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS

(Continued)

Group PRE POST Group PRE POST

Patient 4.88 70.20 Patient 36.80 142.45

Patient 42.75 86.25

Source: Dr. Khashayar Sakhaee. Used with permission.

21. The purpose of a study by Dubuis et al. (A-23) was to determine whether neuropsychological deﬁcit

of children with the severe form of congenital hypothyroidism can be avoided by earlier onset of

therapy and higher doses of levothyroxine. Subjects consisted of 10 infants (ages 3 to 24 days) with

severe and 35 infants (ages 2 to 10 days) with moderate congenital hypothyroidism. Among the

data collected were the following measurements on plasma (nmol/L) levels at screening:

Severe Cases Moderate Cases

Sex (nmol/L) Sex (nmol/L) Sex (nmol/L)

M 16 F 20 F 62

M 57 F 34 M 50

M 40 F 188 F 40

F 50 F 69 F 116

F 57 F 162 F 80

F 38 F 148 F 97

F 51 F 108 F 51

F 38 F 54 F 84

M*F96F51

F 60 M 76 F 94

M 122 M 158

M43F *

F40M47

F 29 M 143

F 83 M 128

F 62 M 112

M 111

F84

M55

22. Kuna et al. (A-24) conducted a study concerned with chemokines in seasonal allergic rhinitis. Sub-

jects included 18 atopic individuals with seasonal allergic rhinitis caused by ragweed pollen.

Among the data collected on these subjects were the following eosinophil cationic protein (ECP)

and histamine measurements:

ECP (ng/ml) Histamine (ng/ml) ECP (ng/ml) Histamine (ng/ml)

511.0 31.2 25.3 5.6

388.0 106.0 31.1 62.7

14.1 37.0 325.0 138.0

314.0 90.0 437.0 116.0

REVIEW QUESTIONS AND EXERCISES 755

(Continued)

Missing data.

Source: Dr. Guy Van Vliet.

Used with permission.

* 

14 158 182 179 34 181 182 208

15 184 177 182 35 191 230 208

16 223 244 234 36 248 284 279

17 154 178 187 37 224 228 199

18 176 137 162 38 229 318 272

19 205 253 288 39 147 199 194

20 167 156 136 40 248 258 302

21 164 176 191 41 160 218 229

22 177 168 185 42 175 187 166

23 140 175 167 43 262 260 247

24 167 186 195 44 189 199 181

Source: Dr. Chee Jeong Kim. Used with permission.

ECP (ng/ml) Histamine (ng/ml) ECP (ng/ml) Histamine (ng/ml)

74.1 29.0 277.0 70.6

8.8 87.0 602.0 184.0

144.0 45.0 33.0 8.6

56.0 151.8 661.0 264.0

205.0 86.0 162.0 92.0

Source: Dr. Allen P. Kaplan. Used with permission.

23. The purpose of a study by Kim et al. (A-25) was to investigate the serial changes in Lp(a) lipopro-

tein levels with the loss of female sex hormones by surgical menopause and with estrogen

replacement therapy in the same women. Subjects were 44 premenopausal women who underwent

a transabdominal hysterectomy (TAH). Thirty-one of the women had a TAH and unilateral salp-

ingo-oophorectomy (USO), and 13 had a TAH and bilateral salpingo-oophorectomy (BSO). The

women ranged in age from 30 to 53 years. Subjects in the BSO group received .625 mg of con-

jugated equine estrogen daily 2 months after the operation. The following were the subjects’ total

cholesterol levels before (TC0), 2 months after (TC2), and 4 months after (TC4) the surgical pro-

cedure and hormone replacement therapy.

USO USO

Subject TC0 TC2 TC4 Subject TC0 TC2 TC4

1 202 203 196 25 134 131 135

2 204 183 203 26 163 190 185

3 206 199 192 27 196 183 192

4 166 180 176 28 181 194 208

5 150 171 154 29 160 162 181

6 137 134 129 30 188 200 181

7 164 168 171 31 172 188 189

8 207 249 223

BSO

9 126 121 140

Subject TC0 TC2 TC4

10 131 141 167

32 224 218 239

11 133 159 149

33 202 196 231

12 142 152 140

13 225 193 180

756 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS