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

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Since no ties occurred in the Y rankings, we have and

From Table 13.10.4 we have From these data we may now

compute by Equation 13.10.4

We see that in this case the correction for ties does not make any difference

in the value of

■

Computer Analysis We may use MINITAB, as well as many other statistical

software packages, to compute the Spearman correlation coefﬁcient. To use MINITAB,

we must first have MINITAB rank the observations and store the ranks in separate

columns, one for the X ranks and one for the Y ranks. If we rank the X and Y values of

Example 13.10.1 and store them in Columns 3 and 4, we may obtain the Spearman rank

correlation coefﬁcient with the procedure shown in Figure 13.10.1. Other software pack-

ages such as SAS

and SPSS, for example, automatically rank the measurements before

computing the coefﬁcient, thereby eliminating an extra step in the procedure.

EXERCISES

For the following exercises perform the test at the indicated level of signiﬁcance and determine

the p value.

13.10.1 The following table shows 15 randomly selected geographic areas ranked by population density

and age-adjusted death rate. Can we conclude at the .05 level of signiﬁcance that population den-

sity and age-adjusted death rate are not mutually independent?

3565.5 + 3570.0 - 1788.5

2113565.52135702

= .75

= 1788.5.

- 35

- 0 = 3570.0

= 0

EXERCISES 737

Dialog box: Session command:

Stat ➤ Basic Statistics ➤ Correlation MTB > CORRELATION C3 C4

Type C3–C4 in Variables. Click OK.

Output:

Correlations (Pearson)

Correlation of (X)Rank and (Y)Rank  -0.759

FIGURE 13.10.1 MINITAB procedure and output for computing Spearman rank correlation

coefﬁcient, Example 13.10.1.

Rank by Rank by

Population Age-Adjusted Population Age-Adjusted

Area Density (X) Death Rate (Y ) Area Density (X) Death Rate (Y )

1 8 10 9 6 8

2 2 14 10 14 5

31241176

4 4 15 12 1 2

5 9 11 13 13 9

63 11415 3

71012151113

85 7

13.10.2 The following table shows 10 communities ranked by decayed, missing, or ﬁlled (DMF) teeth per

100 children and ﬂuoride concentration in ppm in the public water supply:

Rank by Rank by

DMF Teeth Fluoride DMF Teeth Fluoride

per 100 Concentration per 100 Concentration

Community Children (X)(Y ) Community Children (X)(Y )

181647

2937110

374856

439965

52810102

Do these data provide sufﬁcient evidence to indicate that the number of DMF teeth per 100 chil-

dren tends to decrease as ﬂuoride concentration increases? Let

13.10.3 The purpose of a study by Nozawa et al. (A-10) was to evaluate the outcome of surgical repair of

pars interarticularis defect by segmental wire ﬁxation in young adults with lumbar spondylolysis.

The authors cite literature indicating that segmental wire ﬁxation has been successful in the treat-

ment of nonathletes with spondylolysis and point out that no information existed on the results of

this type of surgery in athletes. In a retrospective study of subjects having surgery between 1993

and 2000, the authors found 20 subjects who had undergone the surgery. The following table shows

the age (years) at surgery and duration (months) of follow-up care for these subjects.

Duration of Follow-Up Duration of Follow-Up

(Months) Age (Years) (Months) Age (Years)

103 37 38 27

68 27 36 31

62 12 34 24

60 18 30 23

60 18 19 14

a = .05.

738 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS

(Continued)

Duration of Follow-Up Duration of Follow-Up

(Months) Age (Years) (Months) Age (Years)

54 28 19 23

49 25 19 18

44 20 19 29

42 18 17 24

41 30 16 27

Source: Satoshi Nozawa, Katsuji Shimizu, Kei Miyamoto, and Mizuo Tanaka, “Repair of Pars Interarticu-

laris Defect by Segmental Wire Fixation in Young Athletes with Spondylolysis,” American Journal of Sports

Medicine, 31 (2003), pp. 359–364.

May we conclude, on the basis of these data, that in a population of similar subjects there is an

association between age and duration of follow-up? Let

13.10.4 Refer to Exercise 13.10.3. Nozawa et al. (A-10) also calculated the Japanese Orthopaedic Associ-

ation score for measuring back pain ( JOA). The results for the 20 subjects along with the duration

of follow-up are shown in the following table. The higher the number, the lesser the degree of pain.

Duration of Follow-Up Duration of Follow-Up

(Months) JOA (Months) JOA

103 21 38 13

68 14 36 24

62 26 34 21

60 24 30 22

60 13 19 25

54 24 19 23

49 22 19 20

44 23 19 21

42 18 17 25

41 24 16 21

Source: Satoshi Nozawa, Katsuji Shimizu, Kei Miyamoto, and Mizuo Tanaka, “Repair of

Pars Interarticularis Defect by Segmental Wire Fixation in Young Athletes with Spondylolysis,”

American Journal of Sports Medicine, 31 (2003), pp. 359–364.

Can we conclude from these data that in general there is a relationship between length of follow-

up and JOA score at the time of the operation? Let

13.10.5 Butz et al. (A-11) studied the use of noninvasive positive-pressure ventilation by patients with amy-

otrophic lateral sclerosis. They evaluated the beneﬁt of the procedure on patients’ symptoms, qual-

ity of life, and survival. Two variables of interest are partial pressure of arterial carbon

dioxide, and partial pressure of arterial oxygen. The following table shows, for 30 subjects,

values of these variables (mm Hg) obtained from baseline arterial blood gas analyses.

40 101 54.5 80 34.5 86.5

47 69 54 72 40.1 74.7

PaO

PaCO

PaO

PaCO

PaO

PaCO

PaO

PaCO

a = .05.

EXERCISES 739

(Continued)

34 132 43 105 33 94

42 65 44.3 113 59.9 60.4

54 72 53.9 69.2 62.6 52.5

48 76 41.8 66.7 54.1 76.9

53.6 67.2 33 67 45.7 65.3

56.9 70.9 43.1 77.5 40.6 80.3

58 73 52.4 65.1 56.6 53.2

45 66 37.9 71 59 71.9

Source: M. Butz, K. H. Wollinsky, U. Widemuth-Catrinescu, A. Sperfeld,

S. Winter, H. H. Mehrkens, A. C. Ludolph, and H. Schreiber, “Longitudinal

Effects of Noninvasive Positive-Pressure Ventilation in Patients with Amy-

otrophic Lateral Sclerosis,” American Journal of Medical Rehabilitation, 82

(2003) 597–604.

On the basis of these data may we conclude that there is an association between and

values? Let

13.10.6 Seventeen patients with a history of congestive heart failure participated in a study to assess the

effects of exercise on various bodily functions. During a period of exercise the following data were

collected on the percent change in plasma norepinephrine (Y ) and the percent change in oxygen

consumption (X ):

Subject XY Subject XY

1 500 525 10 50 60

2 475 130 11 175 105

3 390 325 12 130 148

4 325 190 13 76 75

5 325 90 14 200 250

6 205 295 15 174 102

7 200 180 16 201 151

8 75 74 17 125 130

9 230 420

On the basis of these data can one conclude that there is an association between the two variables?

Let

13.11 NONPARAMETRIC

REGRESSION ANALYSIS

When the assumptions underlying simple linear regression analysis as discussed in Chap-

ter 9 are not met, we may employ nonparametric procedures. In this section we present

estimators of the slope and intercept that are easy-to-calculate alternatives to the least-

squares estimators described in Chapter 9.

a = .05.

PaO

PaCO

PaO

PaCO

PaO

PaCO

PaO

PaCO

740 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS

Theil’s Slope Estimator Theil (12) proposes a method for obtaining a point

estimate of the slope coefﬁcient We assume that the data conform to the classic

regression model

where the are known constants, and are unknown parameters, and is an

observed value of the continuous random variable Y at For each value of we assume

a subpopulation of Y values, and the are mutually independent. The are all distinct

(no ties), and we take

The data consist of n pairs of sample observations,

where the ith pair represents measurements taken on the ith unit of association.

To obtain Theil’s estimator of we ﬁrst form all possible sample slopes

where There will be values of The estimator of

which we designate by is the median of values. That is,

(13.11.1)

The following example illustrates the calculation of

EXAMPLE 13.11.1

In Table 13.11.1 are the plasma testosterone (ng/ml) levels (Y ) and seminal citric acid

(mg/ml) levels in a sample of eight adult males. We wish to compute the estimate of

the population regression slope coefficient by Theil’s method.

Solution: The ordered values of are shown in Table 13.11.2.

If we let the indicators of the ﬁrst and second val-

ues of Y and X in Table 13.11.1, we may compute as follows:

When all the slopes are computed in a similar manner and ordered as in

Table 13.11.2, winds up as the tenth value in the ordered array.

The median of the values is .4878. Consequently, our estimate of

the population slope coefﬁcient b

= .4878.

-.3846

= 1175 - 2302>1278 - 4212=-.3846

i = 1 and j = 2,

N =

= 28

= median5S

.N =

i 6 j.1y

- y

2>1x

- x

, y

2, 1x

, y

2, . . . , 1x

, y

6 x

. . .

6 x

= b

+ b

i = 1, . . . , n

13.11 NONPARAMETRIC REGRESSION ANALYSIS 741

TABLE 13.11.1 Plasma Testosterone and Seminal Citric Acid Levels

in Adult Males

Testosterone: 230 175 315 290 275 150 360 425

Citric acid: 421 278 618 482 465 105 550 750

An Estimator of the Intercept Coefﬁcient Dietz (13) recommends

two intercept estimators. The first, designated is the median of the n terms

in which is the Theil estimator. It is recommended when the researcher

is not willing to assume that the error terms are symmetric about 0. If the researcher is

willing to assume a symmetric distribution of error terms, Dietz recommends the esti-

mator which is the median of the pairwise averages of the

terms. We illustrate the calculation of each in the following example.

EXAMPLE 13.11.2

Refer to Example 13.11.1. Let us compute and from the data on testosterone

and citric acid levels.

Solution: The ordered terms are: 13.5396, 24.6362, 39.3916, 48.1730,

54.8804, 59.1500, 91.7100, and 98.7810. The median, 51.5267, is the

estimator .

The ordered pairwise averages of the

are

13.5396 49.2708 75.43

19.0879 51.5267 76.8307

24.6362 52.6248 78.9655

26.4656 53.6615 91.71

30.8563 54.8804 95.2455

32.0139 56.1603 98.781

34.21 57.0152

36.3448 58.1731

.4878x

-818 + 12>2 = 36

1,M

- .4878x

2,M

1,M

- b

n1n + 12>21b

2,M

- b

1,M

742 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS

TABLE 13.11.2 Ordered Values of for

Example 13.11.1

.5037

.1445 .5263

.1838 .5297

.2532 .5348

.2614 .5637

.3216 .5927

.3250 .6801

.3472 .8333

.3714 .8824

.3846 .9836

.4118 1.0000

.4264 1.0078

.4315 1.0227

.4719 1.0294

-.6618

(Continued)

■

36.4046 59.15

39.3916 61.7086

39.7583 65.5508

41.8931 69.0863

43.7823 69.9415

47.136 73.2952

48.173 73.477

The median of these averages, 53.1432, is the estimator The esti-

mating equation, then, is if we are willing to

assume that the distribution of error terms is symmetric about 0. If we are

not willing to make the assumption of symmetry, the estimating equation

■

EXERCISES

13.11.1 The following are the heart rates (HR: beats/minute) and oxygen consumption values (

cal/kg/24 h) for nine infants with chronic congestive heart failure:

HR(X): 163 164 156 151 152 167 165 153 155

53.9 57.4 41.0 40.0 42.0 64.4 59.1 49.9 43.2

Compute

13.11.2 The following are the body weights (grams) and total surface area of nine laboratory

animals:

Body weight (X): 660.2 706.0 924.0 936.0 992.1 888.9 999.4 890.3 841.2

Surface area (Y): 781.7 888.7 1038.1 1040.0 1120.0 1071.5 1134.5 965.3 925.0

Compute the slope estimator and two intercept estimators.

13.12 SUMMARY

This chapter is concerned with nonparametric statistical tests. These tests may be used

either when the assumptions underlying the parametric tests are not realized or when

the data to be analyzed are measured on a scale too weak for the arithmetic procedures

necessary for the parametric tests.

Nine nonparametric tests are described and illustrated. Except for the

Kolmogorov–Smirnov goodness-of-ﬁt test, each test provides a nonparametric alternative

to a well-known parametric test. There are a number of other nonparametric tests avail-

able. The interested reader is referred to the many books devoted to nonparametric

methods, including those by Gibbons (14) and Pett (15).

1cm

, 1b

1,M

, and 1b

2, M

1Y 2:

= 51.5267 + .4878x

= 53.1432 + .4878x

2,M

13.12 SUMMARY 743

SUMMARY OF FORMULAS FOR CHAPTER 13

Formula

Number Name Formula

13.3.1 Sign test statistic

13.3.2 Large-sample

approximation

of the sign test

13.6.1 Mann–Whitney test

statistic

13.6.2 Large-sample

approximation of the

Mann–Whitney test

13.6.3 Equivalence of the

Mann–Whitney and

Wilcoxon two-sample

statistics

13.7.1–13.7.2 Kolmogorov–Smirnov

test statistic

13.8.1 Kruskal–Wallis test

statistic

13.8.2 Kruskal–Wallis test

statistic adjustment

for ties

13.9.2 Friedman test statistic

13.10.1 Spearman rank

correlation test

statistic

= 1 -

n1n

- 12

nk1k + 12

j = 1

- 3n1k + 12

1 -

- n

H =

n1n + 12

j = 1

- 31n + 12

U + W =

m1m + 2n + 12

Z =

T - mn>2

2nm1n + m + 12>12

T = S -

n1n + 12

P1k … x

n, p2=

k = 0

n - k

744 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS

- F

i - 1

= max

max

2- F

i - 1

D = sup

1x2- F

1x2

1…i…r

z =

1k - 0.52- 0.5n

0.52n

k Ú

z =

1k + 0.52+ 0.5n

0.52n

k 6

13.10.2 Large-sample

approximation of

the Spearman

rank correlation

13.10.3–13.10.4 Correction for tied

observations in the

Spearman rank

correlation

13.11.1 Theil’s

estimator of

Symbol Key •  Theil’s estimator of

•  chi-square

• D  Kolmogorov–Smirnov test statistic

• F

(x)  distribution function of i

• H  Friedman test statistic

• k  sign test statistic and the number of columns in the Friedman test

• m  sample size of the smaller of two samples

• n  sample size of the larger of two samples

• p  probability of success

• q  1  p  probability of failure

• R  rank

• r

 Spearman rank correlation coefﬁcient

• S  sum of ranks

• S

 slope between point i and j

• sup  supremum (greatest)

• t  number of tied observations

• T  correction for tied observations

• x and y  data value for variables x and y

• U  Mann–Whitney test statistic

• W  Wilcoxon test statistic

• z  normal variate

1or X

= median{S

}

z = r

2n - 1

REVIEW QUESTIONS AND EXERCISES 745

with

T =

- t

REVIEW QUESTIONS AND EXERCISES

1. Deﬁne nonparametric statistics.

2. What is meant by the term distribution-free statistical tests?

3. What are some of the advantages of using nonparametric statistical tests?

4. What are some of the disadvantages of the nonparametric tests?

5. Describe a situation in your particular area of interest where each of the following tests could be

used. Use real or realistic data and test an appropriate hypothesis using each test.

(a) The sign test

(b) The median test

(d) The Mann–Whitney test

(e) The Kolmogorov–Smirnov goodness-of-ﬁt test

(f) The Kruskal–Wallis one-way analysis of variance by ranks

(g) The Friedman two-way analysis of variance by ranks

(h) The Spearman rank correlation coefﬁcient

(i) Nonparametric regression analysis

6. The following are the ranks of the ages (X ) of 20 surgical patients and the dose (Y ) of an anal-

gesic agent required to block one spinal segment.

Rank of Rank of Dose Rank of Rank of Dose

Age in Requirement Age in Requirement

Years (X )(Y ) Years (X )(Y )

1 1 11 13

2 7 12 5

3 2 13 11

4 4 14 16

5 6 15 20

6 8 16 18

7 3 17 19

8151817

9 9 19 10

10 12 20 14

Compute and test (two-sided) for signiﬁcance. Let Determine the p value for this test.

7. Otani and Kishi (A-12) studied seven subjects with diabetic macular edema. They measured the

foveal thickness in seven eyes pre- and post-unilateral vitrectomy surgery. The results are

shown in the following table:

Subject Pre-op Foveal Thickness Post-op Foveal Thickness

1 690 200

2 840 280

3 470 230

4 690 200

5 730 560

6 500 210

7 440 200

Source: Tomohiro Otani, M.D. Used with permission.

Use the Wilcoxon signed-rank test to determine whether one should conclude that the sur-

gery is effective in reducing foveal thickness. Let What is the p value?a = .05.

(Mm)(Mm)

1mm2

a = .05.r

746 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS