Devore J.L., Berk K.N. Modern Mathematical Statistics with Applications

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Assuming a beta prior for p on [0,1] with parameters a and b and the binomial

distribution Bin(n ¼ 1574, p) for the data, we get for the posterior distribution,

hðpjxÞ¼

f ðx jpÞgðpÞ

1

f ðx jpÞgðpÞdp



ð1  pÞ

nx

Gða þ bÞ

GðaÞGðbÞ

a1

ð1  pÞ

b1



ð1  pÞ

nx

Gða þ bÞ

GðaÞGðbÞ

a1

ð1  pÞ

b1

The numerator can be written as



Gða þ bÞ

GðaÞGðbÞ

Gðx þ aÞG ðn  x þ bÞ

Gðn þ a þ bÞ

Gðx þ aÞG ðn  x þ bÞ

xþa1

ð1  pÞ

nxþb1



Given that the part in square brackets is of the form of a beta pdf on [0, 1], its

integral over this interval is 1. The part in front of the square brackets is shared by

the numerator and denominator, and will therefore cancel. Thus

hðpjxÞ¼

Gðn þ a þ bÞ

Gðx þ aÞGðn  x þ bÞ

xþa1

ð1  pÞ

nxþb1

That is, the posterior distribution of p is itself a beta distribution with parameters

x + a and n  x+b.

If we were using the traditional non-Bayesian frequentist approach to statis-

tics, and we wanted to give an estimate of p for this example, we would give the

usual estimate from Section 8.2, x/n ¼ 803/1574 ¼ .51. The usual Bayesian esti-

mate is the posterior mean, the expected value of p given the data. Recalling that the

mean of the beta distribution on [0, 1] is a=ða þ bÞ, we obtain

EpjxðÞ¼x þ aðÞ= n þ a þ bðÞ¼ð803 þ aÞ=ð1574 þ a þ bÞ

for the posterior mean.

Suppose that a ¼ b ¼ 1, so the beta prior distribution reduces to the uniform

distribution on [0, 1]. Then E(pjx) ¼ (803 + 1)/(1574 + 2) ¼ .51, and in this case

the Bayesian and frequentist results are essentially the same. It should be apparent that,

if a and b are small compared to n, then the prior distribution will not matter much.

Indeed, if a and b are close to 0 and positive, then Ep

jxðÞx=n. We should hesitate

to set a and b equal to 0, because this would make the beta prior pdf not integrable,

but it does nevertheless give a reasonable posterior distribution if x and n  x are

positive. When a prior distribution is not integrable it is said to be improper.

In Bayesian inference, is there an interval corresponding to the confidence

interval for p given in Section 8.2? We have the posterior distribution for p,sowe

can take the central 95% of this distribution and call it a 95% credibility interval, as

mentioned at the beginning of this sectio n. In the case with a beta prior and a ¼ 1,

b ¼ 1, we have a beta posterior with a ¼ 804, b ¼ 772. Using the inverse cumu-

lative beta distribution function from MINITAB (or almost any major statistical

package) evaluated at .025 and .975, we obtain the interval [.4855, .5348]. For

comparison the 95% confidence interval from Equation (8.10)ofSection8.2 is

[.4855, .5348]. The intervals are not exactly the same, although they do agre e to

778 CHAPTER 14 Alternative Approaches to Inference

four decimals. The simpler formula, Equation (8.11), gives the answer [.4855,

.5349], which is very close because of the large sample size.

It is interesting that, although the frequentist and Bayesian intervals agree to

four decimals, they have very different interpretations. For the Bayesian interval we

can say that the probability is 95% that p is in the interval, given the data. However,

this is not correct for the frequentist interval, because p is not random and the

endpoints are not random after they have been specified, and ther efore no proba-

bility statement is appropriate. Here the 95% applies to the aggregate of confidence

intervals, of which in the long run 95% should include the true p .

The confidence intervals and credibility interval all include .5, so they allow

the possi bility that p ¼ .5. Another way to view this possibility in Bayesian terms is

to see whether the posterior distribution is consistent with p ¼ .5. We actually

consider the related hypothesis p  .5. Using a ¼ 1 and b ¼ 1 again, we find from

MINITAB that the beta distributio n with a ¼ 804 and b ¼ 772 has probability

.2100 of being less than or equal to .5. The corresponding one-tailed frequentist P-

value is the probability, assuming p ¼ .5, of at least 803 successes in 1574 trials,

which is .2173. Both the Bayesian and frequentist values are much greater than .05,

and there is no reason to reject .5 as a possible value for p.

To clarify the relations hip between E(pjx) and x/n, we can write E( pjx )asa

weighted average of the prior mean a/(a + b) and x/n.

EðpjxÞ¼

a þ b

n þ a þ b



a þ b

n þ a þ b



The weights can be interpreted in terms of the sum of the two parameters of the beta

distribution, which is often called the concentration parameter. The weights are

proportional to the concentration parameter a+bof the prior distribution and the

number n of observations. The weight of the prior depends on the size of a+bin

relation to n, and the concentration parameter of the posterior distribution is the

total a þ b þ n.

It is also useful to interpret the posterior pdf in terms of the concentration

parameter. Because the first parameter is the sum x + a and the second para-

meter is the sum (n  x)+b, the effect of a is to add to the number of successes

and the effect of b is to add to the number of failures. In particular, setting a to 1 and

b to 1 resulted in a posterior with the equivalent of 803 + 1 successes and

(1574 – 803) + 1 failures, for a total of 1574 + 2 observations. From this view-

point, the total observations are the a+bprovided by the prior plus the n provided

by the data, and this addition also gives the concentration parameter of the posterior

in terms of the concentration para meter of the prior.

How should we specify the prior distribution? The beta distribution is

convenient, because it is easy with this specification to find the posterior distribu-

tion, but what about a and b? Suppose we have asked 10 adults about the effect of

antibiotics on viruses, and it is reasonable to assume that the 10 are a random

sample. If 6 of the 10 say that antibiotics kill viruses, then we set a ¼ 6 and

b ¼ 10 – 6 ¼ 4. That is, we have a beta distributed prior with parameters 6

and 4. Then the posterior distribution is beta with parameters 803 + 6 ¼ 809 and

(1574 – 803) + 4 ¼ 775. The posterior is the same as if we had started with a ¼ 0

and b ¼ 0 and observed 809 who said that antibiotics kill viruses and 775 who

14.4 Bayesian Methods 779

said no. In other words, observations can be incorporated into the prior and count

just as if they were part of the NSF survey. ■

Life in the Bayesian world is sometimes more complicated. Perhaps the prior

observations are not of a quality equivalent to that of the survey, but we would still

like to use them to form a prior distribution. If we regard them as being only half as

good, then we could use the same proportions but cut the a and b in half, using 3 and

2 instea d of 6 and 4. There is certainly a subjective element to this, and it suggests

why some statisticians are hesitant about using Bayesian methods. When everyone

can agree about the prior distribution, there is little controversy about the Bayesian

procedure, but when the prior is very much a matter of opinion people tend to

disagree about its value.

Example 14.8 Assume a random sample X

; X

; ...; X

from the normal distribution with known

variance, and assume a normal prior distribution for m. In particular, consider the IQ

scores of 18 first- grade boys,

113 108 140 113 115 146 136 107 108 119 132 127 118

108 103 103 122 111

from the private speech data introduced in Example 1.2. Because the IQ has

a standard deviation of 15 nationwide, we can assume s ¼ 15 is valid here. For

the prior distribution it is reasonable to use a mean of m

¼ 110, a ballpark figure

for previous years in this school. It is harder to prescribe a standard deviation for

the prior, but we will use s

¼ 7.5. This is the standard deviation for the average of

four independent observations if the individual standard deviation is 15. As a result,

the effect on the posterior mean will turn out to be the same as if there were four

additional observations with average 110.

To compute the posterior distribution of the mean m, we use Bayes’ theorem

hðmjx

; x

; ...; x

Þ¼

f ð x

; x

; ...; x

jmÞgðmÞ

1

f ð x

; x

; ...; x

jmÞgðmÞdm

The numerator is

f ð x

; x

; ...; x

jmÞgðmÞ¼

ﬃﬃﬃﬃﬃﬃ

:5ðx

mÞ



ﬃﬃﬃﬃﬃﬃ

:5ðx

mÞ



ﬃﬃﬃﬃﬃﬃ

:5ðmm

ð2pÞ

ðnþ1Þ=2

:5½ðx

mÞ

þþðx

mÞ

þðmm



The trick here is to complete the square in the exponent, which yields

ð:5=s

Þðm  m

þ C

where C does not involve m and

; m

780 CHAPTER 14 Alternative Approaches to Inference

The posterior is then

hðmjx

; x

; ...; x

Þ¼

ð2pÞ

n=2



ð2pÞ

ð:5=s

Þðmm

ð2pÞ

n=2

1

ð2pÞ

ð:5=s

Þðmm

The integral is 1 because it is the area under a normal pdf, and the part in front of the

integral cancels out, leaving a posterior distribution that is normal with mean m

and

standard deviation s

hðmjx

; x

; ...; x

Þ¼

ð2pÞ

ð:5=s

Þðmm

Notice that the posterior mean m

is a weighted average of the prior mean

and the data mean x, with weights that are the reciprocals of the prior variance

and the variance of

x. It makes sense to define the precision as the reciprocal of

the variance because a lower variance implies a more precise measurement, and the

weights then are the corresponding precisions. Furthermore, the posterior variance

is the reciprocal of the sum of the reciprocals of the two variances, but this can

be described much more simply by saying that the posterior precision is the sum of

the prior precision plus the precision of

Numerically, we have

=18

7:5

¼ :09778 ¼

10:227

3:198

18ð118:28Þ

110

7:5

¼ 116:77

The posterior distribution is normal with mean m

¼ 116.77 and standard deviation

¼ 3.198. The mean m

is a weighted average of x ¼ 118:28 and m

¼ 110, so m

is necessarily between them. As n becomes large the weight given to m

declines,

and m

will be closer to x.

Knowing the mean and standard devi ation, we can use the normal distribu-

tion to find an interval with 95% probability for m. This 95% credibility interval is

[110.502, 123.038]. For comparison the 95% confidence interval using

x ¼ 118:28

and s ¼ 15 is

x  1:96s=

ﬃﬃﬃ

¼½111:35; 125:21. Notice that this interval must

be wider. Because the precisions add to give the posterior precision, the posterior

precision is greater than the prior precision and it is greater than the data precision.

Therefore, it is guaranteed that the posterior standard deviation s

will be less than

and less than the data standard deviation s=

ﬃﬃﬃ

Both the credibility interval and the confidence interval exclude 110, so we

can be pretty sure that m exceeds 110. Another way of looking at this is to calculate

the posterior probability of m bein g less than or equal to 110. Using m

¼ 116.77

and s

¼ 3.198, we obtain the probability .0171, so this too supports the idea that m

exceeds 110.

How should we go about choosing m

and s

for the prior distribution?

Suppose we have four prior observations for which the mean is 110. The standard

14.4 Bayesian Methods 781

deviation of the mean is 15=

ﬃﬃﬃ

. We therefore choose m

¼ 110 and s

¼ 7.5,

the same values used for this example. If the four values are com bined with the 18

values from the data set, then the mean of all 22 is 116.77 ¼ m

and the standard

deviation is 15=

ﬃﬃﬃﬃﬃ

¼ 3:198 ¼ s

. The 95% confidence interval for the mean,

based on the average of all 22 observations, is the same as the Bayesian 95%

credibility interval. This says that if you have some preliminary data values that are

just as good as the regular data values that will be obtained, then base the prior

distribution on the preliminary data. The posterior mean and its standard deviation

will be the same as if the preliminary data were combined with the regular data, and

the 95% credibility interval will be the sam e as the 95% confidence interval.

It should be emphasized that, even if the confidence interval is the same as

the credibility interval, they have dif ferent interpretations. To interp ret the Bayes-

ian credibility interval, we can say that the probability is 95% that m is in the

interval [110.502, 123.038]. However, for the frequentist confidence interval such a

probability statement does not make sense because m and the endpoints of the

interval are all constants after the interval has been calculated. Instead we have the

more complicated interpretation that, in repeated realizations of the confidence

interval, 95% of the intervals will include the true m in the long run.

What should be done if there are no prior observations and there are no strong

opinions about the prior mean m

? In this case the prior standard deviation s

can be

taken as some large number much bigger than s,suchass

¼ 1000 in our example.

The result is that the prior will have essentially no effect, and the posterior distribu-

tion will be based on the data, m

¼ x ¼ 118:28 and s

¼ s ¼ 15. The 95%

credibility interval will be the same as the 95% confidence interval based on the 18

observations, [111.35, 125.21], but of course the interpretation is different.

■

In both examples it turned out that the posterior distribution has the same form as

the prior distribution. When this happens we say that the prior distribution is

conjugate to the data distribution. Exercises 31 and 32 offer additional examples

of conjugate distributions.

Exercises Section 14.4 (23–32)

23. For the data of Example 14.7 assume a beta prior

distribution and assume that the 1574 observa-

tions are a random sample from the Bernoulli

distribution. Use Bayes’ theorem to derive the

posterior distribution, and compare your answer

with the result of Example 14.7.

24. Here are the IQ scores for the 15 first-grade girls

from the study mentioned in Example 14.8.

102 96 106 118 108 122 115 113

109 113 82 110 121 110 99

Assume the same prior distribution used in

Example 14.8, and assume that the data is a ran-

dom sample from a normal distribution with mean

m and s ¼ 15.

a. Find the posterior distribution of m.

b. Find a 95% credibility interval for m.

c. Add four observations with average 110 to the

data and find a 95% confidence interval for m

using the 19 observations. Compare with the

result of (b).

d. Change the prior so the prior precision is very

small but positive, and then recompute (a) and (b).

e. Find a 95% confidence interval for m using the

15 observations and compare with the credibil-

ity interval of (d).

25. Laplace’s rule of succession says that if there

have been n Bernoulli trials and they have all

been successes, then the probability of a success

on the next trial is ðn þ 1Þ=ðn þ 2Þ. For the deri-

vation Laplace used a beta prior with a ¼ 1 and

b ¼ 1 for binomial data, as in Example 14.7.

a. Show that, if a ¼ 1 and b ¼ 1 and there are n

successes in n trials, then the posterior mean of

p is ðn þ 1Þ=ðn þ 2Þ.

b. Explain (a) in terms of total successes and

failures; that is, explain the result in terms of

two prior trials plus n later trials.

782

CHAPTER 14 Alternative Approaches to Inference

c. Laplace applied his rule of succession to

compute the probability that the sun will rise

tomorrow using 5000 years, or n ¼ 1,826,214

days of history in which the sun rose every day.

Is Laplace’s method equivalent to including

two prior days when the sun rose once and

failed to rise once? Criticize the answer in

terms of total successes and failures.

26. For the scenario of Example 14.8 assume the same

normal prior distribution but assume that the data

set is just one observation

x ¼ 118:28 with stan-

dard deviation s

ﬃﬃﬃ

¼ 15

ﬃﬃﬃﬃﬃ



¼ 3:5355. Use

Bayes’ theorem to derive the posterior distribu-

tion, and compare your answer with the result of

Example 14.8.

27. Let X have the beta distribution on [0, 1] with

parameters a ¼ n

/2 and b ¼ n

/2, where n

and n

/2 are positive integers. Define Y ¼

X=aðÞ= 1  XðÞ=b½. Show that Y has the F distri-

bution with degrees of freedom n

, n

28. In a study by Erich Brandt of 70 restaurant bills,

40 of the 70 were paid using cash. We assume a

random sample and estimate the posterior distri-

bution of the binomial parameter p, the population

proportion paying cash.

a. Use a beta prior distribution with a ¼ 2 and

b ¼ 2.

b. Use a beta prior distribution with a ¼ 1 and

b ¼ 1.

c. Use a beta prior distribution with a and b very

small and positive.

d. Calculate a 95% credibility interval for p using

(c). Is your interval compatible with p ¼ .5?

e. Calculate a 95% confidence interval for p using

Equation (8.10) of Section 8.2, and compare

with the result of (d).

f. Calculate a 95% confidence interval for p using

Equation (8.11) of Section 8.2, and compare

with the results of (d) and (e).

g. Compare the interpretations of the credibility

interval and the confidence intervals.

h. Based on the prior in (c), test the hypothesis

p  .5 using the posterior distribution to find

P( p  .5).

29. Exercise 27 gives an alternative way of finding

beta probabilities when software for the beta dis-

tribution is unavailable.

a. Use Exercise 27 together with the F table to

obtain a 90% credibility interval for Exercise

28(c). [Hint: To find c such that .05 is the

probability that F is to the left of c, reverse

the degrees of freedom and take the reciprocal

of the value for a ¼ .05.]

b. Repeat (a) using software for the beta distribu-

tion and compare with the result of (a).

30. If a and

b are large, then the beta distribution can

be approximated by the normal distribution using

the beta mean and variance given in Section 4.5.

This is useful in case beta distribution software is

unavailable. Use the approximation to compute

the credibility interval in Example 14.7.

31. Assume a random sample X

, X

, ... , X

from the

Poisson distribution with mean l. If the prior dis-

tribution for l has a gamma distribution with para-

meters a and b, show that the posterior distribution

is also gamma distributed. What are its parameters?

32. Consider a random sample X

, X

, ..., X

from the

normal distribution with mean 0 and precision t

(use t as a parameter instead of s

¼ 1/t).

Assume a gamma-distributed prior for t and

show that the posterior distribution of t is also

gamma. What are its parameters?

Supplementary Exercises (33–42)

33. The article “Effects of a Rice-Rich Versus Potato-

Rich Diet on Glucose, Lipoprotein, and Cholesterol

Metabolism in Noninsulin-Dependent Diabetics”

(Amer. J. Clin. Nutrit., 1984: 598–606) gives the

accompanying data on cholesterol-synthesis rate

for eight diabetic subjects. Subjects were fed a

standardized diet with potato or rice as the major

carbohydrate source. Participants received both

diets for specified periods of time, with cholesterol-

synthesis rate (mmol/day) measured at the end of

each dietary period. The analysis presented in this

article used a distribution-free test. Use such a test

with significance level .05 to determine whether

the true mean cholesterol-synthesis rate differs sig-

nificantly for the two sources of carbohydrates.

Subject 1 2 3 4 5 6 7 8

Potato 1.88 2.60 1.38 4.41 1.87 2.89 3.96 2.31

Rice 1.70 3.84 1.13 4.97 .86 1.93 3.36 2.15

Supplementary Exercises 783

34. The study reported in “Gait Patterns During Free

Choice Ladder Ascents” (Hum. Movement Sci.,

1983: 187–195) was motivated by publicity con-

cerning the increased accident rate for individuals

climbing ladders. A number of different gait pat-

terns were used by subjects climbing a portable

straight ladder according to specified instructions.

The ascent times for seven subjects who used a

lateral gait and six subjects who used a four-beat

diagonal gait are given.

Lateral .86 1.31 1.64 1.51 1.53 1.39 1.09

Diagonal 1.27 1.82 1.66 .85 1.45 1.24

a. Carry out a test using a ¼ .05 to see whether

the data suggests any difference in the true

average ascent times for the two gaits.

b. Compute a 95% CI for the difference between

the true average gait times.

35. The sign test is a very simple procedure for testing

hypotheses about a population median assuming

only that the underlying distribution is continuous.

To illustrate, consider the following sample of 20

observations on component lifetime (hr):

1.7 3.3 5.1 6.9 12.6 14.4 16.4

24.6 26.0 26.5 32.1 37.4 40.1 40.5

41.5 72.4 80.1 86.4 87.5 100.2

We wish to test the hypotheses H

m ¼ 25:0versus

m > 25:0 The test statistic is Y ¼ the number of

observations that exceed 25.

a. Consider rejecting H

if Y  15. What is the

value of a (the probability of a type I error) for

this test? [Hint: Think of a “success” as a

lifetime that exceeds 25.0. Then Y is the num-

ber of successes in the sample. What kind of a

distribution does Y have when

m ¼ 25:0?]

b. What rejection region of the form Y  c spe-

cifies a test with a significance level as close to

.05 as possible? Use this region to carry out the

test for the given data. [Note: The test statistic

is the number of differences X

 25.0 that

have positive signs, hence the name sign test.]

36. Refer to Exercise 35, and consider a confidence

interval associated with the sign test, the sign

interval. The relevant hypotheses are now

m ¼

versus H

m 6¼

. Let’s use the fol-

lowing rejection region: either Y  15 or Y  5.

a. What is the significance level for this test?

b. The confidence interval will consist of all

values

for which H

is not rejected. Deter-

mine the CI for the given data, and state the

confidence level.

37. The single-factor ANOVA model considered in

Chapter 11 assumed the observations in the ith

sample were selected from a normal distribution

with mean m

and variance s

, that is,

¼ m

þ e

where the e’s are normal with

mean 0 and variance s

. The normality assump-

tion implies that the F test is not distribution-free.

We now assume that the e’s all come from the

same continuous, but not necessarily normal, dis-

tribution, and develop a distribution-free test of

the null hypothesis that all I m

’s are identical. Let

N ¼

, the total number of observations in the

data set (there are J

observations in the ith sam-

ple). Rank these N observations from 1 (the smal-

lest) to N, and let

be the average of the ranks for

the observations in the ith sample. When H

true, we expect the rank of any particular observa-

tion and therefore also

to be (N+1)/2. The

data argues against H

when some of the R

’s

differ considerably from (N+1)/2. The Krus-

kal–Wallis test statistic is

K ¼

NðN þ 1Þ



N þ 1



When H

is true and either (1) I ¼ 3, all J

 6or

(2) I > 3, all J

 5, the test statistic has approxi-

mately a chi-squared distribution with I  1 df.

The accompanying observations on axial stiff-

ness index resulted from a study of metal-plate

connected trusses in which five different plate

lengths—4 in., 6 in., 8 in., 10 in., and 12 in. —

were used (“Modeling Joints Made with Light-

Gauge Metal Connector Plates,” Forest Products

J., 1979: 39–44).

i ¼ 1 (4 in.): 309.2 309.7 311.0 316.8

326.5 349.8 409.5

i ¼ 2 (6 in.): 331.0 347.2 348.9 361.0

381.7 402.1 404.5

i ¼ 3 (8 in.): 351.0 357.1 366.2 367.3

382.0 392.4 409.9

i ¼ 4 (10 in.): 346.7 362.6 384.2 410.6

433.1 452.9 461.4

i ¼ 5 (12 in.): 407.4 410.7 419.9 441.2

441.8 465.8 473.4

Use the K–W test to decide at significance level

.01 whether the true average axial stiffness index

depends somehow on plate length.

784

CHAPTER 14 Alternative Approaches to Inference

38. The article “Production of Gaseous Nitrogen in

Human Steady-State Conditions” (J. Appl. Physiol.,

1972: 155–159) reports the following observations

on the amount of nitrogen expired (in liters) under

four dietary regimens: (1) fasting, (2) 23% protein,

(3) 32% protein, and (4) 67% protein. Use the

Kruskal–Wallis test (Exercise 37) at level .05 to

test equality of the corresponding m

’s.

1. 4.079 4.859 3.540 5.047 3.298

4.679 2.870 4.648 3.847

2. 4.368 5.668 3.752 5.848 3.802

4.844 3.578 5.393 4.374

3. 4.169 5.709 4.416 5.666 4.123

5.059 4.403 4.496 4.688

4. 4.928 5.608 4.940 5.291 4.674

5.038 4.905 5.208 4.806

39. The model for the data from a randomized block

experiment for comparing I treatments was

¼ m þ a

þ b

þ e

, where the a’s are treat-

ment effects, the b’s are block effects, and the

e’s were assumed normal with mean 0 and vari-

ance s

. We now replace normality by the

assumption that the e’s have the same continuous

distribution. A distribution-free test of the null

hypothesis of no treatment effects, called Fried-

man’s test, involves first ranking the observations

in each block separately from 1 to I . The rank

average

is then calculated for each of the I

treatments. If H

is true, the expected value of

each rank average is (I+1)/2. The test statistic is

12J

IðI þ 1Þ



I þ 1



For even moderate values of J, the test statistic has

approximately a chi-squared distribution with

I  1 df when H

is true.

The article “Physiological Effects During

Hypnotically Requested Emotions” (Psychoso-

matic Med., 1963: 334–343) reports the follow-

ing data (x

) on skin potential in millivolts when

the emotions of fear, happiness, depression, and

calmness were requested from each of eight

subjects.

Blocks (Subjects)

1234

Fear 23.1 57.6 10.5 23.6

Happiness 22.7 53.2 9.7 19.6

Depression 22.5 53.7 10.8 21.1

Calmness 22.6 53.1 8.3 21.6

5678

Fear 11.9 54.6 21.0 20.3

Happiness 13.8 47.1 13.6 23.6

Depression 13.7 39.2 13.7 16.3

Calmness 13.3 37.0 14.8 14.8

Use Friedman’s test to decide whether emotion

has an effect on skin potential.

40. In an experiment to study the way in which differ-

ent anesthetics affect plasma epinephrine concen-

tration, ten dogs were selected and concentration

was measured while they were under the influence

of the anesthetics isoflurane, halothane, and

cyclopropane (“Sympathoadrenal and Hemody-

namic Effects of Isoflurane, Halothane, and

Cyclopropane in Dogs,” Anesthesiology , 1974:

465–470). Test at level .05 to see whether there

is an anesthetic effect on concentration. [Hint: See

Exercise 39.]

Dog

12345

Isoflurane .28 .51 1.00 .39 .29

Halothane .30 .39 .63 .38 .21

Cyclopropane 1.07 1.35 .69 .28 1.24

678910

Isoflurane .36 .32 .69 .17 .33

Halothane .88 .39 .51 .32 .42

Cyclopropane 1.53 .49 .56 1.02 .30

41. Suppose we wish to test

: the X and Y distributions are identical

versus

: the X distribution is less spread out than the Y

distribution

The accompanying figure pictures X and Y dis-

tributions for which H

is true. The Wilcoxon

rank-sum test is not appropriate in this situation

because when H

is true as pictured, the Y’s will

tend to be at the extreme ends of the combined

sample (resulting in small and large Y ranks), so

Supplementary Exercises 785

the sum of X ranks will result in a W value that is

neither large nor small.

X distribution

Y distribution

“Ranks” : 5

13 64

Consider modifying the procedure for assigning

ranks as follows: After the combined sample of

m+nobservations is ordered, the smallest obser-

vation is given rank 1, the largest observation is

given rank 2, the second smallest is given rank 3,

the second largest is given rank 4, and so on. Then

if H

is true as pictured, the X values will tend to

be in the middle of the sample and thus receive

large ranks. Let W

denote the sum of the X ranks

and consider rejecting H

in favor of H

when

 c. When H

is true, every possible set of X

ranks has the same probability, so W

has the same

distribution as does W when H

is true. Thus c can

be chosen from Appendix Table A.13 to yield a

level a test. The accompanying data refers to

medial muscle thickness for arterioles from the

lungs of children who died from sudden infant

death syndrome (x’s) and a control group of chil-

dren (y’s). Carry out the test of H

versus H

level .05.

SIDS 4.0 4.4 4.8 4.9

Control 3.7 4.1 4.3 5.1 5.6

Consult the Lehmann book (in the chapter bibli-

ography) for more information on this test, called

the Siegel–Tukey test.

42. The ranking procedure described in Exercise 41

is somewhat asymmetric, because the smallest

observation receives rank 1 whereas the largest

receives rank 2, and so on. Suppose both the

smallest and the largest receive rank 1, the second

smallest and second largest receive rank 2, and so

on, and let W

be the sum of the X ranks. The null

distribution of W

is not identical to the null dis-

tribution of W, so different tables are needed.

Consider the case m ¼ 3, n ¼ 4. List all 35 possi-

ble orderings of the three X values among the

seven observations (e.g., 1, 3, 7 or 4, 5, 6), assign

ranks in the manner described, compute the value

of W

for each possibility, and then tabulate the

null distribution of W

. For the test that rejects if

 c, what value of c prescribes approximately

a level .10 test? This is the Ansari–Bradley test;

for additional information, see the book by Hol-

lander and Wolfe in the chapter bibliography.

Bibliography

Berry, Donald A., Statistics: A Bayesian Perspective,

Brooks/Cole—Cengage Learning, Belmont, CA,

1996. An elementary introduction to Bayesian

ideas and methodology.

Gelman, Andrew, John B. Carlin, Hal S. Stern, and

Donald B. Rubin, Bayesian Data Analysis (2nd ed.),

Chapman and Hall, London, 2003. An up-to-date

survey of theoretical, practical, and computational

issues in Bayesian inference.

Hollander, Myles, and Douglas Wolfe, Nonparametric

Statistical Methods (2nd ed.), Wiley, New York,

1999. A very good reference on distribution-free

methods with an excellent collection of tables.

Lehmann, Erich, Nonparametrics: Statistical Methods

Based on Ranks (revised ed.), Springer, New York,

2006. An excellent discussion of the most impor-

tant distribution-free methods, presented with a

great deal of insightful commentary.

786

CHAPTER 14 Alternative Approaches to Inference

Appendix Tables

787