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9.8 Exercises 349

(d) The data for this exercise come from a study conducted in Canada. If you

wanted to replicate the test in the USA, what sample size would guarantee

a power of 99% if H

were to be rejected whenever the difference from the

true mean was 4, (

|µ

−µ

|=4)? A small pilot study conducted to assess the

variability of hemopexin level estimated the standard deviation as s

=12.

(e) Find the posterior probability of the hypothesis H

: µ > 85 using Win-

BUGS. Use noninformative priors. Also, compare the 95% credible set for

that you obtained with the conﬁdence interval in (a).

Hint:

The commands

%file dmd.mat should be on path

load ’dmd.mat’; hemo = dmd( dmd(:,6)==1, 3);

will distill the levels of hemopexin in carrier cases.

9.17. Retinol and a Copper-Deﬁcient Diet. The liver is the main storage site

of vitamin A and copper. Inverse relationships between copper and vita-

min A liver concentrations have been suggested. In Rachman et al. (1987)

the consequences of a copper-deﬁcient diet on liver and blood vitamin A

storage in Wistar rats was investigated. Nine animals were fed a copper-

deﬁcient diet for 45 days from weaning. Concentrations of vitamin A were

determined by isocratic high-performance liquid chromatography using UV

detection. Rachman et al. (1987) observed in the liver of the rats fed a

copper-deﬁcient diet a mean level of retinol [in micrograms/g of liver] was

X =3.3 and s =1.4. It is known that the normal level of retinol in a rat liver

=1.6.

(a) Find the 95% conﬁdence interval for the mean level of liver retinol in the

population of copper-deﬁcient rats. Recall that the sample size was n

=9.

(b) Test the hypothesis that the mean level of retinol in the population of

copper-deﬁcient rats is

= 1.6 versus a sensible alternative (either one-

sided or two-sided), at the signiﬁcance level

α = 0.05. Use both rejection

region and p-value approaches.

: µ =µ

2.4. Comment.

(d) Suppose that you are designing a new, larger study in which you are

going to assume that the variance of observations is

= 1.4

, as the lim-

ited nine-animal study indicated. Find the sample size so that the power of

rejecting H

when an alternative H

: µ =2.1 is true is 0.80. Use α =0.05.

(e) Provide a Bayesian solution using WinBUGS.

9.18. Aniline. Organic chemists often purify organic compounds by a method

known as fractional crystallization. An experimenter wanted to prepare

and purify 5 grams of aniline. It is postulated that 5 grams of aniline would

yield 4 grams of acetanilide. Ten 5-gram quantities of aniline were individ-

ually prepared and puriﬁed.

(a) Test the hypothesis that the mean dry yield differs from 4 grams if the

mean yield observed in a sample was

X = 4.21. The population is assumed

350 9 Testing Statistical Hypotheses

normal with known variance σ

= 0.08. The signiﬁcance level is set to α =

0.05.

(b) Report the p-value.

X will the null hypothesis be rejected at the level

α =0.05?

(d) What is the power of the test for the alternative H

: µ =3.6 at α =0.05.

(e) If you are to design a similar experiment but would like to achieve a

power of 90% versus the alternative H

: µ = 3.6 at α = 0.05, what sample

size would you recommended?

9.19. DNA Random Walks. DNA random walks are numerical transcriptions

of a sequence of nucleotides. The imaginary walker starts at 0 and goes

one step up (s

= +1) if a purine nucleotide (A, G) is encountered, and one

step down (s

=−1) if a pyramidine nucleotide (C, T) is encountered. Peng et

al. (1992) proposed identifying coding/noncoding regions by measuring the

irregularity of associated DNA random walks. A standard irregularity mea-

sure is the Hurst exponent H, an index that ranges from 0 to 1. Numerical

sequences with H close to 0 are irregular, while the sequences with H close

to 1 appear more smooth.

Figure 9.3 shows a DNA random walk in the DNA of a spider monkey (Ate-

les geoffroyi). The sequence is formed from a noncoding region and has a

Hurst exponent of H

=0.61.

0 1000 2000 3000 4000 5000 6000 7000 8000

−50

100

150

200

250

300

350

400

index

DNA RW

Fig. 9.3 A DNA random walk formed by a noncoding region from the DNA of a spider

monkey. The Hurst exponent is 0.61.

A researcher wishes to design an experiment in which n nonoverlapping

DNA random walks of a ﬁxed length will be constructed, with the goal of

testing to see if the Hurst exponent for noncoding regions is 0.6.

The researcher would like to develop a test so that an effect e

=|µ

−µ

|/σ

will be detected with a probability of 1 −β = 0.9. The test should be two-

sided with a signiﬁcance level of

α = 0.05. Previous analyses of noncoding

regions in the DNA of various species suggest that exponent H is approx-

9.8 Exercises 351

imately normally distributed with a variance of approximately σ

= 0.03

The researcher believes that

|µ

−µ

| = 0.02 is a biologically meaningful

difference. In statistical terms, a 5%-level test for H

: µ = 0.6 versus the

alternative H

: µ = 0.6 ±0.02 should have a power of 90%. The preexperi-

mentally assessed variance

=0.03

leads to an effect size of e =2/3.

(a) Argue that a sample size of n

=24 satisﬁes the power requirements.

The experiment is conducted and the following 24 values for the Hurst ex-

ponent are obtained:

H =[0.56 0.61 0.62 0.53 0.54 0.60 0.56 0.59 ...

0.60 0.60 0.62 0.60 0.58 0.57 0.61 0.64 ...

0.60 0.61 0.58 0.59 0.55 0.59 0.60 0.65 ];

% [mean(H) std(H)] %%% 0.5917 0.0293

(b) Using the t-test, test H

against the two-sided alternative at the level

α =0.05 using both the rejection-region approach and the p-value approach.

noncentral t-distribution and s found from the sample.

9.20. Binding of Propofol. Serum protein binding is a limiting factor in the ac-

cess of drugs to the central nervous system. Disease-induced modiﬁcations

of the degree of binding may inﬂuence the effect of anesthetic drugs.

The protein binding of propofol, an intravenous anaesthetic agent that is

highly bound to serum albumin, has been investigated in patients with

chronic renal failure. Protein binding was determined by the ultraﬁltration

technique using an Amicon Micropartition System, MPS-1.

The mean proportion of unbound propofol in healthy individuals is 0.96,

and it is assumed that individual proportions follow a beta distribution,

B e(96, 4). Based on a sample of size n = 87 of patients with chronic renal

failure, the average proportion of unbound propofol was found to be 0.93

with a sample standard deviation of 0.12.

(a) Test the hypothesis that the mean proportion of unbound propofol in a

population of patients with chronic renal failure is 0.96 versus the one-sided

alternative. Use

α = 0.05 and perform the test using both the rejection-

region approach and the p-value approach. Would you change the decision

α =0.01?

(b) Even though the individual measurements (proportions) follow a beta

distribution, one can use the normal theory in (a). Why?

9.21. Improvement of Surgical Procedure. In a disease in which the post-

operative mortality is usually 10%, a surgeon devises a new surgical tech-

nique. He tries the technique on 15 patients and has no fatalities.

(a) What is the probability of the surgeon having no fatalities in treating 15

patients if the mortality rate is 10%.

(b) The surgeon claims that his new surgical technique signiﬁcantly im-

proves the survival rate. Is his claim justiﬁed?

without a single fatality in order to convince you that his procedure is a

352 9 Testing Statistical Hypotheses

signiﬁcant improvement over the old technique? Specify your criteria and

justify your answer.

9.22. Cancer Therapy. Researchers in cancer therapy often report only the

number of patients who survive for a speciﬁed period of time after treat-

ment rather than the patients’ actual survival times. Suppose that 40%

of the patients who undergo the standard treatment are known to survive

5 years. A new treatment is administered to 200 patients, and 92 of them

are still alive after a period of 5 years.

(a) Formulate the hypotheses for testing the validity of the claim that the

new treatment is more effective than the standard therapy.

(b) Test with

α = 0.05 and state your conclusion; use the rejection-region

method.

9.23. Is the Cloning of Humans Moral? Gallup Poll estimates that 88% Amer-

icans believe that cloning humans is morally unacceptable. Results are

based on telephone interviews with a randomly selected national sample

of n

=1000 adults, aged 18 and older.

(a) Test the hypothesis that the true proportion is 0.9, versus the two-sided

alternative, based on the Gallup data. Use

α =0.05.

(b) Does 0.9 fall in the 95% conﬁdence interval for the proportion.

=0.85?

9.24. Smoking Illegal? In a recent Gallup poll of Americans, less than a third

of respondents thought smoking in public places should be made illegal, a

signiﬁcant decrease from the 39% who thought so in 2001.

The question used in the poll was: Should smoking in all public places be

made totally illegal? In the poll, 497 people responded and 154 answered

yes. Let p be the proportion of people in the US voting population support-

ing the idea that smoking in public places should be made illegal.

(a) Test the hypothesis H

: p =0.39 versus the alternative H

: p <0.39 at

the level

α =0.05.

(b) What is the 90% conﬁdence interval for the unknown population propor-

tion p? In terms of the Gallup pollsters, what is the “margin of error”?

9.25. Spider Monkey DNA. An 8192-long nucleotide sequence segment taken

from the DNA of a spider monkey (Ateles geoffroyi) is provided in the ﬁle

dnatest.m.

Find the relative frequency of adenine

as an estimator of the overall

population proportion, p

Find a 99% conﬁdence interval for p

and test the hypothesis H

: p

=0.2

versus the alternative H

: p

>0.2. Use α =0.05.

Chapter References 353

MATLAB AND WINBUGS FILES AND DATA SETS USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch9.Testing/

chi2test1.m, dnarw.m, dnatest.m FDR.m, hemopexin1.m, hemoragic.m,

moon.m, powers.m, SBB.m, testexa.m

hemopexin.odc, hemorrhagic.odc, hypersplenism.odc, moonillusion.odc,

retinol.odc, spikes.odc, systolic.odc

dnadat.mat|txt, spid.dat

CHAPTER REFERENCES

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for the Student and Research Worker. Springer, Berlin Heidelberg New York.

Benjamini, Y. and Hochberg, Y. (1995) Controlling the false discovery rate: a practical and

powerful approach to multiple testing. J. R. Stat. Soc. B, 57, 289–300.

Berger, J. O. and Sellke, T. (1987). Testing a point null hypothesis: the irreconcilability of

p-values and evidence (with discussion). J. Am. Stat. Assoc., 82, 112–122.

Casella, G. and Berger, R. (1990). Statistical Inference. Duxbury, Belmont.

Edwards, W., Lindman, H., and Savage, L. J. (1963). Bayesian statistical inference for psy-

chological research. Psychol. Rev., 70, 193–242.

Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh.

Fisher, R. A. (1926). The arrangement of ﬁeld experiments. J. Ministry Agricult., 33, 503–

513.

Goodman, S. (1999a). Toward evidence-based medical statistics. 1: The p-value fallacy. Ann.

Intern. Med., 130, 995–1004.

Goodman, S. (1999b). Toward evidence-based medical statistics. 2: The Bayes factor. Ann.

Intern. Med., 130, 1005–1013.

Goodman, S. (2001). Of p-values and Bayes: a modest proposal. Epidemiology, 12, 3, 295–297

Hamilton, L. C. (1990). Modern Data Analysis: A First Course in Applied Statistics. Brooks/

Cole, Paciﬁc Grove.

Hoenig, J. M. and Heisey, D. M. (2001). Abuse of power: the pervasive fallacy of power cal-

culations for data analysis. Am. Statist., 55, 1, 19–24.

Ioannidis, J. P. (2005). Why most published research ﬁndings are false. PLoS Med 2(8): e124.

doi:10.1371/journal.pmed.0020124.

Kaufman, L. and Rock, I. (1962). The moon illusion, I. Science, 136, 953–961.

354 9 Testing Statistical Hypotheses

Katz, S., Lautenschlager, G. J., Blackburn, A. B., and Harris, F. H. (1990). Answering reading

comprehension items without passages on the SAT. Psychol. Sci., 1, 122–127.

Peng, C. K., Buldyrev, S. V., Goldberger, A. L., Goldberg, Z. D., Havlin, S., Sciortino, E., Si-

mons, M., and Stanley, H. E. (1992). Long-range correlations in nucleotide sequences.

Nature, 356, 168–170.

Rachman, F., Conjat, F., Carreau, J. P., Bleiberg-Daniel, F., and Amedee-Maneseme, O.

(1987). Modiﬁcation of vitamin A metabolism in rats fed a copper-deﬁcient diet. Int. J.

Vitamin Nutr. Res., 57, 247–252.

Sampford, M. R. and Taylor, J. (1959). [not cited?] Censored observations in randomized

block experiments. J. R. Stat. Soc. Ser. B, 21, 214–237.

Schervish, M. (1996). P-values: What they are and what they are not. Am. Stat., 50, 203–

206.

Sellke, T., Bayarri, M. J., and Berger, J. O. (2001). Calibration of p values for testing precise

null hypotheses. Am. Stat., 55, 62–71.

Chapter 10

Two Samples

Given a choice between two theories, take the one which is funnier.

– Blore’s Razor

WHAT IS COVERED IN THIS CHAPTER

• Testing the Equality of Normal Means and Variances

• A Bayesian Approach to a Two-Sample Problem

• Paired t-Test

• Testing the Equality of Two Proportions

• Risk Jargon: Risk Differences, Ratios, and Odds Ratios

• Two Poisson Means

• Equivalence Testing

10.1 Introduction

A two-sample inference is one of the most common statistical procedures used

in practice. For example, a colloquial use of “t-test” usually refers to the com-

parison of means from two independent normal populations rather than a

single-sample t-test. In this chapter we will test the equality of two normal

means for independent and dependent (paired) populations as well as the

equality of two variances and proportions. In the context of comparing pro-

portions, we will discuss the risk and odds ratios. In testing the equality of

B. Vidakovic, Statistics for Bioengineering Sciences: With MATLAB and WinBUGS Support, 355

Springer Texts in Statistics, DOI 10.1007/978-1-4614-0394-4_10,

356 10 Two Samples

means in independent normal populations, we will distinguish two cases: (i)

when the underlying population variances are the same and (ii) when no as-

sumption about the variances is made. In this second case the population vari-

ances may be different, or even equal, but simply no assumption about their

equality enters the test. Each of the tests involves the difference or ratio of

the parameters (means, proportions, variances), and for each difference/ratio

we provide the (1

−α)100% conﬁdence interval. For selected tests we will in-

clude the power analysis. This chapter is intertwined with parallel Bayesian

solutions whenever appropriate.

It is important to emphasize that the normality of populations and large

samples for the proportions (for the CLT to hold) are critical for some tests.

Later in the text, in Chap. 12, we discuss distribution-free counterpart tests

that relax the assumption of normality (sign test, Wilcoxon signed rank test,

Wilcoxon Mann Whitney test) at the expense of efﬁciency if the normality

holds.

10.2 Means and Variances in Two Independent Normal

Populations

We start with an example that motivates the testing of two population means.

Example 10.1. Lead Exposure. It is hypothesized that blood levels of lead

tend to be higher for children whose parents work in a factory that uses lead

in the manufacturing process. Researchers examined lead levels in the blood

of 12 children whose parents worked in a battery manufacturing factory. The

results for the “case children” X

, X

,... , X

1,12

were compared to those of the

“control” sample X

, X

,... , X

2,15

consisting of 15 children selected randomly

from families where the parents did not work in a factory that used lead. It is

assumed that the measurements are independent and come from normal pop-

ulations. The resulting sample means and sample standard deviations were

=0.015, s

=0.004, X

=0.006, and s

=0.006.

Obviously, the sample mean for the “case” children is higher than the sam-

ple mean in the control sample. But is this difference signiﬁcant?

To state the problem in more general terms, we assume that two samples

, X

,... , X

1,n

and X

, X

,... , X

2,n

are observed from populations with

normal

N (µ

,σ

) and N (µ

,σ

) distributions, respectively.

We are interested in testing the hypothesis H

: µ

=µ

versus the alterna-

tive H

: µ

>,6=,<µ

at a signiﬁcance level α.

For the lead exposure example, the null hypothesis being tested is that the

parents’ workplace has no effect on their children’s lead concentration, that is,

the two population means will be the same:

: µ

=µ

10.2 Means and Variances in Two Independent Normal Populations 357

Here the populations are deﬁned as all children that are exposed or nonex-

posed. The alternative hypothesis H

may be either one- or two-sided. The

two-sided alternative is simply H

: µ

6= µ

, the population means are not

equal, and the difference can go either way. The choice of one-sided hypothesis

should be guided by the problem setup, and sometimes by the observations. In

the context of this example, it would not make sense to take the one-sided al-

ternative as H

: µ

<µ

stating that the concentration in the exposed group is

smaller than that in the control. In addition,

=.015 and X

=.006 are ob-

served. Thus, the sensible one-sided hypothesis in this context is H

: µ

>µ

There are two testing scenarios of population means that depend on an

assumption about associated population variances.

Scenario 1: Variances unknown but assumed equal. In this case, the

joint

is estimated by both s

and s

. The weighted average of s

and s

with weights w and 1 −w, depending on group sample sizes n

and n

−1)s

+(n

−1)s

−2

−1

−2

−1

−2

=ws

+(1 −w)s

is called the pooled sample variance and it better estimates the population

variance than any individual s

. The square root of s

is called the pooled

sample standard deviation and is denoted by s

. One can show that when H

is true, that is, when µ

=µ

, the statistic

−X

1/n

+1/n

(10.1)

has Student’s t distribution with df

= n

−2 degrees of freedom.

Scenario 2: No assumption about the variances. In this case, when H

is true, i.e., when µ

=µ

, the statistic

−X

has a t distribution with approximately

)

/(n

−1) +(s

)

/(n

−1)

(10.2)

degrees of freedom. This is a special case of the so-called Welch–Satterthwaite

formula, which approximates the degrees of freedom for a linear combination

of chi-square distributions (Satterthwaite 1946, Welch 1948).

For both scenarios:

358 10 Two Samples

Alternative α-level rejection region p-value

: µ

>µ

d f ,1−α

,∞) 1-tcdf(t, df)

: µ

6=µ

(−∞, t

df,α/2

] ∪[t

d f ,1−α/2

,∞) 2

tcdf(-abs(t), df)

: µ

<µ

(−∞, t

d f ,α

] tcdf(t, df)

When the population variances are known, the proper statistic is Z:

−X

+σ

with a normal

N (0,1) distribution, and the proper statistical analysis involves

normal quantiles as in the z-test. Given the fact that in realistic examples the

variances are not known when the means are tested, the z-test is mainly used

as an asymptotic test.

When sample sizes n

and n

are large, the z-statistic can be used instead

of t even if the variances are not known, due to the CLT. This approximation

was more interesting in the past when computing was expensive, but these

days one should always use a t-test for any sample size, as long as the popula-

tion variances are not known.

In Example 10.1, the variances are not known. We may assume that they

are either equal or possibly not equal based on the nature of the experiment,

sampling, and some other nonexperimental factors. However, we may also for-

mally test whether the population variances are equal prior to deciding on the

testing scenario.

We brieﬂy interrupt our discussion of testing the equality of means with an

exposition of how to test the equality of variances in two normal populations.

Testing the Equality of Two Normal Variances. Selecting the “scenario”

for testing the equality of normal means requires an assumption about the

associated variances. This assumption can be guided by an additional test for

the equality of two normal variances prior to testing the means. The variance-

before-the-means testing is criticized mainly on the grounds that tests for vari-

ances are not as robust (with respect to deviations from normality) compared

to the test for means, especially when the sample sizes are small or unbal-

anced. Although we agree with this criticism, it should be noted that choosing

the scenario seldom inﬂuences the resulting inference, except maybe in the

borderline cases. See also discussion on p. 416. When in doubt, one should

not make the assumption about variances and should use a more conservative

scenario 2.

Suppose the samples X

, X

,... , X

1,n

and X

, X

,... , X

2,n

come from

normal populations with distributions of

N (µ

,σ

) and N (µ

,σ

), respec-

tively. To choose the strategy for testing the means, the following test of vari-

ances with the two-sided alternative is helpful: