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

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11.13 Exercises 469

ly wh gr bl

Values

5 10 15 20 25 30 35 40 45 50 55

Click on the group you want to test

2 groups have means significantly different from wh

(a) (b)

Fig. 11.13 (a) Barplots of

ntrap for the four treatments. (b) A snapshot of the interactive

plot for multiple comparisons.

changes or emotional events. Skin discoloration is caused by an abnormal

spasm of the blood vessels, diminishing blood supply to the local tissues.

Kahan et al. (1987) investigated the efﬁcacy of the calcium-channel blocker

nicardipine in the treatment of Raynaud’s phenomenon. This efﬁcacy was

assessed in a prospective, double-blind, randomized, crossover trial in 20

patients. Each patient received 20 mg nicardipine or placebo three times

a day for 2 weeks and then was crossed over for 2 weeks. To suppress

any carryover effect, there was a 1-week washout period between the two

treatments. The researchers were interested in seeing if nicardipine signif-

icantly decreased the frequency and severity of Raynaud’s phenomenon as

compared with placebo. The data consist of a number of attacks in 2 weeks.

Period 1 Period 2 Period 1 Period 2

Subject Nicardipine Placebo Subject Placebo Nicardipine

1 16 12 11 18 12

2 26 19 12 12 4

3 8 20 13 46 37

4 37 44 14 51 58

5 9 25 15 28 2

6 41 36 16 29 18

7 52 36 17 51 44

8 10 11 18 46 14

9 11 20 19 18 30

10 30 27 20 44 4

Download the MATLAB format of these data ( raynaud.m) and perform

an ANOVA test in MATLAB. Select an additive model, with the number of

attacks y

i jk

represented as

i jk

=µ +α

+β

+γ

+²

i jk

470 11 ANOVA and Elements of Experimental Design

Here µ is the grand mean, α

, i = 1,...,20 is the subject effect, β

, j = 1,2

is the drug effect,

, k =1,2 is the period effect, and ²

i jk

is the zero-mean

normal random error.

We are not interested in testing the null hypothesis about the subjects,

: α

=0,

since the subjects are blocks here. We are interested in testing the effect of

the drug,

: β

=β

=0,

and in testing the effect of the washout period,

: γ

=γ

=0.

(a) Provide an ANOVA table for this model using MATLAB. Test hypotheses

and H

. Describe your ﬁndings.

(b) What proportion of variability in the data y

i jk

is explained by the above

ANOVA model?

domized to drug/period treatments. Test H

and H

in this context and

comment.

11.22. Simvastatin. In a quantitative physiology lab II at Georgia Tech, students

were asked to ﬁnd a therapeutic model to test on MC3T3-E1 cell line to en-

hance osteoblastic growth. The students found a drug called Simvastatin,

a cholesterol lowering drug to test on these cells. Using a control and three

different concentrations 10

−9

M, 10

−8

M, and 10

−7

M, cells were treated with

the drug. These cells were plated on four, 24 well plates with each well plate

having a different treatment. To test for osteoblastic differentiation an as-

say, pNPP, was used to test for alkaline phosphatase activity. The higher

the alkaline phosphatase activity the better the cells are differentiating,

and become more bone like. This assay was performed 6 times total within

11 days. Each time the assay was performed, four wells from each plate

were used. The data (

simvastatin.dat) are provided in the following ta-

ble:

11.13 Exercises 471

Time

Concentration Day 1 Day 3 Day 5 Day 7 Day 9 Day 11

0.062 0.055 0.055 1.028 0.607 0.067

A 0.517 0.054 0.059 1.067 0.104 0.093

(Control) 0.261 0.056 0.062 1.128 0.163 0.165

0.154 0.063 0.062 0.855 0.109 0.076

0.071 0.055 0.067 0.075 0.068 0.347

B 0.472 0.060 1.234 0.076 0.143 0.106

(10

−7

M) 0.903 0.057 1.086 0.090 0.108 0.170

0.565 0.056 0.188 1.209 0.075 0.097

0.068 0.059 0.092 0.091 0.098 0.115

C 0.474 0.070 0.096 0.218 0.122 0.085

(10

−8

M) 0.063 0.090 0.123 0.618 0.837 0.076

0.059 0.064 0.091 0.093 0.142 0.085

0.066 0.447 0.086 0.248 0.108 0.290

D 0.670 0.091 0.076 0.094 0.105 0.090

(10

−9

M) 0.076 0.079 0.082 0.215 0.093 0.518

0.080 0.071 0.080 0.401 0.580 0.071

In this design there are two crossed factors Concentration and Time.

(a) Using the two-way ANOVA show that the interaction between factors

Time and Concentration is signiﬁcant.

(b) Since the presence of a signiﬁcant interaction affects the tests for main

effects, conduct a conditional test for the Concentration factor if Time is

ﬁxed at Day 3. Also, conduct the conditional test for Time factor if the Con-

centration level is C.

11.23. Antitobacco Media Campaigns. Since the early 1990s, the US popula-

tion has been exposed to a growing number and variety of televised antito-

bacco advertisements. By 2002, 35 states had launched antitobacco media

campaigns.

Four states (factor A) participated in an antitobacco awareness study. Each

state independently devised antitobacco program. Three cities (factor B)

within each state were selected for participation and 10 households within

each city were randomly selected to evaluate effectiveness of the program.

All members of the selected households were interviewed and a composite

index was formed for each household measuring the extent of antitobacco

awareness. The data are given in the table (larger the index, the greater

the awareness)

472 11 ANOVA and Elements of Experimental Design

State i 1 2 3 4

City j 1 2 3 1 2 3 1 2 3 1 2 3

1 42 26 33 48 56 44 23 47 48 70 56 35

2 56 38 51 54 65 34 31 39 40 61 49 30

3 35 42 43 48 71 39 34 45 43 57 58 41

4 40 29 49 57 66 40 28 51 48 69 55 52

Household k 5 54 44 42 51 64 51 30 38 51 71 47 39

6 49 43 39 51 51 47 36 47 57 49 50 44

7 51 45 41 56 60 34 41 39 40 61 49 30

8 48 39 47 50 72 41 35 44 41 57 61 40

9 52 30 45 50 60 42 28 44 49 69 54 52

10 54 40 51 49 63 49 33 46 49 67 52 45

(a) Discuss and propose the analyzing methodology.

(b) Assume that nested design with ﬁxed factor effects is appropriate. Pro-

vide the ANOVA table.

α =0.05. State the alternatives and conclusion.

11.24. Orthosis. The data

Ortho.dat were acquired and computed by Dr.

Amarantini David and Dr. Martin Luc (Laboratoire Sport et Performance

Motrice, Grenoble University, France). The purpose of recording such data

was the interest to better understand the processes underlying movement

generation under various levels of an externally applied moment to the

knee. In this experiment, stepping-in-place was a relevant task to inves-

tigate how muscle redundancy could be appropriately used to cope with an

external perturbation while complying with the mechanical requirements

related either to balance control and/or minimum energy expenditure. For

this purpose, 7 young male volunteers wore a spring-loaded orthosis of

adjustable stiffness under 4 experimental conditions: a control condition

(without orthosis), an orthosis condition (with the orthosis only), and two

conditions (Spring1, Spring2) in which stepping in place was perturbed by

ﬁtting a spring-loaded orthosis onto the right knee joint.

For each stepping-in-place replication, the resultant moment was computed

at 256 time points equally spaced and scaled so that a time interval corre-

sponds to an individual gait cycle. A typical moment observation is there-

fore a one-dimensional function of normalized time t so that t

∈ [0, 1]. The

data set consists in 280 separate runs and involves the j

= 7 subjects over

= 4 described experimental conditions, replicated k = 10 times for each

subject. Figure 11.14 shows the data set; typical moment plots over gait cy-

cles. Model the data as arising from a ﬁxed-effects FANOVA model with 2

qualitative factors (Subjects and Treatments), 1 quantitative factor (Time)

and 10 replications for each level combination.

11.13 Exercises 473

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

0 0.5 1

−50

Fig. 11.14 Orthosis data set: The panels in rows correspond to Treatments while the panels

in columns correspond to Subjects; there are 10 repeated measurements in each panel.



If the functional ANOVA model is E y

i jk

(t) = µ(t) +α

(t) +β

(t), ﬁnd and

plot functional estimators for the grand mean

µ(t) and treatment effects

(t) −α

(t).

11.25. Bone Screws. Bone screws are the most commonly used type of orthopedic

implant. Their precision is critical and several dimensions are rigorously

checked for conformity (outside and root diameters, pitch, tread angle, etc.).

The screws are produced in lots of 20000. To conﬁrm the homogeneity of

production, 6 lots were selected and from each lot a sample of size 100 was

obtained. The following numbers of nonconforming screws are found in the

six samples: 20, 17, 23, 9, 15, and 14.

(a) Test the hypothesis of homogeneity of proportions of nonconforming

screws in the six lots.

(b) If the null hypothesis in (a) is rejected, ﬁnd which lots differ. Assume

α =0.05.

11.26. R&R Study. Five parts are measured by two appraisers using the same

measuring instrument. Each appraiser measured each part three times.

Find the relevant parameters of the measuring system and assess its capa-

bility (P RR and NDC).

measure =[ ...

217 220 217 214 216 ...

216 216 216 212 219 ...

216 218 216 212 220 ...

216 216 215 212 220 ...

474 11 ANOVA and Elements of Experimental Design

219 216 215 212 220 ...

220 220 216 212 220]’;

part = [ ...

1 2 3 4 5 1 2 3 4 5 ...

1 2 3 4 5 1 2 3 4 5]’;

appraiser =[...

1 1 1 1 1 ...

2 2 2 2 2 ...

2 2 2 2 2]’;

11.27. Additive R&R ANOVA for Measuring Impedance. Repeat the analysis

in Example 11.14 by assuming that parts and operators do not interact,

that is, with an additive ANOVA model. Provide comparisons with results

from

gagerr, where the option (’model’,’linear’) is used.

MATLAB AND WINBUGS FILES AND DATA SETS USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch11.Anova/

anomct.m, anovarep.m, Barspher.m, C677T.m, cardiac.m,

chrysanthemum.m, circularity.m, coagulationtimes.m, cochcdf1.m,

Cochtest.m, dialysis.m, dorsalspines.m, dunn.m, epsGG.m, fanova.m,

fibers.m, insects.m, insulin.m, maternalbehavior.m, Mauspher.m,

Mausphercnst.m, micePO2.m, nest1.m, nest2.m, nesta.m, nestAOV.m,

nicardipine.m, orthosis.m, orthosisraw.m, pedometer1300.m,

poissonmeans.m, powerANOVA.m, qtukey.m, RandR.m, RandR2.m, RandR3.m,

ratsblocking.m, raynaud.m, screws.m, secchim.m, sphertest.m,

ssizerepeated.m, tmcomptest.m

anovacoagulation.odc, dialysis.odc, insulin.odc, vortex.odc

arthritis.dat|mat, clouds.txt, deaf.xls, Ortho.dat, oxygen.dat,

pmr1300.mat, porcinedata.xls, Proliferation.xls, secchi.mat|xls,

secchi.xls, silicone.dat, silicone1.dat, simvastatin.dat

Chapter References 475

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476 11 ANOVA and Elements of Experimental Design

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Chapter 12

Distribution-Free Tests

Assumptions are the termites of relationships.

– Henry Winkler

WHAT IS COVERED IN THIS CHAPTER

• Sign Test

• Ranks

• Wilcoxon Signed-Rank Test and Wilcoxon Sum Rank Test

• Kruskal–Wallis and Friedman Test

• Walsh Test for Outliers

12.1 Introduction

Most of the methods we have covered until now are based on parametric as-

sumptions; the data are assumed to follow some well-known family of distri-

butions, such as normal, exponential, Poisson, and so on. Each of these dis-

tributions is indexed by one or more parameters (e.g., the normal distribution

has

µ and σ

), and at least one is presumed unknown and must be inferred.

However, with complex experiments and messy sampling plans, the generated

data might not conform to any well-known distribution. In the case where the

experimenter is not sure about the underlying distribution of the data, statis-

tical techniques that can be applied regardless of the true distribution of the

B. Vidakovic, Statistics for Bioengineering Sciences: With MATLAB and WinBUGS Support, 477

Springer Texts in Statistics, DOI 10.1007/978-1-4614-0394-4_12,

478 12 Distribution-Free Tests

data are needed. These techniques are called distribution-free or nonparamet-

ric. To quote statistician J.V. Bradley (1968):

The terms nonparametric and distribution-free are not synonymous. . . . Popular us-

age, however, has equated the terms. . . .Roughly speaking, a nonparametric test is

one which makes no hypothesis about the value of a parameter in a statistical den-

sity function, whereas a distribution-free test is one which makes no assumptions

about the precise form of the sampled population.

Many basic statistical procedures, such as the t-test, test for Pearson’s cor-

relation coefﬁcient, analysis of variance, etc., assume that the sampled data

come from a normal distribution or that the sample size is sufﬁciently large

for the CLT to make the normality assumption reasonable.

In this chapter, we will revisit the testing of means, the two-sample prob-

lem, ANOVA, and the block design, without making strict distributional as-

sumptions about the data. The material in the next chapter dealing with

goodness-of-ﬁt tests is also considered a nonparametric methodology.

For a more comprehensive account on the theory and use of nonparametric

methods in engineering research, we direct the reader to Kvam and Vidakovic

(2007).

The following table gives nonparametric counterparts to a few of the most

popular inferential procedures. The acronym WSiRT stands for Wilcoxon’s

Signed Rank Test, while WSuRT stands for Wilcoxon’s Sum Rank Test.

Parametric Nonparametric

One-sample t-test for the location Sign test, WSiRT

Paired t-test Sign test, WSiRT

Two-sample t-test WSuRT, Wilcoxon–Mann–Whitney

One-way ANOVA Kruskal–Wallis test

Block design ANOVA Friedman test

12.2 Sign Test

We start with the simplest nonparametric procedure, the sign test. Suppose we

are interested in testing the hypothesis H

that a population with a continu-

ous CDF has a median m

against one of the alternatives H

: med >6=< m

We assign a

+ sign when X

> m

, i.e., when the difference X

− m

positive, and a

− sign otherwise. The case X

= m

(a tie) is theoretically im-

possible for continuous distributions, although when observations are coded

with limited precision, ties are quite possible and most of the time present.

Henceforth we will assume an ideal situation in which no ties occur.

If m

is the median and if H

is true, then by the deﬁnition of the me-

dian,

P(X

> m

) =P(X

< m

) =

. Thus, the statistic T representing the total