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

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2.11 Exercises 55

as by monitoring various markers inside the leg. These markers provided

data for measures such as joint lengths and joint moments. A variety of

data was collected from three different cats: Natasha, Riga, and Korina.

Natasha (cat = 1) has 47 data entries, Riga (cat = 2) has 39 entries, and

Korina (cat = 3) has 35 entries.

The measurements taken are the number of steps for each trial, the length

of the stance phase (in milliseconds), the hip height (in meters), and the

velocity (in meters/second). The researchers observe these variables for dif-

ferent reasons. They want uniformity both within and between samples (to

prevent confounding variables) for steps and velocity. The hip height helps

monitor the recovery process. A detailed description can be found in Farrell

et al. (2009).

The data set, courtesy of Dr. Boris Prilutsky, School of Applied Physiology

at, Georgia Tech, is given as the MATLAB structure ﬁle

cats.mat.

Form a data matrix

X = [cat.nsteps cat.stancedur cat.hipheight cat.velocity cat.cat];

and ﬁnd its mean and correlation matrix. Form matrix Z by standardiz-

ing the columns of X (use

zscore). Plot the image of the standardized data

matrix.

2.22. BUPA Liver Data. The BUPA liver disorders database (courtesy of

Richard Forsyth, BUPA Medical Research Ltd.) consists of 345 records of

male individuals. Each record has 7 attributes,

Attribute Name Meaning

1 mcv Mean corpuscular volume

2 alkphos Alkaline phosphotase

3 sgpt Alamine aminotransferase

4 sgot Aspartate aminotransferase

5 gammagt Gamma-glutamyl transpeptidase

6 drinks Number of half-pint equivalents

of alcoholic beverages drunk per day

7 selector Field to split the database

The ﬁrst ﬁve variables are all blood tests that are thought to be sensitive to

liver disorders that might arise from excessive alcohol consumption.

The variable

selector was used to partition the data into two sets, very

likely into a training and validation part.

Using

gplotmatrix explore the relationship among variables 1 through 6

(exclude the

selector).

2.23. Cell Circularity Data. In the lab of Dr. Todd McDevitt at Georgia Tech,

researchers wanted to elucidate differences between the “static” and “ro-

tary” culture of embrionic bodies (EBs) that were formed under both con-

ditions with equal starting cell densities. After 2, 4, and 7 days of cul-

ture, images of EBs were acquired using phase-contrast microscopy. Im-

age analysis software was used to determine the circularity (deﬁned as

56 2 The Sample and Its Properties

4π(Area/P erimeter

)) of each EB imaged. A total of n =325 EBs were an-

alyzed from three separate plates for both static and rotary cultures at the

three time points studied. The circularity measures were used to examine

differences in the shape of EBs formed under the two conditions as well as

differences in their variability.

The data set

circ.dat|mat consists of six columns corresponding to six

treatments (2d, rotary), (4d, rotary), (7d, rotary), (2d, static), (4d, static),

and (7d, static). Note that this is not an example of multivariate data since

the columns are freely permutable, but rather six univariate data sets.

(a) For rotation and static 2d measurements, plot back-to-back histograms

(

bihist.m) as well as boxplots.

(b) For static 7d measurements graph by pie chart (

pie) the proportion of

EBs with circularity smaller than 0.75.

MATLAB FILES AND DATA SETS USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch2.Descriptive/

acorr.m, acov.m, ashton.m, balances.m, bat.m, bihist.m,

biomed.m, blowfliesTS.m, BUPAliver.m, carea.m, cats.m, cats1.m,

circular.m, corkrao.m, crouxrouss.m, crouxrouss2.m, diversity.m,

ecg.m, empiricalcdf.m, fisher1.m, grubbs.m, hist2d.m, histn.m,

lightrev.m, limestone.m, mahalanobis.m, meanvarchange.m,

multifat.m, multifatstat.m, mushrooms.m, myquantile.m, mytrimmean.m,

piecharts.m, scattercloud.m, simple2comp.m, smoothhist2D.m, spikes.m,

surveysUKUS.m

ashton.dat, balance2.mat, bat.dat, blowflies.dat|mat,

BUPA.dat|mat|xlsx, cats.mat, cellarea.dat|mat, circ.dat|mat,

coburn.mat, cork.dat|mat, diabetes.xls, dmd.dat|mat|xls, fat.dat,

light.dat, limestone.dat, QT.dat|mat, raman.dat|mat, spikes.dat

Chapter References 57

CHAPTER REFERENCES

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Andrews, F. D. (1972). Plots of high dimensional data. Biometrics, 28, 125–136.

Bowley, A. L. (1920). Elements of Statistics. Scribner, New York.

Brillinger, D. R. (2001). Time Series: Data Analysis and Theory. Classics Appl. Math. 36,

SIAM, pp 540.

Brockwell, P. J. and Davis, R. A. (2009). Introduction to Time Series and Forecasting.

Springer, New York.

Brozek, J., Grande, F., Anderson, J., and Keys, A. (1963). Densitometric analysis of body

composition: revision of some quantitative assumptions. Ann. New York Acad. Sci.,

110, 113–140.

Chernoff, H. (1973). The use of faces to represent points in k-dimensional space graphically.

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Christov, I., Dotsinsky, I. , Simova, I. , Prokopova, R., Trendaﬁlova, E., and Naydenov, S.

(2006). Dataset of manually measured QT intervals in the electrocardiogram. BioMed.

Eng. OnLine, 5, 31

doi:10.1186/1475-925X-5-31. The electronic version of this

article can be found online at:

http://www.biomedical-engineering-online.com/content/5/1/31

David, H. A. (1998). Early sample measures of variability. Stat. Sci., 13, 4, 368–377.

Farrell B., Bulgakova M., Hodson-Tole E.F., Shah S., Gregor R.J., Prilutsky B.I. (2009).

Short-term locomotor adaptations to denervation of lateral gastrocnemius and soleus

muscles in the cat. In: Proceedings of the Society for Neuroscience meeting, 17–21

October 2009, Chicago.

Fisher, R.A. (1936). The use of multiple measurements in taxonomic problems. Ann. Eugen.

7, Pt. II, 179–188.

Gauss, C. F. (1816). Bestimmung der Genauigkeit der Beobachtungen. Zeitschrift Astron., 1,

185–197.

Johnson, R. W. (1996). Fitting percentage of body fat to simple body measurements. J. Stat.

Educ., 4, 1.

http://www.amstat.org/publications/jse/v4n1/datasets.johnson.html

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

Moors, J. J. A. (1988). A Quantile Alternative for Kurtosis. Statistician, 37, 25–32.

Morrison, D. F. (1976). Multivariate Statistical Methods, 2nd edn. McGraw-Hill, New York

Nicholson, A. J. (1954). An Outline of the Dynamics of Animal Populations. Aust. J. Zool. ,

2, 1, 9–65.

Nightingale, F. (1858). Notes on matters affecting the health, efﬁciency, and hospital ad-

ministration of the British army. Founded chieﬂy on the experience of the late war.

Presented by request to the Secretary of State for War. Privately printed for Miss

Nightingale, Harrison and Sons.

Penrose, K., Nelson, A., and Fisher, A. (1985). Generalized body composition prediction equa-

tion for men using simple measurement techniques (abstract). Med. Sc. Sports Exerc.,

17, 2, 189.

Percy, M. E., Andrews, D. F., Thompson,M. W., and Opitz J. M. (1981). Duchenne muscular

dystrophy carrier detection using logistic discrimination: Serum creatine kinase and

hemopexin in combination. Am. J. Med. Genet., 8, 4, 397–409.

Rao, C. R. (1948). Tests of signiﬁcance in multivariate analysis. Biometrika, 35, 58–79.

Shumway, R. H. and Stoffer, D. S. (2005). Time Series Analysis and Its Applications. Springer

Texts in Statistics, Springer, New York.

58 2 The Sample and Its Properties

Siri, W. E. (1961). Body composition from ﬂuid spaces and density: Analysis of methods.

In Techniques for Measuring Body Composition, Eds. J. Brozek and A. Henzchel. Na-

tional Academy of Sciences, Washington, 224–244.

Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103, 2684, 677–680.

PMID 17750512.

Sturges, H. (1926). The choice of a class-interval. J. Am. Stat. Assoc., 21, 65–66.

Velleman, P. F. and Wilkinson, L. (1993). Nominal, ordinal, interval, and ratio typologies are

misleading. Am. Stat., 47, 1, 65–72.

Chapter 3

Probability, Conditional Probability, and

Bayes’ Rule

Misunderstanding of probability may be the greatest of all impediments to scientiﬁc

literacy.

– Stephen Jay Gould

WHAT IS COVERED IN THIS CHAPTER

• Events, Sample Spaces, and Classical Deﬁnition of Probability

• Probability of Unions and Intersections

• Independence of Events and Conditional Probability

• Total Probability and Bayes’ Rule

3.1 Introduction

If statistics can be deﬁned as the science that studies uncertainty, then proba-

bility is the branch of mathematics that quantiﬁes it. One’s intuition of chance

and probability develops at a very early age (Piaget and Inhelder, 1976). How-

ever, the formal, precise deﬁnition of probability is elusive. There are several

competing deﬁnitions for the probability of an event, but the most practical

one uses its relative frequency in a potentially inﬁnite series of experiments.

Probability is a part of all introductory statistics programs for a good rea-

son: It is the theoretical foundation of statistics. The basic statistical concepts,

B. Vidakovic, Statistics for Bioengineering Sciences: With MATLAB and WinBUGS Support, 59

Springer Texts in Statistics, DOI 10.1007/978-1-4614-0394-4_3,

60 3 Probability, Conditional Probability, and Bayes’ Rule

random sample, sampling distributions, statistic, etc., require familiarity with

probability to be understood, explained, and applied.

Probability is critical for the development of statistical concepts. Despite

this fact, it will not be a focal point of this course for reasons of time. There is

a dangerous temptation to dwell on urns, black and white balls, and combina-

torics for so long that more important statistical concepts such as regression

or ANOVA fall into a zeitnot (a term used in chess to describe the pressure felt

from having little remaining time).

Many students taking a university-level introductory statistics course have

already been exposed to probability and statistics previously in their educa-

tion. With this in mind, we will use this chapter as a survey of probability

using a range of examples. The more important concepts of independence,

conditioning, and Bayes’ rule will be covered in more detail and repeatedly

used later in various contexts. Ross (2009) is recommended for a review and

comprehensive coverage.

3.2 Events and Probability

If an experiment has the potential to be repeated an inﬁnite number of times,

then the probability of an outcome can be deﬁned through its relative fre-

quency of appearing. For instance, if we rolled a die a number of times, we

could construct a table showing how many times each face came up. These

individual frequencies (n

) can be transformed into proportions or relative fre-

quencies by dividing them by the total number of tosses n : f

= n

/n. If we

were to see the outcome

in 53 out of 300 tosses, then that face’s propor-

tion, or relative frequency, would be f

= 53/300 =0.1767. As more tosses are

made, we would “expect” the proportion of

to stabilize around

. The “exper-

iments” in the next example are often quoted in the literature on elementary

probability.

Example 3.1. Famous Coin Tosses. Buffon tossed a coin 4,040 times. Heads

appeared 2,048 times. K. Pearson tossed a coin 12,000 times and 24,000 times.

The heads appeared 6,019 times and 12,012, respectively. For these three

tosses the relative frequencies of heads are 2048/4040

≈0.5049, 6019/12000 ≈

0.5016, and 12012/24000 ≈0.5005.

What if the experiments cannot be repeated? For example, what is the

probability that “Squiki” the guinea pig survives its ﬁrst treatment by a par-

ticular drug? Or in the “experiment” of taking a statistics course this semester,

what is the probability of getting an A? In such cases we can deﬁne probability

subjectively as a measure of strength of belief. Here is another example.

3.2 Events and Probability 61

Example 3.2. Tutubalin’s Problem. In a desk drawer in the house of Mr. Jay

Parrino of Kansas City there is a coin, a 1913 Liberty Head nickel (Fig. 3.1).

What is the probability that the coin is heads up? This is an example where

Fig. 3.1 A gem-proof-condition 1913 Liberty Head nickel, one of only ﬁve known, and the

ﬁnest of the ﬁve. Collector Jay Parrino of Kansas City bought the elusive nickel for a record

$1,485,000, the ﬁrst and only time an American coin has sold for over $1 million.

equal levels of uncertainty for the two sides lead to the subjective answer of

1/2.

The symmetry of the experiment led to the classical deﬁnition of proba-

bility. An ideal die is symmetric. All sides are “equiprobable.” When rolling

a fair die, the probability of outcome

is a ratio of the number of favorable

outcomes (in our example only one outcome is favorable) to the number of all

possible outcomes, 1/6.

Among several possible ways to deﬁne probability, three are outlined below.

Frequentist. An event’s probability is the proportion of times that

we would expect the event to occur if the experiment were repeated a

large number of times.

Subjectivist. A subjective probability is an individual’s degree of be-

lief in the occurrence of an event.

Classical. An event’s probability is the ratio of the number of favor-

able outcomes to possible outcomes in a (symmetric) experiment.

A formal deﬁnition of probability is axiomatic (Kolmogorov, 1933) and is a

special case of measure theory in mathematics.

The events that are assigned probabilities can be considered as sets of out-

comes. The following table uses a rolling die experiment to introduce the set

notation among events.

This deﬁnition is criticized by philosophers because of the fallacy called a vicious circle in

deﬁnition (circulus vitiosus in deﬁniendo). One deﬁnes the notion of probability in terms of

equiprobable outcomes.

62 3 Probability, Conditional Probability, and Bayes’ Rule

Term Description Example

Experiment

A phenomenon, action, or pro-

cedure where the outcomes

are uncertain

A single roll of a balanced six-

sided die

Sample space

Set of all possible outcomes in

an experiment

S = { , , , , , }

Event

A collection of outcomes; a

subset of

A = { } (3 dots show), B =

{

, , , } (at least three

dots show), C

= { , }

Probability

A number between 0 and 1 as-

signed to an event.

P(A) =

, P(B) =

P(C)

To understand the probabilities from the above table, consider a simple

MATLAB code that will simulate rolling a fair die. A random number from

(0,1) is generated and multiplied by 6. This becomes a random number be-

tween 0 and 6. When this number is rounded up to the closest integer, the

outcomes

, ,. . . , are simulated. They are all equally likely. For example,

the outcome

comes from the original number, which is in the range (3,4),

and this interval is one-sixth part of (0,6). Formal justiﬁcation of this fact re-

quires the concept of uniform distribution, which will be covered in Chap. 5.

0 20 40 60 80 100

One

Two

Three

Four

Five

Six

Trial Number

Outcome

Fig. 3.2 MATLAB simulation of rolling a fair die. The ﬁrst 100 outcomes

{4,6,6,5,1,...,4,5,3} are shown.

The MATLAB code rollongdie1.m generates 50,000 outcomes and checks

the proportion of those equal to 3,

probA, those outcomes greater than or equal

to 3,

probB, and those smaller than 3, probC. The relative frequencies of these

outcomes tend to their theoretical probabilities of 1/6, 2/3, and 1/3. Figure 3.2

3.2 Events and Probability 63

shows the outcomes of the ﬁrst 100 simulations described in the MATLAB

code below.

% rollingdie1.m

outcomes = []; %keep outcomes here

M=50000 %# of rolls

for i= 1:M

outcomes = [outcomes ceil( 6

rand )];

% ceil(6

rand) rounds up (takes ceiling) of random

% number from (0,6), thus the outcomes 1,2,3,4,5,and 6

% are equally likely

end

probA = sum((outcomes == 3))/M

% probA = 0.1692

probB = sum((outcomes >= 3))/M

% probB = 0.6693

probC = sum((outcomes < 3))/M

% probC = 0.3307

Events in an experiment are sets containing the elementary outcomes, that

is, distinctive outcomes of the experiment. Among all events in an experiment,

two are special: a sure event and an impossible event. A sure event occurs

every time an experiment is repeated and has a probability of 1. It consists

of all outcomes and is equal to the sample space of the experiment,

S . An

impossible event never occurs when an experiment is performed and is usually

denoted as

;. It contains no elementary outcomes and its probability is 0.

For any event A, the probability that A will occur is a number between

0 and 1, inclusive:

≤P(A) ≤1.

Also,

P(;) =0, and P(S ) =1.

The intersection A

∩B of two events A and B occurs if both events A and

B occur. The key word in the deﬁnition of the intersection is and. The inter-

section of two events A

∩B is often written as a product AB. We will use both

the

∩ and product notations.

The product of the events translates into the product of their probabilities

only if the events are independent. We will see later that relationship

P(AB) =

(A)P(B) is the deﬁnition of the independence of events A and B.

64 3 Probability, Conditional Probability, and Bayes’ Rule

Events are said to be mutually exclusive if they have no common elemen-

tary outcomes. In other words, it is impossible for both events to occur in a sin-

gle trial of the experiment. For mutually exclusive events,

P(A ·B) =P(;) =0.

In the die-toss example, events A

={ } and B ={ , , , } are not mutu-

ally exclusive, since the elementary outcome {

} belongs to both of them. On

the other hand, the events A

={ } and C ={ , } are mutually exclusive.

The union A

∪B of two events A and B occurs if at least one of the events

A or B occurs. The key word in the deﬁnition of the union is or.

For mutually exclusive events, the probability that at least one of them

occurs is

P(A ∪C) =P(A) +P(C).

For example, if the probability of event A

= { } is 1/6, and the probability of

the event C

={ , } is 1/3, then the probability of A or C is

P(A ∪C) =P(A) +P(C) =1/6 +1/3 =1/2.

The additivity property is valid for any number of mutually exclusive

events A

, A

,... :

P(A

∪A

∪. ..) =P(A

) +P(A

) +....

What is

P(A ∪B) if events A and B are not mutually exclusive?

For any two events A and B, the probability that either A or B will occur

is given by the inclusion-exclusion rule:

P(A ∪B) =P(A) +P(B)−P(A ·B). (3.1)

If events A and B are exclusive, then

P(A ·B) = 0, and we get the familiar

result

P(A ∪B) =P(A) +P(B).

The inclusion-exclusion rule can be generalized to unions of an arbitrary

number of events. For example, for three events A, B, and C, the rule is

P(A ∪B ∪C) =P(A) +P(B) +P(C)−P(A ·B)−P(A ·C) −P(B ·C) +P(A ·B ·C).

(3.2)

For every event deﬁned on a space of elementary outcomes,

S , we can

deﬁne a counterpart event called its complement. The complement A

of an

event A consists of all outcomes that are in

S but are not in A. The key word

in the deﬁnition of a complement is not. In our example, A

consists of the

outcomes

{ , , , , }.

Events A and A

are mutually exclusive by deﬁnition. Consequently,