King M.R., Mody N.A. Numerical and Statistical Methods for Bioengineering: Applications in MATLAB

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where ðx

 μÞ is the deviation of each data point from the mean. By squaring the

deviations, more weight is placed on points that lie further away from the mean. The

variance, σ

, is a population parameter and quantiﬁes the dispersion in the entire

population (size N). The variance for a sample is denoted by s

and is calculated

differently, as shown below:

i¼1

ðx





xÞ

n  1

: (3:3)

It is important to recognize that the sample variance is often used as an estimate of

the variance of the population. We wish to obtain the most unbiased estimate

possible. Because the sample contains only a subset of the population values, its

variance will tend to be smaller than the population variance. The n – 1 term in the

denominator corrects for the extreme values that are usually excluded from smaller

samples. It can be proven that Equation (3.3) is the unbiased estimator for the

population variance, and that replacing n – 1 with n would produce a smaller and

biased estimate (see Section 3.5.3 for the proof).

The deﬁnition of sampl e variance also involves the concept of degrees of freedom.

The mean



x of a sample is a ﬁxed value calculated from the n data points of the

sample. Out of these nx

numbers, n – 1 can be varied freely about the mean, while

the nth number is constrained to some value to ﬁx the sample mean. Thus, n –1

represents the degrees of freedom available for calculating the variance.

The standard deviation of a sample is the square root of the variance and is

calculated a s follows:

s ¼

ﬃﬃﬃﬃ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

i¼1

ðx





xÞ

ðn  1Þ

: (3:4)

Accordingly, σ ¼

ﬃﬃﬃﬃﬃ

is the population standar d deviation.

Using MATLAB

The measures of dispersion or spread of data discussed in this section are all

available as functions in MATLAB. The function range calculates the difference

between the maximum and minimum values in a data set. The function var calcu-

lates the variance in the data set according to Equation (3.3) (unbiased estimator)

and the function std calculates the square root of the variance in the data set. The

input parameter for these three functions is either a vector or a matrix. If a matrix is

the input parameter for any of these functions, MATLAB outputs a row vector

containing the respective measure of dispersion calculated for each column of the

matrix. For additional information ab out the MATLAB functions range, var,

and std, see MATLAB help or type help range, help var, or help std at the

command line.

3.3 Concepts from probability

Probability theory originated in the seventeenth century when an avid gambler,

puzzled by a series of failed attempts at winning a game of dice, sought the advice

of a mathematician. The gambler was a French nobleman, Antoine Gombau ld, also

known as Chevalier de Me

, who indulged in a popular gambling pastime, betting

on dice. In accordance with his purported winning strategy, de Me

repeatedly bet

147

3.3 Concepts from probability

even money that at least one 12 (two sixes) would occur in 24 roll s of a pair of dice.

He conjectured that the prob ability of obtaining a 12 in one roll of two dice is 1/36,

and therefore that the chances of getting at least one 12 in 24 rolls must be 24/36 or

2/3. However, this method of calculation is ﬂawed. The true odds of getting at least

one 12 in 24 rolls of a pair of dice is actually a little less than 0.5, and de Me

’s

betting strategy failed him. He contacted Blaise Pascal (1623–1662), a proﬁcient

mathematician and philosopher at the time, for help in solving the mystery behind

his gambling fallout. Pascal calculated that the chance of winning such a bet is 49%,

and this of course explained de Me

’s unfortunate losses.

Soon after, Pascal collaborated with another famous mathematician, Pierre de

Fermat, and together they developed a foundation for probability theory. The realm

of probability theory initially focused only on games of chance to address the queries

of gamblers ignorant in the rigors of mathematics. Jacob (also known as James or

Jacques) Bernoulli (1654–1705) and Abraham de Moivre

(1667–1754), both proliﬁc

mathematicians in the seventeenth and eighteenth centuries, made signiﬁcant

contributions to this ﬁeld. Due to the pioneering work of Pierre-Simon Laplace

(1749–1827), probability theory grew to address a variety of scientiﬁc problems in

mathematical statistics, genetics, economics, and engineering, with great advances

also brought about by Kolmogorov, Markov, and Chebyshev.

Getting back to our topic of statistics, we may ask, “How is the theory of

probability related to statistical evaluations of a population parameter?” As dis-

cussed in Sectio n 3.1, statistical values are numerical estimates. Statistical analyses

are accompanied by uncertainty in the results. Why is this so? It is because statistics

are derived from samples that do not usually contain e very individual in the pop-

ulation under study. Because a sample cannot perfectly represent the population’s

properties, it has an inherent deﬁciency. The purpose of deriving any statistic is

mainly to extrapolate our ﬁndings to a much larger population of interest compared

to the sample size under study. Our certainty in the value of a statistic is often

conveyed in terms of conﬁdence limits. In other words, an upper and lower bound is

speciﬁed that demarcates a range of values within which the statistic is known to lie

with some probability, say 95% probability or 95% conﬁdence. A statistical value

implies a probable range of values. Therefore no statistic is known with complete

certainty unless the entire population of individuals is used to calculate the statistic,

in which case the statistical number is no longer an estimate but an exact description

of the population.

Probability theory applied to statistics allows us to draw inferences on population

characteristics. Probabilities are very useful for comparing the properties of two

different populations such as a control (or normal) group and a treated group.

A treated group could consist of living beings, cells, tissues, or proteins that have

been treated with some drug, while a control group refers to the untreated group.

Following treatment procedures, the experimentalist or clinical researcher wishes to

compare the differences in the statistical parameters (indicators), such as the mean

value of some observed characteristic of the population, that the treatment is

believed to alter. By comparing the properties of the treated group with that of the

untreated, one can use statistical methods to discern whether a treatment really had

an effect. One uses “probabilities” to convey the conﬁdence in the observed differ-

ences. For exampl e, we may say that we are 99.9% conﬁdent that drug XYZ causes a

reduction in the blood pressure of adults (or that there is a 0.1% chance that the

Barring numerical error in the calculation of the statistic as discussed in Chapter 1.

148

Probability and statistics

difference observed between the control and treated study group could have arisen

from simple random varia tion).

You will notice that probabilities are usually conveyed in terms of percentages, such

as “the chance of survival of a patient is 50%” or “the chance of a particular medical

treatment being successful is 70%.” Probability is a branch of mathematics that

prescribes the methods used to calculate the chance that a particular event will occur.

Any process or experiment can have one or more possible outcomes. A single outcome

or a collective set of outcomes from a process is called an event.Ifaprocessyieldsonly

one outcome, then this outcome is certain to occur and is assigned a probability of 1. If

an event cannot possibly occur, it is called an impossible event and is given the

probability of 0. The probability P of any event E can take values 0 ≤ P(E) ≤ 1.

If two events cannot occur simultaneously, they are called mutually exclusive

events. For example, a light bulb can either be on or off, or a room can either be

occupied or empty, but not both at the same time. A card picked from a deck of 52 is

either a spade, a diamond, a club, or a heart – four mutually exclusive events. No

playing card can belong to more than one suit. For the case of a well-shufﬂed deck of

cards, the probability of any randomly drawn card belonging to any one of the four

suits is equal and is ¼. The probability that the drawn card will be a heart or a spade

is¼+¼=½.Note that we have simply added the individual probabilities of two

mutually exclusive events to get the total probability of either event occurring. The

probability that either a spade, heart, diamond, or club will turn up is ¼ + ¼ + ¼

+¼=1.Inother words, it is certain that one of these four events will occur.

If a process yields n mutually exclusive events E

, E

, ..., E

, and if the probability of the event E

P(E

) for 1 ≤ i ≤ n, then the probability that any of n mutually exclusive events will occur is

[ E

[[E

ðÞ¼

i¼1

ðÞ: (3:5)

The sign [ denotes “or” or “union of events.”

The probabilities of mutually exclusive events are simply added toget her. If the n

mutually exclusive events are exhaustive, i.e. they account for all the possible out-

comes arising from a process, then

[ E

[[E

ðÞ¼

i¼1

ðÞ¼1:

If the probability of event E is P(E), then the probability that E will not occur is 1 –

P(E). This is called the complementary event. Events E and the complementary event

“not E” are mutually exclusive. Thus, if an event is deﬁned as E: “the card is a spade”

and the probability that the drawn card is a spade is P(E) = 1/4, then the probability

that the card is not a spade is PE

ðÞ= 1 – 1/4 = 3/4.

3.3.1 Random sampling and probability

Sampling is the method of choosing a sample or a group of individuals from a large

population. It is important to obtain a random sample in order to faithfully mimic the

population without bias. If every individual of the population has an equal opportu-

nity in being recruited to the sample, then the resulting sample is said to be random. A

sample obtained by choosing individuals from the population in a completely random

manner, and which reﬂects the variability inherent in the population, is called a

149

3.3 Concepts from probability

representative sample. If a sample does not adequately reﬂect the population from

which it is constructed, then it is said to be biased. A sample that is formed by

haphazard selection of individual units will not form a representative sample, but

will most likely be biased. When the sample is too homogeneous, the variability of the

population is poorly represented, and the sample represents the behavior of only a

speciﬁc segment of the population. Extrapolation of biased results to the larger

population from which the sample was drawn produces misleading predictions.

The importance of random sampling can be appreciated when we apply the con-

cepts of probability to statistics. Probability theory assumes that sampling is random.

If an individual is randomly selected from a group, the probability that the individual possesses a

particular trait is equal to the fraction (or proportion) of the population that displays that trait.

If our sampling method is biased (i.e. certain individuals of the population are

preferred over others when being selected), then the probability that the selected

individual will have a particular trait is either less than or greater than the population

proportion displaying that trait, depending on the direction of the bias.

Example 3.1

We are interested in creating a random sample of a group of 20 healthy adults selected from 40 healthy

human subjects in a new vaccine study. The randomly sampled group of 20 will serve as the treated

population and will receive the vaccination. The other 20 subjects will serve as the control. Out of the 40

subjects who have consented to participate in the trial, half are men and the other half are women.

The

distribution of ages is tabulated in Table 3.2.

The ratio of the number of males to females in each age range is 1:1.

(1) What is the probability that the first person selected from the group of 40 subjects is female?

(2) What is the probability that the first person selected from the group of 40 subjects is female and is between

the ages of 31 and 50?

We define the following events:

: person is female,

: person belongs to the 31–50 age group.

On the first draw, the probability that the first person randomly selected is female is equal to the

proportion of females in the group, and is given by

Table 3.2. Age distribution of healthy adult participants

Age range Number of healthy human subjects

21–30 16

31–40 10

41–50 10

51–60 4

When ascertaining the effects of a drug, outcome on exposure to a toxin, or cause of a disease, two study

groups are employed: one is called the treated group and the other is the control or untreated group.

Often other factors like age, sex, diet, and smoking also inﬂuence the result and can thereby confound or

distort the result. Matching is often used in case-control (retrospective) studies and cohort (prospective)

studies to ensure that the control group is similar to the treated group with respect to the confounding

factor. They are called matched case-control studies and matched cohort studies, respectively.

150

Probability and statistics

ðÞ¼

20 females

40 subjects

The four age groups are mutually exclusive. A person cannot be in two age ranges at the same time.

Therefore, the probability that a randomly selected person is in the age group 31–40 or 41–50 can be

simply added. If

: Person belongs to the 31–40 age group,

: Person belongs to the 41–50 age group,

then PE

ðÞ¼PE

[ E

ðÞ¼PE

ðÞþPE

ðÞ.

Since PE

ðÞ¼10=40 ¼ 1=4; and PE

ðÞ¼10=40 ¼ 1=4,

ðÞ¼

We wish to determine PE

\ E

ðÞ, where ∩ denotes “and.” Now, since the ratio of male to female is 1:1

in each age category, P(E

) = ½ applies regardless of whether the first chosen candidate is male or

female. The probability of E

is not influenced by the fact that E

has occur red. Events E

and E

are

independent of each other.

Events whose probabilities are not influenced by the fact that the other has occurred are called

independent events.

A joint probability is the probability of two different events occurring simultaneously. When two

independent events occur simultaneously, the joint probability is calculated by multiplying the

individual probabilities of the events of interest.

If E

and E

are two independent events, then

\ E

ðÞ¼PE

ðÞPE

ðÞ: (3:6)

The sign \ denotes “and” or “intersection of events.”

Here, we wish to determine the joint probability that the ﬁrst person selected is

both E

: female and E

: within the 31–50 age group:

\ E

ðÞ¼



Example 3.2

We reproduce Pascal’s probability calculations that brought to light de Méré’s true odds of winning (or

losing!) at dice.

In a single roll of one die, the probability of a six turning up is 1/6.

In a single roll of two dice, the probability of a six turning up on one die is independent of a six turning

up on the second die. The joint probability of two sixes turning up on both dice is calculated using

Equation (3.6):

P two sixes on two diceðÞ¼



The probability of two sixes on a pair of dice is equivalent to the probability of a 12 on a pair of dice,

since 6 + 6 is the only two-dice combination that gives 12. The probability of 12 not turning up on a roll of

dice = 1  1=36 ¼ 35=36:

P at least one 12 on a pair of dice in 24 rolls of diceðÞ

¼ 1  P 12 not observed on a pair of dice in 24 rollsðÞ:

151

3.3 Concepts from probability

The outcome of one roll of the dice is independent of the outcome of the previous rolls of the dice.

Using Equation (3.6), we calculate the joint probability of

P 12 does not turn up on a pair of dice in 24 rollsðÞ



 24 times



Then,

Pðat least one 12 on a pair dice in 24 rolls of diceÞ

¼ 1 



¼ 1  0: 5086 ¼ 0:4914:

Thus, the probability of getting at least one 12 in 24 rolls of the dice is 49.14%.

The joint probability of two mutually exclusive events arising from the same

experiment is always 0. Thus, mutually exclusive events are not independent e vents.

If E

and E

are two mutually exclusive events that result from a single process, then

the knowledge of the occurrence of E

indicates that E

did not occur. Thus, if E

has

occurred, then the probabil ity of E

once we know that E

has occurred is 0.

If the probability of any event E

changes when event E

takes place, then the event E

said to be dependent on E

. For example, if it is already known that the ﬁrst person selected

from a study group of 40 people has an age lying between 31 and 50, i.e. E

has occurred,

then the probability of E

: person is in the age group 31–40, is no longer ¼. Instead

ðÞ¼

10 people in group 3140

20 people in group 3150

ðÞdenotes the conditional probabi lity of the event E

, given that E

has

occurred. The unconditional probability of selecting a person from the study group

with age between 31 and 40 is ¼.

The conditional probability is the probability of the event when it is known that some other event has

occurred.

In the example above,

ðÞ6¼ PðE

Þ;

which signiﬁes that event E

is not independent from E

. Once it is known that E

has

occurred, the probability of E

changes.

The joint probability of two dependent events E

and E

is calculated by multiplying the probability

of E

with the conditional probability of E

given that E

has occurred. This is written as

\ E

ðÞ¼PE

ðÞPE

ðÞ: (3:7)

On the other hand, for two independent events,

ðÞ¼PE

ðÞ: (3:8)

The knowledge that E

has occurred does not alter the probability of E

Substituting Equation (3.8) into Equation (3.7) returns Equation (3.6).

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Probability and statistics

Example 3.3

In Example 3.1, it was stated that out of the 40 subjects who have consented to participate in the trial, half

are men and the other half are women, and that the ratio of males to females in each age group was the

same. Assume now that the ratio of males to females varies in each age group as shown in Table 3.3.

(1) What is the probability that the first person selected from the group of 40 subjects is female?

(2) What is the probability that the first person selected from the group of 40 subjects is female and is between

the ages of 31 and 50?

Again we define

: person is female,

: person belongs to age range 31 to 50.

The answer to the first question remains the same. Without prior knowledge of the age of the first

selected candidate, P(E

) = 20/40 or ½.

The answer to the second question changes. The joint probability of E

and E

can be calculated from

either Equation (3.7),

\ E

ðÞ¼PE

ðÞPE

ðÞ;

\ E

ðÞ¼PE

ðÞPE

ðÞ; (3:9)

since E

and E

are not independent events. As shown in Example 3.1,

ðÞ

; PE

ðÞ

Now, PE

ðÞis the probability that the randomly selected person belongs to the age range 31–50, if

we already know that the person is female. Since there are 12 females in the age range 31–50, and

there are 20 females in total, the probability that the female selected belongs to the 31–50 age group is

given by

ðÞ¼

females in age range 3150

total no: of females

Substituting into Equation (3.7), we have PE

\ E

ðÞ¼3=10:

We can also calculate PE

\ E

ðÞusing Equation (3.9):

ðÞ¼

females in age range 3150

total no: in age range 3150

Substituting into Equation (3.9), we have PE

\ E

ðÞ¼3=10:

Table 3.3. Age and gender distribution of healthy adult participants

Age range Number of healthy human subjects Male to female ratio

21–30 16 5:3 (10:6)

31–40 10 2:3 (4:6)

41–50 10 2:3 (4:6)

51–60 4 1:1 (2:2)

Total 40 participants 20 men : 20 women

153

3.3 Concepts from probability

Example 3.4

If two people are randomly selected from the group of 40 people described in Example 3.3, what is the

probability that both are male?

The random selection of more than one person from a group of people can be modeled as a sequential

process. If n people are to be selected from a group containing N ≥ n people, then the sample size is n.

Each individual that constitutes the sample is randomly chosen from the N-sized group one at a time until n

people have been selected. Each person that is selected is not returned to the group from which the

next selection is to be made. The size of the group decreases as the selection process continues. This

method of sampling is called sampling without replacement. Because the properties of the group

change as each individual is selected and removed from the group, the probability of an outcome at the ith

selection is dependent on the outcomes of the previous (i – 1)th selections. To obtain a truly random

(unbiased) sample, it is necessary that each selection made while constructing a sample be independent

of any or all of the previous selections. In other words, selection of any individual from a population should

not influence the probability of selection of other individuals from the population. For example, if the

individual units in a sample share several common characteristics that are not present in the other

individuals within the population represented by the sample, then the individual sample units are not

independent, and the sample is biased (not random).

In our example, the probability of the second selected person being male is dependent on the outcome

of the first selection. The probability that the first person selected is male is 20/40 or ½. The probability

that the second person selected is male depends on whether the first individual selected is male. If this is

the case, the number of men lef t in the group is 19 out of a total of 39. If

: first person selected is male,

: second person selected is male,

then,

\ E

ðÞ¼PE

ðÞPE

ðÞ;

ðÞ¼

;

\ E

ðÞ¼



The calculations performed in Example 3.4 are straightforward and easy because we

are only considering the selection of two people. How would we approach this situation

if we had to determine outcomes pertaining to the selection of 20 people? We would

need to create a highly elaborate tree structure detailing every possible outcome at

every step of the selection process – a cumbersome and laborious affair. This problem is

solved by theories that address methods of counting – permutations and combinations.

3.3.2 Combinatorics: permutations and combinations

A fundamental principle of counting that forms the basis of the theory of combina-

torics is as follows. If

event A can happen in exactly m ways, and

event B can happen in exactly n ways, and

the occurrence of event B follows event A,

then, these two events can happen together in exactly m × n ways.

If we have a g roup of ten people numbered 1 to 10, in how many ways can we line

them up? The ﬁrst person to be placed in line can be chosen in ten ways. Now that the

154

Probability and statistics

ﬁrst person in line is chosen, there remain only nine people in the group who can still

be assigned positions in the line. The second person can be chosen in nine ways.

When one person is remaining, he/she can be chosen in only one way. Thus, the

number of ways in which we can order ten people is 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

= 10! (read as ten factorial; n!=n( n –1)(n –2)

...

3 × 2 × 1; 0! = 1). What if we wish

to select three people out of ten and determine the total number of ordered sets of

three possible? In that case, the total number of ways in which we can select three

ordered sets of individuals, or the total number of permutations of three individuals

selected out of ten, is 10 × 9 × 8. This can also be written as

10!

If r objects are to be selected out of n objects, the total number of permutations possible is given

ðn  rÞ!

: (3:10)

What if we are only interested in obtaining sets of three individuals out of ten, and

the ordering of the individuals within each set is not important? In this case, we

are interested in knowing the number of combinations we can make from ran-

domly choosing any three out of ten. We are not interested in permuting or

arranging the objects within each set. Now, three objects in a set of three can be

ordered in

¼ 3!

ways. And r objects can be ordered or permut ed in

¼ r!

ways. Thus, every set containing r objects can undergo r! permutations or have r!

arrangements.

We can choose r out of n objects in a total of n  n  1ðÞn  2ð Þ

ðn  r þ 1Þ ways. Every possible set of r individuals formed is counted r! times

(but with different ordering).

Since we wish to disregard the ways in which we can permute the individual objects in each set of r

objects, we instead calculate the number of combinations we can make, or ways that we can select

sets of r objects out of n objects as follows:

n  n  1ðÞn  2ð Þðn  r þ 1Þ

r!ðn  rÞ!

: (3:11)

We use Equation (3.11) to determine the number of sets containing three people that

we can make by selecting from a group of ten people:

10!

3!ð10  3Þ!

¼ 120:

Example 3.5

We wish to select 20 people from a group of 40 such that exactly 10 are men and exactly 10 are women.

The group contains 20 men and 20 women. In how many ways can this be done?

155

3.3 Concepts from probability

We wish to select 10 out of 20 men. The order in which they are selected is irrelevant. We want to

determine the total number of combinations of 10 out of 20 men possible, which is

20!

10!ð20  10Þ!

¼ 184 756:

We wish to select 10 out of 20 women. The total number of possible combinations of 10 out of 20 women is also

20!

10!ð20  10Þ!

¼ 184 756:

Since both the selection of 10 out of 20 men and 10 out of 20 women can happen in 184 756 ways,

then the total possible number of combinations that we can make of 10 men and 10 women is

184 576 × 184 576 = 3.41 × 10

(see the fundamental principle of counting). Remember that the ordering

of the 20 individuals within each of the 3.41×10

sets is irrelevant in this example.

Box 3.2 Medical testing

Diagnostic devices are often used in clinical practice or in the privacy of one’s home to identify a condition

or disease, e.g. kits for HIV testing, Hepatitis C testing, drug testing, pregnancy, and genetic testing. For

example, a routine screening test performed during early pregnancy detects the levels of alphetoprotein

(AFP) in the mother’s blood. AFP is a protein secreted by the fetal liver as an excretory product into the

mother’s blood. Low levels of AFP indicate Down’s syndrome, neural tube defects, and other problems

associated with the fetus. This test method can accurately detect Down’s syndrome in 60% of affected

babies. The remaining 40% of babies with Down’s syndrome go undetected in such tests. When a test

incorrectly concludes the result as negative, the conclusion is called a false negative.TheAFPtestis

somewhat controversial because it is found to produce positive results even when the baby is unaffected. If

a test produces a positive result when the disease is not present, the result is called a false positive.

Assume that a diagnostic test for a particular human disease can detect the presence of the disease

90% of the time. The test can accurately determine if a person does not have the disease with 0.8

probability. If 5% of the population has this disease, what is the probability that a randomly selected

person will test positive? What is the conditional probability that, if tested positive, the person is actually

afflicted with the disease?

Let H represent a healthy person and let D represent a person afflicted with the disease.

If Y indicates a positive result, then

PYjDðÞ¼0:9 and PYjHðÞ¼0:2:

If a person is randomly chosen from the population, the probability that the person has the disease is

P(D) = 0.05 and the probability that the person does not have the disease is P(H) = 0.95. Also,

probability that the person has the disease and tests positive ¼ PD\ YðÞ

¼ PDðÞPYjDðÞ¼0:05  0:9 ¼ 0:045;

probability that the person does not have the disease and tests positive ðfalse positiveÞ

¼ PH\ YðÞ¼PHðÞPYj

HðÞ

¼0:

95 0:2 ¼ 0:19;

probability that a random person will test positive ¼ PD\ YðÞþPH\ YðÞ

¼ 0:045 þ 0:19 ¼ 0:235;

conditional probability that person actually has the disease if tested positive

PD\ YðÞ

PD\ YðÞþPH\ YðÞ

0:045

0:235

¼ 0:191:

So, 19% of the people that test positive actually have the disease. The test appears to be much less

reliable than one would intuitively expect.

156

Probability and statistics