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126 4 Sensitivity, Speciﬁcity, and Relatives

4.6. Blood Tests in Diagnosis of Inﬂammatory Bowel Disease. Cabrera-

Abreu et al. (2004) explored the reliability of a panel of blood tests in screen-

ing for ulcerative colitis and Crohn’s disease. The subjects were 153 chil-

dren who were referred to a pediatric gastroenterology department with

possible inﬂammatory bowel disease (IBD). Of these, 103 were found to

have IBD (Crohn’s disease 60, ulcerative colitis 37, indeterminate colitis

6). The 50 without IBD formed the controls. Blood tests evaluated sev-

eral parameters including hemoglobin, platelet count, ESR, CRP, and al-

bumin. The optimal screening strategy used a combination of hemoglobin

and platelet count and “one of two abnormal” as the criterion for positivity.

This was associated with a sensitivity of 90.3% and a speciﬁcity of 80.0%.

(a) Construct a table with TP, FP, FN and TN rounded to the nearest inte-

ger.

(b) Find the prevalence and PPV if the prevalence can be assessed from the

table (the table is obtained from a random sample from the population of

interest).

4.7. Carpal Tunnel Syndrome Tests. Three commonly used tests for carpal

tunnel syndrome are Tinel’s sign, Phalen’s test, and the nerve conduction

velocity test. Tinel’s sign and Phalen’s test are both highly sensitive (0.97

and 0.92, respectively) and speciﬁc (0.91 and 0.88, respectively). The sensi-

tivity and speciﬁcity of the nerve conduction velocity test are 0.93 and 0.87,

respectively. Assume that the tests are independent.

Calculate the sensitivity and speciﬁcity of a combined test if combining is

done

(a) in a serial manner;

(b) in a parallel manner.

drome is approximately 50 cases per 1000 subjects in the general popula-

tion.

4.8. Hepatitic Scintigraphy. A commonly used imaging procedure for detect-

ing abnormalities in the liver is hepatitic scintigraphy. Drum and Christa-

copoulos (1972) reported data on 344 patients who underwent scintigraphy

and were later examined by autopsy, biopsy, or surgical inspection for a

gold-standard determination of the presence of liver pathology (parenchy-

mal, focal, or inﬁltrative disease). The table summarizes the experimental

results. Assume that this table is representative of the population of inter-

est for this study.

Liver disease (D) No liver disease (C) Total

Abnormal liver scan (P) 231 32 263

Normal liver scan (N) 27 54 81

total 258 86 344

4.5 Exercises 127

Find the sensitivity, speciﬁcity, prevalence, PPV, NPV, LRP, LRN, and con-

cordance. Interpret the meaning of a LRP.

4.9. Apparent Prevalence. When the disease status in a sample is not known,

the prevalence cannot be estimated directly. It is estimated using apparent

prevalence. There is a distinction between the true prevalence (Pre – the

proportion of a population with the disease) and apparent prevalence (APre

– the proportion of the population that tests positive for the disease). If the

estimators of sensitivity, speciﬁcity, and apparent prevalence are available,

show that the estimator of prevalence is

Pre

APre +Sp −1

Se +Sp −1

4.10. HAAH Improves the Test for Prostate Cancer. A new procedure

based on a protein called human aspartyl (asparaginyl) beta-hydroxylase,

or HAAH, adds to the accuracy of standard prostate-speciﬁc antigen (PSA)

testing for prostate cancer. The ﬁndings were presented at the 2008 Geni-

tourinary Cancers Symposium (Keith et al. 2008).

The research involved 233 men with prostate cancer and 43 healthy men,

all over 50 years old. Results showed that the HAAH test had an overall

sensitivity of 95% and speciﬁcity of 93%.

Compared to the sensitivity and speciﬁcity of PSA (about 40%), this test

may prove particularly useful for men with both low and high PSA scores.

In men with high PSA scores (4 to 10), the addition of HAAH information

could substantially decrease the number of unnecessary biopsies, according

to the authors.

(a) From the reported percentages, construct a table with true positives,

false positives, true negatives, and false negatives (

tp, fp, tn, and fn).

You will need to round to the nearest integer since the speciﬁcity and sen-

sitivity were reported as integer percents.

(b) Suppose that for the men aged 50+ in the USA, the prevalence of

prostate cancer is 7%. Suppose Jim Smith is randomly selected from this

group and tested positive on the HAAH test. What is the probability that

Jim has prostate cancer?

ple of n

=276 (=233+43) subjects involved in the HAAH study. What is the

probability that Bill has prostate cancer if he tests positive and no other in-

formation is available? What do you call this probability? What is different

here from (b)?

4.11. Creatinine Kinase and Acute Myocardial Infraction. In a study of

773 patients, Radack et al. (1986) used an elevated serum creatinine ki-

nase concentration as a diagnostic test for acute myocardial infraction. The

following thresholds of a diagnostic test have been suggested: 481, 361,

241, and 121 IU/l; if the creatine kinase concentration exceeds the selected

128 4 Sensitivity, Speciﬁcity, and Relatives

threshold, the test for myocardial infraction is considered positive. The gold

standard is dichotomized: myocardial infraction present (MIP) and myocar-

dial infraction not present (MINP). Assume that the sample of 773 subjects

is randomly selected from the population, so that the prevalence of the dis-

ease is estimated as 51/773.

MIP MINP Total

≥481 IU/l 9 14 23

<481 IU/l 42 708 750

≥361 IU/l 15 26 41

<361 IU/l 36 696 732

≥241 IU/l 22 50 72

<241 IU/l 29 672 701

≥121 IU/l 28 251 279

<121 IU/l 23 471 494

Total 51 722 773

(a) For the test that uses 361 IU/l as a threshold, ﬁnd the sensitivity, speci-

ﬁcity, PPV, NPV, LRP, and degree of agreement.

(b) Using given thresholds plot the ROC curve. What threshold would you

suggest?

4.12. Asthma. A medical research team wished to evaluate a proposed screening

test for asthma. The test was given to a random sample of 100 patients with

asthma and to an independent sample of 200 subjects without symptoms of

the disease.

The two samples were drawn from a population of subjects who were 50

years old or older. The results are as follows.

Test result Asthma, D No asthma, D

Total

Positive test T 92 13 105

Negative test T

8 187 195

Total 100 200 300

(a) Using the numbers from the table, estimate the sensitivity and speci-

ﬁcity. Interpret these proportions in terms of the problem, one sentence for

each.

(b) The probability of D (prevalence) as the rate of the disease in the rel-

evant population (

≥ 50 y.o.) is estimated to be 6.3%. Find the PPV using

Bayes’ rule.

Chapter References 129

MATLAB FILES AND DATA SETS USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch4.ROC/

auc.m, hypothyroidism.m, Kinaseandmi.m rocada.m, RocDdimer.m,

sesp.m, simulatetesting.m, simulatetseting2.m

ADA.mat, pasi.dat, roccreatine.vi, ROCTBCA.XLS

CHAPTER REFERENCES

Cabrera-Abreu, J., Davies, P., Matek, Z., and Murphy, M. (2004). Performance of blood tests

in diagnosis of inﬂammatory bowel disease in a specialist clinic. Arch. Dis. Child., 89,

69–71.

Casscells, W., Schoenberger, A., and Grayboys, T. (1978). Interpretation by physicians of

clinical laboratory results. New Engl. J. Med., 299, 999–1000.

Don, M., Kwong, B., Tanaka, C., Brackmann, D., and Nelson, R. (2005). The stacked ABR:

A sensitive and speciﬁc screening tool for detecting small acoustic tumors. Audiol.

Neurotol., 10, 274–290.

Drum, D. E. and Christacopoulos, S. (1972). Hepatic seintigraphy in clinical decision mak-

ing. J. Nucl. Med., 13, 908–915.

Evans, D. A. (1990). Estimated prevalence of Alzheimer’s disease in the United States. Mil-

bank Q., 68, 267–289.

Goldhaber, S. Z., Simons, G. R., Elliott, C. G., Haire, W. D., Toltzis, R., Blacklow, S. C., Doolit-

tle M. H., and Weinberg, D. S. (1993). Quantitative plasma D-dimer levels among pa-

tients undergoing pulmonary angiography for suspected pulmonary embolism. J. Am.

Med. Assoc., 270, 23, 819–822.

Goldstein, B. J. and Mushlin, A. I. (1987). Use of a single thyroxine test to evaluate ambu-

latory medical patients for suspected hypothyroidism. J. Gen. Intern. Med., 2, 20–24.

Hawkins, D. M., Garrett, J. A., and Stephenson, B. (2001). Some issues in resolution of

diagnostic tests using an imperfect gold standard. Stat. Med., 20, 1987–2001.

Keith, S. N., Repoli, A. F., Semenuk, M., Harris, P. J., Ghanbari, H. A., Lebowitz, M. S.

(2008). HAAH identiﬁes prostate cancer, regardless of PSA level. Presentation at 2008

Genitourinary Cancers Symposium.

Mavromatis, B. H. and Kessler, C. M. (2001). D-Dimer testing: the role of the clinical labo-

ratory in the diagnosis of pulmonary embolism. J. Clin. Pathol., 54, 664–668.

Parikh, R., Mathai, A., Parikh, S., Sekhar, G. C., Thomas, R. (2008). Understanding and

using sensitivity, speciﬁcity and predictive values. Indian J. Ophthalmol., 56, 1, 45–

50.

130 4 Sensitivity, Speciﬁcity, and Relatives

Qu, Y. and Hadgu, A. (1998). A model for evaluating sensitivity and speciﬁcity for correlated

diagnostic tests in efﬁcacy studies with an imperfect reference test. J. Am. Stat. Assoc.,

93, 920–928.

Radack, K. L., Rouan, G., Hedges, J. (1986). The likelihood ratio: An improved measure for

reporting and evaluating diagnostic test results. Arch. Pathol. Lab. Med., 110, 689–

693.

Schipper, H., Kwok, C.-S., Rosendahl, S. M., Bandilla, D., Maes, O., Melmed, C., Rabinovitch,

D., Burns, D. H. (2008). Spectroscopy of human plasma for diagnosis of idiopathic

Parkinson’s disease. Biomark. Med., 2, 3, 229–238.

Staquet, M., Rozencweig, M., Lee, Y.J., Muggia, F.M. (1981). Methodology for assessment of

new dichotomous diagnostic tests. J. Chronic. Dis., 34, 599–610.

Chapter 5

Random Variables

The generation of random numbers is too important to be left to chance.

– Robert R. Coveyou

WHAT IS COVERED IN THIS CHAPTER

• Deﬁnition of Random Variables and Their Basic Characteristics

• Discrete Random Variables: Bernoulli, Binomial, Poisson, Hyperge-

ometric, Geometric, Negative Binomial, and Multinomial

• Continuous Random Variables: Uniform, Exponential, Gamma, In-

verse Gamma, Beta, Double Exponential, Logistic, Weibull, Pareto, and

Dirichlet

• Transformation of Random Variables

• Markov Chains

5.1 Introduction

Thus far we have been concerned with random experiments, events, and their

probabilities. In this chapter we will discuss random variables and their prob-

ability distributions. The outcomes of an experiment can be associated with

numerical values, and this association will help us arrive at the deﬁnition of a

random variable.

B. Vidakovic, Statistics for Bioengineering Sciences: With MATLAB and WinBUGS Support, 131

Springer Texts in Statistics, DOI 10.1007/978-1-4614-0394-4_5,

132 5 Random Variables

A random variable is a variable whose numerical value is determined by

the outcome of a random experiment.

Thus, a random variable is a mapping from the sample space of an experi-

ment,

S , to a set of real numbers. In this respect, the term random variable is

a misnomer. The more appropriate term would be random function or random

mapping, given that X maps a sample space

S to real numbers. We generally

denote random variables by capital letters X ,Y , Z,....

Example 5.1. Three Coin Tosses. Suppose a fair coin is tossed three times.

We can deﬁne several random variables connected with this experiment. For

example, we can set X to be the number of heads, Y the difference between

the number of heads and the number of tails, and Z an indicator that heads

appeared, etc.

Random variables X , Y , and Z are fully described by their probability dis-

tributions, associated with the sample space on which they are deﬁned.

For random variable X the possible realizations are 0 (no heads in three

ﬂips), 1 (exactly one head), 2 (exactly two heads), and 3 (all heads). Fully de-

scribing random variable X amounts to ﬁnding the probabilities of all pos-

sible realizations. For instance, the realization

{X = 2} corresponds to either

outcome in the event

{HHT, HTH, THH}. Thus, the probability of X taking

value 2 is equal to the probability of the event

{HHT, HTH, THH}, which is

equal to 3/8. After ﬁnding the probabilities for other outcomes, we determine

the distribution of random variable X :

0 1 2 3

Prob 1/8 3/8 3/8 1/8

The probability distribution of a random variable X is a table (assign-

ment, rule, formula) that assigns probabilities to realizations of X , or

sets of realizations.

Most random variables of interest to us will be the results of a random

sampling. There is a general classiﬁcation of random variables that is based

on the nature of realizations they can take. Random variables that take values

from a ﬁnite or countable set are called discrete random variables. Random

variable X from Example 5.1 is an example of a discrete random variable.

Another type of random variable can take any value from an interval on a

real line. These are called continuous random variables. The results of mea-

surements are usually modeled by continuous random variables. Next, we will

5.2 Discrete Random Variables 133

describe discrete and continuous random variables in a more structured man-

ner.

5.2 Discrete Random Variables

Let random variable X take discrete values x

, x

,... , x

,... with probabilities

, p

,... , p

,... ,

= 1. The probability distribution function (PDF) is

simply an assignment of probabilities to the realizations of X and is given by

the following table.

··· x

···

Prob p

··· p

···

The probabilities p

sum up to 1:

= 1. It is important to emphasize

that discrete random variables can have an inﬁnite number of realizations,

as long as the inﬁnite sum of the probabilities converges to 1. The PDF for

discrete random variables is also called the probability mass function (PMF).

The cumulative distribution function (CDF)

F(x)

=P(X ≤ x) =

n:x

≤x

sums the probabilities of all realizations smaller than or equal to x. Fig-

ure 5.1a shows an example of a discrete random variable X with four values

and a CDF as the sum of probabilities in the range X

≤ x shown in yellow.

(a) (b)

Fig. 5.1 (a) An example of a cumulative distribution function for discrete random vari-

able X. The CDF is the sum of probabilities in the region X

≤ x (yellow). (b) Expectation as

a point of balance for “masses” p

,..., p

located at the points x

,..., x

The expectation of X is given by

EX = x

+···+x

+···=

and is a weighted average of all possible realizations with their probabilities

as weights. Figure 5.1b illustrates the interpretation of the expectation as the

134 5 Random Variables

point of balance for a system with weights p

,... , p

located at the locations

,... , x

The distribution and expectation of a function g(X ) are simple when X is

discrete: one applies function g to realizations of X and retains the probabili-

ties:

g(X )

g(x

) g(x

) ··· g(x

) ···

Prob p

··· p

···

and

Eg(X ) = g(x

+···+ g(x

+···=

g(x

The kth moment of a discrete random variable X is deﬁned as

, and the kth central moment is E(X −EX )

−EX )

. The

ﬁrst moment is the expectation and the second central moment is the vari-

ance,

Var (X ) = E(X −EX )

. Thus, the variance for a discrete random variable

Var (X ) =

−EX )

The following properties are common for both discrete and continuous ran-

dom variables.

For any set of random variables X

, X

,... , X

E(X

+···+X

) =EX

+EX

+···+EX

. (5.1)

For any constant c,

E(c) = c and EcX = cEX .

The independence of two random variables is deﬁned via the indepen-

dence of events. Two random variables X and Y are independent if for

arbitrary intervals A and B, the events

{X ∈ A} and {Y ∈B} are indepen-

dent, that is, when

P(X ∈ A, Y ∈B) =P(X ∈ A) ·P(Y ∈ B),

holds.

If the random variables X

, X

,... , X

are independent, then

E(X

·X

·····X

) = EX

·EX

·····EX

, and

Var (X

+···+X

) = Var X

+Var X

+···+V ar X

. (5.2)

For a constant c,

Var (c)=0, and V ar (cX ) = c

Var X.

If X

, X

,... , X

,... are independent and identically distributed random

variables, we will refer to them as i.i.d. random variables.

5.2 Discrete Random Variables 135

The arguments behind these properties involve the linearity of the sums

(for discrete variables) and integrals (for continuous variables). The indepen-

dence of the X

s is critical for (5.2).

Moment-Generating Function. A particularly useful function for ﬁnd-

ing moments and for more advanced operations with random variables is the

moment-generating function. For a random variable X, the moment-generating

function is deﬁned as

(t) =Ee

, (5.3)

which for discrete random variables has the form m

(t) =

. When the

moment generating function exists, it uniquely determines the distribution. If

X has distribution F

and Y has distribution F

, and if m

(t) = m

(t) for all

t, then it follows that F

The name “moment generating” is motivated by the fact that the kth

derivative of m

(t) evaluated at 0 results in the kth moment of X , that is,

(k)

(t) =

, and EX

= m

(k)

(0). For example, if

0 1 3

Prob 0.2 0.3 0.5

then m

(t) =0.2+0.3 e

+0.5 e

. Since m

(t) =0.3 e

+1.5 e

, the ﬁrst moment

EX = m

(0) =0.3 +1.5 =1.8. The second derivative is m

(t) =0.3 e

+4.5 e

the second moment is

= m

(0) =0.3 +4.5 =4.8, and so on.

In addition to generating the moments, moment generating functions sat-

isfy

X +Y

(t) = m

(t) m

(t), (5.4)

(t) = m

(ct),

which helps in identifying distributions of linear combinations of random vari-

ables whenever their moment-generating functions exist.

The properties in (5.4) follow from the properties of expectations. When

X and Y are independent, e

and e

are independent as well, and by (5.2)

t(X +Y )

=Ee

·Ee

Example 5.2. Apgar Score. In the early 1950s, Dr. Virginia Apgar proposed

a method to assess the health of a newborn child by assigning a grade referred

to as the Apgar score (Apgar 1953). It is given twice for each newborn, once at

1 min after birth and again at 5 min after birth.

Possible values for the Apgar score are 0, 1, 2,

···, 9, and 10. A child’s

score is determined by ﬁve factors: muscle tone, skin color, respiratory effort,