Introduction to Statistics for Biomedical Engineers doc

Two elements of expermental desgn that are crtcal to prevent basng the data or selectng samples that do not farly represent the underlyng populaton are randomzaton and blockng.Randomzato

Introduction to Statistics

for Biomedical Engineers

Copyrght © 2007 by Morgan & Claypool

All rghts reserved No part of ths publcaton may be reproduced, stored n a retreval system, or transmtted n any form or by any means—electronc, mechancal, photocopy, recordng, or any other except for bref quotatons n prnted revews, wthout the pror permsson of the publsher.

Introducton to Statstcs for Bomedcal Engneers

A Publcaton n the Morgan & Claypool Publshers seres

SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING #14

This text is dedicated to all the students who have completed my BIEN 084 statistics course for biomedical engineers and have taught me how to be more effective in communicating the subject matter and making statistics come alive for them I also thank J Claypool for his patience and for encouraging me to finally put this text together

Finally, I thank my family for tolerating my time at home on the laptop.

There are many books wrtten about statstcs, some bref, some detaled, some humorous, some colorful, and some qute dry Each of these texts s desgned for a specfic audence Too often, texts about statstcs have been rather theoretcal and ntmdatng for those not practcng statstcal analyss on a routne bass Thus, many engneers and scentsts, who need to use statstcs much more frequently than calculus or dfferental equatons, lack sufficent knowledge of the use of statstcs The audence that s addressed n ths text s the unversty-level bomedcal engneerng student who needs a bare-bones coverage of the most basc statstcal analyss frequently used n bomedcal engneerng practce The text ntroduces students to the essental vocabulary and basc concepts of probablty and statstcs that are requred to perform the numercal summary and sta-tstcal analyss used n the bomedcal field Ths text s consdered a startng pont for mportant

ssues to consder when desgnng experments, summarzng data, assumng a probablty model for the data, testng hypotheses, and drawng conclusons from sampled data

A student who has completed ths text should have sufficent vocabulary to read more vanced texts on statstcs and further ther knowledge about addtonal numercal analyses that are used n the bomedcal engneerng field but are beyond the scope of ths text Ths book s desgned

ad-to supplement an undergraduate-level course n appled statstcs, specfically n bomedcal neerng Practcng engneers who have not had formal nstructon n statstcs may also use ths text

eng-as a smple, bref ntroducton to statstcs used n bomedcal engneerng The empheng-ass s on the applcaton of statstcs, the assumptons made n applyng the statstcal tests, the lmtatons of these elementary statstcal methods, and the errors often commtted n usng statstcal analyss

A number of examples from bomedcal engneerng research and ndustry practce are provded to assst the reader n understandng concepts and applcaton It s benefical for the reader to have some background n the lfe scences and physology and to be famlar wth basc bomedcal n-strumentaton used n the clncal envronment

KEywoRdS

probablty model, hypothess testng, physology, ANOVA, normal dstrbuton,

confidence nterval, power test

1 Introduction .1

2 Collecting data and Experimental design .5

3 data Summary and descriptie Statistics .9

3.1 Why Do We Collect Data? 9

3.2 Why Do We Need Statstcs? 9

3.3 What Questons Do We Hope to Address Wth Our Statstcal Analyss? 10

3.4 How Do We Graphcally Summarze Data? 11

3.4.1 Scatterplots 11

3.4.2 Tme Seres 11

3.4.3 Box-and-Whsker Plots 12

3.4.4 Hstogram 13

3.5 General Approach to Statstcal Analyss 17

3.6 Descrptve Statstcs 20

3.6.1 Measures of Central Tendency 21

3.6.2 Measures of Varablty 22

4 Assuming a Probability Model From the Sample data 25

4.1 The Standard Normal Dstrbuton 29

4.2 The Normal Dstrbuton and Sample Mean 32

4.3 Confidence Interval for the Sample Mean 33

4.4 The t Dstrbuton 36

4.5 Confidence Interval Usng t Dstrbuton 38

5 Statistical Inference 41

5.1 Comparson of Populaton Means 41

5.1.1 The t Test 42

5.1.1.1 Hypothess Testng 42

5.1.1.2 Applyng the t Test 43

5.1.1.3 Unpared t Test 44

5.1.1.4 Pared t Test 49

5.1.1.5 Example of a Bomedcal Engneerng Challenge 50

5.2 Comparson of Two Varances 54

5.3 Comparson of Three or More Populaton Means 59

5.3.1 One-Factor Experments 60

5.3.1.1 Example of Bomedcal Engneerng Challenge 60

5.3.2 Two-Factor Experments 69

5.3.3 Tukey’s Multple Comparson Procedure 73

6 Linear Regression and Correlation Analysis 75

7 Power Analysis and Sample Size 81

7.1 Power of a Test 82

7.2 Power Tests to Determne Sample Sze 83

8 Just the Beginning 87

Bibliography 91

Author Biography 93

iii INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

C H A P T E R 1

Bomedcal engneers typcally collect all sorts of data, from patents, anmals, cell counters, assays, magng systems, pressure transducers, bedsde montors, manufacturng processes, materal testng systems, and other measurement systems that support a broad spectrum of research, desgn, and manufacturng envronments Ultmately, the reason for collectng data s to make a decson That decson may concern dfferentatng bologcal characterstcs among dfferent populatons

mcro-of people, determnng whether a pharmacologcal treatment s effectve, determnng whether t s cost-effectve to nvest n multmllon-dollar medcal magng technology, determnng whether a manufacturng process s under control, or selectng the best rehabltatve therapy for an ndvdual patent

The challenge n makng such decsons often les n the fact that all real-world data contans some element of uncertanty because of random processes that underle most physcal phenomenon These random elements prevent us from predctng the exact value of any physcal quantty at any moment of tme In other words, when we collect a sample or data pont, we usually cannot predct the exact value of that sample or expermental outcome For example, although the average restng heart rate of normal adults s about 70 beats per mnute, we cannot predct the exact arrval tme

of our next heartbeat However, we can approxmate the lkelhood that the arrval tme of the next heartbeat wll fall n a specfic tme nterval f we have a good probablty model to descrbe the random phenomenon contrbutng to the tme nterval between heartbeats The tmng of heart-beats s nfluenced by a number of physologcal varables [1], ncludng the refractory perod of the ndvdual cells that make up the heart muscle, the leakness of the cell membranes n the snus node (the heart’s natural pacemaker), and the actvty of the autonomc nervous system, whch may speed up or slow down the heart rate n response to the body’s need for ncreased blood flow, oxygen, and nutrents The sum of these bologcal processes produces a pattern of heartbeats that we may measure by countng the pulse rate from our wrst or carotd artery or by searchng for specfic QRS waveforms n the ECG [2] Although ths sum of events makes t dfficult for us to predct exactly when the new heartbeat wll arrve, we can guess, wth a certanty amount of confidence when the next beat wll arrve In other words, we can assgn a probablty to the lkelhood that the next heartbeat wll arrve n a specfied tme nterval If we were to consder all possble arrval tmes and

Introduction

2 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

assgned a probablty to those arrval tmes, we would have a probablty model for the heartbeat

ntervals If we can find a probablty model to descrbe the lkelhood of occurrence of a certan event or expermental outcome, we can use statstcal methods to make decsons The probablty models descrbe characterstcs of the populaton or phenomenon beng studed Statstcal analyss then makes use of these models to help us make decsons about the populaton(s) or processes.The conclusons that one may draw from usng statstcal analyss are only as good as the underlyng model that s used to descrbe the real-world phenomenon, such as the tme nterval between heartbeats For example, a normally functonng heart exhbts consderable varablty n beat-to-beat ntervals (Fgure 1.1) Ths varablty reflects the body’s contnual effort to mantan homeostass so that the body may contnue to perform ts most essental functons and supply the body wth the oxygen and nutrents requred to functon normally It has been demonstrated through bomedcal research that there s a loss of heart rate varablty assocated wth some dseases, such

as dabetes and schemc heart dsease Researchers seek to determne f ths dfference n varablty between normal subjects and subjects wth heart dsease s sgnficant (meanng, t s due to some underlyng change n bology and not smply a result of chance) and whether t mght be used to predct the progresson of the dsease [1] One wll note that the probablty model changes as a consequence of changes n the underlyng bologcal functon or process In the case of manufactur-

ng, the probablty model used to descrbe the output of the manufacturng process may change as

be-a functon of mbe-achne operbe-aton or chbe-anges n the surroundng mbe-anufbe-acturng envronment, such be-as temperature, humdty, or human operator.

Besdes helpng us to descrbe the probablty model assocated wth real-world phenomenon, statstcs help us to make decsons by gvng us quanttatve tools for testng hypotheses We call

ths inferential statistics, whereby the outcome of a statstcal test allows us to draw conclusons or

make nferences about one or more populatons from whch samples are drawn Most often, tsts and engneers are nterested n comparng data from two or more dfferent populatons or from two or more dfferent processes Typcally, the default hypothess s that there s no dfference n the dstrbutons of two or more populatons or processes, and we use statstcal analyss to determne whether there are true dfferences n the dstrbutons of the underlyng populatons to warrant df-ferent probablty models be assgned to the ndvdual processes

scen-In summary, bomedcal engneers typcally collect data or samples from varous phenomena, whch contan some element of randomness or unpredctable varablty, for the purposes of makng decsons To make sound decsons n the context of the uncertanty wth some level of confidence,

we need to assume some probablty model for the populatons from whch the samples have been collected Once we have assumed an underlyng model, we can select the approprate statstcal tests for comparng two or more populatons and then use these tests to draw conclusons about

our hypotheses for whch we collected the data n the first place Fgure 1.2 outlnes the steps for performng statstcal analyss of data.

In the followng chapters, we wll descrbe methods for graphcally and numercally marzng collected data We wll then talk about fittng a probablty model to the collected data by brefly descrbng a number of well-known probablty models that are used to descrbe bologcal phenomenon Fnally, once we have assumed a model for the populatons from whch we have col-lected our sample data, we wll dscuss the types of statstcal tests that may be used to compare data from multple populatons and allow us to test hypotheses about the underlyng populatons

sum-• sum-• sum-• sum-•

4 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

col-f we are tryng to determne whether men between the ages of 20 and 50 years respond postvely

to a drug that reduces cholesterol level, we need to carefully select the populaton of subjects for whom we admnster the drug and take measurements In other words, we have to have enough samples to represent the varablty of the underlyng populaton There s a great deal of varety n the weght, heght, genetc makeup, det, exercse habts, and drug use n all men ages 20 to 50 years who may also have hgh cholesterol If we are to test the effectveness of a new drug n lowerng cholesterol, we must collect enough data or samples to capture the varablty of bologcal makeup and envronment of the populaton that we are nterested n treatng wth the new drug Capturng ths varablty s often the greatest challenge that bomedcal engneers face n collectng data and usng statstcs to draw meanngful conclusons The expermentalst must ask questons such as the followng:

What type of person, object, or phenomenon do I sample?

What varables that mpact the measure or data can I control?

How many samples do I requre to capture the populaton varablty to apply the prate statstcs and draw meanngful conclusons?

appro-How do I avod basng the data wth the expermental desgn?

Expermental desgn, although not the prmary focus of ths book, s the most crtcal step to port the statstcal analyss that wll lead to meanngful conclusons and hence sound decsons.One of the most fundamental questons asked by bomedcal researchers s, “What sze sam-ple do I need?” or “How many subjects wll I need to make decsons wth any level of confidence?”

We wll address these mportant questons at the end of ths book when concepts such as varablty, probablty models, and hypothess testng have already been covered For example, power tests wll

be descrbed as a means for predctng the sample sze requred to detect sgnficant dfferences n

two populaton means usng a t test.

Two elements of expermental desgn that are crtcal to prevent basng the data or selectng samples that do not farly represent the underlyng populaton are randomzaton and blockng.Randomzaton refers to the process by whch we randomly select samples or expermental unts from the larger underlyng populaton such that we maxmze our chance of capturng the varablty n the underlyng populaton In other words, we do not lmt our samples such that only a fracton of the characterstcs or behavors of the underlyng populaton are captured n the samples More mportantly, we do not bas the results by artfically lmtng the varablty n the samples such that we alter the probablty model of the sample populaton wth respect to the prob-ablty model of the underlyng populaton

In addton to randomzng our selecton of expermental unts from whch to take samples, we mght also randomze our assgnment of treatments to our expermental unts Or, we may random-

ze the order n whch we take data from the expermental unts For example, f we are testng the effectveness of two dfferent medcal magng methods n detectng bran tumor, we wll randomly assgn all subjects suspect of havng bran tumor to one of the two magng methods Thus, f we have

a mx of sex, age, and type of bran tumor partcpatng n the study, we reduce the chance of havng all one sex or one age group assgned to one magng method and a very dfferent type of populaton assgned to the second magng method If a dfference s noted n the outcome of the two magng methods, we wll not artfically ntroduce sex or age as a factor nfluencng the magng results

As another example, f one are testng the strength of three dfferent materals for use n hp mplants usng several strength measures from a materals testng machne, one mght random-

ze the order n whch samples of the three dfferent test materals are submtted to the machne Machne performance can vary wth tme because of wear, temperature, humdty, deformaton, stress, and user characterstcs If the bomedcal engneer were asked to find the strongest materal for an artfical hp usng specfic strength crtera, he or she may conduct an experment Let us assume that the engneer s gven three boxes, wth each box contanng five artfical hp mplants made from one of three materals: ttanum, steel, and plastc For any one box, all five mplant samples are made from the same materal To test the 15 dfferent mplants for materal strength, the engneer mght randomze the order n whch each of the 15 mplants s tested n the mater-als testng machne so that tme-dependent changes n machne performance or machne-materal

nteractons or tme-varyng envronmental condton do not bas the results for one or more of the materals Thus, to fully randomze the mplant testng, an engneer may lterally place the numbers 1–15 n a hat and also assgn the numbers 1–15 to each of the mplants to be tested The engneer

6 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

that number Ths way the engneer s not testng all of one materal n any partcular order, and we avod ntroducng order effects nto the data.

The second aspect of expermental desgn s blockng In many experments, we are nterested

n one or two specfic factors or varables that may mpact our measure or sample However, there may be other factors that also nfluence our measure and confound our statstcs In good exper-mental desgn, we try to collect samples such that dfferent treatments wthn the factor of nterest are not based by the dfferng values of the confoundng factors In other words, we should be cer-tan that every treatment wthn our factor of nterest s tested wthn each value of the confoundng factor We refer to ths desgn as blockng by the confoundng factor For example, we may want to study weght loss as a functon of three dfferent det plls One confoundng factor may be a person’s startng weght Thus, n testng the effectveness of the three plls n reducng weght, we may want

to block the subjects by startng weght Thus, we may first group the subjects by ther startng weght and then test each of the det plls wthn each group of startng weghts

In bomedcal research, we often block by expermental unt When ths type of blockng s part of the expermental desgn, the expermentalst collects multple samples of data, wth each sample representng dfferent expermental condtons, from each of the expermental unts Fg-ure 2.1 provdes a dagram of an experment n whch data are collected before and after patents receves therapy, and the expermental desgn uses blockng (left) or no blockng (rght) by exper-mental unt In the case of blockng, data are collected before and after therapy from the same set of human subjects Thus, wthn an ndvdual, the same bologcal factors that nfluence the bologcal response to the therapy are present before and after therapy Each subject serves as hs or her own control for factors that may randomly vary from subject to subject both before and after therapy

In essence, wth blockng, we are elmnatng bases n the dfferences between the two populatons

Block (Repeated Measures) No Block (No repeated measures)

Subject Measure

before

treatment

Measure after treatment

Subject Measure

before treatment

Subject Measure

after treatment

exper-8 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

(before and after) that may result because we are usng two dfferent sets of expermental unts For example, f we used one set of subjects before therapy and then an entrely dfferent set of subjects after therapy (Fgure 2.1, rght), there s a chance that the two sets of subjects may vary enough n sex, age, weght, race, or genetc makeup, whch would lead to a dfference n response to the therapy that has lttle to do wth the underlyng therapy In other words, there may be confoundng factors that contrbute to the dfference n the expermental outcome before and after therapy that are not only a factor of the therapy but really an artfact of dfferences n the dstrbutons of the two dffer-ent groups of subjects from whch the two samples sets were chosen Blockng wll help to elmnate the effect of ntersubject varablty

However, blockng s not always possble, gven the nature of some bomedcal research

stud-es For example, f one wanted to study the effectveness of two dfferent chemotherapy drugs n reducng tumor sze, t s mpractcal to test both drugs on the same tumor mass Thus, the two drugs are tested on dfferent groups of ndvduals The same type of desgn would be necessary for testng the effectveness of weght-loss regmens

Thus, some mportant concepts and defintons to keep n mnd when desgnng experments

nclude the followng:

experimental unit: the tem, object, or subject to whch we apply the treatment and from

whch we take sample measurements;

randomization: allocate the treatments randomly to the expermental unts;

blocking: assgnng all treatments wthn a factor to every level of the blockng factor

Often, the blockng factor s the expermental unt Note that n usng blockng, we stll randomze the order n whch treatments are appled to each expermental unt to avod orderng bas

Fnally, the expermentalst must always thnk about how representatve the sample populaton s wth respect to the greater underlyng populaton Because t s vrtually mpossble to test every member of a populaton or every product rollng down an assembly lne, especally when destruc-tve testng methods are used, the bomedcal engneer must often collect data from a much smaller sample drawn from the larger populaton It s mportant, f the statstcs are gong to lead to useful conclusons, that the sample populaton captures the varablty of the underlyng populaton What

s even more challengng s that we often do not have a good grasp of the varablty of the

underly-ng populaton, and because of expense and respect for lfe, we are typcally lmted n the number of samples we may collect n bomedcal research and manufacturng These lmtatons are not easy to address and requre that the engneer always consder how far the sample and data analyss s and how well t represents the underlyng populaton(s) from whch the samples are drawn

• • • •

•

•

•

C H A P T E R 3

We assume now that we have collected our data through the use of good expermental desgn We now have a collecton of numbers, observatons, or descrptons to descrbe our data, and we would lke to summarze the data to make decsons, test a hypothess, or draw a concluson

The world s full of uncertanty, n the sense that there are random or unpredctable factors that

nfluence every expermental measure we make The unpredctable aspects of the expermental comes also arse from the varablty n bologcal systems (due to genetc and envronmental fac-tors) and manufacturng processes, human error n makng measurements, and other underlyng processes that nfluence the measures beng made

out-Despte the uncertanty regardng the exact outcome of an experment or occurrence of a ture event, we collect data to try to better understand the processes or populatons that nfluence an expermental outcome so that we can make some predctons Data provde nformaton to reduce uncertanty and allow for decson makng When properly collected and analyzed, data help us solve problems It cannot be stressed enough that the data must be properly collected and analyzed

fu-f the data analyss and subsequent conclusons are to have any value

We have three major reasons for usng statstcal data summary and analyss:

The real world s full of random events that cannot be descrbed by exact mathematcal expressons

Varablty s a natural and normal characterstc of the natural world

We lke to make decsons wth some confidence Ths means that we need to find trends wthn the varablty

10 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

ouR STATISTICAL ANALySIS?

There are several basc questons we hope to address when usng numercal and graphcal summary

of data:

Can we dfferentate between groups or populatons?

Are there correlatons between varables or populatons?

Are processes under control?

Fndng physologcal dfferences between populatons s probably the most frequent am

of bomedcal research For example, researchers may want to know f there s a dfference n lfe expectancy between overweght and underweght people Or, a pharmaceutcal company may want

to determne f one type of antbotc s more effectve n combatng bactera than another Or, a physcan wonders f dastolc blood pressure s reduced n a group of hypertensve subjects after the consumpton of a pressure-reducng drug Most often, bomedcal researchers are comparng populatons of people or anmals that have been exposed to two or more dfferent treatments or d-agnostc tests, and they want to know f there s dfference between the responses of the populatons that have receved dfferent treatments or tests Sometmes, we are drawng multple samples from the same group of subjects or expermental unts A common example s when the physologcal data are taken before and after some treatment, such as drug ntake or electronc therapy, from one group

of patents We call ths type of data collecton blocking n the expermental desgn Ths concept of

blockng s dscussed more fully n Chapter 2

Another queston that s frequently the target of bomedcal research s whether there s a relaton between two physologcal varables For example, s there a correlaton between body buld and mortalty? Or, s there a correlaton between fat ntake and the occurrence of cancerous tumors

cor-Or, s there a correlaton between the sze of the ventrcular muscle of the heart and the frequency of abnormal heart rhythms? These type of questons nvolve collectng two set of data and performng

a correlaton analyss to determne how well one set of data may be predcted from another When

we speak of correlaton analyss, we are referrng to the lnear relaton between two varables and the ablty to predct one set of data by modelng the data as a lnear functon of the second set of data Because correlaton analyss only quantfies the lnear relaton between two processes or data sets, nonlnear relatons between the two processes may not be evdent A more detaled descrpton of correlaton analyss may be found n Chapter 7

Fnally, a bomedcal engneer, partcularly the engneer nvolved n manufacturng, may be

nterested n knowng whether a manufacturng process s under control Such a queston may arse

f there are tght controls on the manufacturng specficatons for a medcal devce For example,

1

2

3

f the engneer s tryng to ensure qualty n producng ntravascular catheters that must have ameters between 1 and 2 cm, the engneer may randomly collect samples of catheters from the assembly lne at random ntervals durng the day, measure ther dameters, determne how many of the catheters meet specficatons, and determne whether there s a sudden change n the number

d-of catheters that fal to meet specficatons If there s such a change, the engneers may look for elements of the manufacturng process that change over tme, changes n envronmental factors, or user errors The engneer can use control charts to assess whether the processes are under control These methods of statstcal analyss are not covered n ths text, but may be found n a number of references, ncludng [3]

We can summarze data n graphcal or numercal form The numercal form s what we refer to as statstcs Before blndly applyng the statstcal analyss, t s always good to look at the raw data, usually n a graphcal form, and then use graphcal methods to summarze the data n an easy to

3.4.2 Time Series

A tme seres s used to plot the changes n a varable as a functon of tme The varable s usually

a physologcal measure, such as electrcal actvaton n the bran or hormone concentraton n the blood stream, that changes wth tme Fgure 3.2 llustrates an example of a tme seres plot In ths figure, we are lookng at a smple snusod functon as t changes wth tme

12 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

3.4.3 Box-and-whisker Plots

These plots llustrate the first, second, and thrd quartles as well as the mnmum and maxmum values of the data collected The second quartle (Q2) s also known as the medan of the data Ths quantty, as defined later n ths text, s the mddle data pont or sample value when the samples are lsted n descendng order The first quartle (Q1) can be thought of as the medan value of the samples that fall below the second quartle Smlarly, the thrd quartle (Q3) can be thought of as the medan value of the samples that fall above the second quartle Box-and-whsker plots are use-ful n that they hghlght whether there s skew to the data or any unusual outlers n the samples (Fgure 3.3)

-2 -1 0 1 2

0

10 9 8 7 6 5 4 3 2 1 0

10 9 8 7 6 5 4 3 2 1 0

Category

Q1 Q2 Q3

Box and Whisker Plot

FIguRE 3.3: Illustraton of a box-and-whsker plot for the data set lsted The first (Q1), second (Q2), and thrd (Q3) quartles are shown In addton, the whskers extend to the mnmum and maxmum values of the sample set

3.4.4 Histogram

The hstogram s defined as a frequency dstrbuton Gven N samples or measurements, x i, whch

range from Xmn to Xmax, the samples are grouped nto nonoverlappng ntervals (bns), usually of equal wdth (Fgure 3.4) Typcally, the number of bns s on the order of 7–14, dependng on the nature of the data In addton, we typcally expect to have at least three samples per bn [7] Stur-gess’ rule [6] may also be used to estmate the number of bns and s gven by

k = 1 + 3.3 log(n).

where k s the number of bns and n s the number of samples.

Each bn of the hstogram has a lower boundary, upper boundary, and mdpont The gram s constructed by plottng the number of samples n each bn Fgure 3.5 llustrates a hstogram for 1000 samples drawn from a normal dstrbuton wth mean (µ) = 0 and standard devaton (σ) = 1.0 On the horzontal axs, we have the sample value, and on the vertcal axs, we have the number

hsto-of occurrences hsto-of samples that fall wthn a bn

Two measures that we find useful n descrbng a hstogram are the absolute frequency and relatve frequency n one or more bns These quanttes are defined as

f i = absolute frequency n ith bn;

f i /n = relatve frequency n th bn, where n s the total number of samples beng summarzed

n the hstogram

a)

b)

14 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

A number of algorthms used by bomedcal nstruments for dagnosng or detectng normaltes n bologcal functon make use of the hstogram of collected data and the assocated relatve frequences of selected bns [8] Often tmes, normal and abnormal physologcal functons (breath sounds, heart rate varablty, frequency content of electrophysologcal sgnals) may be df-ferentated by comparng the relatve frequences n targeted bns of the hstograms of data repre-sentng these bologcal processes

ab-Lower Bound Upper Bound

hor-The hstogram can exhbt several shapes hor-The shapes, llustrated n Fgure 3.6, are referred

to as symmetrc, skewed, or bmodal

A skewed hstogram may be attrbuted to the followng [9]:

mechansms of nterest that generate the data (e.g., the physologcal mechansms that determne the beat-to-beat ntervals n the heart);

an artfact of the measurement process or a shft n the underlyng mechansm over tme (e.g., there may be tme-varyng changes n a manufacturng process that lead to a change

n the statstcs of the manufacturng process over tme);

a mxng of populatons from whch samples are drawn (ths s typcally the source of a bmodal hstogram)

The hstogram s mportant because t serves as a rough estmate of the true probablty sty functon or probablty dstrbuton of the underlyng random process from whch the samples are beng collected

den-The probablty densty functon or probablty dstrbuton s a functon that quantfies the

probablty of a random event, x, occurrng When the underlyng random event s dscrete n nature,

we refer to the probablty densty functon as the probablty mass functon [10] In ether case, the functon descrbes the probablstc nature of the underlyng random varable or event and allows us

to predct the probablty of observng a specfic outcome, x (represented by the random varable),

of an experment The cumulatve dstrbuton functon s smply the sum of the probabltes for a

group of outcomes, where the outcome s less than or equal to some value, x.

Let us consder a random varable for whch the probablty densty functon s well defined (for most real-world phenomenon, such a probablty model s not known.) The random varable s the outcome of a sngle toss of a dce Gven a sngle far dce wth sx sdes, the probablty of rollng

a sx on the throw of a dce s 1 of 6 In fact, the probablty of throwng a one s also 1 of 6 If we consder all possble outcomes of the toss of a dce and plot the probablty of observng any one of those sx outcomes n a sngle toss, we would have a plot such as that shown n Fgure 3.7

Ths plot shows the probablty densty or probablty mass functon for the toss of a dce Ths type of probablty model s known as a unform dstrbuton because each outcome has the exact same probablty of occurrng (1/6 n ths case)

For the toss of a dce, we know the true probablty dstrbuton However, for most world random processes, especally bologcal processes, we do not know what the true probablty densty or mass functon looks lke As a consequence, we have to use the hstogram, created from a small sample, to try to estmate the best probablty dstrbuton or probablty model to descrbe the real-world phenomenon If we return to the example of the toss of a dce, we can actually toss the dce a number of tmes and see how close the hstogram, obtaned from expermental data, matches

real-1

2

3

16 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

-4 -3 -2 -1 0 1 2 3 0

100 200

Measure

Symmetric

0 5

0 0 100 200 300 400

the true probablty mass functon for the deal sx-sded dce Fgure 3.8 llustrates the hstograms for the outcomes of 50 and 1000 tosses of a sngle dce Note that even wth 50 tosses or samples, t

s dfficult to determne what the true probablty dstrbuton mght look lke However, as we proach 1000 samples, the hstogram s approachng the true probablty mass functon (the unform dstrbuton) for the toss of a dce But, there s stll some varablty from bn to bn that does not look as unform as the deal probablty dstrbuton llustrated n Fgure 3.7 The message to take away from ths llustraton s that most bomedcal research reports the outcomes of a small number

ap-of samples It s clear from the dce example that the statstcs ap-of the underlyng random process are very dfficult to dscern from a small sample, yet most bomedcal research reles on data from small samples

We have now collected our data and looked at some graphcal summares of the data Now we wll use numercal summary, also known as statstcs, to try to descrbe the nature of the underlyng populaton or process from whch we have taken our samples From these descrptve statstcs, we assume a probablty model or probablty dstrbuton for the underlyng populaton or process and then select the approprate statstcal tests to test hypotheses or make decsons It s mportant to note that the conclusons one may draw from a statstcal test depends on how well the assumed probablty model fits the underlyng populaton or process

0 1/6

Result of Toss of Single Dice

Probability Mass Function

FIguRE 3.7: The probablty densty functon for a dscrete random varable (probablty mass ton) In ths case, the random varable s the value of a toss of a sngle dce Note that each of the sx pos-sble outcomes has a probablty of occurrence of 1 of 6 Ths probablty densty functon s also known

func-as a unform probablty dstrbuton

18 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

6 5 4

3 2

3 2

Histogram of 2000 Dice Tosses

FIguRE 3.8: Hstograms representng the outcomes of experments n whch a sngle dce s tossed

50 (top) and 2000 tmes (lower), respectvely Note that as the sample sze ncreases, the hstogram proaches the true probablty dstrbuton llustrated n Fgure 3.7

ap-As stated n the Introducton, bomedcal engneers are tryng to make decsons about latons or processes to whch they have lmted access Thus, they desgn experments and collect samples that they thnk wll farly represent the underlyng populaton or process Regardless of what type of statstcal analyss wll result from the nvestgaton or study, all statstcal analyss should follow the same general approach:

popu-Measure a lmted number of representatve samples from a larger populaton.

Estmate the true statstcs of larger populaton from the sample statstcs

Some mportant concepts need to be addressed here The first concept s somewhat obvous It s often mpossble or mpractcal to take measurements or observatons from an entre populaton Thus, the bomedcal engneer wll typcally select a smaller, more practcal sample that represents the underlyng populaton and the extent of varablty n the larger populaton For example, we cannot possbly measure the restng body temperature of every person on earth to get an estmate of normal body temperature and normal range We are nterested n knowng what the normal body temperature s, on average, of a healthy human beng and the normal range of restng temperatures

as well as the lkelhood or probablty of measurng a specfic body temperature under healthy,

rest-ng condtons In tryrest-ng to determne the characterstcs or underlyrest-ng probablty model for body temperature for healthy, restng ndvduals, the researcher wll select, at random, a sample of healthy, restng ndvduals and measure ther ndvdual restng body temperatures wth a thermometer The researchers wll have to consder the composton and sze of the sample populaton to adequately represent the varablty n the overall populaton The researcher wll have to define what character-

zes a normal, healthy ndvdual, such as age, sze, race, sex, and other trats If a researcher were to collect body temperature data from such a sample of 3000 ndvduals, he or she may plot a hsto-gram of temperatures measured from the 3000 subjects and end up wth the followng hstogram (Fgure 3.9).The researcher may also calculate some basc descrptve statstcs for the 3000 samples, such as sample average (mean), medan, and standard devaton

1

2

95 96 97 98 99 100 101 102 0.0

0.1 0.2 0.3 0.4 0.5

20 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

Once the researcher has estmated the sample statstcs from the sample populaton, he or she wll try to draw conclusons about the larger (true) populaton The most mportant queston to ask when revewng the statstcs and conclusons drawn from the sample populaton s how well the sample populaton represents the larger, underlyng populaton

Once the data have been collected, we use some basc descrptve statstcs to summarze the data These basc descrptve statstcs nclude the followng general measures: central tendency, varablty, and correlaton

There are a number of descrptve statstcs that help us to pcture the dstrbuton of the underlyng populaton In other words, our ultmate goal s to assume an underlyng probablty model for the populaton and then select the statstcal analyses that are approprate for that probablty model.When we try to draw conclusons about the larger underlyng populaton or process from our smaller sample of data, we assume that the underlyng model for any sample, “event,” or measure (the outcome of the experment) s as follows:

X = µ ± ndvdual dfferences ± stuatonal factors ± unknown varables,

where X s our measure or sample value and s nfluenced by µ, whch s the true populaton mean;

ndvdual dfferences such as genetcs, tranng, motvaton, and physcal condton; stuaton factors, such as envronmental factors; and unknown varables such as undentfied/nonquantfied factors that behave n an unpredctable fashon from moment to moment

In other words, when we make a measurement or observaton, the measured value represents

or s nfluenced by not only the statstcs of the underlyng populaton, such as the populaton mean, but factors such as bologcal varablty from ndvdual to ndvdual, envronmental factors (tme, temperature, humdty, lghtng, drugs, etc.), and random factors that cannot be predcted exactly from moment to moment All of these factors wll gve rse to a hstogram for the sample data, whch may or may not reflect the true probablty densty functon of the underlyng popula-ton If we have done a good job wth our expermental desgn and collected a sufficent number of samples, the hstogram and descrptve statstcs for the sample populaton should closely reflect the true probablty densty functon and descrptve statstcs for the true or underlyng populaton If ths s the case, then we can make conclusons about the larger populaton from the smaller sample populaton If the sample populaton does not reflect varablty of the true populaton, then the conclusons we draw from statstcal analyss of the sample data may be of lttle value

There are a number of probablty models that are useful for descrbng bologcal and facturng processes These nclude the normal, Posson, exponental, and gamma dstrbutons [10]

manu-In ths book, we wll focus on populatons that follow a normal dstrbuton because ths s the most frequently encountered probablty dstrbuton used n descrbng populatons Moreover, the most frequently used methods of statstcal analyss assume that the data are well modeled by a normal (“bell-curve”) dstrbuton It s mportant to note that many bologcal processes are not well mod-eled by a normal dstrbuton (such as heart rate varablty), and the statstcs assocated wth the normal dstrbuton are not approprate for such processes In such cases, nonparametrc statstcs, whch do not assume a specfic type of dstrbuton for the data, may serve the researcher better n understandng processes and makng decsons However, usng the normal dstrbuton and ts asso-cated statstcs are often adequate gven the central lmt theorem, whch smply states that the sum

of random processes wth arbtrary dstrbutons wll result n a random varable wth a normal trbuton One can assume that most bologcal phenomena result from a sum of random processes

ds-3.6.1 Measures of Central Tendency

There are several measures that reflect the central tendency or concentraton of a sample populaton: sample mean (arthmetc average), sample medan, and sample mode

The sample mean may be estmated from a group of samples, x i , where i s sample number,

usng the formula below

Gven n data ponts, x1, x2,…, x n:

x

n x i i

mean, x, should approach the true mean, µ, assumng that the statstcs of the underlyng populaton

or process do not change over tme or space

One of the problems wth usng the sample mean to represent the central tendency of a populaton s that the sample mean s susceptble to outlers Ths can be problematc and often decevng when reportng the average of a populaton that s heavly skewed For example, when reportng ncome for a group of new college graduates for whch one s an NBA player who has just sgned a multmllon-dollar contract, the estmated mean ncome wll be much greater than what most graduates earns The same msrepresentaton s often evdent when reportng mean value for homes n a specfic geographc regon where a few homes valued on the order of a mllon can hde the fact that several hundred other homes are valued at less than $200,000

22 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

Another useful measure for summarzng the central tendency of a populaton s the sample

medan The medan value of a group of observatons or samples, x i, s the mddle observaton when

samples, x i, are lsted n descendng order

For example, f we have the followng values for tdal volume of the lung:

2, 1.5, 1.3, 1.8, 2.2, 2.5, 1.4, 1.3,

we can find the medan value by first orderng the data n descendng order:

2.5, 2.2, 2.0, 1.8, 1.5, 1.4, 1.3, 1.3,and then we cross of values on each end untl we reach a mddle value:

pared wth the sample mean, the sample medan s less susceptble to outlers It gnores the skew n

a group of samples or n the probablty densty functon of the underlyng populaton In general,

to farly represent the central tendency of a collecton of samples or the underlyng populaton, we use the followng rule of thumb:

If the sample hstogram or probablty densty functon of the underlyng populaton s symmetrc, use mean as a central measure For such populatons, the mean and medan are about equal, and the mean estmate makes use of all the data

If the sample hstogram or probablty densty functon of the underlyng populaton s skewed, medan s a more far measure of center of dstrbuton

Another measure of central tendency s mode, whch s smply the most frequent observaton n

a collecton of samples In the tdal volume example gven above, 1.3 s the most frequently occurrng sample value Mode s not used as frequently as mean or medan n representng central tendency

3.6.2 Measures of variability

Measures of central tendency alone are nsufficent for representng the statstcs of a populaton or process In fact, t s usually the varablty n the populaton that makes thngs nterestng and leads 1

2

to uncertanty n decson makng The varablty from subject to subject, especally n physologcal functon, s what makes findng fool-proof dagnoss and treatment often so dfficult What works for one person often fals for another, and, t s not the mean or medan that pcks up on those subject-to-subject dfferences, but rather the varablty, whch s reflected n dfferences n the prob-ablty models underlyng those dfferent populatons.

When summarzng the varablty of a populaton or process, we typcally ask, “How far from the center (sample mean) do the samples (data) le?” To answer ths queston, we typcally use the followng estmates that represent the spread of the sample data: nterquartle ranges, sample var-ance, and sample standard devaton

The nterquartle range s the dfference between the first and thrd quartles of the sample data For sampled data, the medan s also known as the second quartle, Q2 Gven Q2, we can find the first quartle, Q1, by smply takng the medan value of those samples that le below the second quartle We can find the thrd quartle, Q3, by takng the medan value of those samples that le above the second quartle As an llustraton, we have the followng samples:

1, 3, 3, 2, 5, 1, 1, 4, 3, 2

If we lst these samples n descendng order,

5, 4, 3, 3, 3, 2, 2, 1, 1, 1,the medan value and second quartle for these samples s 2.5 The first quartle, Q1, can be found

by takng the medan of the followng samples,

2.5, 2, 2, 1, 1, 1,whch s 1.5 In addton, the thrd quartle, Q3, may be found by takng the medan value of the followng samples:

5, 4, 3, 3, 3, 2.5,whch s 3 Thus, the nterquartle range, Q3 − Q1 = 3 − 1.5 = 2

Sample varance, s2, s defined as the “average dstance of data from the mean” and the formula

for estmatng s2 from a collecton of samples, x i, s

=

− ∑− ( − )

24 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

Sample standard devaton, s, whch s more commonly referred to n descrbng the varablty of

the data s

= 2

s s (same unts as orgnal samples).

It s mportant to note that for normal dstrbutons (symmetrcal hstograms), sample mean and sample devaton are the only parameters needed to descrbe the statstcs of the underlyng phenomenon Thus, f one were to compare two or more normally dstrbuted populatons, one only need to test the equvalence of the means and varances of those populatons

• • • •

Now that we have collected the data, graphed the hstogram, estmated measures of central dency and varablty, such as mean, medan, and standard devaton, we are ready to assume a probablty model for the underlyng populaton or process from whch we have obtaned samples

ten-At ths pont, we wll make a rough assumpton usng smple measures of mean, medan, standard devaton and the hstogram But t s mportant to note that there are more rgorous tests, such as the c2 test for normalty [7] to determne whether a partcular probablty model s approprate to assume from a collecton of sample data

Once we have assumed an approprate probablty model, we may select the approprate statstcal tests that wll allow us to test hypotheses and draw conclusons wth some level of con-fidence The probablty model wll dctate what level of confidence we have when acceptng or rejectng a hypothess

There are two fundamental questons that we are tryng to address when assumng a ablty model for our underlyng populaton:

prob-How confident are we that the sample statstcs are representatve of the entre

There are a number of probablty models that are frequently assumed to descrbe bologcal processes For example, when descrbng heart rate varablty, the probablty of observng a specfic tme nterval between consecutve heartbeats mght be descrbed by an exponental dstrbuton [1, 8] Fgure 3.6 n Chapter 3 llustrates a hstogram for samples drawn from an exponental dstrbuton

1

2

Assuming a Probability Model

From the Sample data

C H A P T E R 4

26 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

Note that ths dstrbuton s hghly skewed to the rght For R-R ntervals, such a probablty ton makes sense physologcally because the ndvdual heart cells have a refractory perod that pre-vents them from contractng n less that a mnmum tme nterval Yet, a very prolonged tme nterval may occur between beats, gvng rse to some long tme ntervals that occur nfrequently

func-The most frequently assumed probablty model for most scentfic and engneerng tons s the normal or Gaussan dstrbuton Ths dstrbuton s llustrated by the sold black lne n Fgure 4.1 and often referred to as the bell curve because t looks lke a muscal bell

applca-The equaton that gves the probablty, f (x), of observng a specfic value of x from the derlyng normal populaton s

1 2

2

µ σ

e



∞ < x < ∞

where µ s the true mean of the underlyng populaton or process and σ s the standard devaton

of the same populaton or process A graph of ths equaton s gven llustrated by the sold, smooth curve n Fgure 4.1 The area under the curve equals one

Note that the normal dstrbuton s

a symmetrc, bell-shaped curve completely descrbed by ts mean, µ, and standard ton, σ

deva-by changng µ and σ, we stretch and slde the dstrbuton

1

2

0 0.05 0.1

FIguRE 4.1: A hstogram of 1000 samples drawn from a normal dstrbuton s llustrated

Super-mposed on the hstogram s the deal normal curve representng the normal probablty dstrbuton functon

Fgure 4.1 also llustrates a hstogram that s obtaned when we randomly select 1000 samples from a populaton that s normally dstrbuted and has a mean of 0 and a varance of 1 It s mpor-

tant to recognze that as we ncrease the sample sze n, the hstogram approaches the deal normal

dstrbuton shown wth the sold, smooth lne But, at small sample szes, the hstogram may look very dfferent from the normal curve Thus, from small sample szes, t may be dfficult to determne

f the assumed model s approprate for the underlyng populaton or process, and any statstcal tests that we perform may not allow us to test hypotheses and draw conclusons wth any real level

of confidence

We can perform lnear operatons on our normally dstrbuted random varable, x, to produce another normally dstrbuted random varable, y These operatons nclude multplcaton of x by a constant and addton of a constant (offset) to x Fgure 4.2 llustrates hstograms for samples drawn

from each of populatons x and y We note that the dstrbuton for y s shfted (the mean s now equal to 5) and the varance has ncreased wth respect to x.

One test that we may use to determne how well a normal probablty model fits our data

s to count how many samples fall wthn ±1 and ±2 standard devatons of the mean If the data and underlyng populaton or process s well modeled by a normal dstrbuton, 68% of the samples should le wthn ±1 standard devaton from the mean and 95% of the samples should le wthn

105

ln-28 INTRoduCTIoN To STATISTICS FoR BIoMEdICAL ENgINEERS

±2 standard devatons from the mean These percentages are llustrated n Fgure 4.3 It s tant to remember these few numbers, because we wll frequently use ths 95% nterval when draw-

mpor-ng conclusons from our statstcal analyss

Another means for determnng how well our sampled data, x, represent a normal

dstrbu-ton s the estmate Pearson’s coefficent of skew (PCS) [5] The coefficent of skew s gven by

PCS =3 x x−smedian

If the PCS > 0.5, we assume that our samples were not drawn from a normally dstrbuted populaton.When we collect data, the data are typcally collected n many dfferent types of physcal unts (volts, celsus, newtons, centmeters, grams, etc.) For us to use tables that have been developed for probablty models, we need to normalze the data so that the normalzed data wll have a mean of

0 and a standard devaton of 1 Such a normal dstrbuton s called a standard normal dstrbuton and s llustrated n Fgure 4.1

32

10

-1-2

-3

9080706050403020100

Normalized value ( Z score)

ds-The standard normal dstrbuton has a bell-shaped, symmetrc dstrbuton wth µ = 0 and

For an ndvdual sample, the z score s a “normalzed” or “standardzed” value We can use ths value

wth our equatons for probablty densty functon or our standardzed probablty tables [3] to termne the probablty of observng such a sample value from the underlyng populaton

de-The z score can also be thought of as a measure of the dstance of the ndvdual sample, x i,

from the sample average, x− , n unts of standard devaton For example, f a sample pont, x i has a z score of z i = 2, t means that the data pont, x i, s 2 standard devatons from the sample mean

We use normalzed z scores nstead of the orgnal data when performng statstcal analyss

because the tables for the normalzed data are already worked out and avalable n most statstcs texts or statstcal software packages In addton, by usng normalzed values, we need not worry about the absolute ampltude of the data or the unts used to measure the data

The standard normal dstrbuton s llustrated n Table 4.1

The z table assocated wth ths figure provdes table entres that gve the probablty that z ≤

a, whch equals the area under the normal curve to the left of z = a If our data come from a normal

dstrbuton, the table tells us the probablty or “chance” of our sample value or expermental

out-comes havng a value less than or equal to a.

Thus, we can take any sample and compute ts z score as descrbed above and then use the

z table to find the probablty of observng a z value that s less than or equal to some normalzed

value, a For example, the probablty of observng a z value that s less than or equal to 1.96 s 97.5% Thus, the probablty of observng a z value greater than 1.96 s 2.5% In addton, because of symmetry n the dstrbuton, we know that the probablty of observng a z value greater than −1.96

s also 97.5%, and the probablty of observng a z value less than or equal to −1.96 s 2.5% Fnally,

1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

…

2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

…

the probablty of observng a z value between −1.96 and 1.96 s 95% The reader should study the

z table and assocated graph of the z dstrbuton to verfy that the probabltes (or areas under the

probablty densty functon) descrbed above are correct

Often, we need to determne the probablty that an expermental outcome falls between two

values or that the outcome s greater than some value a or less or greater than some value b To find

these areas, we can use the followng mportant formulas, where Pr s the probablty:

Pr(a ≤ z ≤ b) = Pr(z ≤ b) – Pr(z ≤ a)

= area between z = a and z = b.

Pr(z ≤ a) = 1 – Pr(z < a)

= area to rght of z = a

= area n the rght “tal”

Thus, for any observaton or measurement, x, from any normal dstrbuton:









where µ s the mean of normal dstrbuton and σ s the standard devaton of normal dstrbuton

In other words, we need to normalze or find the z values for each of our parameters, a and b,

to find the area under the standard normal curve (z dstrbuton) that represents the expresson on

the left sde of the above equaton

Example 4.1 The mean ntake of fat for males 6 to 9 years old s 28 g, wth a standard devaton

of 13.2 g Assume that the ntake s normally dstrbuted Steve’s ntake s 42 g and Ben’s ntake s