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Hack 34. Make Wise Medical Decisions

Hack 34. Make Wise Medical Decisions

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thecondition(A)

Testresultindicatespatient

hascondition

Testresultindicatespatient

doesnothavecondition



Truepositive(scoreis

correct)

Falsenegative(scoreis

wrong)



Falsepositive(scoreis

wrong)

Truenegative(scoreis

correct)



Thereliability[Hack#6]ofmedicalscreeningtestsis

summarizedbytwoproportionscalledsensitivityandspecificity.

Essentially,thosewhorelyonthesetestsareconcernedwith

threequestionsofaccuracy:

Ifapersonhasthedisease,howlikelyisthepersonto

scoreapositivetestresult?Thislikelihoodissensitivity.Of

thosepeopleincolumnA,whatpercentwillreceivea

positivetestresult?

Ifthepersondoesnothavethedisease,howlikelyisthe

persontoscoreanegativetestresult?Thislikelihoodis

specificity.OfthosepeopleincolumnB,whatpercentwill

receiveanegativetestresult?

Ifapersonscoresapositivetestresult,howlikelyisthe

persontohavethedisease?Fromthepatient'sperspective,

thisistheultimatequestion,anditcanbethoughtofasthe

basicvalidityconcernwiththesetests.Doctor,canItrust

thesetestresults,orcouldtherebesomemistake?



NoticeinTable3-12thattherearedifferentpeopleincolumnsAandB.

PeoplewiththediseaseareincolumnAandpeoplewithoutthedisease

areincolumnB.IfyouareincolumnA,youcannotscoreafalse

positiveonthetest,becauseapositiveresultiscorrect.Ifyouarein

columnB,youcannotscoreafalsenegative,becauseanegativeresult

iscorrect.



Whichcolumnanyoneisindependsonthenaturaldistribution

ofthedisease.ThechancethatsomeonewillbeincolumnA

(thechancethepersonactuallyhasthedisease)dependson

thebaserateofthedisease.If5percentofthepopulationhas

thedisease,5percentofthepopulationwouldfindthemselves

incolumnA.



UnderstandingBreastCancerScreening

Breastcancerisanexampleofaseriousconditionforwhich

therearediagnosticscreeningtests.Breastcancerscreening

beginswithamammogramtest.Apositiveresultonthistest

resultsinfurthertesting:anothermammogram,ultrasound,or

biopsy.

Wearefirstinterestedinansweringthequestionsregardingthe

sensitivityandspecificityofbreastcancerscreening.Withthat

informationandknowledgeofthebaserateforbreastcancer,

wecananswerthemostimportantquestion:

Ifawomanscoresapositivetestresult,howlikelyissheto

havebreastcancer?

Byaskingyourdoctorordoingsomeresearch,youmight

discoverthatsensitivityformammogramsisabout90percent.

Specificityisabout92percent.



Theexactsensitivityandspecificityforbreastcancerscreeningschange

overtimeasdifferentpopulationstakethetest.Youngerwomennow

havemammogramsmorecommonlythaninthepast,andthetestis

lesssensitiveandlessspecificforyoungerwomen.Ofcourse,you

shouldcheckwithaphysicianorexpertforcurrentlevelsofprecision.



Table3-13showsthosenumbersinthelayoutusedinTable312.BecausecolumnsAandBmustbothindependentlyaddto

100percent,wecanalsoestimatetherateoffalsenegatives

andrateoffalsepositives.

TableTheoreticalmammogramresultsfor10,000women



Patientactuallyhas

breastcancer(A)

N=120





Mammogram

indicatescancer

Mammogramdoes

notindicatecancer



Sensitivity90

percentN=108

Falsenegatives10

percentN=12



Patientactuallydoesnot

havebreastcancer(B)

N=9,880

Falsepositives8percentN=790

Specificity92percentN=9,090



Table3-13alsoshowstheoutcomesfor10,000hypothetical

women,basedonthebaserateofbreastcancerinthe

population,whichisabout1.2percent.



Itturnsoutthatitisdifficulttoidentifyanaccurateincidentratefor

breastcancerbecauseofthedifferentwaysonecandefinetherelevant

populationand,ofcourse,limitationsintheaccuracyofbreastcancer

testing.I'musinganoften-reportedandfairlywell-acceptedestimate

ofthecurrentpercentageofwomenaged40to84thathavebreast

cancer.



Let'sreturnnowtothethirdquestioninourlistofimportant

questionstoaskbeforeinterpretingtheresultsofamedical

test.Ifapersonscoresapositivetestresult,howlikelyisthe

persontohavethedisease?Outof10,000womenwhohavea

breastcancerscreening,898willreceiveapositivescore.For



790ofthosewomen,thescoreiswrong;theydonotactually

havebreastcancer.For108ofthosewomen,thetestwasright;

theydohavecancer.Inotherwords,ifapersonscoresa

positiveresult,itisonly12percentlikelythattheyhavethe

disease.Themostcommonresultforfollow-uptestingtoa

positivemammogramisthatthepatientis,infact,cancerfree.

Whatabouttheaccuracyofanegativeresult?Ofthe9,102

womenwhowillscorenegativeonthescreening,12actually

havecancer.Thisisarelativelysmall1/10of1percent,butthe

testingwillmissthosepeoplealtogether,andtheywillnot

receivetreatment.



WhyItWorks

Medicalscreeningaccuracyusesaspecificapplicationofa

generalizedapproachtoconditionalprobabilityattributedto

ThomasBayes,aphilosopherandmathematicianinthe1700s.

"Ifthis,thenwhatarethechancesthat..."isaconditional

probabilityquestion.

Bayes'sapproachtoconditionalprobabilitieswastolookatthe

naturallyoccurringfrequenciesofevents.Thebasicformulafor

estimatingthechancethatonehasadiseaseifonehasa

positivetestresultis:

Expressedasconditionalprobabilities,theformulais:

Toanswertheall-importantquestioninourbreastcancer

example("Ifawomanscoresapositivetestresult,howlikelyis

shetohavebreastcancer?"),themammogramequationtakes

onthesevalues:



MakingInformedDecisions



Medicaltestsareusedtoindicatewhetherpatientsmighthave

adiseaseorbeatriskforgettingone.Identifyingthepresence

orabsenceofadiseasesuchascancerisaprocessthatusually

hasatleasttwosteps.Instepone,apatientisadministereda

screeningtest,typicallyarelativelysimpleandnoninvasivetest

thatlooksforindicationsthatapersonmighthaveacertain

medicalcondition.Iftheresultispositive,thesecondstepisto

conductasecondtest(orseriesoftests)thatistypicallymore

complex,invasive,andexpensive,butalsomuchmore

accurate,toconfirmordisconfirmtheoriginalfinding.

Medicaltestsarenotperfectlyreliableandvalid.Testresults

canbewrong.Therearefourpossibilitiesforanyonewho

undergoesmedicaltesting.Apatientmighthavethedisease

andthetestindicatesthis,orthepatientdoesnothavethe

diseaseandthetestfindsnopresenceofit.Inthesecases,the

testworkedrightandthescoresarevalid.

Conversely,thetestresultsmightreflecttheoppositeofthe

truemedicalcondition,withapositiveresultwronglyindicating

presenceofadiseasethatisnotthere,oranegativeresult

wronglyindicatingthatthepatientisdisease-free.Inthese

cases,thetestdidnotworkrightandtheresultsarenotvalid.

Thistableofoutcomesissimilartothepossibilitieswhenone

acceptsorrejectsahypothesisinstatisticaldecisionmaking

[Hack#4].

Breastcancerscreeningisverygoodatfindingbreastcancer

whenitistheretofind.However,onedrawbacktosucha

sensitivetestforalow-incidencediseaseisthatmanymore

peoplewillbetoldthattheymighthavethediseasethan

actuallydo.Thereisatrade-offinmedicaltestingbetweentest

sensitivityandtestspecificity.Moresensitiveteststendto

resultinmorefalsepositives,butinserioussituationslikelife

anddeath,thisseemstobearesultwecanlivewith.



SeeAlso



Gigerenzer,G.(2002).Calculatedrisks.Howtoknowwhen

numbersdeceiveyou.NewYork:SimonandSchuster.







Chapter4.BeatingtheOdds

Whyriskmorethanyouhavetowhenyoutakerisks?Casino

gamesrequireyoutotakesomechances,butthischapterof

real-worldstathackswillhelpyoukeepyouredgeandperhaps

evenovercomethehouse'sedge.

StartwithTexasHold'Empoker[Hack#36].(Maybeyou've

heardofit?)Whenyouplaypoker[Hack#37],playtheodds

[Hack#38].

Makesure,ofcourse,toalwaysgamblesmart[Hack#35],

regardlessofwhatyouplay,thoughwhenitcomestothelevel

ofriskyoutake,somegames[Hacks#39and#40]arebetter

thanothers[Hack#41].

Ifyouliketomakefriendlywagerswithfriendsorstrange

wagerswithstrangers,youcanusethepowerofstatisticsto

winsomesurprisinglywinnablebarbetswithcards[Hacks#42

and#44],dice[Hack#43],orjustaboutanythingelseyoucan

thinkof[Hack#46],includingyourfriends'birthdays[Hack

#45].

Speakingofweirdgamblinggames(andIthinkwewere),there

aresomeoddstatisticalquirks[Hacks#47and#49]you'll

needtoknowwhenyouplaythem,evenifitisjustflippinga

coin[Hack#48].







Hack35.GambleSmart



Whateverthegame,ifmoneyandchanceareinvolved,

therearesomebasicgamblingtruthsthatcanhelpthe

happystatisticianstayhappy.

Althoughthischapterisfullofhacksaimedatparticulargames,

manyofthemgamesofchance,thereareavarietyoftipsand

toolsthatareusefulacrosstheboardforallgamblers.Much

mystery,superstition,andmathematicalconfusionpervadethe

worldofgambling,andknowingalittlemoreaboutthe

geographyofthisworldshouldhelpyougetaround.Thishack

showshowtogamblesmarterbyteachingyouaboutthe

followingthings:

TheGambler'sFallacy,anintuitiveyetfalsebeliefsystem

thathascostmanyanotherwisewell-informedgamer

Casinosandmoney

Systems,sophisticatedmoneymanagement,andwagering

proceduresthatdonotwork



TheGambler'sFallacy

Didyoueverhavesomanybadblackjackhandsinarowthat

youincreasedyourbet,knowingthatthingsweredueto

changeanytimenow?Ifso,yousuccumbedtothegambler's

fallacy,abeliefthatbecausetherearecertainprobabilities

expectedinthelongrun,ashort-termstreakofbadluckis



likelytochangesoon.

Thegambler'sfallacyisthatthereisaswingingpendulumof

chanceanditswingsintheregionofbadoutcomesforawhile,

losesmomentum,andswingsbackintoaregionofgood

outcomesforawhile.Theproblemwithfollowingthismindsetis

thatluck,asitappliestogamesofpurechance,isaseriesof

independentevents,witheachindividualoutcomeunrelatedto

theoutcomethatcamebefore.Inotherwords,thelocationof

thependuluminagoodregionorbadregionisunrelatedto

whereitwasasecondbefore,andhere'stherubthereisn'teven

apendulum.Theficklefingeroffatepopsrandomlyfrom

possibleoutcometopossibleoutcome,andtheprobabilityofit

appearingatanyoutcomeistheprobabilityassociatedwith

eachoutcome.Thereisnomomentum.Thistruthisoften

summarizedas"thedicehavenomemory."

Examplesofbeliefsconsistentwiththegambler'sfallacy

include:

Aslotmachinethathasn'tpaidoutinawhileisdue.

Apokerplayerwhohashadnothingbutbadhandsall

eveningwillsoongetasupercolossalhandtoeventhings

out.

Alosingbaseballteamthathaslostthelastthreegamesis

morelikelytowinthefourth.

Becauserollingdiceandgettingthree7sinarowisunlikely

tooccur,rollingafourthafterhavingjustrolledthree

straightmustbebasicallyimpossible.

Arouletteballthathaslandedoneightrednumbersina

rowprettymuchmusthitablacknumbernext.



Avoidfallacieslikethisatallcosts,andgamblingshouldcost

youless.



CasinosandMoney

Casinosmakemoney.Onereasontheymakeaprofitisthatthe

gamesthemselvespayoffamountsofmoneythatareslightly

lessthantheamountofmoneythatwouldbefair.Inagameof

chance,afairpayoutisonethatmakesbothparticipants,the

casinoandtheplayer,breakeveninthelongrun.

Anexampleofafairpayoutwouldbeforcasinostouseroulette

wheelswithonly36numbersonthem,halfredandhalfblack.

Thecasinowouldthendoublethemoneyofthosewhobeton

redafterarednumberhits.Halfthetimethecasinowouldwin,

andhalfthetimetheplayerwouldwin.Inreality,American

casinosuse38numbers,twoofthemneitherrednorblack.

Thisgivesthehousea2/38edgeoverafairpayout.Ofcourse,

it'snotunfairinthegeneralsenseforacasinotomakeaprofit

thisway;it'sexpectedandpartofthesocialcontractthat

gamblershavewiththecasinos.Thetruthis,though,thatif

casinosmademoneyonlybecauseofthisedge,fewwould

remaininbusiness.

Thesecondreasonthatcasinosmakemoneyisthatgamblers

donothaveinfinitelydeeppockets,andtheydonotgamblean

infiniteperiodoftime.Theedgethatacasinohasthe5.26

percentonroulette,forexampleisonlytheamountofmoney

theywouldtakeifagamblerbetaninfinitenumberoftimes.

Thisinfinitegamblerwouldbeupforawhile,downforawhile,

andatanygiventime,onaverage,wouldbedown5.26percent

fromherstartingbankroll.

Whathappensinreallife,though,isthatmostplayersstop

playingsometime,usuallywhentheyareoutofchips.Most

playerskeepbettingwhentheyhavemoneyandstopbetting



whentheydon't.Someplayers,ofcourse,walkawaywhen

theyareahead.Noplayer,though,keepsplayingwhenthey

havenomoney(andnocredit).

ImaginethatTable4-1represents1,000playersofanycasino

game.Allplayersstartedwith$100andplannedtospendan

evening(fourhours)playingthegames.We'llassumeahouse

edgeof5.26percent,asroulettehas,thoughothergameshave

higherorloweredges.

TableFateof1,000hypotheticalgamblers



Timespent

playing

Afteranhourof

play

Aftertwohours

ofplay

Afterthree

hoursofplay

Afterfourhours

ofplay



Havesome

moneyleft



Mean

bankrollleft



Havelostall

theirmoney



Still

playing



900



$94.74



100



900



800



$94.74



200



800



700



$94.74



300



700



600



$94.74



400



600



Inthisexamplewhichusesmade-upbut,Ibet,conservative

dataafterfourhours,theplayersstillhave$56,844,thecasino

has$43,156,andfromthetotalamountofmoneyavailable,the

casinotook43.16percent.That'ssomewhatmorethanthe

official5.26percenthouseedge.

Itishumanbehaviorthetendencyofplayerstokeepplayingnot

theprobabilitiesassociatedwithaparticulargame,thatmakes

gamblingsoprofitableforcasinos.Becausethehouserulesare

publishedandreported,statisticianscanfigurethehouseedge

foranyparticulargame.

Casinosarenotrequiredtoreporttheactualmoneytheytake

infromtablegames,however.Basedonthedepthoftheshag



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