Biostatistics. Biostatistics Introduction & Data Presentation (Short Version) Intro - 1. Variable Types. Basic Terms In Statistics

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1 Biostatistics Itroductio & Data Presetatio (Short Versio) Biostatistics Ady Chag Yougstow State Uiversity Statistics i Broader Sese Statistics is a field of study cocered with the ) data collectio, [Producig data] ) orgaizatio, summarizatio, examiatio ad providig a overview of the geeral features of data, [Explorig Data] 3) ad the drawig of ifereces about a body of data (populatio) based o the properties of a part of the data (sample) observed. [Statistical Iferece] Producig Data Explorig Data Statistical Iferece I health ad medical (or cliical) study, researchers ivestigate a sample of subjects to uderstad the effectiveess of a treatmet or a itervetio o target populatio. Public health is fudametally cocered with prevetig disease, disability, ad premature death i huma populatio or commuity. Therefore, statistical iferece is very importat i public health research. Therefore, statistical iferece is very importat i health ad medical research. Goal of the Healthy People : Elimiate Health Disparities 3 4 Basic Terms I Statistics Idividuals (subjects, experimetal uit): the etities o which data are collected. Variable: a characteristic of iterest for the idividual which takes o differet values i differet idividual. Purpose of Statistics Examie a variable Examie correlatio betwee two or more variables Variable Types Quatitative Variables (umeric) [height, umber of subscriptios,...] Cotiuous: a variable that has a ucoutable umber of possible values. (measuremets) Discrete: a variable that has a coutable umber of possible values.(couts) Qualitative (Categorical) Variables [hair color, geder,...] 5 6 Itro -

2 Biostatistics Itroductio & Data Presetatio (Short Versio) Measuremet Scales Nomial: cosists of labels, ames or categories. Ordial: data that the order or rak is meaigful. Iterval: umerical data that arithmetic operatios are meaigful. Ratio: data that the ratio of two data is meaigful. Producig Data 7 8 Data i Public Health : Vital Statistics ad the Cesus Public Health Surveillace Survey Registries Epidemic Ivestigatios Research Program Evaluatios Descriptive Statistics Data Presetatioi Groupig tables graphical summary Numerical Summary Ceter Dispersio (Spread) Exploratory Data Aalysis 9 Data Presetatio What type of statistical techique is appropriate for Data Presetatio? Categorical variable? Quatitative variable? Data Sheet (Raw data) ID Height(i) Weight(lb) BirthMoth Exp. Geder H F H F T M H F T F H M H F H M... Itro -

3 Biostatistics Itroductio & Data Presetatio (Short Versio) (A complete list) ID Height Weight BirthMoth Exp. Geder H F H F T M H F T F H M H F H M T M H M T M T F H M T F H M H F T F T M H M T M 7 7 H M H M 3 Groupig ad Displayig A Categorical Variable 4 7 Frequecy Table ad Charts (Oe Categorical Variable) Class Frequecy Relative Frequecy Female 9 9/ =.49 = 4.9% Male 3 3/ =.59 = 59.% Total % Cout Pareto chart Percet 6585 Female 4.9% White Asia 3 Male 59.% Pe rcet sex Female Male 5 Black or Africa Ame Hispaic or Latio Multiple - No-hispa Multiple - Hispaic How do you describe yourself Am. Idia or Alaska * Bars arraged accordig to their frequecies. N. Hawaii/Other Pac 6 Frequecy Distributio Table Data: 35, 9, 75, 6, 35, 7, 8, 5, 95, Groupig ad Displayig A Quatitative Data Class Tally Frequecy - < - <4 4 - < - <8 8 - < - < - <4 4 - < - <8 8 - <3 Total 7 8 to less tha 3 8 Itro - 3

4 Biostatistics Itroductio & Data Presetatio (Short Versio) Frequecy Distributio Table Data: 35, 9, 75, 6, 35, 7, 8, 5, 95, Class Tally Frequecy - < 3 - < < - < < 7 - < - <4 4 - < - <8 8 - <3 Total Frequecy Distributio Table (From data sheet) Class Frequecy Relative Freq. Cumulative R.F. - < 3 3/ =.36 3/ - <4 3 3/ =.36 6/ 4 - < / =.9 8/ - <8 4 4/ =.8 / 8 - < 7 7/ =.38 9/ - < / =.45 / - <4 / =.45 / 4 - < / =. / - <8 / =. / 8 - <3 / =.45 / Total. 8 to less tha 3 9 Classes: Categories for groupig data. Frequecy (class frequecy): The umber of data values i a class. Relative frequecy: The ratio of the frequecy of a class to the total umber of pieces of data. Frequecy distributio: A listig of classes ad their frequecies. Relative Frequecy distributio: A listig of classes ad their relative frequecies. Upper class limit: The largest value that ca go i a class. Lower class limit: The smallest value that ca go i a class. Class width: The differece betwee the lower class limit of the give class ad the lower class limit of the ext higher class. Class midpoit (class mark): The midpoit of a class. Guidelies for groupig data: (for quatitative variable) There should be betwee five ad twety classes. Each piece of data must belog to oe, ad oly oe, class.(mutually Exclusive) Wheever feasible, all classes should have the same width. To build a Frequecy Table: Fid the rage of the data: Rage = Largest value smallest value Use the rage ad try differet class width to determie how may classes you eed to make frequecy table or histogram. Studet data example: Rage = 85 6 = 79/ 9 If usig a class width of, there ll be about 9 classes which is good. 3 Frequecy Distributio Table (From data sheet) Class Frequecy Relative Freq. Cumulative R.F. < - 3 3/ =.36 3/ < / =.36 6/ 4< - 3 3/ =.36 9/ < / =.7 4/ 8< - 5 5/ =.7 9/ < - / =.9 / < - 4 / =. / 4< - / =. / < - 8 / =. / 8< - 3 / =.45 / Total. 4 Itro - 4

5 Biostatistics Itroductio & Data Presetatio (Short Versio) Histogram (SPSS) Polygo (SPSS) 5 6 Polygo (SPSS) Cumulative R. F. Histogram % % Cumulative R. F. Polygo (Ogive) What to observe i Histograms? % % Outliers: observatios that stad out from the rest for some reaso. Ceter: the middle of the data. Spread: the rage; the extet of the data; how far the values are from each other. Shape: distributio patter. [Skewess, symmetry, uiform, Normal,...] Itro - 5

6 Biostatistics Itroductio & Data Presetatio (Short Versio) Histogram & Desity Curve Symmetric (Bell) shape Skewed to the right, or positively skewed Percet A smooth curve that describes the distributio Bimodal Desity fuctio, f (x) Uiform Skewed to the left, or egatively skewed Use a mathematical model to describe the variable. 3 3 Stemplots (or Stem-ad-leaf plots) -- leadig digits are called stems -- fial digits are called leaves Example: (umber of hysterectomies performed by 5 male doctors) 7,, 33, 5, 86, 5, 85, 3, 37, 44,, 36, 59, 34, Stemplot 34 Example: (umber of hysterectomies performed by 5 male doctors) 7,, 33, 5, 86, 5, 85, 3, 37, 44,, 36, 59, 34, Ordered Stemplot 35 Example: [Back-to-back stem-plot] Number of hysterectomies performed by 5 male doctors: 7,, 33, 5, 86, 5, 85, 3, 37, 44,, 36, 59, 34, 8 by female doctors, the umbers are: 5, 7,, 4, 8, 9, 5, 9, 3, 33 (Female) (Male) Itro - 6

7 Biostatistics Itroductio & Data Presetatio (Short Versio) Box Plot 8 7 Examie Bivariate Data (Bivariate Aalysis) Examie the relatio betwee two variables N = HEIGHT Two Categorical Variables Cotigecy Table Two Categorical Variables Cluster bar chart Smoker No- Smoker Colum Total Cacer No cacer Row Total (%) 5 (5%) 5 (5%) 3 (3%) 45 (45%) 75 (75%) (%) (%) Odds of smoker to have cacer: /3 = 6/9 Odds of osmoker to have cacer: 5/45 = /9 Odds Ratio = (6/9)/(/9) = Two Quatitative Variables Two Quatitative Variables Data: Temperature Mortality Idex Average aual temperature ad the mortality idex for a type of breast cacer i wome i certai regio of Europe. Mortality Idex Average Temperature 4 Itro - 7

8 Biostatistics Itroductio & Data Presetatio (Short Versio) Respose/Explaatory Variables Respose (Depedet, Outcome) Variable Lug Cacer, Mortality Idex Explaatory (Idepedet, Predictor) Variable Smokig, Average Temperature A Categorical & A Quatitative Variables 8 7 Side-by-side Boxplot HEIGHT N = 8 3 Female Male 43 sex 44 Time Plot Rate Time Rate Time Value HRATE Time Plot Yougstow Homicide Rate by Year YEAR Time Plot Misleadig Chart Natioal Homicide Rate By Year Natioal Homicide Rate By Year Natioal Homicide Rate Natioal Homicide Rate Cout Cout 3 YEAR YEAR Female Male Female Male Geder Geder Differet Scales Differet Scales Itro - 8

9 Biostatistics Itroductio & Data Presetatio (Short Versio) Icorrect ad Misleadig Chart Type of Statistical Studies Observatioal Study: coditios to which subjects are exposed are ot cotrolled by the ivestigator. (o attempt is made to cotrol or ifluece the variables of iterest) 49 Experimetal (Cotrolled) Study: coditios to which subjects are exposed to are cotrolled by the ivestigator. (treatmets are used i order to observe the respose) (Radomizatio, Replicatios) Results from observig behavior ad outcomes from the use of medicie for radomly selected patiets. (Patiets chose their medicie) Treatmet Drug A Drug B Hypertesio Yes 44 9 No 56 7 Total Treatmet Drug A Drug B Simpso s Paradox Yes 5 7 Below 65 No 8 Hypertesio Total 3 77 Yes No 38 Total 77 3 Total 73 7 Drug A: 44/ = 44% Drug B: 9/ = 9% 5 * Older patiets prefer Drug A OR <65: Drug A: 5/3 = % Drug B: 7/77 = % OR 65+: Drug A: 39/77 = 5% Drug B: /3 = 5% 5 Cofoudig Effect Treatmet & Treatmet Cofoudig variables Cause? Patiet s Age & Health Coditio Patiet s Survival Variables, whether part of a study or ot, are said to be cofouded whe their effects o the outcome caot be distiguished from each other Age may affect the reactio to drug ad may also affect drug choosig decisio Itro - 9

10 Descriptive Statistics (Short Versio) Example: Birth weights (i lb) of 5 babies bor from two groups of wome uder differet care programs. Group : 7, 6, 8, 7, 7 Group : 3, 4, 8, 9, Numerical Summary Measures Describe Distributio with Numbers Measure of Ceter Measure of Variatio Measure of Positio Measure of Cetral Tedecy Mea: the average value of the data. Example: Birth weights (i lb) of 5 babies bor from a group of wome uder certai diet. 7, 6, 8, 7, 7 If observatios are deoted by x, x,..., x, their (sample) mea is x x + x x i= = = x i Sol: mea = = = [ear the ceter of the data set] 3 4 Media: of a data set is the data value exactly i the middle of its ordered list if the umber of pieces of data is odd, the mea of the two middle data values i its ordered list if the umber of pieces of data is eve. [media is ot iflueced by outliers ad is best for o-symmetric distributio] Example: (umber of hysterectomies performed by 5 doctors) 7,, 33, 5, 86, 5, 85, 3, 37, 44,, 36, 59, 34, 8 ordered list =>, 5, 5, 7, 8, 3, 33, 34, 36, 37, 44,, 59, 85, 86 media = Descriptive Stat -

11 Descriptive Statistics (Short Versio) Example: (Birth weights for 6 ifats.) 5, 7, 6, 8, 5, 9 ordered list => 5, 5, 6, 7, 8, 9 Mode: of a data set is the observatio that occurs most frequetly. media = (6+7) / = Example : (umber of times visited class website by 5 studets) 7,, 33, 5, 86, 5, 85, 3, 37, 44,, 36, 59, 34, 8 ordered list =>, 5, 5, 7, 8, 3, 33, 34, 36, 37, 44,, 59, 85, 86 Mode = 5 Example : (Blood type of 5 studets) A, B, A, A, O, AB, A, A, B, B, O, O, A, A, A Mode = A A 8 B 3 O 3 Mea? Media? Mode? Skewed to the Right AB 9 Measure of Dispersio (Variability) Rage = largest data value smallest data value Sample from group I (diet program I): 7, 6, 8, 7, 7 => mea = ( ) / 5 = 35/5 = 7 Sample from group II (diet program II): 3, 4, 8, 9, => mea = ( ) / 5 = 35/5 = 7 Is there ay differece betwee the two samples? rage of sample I = 8-6 = rage of sample II = - 3 = 8 Does the mother s diet program affect the birth weights of babies? Descriptive Stat -

12 Descriptive Statistics (Short Versio) Example: Birth weights (i lb) of 5 babies bor from a group of wome uder diet program II. 3, 4, 8, 9, mea = x = 7 Variace ad Stadard Deviatio Measure the spread of the data aroud the ceter of the data. Data Value x i Total Deviatio from mea x i x 3 7 = = = 9 7 = 7 = 4 Squared Dev. ( x i x) Sample Variace = 46/4 =.5 lb, Sample Stadard Deviatio = 46/ 4 = 3.39 lb. 4 s A Short Cut formula: i = = i x xi i= 35 9 = 5 =.5 4 Data, x x What is the stadard deviatio of the weights of babies from the sample of mothers who received diet program I? Data: 7, 6, 8, 7, 7 s = (++++)/(5-) = ½ s = =.7 Does the mother s diet program affect the birth weights of babies? Diet I: mea = 7, s =.7 Diet II: mea = 7, s = If observatios are deoted by x, x,..., x, their variace ad stadard deviatio are ( xi x) i= Sample Variace: s = (ubiased estimator for variace of a ifiite populatio.) Sample Mea: Sample Stadard Deviatio: s = x + x x = i= x ( x x) i = i= x i 7 Populatio Parameters If N observatios are deoted by x, x,..., x, are all the observatio i a fiite populatio, their mea, μ, variace σ, ad stadard deviatio, σ, are x + x x Populatio Mea: μ = = x N N i= ( xi μ) i= Populatio Variace: σ = Populatio Stadard Deviatio: N σ i= = ( x μ) i N i 8 Descriptive Stat - 3

13 Descriptive Statistics (Short Versio) About s (sample stadard deviatio) : s measures the spread aroud the mea. the larger s is, the more spread out the data are. if s =, the all the observatios must be equal. s is strogly iflueced by outliers. The Use of Mea ad Stadard Deviatio Describe distributio Uderstad the ceter ad the spread of the distributio 9 Uit: mg/ml Female Boe Desity Data Mea,. x Stadard Deviatio, s 5 May distributios ca be described by a mathematical fuctio with specific parameters, such as mea ad stadard deviatio. Example: Normal Distributio (Bell-shaped) Male 8.4 σ μ Empirical Rule Properties of a symmetric ad bell-shaped (Normal) distributio: The distributio is symmetric about it mea (μ), 68% of the area betwee μ σ ad μ + σ, 95% of the area betwee μ σ ad μ + σ, 99.7% of the area betwee μ 3σ ad μ + 3σ. Heart rates for a certai populatio at a certai coditio follow a bell shape symmetric distributio with mea 7 ad stadard deviatio. What percetage of people i this populatio will have heart rates betwee 66 ad 74? 95%?% μ 3σ μ μ + 3σ Descriptive Stat - 4

14 Descriptive Statistics (Short Versio) Chebychev s Rule Chebychev s iequality There is at least (/k ) of the data i a data set lie withi k stadard deviatio of their mea. Example: Heart rates for asthmatic patiets i a state of respiratory arrest has a mea of 4 beats per miute ad a stadard deviatio of 35.5 beats per miute. What percetage of the populatio of this type of patiets have heart rates lie betwee two stadard deviatios of the mea i a state of respiratory arrest? It will be at least 75%, because k =, ad (/ ) = ¾ = 75%. 5 6 Heart rates example: mea=44, s.d.=35.5 k = 75% = (/ ) What about withi three stadard deviatios? Heart rates example: mea=44, s.d.=35.5 k = 3?% 89% (/3 ) ) At least 75% At At least 89%?% x35.5 = x35.5 = x35.5 = x35.5 = Measure of Positio Z-score (Stadard Score) If x is a observatio from a distributio that has mea μ, ad stadard deviatio σ, the stadardized value of x is, Stadard Score, Percetile, Quartile z-score of x : Populatio z-score x μ x mea z = = σ stadard deviatio μ + 3σ has a z-score 3, sice it is 3 s.d. from mea. 9 3 Descriptive Stat - 5

15 Descriptive Statistics (Short Versio) If a distributio has a mea ad a s.d., the value 7 has a z-score.5. z-score = (7 )/ =.5. Sample z-score z = x x s Example: If the mea of a radom sample is 5 ad the stadard deviatio is, what would be the sample z-score of the value 6? x = 5, s =, x = 6.5 s.d z = = =.5 3 Example: Boe Mieral Desity DEXA BMD Values T score > -. S.D Defiitio Normal boe mieral desity The WHO Workig Group defies osteoporosis accordig to measuremets of boe mieral desity (BMD) usig dualeergy X-ray absorptiometry (DEXA). Thus osteoporosis is defied as a boe desity T score at or below.5 stadard deviatios (T score) below ormal peak values for youg adults. T score betwee. ad.5 SD T score < -.5 SD T score < -.5 SD with or more fragility fractures Osteopaeia Osteoporosis Severe osteoporosis These criteria were iitially established for the assessmet of osteoporosis i Caucasia wome. BMD reports may iclude a Z score which is the umber of stadard deviatios by which the subject of iterest differs from the mea for their age Quartiles: (Measure of Positio) The first quartile, Q, or 5 th percetile, is the media of the lower half of the list of ordered observatios. The third quartile, Q 3, or 75 th percetile, is the media of the upper half of the list of ordered observatios. Example: [odd umber of data values] ( = ),6,63,64,64,65,65,65,66,67,69,7,7,7,7,7,7,7,73,74,75 Q =? 64.5 Media = 69 Q 3 =? 7 Measure of spread: Iterquartile rage (IQR) = Q 3 Q IQR = = Descriptive Stat - 6

16 Descriptive Statistics (Short Versio) Example: [eve umber of data] ( = ) 6,,6,63,64,64,65,65,65,66,67,69,7,7,7,7,7,7,7,73,74,75 Q = 64? Media = 68? Q 3 =? 7 Measure of spread: Iterquartile rage (IQR) = Q 3 Q The five-umber summary.miimum value.q.media.q 3.Maximum value IQR = 7-64 = Example: (data sheet without outlier 6 ),6,63,64,64,65,65,65,66,67,69,7,7,7,7,7,7,7,73,74,75 Mi =, Q = 64.5, Media = 69, Q 3 = 7, Max = With 6 i the data: 6,,6,63,64,64,65,65,65,66,67,69,7,7,7,7,7,7,7,73,74,75 Q = 64 Media = 68 Q 3 = 7 IQR = 7-64 = N = HEIGHT 39 N = HEIGHT 4 Ier ad outer feces for outliers IQR = 7 64 = 8; Q = 64; Q 3 = 7 The ier feces are located at a distace of.5 IQR below Q (lower ier fece = Q -.5 x IQR ) ad at a distace of.5 IQR above Q 3 (upper ier fece = Q x IQR ). The outer feces are located at a distace of 3 IQR below Q (lower outer fece = Q 3 x IQR ) ad at a distace of 3 IQR above Q 3 (upper outer fece = Q x IQR ). The ier feces are located at a distace of.5 IQR below Q (lower ier fece = x 8 = 5 ) ad at a distace of.5 IQR above Q 3 (upper ier fece = x 8 = 84). The outer feces are located at a distace of 3 IQR below Q (lower outer fece = 64 3 x 8 = 4) ad at a distace of 3 IQR above Q 3 (upper outer fece = x 8 = 96). 4 4 Descriptive Stat - 7

17 Descriptive Statistics (Short Versio) 84 8 UIF: x 8 = 84 Ier fece 96 8 UOF:7 + 3 x 8 = 96 Outer fece Ier fece IQR IQR 5 4 LIF: x 8 = 5 Ier fece 4 4 LOF: 64-3 x 8 = 4 Ier fece Outer fece Q = 64; Q 3 = 7; IQR = 7 64 = 8 N = N = HEIGHT HEIGHT Mild ad Extreme outliers Side-by-side Box Plot Data values fallig betwee the ier ad outer feces are cosidered mild outliers. Data values fallig outside the outer feces are cosidered extreme outliers Whe outliers exist, the whisker exteded to the smallest ad largest data values withi the ier fece. HEIGHT N = Female Male 45 sex 46 Remarks: If the distributio of the data is symmetric, the the mea ad media will be about the same. The five-umber summary is best for o-symmetric data. The media, quartiles, iter-quartile rage are ot iflueced by outliers. The mea ad stadard deviatio are most appropriate to use oly if the data are symmetric because both of these measures are easily iflueced by outliers. Boxplot For the followig data: Fid the five-umber-summary & IRQ Make a boxplot Fid the th percetile Descriptive Stat - 8

18 Probability (Short Versio) Probability ad Coutig Rules A researcher claims that % of a large populatio have disease H. A radom sample of people is take from this populatio ad examied. If people i this radom sample have the disease, what does it mea? How likely would this happe if the researcher is right? Sample Space ad Probability Radom Experimet: (Probability Experimet) a experimet whose outcomes deped o chace. Sample Space (S): collectio of all possible outcomes i radom experimet. Evet (E): a collectio of outcomes of iterest i a radom experimet. Sample Space ad Evet Sample Space: S = {Head, Tail} S = {Life spa of a huma} = {x x, x R} Evet: E = {Head} E = {Life spa of a huma is less tha 3 years} 3 A Simple Example What s the probability of gettig a head o the toss of a sigle fair coi? Use a scale from (o way) to (sure thig). So toss a coi twice. Do it! Did you get oe head & oe tail? What s it all mea? 4 Defiitio of Probability A rough defiitio: (frequetist defiitio) Probability of a certai outcome to occur i a radom experimet is the proportio of times that the this outcome would occur i a very log series of repetitios of the radom experimet. Total Heads / Number of Tosses Number of Tosses 5 5 Determiig Probability How to determie probability? Empirical Probability Theoretical Probability (Subjective approach) 6 Probability -

19 Probability (Short Versio) Empirical Probability Assigmet Empirical study: (Do t kow if it is a balaced Coi?) Outcome Head Tail Total Frequecy Empirical Probability Assigmet Empirical probability assigmet: Number of times evet E occurs P(E) = Number of times experimet is repeated m = Probability of Head: P(Head) = 5 =.5 = 5.% 7 8 Empirical Probability Distributio Empirical study: Theoretical Probability Assigmet Make a reasoable assumptio: Outcome Frequecy Probability Head 5.5 Tail Total. Empirical Probability Distributio 9 What is the probability distributio i tossig a coi? Assumptio: We have a balaced coi! Theoretical Probability Assigmet Theoretical probability assigmet: Theoretical Probability Distributio (Model) Empirical study: Number of equally likely outcomes i evet E P(E) = Size of the sample space ( E) = ( S) Probability of Head: P(Head) = =.5 = % Outcome Probability Head. Tail. Total. Empirical Probability Distributio Probability -

20 Probability (Short Versio) Relative Frequecy ad Probability Distributios Number of times visited a doctor from a radom sample of 3 idividuals from a commuity Class Frequecy Relative Frequecy 54.8 P() = P() = P() = P(3) = P(4) = P(5) =. Total Discrete Distributio Relative Frequecy Distributio Relative Frequecy ad Probability Whe selectig oe idividual at radom from a populatio, the probability distributio ad the relative frequecy distributio are the same. 5 Probability for the Discrete Case If a idividual is radomly selected from this group 3, what is the probability that this perso visited doctor 3 times? P(3 times) = (4)/3 Class Frequecy Relative Frequecy Total 3. =.4 or 4% 6 Discrete Distributio If a idividual is radomly selected from this group 3, what is the probability that this perso visited doctor 4 or 5 times? Class Frequecy Total 3. P(4 or 5 times) = P(4) + P(5) Relative =.4 +. Frequecy =.5 It would be a empirical probability distributio, if the sample of 3 idividuals is utilized for uderstadig a large populatio. 7 Properties of Probability Probability is always a value betwee ad. Total probability (all outcomes together) equals. Probability of either oe of the disjoit evets A or B to occur is the sum of their idividual probabilities. P(A or B) = P(A) + P(B) 8 Probability - 3

21 Probability (Short Versio) Complemetatio Rule For ay evet E, P(E does ot occur) = P(E) Complemetatio Rule If a ubalaced coi has a probability of.7 to tur up Head each time tossig this coi. What is the probability of ot gettig a Head for a radom toss? Complemet of E = E * Some places use E c or E EP(E) P(E) E P(ot gettig Head) =.7 =.3 9 Complemetatio Rule Birthday Problem If the chace of a radomly selected idividual livig i commuity A to have disease H is., what is the probability that this perso does ot have disease H? P(havig disease H) =. P(ot havig disease H) = P(havig disease H) =. =.999 I a group of radomly select 3 people, what is the probability that at least two people have the same birth date? (Assume there are 365 days i a year.) P(at least two people have the same birth date) Too hard!!! = P(everybody has differet birth date) = [365x364x x(365-3+)] / S = {,, 3, 4, 5, 6} A = {,, 3} B = {3, 6} Itersectio of evets: A B <=> A ad B Example: A B = {3} Uio of evets: A B <=> A or B Example: A B = {,, 3, 6} A B S Ve Diagram (with elemets listed) 3 Ve Diagram (with couts) Give total of subjects 3 A A=Smokers, (A) = B=Lug Cacer, (B) = 5 5 B (A B) =?55 A B (A B) =? 45 Joit Evet 4 Probability - 4

22 Probability (Short Versio) Ve Diagram (with relative frequecies) Give a sample space.3. A. 5 B P(A B) =.55 Cotigecy Table Cacer, B No Cacer, B c Total Smoke, A 3 Not Smoke, A c ?.45 Ve Diagram A B A=Smokers, P(A) =. B=Lug Cacer, P(B) =.5 A B P(A B) =. Joit Evet 5 6 Coditioal Probability The coditioal probability of evet A to occur give evet B has occurred (or give the coditio B) is deoted as P(A B) ad is, if P(B) is ot zero, (E) = # of equally likely outcomes i E, P( A B) ( A B) P( A B) = or P( A B) = P( B) ( B) A B Coditioal Probability Smoke S Not Smoke S Cacer C 5 5 No Cacer C ( A B) P( A B) = ( B) Total 7 P(C S) = / =.4 P(C S ' ) = 5/ =. 8 Coditioal Probability Smoke S Not Smoke S Cacer C (.) 5 (.5) 5 P(C) =(.5) P(C S) =./.5 =.4 P(C S ' ) =.5/.5 =. No Cacer C 3 (.3) 45 (.45) 75 P(C)=(.75) P( A B) P( A B) = P( B) Total P(S) =(.5) P(S ) =(.5) (.) P(C S) What is P(C S ) =? 4 (Relative Risk ) 9 Idepedet Evets Evets A ad B are idepedet if P(A B) = P(A) or P(B A) = P(B) or P(A ad B) = P(A) P(B) 3 Probability - 5

23 Probability (Short Versio) Example If a balaced die is rolled twice, what is the probability of havig two 6 s? 6 = the evet of gettig a 6 o the st trial 6 = the evet of gettig a 6 o the d trial P(6 ) = /6, P(6 ) = /6, 6 ad 6 are idepedet evets P(6 ad 6 ) = P(6 ) P(6 ) = (/6)(/6) = /36 3 Idepedet Evets % of the people i a large populatio has disease H. If a radom sample of two subjects was selected from this populatio, what is the probability that both subjects have disease H? H i : Evet that the i-th radomly selected subject has disease H. P(H H ) = P(H ) [Evets are almost idepedet] P(H H ) =? P(H ) P(H ) =. x. =. 3 Idepedet Evets If evets A, A,, A k are idepedet, the P(A ad A ad ad A k ) = P(A ) P(A ) P(A k ) What is the probability of gettig all heads i tossig a balaced coi four times experimet? P(H ) P(H ) P(H 3 ) P(H 4 ) = (.5) 4 = Biomial Probability What is the probability of gettig two 6 s i castig a balaced die 5 times experimet? P(S S S S S ) = (/6) x (5/6) 3 =.6 P(S S S S S ) = (/6) x (5/6) 3 P(S S S S S ) = (/6) x (5/6) 3 5 5! How may of them? = =!3! Probability (two 6 s) =.6 x =.6 34 Multiplicatio Rule (Geeral Multiplicatio Rule) For ay two evets A ad B, P(A ad B) = P(A B) P(B) = P(B A) P(A) Multiplicatio Rule If i the populatio, % of the people smoked, ad 4% of the smokers have lug cacer, what percetage of the populatio that are smoker ad have lug cacer? P(A B) = P(B A) = P(A ad B) P(B) P(A ad B) P(A) 35 P(S) = % of the subjects smoked P(C S) = 4% of the smokers have cacer P(C ad S) = P(C S) P(S) =.4 x.5 =. 36 Probability - 6