Conditional Probabilities

Transcription

1 Lecture Outline BIOST 514/517 Biostatistics I / pplied Biostatistics I Kathleen Kerr, Ph.D. ssociate Professor of Biostatistics University of Washington Probability Diagnostic Testing Random variables: expectation and variance Normal distribution Some discrete distributions: Bernoulli, Binomial, Poisson Lecture 8: Probability, Diagnostic Testing, Random Variables, Parametric Distributions October 23, Probability Biostatisticians mostly use the long-run frequency definition of probability: If we do lots of coin tosses, half of them will come up heads. The long-run frequency of heads is ½, so heads)=1/2. There are other definitions of probability subjective probability definitions based on symmetry We care most about the properties of probability. Conditional Probabilities ll probabilities are conditional all probabilities depend on the context. heads)=0.5 depends on the coin being fair. If E is an event or proposition of interest, and I is information or data, then E I ) is the probability of E conditional on I or probability of E given I

2 Conditional Probability Example Role a fair die. E=get a 4 I=get an even number E Sample space: set of all possible outcomes E role a fair die)=1/6 E role a fair die and I)=1/3 Event: subset of outcomes of interest Probability of event E: 0 E) Combining Probability Example: Events or = ) + and and = B ) B ) and = B ) ) )= and + and not father s blood type is and a mother s blood type is B. Event: child has blood type Event: child has an O allele 7 8 2

3 Complement event c C S B Union event B [ or B] Intersection event B [ and B] [B] " or B" B "both and B" B S S Disjoint or Mutually Exclusive events [ B = ( empty set )] Example: father s blood type is and a mother s blood type is B. = first child has blood type B= first child has blood type B Example: and c are always mutually exclusive events 9 10 Let and B denote events and S denote the sample space Rule 1: 0 ) 1 Rule 2: S) = 1 Rule 3: ddition rule for disjoint events or = ) + Rule 4: Complement rule c ) = 1 ) General addition rule: What is or? " or B" B or #( or (# ) (# (# # S # S ) S 11 If events are disjoint, then = 0 and or = )

4 Conditional probability: What is? given B the probability of given B B = updated sample space original sample space S Equivalently: B Definition: Two events and B are independent if =) or, equivalently, if B ) =. Caution Independent and mutually exclusive sound similar in everyday language. But in probability they mean VERY different things. Interpretation: If two events do not influence each other (that is, if knowing one has occurred does not change the probability of the other occurring), the events are independent. Using the multiplicative rule, if two events and B are independent, then = )

5 S B Partitioning B B c Independence is harder to visualize Independence means that =) [area of in relation to S is the same as the area of B in relation to B] 17 ( P and or ( and B c c B ) B ) c ) Disjoint! 18 Bayes rule Not just for Bayesians! Bayes theorem/bayes rule and B ) ) not not The term controversial theorem sounds like an oxymoron, but Bayes' theorem has played this part for two-and-a-half centuries. Twice it has soared to scientific celebrity, twice it has crashed, and it is currently enjoying another boom. -Bradley Efron, Science 2013 Definition of conditional probability Partitioning

6 Diagnostic Testing We characterize the accuracy of a diagnostic test with the sensitivity and specificity Sensitivity of test: Positivity in diseased + D ) Specificity of test: Negativity in healthy (non-diseased) - H ) Predictive Values Neither sensitivity nor specificity directly answers the patient s question: I got a positive test result. m I diseased? Or: I got a negative test result. m I not diseased? Positive predictive value: Probability of disease when test is positive D + ) 21 Negative predictive value: Probability of healthy when test is negative H -) 22 Computing Predictive Values PPV: Relationship to Prevalence pply Bayes Rule to relate PPV and NPV to Sensitivity and Specificity Pr D Pr Pr DPrD DPrD Pr H PrH Need to know sensitivity, specificity, ND prevalence of disease Pr D Pr Pr DPrD DPrD Pr H PrH Pr H Pr Pr H PrH H PrH Pr DPrD PPV Sens Prev Sens Prev (1 Spec) (1 Prev)

7 NPV: Relationship to Prevalence Need to know sensitivity, specificity, ND prevalence of disease Ex: Syphilis and VDRL Typical study: Sample by disease Pr H Pr Pr H PrH H PrH Pr DPrD Sensitivity of test: Probability of positive in diseased 90% of subjects with syphilis test positive (ctually depends on duration of infection) Specificity of test: Probability of negative in healthy NPV Spec (1 Prev) Spec (1 Prev) (1 Sens) Prev 98% of subjects without syphilis test negative (ctually depends on age, and presence/absence of certain other diseases) Ex: PPV, NPV at STD Clinic Ex: 1000 patients at STD clinic Prevalence of syphilis 30% PPV: 95% with positive VDRL have syphilis (270/284) Ex: PPV, NPV in Marriage License Ex: Screening for marriage license Prevalence of syphilis 2% PPV: 47% with positive VDRL have syphilis (18/38) Syphilis Yes No Tot VDRL Pos Neg Total Syphilis Yes No Tot VDRL Pos Neg Total

8 Bottom Line The Positive and Negative Predictive Values are clinically important, but the Sensitivity and Specificity are more likely to be generalizable. The relationship between (Sensitivity, Specificity) and (PPV, NPV) depends heavily on the prevalence. When a disease or condition is rare, even a test with excellent sensitivity can have extremely poor PPV. Rare Disease Consider testing 10,000 people, ten of whom have the disease, using a test with 95% sensitivity and 95% specificity. 95% sensitivity means that we expect 9-10 of the people with the disease to test positive. 95% specificity means that about 9500 of the 9990 people without the disease will test negative. So roughly 500 people test positive, roughly 10 of them have the disease, so PPV 10/500=2% Thresholds The relationship between false positives and prevalence means that the same test may be used with different thresholds in different circumstances. For example, the Mantoux skin test for tuberculosis involves injecting TB antigen under the skin and measuring the resulting indurated area 2 3 days later. The recommended thresholds are 15mm induration in healthy people with no risk factors 10mm in people with occupational or other potential exposure 5mm in people for whom TB would be especially dangerous. E.g., HIV infection or immunosuppression identify a population for whom a false negative is relatively more serious, so a lower PPV can be tolerated in favor of a better NPV. higher threshold means a test with lower sensitivity and higher specificity

9 Random Variables Mathematical definition: a function from the sample space to the real numbers. Rule that assigns a numerical value to each point in the sample space Working definition: some measurement that has a distribution across subjects Denoted with a capital letter or a mnemonic X, Y,, GE, We usually use lower-case letters to represent the values a random variable can attain. We talk about events such as = a 1, > a 2, a 3 a 4 Random Variables: Examples Sampling space is our population Random variables examples: HEIGHT: Height CHOL: Cholesterol level BP: Blood pressure H=1 if person is hypertensive, 0 otherwise STGE: Stage of cancer Expectation The expected value of a random variable is the average (mean) of that variable over the population. For a discrete variable there is a simple formula: Example: roll two fair dice. Let X=the number of 6 s. The expected value of X is 0 25/ / /36=12/36=1/3 The expected number of 6 s we get when we roll two fair dice is 1/3. 35 Example: Experiment: Roll 2 fair dice. Sample space = { (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) } X=0 X=1 Random Variable: X=1 X: number of 6 that occur when rolling two fair dice. Takes values 0, 1 and 2 X X=x) 25/36 10/36 1/36 X=2 36 9

10 Expectation For continuous variables the concept is the same but the formula requires calculus. Properties of expectations: E[X+Y]=E[X]+E[Y] For a number c, E[cX] = c E[X] Variability We can also talk about the variance of a random variable. Let E[X]=μ. Then var[x]=e[x μ] 2. Properties of variance: var[cx] = c 2 var[x] Var[X+Y] = var[x]+var[y]+2cov[x,y] Only when X and Y are independent: Var[X+Y] = var[x]+var[y] Example We can use these formulas to work out the expected value of the mean for independent*, identically distributed X 1, X 2,, X n : Example We can use these formulas to work out the variance of the mean for independent, identically distributed X 1, X 2,, X n. *do not need to assume independence 39 Therefore, SD 40 10

11 Continuous Normal Discrete Bernoulli Binomial Poisson Some parametric distributions 41 Continuous Random Variables: Normal Notation: X ~ N(, ) Continuous quantitative random variable X is normally distributed if the density of its probability distribution is f ( x) 1 exp 2 The normal distribution corresponds to the familiar bell-shaped curve Symmetric around Expected Value: Variance is 2 and standard deviation is x

12 Standard Normal Random Variable Notation: Z ~ N(0, 1) Normal random variable with mean 0 and standard deviation 1. Standard Normal Random Variables 2 1 z f ( z) exp 2 2 Symmetric around 0 Expected Value: 1 Variance: 1, Standard Deviation: 1 (a standard Normal random variable is often denoted with Z) 45 Z ~ N(0,1) 46 Properties of Normal Random Variables Let Y= ax + b. If X ~ N (mean, standard deviation ), then Y ~ N( Y, Y ), where Y a b, Y a Select Discrete Random Variables Bernoulli Binomial Poisson Let W = ax + by. If X ~ N (, ) is independent of Y ~ N (, ) then W ~ N( W, W ), where W a b, W a b

13 Discrete Random Variable: Bernoulli Bernoulli: takes only two possible values Examples: Patient has a disease or not Patient responds to a treatment or not Patient dies or not Bernoulli random variable X is complete characterized by a single parameter p=x=1) Because X=0)=1-p 49 Discrete Random Variable: Binomial number of times that a particular binary event occurs in a sequence of n independent observations with the same probability p of occurrence Related to Bernoulli Example: Roll a fair die 60 times, event of interest is getting a 4. So 60 trials with probability p=1/6 of success each time. Let X be the total number of 4 s. X has a binomial distribution. Mathematically, X can be any number between 0 and 60, but E[X]=10 Example: n patients participate in a study and take aspirin; each patient is followed for 2 years on aspirin; the number of patients who experience a heart attack during the study period is a Binomial random variable. Number of trials n is known, interest is in p=heart attack) 50 BINOMIL(n,p) n=5 n=10 n=100 Probability Probability obab ty Probability Probability obab ty Probability p=0.2 p=0.5 p= Probability obab ty Discrete Random Variable: Poisson Poisson: number of events in a specific time period or area. Examples: number of telephone calls during business hours number of individuals with heat exhaustion at UWMC emergency room Poisson random variable is characterized by a single parameter, the rate, often denoted with the Greek letter lambda λ Poisson random variable can take on any non-negative integer value (0, 1, 2, 3,.). However, the most likely values are the values around λ

14 POISSON() =0.8 = =5 =15 Better description: Binomial or Poisson? X= Number of HPV infected women among UW female sophomores Y= number of bicycling accidents in Seattle per year Normal pproximations to some discrete variables.05 Poisson Some distributions that are not Normal (may not even be continuous) can be well approximated with a Normal distribution. Fraction X.12 Binomial Fraction X

15 Normal pproximations to some discrete variables Summary: Parametric Distributions 1. Binomial When both np and n(1-p) are large a Normal may be used to approximate the binomial. [ Rule of thumb: np>5, n(1-p) > 5] : X ~ Bin(n,p) E(X) = np V(X) = np(1-p) Bernoulli(p) E[X]=p Var[X]=p(1-p) Discrete Random Variable Continuous Normal(,) E[X]= Var[X]= 2 pproximation: X ~ N( np, np(1 p)) 2. Poisson When is large a Normal may be used to approximate the Poisson [ Rule of thumb: > 10]. : X ~ Po() E(X) = V(X) = Binomial(n,p) E[X] = np Var[X] = np(1-p) Poisson() E[X]= Var[X]= pproximation: X ~ N(, )