Table of contentsOutlineThe Poisson distributionPoisson approximation to the binomialPerson-time analysisExact testsTime-to-event modelingLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingLecture 27Brian CaffoDepartment of BiostatisticsJohns Hopkins Bloomberg School of Public HealthJohns Hopkins UniversityDecember 19, 2007Lecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingTable of contents1 Table of contents2 Outline3 The Poisson distribution4 Poisson approximation to the binomial5 Person-time analysis6 Exact tests7 Time-to-event modelingLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingOutline1 Poisson distribution2 Tests of hypothesis for a single Poisson mean3 Comparing multiple Poisson means4 Likelihood equivalence with exponential modelLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingPump failure dataPump 1 2 3 4 5Failures 5 1 5 14 3Time 94.32 15.72 62.88 125.76 5.24Pump 6 7 8 9 10Failures 19 1 1 4 22Time 31.44 1.05 1.05 2.10 10.48From Casella and Robert, Monte Carlo Statistical Methods;first editionLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingThe Poisson distribution•Used to model counts•The Poisson mass function isP(X = x; λ) =λxe−λx!for x = 0, 1, . . .•The mean of this distribution is λ•The variance of this distribution is λ•Notice that x ranges from 0 to ∞Lecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingSome uses for the Poissondistribution•Modeling event/time data•Modeling radioactive decay•Modeling survival data•Modeling unbounded count data•Modeling contingency tables•Approximating binomials when n is large and p is smallLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingDefinition•λ is the mean number of events per unit time•Let h be very small•Suppose we assume that•Prob. of an event in an interval of length h is λh while theprob. of more than one event is negligible•Whether or not an event occurs in one small interval doesnot impact whether or not an event occurs in anothersmall intervalthen, the number of events per unit time is Poisson withmean λLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingPoisson approximation to thebinomial•When n is large and p is small the Poisson distribution isan accurate approximation to the binomial distribution•Notation•λ = np•X ∼ Binomial(n, p), λ = np and•n gets large•p gets small•λ stays constantLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingProofRice Mathematical Statistics and Data Analysis page 41P(X = k) =n!k!(n − k)!pk(1 −p)n−k=n!k!(n − k)!λnk1 −λnn−k=n!(n − k)!nk×1 −λn−k×λkk!1 −λnn→ 1 ×1 ×λkk!e−λ=λkk!e−λLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingNotes•That (1 −λ/n)nconverges to e−λfor large n is a very oldmathematical fact•We can show thatn!(n−k)!nkgoes to one easily becausen!(n − k)!nk= 1×1 −1n×1 −2n×. . .×1 −k − 1neach term goes to 1Lecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingExamplesSome example uses of the Poisson distribution•deaths per day in a city•homicides witnessed in a year•teen pregnancies per month•Medicare claims per day•cases of a disease per year•cars passing an intersection in a day•telephone calls received by a switchboard in an hourLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingSome results•If X ∼ Poisson(tλ) thenX − tλ√tλ=X − MeanSDconverges to a standard normal as tλ → ∞•Hence(X − tλ)2tλ=(O − E )2Econverges to a Chi-squared with 1 degree of freedom forlarge tλ•If X ∼ Poisson(tλ) then X /t is the ML estimate of λLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingProof•Likelihood is(tλ)xe−tλx!•So that the log-likelihood isx log(λ) − tλ + constants in λ•The derivative of the log likelihood isx/λ − t•Setting equal to 0 we get thatˆλ = x/tLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingPump failure data•Failures for Pump 1: 5, monitoring time: 94.32 days•Estimate of λ, the mean number of failures per day= 5/94.32 = .053•Test the hypothesis that the mean number of failures perday is larger than the industry standard, .15 events perday: H0: λ = .15 versus Ha: λ > .15•TS = (5 − 94.32 × .15)/√94.2 ×.15 = −2.433•Hence P-value is very large (.99)•HW: Obtain a confidence interval for λLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingPump failure data•Exact P-value can be obtained by using the Poissondistribution directly•P(X ≥ 5) where X ∼ Poisson(.15 × 94.32)ppois(5, .15 * 94.32, lower.tail = FALSE) = .995very little evidence to suggest that this pump ismalfunctioning•To obtain a P-value for a two-sided alternative, double thesmaller of the two one sided P-valuesLecture 27Brian CaffoTable ofcontentsOutlineThe PoissondistributionPoissonapproximationto thebinomialPerson-timeanalysisExact testsTime-to-eventmodelingPump 1 2 3 4 5Failures 5 1 5 14 3Time 94.32 15.72 62.88 125.76 5.24ˆλ .053 .064 .080 .111 .573P-value .995