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lec9_jls - BE.104 Spring Biostatistics Poisson Analyses and...

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BE.104 Spring Biostatistics: Poisson Analyses and Power J. L. Sherley Outline 1) Poisson analyses 2) Power What is a Poisson process? Rare events Values are observational (yes or no) Random distributed over time or place Observations do not affect the frequency of future observations (independent) Much of the variance is due to statistical variation, sampling variation Many natural processes can be fit to a Poisson distribution Consider this case: Incidence for leukemia in MA in 1996: 680 cases distributed over 351 towns & cities Let us assume that 1) the cases are independent and randomly distributed 2) they are sufficiently infrequent as to not effect the total population ( >5 million) We can expect: 1) many towns & cities with no cases 2) many towns & cities with number of cases near the mean- “the expected number” # of cases: 680/351 2.0 per town & city 3) few towns & cities with a number of cases that greatly exceed the mean. 4) As the number of cases increases, the number of towns & cities with that number will approach 0.0 1
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Graphically- Poisson Distribution <Graph> If it were ideal: Properties- 1) the most probably number of events = 1 st integer < μ, the distribution mean, unless μ is an integer, in which case there are two equally probable maxima at μ and μ - 1 2) μ = σ 2 , the variance; therefore μ = σ So, μ is the parameter that completely defines the Poisson distribution What questions can be addressed with Poisson stats? 1) You are notified of a town in MA with 12 cases of leukemia. Is it significantly different than the mean for MA of 2? Is it unique, or could 12 be expected by chance? We calculate confidence intervals for μ , given an observed number of events = x , assuming a Poisson distribution: 95%CI for µ about x = x + 1.92 ± 1.960 x + 1.0 for x = 12, 95%CI = 6.8 to 21 Therefore, we have greater than 95% confidence that the Poisson distribution to which 12 events belongs is not equivalent to the Poisson distribution that has µ 2.0. If we conclude that 12 events is a part of a different Poisson distribution (i.e., it is not expected by chance), we will be wrong < 5% of the time. 2
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If our chance of error, in thinking that 12 is not expected by chance as a part of the Poisson distribution with µ 2.0, is > 5% then 2 will reside in the 95% CI about 12. We can then say that our error for saying that “ something is going on ” in the town with 12 leukemias is less than 5%. 99%CI for µ about x = x + 3.32 ± 2.567 x + 1.7 for x = 12, 99%CI = 5.79 to 24.9 Based on the same reasoning as for the 95% CI, we have >99% confidence that 12 did not occur by chance when the observed population mean, µ (i.e., the most expected value of x), is approximately 2. Poisson Probability Mass Function Given a known Poisson distribution, we can estimate the probability of an event occurring… Pr {X obs = x} = (e ) μ x x = 0, 1, 2,… x!
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lec9_jls - BE.104 Spring Biostatistics Poisson Analyses and...

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