The Poisson Distribution and its Applications

The Poisson Distribution Nina Gunnes January 30, 2019 Lecture 11 1

Poisson distribution https://sco.m.wikipedia.org/wiki/File:Simeon_Poisson.jpg Named after the French mathematician Poisson (1781 1840) Commonly used to describe rare events in a large population Probability distribution: ? ? = ? =??? ? ?: number of cases ?: expected number of cases Expected value equal to the variance E ? = ? = Var ? for ? = 0,1,2, ?! Lecture 11 2

Poisson distribution, cont. Lecture 11 3

Poisson distribution, cont. Two approaches to the Poisson distribution As an approximation to the binomial distribution Through a Poisson process Lecture 11 4

Approximation to the binomial distribution Binomial distribution approximated by the Poisson distribution ?~Bin ?,? ?~Poi ? Expected number of events: E ? = ? = ?? Two key assumptions for the approximation to be valid Small probability of an event of interest ( success ): ? 0.05 Large number of trials: ? 50 Probability calculations easier with the Poisson distribution Quite demanding using the formula for the binomial distribution Lecture 11 5

Spina bifida (Aalen et al., 2006) Considering a binomial experiment of ? = 5000 births Probability of spina bifida: ? = Number of cases of spina bifida: ? Checking the assumptions for the Poisson approximation ? = 0.001 < 0.05 ? = 5000 > 50 Calculating the expected number of cases of spina bifida ? = ?? = 5000 0.001 = 5 1 1000 Lecture 11 6

Spina bifida (Aalen et al., 2006), cont. Calculating the probability of four cases of spina bifida ? ? = 4 =54? 5 4! = 0.175 = 17.5% Calculating the probability of five cases of spina bifida ? ? = 5 =55? 5 5! = 0.175 = 17.5% Calculating the probability of six cases of spina bifida ? ? = 6 =56? 5 6! = 0.146 = 14.6% Lecture 11 7

Poisson process Objects (or events) distributed randomly over a continuum Time Surface Volume Three fundamental assumptions Expected number of objects per unit (time, surface, volume) being constant Objects appearing randomly and independently of each other No two objects completely coincident in time/space Lecture 11 8

Poisson process, cont. Several examples in real life Childbirths at a maternity ward during a period of time Red blood cells in blood plasma Bacteria in water Implying a Poisson distribution Number of objects (events): ? Expected number of objects (events): ? ?~Poi ? www.publicdomainpictures.net/en/view-image.php?image=127965%26picture=bacteria Lecture 11 9

Anencephaly (Aalen et al., 2006) Considering anencephaly among children who are born A potential increase over time may have different causes Environmental pollutants Radioactivity Medicaments Using data on anencephaly in Edinburgh in 1956 1966 (Osborn, 1979) Number of cases per month 0 1 2 3 4 5 6 7 8 Total Observed number of months 18 42 34 18 11 6 0 2 1 132 Lecture 11 10

Anencephaly (Aalen et al., 2006), cont. Comparing the observed distribution to the Poisson distribution Number of cases per month: ?~Poi ? Estimating the expected number of cases per month ? = 0 18 + 1 42 + 2 34 + + 8 1 Calculating probabilities using the formula for the Poisson distribution ? ? = 0 = 1.970 ? 1.970! = 0.1395 ? ? = 1 = 1.971 ? 1.971! = 0.2748 132 = 260 132 = 1.97 Lecture 11 11

Anencephaly (Aalen et al., 2006), cont. Calculating the expected number of months with 0, 1, 2, etc. cases 0 cases: ? ? = 0 132 = 0.1395 132 = 18.4 1 case: ? ? = 1 132 = 0.2748 132 = 36.3 Number of cases per month 0 1 2 3 4 5 6 7 8 Observed number of months 18 42 34 18 11 6 0 2 1 Expected number of months 18.4 36.3 35.7 23.5 11.5 4.5 1.5 0.4 0.1 Good agreement between observed and expected numbers Lecture 11 12

Malformations (Aalen et al., 2006) Three cases of congenital malformations in B mlo within six months More than what would be expected Investigation carried out to find causes of the overrepresentation Concluded to be the result of coincidence Assuming a Poisson distribution for the number of cases: ?~Poi ? Approximate number of babies born in B mlo during six months: 80 Overall risk of malformations in Norway: 1.6 per 1000 births Expected number of cases in B mlo: ? = 1.6 1000 80 = 0.13 Lecture 11 13

Malformations (Aalen et al., 2006), cont. Calculating probabilities for three or more cases during six months ? ? = 3 = 0.133 ? 0.133! = 0.00032 ? ? = 4 = 0.134 ? 0.134! = 0.00001 ? ? 3 = 0.00032 + 0.00001 + 0.033% Considering 50 municipalities of the same size as B mlo over 5 years 500 six-month periods among all municipalities Probability of at least three cases during six months: 1 1 0.00033500= 0.15 = 15% (not so small anymore!) Lecture 11 14

Lymphoma (Aalen et al., 2006) Considering cases of lymphoma in Seascale in Great Britain Observing increased incidence 4 out of 411 children diagnosed before the age of 15 years Assuming a Poisson distribution for the cases Number of cases: ? Expected number of cases (based on data from around the district): ? = 0.25 ?~Poi ? Excess of cases due to the nuclear power plant in Windscale? Lecture 11 15

Lymphoma (Aalen et al., 2006), cont. Defining the hypotheses for a one-sided test Null hypothesis, ?0: ? = 0.25 One-sided alternative hypothesis, ??: ? > 0.25 Choosing a significance level of 1% Evidence must be compelling before rejecting the null hypothesis Calculating the p value ? ? 4|? = 0.25 0.254 ? 0.254! + Rejecting the null hypothesis and claiming increased risk of lymphoma 0.255 ? 0.255! = 0.00014 Lecture 11 16

Normal approximation to Poisson Poisson distribution resembling a normal distribution when ? is large Expected value (mean) of ?: E ? = ? Variance of ?: Var ? = ? Standard deviation of ?: SD ? = Using ? = 5 as a lower limit for the approximation The larger ?, the better the approximation Poor approximation when ? < 5 The two distributions coinciding when ? ? http://www.thescientificcartoonist.com/?paged=109 Lecture 11 17

Normal approximation to Poisson, cont. Lecture 11 18

Death in traffic (Aalen et al., 2006) Considering traffic deaths among people aged 15 24 years Approximately 120 persons killed in traffic each year for several years Assuming a Poisson distribution Number of persons killed: ? Expected number of persons killed: ? ?~Poi ? Imagining the number of persons killed being reduced to 90 one year Coincidence or an indication of a real risk reduction? Lecture 11 19

Death in traffic (Aalen et al., 2006), cont. Defining hypotheses for a two-sided test Null hypothesis, ?0: ? = 120 Two-sided alternative hypothesis, ??: ? 120 Choosing a significance level of 5% Defining the p value: ? ? 90 or ? 150|?0 Probability of a result at least as extreme as that observed in either direction Using the normal approximation under the null hypothesis ?~N 120, 120 Lecture 11 20

Death in traffic (Aalen et al., 2006), cont. Calculating the p value by standardizing ? ? 90 + ? ? 150 = 2 ? ? 90 = 2 ? 2 ? ? 2.74 = 2 0.0031 = 0.0062 = 0.62% Rejecting the null hypothesis based on the p value Real reduction in risk of being killed in traffic More exact result by using continuity correction Improving the normal approximation Calculating 2 ? ? 90.5 instead of 2 ? ? 90 ? 120 120 90 120 120 Lecture 11 21

Death in traffic (Aalen et al., 2006), cont. Calculating the interval covering 95% of the normal distribution 120 1.96 120 = 99,141 Lecture 11 22

Comparison of two Poisson variables Considering two independent Poisson distributed variables Numbers of cases: ?1and ?2 Expected values: ?1and ?2 Do the expected values differ? Null hypothesis, ?0: ?1= ?2 Test statistic: ? = ?1 ?2 Follows a standard normal distribution, i.e., ?~N 0,1 ?1+ ?2 Lecture 11 23

Traffic fatalities (Aalen et al., 2006) Considering traffic fatalities over two years First year: ?1= 338 Second year: ?2= 401 Expected values: ?1and ?2 Real change in risk or just a coincidence? Defining the hypothesis for a two-sided test Null hypothesis, ?0: ?1= ?2 Two-sided alternative hypothesis, ??: ?1 ?2 Lecture 11 24

Traffic fatalities (Aalen et al., 2006), cont. Choosing a significance level of 5% Calculating the observed value of the test statistic ?obs= 338 401 Calculating the p value 2 ? ? 2.32 = 2 0.0102 = 2% Rejecting the null hypothesis based on the p value The observed increase in traffic fatalities indicating a real change 338 + 401 = 2.32 Lecture 11 25

References Aalen OO, Frigessi A, Moger TA, Scheel I, Skovlund E, Veier d MB. 2006. Statistiske metoder i medisin og helsefag. Oslo: Gyldendal akademisk. Lecture 11 26