Tests for One Poisson Mean
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1 Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution is often used to fit count dt, such s the number of defects on n item, the number of ccidents t n intersection during yer, the number of clls to cll center during n hour, or the number of meteors seen in the evening sky during n hour. The Poisson distribution is chrcterized by single prmeter, λ, which is the men number of occurrences during the intervl. This procedure clcultes the power or smple size for testing whether λ is less thn or greter thn specified vlue. This test is usully clled the test of the Poisson men. The test is described in Ostle (1988) nd the power clcultion is given in Guenther (1977). Test Procedure Assume tht the men is λ 0. To test H : λ λ vs. 0 0 H : λ > λ0, you would tke the following steps. 1. Find the criticl vlue. Choose the criticl vlue X so tht the probbility of rejecting H 0 when it is true is equl to α. This is done by solving the following inequlity for X. x= X ( nλ ) nλ0 0 e α. x! Note tht becuse X is n integer, equlity will seldom occur. Therefore, the minimum vlue of for which the inequlity holds. 2. Select smple of n items compute the totl number of events X = of H. x n x i i= 1 The test in the other direction ( H : λ λ vs. 0 0 H : λ < λ0 ) is computed similrly.. If X is found X > X reject H0 in fvor 412-1
2 Assumptions The ssumptions of the one-smple Poisson test re: 1. The dt re counts (discrete) tht follow the Poisson distribution. 2. The smple is simple rndom smple from its popultion. Ech individul in the popultion hs n equl probbility of being selected in the smple. Limittions There re few limittions when using these tests. As long s the ssumption tht the men occurrence rte is constnt is met, the test is vlid. Technicl Detils Computing Power The power is computing for specific lterntive vlue λ1 using the following formul. Power = 1 β = x= X e nλ 1 ( nλ ) 1 x! x Computing Smple Size Following Guenther (1977), the smple size, n, is found by incresing the vlue of d in the following expression until the left-hnd end-point is less thn the right-hnd end-point nd the intervl contins t lest one integer. Χ 2 2 2d ;1 Χ β 2 ; n d α, d = 1,2,3, λ 2λ Here Χ 2 v;p is percentge point of the chi-squre distribution with v degrees of freedom. Procedure Options This section describes the options tht re specific to this procedure. These re locted on the Design tb. For more informtion bout the options of other tbs, go to the Procedure Window chpter. The Design tb contins most of the prmeters nd options tht you will be concerned with. Solve For Solve For This option specifies the prmeter to be clculted from the vlues of the other prmeters. Under most conditions, you would select either Power or Smple Size
3 Select Smple Size when you wnt to determine the smple size needed to chieve given power nd lph error level. Select Power when you wnt to clculte the power of n experiment tht hs lredy been run. Test H (Alterntive Hypothesis) This option specifies the lterntive hypothesis. This implicitly specifies the direction of the hypothesis test. The null hypothesis is lwys H : λ = λ Possible selections for the lterntive hypothesis re: 1. H : λ1 λ0. This option yields one-tiled t test. 2. H : λ1 λ0. This option yields one-tiled t test. Power nd Alph Power This option specifies one or more vlues for power. Power is the probbility of rejecting flse null hypothesis, nd is equl to one minus bet. Bet is the probbility of type-ii error, which occurs when flse null hypothesis is not rejected. Vlues must be between zero nd one. Historiclly, the vlue of 0.80 (bet = 0.20) ws used for power. Now, 0.90 (bet = 0.10) is lso commonly used. A single vlue my be entered here or rnge of vlues such s 0.8 to 0.95 by 0.05 my be entered. Alph This option specifies one or more vlues for the probbility of type-i error. A type-i error occurs when true null hypothesis is rejected. Vlues must be between zero nd one. For one-sided tests such s this, the vlue of is recommended for lph. You my enter rnge of vlues such s or to 0.05 by Smple Size n (Smple Size) This option specifies one or more vlues of the smple size, the number of individuls in the study. This vlue must be n integer greter thn one. Note tht you my enter list of vlues using the syntx 50,100,150,200,250 or 50 to 250 by 50. Effect Size Mens λ0 (Null or Bseline Men) This option specifies one or more vlues of the men occurrence rte corresponding to the null hypothesis. This vlue must be greter thn zero. λ1 (Alterntive Men) This option specifies one or more vlues of the men occurrence rte corresponding to the lterntive hypothesis. This vlue must be greter thn zero
4 Exmple 1 Power fter Study This exmple demonstrtes how to clculte the power for specific vlues of the other prmeters. Suppose tht ccidents hve occurred t n intersection t n verge rte of 1 per month for the lst severl yers. Recently, distrction hs been constructed ner the intersection tht ppers to hve incresed the ccident rte. Suppose the smple sizes re 12 nd 24 months nd lph is Wht is the power to detect lterntives of 1.1, 1.5, 2.0, nd 2.5? Setup This section presents the vlues of ech of the prmeters needed to run this exmple. First, from the PASS Home window, lod the procedure window by expnding Mens, then One Men, then clicking on Test (Inequlity), nd then clicking on. You my then mke the pproprite entries s listed below, or open Exmple 1 by going to the File menu nd choosing Open Exmple Templte. Option Vlue Solve For... Power H (Alterntive Hypothesis)... H: λ0 < λ1 Alph n (Smple Size) λ0 (Null or Bseline) λ1 (Alterntive) Annotted Output Click the Clculte button to perform the clcultions nd generte the following output. Numeric Results Numeric Results for One-Smple Poisson Test Null Hypothesis: λ0 = λ1 Alterntive Hypothesis: λ0 < λ1 Trget Actul Diff Effect Power n Alph Alph λ0 λ1 (λ0-λ1) Size Bet References Guenther, Willim C Smpling Inspection in Sttisticl Qulity Control. Griffin's Sttisticl Monogrphs. Mcmilln, NY. Pges Ostle, B. nd Mlone, L Sttistics in Reserch, 4th Edition. Iow Stte University Press. Iow. Pges
5 Report Definitions Power is the probbility of rejecting flse null hypothesis. It should be close to one. n is the size of the smple drwn from the popultion. To conserve resources, it should be smll. Alph is the probbility of rejecting true null hypothesis. It should be smll. Diff is the vlue of λ0 - λ1, the difference being tested. λ0 is the vlue of the popultion men under the null hypothesis. λ1 is the vlue of the popultion men under the lterntive hypothesis. Effect Size is the vlue of (λ0 - λ1) / (λ1). Bet is the probbility of ccepting flse null hypothesis. It should be smll. Summry Sttements A smple size of 12 chieves 5% power to detect difference of between the null hypothesis men of 1.00 nd the lterntive hypothesis men of 1.10 nd with significnce level (lph) of using one-sided one-smple Poisson test. This report shows the vlues of ech of the prmeters, one scenrio per row. The vlues of power nd bet were clculted from the other prmeters. Note tht the ctul power chieved is greter thn the trget power. Similrly, the ctul lph is less thn the trget lph. These differences occur becuse only integer vlues of the count vrible occur. Plots Section 412-5
6 These plots show the reltionship between smple size nd power for vrious vlues of the lterntive men nd the smple size
7 Exmple 2 Finding the Smple Size This exmple will extend Exmple 1 to the cse in which we wnt to find the necessry smple size to chieve t lest 90% power. This is done s follows. Setup This section presents the vlues of ech of the prmeters needed to run this exmple. First, from the PASS Home window, lod the procedure window by expnding Mens, then One Men, then clicking on Test (Inequlity), nd then clicking on. You my then mke the pproprite entries s listed below, or open Exmple 2 by going to the File menu nd choosing Open Exmple Templte. Option Vlue Solve For... Smple Size H (Alterntive Hypothesis)... H: λ0 < λ1 Power Alph λ0 (Null or Bseline) λ1 (Alterntive) Annotted Output Click the Clculte button to perform the clcultions nd generte the following output. Numeric Results Numeric Results for One-Smple Poisson Test Null Hypothesis: λ0 = λ1 Alterntive Hypothesis: λ0 < λ1 Trget Actul Diff Effect Power n Alph Alph λ0 λ1 (λ0-λ1) Size Bet This report shows the smple sizes tht re necessry to chieve the required power
8 Exmple 3 Finding the Minimum Detectble Difference Continuing with the previous exmple, suppose only 10 months of dt re vilble. Wht is the minimum detectble difference tht cn be detected by this design? Setup This section presents the vlues of ech of the prmeters needed to run this exmple. First, from the PASS Home window, lod the procedure window by expnding Mens, then One Men, then clicking on Test (Inequlity), nd then clicking on. You my then mke the pproprite entries s listed below, or open Exmple 3 by going to the File menu nd choosing Open Exmple Templte. Option Vlue Solve For... λ1 H (Alterntive Hypothesis)... H: λ0 < λ1 Power Alph n (Smple Size) λ0 (Null or Bseline) Output Click the Clculte button to perform the clcultions nd generte the following output. Numeric Results Numeric Results for One-Smple Poisson Test Null Hypothesis: λ0 = λ1 Alterntive Hypothesis: λ0 < λ1 Trget Actul Diff Effect Power n Alph Alph λ0 λ1 (λ0-λ1) Size Bet This report shows tht the minimum detectble difference is =
9 Exmple 4 Vlidtion using Guenther Guenter (1977) pge 27 gives n exmple in which λ0 = 0.05, λ1 =.2, α = 0.05, β = 0.10, nd n = 47. We will now run this exmple. Setup This section presents the vlues of ech of the prmeters needed to run this exmple. First, from the PASS Home window, lod the procedure window by expnding Mens, then One Men, then clicking on Test (Inequlity), nd then clicking on. You my then mke the pproprite entries s listed below, or open Exmple 4 by going to the File menu nd choosing Open Exmple Templte. Option Vlue Solve For... Smple Size H (Alterntive Hypothesis)... H: λ0 < λ1 Power Alph λ0 (Null or Bseline) λ1 (Alterntive) Output Click the Clculte button to perform the clcultions nd generte the following output. Numeric Results Numeric Results for One-Smple Poisson Test Null Hypothesis: λ0 = λ1 Alterntive Hypothesis: λ0 < λ1 Trget Actul Diff Effect Power n Alph Alph λ0 λ1 (λ0-λ1) Size Bet Note tht the vlue of n is indeed
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