Tests for the Ratio of Two Poisson Rates
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1 Chpter 437 Tests for the Rtio of Two Poisson Rtes 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 which is the men number of occurrences during the specified intervl. The procedure documented in this chpter clcultes the power or smple size for testing whether the rtio of two Poisson mens is different from specified vlue (usully one). The test procedure is described in Gu et l. (8). Test Procedure Assume tht ll subjects in ech group re observed for fixed time period nd the number of events, X, (outcomes or defects) is recorded. The following tble presents the vrious terms tht re used. Group Fixed time intervl t t Smple Size N N Number of events X X Individul event rtes λ λ Distribution of X Poisson ( λ t ) Poisson ( λ t ) Define the rtio of event rtes,, s λ λ Gu (8) considered severl test sttistics tht cn be used to test hypotheses bout the rtio. For exmple, or equivlently, λ H : λ λ versus : >. λ H : versus : >. 437-
2 where is the rtio of event rtes under the null hypothesis. Two test sttistics re vilble in this cse. The first is bsed on unconstrined mximum likelihood estimtes with d t. N / tn W X X X + X d d The second test is bsed on constrined mximum likelihood estimtes W X X d d ( X + X ) An equivlent pir of test sttistics re vilble if logrithms re used. The sttisticl hypothesis is or equivlently, H λ ln ln( ) λ : λ λ : versus ln ln( ) > H ln( ) ln( ) versus : ln( ) ln( ) > : Two test sttistics re vilble in this cse s well. The first is bsed on unconstrined mximum likelihood estimtes W 3 X ln X ln d + X X The second test is bsed on constrined mximum likelihood estimtes W 4 X ln ln X d d + + d X + X 437-
3 After extensive simultion, they recommend the following extension of the vrince-stbilized test proposed by Huffmn (984) for the cse when / d. W 5 > X d + d ( X + 3 8) Gu et l. (8) show tht ll of these test sttistics re pproximtely distributed s stndrd norml nd thus use the norml distribution s the bsis of significnce testing nd power nlysis. Assumptions The ssumptions of the two-smple Poisson test re:. The dt in ech group re counts (discrete) tht follow the Poisson distribution.. Ech smple is simple rndom smple from its popultion. Unlike most designs, in this design the smple size involves fixed time prmeter. Tht is, insted of specifying the number of people in study, the number of mn-hours is wht is importnt. Hence, smple size of hours could be chieved by ten people being observed for one hour or two people being observed for five hours. Technicl Detils Computing Power If we define s the rtio of event rtes under the lterntive hypothesis t which the power is clculted, the power nlysis for testing the hypothesis H : versus : >. using the test sttistics defined bove is completed s follows.. Find the criticl vlue. Choose the criticl vlue z α using the stndrd norml distribution so tht the probbility of rejecting H when it is true is α.. Compute the power. Compute the power for ech test s follows. For W, W 3, nd W 4, the power is given by where Power( W ) i z Φ ασ i µ i σ i Φ z ( z) Norml(, ) µ d tnλ, σ + t Nλ d d d 437-3
4 µ 3 ln, σ 3 d + t N λ µ 4 ln, σ 4 d + + d t N + λ d For W the power is computed using where Power( W Ez Φ ) α F G E +, d G + d F λtn, d For W5 the power is computed using where A Power( W Φ 5) B z D α C, B λ tn + 3 8, A C + d, D + d Computing Smple Size The smple size is found using the formul N z αc + zpowerd A λ t
5 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. Design Tb 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 solved for from the other prmeters. Select Power when you wnt to clculte the power of n experiment. Select Smple Size when you wnt to clculte the smple size needed to chieve given power nd lph level. Select t nd t (Exposure Time with t t) to clculte the required exposure time for set number of subjects to chieve desired power. The solution is found for the cse where t t. Select (Rtio of Event Rtes) to find the detectble rtio for given power nd smple size. Test Alterntive Hypothesis Specify whether the lterntive hypothesis of the test is one-sided or two-sided. If one-sided test is chosen, the hypothesis test direction is chosen bsed on whether the ssumed event rte rtio () is greter thn or less thn the null hypothesized event rtio (). The options re One-Sided The upper nd lower null nd lterntive hypotheses re Upper: H : versus : > Lower: H : versus : < Two-Sided The null nd lterntive hypotheses re H : versus : Test Sttistic This option llows you to select which test sttistic will be used. Possible choices re W MLE This is the unconstrined mximum likelihood test in which λ nd λ re estimted seprtely. W CMLE This is the constrined mximum likelihood test in which λ nd λ re estimted jointly
6 W3 Ln(MLE) This is the logrithmic version of the unconstrined mximum likelihood test in which λ nd λ re estimted seprtely. W4 Ln(CMLE) This is the logrithmic version of the constrined mximum likelihood test λ nd λ re estimted jointly. W5 Vrince Stbilized This is the vrince stbilized version of the unconstrined mximum likelihood test. This is the defult vlue. 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.8 (bet.) ws used for power. Now,.9 (bet.) is lso commonly used. A single vlue my be entered here or rnge of vlues such s.8 to.95 by.5 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.5 is recommended for lph. You my enter rnge of vlues such s.5.5. or.5 to.5 by.5. Smple Size t (Exposure Time of Group ) Enter vlue (or rnge of vlues) for the fixed exposure (observtion) time for group. Ech subject in group is observed for this mount of time. If the exposure times re vrible, this is the verge exposure time per subject in group. The vlue must be greter thn. A single vlue my be entered here or rnge of vlues such s.8 to. by.5 my be entered. t (Exposure Time of Group ) Enter vlue (or rnge of vlues) for the fixed exposure (observtion) time for group. Ech subject in group is observed for this mount of time. If the exposure times re vrible, this is the verge exposure time per subject in group. The vlue must be greter thn. A single vlue my be entered here or rnge of vlues such s.8 to. by.5 my be entered
7 Smple Size (When Solving for Smple Size) Group Alloction Select the option tht describes the constrints on N or N or both. The options re Equl (N N) This selection is used when you wish to hve equl smple sizes in ech group. Since you re solving for both smple sizes t once, no dditionl smple size prmeters need to be entered. Enter N, solve for N Select this option when you wish to fix N t some vlue (or vlues), nd then solve only for N. Plese note tht for some vlues of N, there my not be vlue of N tht is lrge enough to obtin the desired power. Enter N, solve for N Select this option when you wish to fix N t some vlue (or vlues), nd then solve only for N. Plese note tht for some vlues of N, there my not be vlue of N tht is lrge enough to obtin the desired power. Enter R N/N, solve for N nd N For this choice, you set vlue for the rtio of N to N, nd then PASS determines the needed N nd N, with this rtio, to obtin the desired power. An equivlent representtion of the rtio, R, is N R * N. Enter percentge in Group, solve for N nd N For this choice, you set vlue for the percentge of the totl smple size tht is in Group, nd then PASS determines the needed N nd N with this percentge to obtin the desired power. N (Smple Size, Group ) This option is displyed if Group Alloction Enter N, solve for N N is the number of items or individuls smpled from the Group popultion. N must be. You cn enter single vlue or series of vlues. N (Smple Size, Group ) This option is displyed if Group Alloction Enter N, solve for N N is the number of items or individuls smpled from the Group popultion. N must be. You cn enter single vlue or series of vlues. R (Group Smple Size Rtio) This option is displyed only if Group Alloction Enter R N/N, solve for N nd N. R is the rtio of N to N. Tht is, R N / N. Use this vlue to fix the rtio of N to N while solving for N nd N. Only smple size combintions with this rtio re considered. N is relted to N by the formul: N [R N], where the vlue [Y] is the next integer Y
8 For exmple, setting R. results in Group smple size tht is double the smple size in Group (e.g., N nd N, or N 5 nd N ). R must be greter thn. If R <, then N will be less thn N; if R >, then N will be greter thn N. You cn enter single or series of vlues. Percent in Group This option is displyed only if Group Alloction Enter percentge in Group, solve for N nd N. Use this vlue to fix the percentge of the totl smple size llocted to Group while solving for N nd N. Only smple size combintions with this Group percentge re considered. Smll vritions from the specified percentge my occur due to the discrete nture of smple sizes. The Percent in Group must be greter thn nd less thn. You cn enter single or series of vlues. Smple Size (When Not Solving for Smple Size) Group Alloction Select the option tht describes how individuls in the study will be llocted to Group nd to Group. The options re Equl (N N) This selection is used when you wish to hve equl smple sizes in ech group. A single per group smple size will be entered. Enter N nd N individully This choice permits you to enter different vlues for N nd N. Enter N nd R, where N R * N Choose this option to specify vlue (or vlues) for N, nd obtin N s rtio (multiple) of N. Enter totl smple size nd percentge in Group Choose this option to specify vlue (or vlues) for the totl smple size (N), obtin N s percentge of N, nd then N s N - N. Smple Size Per Group This option is displyed only if Group Alloction Equl (N N). The Smple Size Per Group is the number of items or individuls smpled from ech of the Group nd Group popultions. Since the smple sizes re the sme in ech group, this vlue is the vlue for N, nd lso the vlue for N. The Smple Size Per Group must be. You cn enter single vlue or series of vlues. N (Smple Size, Group ) This option is displyed if Group Alloction Enter N nd N individully or Enter N nd R, where N R * N. N is the number of items or individuls smpled from the Group popultion. N must be. You cn enter single vlue or series of vlues
9 N (Smple Size, Group ) This option is displyed only if Group Alloction Enter N nd N individully. N is the number of items or individuls smpled from the Group popultion. N must be. You cn enter single vlue or series of vlues. R (Group Smple Size Rtio) This option is displyed only if Group Alloction Enter N nd R, where N R * N. R is the rtio of N to N. Tht is, R N/N Use this vlue to obtin N s multiple (or proportion) of N. N is clculted from N using the formul: where the vlue [Y] is the next integer Y. N[R x N], For exmple, setting R. results in Group smple size tht is double the smple size in Group. R must be greter thn. If R <, then N will be less thn N; if R >, then N will be greter thn N. You cn enter single vlue or series of vlues. Totl Smple Size (N) This option is displyed only if Group Alloction Enter totl smple size nd percentge in Group. This is the totl smple size, or the sum of the two group smple sizes. This vlue, long with the percentge of the totl smple size in Group, implicitly defines N nd N. The totl smple size must be greter thn one, but prcticlly, must be greter thn 3, since ech group smple size needs to be t lest. You cn enter single vlue or series of vlues. Percent in Group This option is displyed only if Group Alloction Enter totl smple size nd percentge in Group. This vlue fixes the percentge of the totl smple size llocted to Group. Smll vritions from the specified percentge my occur due to the discrete nture of smple sizes. The Percent in Group must be greter thn nd less thn. You cn enter single vlue or series of vlues. Effect Size λ (Event Rte of Group ) Enter vlue (or rnge of vlues) for the men event rte per time unit in group (control). Exmple of Estimting λ If ptients were exposed for yer (i.e. t yer) nd 4 experienced the event of interest, then the men event rte would be λ 4/(*). per ptient-yer If ptients were exposed for yers (i.e. t yers) nd 4 experienced the event of interest, then the men event rte would be λ 4/(*). per ptient-yer 437-9
10 Event Rte Rtio λ is used with λ to clculte the event rte rtio s λ/λ such tht λ λ/ The rnge of cceptble vlues is λ >. You cn enter single vlue such s or series of vlues such s to by.5. (Rtio of Event Rtes under H) Enter vlue (or rnge of vlues) for the rtio of the two men event rtes ssumed by the null hypothesis, H. Usully,. which implies tht the two rtes re equl (i.e. λ λ). However, you my test other vlues of s well. One-Sided The upper nd lower null nd lterntive hypotheses re Upper: H : versus : > Lower: H : versus : < Two-Sided The null nd lterntive hypotheses re H : versus : Enter λ or Rtio for Group Indicte whether to enter the group event rte (λ) or the event rte rtio () to specify the effect size. The event rte rtio is clculted from λ nd λ s λ/λ (Rtio of Event Rtes under H) This option is displyed only if Enter λ or Rtio for Group (Rtio of Event Rtes under H). This is the vlue of the rtio of the two event rtes, λ nd λ, t which the power is to be clculted. The event rte rtio is clculted from λ nd λ s λ/λ The rnge of cceptble vlues is > nd. You cn enter single vlue such s.5 or series of vlues such s.5 to. by.5. (Rtio of Event Rtes under H) [when solving for (Rtio of Event Rtes)] Specify whether to solve for event rte rtios tht re less thn one or greter thn. 437-
11 λ (Event Rte of Group under H) This option is displyed only if Enter λ or Rtio for Group λ (Event Rte of Group under H). Enter vlue (or rnge of vlues) for the men event rte per time unit in group (tretment) under H, the lterntive hypothesis. Exmple of Estimting λ If ptients were exposed for yer (i.e. t yer) nd 4 experienced the event of interest, then the men event rte would be λ 4/(*). per ptient-yer If ptients were exposed for yers (i.e. t yers) nd 4 experienced the event of interest, then the men event rte would be λ 4/(*). per ptient-yer Event Rte Rtio λ is used with λ to clculte the event rte rtio s λ/λ such tht λ *λ The rnge of cceptble vlues is λ >. You cn enter single vlue such s or series of vlues such s to by
12 Exmple Finding the Smple Size We will use the exmple of Gu (8) in which epidemiologist wish to exmine the reltionship of postmenopusl hormone use nd coronry hert disese (CHD). The incidence rte for those not using the hormone is.5 ( λ.5). How lrge of smple is needed to detect chnge in the incidence rtio from to, 3, 4, 5, or 6. Assume tht 9% power is required nd α.5. Assume tht ech subject will be observed for two yers nd tht the design clls for n equl number of subjects in both groups. 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. You my then mke the pproprite entries s listed below, or open Exmple by going to the File menu nd choosing Open Exmple Templte. Option Vlue Design Tb Solve For... Smple Size Alterntive Hypothesis... One-Sided Test Sttistic... W5 Vrince Stbilized Power....9 Alph....5 t (Exposure Time of Group )... t (Exposure Time of Group )... Group Alloction... Equl (N N) λ (Group Event Rte)....5 (Rtio of Event Rtes under H)... Enter λ or Rtio for Group... (Rtio of Event Rtes under H) (Rtio of Event Rtes under H) Reports Tb Decimls for Event Rtes
13 Annotted Output Click the Clculte button to perform the clcultions nd generte the following output. Numeric Results Numeric Results for Testing the Rtio of Two Poisson Rtes Alterntive Hypothesis: One-Sided (H: vs. H: > ) Test Sttistic: W5 Vrince Stbilized Grp Grp Grp Grp Alt Null Expos Expos Event Event Rte Rte Time Time Rte Rte Rtio Rtio Power N N N t t λ λ Alph References Gu, K., Ng, H.K.T., Tng, M.L., nd Schucny, W. 8. 'Testing the Rtio of Two Poisson Rtes.' Biometricl Journl, 5,, Huffmn, Michel 'An Improved Approximte Two-Smple Poisson Test.' Applied Sttistics, 33,, 4-6. Report Definitions Power is the probbility of rejecting the null hypothesis when it is flse. It should be close to one. N nd N re the number of subjects in groups nd, respectively. N is the totl smple size. N N + N. t nd t re the exposure (observtion) times in groups nd, respectively. λ is the men event rte per time unit in group (control). This is often the bseline event rte. λ is the men event rte per time unit in group (tretment) under the lterntive hypothesis. is the rtio of the two event rtes under the lterntive hypothesis. λ/λ. is the rtio of the two event rtes under the null hypothesis. Alph is the probbility of rejecting the null hypothesis when it is true. It should be smll. Summry Sttements For one-sided test of the null hypothesis H:. vs. the lterntive H: >. using the W5 Vrince Stbilized test sttistic, smples of 9737 subjects in group with exposure time of. nd 9737 subjects in group with exposure time of. chieve 9.% power to detect n event rte rtio () of. when the event rte in group (λ) is.5 nd the significnce level (lph) is.5. 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. Power Power is the probbility of rejecting the null hypothesis when it is flse. It should be close to one. N nd N N nd N re the number of subjects in groups nd, respectively. To clculte the smple size in terms of person-time, you would multiply the smple size by the exposure time, t i. For exmple, the group smple size in the first row clculted in person-yers is 9737 x person-yers. N N is the totl smple size. N N + N. t nd t t nd t re the exposure (observtion) times in groups nd, respectively. λ λ is the men event rte per time unit in group (control). This is often the bseline event rte
14 λ λ is the men event rte per time unit in group (tretment) under the lterntive hypothesis. is the rtio of the two event rtes under the lterntive hypothesis. λ/λ. is the rtio of the two event rtes under the null hypothesis. Alph Alph is the probbility of rejecting the null hypothesis when it is true. It should be smll. Plots Section This plot shows the reltionship between group smple size nd
15 Exmple Vlidtion using Gu (8) Gu et l. (8) present n exmple tht we will use to vlidte this procedure. Using the scenrio cited in Exmple bove, they give smple size clcultion on pge 95. In this exmple, λ.5, ρ, ρ 4, ( A) t t, α.5, R.5, nd power.9. In their Tble 6, they list the smple size for p5 in this scenrio s 867. However, this number is inccurte becuse of the two-deciml plce rounding tht ws done during their clcultion. In privte communiction, they greed tht the more ccurte number is 859. 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. You my then mke the pproprite entries s listed below, or open Exmple by going to the File menu nd choosing Open Exmple Templte. Option Vlue Design Tb Solve For... Smple Size Alterntive Hypothesis... One-Sided Test Sttistic... W5 Vrince Stbilized Power....9 Alph....5 t (Exposure Time of Group )... t (Exposure Time of Group )... Group Alloction... Enter R N/N, solve for N nd N R....5 λ (Group Event Rte)....5 (Rtio of Event Rtes under H)... Enter λ or Rtio for Group... (Rtio of Event Rtes under H) (Rtio of Event Rtes under H)... 4 Reports Tb Decimls for Event Rtes... 4 Output Click the Clculte button to perform the clcultions nd generte the following output. Numeric Results Numeric Results for Testing the Rtio of Two Poisson Rtes Alterntive Hypothesis: One-Sided (H: vs. H: > ) Test Sttistic: W5 Vrince Stbilized Grp Grp Grp Grp Alt Null Expos Expos Event Event Rte Rte Time Time Rte Rte Rtio Rtio Power N N N R t t λ λ Alph These results mtch the more ccurte vlue of
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