Pairwise comparisons in the analysis of carcinogenicity data *

Transcription

1 Vol4, No1, (1) ealth Pawse compasos the aalyss of cacogecty ata Mohamma A Rahma 1#, Ram C Twa 1 Dvso of Bometcs-6, Offce of Bostatstcs, Cete fo Dug Evaluato a Reseach, Foo a Dug Amstato, Slve Spg, USA; # Coespog Autho: mohammaahma@fahhsgov, Atasemal@gmalcom Offce of Bostatstcs, Cete fo Dug Evaluato a Reseach, Foo a Dug Amstato, Slve Spg, USA Receve 5 Jue 1; evse 3 July 1; accepte 6 August 1 ABSTRACT Aalyss of cacogecty ata geeally volves a te test acoss all ose goups a a pawse compaso of the hgh ose goup wth the cotol The most commoly use test fo a postve te s the Cocha-Amtage test Ths test s asymptotcally omal Fo the pawse compaso of the hgh ose goup wth the cotol goup, we popose two mofcatos: the fst mofcato s to apply the test o the ata fom hgh ose a cotol goups afte oppg the ata fom the low a the meum ose goups; the seco mofcato s to ajust the test cotoal o ata fom all ose goups We compae the powe pefomace of these two mofcatos fo the pawse compasos Keywos: Cacogecty Stuy; Te Test; Pawse Test; Exact Test 1 INTRODUCTION Dsclame: Ths atcle eflects the vews of the authos a shoul ot be costue to epeset FDA s vews o polces The staa esg fo a log tem cacogecty stuy of a ew ug evelopmet clcal eseach clues thee teatmet goups of ceasg oses of the stuy ug (low, meum, a hgh) a oe uteate cotol The goup szes ae about 5 amals pe goup The statstcal aalyses clue a te test fo postve ose espose elatoshp tumo cece ates acoss all ose goups a pawse compasos of teate goups wth the cotol goup by oga/tumo combato The most commo test fo postve te s the Cocha-Amtage [CA] test, see eg Cocha [1] a Amtage [] Thee ae seveal extesos of the CA test see eg Taoe [3,4], oel a Yaagawa [5], a Tamua a Youg [6] amog othes Sce ffeece motaltes amog teatmet goups s a coce, thee ae vaous motalty ajuste tests suggeste by ffeet authos see eg Peto et al [7], Bale a Pote [8] Both of these motalty ajuste tests ca be appoache fom CA test The CA test s asymptotcally omal A exact test was popose by Mehta et al [9] Fo the pawse compaso of a teate goup eg hgh ose goup wth the cotol goup, both the asymptotc CA test a the exact test ca be mofe two ffeet ways The fst way s to op the ata fom the low a meum ose goups a apply the te tests to the emag ata fom the hgh ose goup a the cotol goup The seco way s to mofy the tests fo pawse compaso of the hgh ose goup a the cotol goup cotoal o the ata fom all ose goups We shall efe to these tests as ucotoal pawse test a cotoal pawse tests, espectvely The pupose of ths wok s to compae the powe pefomace of these two mofcatos of pawse tests It may be ote that a sgfcat te test may ot ecessaly cate oe of the pawse tests to be statstcally sgfcat (see Table 1) a also a o-sgfcat te test may ot ecessaly cate o pawse test to be sgfcat (see Table ) These tables show that t s mpotat to check the pawse tests afte sgfcat o o-sgfcat te test The est of the pape s ogaze as follows I Secto, we evew the CA a exact te tests a peset the mofcatos fo the pawse compasos I Table 1 Asymptotc a exact p-values showg sgfcat Te wth o-sgfcat pawse compasos at α = 5 Goup sze p-value Dose 1 3 Te Pa C #TBA 9 (Asymp) 6 (Exact) Goup sze p-value 78 (Asymp) 48 (Exact) Dose 1 3 Te Pa C #TBA 3 1 (Asymp) 15 (Exact) 4 (Asymp) 11 (Exact) Copyght 1 ScRes

2 M A Rahma, R C Twa / ealth 4 (1) Table Asymptotc a exact p-values showg o-sgfcat te wth sgfcat pawse compaso at α = 5 Goup sze p-value Dose 1 3 Te Pa LC #TBA (Asymp) 44 (Exact) Goup sze p-value 38 (Asymp) 117 (Exact) Dose 1 3 Te Pa C #TBA (Asymp) 3846 (Exact) 11 (Asymp) 8 (Exact) Secto 3, we llustate the applcato of the two mofe pawse tests o a ataset, a cay out a smulato stuy Secto 4 to evaluate the powe pefomaces I Secto 5, we make some coclug emaks TEORETICAL DEVELOPMENT Cose a cacogecty stuy wth + 1 ose goups cosstg of oe cotol a teate goups Let be the umbe of amals assge to the th teatmet goup, x be the umbe of tumo beag amals obseve the th teatmet goup, a be the ose level fo the th teatmet goup, wth = fo cotol goup Assume that x has a Bomal stbuto as x B, p, whee p s the pobablty of evelopg tumo by a amal the th ose goup The value of p s geeally moele as p ab wth xexpx 1 expx, the logstc stbuto The value p a wth = coespos to the cotol goup ee, a s a usace paamete a b s the paamete of teest 1 Test fo Postve Te The postve te s teste by the hypothess : b vesus the alteatve hypothess : b 1, o equvaletly by testg : p p1 p p, vs 1 :p p +1 fo all wth stct equalty fo at least oe The value of p s the oveall pobablty of evelopg tumo by a amal o oveall popoto of tumo beag amals the populato ue ull hypothess It ca be easly show that the pa, x s jotly suffcet, whee s the total umbe of subjects o test 1 goups The CA test s base o the suffcet statstc The CA test fo testg the ull hypothess that thee s o te, : b, vesus the alteatve hypothess,, s gve by : b 1 x Tte p 1 p 1 N (, 1), whee a I applcatos, we eplace p x by, wth x x The CA test s a asymptotcally omal The exact test, as eve by Mehta et al, s as follows: Let x x~ : x~ x,, x, x l x be the sample space whch s the collecto of all pemutatos of, x x such that x x, the obseve total umbe of tumo beag amals Defe the ctcal ego fo te test: te ~ : 1 1 obs,te C t x x x x t, x x whee tobs,te x x statstc Tte X s the ealzato of the, base o the obseve ata x,, x Usg the hype geometc stbuto, the pobablty of each ealzato of l l ~,, s x x x x x Ths pobablty s also kow as the table pobablty, sgfyg the pobablty of each table the all possble pemutato of the obseve umbe of tumo beag amals The exact p-value fo testg (ght ha tal) s the PT te tobs,te x, x PC x tte x, xc xte x Pawse Compasos Sce the hghest ose fo a egulatoy cacogecty stuy s selecte mostly base o the maxmum toleate ose (MTD) cteo, the pawse compasos betwee the hgh ose goup wth the cotol goup has specal egulatoy teest I ths pape we peset some esults elate to pawse compasos of hgh ose goup wth the cotol goup The esults, howeve, ca be use fo the pawse compaso of ay teatmet goup wth cotol If we wee teeste testg smultaeous multple cotasts, such as Wllams type cotast, the appoach escbe otho et al [1] ca be use These methos ae base o the quatles of multvaate omal stbuto takg cae of the coelato to accout as the package MVTNORM Fo pawse compaso of the hghest ose Goup Copyght 1 ScRes

3 91 M A Rahma, R C Twa / ealth 4 (1) wth cotol wth the ull hypothess : p p, a the alteatve hypothess 1 : p p, we escbe the followg two appoaches Fst ote that b, a V p V b p 1 p I ou fst appoach, we elete the ata fom all ose goups except ata fom ose goups a, estmate oveall popoto of tumo beag amals as x,, a efe the test statstc as, T pawse1 p V p p,,, 1 (1) ucotoal test oweve, ue the ull hypothess of o ose effect, a bette estmate of vaace of p ca be obtae fom the complete ata e 1 p p V p p o whee ow p s estmate as all x, base o ata fom all ose goups Usg ths estmate the eomato, ou seco appoach s to efe the test statstc as p Tpawse () all 1 all o We wll efe to ths test as the cotoal test It shoul be ote that ue the leaty assumpto of p wth (the eomato of) the above test s same as the Cocha-Amtage te test 3 Asymptotc Relatve Effcecy of the Cotoal a the Ucotoal Pawse Tests As metoe, the evato of the above test, the vaace of p s estmate base o the ata fom The asymptotc elatve effcecy (ARE) of T pawse Goup a cotol oly We wll efe to ths test as the a T s ARE T pawse1 pawse pawse 33 Va T pawse, Tpawse1 Va T showg that T pawse s asymptotcally moe effcet tha T pawse1 otho a Betz [11] popose (asymptotc) tests fo postve te base o sgle a multple cotasts ue the assumpto of equally space ose-levels Fo sgle cotast, test s efe as T B p, c 1 p, c, whee c Fo pawse compaso of Goup a cotol ( = ) wth c = 1 a c = 1, ths test statstc s T pawseb p p p,, The p, s estmate by p,, as efe eale If the goup szes ae equal (e f = 1 = = 3 ) the t ca be show that the statstcs T pawse1 a T pawseb ae etcal 4 Exact Pawse Test We ow cose the evato of the exact pawse tests Followg Mehta et al, the exact p-value fo ucotoal exact test T exact,pawse1 base o the ata fom the Goup a Cotol, fo testg (3), s cal- Copyght 1 ScRes

4 M A Rahma, R C Twa / ealth 4 (1) culate as pt t x x, x x xcx x ( tpawse1) j x x exact,pawse1 obs,pawse1 The exact test fo ou seco appoach s as follows As befoe, let : (,, ), x x x x x x x, a efe the ctcal ego C t x : x x t, x exact,pawse ~ x obs,pawse whee tobs, pawse x x s the ealzato of TExact, Pawse x x base o the obseve ata x, x The exact p-value fo testg s calculate as pt t x, exact,pawse obs,pawse xcx t pawse l l x We wll efe to ths test as the cotoal exact pawse test Poceeg alog the les of Mehta et al, the powe of pawse test s calculate as follows Let t x be such that PT exact,pawse t x x, The the powe of the pawse test cotoal o x + s x pt t x, l x l l p 1 p l l p 1 p l x x t x l l l x x l x p 1 l pl x x t x exact,pawse obs,pawase 1 l x l l p X X x, 1 p x l l l whee p x p p xtx x l x 1 l l l The above powe ca be evaluate fo exact test usg the hype geometc stbuto a appopate ctcal egos ue cotoal a ucotoal stuatos We compae the elatve powe of thee asymptotc pawse tests, as well as that of the two exact tests T Exact, pawse1 a T Exact, pawse Fo evaluato of the powe fuctos, we pefome smulatos a calculate the pecetage of tmes the ull hypothess was ejecte whe the alteatve hypothess was tue The SAS poc Statfy a SAS poc Multtest [1] ae vey coveet fo the calculato of these exact pobabltes 3 EXAMPLE Cose a cacogecty stuy wth fou teatmet goups amely, cotol, low, meum, a hgh ose goups each wth 5 amals, a ose scoes, 1,, a 3, espectvely Suppose we obseve a total of 1 amals evelope a ceta tumo type wth,, 3 a 5 tumo beag amals cotol, low, meum, a hgh ose goups, espectvely We woul lke to pefom a pawse compaso of the hgh ose goup wth the cotol The ull hypothess : p3 p agast alteatve 1: p3 p The esults usg the omal appoxmato test ae T pawse1 = 94, T pawse = 418, a T pawseb = 94 wth coespog p-values as 19, 78, a 19, espectvely Fo exact test we have t obs,pawse1 = t obs,pawse = x + x 3 3 = 15 Table 3, gve below, shows all possble values of T pawse1 alog wth the table pobabltes a the ght tal pobabltes fo pawse compaso of hgh ose wth cotol calculate fom ata afte oppg low a meum ose goups usg SAS poc Statfy Table 4 gve below shows all possble values of T pawse alog wth the table pobabltes a the ght tal pobabltes fo pawse compaso of hgh ose wth cotol calculate fom all ata usg the scoes,, a 3 SAS poc Multtest The esults fom Tables 3 a 4 show that both the table- a ght-tal pobabltes fo the two pawse exact Table 3 Pa compaso of cotol wth hgh ose goup afte eletg the low a meum ose goups Table Rt a T pawse1 Pobablty Tal (obseve p < 5) Table 4 Pawse compaso of hgh ose wth cotol usg all ata Table Rt a T pawse Pobablty Tal (obseve p > 5) Copyght 1 ScRes

5 914 M A Rahma, R C Twa / ealth 4 (1) tests may go ethe ecto Fo example fo the obseve umbe of,, 3 a 5 tumo beag amals, we have t obs,pawse1 = t obs,pawse = x + x 3 3 = 15, a the p-value afte eletg the low a meum ose goups s p pawse1 = 81, a that usg ata fom all ose goups s p pawse = 79 e the p-value afte eletg the low a meum ose goups s smalle tha the p-value usg ata fom all ose goups O the othe ha f the obseves umbe of tumo beag amals wee,, 3, a 3, the t obs,pawse1 = t obs,pawse = x + x 3 3 = 9 The p-value afte eletg the low a meum ose goups woul be p pawse1 = 5, a that usg ata fom all ose goups woul be p pawse = 4763 I ths case the p-value fo pawse exact test afte eletg the low a meum ose goups woul be lage tha the p-value fo the pawse exact test usg the ata fom all ose goups 4 SIMULATION STUDY OF POWER CALCULATION Cose a cacogecty stuy wth fou teatmet goups amely, cotol, low, meum, a hgh ose goups each wth 5 amals, a ose scoes, 1,, a 3, espectvely The powe was calculate fo ffeet choces backgou cece ate the cotol goup (p ) The cece ate fo the hgh ose goup (p 3 ) was the chose by a ceta cemet (δ) ove p The cece ate fo the low ose goup (p 1 ) a that fo meum ose goup (p ) wee calculate usg a logstc moel as follows: a If a log p 1 p p3 p3 p p log 1 log 1 b, the ab1 ab e e p1 a p ab1, 1 ab e 1 e wth = a =, 1,, a 3 The values of the powe wee calculate by fg the pecetages of tmes the ull hypothess was ejecte whe the alteatve was tue a smulato wth 1 loops Table 5 shows the calculate powe usg the asymptotc omal appoxmato a Fgue 1 gves the gaphcal epesetato of the esults Table 6 shows the calculate powe usg the exact test a Fgue gves the gaphcal epesetato of the esults The smulato esults show that asymptotc omal test T pawse s always a moe poweful compae to T pawse1 o T pawseb The two tests T pawse1 a T pawseb have smla powe (as sample szes ae take to be same) The 3 pawse exact test usg ata fom all ose goups has moe powe compae to test base o ata eletg the two mle ose goups fo small values of p a δ Table 5 Powe calculate usg the omal appoxmato fo ucotoal, cotoal, a otho Betz tests Tumo ates cotol goup Delta = p 3 p Powe of Powe of ucotoal test cotoal test Powe of B test Copyght 1 ScRes

6 M A Rahma, R C Twa / ealth 4 (1) Whee : powe of ucotoal test a : powe of cotoal test Fgue 1 Gaphcal epesetato of powe vs elta fo gve p usg omal appoxmato fo pawse compaso of cotol a hgh ose goup Copyght 1 ScRes

7 916 M A Rahma, R C Twa / ealth 4 (1) whee : Powe of ucotoal test a : Powe of cotoal test Fgue Gaphcal epesetato of powe vs elta fo gve p usg exact test fo pawse compaso of cotol a hgh ose goup Copyght 1 ScRes

8 M A Rahma, R C Twa / ealth 4 (1) Table 6 Powe calculate usg the exact test fo ucotoal a cotoal tests Tumo ates cotol goup Delta = p 3 p Powe of ucotoal test Powe of cotoal test CONCLUSIONS I ths pape, we scusse the topc of pawse compaso of the hgh ose goup wth cotol a typcal cacogecty stuy We popose two tests poceue, oe base o ata oly fom the two ose goups to be compae a oe base o ata fom all ose goups We elaboate both exact a omal appoxmato veso of ou popose tests Though a smulato, we compae the powe pefomaces of these tests Fo the compaso of hgh ose goup wth cotol goup a typcal fou ose goup cacogecty stuy, the smulato esults showe that the powe of the asymptotc omal test usg ata fom all ose goups s asymptotcally moe effcet a hece s always moe poweful tha that of the test usg ata fom hgh a cotol goups oly Fo exact test, ethe of the two tests showe ufomly bette powe tha the othe The pawse exact test usg ata fom all ose goups showe moe powe tha that of the test base o ata eletg the two mle ose goups fo tumo types wth low backgou ate a/o ug wth small cacogec effect, whle the pawse exact test usg ata fom all ose goups showe less powe tha that of the test base o ata eletg the two mle ose goups fo tumo types wth hgh backgou ate a/o ug wth lage cacogec effect oweve, sce a test that ops pat of the ata s asymptotcally less effcet, we ecomme that fo the pawse compaso oe uses tests that use the ata fom all ose goups 6 ACKNOWLEDGEMENTS The authos ae eeply ebte to Ds Stella G Machao a Kal K L, Dvso of Bometcs-6, US Foo a Dug Amstato, fo the helpful avces a commets to mpove a complete ths wok REFERENCES [1] Cocha, WG (1954) Some methos of stegtheg the commo χ tests Bometcs, 1, o:137/31616 [] Amtage, P (1955) Tests fo lea te popotos a fequeces Bometcs, 11, o:137/31775 [3] Taoe, RE (1975) Test fo te lfe table aalyss Bometka, 6, o:1193/bomet/63679 [4] Taoe, RE (198) The use of hstocal cotol fomato testg fo a te Posso meas Bometcs, 38, o:137/53459 [5] oel, DG a Yaagawa, T (1986) Icopoatg hstocal cotols testg fo a te popotos Joual of the Ameca Statstcal Assocato, 81, o:118/ Whe p a/o δ become (s) lage the pawse exact test that eletes the ata fom two mle ose goups showe bette powe [6] Tamua, RN a Youg, SS (1986) The copoato Copyght 1 ScRes

9 918 M A Rahma, R C Twa / ealth 4 (1) of hstocal cotol fomato tests a popotos: Smulato stuy of Taoe s Poceue Bometcs, 4, o:137/53154 [7] Peto, R, Pke, MC, Day, NE, Gay, RGK, Lee, PN Pash, S, Peto, J, Rchas, S a Waheof, J (198) Gueles fo sample sestve sgfcace test fo cacogec effects log-tem amal expemets IARC Moogaphs o the Evaluato of the Cacogec Rsk of Chemcals to umas, Suppl, Log-Tem a Shot- Tem Sceeg Assays fo Cacoges: A Ctcal Appasal, IARC, Lyo, [8] Bale, AJ a Pote, CJ (1988) Effects of teatmetuce motalty a tumo-uce motalty o tests fo cacogecty small samples Bometcs, 44, o:137/ [9] Mehta, CR a Patel, NR a Sechauhu, P (1998) Exact powe a sample-sze computatos fo the Cocha-Amtage te test Bometcs, 54, o:137/ [1] otho, LA, Sll, M a Schaaschmt, F (1) Evaluato of cece ates pe-clcal stues usg a Wllams-type poceue The Iteatoal Joual of Bostatstcs, 6, o:1/ [11] otho, LA a Betz, F () Evaluato of amal cacogecty stues: Cocha-amtage te test vs multple cotast tests Bometcal Joual, 4, o:11/ (9)4:5<553::aid-bimj55 3>3CO;-R [1] SAS Isttute Ic, (9) Multtest poceue SAS use s gue 9 Eto, SAS Isttute Ic, Cay, Copyght 1 ScRes