Julian J. Faraway Department of Statistics, University of Michigan, Ann Arbor, Michigan 48109, USA. Phone: (734)

Transcription

1 Ditributio of the Admixture tet for the Detectio of Likage uder Heterogeeity Julia J. Faraway Departmet of Statitic, Uiverity of Michiga, A Arbor, Michiga 48109, USA. Phoe: (734) Ruig Title: Admixture Tet Abtract The admixture tet for the detectio of likage uder heterogeeity i coidered. We how that the ull ditributio of thi tet tatitic ha half it weight cocetrated o zero ad the other half o a complicated ditributio that ca be approximated by max(x 1,X 2 ) where X 1 ad X 2 are idepedet χ 2 1 variable. We alo give exact critical value for mall ample ad how that the power of thi tet to detect likage i geerally greater tha the tadard tet that aume homogeeity. 1

2 1 INTRODUCTION We coider the detectio of likage whe likage heterogeeity exit, that i whe oly a fractio of ibhip may be liked to a give geetic marker. Smith(63) itroduced the admixture model baed o the recombiatio fractio ad the proportio of liked familie. Ott(83,85) ad Rich(88) coider tet for heterogeeity baed o thi model, wherea Hodge et. al.(83) ad Rich(89) coider tet for likage baed o thi ame model. The latter i dicued here. Martiez & Goldi (89) dicu ample ize eeded for uch tet. The tet for likage i oe-ided ice recombiatio fractio greater tha oe half make o biological ee ad hould it etimated value be greater tha oe half, oe would ot take thi a evidece of likage. Hece, the true ull ditributio of the admixture tatitic ha half it weight cocetrated at zero ad the other half o ome other ditributio which i the ubject of our iteret here. Becaue of the ymmetry of the problem, it coveiet ad otatioally impler to jut compute the ull ditributio for the two-ided tet tatitic to dicover the aforemetioed ditributio. Bear i mid that, although we hall be cocered with the two-ided tet tatitic i what follow, the true ull ditributio i a above. Hodge claimed that the aymptotic (a the umber of ibhip become large) ull ditributio of the (two-ided) admixture tet tatitic wa χ 2 1 but Rich cojectured it wa χ2 2. We claim here that either i correct ad that the true aymptotic ditributio i quite complicated but ca be adequately approximated by the max(x 1,X 2 ) where X 1 ad X 2 are idepedet χ 2 1 variable. Thi ditributio lie omewhere betwee the two previou claim ad thu thi reult i of more tha jut techical iteret give the popularity of the tet. Ghoh & Se(85) tudy the aymptotic ditributio of the likelihood ratio tet tatitic for a mixture model that i imilar to the oe here ad obtaied a reult imilar i form to our. 2 DISTRIBUTION OF THE TEST STATISTIC Let the recombiatio fractio be θ, the proportio of liked ibhip be α ad the ibhip ize be. Let the umber of ibhip be ad let X i be the umber of recombiat gamete out of for ibhip i. Thu the likelihood for thi et of ibhip would be L(θ,α) = i=1 [ αθ X i (1 θ) X i + (1 α)(1/2) ] Note that if we map X i X i (producig a outcome that ha equal probability uder the hypothei of o likage) ad θ 1 θ the the likelihood tay the ame. Thi ymmetry allow u to coider the two-ided tet tatitic i our computatio of the actual oe-ided admixture tet. If we wih to tet for likage, the atural ull ad alterative hypothee are H 0 : θ = 1/2 H A : θ < 1/2 ad the maximum likelihood-ratio tet tatitic i T = 2log(L(ˆθ, ˆα)/L(1/2, α)) 2

3 where ˆθ ad ˆα are the maximum likelihood etimate(m.l.e) uder the alterative hypothei ad α i the m.l.e. uder the ull. Note that whe θ = 1/2, α i uidetifiable i.e ay value of α produce the ame likelihood o the actual value of α i immaterial, although thi uidetifiability i the ource of the difficulty i determiig the ditributio of T. We ue atural log here for tatitical coveiece; lod core will be dicued later. So T = 2max T () = 2max i=1 log [ α(2 θ X i (1 θ) X i 1) + 1 ] where 0 α 1, 0 θ 1 Ufortuately, the aymptotic ditributio uder the ull i ot imply χ 2 1 a it would be if the uual theory were applicable. Thi i becaue a regularity coditio regardig the idetifiability of the parameter i ot atified; ee Wald(1949). Thi mea the aymptotic ditributio of T mut be derived by other mea. We give a heuritic jutificatio of our reult ad verify it by imulatio. Sice α i uidetifiable at the ull, the likelihood will be rather flat i the α directio ad ice the rage of α i retricted, the value of α maximizig T will ted to occur at the boudary of the rage of α for large. To ee thi, expad T i θ about 1/2, with α bouded away from 0, T () 8α i /2)(θ i=1(x 1 2 ) + 4α[(1 α) (2X i ) 2 ](θ 1 i=1 2 )2 (where mea approximately) Maximizig over θ give max θ T () 4α[ i=1 (X i /2)] 2 4(1 α) i=1 (X i /2) 2 ( ) Let Z = i=1 (X i /2) ad S 2 = i=1 (X i /2) 2 ad ow differetiatig with repect to α d dα max θ T () 8Z2 (4S 2 ) ( 4S 2 (1 α)) 2 which will be poitive or egative depedig o whether S 2 i le or more tha /4, idepedet of the value of α o for ufficietly large T will be maximized at α = 1 or for α mall (T = 0 whe α = 0). Sice S 2 /4 a, both cae will be roughly equally likely. So we coider the ditributio of max θ T () for α = 1 ad for α mall. Whe α = 1, max θ T (1,θ) 4 [ i=1 (X i /2)] 2 uig ( ). Sice EX i = /2 ad Var X i = /4, max θ T (1,θ) i aymptotically χ 2 1, jut applyig the cetral limit theorem. However, whe α i mall the ditributio of T i ot o clear: Write k j = {umber of X i = j} for j = 0,1,... the T = 2max i=0 i=0 k i log [ α(2 θ i (1 θ) i 1) + 1 ] Now ice α i mall, we ca expad log i term of α (log(1 + x) x x 2 /2): [ T 2max k i α(2 θ i (1 θ) i 1) 1 ] 2 α2 (2 θ i (1 θ) i 1) 2 3

4 ad maximizig thi over α give T max θ [ i=0 k i (2 θ i (1 θ) i 1) ] 2 i=0 k i [2 θ i (1 θ) i 1] 2 Now uder the ull θ = 1/2, ad o Ek i = ( i) 2 o replacig k i by it expectatio ad the by applyig the biomial theorem, we ee that the umerator i approximately i=0 ( i ) 2 [ 2 ˆθ i (1 ˆθ) i 1 ] 2 = [ (θ 2 + (1 θ) 2 ) 2 1 ] where Hece c i (θ) = T max θ [ ] 2 k i c i (θ) i=0 (2 θ i (1 θ) i 1) [(θ 2 + (1 θ) 2 ) 2 1] The ditributio of thi caot be explicitly tated for geeral, but give that k i i aymptotically ormal a, i=0 k ic i (θ) i aymptotically a weighted um of ormal ad i hece ormal for give θ. Thi might ugget a χ 2 1 a a poible approximatio ad imulatio how that thi i ideed a good fit but it hould be emphaized that thi i ot the exact ditributio. Now whe T i maximized for α mall, T i a weighted um of the k i ad whe maximized for α = 1, T i a fuctio of the ample mea, o the maximizig value at thee two poit will be ted to be idepedet epecially for large. Thi ugget a ditributio for T a the maximum of two idepedetly ditributed χ 2 1 variable. Agai, thi i ot a exact reult but imulatio idicate that it i a good approximatio. The true aymptotic ditributio a fuctio of the maximum of a particular Gauia proce but ice thi caot be explicitly calculated, the uggeted approximatio will be of more practical utility. To check the validity of thi uggeted approximatig ditributio, coider the followig imulatio reult: With ibhip ize et to 3 ad the umber of ibhip et to 100, 100,000 dataet were geerated with the true θ = 1/2. The likelihood wa maximized by firt traformig α ad θ to a logit cale (x log(x/(1 x))) o that the cotrait o α ad θ ca be removed ad the uig the Nelder-Mead implex method decribed i Pre et al. (1988) to fid the maximum. The maximum at α = 1 ad α mall a idicated i the dicuio above wa alo calculated. The quatile of the uggeted ditributio of T may be imply calculated by otig that P(T < q) = P(χ 2 1 < q) 2 I figure 1, we how a quatile-quatile plot of the imulated tet-tatitic agait it claimed ditributio. We have coverted to a lod cale ad focued oly o the upper tail (the larget 284 obervatio) of the ditributio ice thi i the area of mot iteret ad the fit i good for the ret of the ditributio ayway. 4

5 Figure 1 Theoretical Simulated Figure 1: A Q-Q plot howig the upper tail of the ditributio of the imulated ad theoretical T o a lod cale The agreemet betwee the imulated ad theoretical ditributio i good. Ay divergece from a perfect match ca reaoably be attributed to imulatio amplig error ad that for fiite, T i dicrete. Similar reult have bee oberved for other mall value of ad the fit improve a get larger. 3 CRITICAL VALUES Recall that if ˆθ 0.5 we have o evidece for likage, otherwie we ca determie the igificace of the oberved T by referrig to the approximate ull ditributio that we have calculated. If lod core are preferred, oe would ue T = 2log(10)T If the ame level of tet i deired a for a χ 2 1 ditributed tatitic, lod core of 2,3 ad 5 correpod to core for T of 2.28,3.28 ad 5.27 repectively. (Compare the value give by Rich(89) of 2.62,3.70 ad 5.80 repectively). Thi reult i aymptotic i ature ad may ot be good for the mall ample ued i practice. It hould be oted that it i computatioally feaible to calculate exact critical value for mall ample. To guaratee at leat the ame level of tet correpodig to uig a lod core of 3 a a criterio (a igificace level of approximately %), the ull hypothei hould be rejected whe T exceed the critical value give i the followig table: 5

6 Table I ibhip Number of familie ize Table I: Critical lod core for the admixture tatitic correpodig to a omial lod core of 3 No etry (-) idicate that the maximum poible value of T ha a probability exceedig the tated igificace level ad o uder thee coditio there i iufficiet data - the ull hypothei will ever be rejected. The critical value fluctuate but approach the expected 3.28 a get larger. Note that there i oe value le tha 3, which may eem odd, but remember thee are exact value, ad thi happe to be the critical value correpodig to the tated level of igificace. 4 POWER Rich(89) compared the power of the heterogeeity tet agait the tadard tet where homogeeity i aumed (α = 1), ad cocluded that the homogeou tet wa more powerful i mot circumtace. Cotrary to thi, we demotrate here, by uig the correct critical value ad computig the power exactly, that the heterogeou tet i geerally preferable. Exact critical value for a igificace level correpodig to a lod of 3 for both tet were computed ad the exact power to detect likage wa calculated for a rage of value of α from 0 to 1 ad of θ from 0 to 1/2. Figure 2 how the power of the heterogeou tet miu that of the homogeou tet. The lie how cotour of equal differece i power (probability expreed a a percetage) ad = deote the regio where there i a le tha a 0.01 differece i the power. 6

7 =2 =50 =5 =20 α = α = θ θ Figure 2: Cotour plot howig the differece i power betwee the heterogeou tet ad the homogeeou tet I the cae of ibhip ize 2 ad 50 ibhip, the homogeou tet exceed the power of heterogeou tet by o more tha 0.01 ad ca be 0.1 le powerful whe α=0.4 ad θ=0. Whe the ibhip ize i 5 ad with 20 ibhip, the heterogeou tet exceed the power of homogeou tet by 0.37 whe α=0.3 ad θ=0. The regio where the homogeou tet i mildly preferable i cofied to a area of low mixig ad moderate likage. Other compario how that the regio where the heterogeou tet i clearly preferable expad with ibhip ize ad umber of ibhip. Eve whe there i o mixig the homogeou tet i oly mildly more ( at bet) powerful tha the heterogeou tet, but if there i ome mixig the heterogeou tet ca be ubtatially more powerful. 5 DISCUSSION We have approximated the ull ditributio of the admixture tet for the detectio of likage ad demotrated that if the poibility of heterogeeity exit, thi admixture tet i geerally more powerful tha the uual tet which take o accout of heterogeeity. We have coidered cotat ibhip ize here for implicity of the expoitio but thi i ot crucial ad the ame aymptotic reult would follow eve if the ibhip ize were allowed to vary. Furthermore, the ame reult hold eve whe the meioe are ot completely iformative. For the leat iformative, phae ukow, cae, the tet tatitic i T = max i=1 log [ α(2 1 {θ X i (1 θ) X i + θ X i (1 θ) X i } 1) + 1 ] 7

8 ad a imilar reaoig to the oe above may be ued to get the ame reult. ACKNOWLEDGEMENT Thak to Michael Boehke of the Departmet of Biotatitic, Uiverity of Michiga for brigig my attetio to thi problem ad offerig helpful commet ad thak alo to two referee for improvig the iitial draft. REFERENCES Ghoh J.K. & Se P.K. (1986) O the Aymptotic performace of the log-likelihood ratio tatitic for the mixture model ad related reult Proceedig of the Berkeley Coferece i Hoor of Jerzy Neyma ad Jack Kiefer Volume II Wadworth Hodge SE, Adero CE, Neiwager K, Sparke RS, Rimoi DL (1983) The earch for heterogeeity i iuli-depedet diabete mellitu (IDDM): Likage tudie, two-locu model, ad geetic heterogeeity Am J Hum Geet Martiez M. & Goldi L. (1989) The detectio of likage ad heterogeeity i uclear familie for complex diorder: Oe veru two marker loci Am J Hum Geet Ott J (1983) Likage aalyi ad family claificatio uder heterogeeity A Hum Geet Ott J (1985) Aalyi of Huma Geetic Likage Baltimore, The Joh Hopki Uiverity Pre Pre W.H., Flaery B.P., Teukolky S.A. & Vetterlig W.T. (1988) Numerical Recipe. Cambridge Uiverity Pre Rich N (1988) A ew tatitical tet for likage heterogeeity Am J Hum Geet Rich N (1989) Likage detectio tet uder heterogeeity Geet Epidem Smith CAB (1963) Tetig for heterogeeity of recombiatio value i huma geetic A Hum Geet Wald A. (1949) Note o the coitecy of the maximum likelihood etimate A Math Statit