BACKGROUND
for linkage and/or association studies based on haplotypes, molecular haplotyping can be done for each individual, but at a high cost.
instead, statistical methods such as clark's algorithm  <cit> , the em algorithm  <cit> , or gibb's sampler method  <cit> , are commonly used to reconstruct haplotypes.
the likelihood ratio test, which is based on the difference between the sum of the log-likelihoods of the case and control groups and the log-likelihood of the combined data, is usually used for case-control association studies  <cit> , while the tdt is used for linkage and/or association studies in nuclear families.
the latter compares the frequency of transmitted parental haplotypes with that of non-transmitted parental haplotypes to an affected offspring.
one of the difficulties with the tdt is that there is uncertainty in estimating haplotype frequencies from parental genotypes.
wilson  <cit>  and clayton and jones  <cit>  proposed that these uncertain families be discarded from the analysis, but this drop in families leads to a low power.
clayton  <cit>  proposed an estimating procedure based on likelihood, but it is no longer robust for population admixture or population stratification.
zhao et al  <cit>  extended spielman and ewens' method  <cit>  to test for marginal homogeneity in the transmission and non-transmission of haplotypes.
their method is advantageous over those of wilson  <cit>  and clayton and jones  <cit>  because there is no discarding of unresolved families, and it is still robust for cases of population admixture or population stratification.
cordell and clayton  <cit>  modeled the nuclear-family data via a conditional logistic regression in which they based on either haplotypes if parental phase may be inferrable or genotypes otherwise.
horvath et al  <cit>  proposed family-based association tests with tightly linked markers where the phase may be ambiguous and the parental genotype data is missing.
their weighted conditional approach extended the method provided by rabinowitz and laird  <cit>  to multiple markers.
here, assuming that the haplotype block is very tight so that there is no recombination, we propose a score test for linkage and investigate its performance through simulations.
we also illustrate the use of the proposed test with a data set taken from the kangwha study  <cit> .
methods and
RESULTS
score test
let {hshu, ht hv} denote the event in which the transmitted haplotype in the father is hs and the non-transmitted haplotype is ht, and simultaneously the transmitted haplotype in the mother is hu and the non-transmitted haplotype is hv.
we designate {hshu, hthv} as one haplotype group and define {hsghug, htghvg} as haplotype groups corresponding to the set of genotypes g =  <dig>  ..., g, where g is the number of distinct sets of genotypes across all markers.
here each set of genotypes refers to the observed genotypes for the individual markers of the two parents and the affected offspring.
suppose the total number of possible haplotypes is k.
zhao et al  <cit>  proposed a tdt statistic to test linkage between markers and disease genes.
they obtained a set of the estimated haplotype frequencies, {h˜i
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgobaagaacamaabaaaleaacqwgpbqaaeqaaaaa@2f9b@}, using the em algorithm based on the parental genotypes only, and for the haplotype group {hshu, hthv} compatible with the set of genotypes g, defined t˜gsu,tv
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaadaaaleaacqwgnbwzaeaacqwgzbwccqwg1bqdcqggsaalcqwg0badcqwg2bgdaaaaaa@3658@ as the estimated number of families in which the father with haplotypes {hs, ht} transmits hs and the mother with haplotypes {hu, hv} transmits hu.
here the value of t˜gsu,tv
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaadaaaleaacqwgnbwzaeaacqwgzbwccqwg1bqdcqggsaalcqwg0badcqwg2bgdaaaaaa@3658@ is given by
t˜gsu,tv=ngh˜sh˜th˜uh˜v∑h˜sgh˜tgh˜ugh˜vg,
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=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@5c13@
where the summation is over all haplotype groups compatible with g, and ng is the total number of families compatible with g. then they constructed a k × k transmission/non-transmission table t˜
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaieqacuwfubavgaacaaaa@2df2@ = {t˜ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3110@}, where t˜ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3110@ is the estimated number of parents who have haplotypes {hi, hj} and who transmit hi to the affected offspring across all sets of genotypes.
that is,
t˜ij=∑g{∑u∑vt˜giu,jv+∑s∑tt˜gsi,tj},
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaabaaaleaacqwgpbqacqwgqbgaaeqaaogaeyypa0zaaabuaeaadagadeqaamaaqafabawaaabuaeaacuwg0badgaacamaadaaaleaacqwgnbwzaeaacqwgpbqacqwg1bqdcqggsaalcqwgqbgacqwg2bgdaaaabagaemodayhabeqdcqghris5aogaey4kasyaaabuaeaadaaeqbqaaiqbdsha0zaaiawaa0baasqaaiabdeganbqaaiabdohazjabdmgapjabcycasiabdsha0jabdqgaqbaaaeaacqwg0badaeqaniabgghildaaleaacqwgzbwcaeqaniabgghildaaleaacqwg1bqdaeqaniabgghildaakiaawuhacagl9baaasqaaiabdeganbqab0gaeyyeiuoakiabcycasaaa@5aea@
where the summation over g ranges from  <dig> to g and the summations over s, t, u, and v range from  <dig> to k. they derived a test applying spielman and ewens' method to the table t˜
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaieqacuwfubavgaacaaaa@2df2@.
in this work we propose a score test that extends sham's method  <cit>  by modifying the transmission/non-transmission table t˜
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaieqacuwfubavgaacaaaa@2df2@.
as mentioned by rohde and fuerst  <cit> , the haplotype frequency estimation done using the em algorithm over independent parents may derive possible haplotype pairs for parents that are contradictory to the ones for the offspring.
however, including the offspring's genotype in the haplotype frequency estimation makes it possible to exclude these misleading haplotype pairs, thereby improving the accuracy of the underlying haplotype pair and the accuracy of a transmission/non-transmission table.
this is why we adopted the em method proposed by rohde and fuerst  <cit>  and becker and knapp  <cit>  rather than the method proposed by long, williams, and urbanek  <cit> .
we first obtained estimated haplotype frequencies, {h^i
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgobaagaqcamaabaaaleaacqwgpbqaaeqaaaaa@2f9c@}, based on family trios instead of on independent parental genotypes.
replacing h˜i
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgobaagaacamaabaaaleaacqwgpbqaaeqaaaaa@2f9b@, i =  <dig>  ..., k, by h^i
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgobaagaqcamaabaaaleaacqwgpbqaaeqaaaaa@2f9c@ in equations  and , we denote t˜gsu,tv
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaadaaaleaacqwgnbwzaeaacqwgzbwccqwg1bqdcqggsaalcqwg0badcqwg2bgdaaaaaa@3658@, t˜ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3110@, and t˜
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaieqacuwfubavgaacaaaa@2df2@ by t^gsu,tv
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaadaaaleaacqwgnbwzaeaacqwgzbwccqwg1bqdcqggsaalcqwg0badcqwg2bgdaaaaaa@3659@, and t^ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3111@, and t^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbamxvp5wqsxmqhnxajn0bkvguhdwzzbqegyvzyrwyufgarqqtubsr4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq=jc9vqaqpepm0xbba9pwe9q8fs0=yqaqpepae9pg0firpepekkfr0xfr=xfr=xb9adbaqaaegacagaaiaabeqaamqadiabaagcbagafmivaqlbakaaaaa@37b9@, respectively.
we let t^i.
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqaaeqaaogaeiola4caaa@30a2@, i =  <dig>  ..., k, be the row marginal totals, defined by t^i.=∑j=1kt^ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqaaeqaaogaeiola4iaeyypa0zaaabmaeaacuwg0badgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaqaaiabdqgaqjabg2da9iabigdaxaqaaiabdugarbqdcqghris5aaaa@3cb6@ and t^.j
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcaiabc6caumaabaaaleaacqwgqbgaaeqaaaaa@309a@, j =  <dig>  ..., k, the column marginal totals, defined by t^.j=∑i=1kt^ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcaiabc6caumaabaaaleaacqwgqbgaaeqaaogaeyypa0zaaabmaeaacuwg0badgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaqaaiabdmgapjabg2da9iabigdaxaqaaiabdugarbqdcqghris5aaaa@3cb6@.
we denote a vector of the marginal discrepancies for haplotypes  <dig> to k -  <dig> by Δ^=′
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfuoargaqcaiabg2da9iabcicaoiqbdsha0zaajawaasbaasqaaiabigdaxaqabagccqgguaglcqghsislcuwg0badgaqcaiabc6caumaabaaaleaacqaixaqmaeqaaogaeiilawiaeiola4iaeiola4iaeiola4iaeiilawiafmidaqnbakaadawgaawcbagaem4aasmaeyoei0iaegymaedabeaakiabc6cauiabgkhitiqbdsha0zaajagaeiola4yaasbaasqaaiabdugarjabgkhitiabigdaxaqabagccuggpaqkgaqbaaaa@49f5@.
note that when there is no linkage, e = ' because the matrix t^i,j
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqacqggsaalcqwgqbgaaeqaaaaa@31f1@ is symmetrical under the null hypothesis of no linkage irrespective of a set of haplotype frequencies  <cit> .
letting Σ^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfjowugaqcaaaa@2e45@ = {σ^ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@316a@} with diagonal elements σ^ii=t^i.+t^.i−2t^ii
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgpbqaaeqaaogaeyypa0jafmidaqnbakaadawgaawcbagaemyaakgabeaakiabc6cauiabgucariqbdsha0zaajagaeiola4yaasbaasqaaiabdmgapbqabagccqghsislcqaiyagmcuwg0badgaqcamaabaaaleaacqwgpbqacqwgpbqaaeqaaaaa@4188@ and off-diagonal elements σ^ij=−;
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaogaeyypa0jaeyoei0iaeiikagiafmidaqnbakaadawgaawcbagaemyaakmaemoaaogabeaakiabgucariqbdsha0zaajawaasbaasqaaiabdqgaqjabdmgapbqabagccqggpaqkcqgg7awocqggoaakcqwgpbqacqghgjsucqwgqbgacqggpaqkaaa@4608@, we propose a score test statistic ts defined by
ts=Δ^′Σ^−1Δ^.
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgubavdawgaawcbagaem4camhabeaakiabg2da9ggabiqb=r5aezaajyaafagaf83odmlbakaadaahaawcbeqaaiabgkhitiabigdaxaaakiqb=r5aezaajagaeiola4caaa@3804@
whenever all the parental haplotype phases are uniquely determined, ts asymptotically follows a chi-square distribution with  degrees of freedom because Σ^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfjowugaqcaaaa@2e45@ is the covariance matrix of Δ^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfuoargaqcaaaa@2e27@ under no linkage .
we note, however, that the cells in t^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbamxvp5wqsxmqhnxajn0bkvguhdwzzbqegyvzyrwyufgarqqtubsr4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq=jc9vqaqpepm0xbba9pwe9q8fs0=yqaqpepae9pg0firpepekkfr0xfr=xfr=xb9adbaqaaegacagaaiaabeqaamqadiabaagcbagafmivaqlbakaaaaa@37b9@ are not independent because the contribution of a father or a mother is not limited to a cell because of its uncertainty as a haplotype pair, but reaches out to some cells according to the number of possible haplotype pairs and corresponding probabilities.
that is why ts may not have a chi-square distribution with  degrees of freedom.
thus, we adopted the randomization test procedure introduced by zhao et al  <cit>  to estimate the statistical significance of the proposed test instead of relying on an asymptotic chi-square test.
simulation studies
in our simulations, a hypothetical genealogy with three biallelic genetic markers was considered.
we denoted alleles at the unobservable disease-susceptibility locus as d and d. there are eight haplotypes consisting of bialleles at three loci, l <dig>  l <dig>  and l3: h <dig> = , h <dig> = , h <dig> = , h <dig> = , h <dig> = , h <dig> = , h <dig> = , and h <dig> = .
here both h <dig> and h <dig> include the disease-susceptibility allele d, whereas the other six haplotypes include the wild-type allele d. for each i =  <dig>  ...,  <dig>  let hi be the frequency of hi.
we set  <dig> types of haplotype distribution:  = , , , and , denoted as 'e', 'ue1', 'ue2', and 'ue3', respectively.
note that the degree of ld between each marker at loci, l <dig> and l <dig>  and the disease marker is high , while the ld between the marker at a locus l <dig> and the marker for the disease is relatively low .
the subjects with the d allele have experienced mutations at loci, l <dig> and l <dig>  whereas the mutation at a locus l <dig> happened irrespective of the disease marker allele.
both dominant and recessive models were considered for the mode of inheritance.
we assumed hardy-weinberg equilibrium and random-mating and that the families were ascertained through one affected offspring.
for the dominant genetic models, if an offspring has at least one of the high risk haplotypes, h <dig> and h <dig>  its penetrance equals to c × rr; otherwise, the individual who does not carry the disease-susceptibility allele d may be affected with a probability of c. for the recessive genetic models, only subjects having both h <dig> and h <dig>  h <dig> homozygote, or h <dig> homozygote may have the penetrance of c × rr, and the rest may be affected with a probability of c. we set c as  <dig>  and rr as  <dig>   <dig>   <dig>  and  <dig>  here rr =  <dig> corresponds to level and the rest to power.
we ascertained  <dig> families and  <dig> independent samples were generated for each simulation model.
we estimated the significance level of the tests based on  <dig> randomly generated samples from each sample in the study of type i error rate and power.
table  <dig> summarizes the estimated type i error rates and powers of the proposed and zhao et al's tests at a significance level of  <dig> .
note that the standard error of the type i error rate estimate is  <dig> .
additionally, entries in columns denoted by 'pf ' and 'po' represent the average percentages of perfectly estimated haplotypes for  <dig> families, respectively, when based on family trios and based on parents alone in estimating haplotype frequencies.
the rows corresponding to rr =  <dig> are for type i error rate, and the rows corresponding to rr =  <dig>   <dig> or  <dig> are for power.
the columns corresponding to pf and po are the average percentages of perfectly estimated haplotypes, respectively, when based on family trios and based on parents alone in estimating haplotype frequencies.
as mentioned by rohde and fuerst  <cit> , the percentage of perfectly estimated haplotypes is higher when estimates are based on family trios rather than on parents alone.
therefore, using the offspring's genotype can reduce uncertainty in parental haplotypes.
as expected, the lower the diversity of haplotypes, the higher the percentage of perfect haplotype estimates.
for example, the percentage of perfectly estimated haplotypes for the ue <dig> case was higher than for the e case, and the percentage was higher for the ue <dig> or ue <dig> case than for the ue <dig> case.
however, there was no difference in percentages between dominant and recessive models.
based on our results with the proposed test, shown in entries corresponding to rr =  <dig>  the estimated type i error rate seems to satisfy a nominal level of  <dig> , regardless of the mode of inheritance and type of haplotype distribution.
for the dominant models, the power of the test increases as rr increases from  <dig> to  <dig>  irrespective of the type of haplotype distribution.
in the recessive models, the power increases as rr increases for the e case, while the power equals to the nominal level for the ue <dig>  ue <dig>  and ue <dig> cases.
this explains why the probability of a proband having the mutant homozygotes is very low at  <dig>  for the ue <dig>  ue <dig>  and ue <dig> cases relative to  <dig>  for the e case.
we also compared the proposed test and zhao et al's test in terms of type i error rate and power.
both tests showed comparable performance at a significance level of  <dig> .
for the e case, zhao et al's test was more powerful than our test; for the ue <dig>  ue <dig>  and ue <dig> cases, however, the latter had better power than the former.
it seems that as the diversity of haplotypes decreases, the off-diagonal entries of t^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbamxvp5wqsxmqhnxajn0bkvguhdwzzbqegyvzyrwyufgarqqtubsr4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq=jc9vqaqpepm0xbba9pwe9q8fs0=yqaqpepae9pg0firpepekkfr0xfr=xfr=xb9adbaqaaegacagaaiaabeqaamqadiabaagcbagafmivaqlbakaaaaa@37b9@ correspondingly become more unbalanced and thereby are as informative as the marginal totals and this results in our test being more powerful than zhao et al's test.
finally, we performed simulations to investigate the robustness of the proposed test in cases of population admixture and to determine the conservativeness of the asymptotic chi-squared test.
to this end, the haplotype distribution of a new population to be mixed with e and ue <dig>  ue <dig> or ue <dig>  respectively, was taken as  =  and , denoted as 'e-3' and 'ue-3'.
these new populations have no mutation at a locus l <dig> in common.
we set c as  <dig>  for the 'e-3' and 'ue-3' cases.
we considered three types of proportion for the two populations in the sample size; 1: <dig>  1: <dig>  and 3: <dig>  denoted as 'p11', 'p13', and 'p31', respectively.
the total number of ascertained families in the mixed population was  <dig>  table  <dig> presents the empirical and asymptotic type i error rates of our test along with zhao et al's test with the dominant model at a nominal level of  <dig> .
based on various situations incorporated in the simulations, we have shown that both our proposed test and zhao et al's test are robust to population admixture.
as mentioned in section score test, we also determined that the asymptotic chi-squared tests were conservative through simulations.
p <dig>  p <dig>  and p <dig>  respectively, combine two populations at the ratio of  <dig> to  <dig>   <dig> to  <dig>  and  <dig> to  <dig>
kangwha study
the kangwha study was performed to analyze the natural history of bp in koreans in order to determine important factors associated with changes in bp  <cit> .
in  <dig>  we initially constructed a cohort of  <dig> 6-year old children who were living in kangwha province, korea.
the size of the cohort increased to  <dig> in  <dig> and  <dig> in  <dig>  our case group included blood samples from  <dig> students who experienced at least once systolic bp measurement of more than  <dig> mmhg or diastolic bp measurement of more than  <dig> mmhg between the ages of  <dig> and  <dig>  among the  <dig> samples from the case group, we were also able to obtain blood samples from the parents of  <dig> subjects and our analysis was based on these  <dig> probands and their parents.
angiotensins are substances smaller than proteins that act as vasoconstricting agents, i.e., they cause blood vessels to narrow.
narrowing the diameter of blood vessels increases the blood pressure.
ace, which is located on chromosome 17q <dig> and has a length of  <dig> kb, converts angiotensin to its activated functional form, angiotensin-ii.
there have been several studies investigating the relationship between the ace gene and high bp  <cit> .
in our analysis we included  <dig> snps, a-240t and c-93t on the promoter, i/d on intron  <dig>  and a2350g on exon  <dig>
we used the long, williams, and urbanek's method to determine the frequencies of haplotypes comprised of these  <dig> snps and estimated the frequencies of haplotypes, ttdg, acia, ttia, tcia, atia, and acig as  <dig> ,  <dig> ,  <dig> ,  <dig> ,  <dig> , and  <dig> %, respectively.
when estimated by rohde and fuerst's method, there were no differences in the frequencies, except that ttdg and acia had frequencies of  <dig>  and  <dig> %, respectively.
table  <dig> shows the results of our proposed test and zhao et al's test.
the values in the row denoted by 'all snps' in the first column of the table represent test statistics and their p-values when all  <dig> snps were considered.
the p-values were obtained by  <dig>  randomizations.
neither test showed genetic linkage between ace and high bp at a significance level of  <dig> , but the observed values for each test were quite different.
in addition we examined whether any of the  <dig> snps were redundant and if exclusion of those snps could affect the outcomes of the two tests.
we used a measure of pde  proposed by clayton  <cit>  to select so-called htsnps from the  <dig> snps being considered.
this pde acts like the coefficient of determination in an ordinary regression model.
using the htsnp procedure  <cit>  on  <dig> samples of the control group, we identified the htsnp set of c-93t and a2350g with a pde of  <dig> .
we repeated the two tests using only the two htsnps and listed the corresponding results in the row denoted by 'htsnps' in the first column of table  <dig>  as was in the case when using all snps, using just the htsnps resulted in no evidence of genetic linkage between ace and hypertension at a level of  <dig> .
discussion
here we propose a score test for linkage between genetic markers and disease-susceptibility genes based on haplotypes.
first, we estimated haplotype frequencies using genotypes from affected offspring and their parents.
as in  <cit> , we constructed a transmission/non-transmission table of parents' haplotypes to the offspring.
then we proposed a test which mimics sham's method  <cit>  to test a marginal homogeneity in the transmission/non-transmission table.
simulations indicate that our test works well at a nominal significant level.
further, we found that the power of the test is monotone and increases as the relative risk increases for dominant models.
for recessive models with an unequal haplotype distribution, however, the test was highly conservative due to a low probability of a proband having the mutant homozygotes.
in comparison with zhao et al's test, their test had better power than our test for e case, while our test was more powerful than their test for the ue <dig>  ue <dig>  and ue <dig> cases.
this implies our test has an advantage over zhao et al's test in the situations where the diversity of haplotypes is low.
we also found that our proposed test as well as zhao et al's test is robust to population admixture.
although both asymptotic chi-squared tests were conservative, zhao et al's test seemed to be less conservative than our test.
CONCLUSIONS
we propose a score test for linkage between genetic markers and disease-susceptibility genes and show through simulations that our test performed well at a nominal significant level.
our test showed the better performance in power than zhao et al's test under some situations where the diversity of haplotypes is low.
for an application to a kangwha adolescent cohort, there was no remarkable evidence of genetic linkage between ace and hypertension.
authors' contributions
jk and cn contributed to the formulation of the score test and performed the simulations to investigate its performance.
dk applied our test to the kangwha study data and interpreted the results.
all authors read and approved the final manuscript.
abbreviations
ace: angiotensin-i converting enzyme
bp: blood pressure
em: expectation-maximization
htsnp: haplotype tagging single nucleotide polymorphism
rr: relative risk
tdt: transmission/disequilibrium test
appendix
if the haplotype phases for each parent were identified, the observed set of genotypes is compatible with only one haplotype group.
then t^ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3111@ = tij and thereby Δ^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfuoargaqcaaaa@2e27@ = ', where tij is the number of parents with haplotypes hihj who transmit hi to the affected offspring.
thus the table t^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbamxvp5wqsxmqhnxajn0bkvguhdwzzbqegyvzyrwyufgarqqtubsr4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq=jc9vqaqpepm0xbba9pwe9q8fs0=yqaqpepae9pg0firpepekkfr0xfr=xfr=xb9adbaqaaegacagaaiaabeqaamqadiabaagcbagafmivaqlbakaaaaa@37b9@ = {tij} can be simply considered as a transmission/non-transmission table corresponding to the case of a locus with k marker alleles because there are no ambiguities in the parental haplotypes.
let n=2∑g=1gng
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgubgbcqgh9aqpcqaiyagmdaaewaqaaiabd6gaunaabaaaleaacqwgnbwzaeqaaaqaaiabdeganjabg2da9iabigdaxaqaaiabdeeahbqdcqghris5aaaa@394c@.
define pij as the probability of each parent with haplotypes hihj transmitting hi to the affected offspring, and the marginal probabilities as pi.=∑j=1kpij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgwbacdawgaawcbagaemyaakgabeaakiabc6cauiabg2da9maaqadabagaemicaa3aasbaasqaaiabdmgapjabdqgaqbqabaaabagaemoaaomaeyypa0jaegymaedabagaem4aasganiabgghildaaaa@3c86@ and p.j=∑i=1kpij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgwbaccqgguagldawgaawcbagaemoaaogabeaakiabg2da9maaqadabagaemicaa3aasbaasqaaiabdmgapjabdqgaqbqabaaabagaemyaakmaeyypa0jaegymaedabagaem4aasganiabgghildaaaa@3c86@.
following the arguments of stuart  <cit> , n
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaadagcaaqaaiabd6gaubwcbeaakiabcufabhgabiqb=r5aezaajagaeyoei0iaemyraukaeiikagiaf8hldqkbakaacqggpaqkcqggdbqxaaa@3753@ asymptotically follows a -variate normal distribution with covariance nΣ = {nσij}, where e = e, ..., n)' and σii = n and for i † j, σij = -n. from sham and curtis  <cit> , under no linkage, pi.
= p.i, i =  <dig>  ..., k. therefore, to test linkage we can use the marginal homogeneity test statistic.
under no linkage, we have e = ' and, plugging the maximum likelihood estimates, {tij/n}, {ti./n}, and {t.j/n}, of {pij}, {pi.}
and {p.j}, respectively into the entries of Σ, the estimated covariance matrix nΣ^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfjowugaqcaaaa@2e45@ = {nσ^ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@316a@}, where σ^ii
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgpbqaaeqaaaaa@3168@ = ti.
+ t.i - 2tii and for i ≠ j, σ^ij
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@316a@ = -.
hence ts asymptotically follows a chi-square distribution with  degrees of freedom.

 for linkage and/or association studies based on haplotypes, molecular haplotyping can be done for each individual, but at a high cost. instead, statistical methods such as clark's algorithm, the em algorithm, or gibb's sampler method, are commonly used to reconstruct haplotypes. the likelihood ratio test, which is based on the difference between the sum of the log-likelihoods of the case and control groups and the log-likelihood of the combined data, is usually used for case-control association studies, while the tdt is used for linkage and/or association studies in nuclear families. the latter compares the frequency of transmitted parental haplotypes with that of non-transmitted parental haplotypes to an affected offspring. one of the difficulties with the tdt is that there is uncertainty in estimating haplotype frequencies from parental genotypes. wilson and clayton and jones proposed that these uncertain families be discarded from the analysis, but this drop in families leads to a low power. clayton proposed an estimating procedure based on likelihood, but it is no longer robust for population admixture or population stratification. zhao et al extended spielman and ewens' method to test for marginal homogeneity in the transmission and non-transmission of haplotypes. their method is advantageous over those of wilson and clayton and jones because there is no discarding of unresolved families, and it is still robust for cases of population admixture or population stratification. cordell and clayton modeled the nuclear-family data via a conditional logistic regression in which they based on either haplotypes if parental phase may be inferrable or genotypes otherwise. horvath et al proposed family-based association tests with tightly linked markers where the phase may be ambiguous and the parental genotype data is missing. their weighted conditional approach extended the method provided by rabinowitz and laird to multiple markers. here, assuming that the haplotype block is very tight so that there is no recombination, we propose a score test for linkage and investigate its performance through simulations. we also illustrate the use of the proposed test with a data set taken from the kangwha study. methods and  score test let {hshu, ht hv} denote the event in which the transmitted haplotype in the father is hs and the non-transmitted haplotype is ht, and simultaneously the transmitted haplotype in the mother is hu and the non-transmitted haplotype is hv. we designate {hshu, hthv} as one haplotype group and define {hsghug, htghvg} as haplotype groups corresponding to the set of genotypes g =..., g, where g is the number of distinct sets of genotypes across all markers. here each set of genotypes refers to the observed genotypes for the individual markers of the two parents and the affected offspring. suppose the total number of possible haplotypes is k. zhao et al proposed a tdt statistic to test linkage between markers and disease genes. they obtained a set of the estimated haplotype frequencies, {h˜i mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgobaagaacamaabaaaleaacqwgpbqaaeqaaaaa@2f9b@}, using the em algorithm based on the parental genotypes only, and for the haplotype group {hshu, hthv} compatible with the set of genotypes g, defined t˜gsu,tv mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaadaaaleaacqwgnbwzaeaacqwgzbwccqwg1bqdcqggsaalcqwg0badcqwg2bgdaaaaaa@3658@ as the estimated number of families in which the father with haplotypes {hs, ht} transmits hs and the mother with haplotypes {hu, hv} transmits hu. here the value of t˜gsu,tv mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaadaaaleaacqwgnbwzaeaacqwgzbwccqwg1bqdcqggsaalcqwg0badcqwg2bgdaaaaaa@3658@ is given by t˜gsu,tv=ngh˜sh˜th˜uh˜v∑h˜sgh˜tgh˜ugh˜vg, mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=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@5c13@ where the summation is over all haplotype groups compatible with g, and ng is the total number of families compatible with g. then they constructed a k × k transmission/non-transmission table t˜ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaieqacuwfubavgaacaaaa@2df2@ = {t˜ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3110@}, where t˜ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3110@ is the estimated number of parents who have haplotypes {hi, hj} and who transmit hi to the affected offspring across all sets of genotypes. that is, t˜ij=∑g{∑u∑vt˜giu,jv+∑s∑tt˜gsi,tj}, mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaabaaaleaacqwgpbqacqwgqbgaaeqaaogaeyypa0zaaabuaeaadagadeqaamaaqafabawaaabuaeaacuwg0badgaacamaadaaaleaacqwgnbwzaeaacqwgpbqacqwg1bqdcqggsaalcqwgqbgacqwg2bgdaaaabagaemodayhabeqdcqghris5aogaey4kasyaaabuaeaadaaeqbqaaiqbdsha0zaaiawaa0baasqaaiabdeganbqaaiabdohazjabdmgapjabcycasiabdsha0jabdqgaqbaaaeaacqwg0badaeqaniabgghildaaleaacqwgzbwcaeqaniabgghildaaleaacqwg1bqdaeqaniabgghildaakiaawuhacagl9baaasqaaiabdeganbqab0gaeyyeiuoakiabcycasaaa@5aea@ where the summation over g ranges from to g and the summations over s, t, u, and v range from to k. they derived a test applying spielman and ewens' method to the table t˜ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaieqacuwfubavgaacaaaa@2df2@. in this work we propose a score test that extends sham's method by modifying the transmission/non-transmission table t˜ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaieqacuwfubavgaacaaaa@2df2@. as mentioned by rohde and fuerst, the haplotype frequency estimation done using the em algorithm over independent parents may derive possible haplotype pairs for parents that are contradictory to the ones for the offspring. however, including the offspring's genotype in the haplotype frequency estimation makes it possible to exclude these misleading haplotype pairs, thereby improving the accuracy of the underlying haplotype pair and the accuracy of a transmission/non-transmission table. this is why we adopted the em method proposed by rohde and fuerst and becker and knapp rather than the method proposed by long, williams, and urbanek. we first obtained estimated haplotype frequencies, {h^i mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgobaagaqcamaabaaaleaacqwgpbqaaeqaaaaa@2f9c@}, based on family trios instead of on independent parental genotypes. replacing h˜i mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgobaagaacamaabaaaleaacqwgpbqaaeqaaaaa@2f9b@, i =..., k, by h^i mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgobaagaqcamaabaaaleaacqwgpbqaaeqaaaaa@2f9c@ in equations and, we denote t˜gsu,tv mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaadaaaleaacqwgnbwzaeaacqwgzbwccqwg1bqdcqggsaalcqwg0badcqwg2bgdaaaaaa@3658@, t˜ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaacamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3110@, and t˜ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaieqacuwfubavgaacaaaa@2df2@ by t^gsu,tv mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaadaaaleaacqwgnbwzaeaacqwgzbwccqwg1bqdcqggsaalcqwg0badcqwg2bgdaaaaaa@3659@, and t^ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3111@, and t^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbamxvp5wqsxmqhnxajn0bkvguhdwzzbqegyvzyrwyufgarqqtubsr4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq=jc9vqaqpepm0xbba9pwe9q8fs0=yqaqpepae9pg0firpepekkfr0xfr=xfr=xb9adbaqaaegacagaaiaabeqaamqadiabaagcbagafmivaqlbakaaaaa@37b9@, respectively. we let t^i. mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqaaeqaaogaeiola4caaa@30a2@, i =..., k, be the row marginal totals, defined by t^i.=∑j=1kt^ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqaaeqaaogaeiola4iaeyypa0zaaabmaeaacuwg0badgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaqaaiabdqgaqjabg2da9iabigdaxaqaaiabdugarbqdcqghris5aaaa@3cb6@ and t^.j mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcaiabc6caumaabaaaleaacqwgqbgaaeqaaaaa@309a@, j =..., k, the column marginal totals, defined by t^.j=∑i=1kt^ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcaiabc6caumaabaaaleaacqwgqbgaaeqaaogaeyypa0zaaabmaeaacuwg0badgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaqaaiabdmgapjabg2da9iabigdaxaqaaiabdugarbqdcqghris5aaaa@3cb6@. we denote a vector of the marginal discrepancies for haplotypes to k - by Δ^=′ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfuoargaqcaiabg2da9iabcicaoiqbdsha0zaajawaasbaasqaaiabigdaxaqabagccqgguaglcqghsislcuwg0badgaqcaiabc6caumaabaaaleaacqaixaqmaeqaaogaeiilawiaeiola4iaeiola4iaeiola4iaeiilawiafmidaqnbakaadawgaawcbagaem4aasmaeyoei0iaegymaedabeaakiabc6cauiabgkhitiqbdsha0zaajagaeiola4yaasbaasqaaiabdugarjabgkhitiabigdaxaqabagccuggpaqkgaqbaaaa@49f5@. note that when there is no linkage, e =' because the matrix t^i,j mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqacqggsaalcqwgqbgaaeqaaaaa@31f1@ is symmetrical under the null hypothesis of no linkage irrespective of a set of haplotype frequencies. letting Σ^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfjowugaqcaaaa@2e45@ = {σ^ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@316a@} with diagonal elements σ^ii=t^i.+t^.i−2t^ii mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgpbqaaeqaaogaeyypa0jafmidaqnbakaadawgaawcbagaemyaakgabeaakiabc6cauiabgucariqbdsha0zaajagaeiola4yaasbaasqaaiabdmgapbqabagccqghsislcqaiyagmcuwg0badgaqcamaabaaaleaacqwgpbqacqwgpbqaaeqaaaaa@4188@ and off-diagonal elements σ^ij=−; mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaogaeyypa0jaeyoei0iaeiikagiafmidaqnbakaadawgaawcbagaemyaakmaemoaaogabeaakiabgucariqbdsha0zaajawaasbaasqaaiabdqgaqjabdmgapbqabagccqggpaqkcqgg7awocqggoaakcqwgpbqacqghgjsucqwgqbgacqggpaqkaaa@4608@, we propose a score test statistic ts defined by ts=Δ^′Σ^−1Δ^. mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgubavdawgaawcbagaem4camhabeaakiabg2da9ggabiqb=r5aezaajyaafagaf83odmlbakaadaahaawcbeqaaiabgkhitiabigdaxaaakiqb=r5aezaajagaeiola4caaa@3804@ whenever all the parental haplotype phases are uniquely determined, ts asymptotically follows a chi-square distribution with degrees of freedom because Σ^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfjowugaqcaaaa@2e45@ is the covariance matrix of Δ^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfuoargaqcaaaa@2e27@ under no linkage. we note, however, that the cells in t^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbamxvp5wqsxmqhnxajn0bkvguhdwzzbqegyvzyrwyufgarqqtubsr4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq=jc9vqaqpepm0xbba9pwe9q8fs0=yqaqpepae9pg0firpepekkfr0xfr=xfr=xb9adbaqaaegacagaaiaabeqaamqadiabaagcbagafmivaqlbakaaaaa@37b9@ are not independent because the contribution of a father or a mother is not limited to a cell because of its uncertainty as a haplotype pair, but reaches out to some cells according to the number of possible haplotype pairs and corresponding probabilities. that is why ts may not have a chi-square distribution with degrees of freedom. thus, we adopted the randomization test procedure introduced by zhao et al to estimate the statistical significance of the proposed test instead of relying on an asymptotic chi-square test. simulation studies in our simulations, a hypothetical genealogy with three biallelic genetic markers was considered. we denoted alleles at the unobservable disease-susceptibility locus as d and d. there are eight haplotypes consisting of bialleles at three loci, l l and l3: h =, h =, h =, h =, h =, h =, h =, and h =. here both h and h include the disease-susceptibility allele d, whereas the other six haplotypes include the wild-type allele d. for each i =..., let hi be the frequency of hi. we set types of haplotype distribution: =,,, and, denoted as'e','ue1','ue2', and'ue3', respectively. note that the degree of ld between each marker at loci, l and l and the disease marker is high, while the ld between the marker at a locus l and the marker for the disease is relatively low. the subjects with the d allele have experienced mutations at loci, l and l whereas the mutation at a locus l happened irrespective of the disease marker allele. both dominant and recessive models were considered for the mode of inheritance. we assumed hardy-weinberg equilibrium and random-mating and that the families were ascertained through one affected offspring. for the dominant genetic models, if an offspring has at least one of the high risk haplotypes, h and h its penetrance equals to c × rr; otherwise, the individual who does not carry the disease-susceptibility allele d may be affected with a probability of c. for the recessive genetic models, only subjects having both h and h h homozygote, or h homozygote may have the penetrance of c × rr, and the rest may be affected with a probability of c. we set c as and rr as and here rr = corresponds to level and the rest to power. we ascertained families and independent samples were generated for each simulation model. we estimated the significance level of the tests based on randomly generated samples from each sample in the study of type i error rate and power. table summarizes the estimated type i error rates and powers of the proposed and zhao et al's tests at a significance level of. note that the standard error of the type i error rate estimate is. additionally, entries in columns denoted by'pf' and'po' represent the average percentages of perfectly estimated haplotypes for families, respectively, when based on family trios and based on parents alone in estimating haplotype frequencies. the rows corresponding to rr = are for type i error rate, and the rows corresponding to rr = or are for power. the columns corresponding to pf and po are the average percentages of perfectly estimated haplotypes, respectively, when based on family trios and based on parents alone in estimating haplotype frequencies. as mentioned by rohde and fuerst, the percentage of perfectly estimated haplotypes is higher when estimates are based on family trios rather than on parents alone. therefore, using the offspring's genotype can reduce uncertainty in parental haplotypes. as expected, the lower the diversity of haplotypes, the higher the percentage of perfect haplotype estimates. for example, the percentage of perfectly estimated haplotypes for the ue case was higher than for the e case, and the percentage was higher for the ue or ue case than for the ue case. however, there was no difference in percentages between dominant and recessive models. based on our results with the proposed test, shown in entries corresponding to rr = the estimated type i error rate seems to satisfy a nominal level of, regardless of the mode of inheritance and type of haplotype distribution. for the dominant models, the power of the test increases as rr increases from to irrespective of the type of haplotype distribution. in the recessive models, the power increases as rr increases for the e case, while the power equals to the nominal level for the ue ue and ue cases. this explains why the probability of a proband having the mutant homozygotes is very low at for the ue ue and ue cases relative to for the e case. we also compared the proposed test and zhao et al's test in terms of type i error rate and power. both tests showed comparable performance at a significance level of. for the e case, zhao et al's test was more powerful than our test; for the ue ue and ue cases, however, the latter had better power than the former. it seems that as the diversity of haplotypes decreases, the off-diagonal entries of t^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbamxvp5wqsxmqhnxajn0bkvguhdwzzbqegyvzyrwyufgarqqtubsr4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq=jc9vqaqpepm0xbba9pwe9q8fs0=yqaqpepae9pg0firpepekkfr0xfr=xfr=xb9adbaqaaegacagaaiaabeqaamqadiabaagcbagafmivaqlbakaaaaa@37b9@ correspondingly become more unbalanced and thereby are as informative as the marginal totals and this results in our test being more powerful than zhao et al's test. finally, we performed simulations to investigate the robustness of the proposed test in cases of population admixture and to determine the conservativeness of the asymptotic chi-squared test. to this end, the haplotype distribution of a new population to be mixed with e and ue ue or ue respectively, was taken as = and, denoted as'e-3' and'ue-3'. these new populations have no mutation at a locus l in common. we set c as for the'e-3' and'ue-3' cases. we considered three types of proportion for the two populations in the sample size; 1: 1: and 3: denoted as'p11','p13', and'p31', respectively. the total number of ascertained families in the mixed population was table presents the empirical and asymptotic type i error rates of our test along with zhao et al's test with the dominant model at a nominal level of. based on various situations incorporated in the simulations, we have shown that both our proposed test and zhao et al's test are robust to population admixture. as mentioned in section score test, we also determined that the asymptotic chi-squared tests were conservative through simulations. p p and p respectively, combine two populations at the ratio of to to and to kangwha study the kangwha study was performed to analyze the natural history of bp in koreans in order to determine important factors associated with changes in bp. in we initially constructed a cohort of 6-year old children who were living in kangwha province, korea. the size of the cohort increased to in and in our case group included blood samples from students who experienced at least once systolic bp measurement of more than mmhg or diastolic bp measurement of more than mmhg between the ages of and among the samples from the case group, we were also able to obtain blood samples from the parents of subjects and our analysis was based on these probands and their parents. angiotensins are substances smaller than proteins that act as vasoconstricting agents, i.e., they cause blood vessels to narrow. narrowing the diameter of blood vessels increases the blood pressure. ace, which is located on chromosome 17q and has a length of kb, converts angiotensin to its activated functional form, angiotensin-ii. there have been several studies investigating the relationship between the ace gene and high bp. in our analysis we included snps, a-240t and c-93t on the promoter, i/d on intron and a2350g on exon we used the long, williams, and urbanek's method to determine the frequencies of haplotypes comprised of these snps and estimated the frequencies of haplotypes, ttdg, acia, ttia, tcia, atia, and acig as,,,,, and %, respectively. when estimated by rohde and fuerst's method, there were no differences in the frequencies, except that ttdg and acia had frequencies of and %, respectively. table shows the results of our proposed test and zhao et al's test. the values in the row denoted by'all snps' in the first column of the table represent test statistics and their p-values when all snps were considered. the p-values were obtained by randomizations. neither test showed genetic linkage between ace and high bp at a significance level of, but the observed values for each test were quite different. in addition we examined whether any of the snps were redundant and if exclusion of those snps could affect the outcomes of the two tests. we used a measure of pde proposed by clayton to select so-called htsnps from the snps being considered. this pde acts like the coefficient of determination in an ordinary regression model. using the htsnp procedure on samples of the control group, we identified the htsnp set of c-93t and a2350g with a pde of. we repeated the two tests using only the two htsnps and listed the corresponding results in the row denoted by'htsnps' in the first column of table as was in the case when using all snps, using just the htsnps resulted in no evidence of genetic linkage between ace and hypertension at a level of. discussion here we propose a score test for linkage between genetic markers and disease-susceptibility genes based on haplotypes. first, we estimated haplotype frequencies using genotypes from affected offspring and their parents. as in, we constructed a transmission/non-transmission table of parents' haplotypes to the offspring. then we proposed a test which mimics sham's method to test a marginal homogeneity in the transmission/non-transmission table. simulations indicate that our test works well at a nominal significant level. further, we found that the power of the test is monotone and increases as the relative risk increases for dominant models. for recessive models with an unequal haplotype distribution, however, the test was highly conservative due to a low probability of a proband having the mutant homozygotes. in comparison with zhao et al's test, their test had better power than our test for e case, while our test was more powerful than their test for the ue ue and ue cases. this implies our test has an advantage over zhao et al's test in the situations where the diversity of haplotypes is low. we also found that our proposed test as well as zhao et al's test is robust to population admixture. although both asymptotic chi-squared tests were conservative, zhao et al's test seemed to be less conservative than our test.  we propose a score test for linkage between genetic markers and disease-susceptibility genes and show through simulations that our test performed well at a nominal significant level. our test showed the better performance in power than zhao et al's test under some situations where the diversity of haplotypes is low. for an application to a kangwha adolescent cohort, there was no remarkable evidence of genetic linkage between ace and hypertension. authors' contributions jk and cn contributed to the formulation of the score test and performed the simulations to investigate its performance. dk applied our test to the kangwha study data and interpreted the results. all authors read and approved the final manuscript. abbreviations ace: angiotensin-i converting enzyme bp: blood pressure em: expectation-maximization htsnp: haplotype tagging single nucleotide polymorphism rr: relative risk tdt: transmission/disequilibrium test appendix if the haplotype phases for each parent were identified, the observed set of genotypes is compatible with only one haplotype group. then t^ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg0badgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@3111@ = tij and thereby Δ^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfuoargaqcaaaa@2e27@ =', where tij is the number of parents with haplotypes hihj who transmit hi to the affected offspring. thus the table t^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbamxvp5wqsxmqhnxajn0bkvguhdwzzbqegyvzyrwyufgarqqtubsr4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq=jc9vqaqpepm0xbba9pwe9q8fs0=yqaqpepae9pg0firpepekkfr0xfr=xfr=xb9adbaqaaegacagaaiaabeqaamqadiabaagcbagafmivaqlbakaaaaa@37b9@ = {tij} can be simply considered as a transmission/non-transmission table corresponding to the case of a locus with k marker alleles because there are no ambiguities in the parental haplotypes. let n=2∑g=1gng mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgubgbcqgh9aqpcqaiyagmdaaewaqaaiabd6gaunaabaaaleaacqwgnbwzaeqaaaqaaiabdeganjabg2da9iabigdaxaqaaiabdeeahbqdcqghris5aaaa@394c@. define pij as the probability of each parent with haplotypes hihj transmitting hi to the affected offspring, and the marginal probabilities as pi.=∑j=1kpij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgwbacdawgaawcbagaemyaakgabeaakiabc6cauiabg2da9maaqadabagaemicaa3aasbaasqaaiabdmgapjabdqgaqbqabaaabagaemoaaomaeyypa0jaegymaedabagaem4aasganiabgghildaaaa@3c86@ and p.j=∑i=1kpij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgwbaccqgguagldawgaawcbagaemoaaogabeaakiabg2da9maaqadabagaemicaa3aasbaasqaaiabdmgapjabdqgaqbqabaaabagaemyaakmaeyypa0jaegymaedabagaem4aasganiabgghildaaaa@3c86@. following the arguments of stuart, n mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaadagcaaqaaiabd6gaubwcbeaakiabcufabhgabiqb=r5aezaajagaeyoei0iaemyraukaeiikagiaf8hldqkbakaacqggpaqkcqggdbqxaaa@3753@ asymptotically follows a -variate normal distribution with covariance nΣ = {nσij}, where e = e,..., n)' and σii = n and for i † j, σij = -n. from sham and curtis, under no linkage, pi. = p.i, i =..., k. therefore, to test linkage we can use the marginal homogeneity test statistic. under no linkage, we have e =' and, plugging the maximum likelihood estimates, {tij/n}, {ti./n}, and {t.j/n}, of {pij}, {pi.} and {p.j}, respectively into the entries of Σ, the estimated covariance matrix nΣ^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiiqacuwfjowugaqcaaaa@2e45@ = {nσ^ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@316a@}, where σ^ii mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgpbqaaeqaaaaa@3168@ = ti. + t.i - 2tii and for i ≠ j, σ^ij mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacuwfdpwcgaqcamaabaaaleaacqwgpbqacqwgqbgaaeqaaaaa@316a@ = -. hence ts asymptotically follows a chi-square distribution with degrees of freedom. 
we developed a score test for linkage and used simulations to demonstrate that our test performs well at a nominal level. we then proposed a test for checking the marginal homogeneity in the table. simulations show that our test performs well on a nominal level and has a monotone power, which increases as the relative risk increases. with our test, there was no evidence of genetic linkage between the ace gene and hypertension in the korean adolescent cohort. we estimated haplotype frequencies using the parental and affected offspring's genotypes and then constructed a transmission/non-transmission table for the parental haplotypes transmitted to the offspring. to look for genetic linkage between angiotensin-i converting enzyme gene and hypertension in a korean adolescent cohort, we developed a powerful test using the covariances between marginal differences and their variances in a transmission/non-transmission table. under some situations where the diversity of haplotypes is low, the proposed test gained a little power over the method based on only variances between marginal differences in a transmission/non-transmission table. because the cells in the table were dependent due to the uncertainty of the parental haplotypes, we adopted a randomization procedure to estimate the significance of the observed test statistic.
 to look for genetic linkage between angiotensin-i converting enzyme gene and hypertension in a korean adolescent cohort, we developed a powerful test using the covariances between marginal differences and their variances in a transmission/non-transmission table.  we estimated haplotype frequencies using the parental and affected offspring's genotypes and then constructed a transmission/non-transmission table for the parental haplotypes transmitted to the offspring. we then proposed a test for checking the marginal homogeneity in the table. because the cells in the table were dependent due to the uncertainty of the parental haplotypes, we adopted a randomization procedure to estimate the significance of the observed test statistic. simulations show that our test performs well on a nominal level and has a monotone power, which increases as the relative risk increases. with our test, there was no evidence of genetic linkage between the ace gene and hypertension in the korean adolescent cohort.  we developed a score test for linkage and used simulations to demonstrate that our test performs well at a nominal level. under some situations where the diversity of haplotypes is low, the proposed test gained a little power over the method based on only variances between marginal differences in a transmission/non-transmission table. 