Louisiana’s Wetlands: A Lesson in Nature Appreciation

Most methods for testing association in the presence of linkage, using family-based studies, have been developed for continuous traits. FBAT (family-based association tests) is one of few methods appropriate for discrete outcomes. In this article we describe a new test of association in the presence of linkage for binary traits. We use a gamma random effects model in which association and linkage are modelled as fixed effects and random effects, respectively. We have compared the gamma random effects model to an FBAT and a generalized estimating equation-based alternative, using two regions in the Genetic Analysis Workshop 14 simulated data. One of these regions contained haplotypes associated with disease, and the other did not. Background Testing association in a region with confirmed linkage may increase the rate of false positives in family-based studies. In a linked region one expects similarity between related individuals. If unaccounted for, this similarity may be mistaken for association. Different remedies have been suggested, ranging from using a robust variance estimator [1] for the general test statistic FBAT (family-based association tests) [2] to a model-based approach in which the linkage is modelled in the covariance structure [3] (VCM, variance components model). The VCM has been developed for continuous traits, while FBAT can be used with both binary and continuous traits. In this article we concentrate on methods for testing association in the presence of linkage, using binary traits. We compare the program FBAT for binary traits to both the gamma random effects (GRE) method and also a GEE (generalized estimating equation) [4] approach. For the purpose of our comparisons we have used the simulated Genetic Analysis Workshop 14 (GAW14) data. We have compared the three methods' ability to pick up a signal in a region with association, as well as their ability to avoid signalling in a region with no association. Methods We consider a random effects model for binary events, which is similar in spirit to the multivariate survival model in Zhong and Li [5], which models association and linkage as fixed effects and random effects, respectively. We use a result for random effects models for binary outcomes, which has been described by Conaway [6]. It is shown that for gamma distributed random effects, the unconditional distribution of the outcome using a log-log link can be written as a sum of easily calculated terms. Analytical tractability is only achievable for a few other combinations of random effects distributions and link functions, such as the beta distribution with a log(-log) link [6]. The random effects model in Zhong and Li [5] assigns one random effect for each of the two alleles of the father and one random effect for each of the two alleles of the mother. The notion of inheritance vector is used to describe the alleles for all family members jointly. The from Genetic Analysis Workshop 14: Microsatellite and single-nucleotide polymorphism Noordwijkerhout, The Netherlands, 7-10 September 2004 Published: 30 December 2005 BMC Genetics 2005, 6(Suppl 1):S92 doi:10.1186/1471-2156-6-S1-S92 Genetic Analysis Workshop 14: Microsatellite and single-nucleotide polymorphism Joan E Bailey-Wilson, Laura Almasy, Mariza de Andrade, Julia Bailey, Heike Bickeböller, Heather J Cordell, E Warwick Daw, Lynn Goldin, Ellen L Goode, Courtney GrayMcGuire, Wayne H ning, ail Jarvik, Brion S Maher, Nancy Mendell, Andrew D Paterson, John Rice, Glen Satten, Brian Suar z, Veronica Vieland, Marsha Wilcox, Heping Zhang, Andre s Ziegler and Jean W MacCluer Proceedings Page 1 of 4 (page number not for citation purposes) BMC Genetics 2005, 6:S92 method presented here works for all sizes of sibships, and may also be easily adapted to extended pedigrees. GRE model Let (Yi1, Yi2, ..., ) be the binary trait vector for family i and let j denote offspring (j = 1, 2, ..., Ji). We allow for different family sizes Ji. We use θmj and θpj to denote the effect of the transmitted alleles to offspring j, with mj = 1, 2 the maternal alleles and pj = 3, 4 the paternal alleles, respectively. Conditional on the transmitted alleles, we write the probability of the trait for offspring j in family i as P(Yij = 1|θmj, θpj). We consider a model with a log(-log) link of the form log(-log(P(Yij = 1|θmj, θpj))) = log(θmj + θpj) + Xjβ, or equivalently The effects θ of the transmitted alleles act multiplicatively on the offspring trait probability, and the effect of each transmitted allele is multiplied by a term involving the parameter vector β describing the fixed genetic effects. Following Li [7] and Li and Zhong [8] we assume that the maternal and paternal alleles are independent and that each allele contributes an effect to the trait which is random and follows a gamma distribution with scale α/2 and shape λ. The model has a tractable closed form for the joint unconditional trait probabilities for the offsprings in a sibship. Let Ψ denote all ordered subsets of 1, 2, ..., Ji, Ψ = {{0}, {1}, {2}, {1, 2}, {3}, ..., {1, 2, ..., Ji}. Let denote the joint unconditional probability of Yij = 1 for all j ∈ T, where T ∈ Ψ. Calculating the probability requires integrating over θ1, θ2, θ3 and θ4. There is a tractable solution [6]. It turns out that The elements of vector ak, ajk, indicate whether allele k has been transmitted to offspring j, j = 1, 2, ..., Ji. The probabilities for all T ∈ Ψ can be placed in a vector π*. It has been shown [6] that the unconditional probability for all possible outcomes of Y can be written as π = Z-1π*. The matrix Z indicates all subsets of T. In order to get the probability of the observed Yij one needs only to pick the corresponding row in π.. In Table 1, an example of T, matrix Z and vector π for three sibs is given. The likelihood for the observed data, for families i (i = 1, 2, ..., n), is