## Introduction

Two common problems in case-control genetic association studies are: (a) it is often difficult, a priori, to determine the proper model of risk (dominant, recessive, etc.) and (b) controls may not be a good match for the cases. Wang & Sheffield (2005) developed a method to deal with (a). They showed how likelihood methods can be used to fit models that only assume a monotone relationship between the genotypic relative risks without needing to correctly specify the exact risk model. The use of triads (case, mother, father) avoids the problems of control selection, (b). However, Wang and Sheffield's method does not apply to studies based on triads.

Triad studies are important because the results are more robust than those of case-control studies. This is because case-control studies require the case and control groups to be representative of the diseased and non-diseased population. There are many ways that the case or control groups can be skewed. Ascertainment and selection bias are two of the most common (Schlesselman, 1982). Another problem for case-control studies is confounding due to population stratification. Case control studies often face the problem that several ethnic/racial groups are studied. If the case group contains different proportions of the ethnic groups than the control group, false positive results may appear. That is to say that differences between the groups that are actually due to ethnic differences in gene frequencies may be mistakenly considered to be risk factors for disease. While race is an obvious example of the problems in matching cases to controls, other more subtle differences may be difficult, or even impossible, to detect. If in a genetic association study, the population consists of strata with different allele frequencies and different disease risks, then the cases and controls will appear different in allele frequency if the stratification variable is not controlled for (Lee & Wang, 2008). Triad studies eliminate the danger that ethnic, or other non-disease related, case control differences will mistakenly be interpreted as risk factors for the disease being studied because the comparisons in triad studies are made between the mathematically expected transmission rates and those observed in the case families. Triad studies have the additional advantages on the practical level that it is often simpler to identify case parents than to find and match appropriate controls. Identifying case mothers has the benefit that in pregnancy studies possible maternal risk factors can be investigated as well. In this paper, we develop a method of testing for use in triad studies that is based on the same model considered by Wang and Sheffield. We show that in triad studies likelihood methods can be used to fit models that only assume a monotone relationship between the genotypic relative risks, thereby yielding tests that are powerful across a wide range of genetic risk models.

Powerful methods to test for association between disease and the number of risk alleles using triads are obtained by specifying a genetic risk model and using the likelihood ratio test (conditional on the parental mating genotype) for the null hypothesis that all genotypes have the same risk. Examples are the transmission/disequilibrium test (TDT) of Spielman et al.(1993), the unrestricted likelihood ratio test conditional on the parental genes of Schaid & Sommer (1993), and tests based on a dominant or recessive genetic risk model. The unrestricted test is the most general and uses two parameters (ψ_{1} and ψ_{2}) to model the genotypic relative risk for one or two copies of the risk allele compared to no copies. The null hypothesis of no genetic association is ψ_{1}=ψ_{2}= 1.0. The unrestricted test makes no restriction on the model of genetic risk. The TDT uses a multiplicative model, where risk of two copies is the square of the risk of one copy (ψ_{2}=ψ^{2}_{1}). A dominant model is obtained by setting ψ_{1}=ψ_{2} and a recessive model is obtained by setting ψ_{1}= 1.0. The dominant and recessive models are quite powerful when those models are correctly specified, but suffer greatly when the true model is different than that used in likelihood ratio test construction.

A monotone restricted model is obtained by enforcing that either 1.0 ≤ψ_{1}≤ψ_{2} or ψ_{2}≤ψ_{1}≤ 1.0. Although this restriction is not true for all genetic risk models, there are relatively few known exceptions and the investigator may know ahead of time if the true model has the potential to contradict this restriction. As will be seen, the power gained by using a model that uses this restriction may make it worthwhile even for investigators uncertain about its truth. Moreover, the test based on the monotone restriction is more robust to that assumption than the TDT, which is based on a model even more restrictive (ψ_{2}=ψ^{2}_{1} implies either 1.0 ≤ψ_{1}≤ψ_{2} or ψ_{2}≤ψ_{1}≤ 1.0).

The paper proceeds as follows. Section 2 reviews the previous methods of association testing with triads and describes the monotone constrained likelihood method. Section 3 presents simulations comparing the type I error rates and power under various true disease models and risk allele proportions. Section 4 applies the tests to two SNPs on the *MTHFR* gene with NTD triads. Finally, Section 5 contains recommendations.