Assume that one is interested in testing for association between disease and the number of risk alleles at a single bi-allelic locus. It is not important which allele actually confer risk; the name risk allele is merely for reference. Triad data can be represented as an ordered triple, (

*M*,

*F*,

*C*), where

Each of the methods considered in this paper can be written as a likelihood ratio test (LRT) for a certain parameterization of the likelihood conditional on the mating type (MT), specified by knowledge of

*M* and

*F* without knowledge of which is which. Thus, MT is a set of possible (

*M, F*). The derivation of the distribution of case genotype given MT is given in

Schaid and Sommer (1993) and summarized in .

| **Table I**Notation and Probability of Case Genotype Conditional on MT |

The conditional log-likelihood is

The unrestricted test of

Schaid and Sommer (1993) is a LRT based on (

1). Maximization of (

1) can be achieved by numerical means quite reliably. Finding a numerical solution is facilitated by noticing that the second score equation leads to a quadratic equation in ψ

_{2}. Thus, ψ

_{2} can effectively be eliminated from the numerical search by considering for any give ψ

_{1} the two possible roots for ψ

_{2}. This enables use of reliable univariate root finding algorithms to solve for the maximum likelihood estimates (MLE) of (

1). Let

_{1} and

_{2} be the MLE of ψ

_{1} and ψ

_{2}. The test statistic for a LRT of association is then

The

*UNR* test statistic is asymptotically distributed as chi-squared with 2 degrees of freedom (DF) under the null hypothesis (

Kendall and Stuart, 1973). A critical value for a 5% level test is therefore the 95% quantile of the chi-squared distribution with 2 DF.

If a dominant disease model is desired, one imposes the restriction ψ

_{1} = ψ

_{2} = ψ The resulting likelihood,

_{D}(ψ), is then a function of only one parameter, and maximization is achieved in closed form (the solution is a root of a quadratic equation). Let

be the maximizer of

_{D}. The test statistic for a LRT of association is then

The

*DOM* test statistic is asymptotically distributed as chi-squared with 1 DF under the null hypothesis. A critical value for a 5% level test is therefore the 95% quantile of the chi-squared distribution with 1 DF.

If a recessive disease model is desired, one imposes the restriction ψ

_{1} = 1. The resulting likelihood,

_{R}(ψ

_{2}), is then a function of only one parameter, and maximization is achieved in closed form (the solution is a root of a quadratic equation). Let

_{2} be the maximizer of

_{R}. The test statistic for a LRT of association is then

The

*REC* test statistic is asymptotically distributed as chi-squared with 1 DF under the null hypothesis. A critical value for a 5% level test is therefore the 95% quantile of the chi-squared distribution with 1 DF.

If a multiplicative model is desired, one may impose the restriction

. This model has the nice feature that the effect of the risk allele is linear on the log scale, analogous to what one would get in a case-control analysis when using a logistic regression model with a continuous term for the number of risk alleles. The resulting likelihood,

_{MULT}(ψ

_{1}), is then a function of only one parameter, and maximization is achieved in closed form. Let

_{1} be the maximizer of

_{MULT}. The test statistic for a LRT of association is then

This test is equivalent to the TDT test of

Spielman (1993). The

*MULT* test statistic is asymptotically distributed as chi-squared with 1 DF under the null hypothesis. A critical value for a 5% level test is therefore the 95% quantile of the chi-squared distribution with 1 DF.

If a model is desired that assumes a monotone risk relationship, one may impose the restriction that either

This model has the flexibility to be correctly specified in cases between dominant, recessive, or multiplicative models, without allowing for unlikely non-monotone patterns. To obtain a LRT for association under this model, one must obtain restricted MLE of

(ψ

_{1},ψ

_{2}). The possible maximizers of

under the restriction (

2) are the critical points for the unrestricted problem or those from either the dominant or recessive problems described above. Thus, it is quite easy to compare the likelihood (

1) at the critical points from the dominant and recessive problems to the likelihood at the maximizer of the unrestricted problem. The maximizer of the monotone restricted problem, (

_{1},

_{2}), is the critical point with maximum likelihood. The test statistic for a LRT of association is then

The asymptotic distribution of the

*MONO* test statistic under the null hypothesis is derived in the

appendix.