## Abstract

I show that fine-scale localization of a survival-related locus can be accomplished on the basis of deviations from Hardy–Weinberg equilibrium and linkage disequilibrium at closely linked marker loci. The method is based on χ^{2}-tests and they can be performed for age-specific samples of alive (or dead) individuals, as for combined samples of alive and dead individuals.

CONVENTIONAL tools for the analysis of QTL can locate loci underlying the variation of continuous quantitative traits to a genomic region of ∼30 cM (Deng *et al*. 2000). Fine-scale mapping (∼1 cM) is required to reduce the range of these candidate genomic regions, and some appropriate techniques have been developed for complex diseases and quantitative traits under Gaussian distributions (Spielman *et al*. 1993; Feder *et al*. 1996; Nielsen *et al*. 1998; Deng *et al*. 2000, 2003). Although survival has become an emergent research field in human health (Puca *et al*. 2001) and animal breeding (Kleinbaum 1996), there are not appropriate fine-mapping techniques for survival traits. The objective of this article is to adapt Deng's *et al*. (2000) QTL fine-mapping method to survival data.

Take as starting point a survival-related QTL locus with two alleles, *A*_{1} and *A*_{2}, and allelic frequencies *p* and *q* = 1 − *p*, respectively. Under the proportional hazards framework (Cox 1972), is the survival probability at time *t* for an individual with genotype *A*_{1}*A*_{1}, where *S*_{0}(*t*) is the baseline survival function and *a* is the genotypic value of the *A*_{1}*A*_{1} genotype. In a similar way, we define and *d* and −*a* being the genotypic values for the *A*_{1}*A*_{2} and *A*_{2}*A*_{2} genotypes, respectively. Without loss of generality, we can assume that *S*_{0}(*t*) represents a random variable for the combined effects of all the rest of the polymorphic loci and all random environmental effects. As in the original research of Feder *et al*. (1996), Nielsen *et al*. (1998), and Deng *et al*. (2000), a large population under random mating is assumed and thus Hardy–Weinberg (HW) equilibrium holds in each generation of individuals at birth. The proportion of survivors at time *t* (π_{t}) is stated as and being the allelic (*A*_{1}) and genotypic (*A*_{1}*A*_{1}) frequency within the group of alive individuals at time *t* (ALIVE_{t}), respectively (the remaining frequencies can be easily derived following Deng *et al*. 2000). Deviation from HW equilibrium at the survival QTL can be measured by the disequilibrium coefficient by Weir (1996), or, following Deng *et al*. (2000), by the function between observed and expected homozygosities,

Previous derivations can be easily adapted to a marker locus closely located near the survival QTL, with alleles *M*_{1} and *M*_{2}, and allelic frequencies *r* and *s* = 1 − *r*. As in Deng *et al*. (2000), is the allelic frequency of *M*_{1} and is the genotypic frequency of *M*_{1}*M*_{1}, where is the linkage disequilibrium (LD) measure between *A*_{1} and *M*_{1} (Crow and Kimura 1970) and *P*_{A1M1} is the frequency of haplotypes carrying both *A*_{1} and *M*_{1}. According to Deng *et al*. (2000), the HW disequilibrium among ALIVE_{t} individuals at the marker locus is it being nonzero when and A wide range of combinations of ϕ_{11}, ϕ_{12}, and ϕ_{22} provide a value different from zero and, in practice, the HW disequilibrium at the marker locus solely reflects the LD in the whole generation (Deng *et al*. 2000). In a similar way, HW disequilibrium for the marker locus among alive individuals can be derived as as described by Feder *et al*. (1996) and Nielsen *et al*. (1998) for affected individuals of complex traits. Both *F*_{M1} and *D*_{M1M1} statistics converge to the key point that HW disequilibrium at a marker locus corresponds to the whole-generation LD between the marker locus and the QTL (Deng *et al*. 2000). Alternatively, one could use a direct measure of LD like the *p*_{excess} statistic proposed by Bengtsson and Thomson (1981). For a survival QTL, *p*_{excess} becomes where was the allelic frequency of *M*_{1} within the group of dead individuals at time *t* (DEAD_{t}). Therefore, *p*_{excess} is proportional to *D*_{A1M1} and reaches its maximum at the marker with the greatest LD with the QTL (Nielsen *et al*. 1998).

To test for the statistical significance of the HW disequilibrium measures ( and ) and the LD measure (*p*_{excess}), two χ^{2}-tests can be easily applied. Following Deng *et al*. (2000), the χ^{2}-test statistic for HW disequilibrium is derived aswhere the tilde (∼) denotes an estimated value from the sample and 2*n* is the total sample size of individuals. The test has d.f., *m* being the number of alleles at the marker locus being tested (*k* = 2). On the other hand, the χ^{2} for *p*_{excess} (Weir 1996; Deng *et al*. 2000) is stated aswith *m* − 1 = 1 d.f.

To illustrate the tests outlined above, extensive computer simulations were performed for a biallelic survival QTL and several biallelic markers. These computer simulations were carried out under a wide range of inheritance models (additive, dominant, recessive, partial dominant, and partial recessive), sample sizes, and ages and under a Weibull assumption for the baseline survival function (Ducrocq *et al*. 1988a,b; Ibrahim *et al*. 2001). For the five genetic models, both tests showed reduced power at greater distances between the QTL and the marker loci, although the power decayed more quickly for than for and it was higher for than for (Figure 1). These results agree with previous QTL fine-mapping research (Nielsen *et al*. 1998; Deng *et al*. 2000) and they are not surprising because, in models where both the survival QTL and the marker locus have only two alleles, HW disequilibrium is proportional to the square of LD (Nielsen *et al*. 1998). Whereas provided a similar power for all genetic models, showed substantial discrepancies. Within this context, seemed preferable if samples of both alive and dead individuals were accessible, although they could be unavailable if the study was not previously scheduled. On the other hand, the average type I error of both tests was close to the expected level of 0.05, slightly higher for than for (Figure 2). This larger variation in was consistent with previous research (Nielsen *et al*. 1998; Deng *et al*. 2000). As was expected, the average power of both tests at the different marker positions increased as the amount of available information increased (*e.g*., the number of sampled individuals increases; Figure 3) or the selection criteria became more strict (*e.g*., elderly ages to differentiate between alive and dead individuals; Figure 4). These results agreed with those of Deng *et al*. (2000).

In conclusion, LD is captured and magnified in extreme samples of elderly individuals, where QTL genotypes and alleles are disproportionately represented. The disequilibrium must be the highest at the QTL locus, since it is the underlying factor that determines the selection criterion, and it decreases as the degree of linkage between the QTL and the markers decreases. This relation between the HW equilibrium and/or LD and the physical distance between a panel of linked marker loci and a QTL is the key point that provides a straightforward basis for QTL fine mapping with use of the peaks of the disequilibrium measures and/or test statistics.

## Footnotes

Communicating editor: J. B. Walsh

- Received October 24, 2006.
- Accepted February 25, 2007.

- Copyright © 2007 by the Genetics Society of America