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Recent advances in molecular biology have provided geneticists with ever-increasing numbers of highly polymorphic genetic markers that have made possible linkage mapping of loci responsible for many human diseases. However, nearly all diseases mapped to date follow clear Mendelian, single-locus segregation patterns. In contrast, many common familial diseases such as diabetes, psoriasis, several forms of cancer, and schizophrenia are familial and appear to have a genetic component but do not exhibit simple Mendelian transmission. More complex models are required to explain the genetics of these important diseases. In this paper, we explore two-trait-locus, two-marker-locus linkage analysis in which two trait loci are mapped simultaneously to separate genetic markers. We compare the utility of this approach to standard one-trait-locus, one-marker-locus linkage analysis with and without allowance for heterogeneity. We also compare the utility of the two-trait-locus, two-marker-locus analysis to two-trait-locus, one-marker-locus linkage analysis. For common diseases, pedigrees are often bilineal, with disease genes entering via two or more unrelated pedigree members. Since such pedigrees often are avoided in linkage studies, we also investigate the relative information content of unilineal and bilineal pedigrees. For the dominant-or-recessive and threshold models that we consider, we find that two-trait-locus, two-marker-locus linkage analysis can provide substantially more linkage information, as measured by expected maximum lod score, than standard one-trait-locus, one-marker-locus methods, even allowing for heterogeneity, while, for a dominant-or-dominant generating model, one-locus models that allow for heterogeneity extract essentially as much information as the two-trait-locus methods. For these three models, we also find that bilineal pedigrees provide sufficient linkage information to warrant their inclusion in such studies. We also discuss strategies for assessing the significance of the two linkages assumed in two-trait-locus, two-marker-locus models.

Many investigators who have been searching for linkage to complex diseases have by now accumulated a drawer full of negative results. If disease is actually caused by genes at several loci, these data might contain multiple-locus system (MLS) information that the investigator does not realize. Trying to obtain this information formally, through the MLS likelihood, leads to severe computational and statistical difficulties. Therefore, we propose a scheme of inference based on single-locus (SL) statistics, considered jointly. By simulation, we find that the MLS lod score is closely approximated by the sum of SL lod scores. However, we also find that for moderately large systems, say three of four loci, both MLS and SL lod scores are likely to be inconclusive. Nonetheless, MLS can often be detected through the correlation of individual pedigree SL lod scores. Significant correlation is itself evidence of an MLS, because, in the absence of linkage, false-positive lod scores are necessarily random. Under epistasis SL lod scores tend to be positively correlated among pedigrees, while under independent action SL lod scores from high-density samples tend to be negatively correlated.

We consider probability distributions of alternative lod statistics, differing in their treatment of segregation parameters when mode of inheritance is uncertain. A particular pedigree structure and a dominant genetic system displaying incomplete penetrance are analyzed. Lod scores calculated assuming an incorrect segregation model appear to conform quite well to the chi 2 distribution in the absence of linkage. In the presence of linkage, some power is lost. However, if lod scores are calculated under several different segregation models and the best one is accepted, opportunity for chance occurrence of high lod scores is enhanced. The distribution is still chi 2, but with extra degrees of freedom. These results hold over a wide range of sample sizes and segregation models, including small samples and low levels of penetrance.

The detection of linkage in complex traits, although potentially of the greatest value, has proved very difficult. One reason may be the drastic effect that locus heterogeneity has on statistical power. We propose a new test for linkage in the presence of heterogeneity, based upon the sum of individual pedigree maximum lod scores, combined with a bootstrap method for estimating the null-hypothesis distribution. The technique is designed to exploit modern computer capability and to avoid reliance on asymptotic-distribution theory. Numerical comparisons indicate that for small pedigrees this new test can detect linkage with 30%-50% less data than are required by standard methods. A computer program for simulating the distribution and for performing the test of linkage is available from the authors.