The completion of a draft sequence of the human genome and the promise of rapid single-nucleotide-polymorphism-genotyping technologies have resulted in a call for the abandonment of linkage studies in favor of genome scans for association. However, there exists a large class of genetic models for which this approach will fail: purely epistatic models with no additive or dominance variation at any of the susceptibility loci. As a result, traditional association methods (such as case/control, measured genotype, and transmission/disequilibrium test [TDT]) will have no power if the loci are examined individually. In this article, we examine this class of models, delimiting the range of genetic determination and recurrence risks for two-, three-, and four-locus purely epistatic models. Our study reveals that these models, although giving rise to no additive or dominance variation, do give rise to increased allele sharing between affected sibs. Thus, a genome scan for linkage could detect genomic subregions harboring susceptibility loci. We also discuss some simple multilocus extensions of single-locus analysis methods, including a conditional form of the TDT.