Calculating exact P-values for the Transmission/Disequilibrium Test

The transmission/disequilibrium test (TDT) has become a popular method for analyzing genetic association studies. It is usually assumed that p-values for the test can be calculated using the asymptotic, chi-squared distribution. A formula for the exact p-value is derived and shown to yield quite different values, even when sample sizes are relatively large. Notably, the asymptotic p-value can be either too large or too small, leading to both false positive and false negative results. Since the exact p-value is simple to calculate, it will be preferable to do so, especially when samples of modest sizes are being used in a study. Genetic association studies have become increasingly common in recent years. These studies aim to detect an increase in the frequency of a disease predisposing variant in a population of affecteds as compared to a control population. In most cases, the predisposing allele cannot be interrogated directly, and instead a dense set of genetic markers is used as a surrogate. The association study then aims to detect a significant difference in the frequency of one or more alleles at the markers. Such an increase depends on the existence of linkage disequilibrium between a predisposing allele and one or more genetic markers. Because linkage disequilibrium only extends over a short distance, the most commonly used genetic markers are single nucleotide polymorphisms (SNPs) as they are numerous enough to provide a dense coverage of the genome (Reich et al. 2001; Reich et al. 2002). The simplest experimental design for a genetic association study is to compare a population of cases (patients with the disease being studied) with a population of controls (unaffected individuals). This classic case-control design has been extensively studied in the field of epidemiology with many refinements being incorporated (unequal numbers of cases and controls, different methods for “matching”, related cases and unrelated controls, etc) (Breslow and Day 1980; Risch and Teng 1998; Teng and Risch 1999; Slager and Schaid 2001). A significant drawback of the case-control design is the potential for confounding which can lead to false positive and false negative results. This arises when an unknown factor causes the populations to differ, even though it may not contribute to the phenotype being examined. In terms of genetics, this may arise when one population is more homogenous than the other. For example, suppose that cases and controls are sampled from different geographic locations exhibiting different genetic histories. Allele frequency discrepancies at a particular marker between cases and controls may be due to the sampling bias rather than disease status. In this simple example, it would likely be relatively easy to tell that the two populations were not well matched (that is, that the background level of relatedness was not equal in the two populations). A simple analysis of markers from across the genome would show that the cases showed greater genetic homogeneity for all the markers (Devlin, Roeder et al. 2001; Pritchard et al. 2000; Ardlie et al. 2002). However, this confounding (or stratification) can exist in much subtler forms and can lead to spurious results arising from case-control studies. Debate as to the extent of this bias between cases and controls is ongoing (Pritchard and Donnelly 2001). An alternative design that has gained popularity is the Transmission/Disequilibrium Test (TDT) (Spielman, McGinnis et al. 1993). This is a family based method that requires the parents of the affected individual to be available for genotyping. The idea is qualitatively similar to the case-control design except that the population of controls comprises of the non-transmitted alleles from the parents. That is, of the four parental alleles, two are transmitted to the affected child. The other two are not transmitted and hence should be a random sample from the population from which the cases were selected. These two alleles are used as the control genotype. In this way, the case and control population are well matched. Importantly, only heterozygous parents are informative and so the effective sample size may be much smaller than the total number of families in the study. Data from a TDT association study are analyzed in a similar way to a case-control study. Because homozygous parents are uninformative, a McNemar Test is used (Spielman, McGinnis et al. 1993). Assuming a bi-allelic marker, the test statistic takes the form of,