Transmission disequilibrium test
The transmission disequilibrium test (TDT) was proposed by Spielman, McGinnis and Ewens (1993) as a family-based association test for the presence of genetic linkage between a genetic marker and a trait. It is an application of McNemar's test. A specificity of the TDT is that it will detect genetic linkage only in the presence of genetic association. While genetic association can be caused by population structure, genetic linkage will not be affected, which makes the TDT robust to the presence of population structure. The case of trios: one affected child per family Description of the test We first describe the TDT in the case where families consist of trios (two parents and one affected child). Our description follows the notations used in Spielman, McGinnis & Ewens (1993). The TDT measures the over-transmission of an allele from heterozygous parents to affected offsprings. The n affected offsprings have 2n parents. These can be represented by the transmitted and the non-transmitted alleles and at some genetic locus. Summarizing the data in a 2 by 2 table gives: The derivation of the TDT shows that one should only use the heterozygous parents (total number b+c). The TDT tests whether the proportions b/(b+c) and c/(b+c) are compatible with probabilities (0.5, 0.5). This hypothesis can be tested using a binomial (asymptotically chi-square) test with one degree of freedom: Outline of the test derivation A derivation of the test consists of using a population genetics model to obtain the expected proportions for the quantities and in the table above. In particular, one can show that under nearly all disease models the expected proportion of and are identical. This result motivates the use of a binomial (asymptotically ) test to test whether these proportions are equal. On the other hand, one can also show that under such models the proportions and are not equal to the product of the marginals probabilities , and , . A rewording of this statement would be that the type of the transmitted allele is not, in general, independent of the type of the non-transmitted allele. A consequence is that a test for homogeneity/independence does not test the appropriate hypothesis, and thus, only heterozygous parents are included. Extension to two affected child per family Extension of the test The TDT can be readily extended beyond the case of trios. We keep following the notations of Spielman, McGinnis & Ewens (1993). Consider a total of heterozygous parents. We use the fact that the transmission to different children are independent. The information can be then summarized in three categories: = number of parents who transmit to both children. = number of parents who transmit to one child and to another. = number of parents who transmit to both children. Using the notations of the previous paragraph we have: leading to the chi-squared test statistic: Relation with another linkage statistic The comparison with the more traditional (at least at the time when the TDT was proposed) linkage test proposed by Blackwelder and Elston 1985 is informative.