The transmission disequilibrium test was proposed by Spielman, McGinnis and Ewens 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. Our description follows the notations used in Spielman, McGinnis & Ewens. 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. The TDT tests whether the proportionsb/ and c/ are compatible with probabilities. This hypothesis can be tested using a binomial test with one degree of freedom:
Outline of the test derivation
A derivation of the test consists of using a population geneticsmodel 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 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. 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 linkage test proposed by Blackwelder and Elston 1985 is informative. The Blackwelder and Elston approach uses the total number of haplotypes identical by descent. This measure ignores the allelic state of a marker and simply compares the number of times a parent transmits the same allele to both affected children with the number of times a different allele is transmitted. The test statistic is: Under the null hypothesis of no linkage the expected proportions of are. One can derive a simple chi-square statistic with 2 degrees of freedom: It clearly appears that the total statistic is the sum of two independent components: one is the traditional linkage measure and the other is the TDT statistic.
A modified version of the TDT
More recently, Wittkowski KM, Liu X. proposed a modification to the TDT that can be more powerful under some alternatives, although the asymptotic properties under the null hypothesis are equivalent. The motivating idea for this modification is the fact that, while the transmissions of both allele from parents to a child are independent, the effects of other filial genetic or environmental covariates on penetrance are the same for both alleles transmitted to the same child. This situation can be important if, for example, the genetic marker is linked to a disease locus with a strong selection against heterozygous individuals. This observation suggests to shift the statistical model from a set of independent transmissions to a set of independent children. While this observation does not affect the distribution under the null hypothesis of no linkage, it allows, for some disease models, to design a more powerful test. In this modified TDT test the children are stratified by parental type and the modified test statistic becomes: where is the number of PQ children from parents with the PQ and QQ types.
