| Total |
a + c |
b + d |
2n |
</tr>
</table>
</center>
The derivation of the TDT shows that one should only use the heterozygous parents (total number b+c).
The TDT tests whether the proportions <math> b/(b+c) </math> and <math> c/(b+c) </math> are compatible with probabilities <math> (0.5, 0.5) </math>.
This hypothesis can be tested using a binomial (asymptotically chisquare) test with one degree of freedom:
<math> \chi^2 = \frac{ [b - (b+c)/2]^2}{(b+c)/2} + \frac{ [c - (b+c)/2]^2}{(b+c)/2} = \frac{(b-c)^2}{b+c} </math>
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 <math> a,b,c </math> and <math> d </math> in the table above. In particular, one can show that under nearly all disease models the expected proportion of <math> b </math> and <math> c </math> are identical. This result motivates the use of a binomial (asymptotically <math> \chi^2 </math>) test to test whether these proportions are equal.
On the other hand, one can also show that under such models the proportions <math> a,b,c </math> and <math> d </math> are not equal to the product of the marginals probabilities <math> a+b/2n</math>, <math> c+d/2n </math> and <math> a+c/2n </math>, <math> b+d/2n </math>. 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 <math> \chi^2 </math> 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). Let us consider a total of <math> h </math> heterozygous parents. We use the fact that the transmissions to different children are independent. The information can be then summarized in three categories:
<math> i </math> = number of parents who transmit <math> M_1 </math> to both children.
<math> h-i-j </math> = number of parents who transmit <math> M_1 </math> to one child and <math> M_2 </math> to another.
<math> j </math> = number of parents who transmit <math> M_2 </math> to both children.
Using the notations of the previous paragraph we have:
<math> b = 2i + (h-i-j) = h + i - j </math>
<math> c = 2j + (h-i-j) = h - i + j </math>
leading to the <math> \chi^2 </math> test statistic:
<math> \chi_{tdt}^2 = \frac{2(i-j)^2}{h} </math>
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.
The Blackwelder and Elston approach uses the total number of haplotypes identical by descent (mean haplotype sharing). This measure ignores the allelic state of a marker and simply compares the number of time a parent transmits the same allele to both affected children with the number of times a different allele is transmitted.
The test statistic is:
<math> \chi^2_{hs} = \frac{(2i+2j-j)^2}{h} </math>
Under the null hypothesis of no linkage the expected proportions of <math> (i, h-i-j, j) </math> are <math> (0.25, 0.5, 0.25) </math>.
One can derive a simple <math> \chi^2 </math> statistic with 2 degrees of freedom:
<math> \chi^2_{total} = \frac{(i - h/4)^2}{h/4} + \frac{(h-i-j-h/2)^2}{h/2} + \frac{(j-h/4)^2}{h/4} = \chi^2_{tdt} + \chi^2_{hs}</math>
It clearly appears that the total statistic (with two degree of freedom) 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. (2002/2004) 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 (see Sasieni (1997) for the corresponding problem in case-control association tests). 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:
<math> \chi^2 = \frac{ \left[ [n_{PQ} - n_{QQ}]_{PQ \sim QQ} + 2\times[n_{PP} - n_{QQ}]_{PQ \sim PQ} + [n_{PQ} - n_{QQ}]_{PP \sim PQ} \right]^2}{[n_{PQ} + n_{QQ}]_{PQ \sim QQ} + 4\times[n_{PP} + n_{QQ}]_{PQ \sim PQ} + [n_{PQ} + n_{QQ}]_{PP \sim PQ}}
</math>
where <math> [n_{PQ}]_{PQ \sim QQ} </math> is the number of PQ children from parents with the PQ and QQ types.
References
- Ewens WJ, Spielman RS. (2005) What Is the Significance of a Significant TDT? Hum Hered. 60(4):206-10.
- Wittkowski KM, Liu X. (2002/2004) A statistically valid alternative to the TDT Hum Hered. 54(3):157-64. Comment, author reply, discussion Hum Hered. 58(1):59-62.
- Spielman RS, Ewens WJ. (1998) A sibship test for linkage in the presence of association: the sib transmission/disequilibrium test. Am J Hum Genet. 62(2):450-8.
- Sasieni PD. (1997) From genotypes to genes: doubling the sample size. Biometrics. 53 (4):1253-61.
- Spielman RS, Ewens WJ. (1996) The TDT and other family-based tests for linkage disequilibrium and association. Am J Hum Genet 59:983-9
- Ewens WJ, Spielman RS. (1995) The transmission/disequilibrium test: history, subdivision, and admixture. Am J Hum Genet. 57(2):455-64.
- McGinnis RE, Ewens WJ, Spielman RS. (1995) The TDT reveals linkage and linkage disequilibrium in a rare disease. Genet Epidemiol. 12(6):637-40.
- Spielman RS, McGinnis RE, Ewens WJ. (1993) Transmission test for linkage disequilibrium: the insulin gene region and insulin-dependent diabetes mellitus (IDDM). Am J Hum Genet. 52(3):506-16.