Page 88 - Applied Probability

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4. Hypothesis Testing and Categorical Data
of Britain at the time of the Clarke et al. study [7], this is a reasonable pro-
cedure. However, in racially mixed societies like that of the United States,
associations can result from population stratification rather than direct
causation of alleles at a candidate locus or linkage disequilibrium between
the alleles at a marker locus and deleterious alleles at a nearby disease-
predisposing locus. Thus, if a disease is concentrated in one racial or ethnic
group, then that group’s allele frequencies at a marker will predominate in
the affecteds regardless of whether or not the marker is linked to a disease
locus. If normal controls are not matched by ethnicity to affecteds, then
transmission association can be easily confused with ethnic association.
The transmission/disequilibrium test neatly circumvents these mislead-
ing ethnic associations by exploiting the internal controls provided by par-
ents [13, 39, 41]. If marker data are collected on the parents of an affected
as well as on the affected himself, then one can determine for a codominant
marker which of the maternal and paternal alleles are passed to the affected
and which are not. The only ambiguity arises when both parents and the
child share the same heterozygous genotype. Even in this case one can still
count the number of alleles of each type passed to the affected. In the trans-
mission/disequilibrium test, the marker alleles potentially contributed by
heterozygous parents to sampled affecteds are arranged in a 2 × m con-
tingency table, with one row counting parental alleles passed to affecteds
and the other row counting parental alleles not passed to affecteds. The m
columns correspond to the m different alleles seen among the parents. It
seems reasonable in this scheme to exclude contributions from homozygous
parents because these tell us nothing about transmission distortion.
In analyzing contingency table data of this sort, we should explicitly
condition on the parental genotypes. This eliminates ethnic association.
Once we have done this, there is no harm in counting alleles transmitted
to affected siblings or to related, affected individuals scattered throughout
an extended pedigree. The two inviolable rules to observe are that both
parents of an affected must be typed and that marker typing should done
in one part of a family without regard to the outcomes of marker typing in
another part of the family.
The transmission/disequilibrium test for two alleles permits exact cal-
culation of p-values [39]. In generalizing the test to multiple alleles, this
convenience is sacrificed, but one can approach the problem of calculating
approximate p-values by standard permutation techniques [37, 20, 25]. The
question of an appropriate test statistic also becomes murky unless we con-
sider rather simple, and probably unrealistic, alternative hypotheses. We
will suggest two statistics that are intuitively reasonable. Both are based on
computing a standardized residual for each cell of the 2×m table. Let c ij be
the count appearing in row i and column j of the table. If h j heterozygous
parents carry allele j, then under the null hypothesis of Mendelian trans-
mission, c ij is binomially distributed with h j trials and success probability