Page 89 - Applied Probability

4. Hypothesis Testing and Categorical Data
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1/2. The standardized residual corresponding to c ij is therefore
h j
2
.
Z ij
$
h j
4
m
2
2
The chi-square statistic χ = 2 = c ij − Z furnishes an omnibus test for
i=1 j=1 ij
departure from the null hypothesis of Mendelian segregation to affecteds.
The maximum standardized residual Z max = max i,j Z ij should be sensitive
to preferential transmission of a single allele to affecteds just as in the in
situ hybridization and the case-control association problems.
The conditional probability space involved in testing the null hypothesis
is complicated. Verbally we can describe its sample points as those tables
that could have been generated by transmission from the parents with their
given genotypes to their affected offspring. The constraints imposed by
conditioning on parental genotypes not only fix the margins on the table but
also couple the fate of alleles shared by parents. Under the null hypothesis,
each relevant transmission event is independent and equally likely to involve
either gene of the transmitting parent. Except for biallelic markers [39], it is
difficult to compute the exact distribution of either proposed test statistic in
this setting. However, we can easily sample from the underlying probability
space by randomly selecting for each affected what maternal and paternal
genes are transmitted to him. Once these random segregation choices are
made, then a new table is constructed by counting the number of alleles of
each type transmitted to affecteds. If we let T i be the value of the statistic T
for the ith randomly generated table from a sample of n such independent
tables, then the p-value of the observed statistic T obs can be approximated
by the sample proportion
n
1
Pr(T ≥ T obs )= 1 {T i ≥T } .
%
n obs
i=1
TABLE 4.3. Transmission/Disequilibrium Test for Costa Rican AT Families
Transmission Allele
Pattern 1 3 4 5 7 8 10 11 20 21
Transmitted 3 0 22 0 1 0 0 0 0 2
Not Transmitted 0 4 0 4 3 4 1 1 2 9

Example 4.8.1 Ataxia-telangiectasia (AT) in Costa Rica
Table 4.3 summarizes marker data on 16 Costa Rican children afflicted
with the recessive disease ataxia-telangiectasia (AT). At the chromosome-
11 marker D11S1817, 28 of their 32 fully-typed parents are heterozygous.