Page 165 - Applied Probability

8. The Polygenic Model
150
TABLE 8.2. Maximum Likelihood Estimates for the Ridge Count Data
Estimate
Parameter
66.6 ± 2.8
.992 ± .008
µ ml
ρ alr
2
σ
59.0 ± 2.8
µ fl
ar
2
68.9 ± 2.9
30.3 ± 7.5
σ
µ mr Estimate Parameter 657.5 ± 69.2
el
µ fr 62.7 ± 2.8 ρ elr −.146 ± .178
σ 2 638.8 ± 65.7 σ 2 35.6 ± 9.9
al er
table we can draw several tentative conclusions. First, ridge counts tend
to be higher for males than females and for right hands than left hands.
Second, left and right-hand ridge counts are highly heritable traits, as re-
flected in the ratio of the additive genetic variances to the corresponding
random environmental variances. Third, the additive genetic correlation is
surprisingly strong and the environmental correlation is surprisingly weak.
If these data are credible, then ridge counts on the left and right hands are
basically determined by the same set of genes. Furthermore, the environ-
mental determinants for the two hands may act independently; indeed, the
estimate of the environmental correlation is less than one standard devia-
tion from 0. Although we omit them here, overall goodness of fit statistics
suggest that the model is reasonably accurate [21].
8.6 QTL Mapping
QTL mapping is predicated on the assumption that one locus contributes
disproportionately to a quantitative trait [1, 3, 13, 15, 32]. We can estimate
the extent of that contribution if we quantify more accurately the allele
sharing between each pair of relatives at the quantitative trait locus
(QTL). The kinship coefficient Φ jk is an average value depending only on
the pedigree connecting j and k. If we track the transmission of marker
genes in the vicinity of the QTL, then we can use this information to
ˆ
estimate a conditional kinship coefficient Φ jk that provides a much better
idea of the extent of allele sharing at the QTL. We can extend this line
of reasoning to an arbitrary number n of QTL loci floating in a polygenic
sea. If we denote the conditional kinship matrix corresponding to the ith
ˆ
QTL by Φ i , then the overall covariance matrix for a univariate trait with
no dominance or household effects becomes
n

2 ˆ
2
2
Ω= 2 σ Φ i +2σ Φ+ σ I,
e
ai
a
i=1
One can test the null hypothesis that the additive genetic variance σ 2 ai
equals zero by comparing the likelihood of this restricted model with the