Page 257 - Applied Probability

11. Radiation Hybrid Mapping
244
Similar reasoning implies that
∂
∂

P
α 1 (j 1 ) β 1 (j 1 )
=
∂r
∂r
j 1 ∈O 1
m−1
(11.17)
∂
+ α i (j i ) t c,i (j i ,j i+1 ) β i+1 (j i+1 ).
∂r
i=1 j i ∈O i j i+1 ∈O i+1
The partial derivatives appearing on the right-hand sides of (11.16) and
(11.17) are tedious but straightforward to evaluate. An efficient evaluation
of P and its partial derivatives can therefore be orchestrated by carrying out
the backward algorithm first, followed by the forward algorithm performed
simultaneously with the computation of all partial derivatives. Given a
partial derivative ∂ P of the likelihood P, one forms the corresponding
∂γ i
entry in the score vector by taking the quotient ( ∂ P)/P.
∂γ i
Finally, we note that the EM algorithm for maximum likelihood esti-
mation generalizes easily to the polyploid case. The only differences are
that now the expected number of trials for a breakage parameter is cH
and the expected number of trials for the common retention probability is
m−1
cH(1 + θ new,i ), where H is again the total number of clones.
i=1
11.8 Obligate Breaks Under Polyploidy
An obligate break is scored between two loci i and i+1 of a clone whenever
X i = 1 and X i+1 = 0 or vice versa. According to equation (11.9), the
probability of this event is
Pr(X i =1,X i+1 = 0) + Pr(X i =0,X i+1 =1)
c
c
= 2(1 − r) [1 − (1 − θ i r) ].
c
c
Because the probability 2(1−r) [1−(1−θ ir) ] has a positive partial deriv-
c
ative 2cr(1−r) (1−θ i r) c−1 with respect to θ i , it is increasing as a function
of θ i . Monotonicity of the obligate breakage probability was the only prop-
erty used in establishing the statistical consistency of the minimum breaks
criterion for ordering loci. Thus, the minimum breaks criterion is applicable
to polyploid radiation hybrids and can form the basis of a quick method
for ranking locus orders.
Outlier detection by counting obligate breaks is also feasible. With the
probabilities p k (i, j) defined as in the haploid case, we can again compute
the distribution of the number of obligate breaks per clone assuming no
missing data. Note that now the index i specifying the number of marker
copies present at locus k ranges from 0 to c instead of from 0 to 1. The
initial conditions are
c i c−i

p 1 (i, 0) = r (1 − r) .
i