On the Analysis of Optical Mapping Data - University of Wisconsin ...

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and rate parameters
Λ i = E(N i ) =
Λ 0i
∫ bi
a i
∫ bi
= κ λ 0 (t) dt
a i
λ(t) dt
In terms of these parameters, lack of CNP in G i corresponds to the null hypothesis H i : Λ i =
Λ 0i . N i ∼ Poisson (Λ 0i ) under H i , forming the basis for a test for each H i . Under departures
from H i , N i ∼ Poisson (Λ i ), with power given by
β(Λ i |Λ 0i ) = P Λi (H i rejected)
Figure 4.1 shows that the power to detect a given copy number change depends on the
choice of Λ 0i , providing us a prescription for the choice of the intervals G i . Specifically, to
obtain tests with a common desired power, obtain the corresponding value of Λ 0 and choose
G i = (a i , b i ] so that Λ 0i = Λ 0 . In other words, the intervals are chosen to have constant
expected counts under the null of no CNP. The non-homogeneity of λ 0 implies that the
G i ’s have variable length. The choice of Λ 0 , the expected number of Y i ∈ G i under the
null, represents a trade-off that is fairly intuitive, namely, the more data (number of aligned
optical maps) we have, the shorter the intervals and hence higher the resolution with which
we can detect copy number changes with a given power. An implicit assumption here is that
copy number does not change within an interval. Long intervals give more power to detect
CNP, but only if the alteration holds throughout the interval.
Negative binomial: To determine intervals G i that all have Λ 0 expected hits under H i ,
we need to know λ 0 (·). In practice, we only have control data {X k } that are event times of
one realization of a NHPP with intensity λ 0 (·). To address the problem, define quantities
M i = ∑ k I {a i