67
A shortcoming of the rule d* is that it implicitly weights all true positives and true
negatives equally (and then t = 1/2). But not all true positives are equally desirable
to the investigators in a phage display experiment.
Besides, the FDR criterion is a Bayes rule of the form of (3.31) when considering a
particular utility function of the form (3.32). Under this criterion, the selected pairs
conform the largest list such that the expected FDR (proportion of false positives in
this list) is bounded by a quantity 0 < a < 1. The same list is obtained when choosing
к in (3.32) such that the set of p'is greater than t k∕{k + 1) and 7⅛ := {i : pi > t},
satisfies:
S := a — (1 — Pi) > 0 and S + a — (1 — pβ < 0, for any j £ Ik. (3.33)
ieʃfe
Ji et al. (2007) selected the pairs with increasing means across the three stages
according to the FDR-based criterion (3.31) where Pi is the probability of (common)
positive slope (across the stages 1 and 2 and 2 and 3) according to their model. It
can be seen from the form of (3.32) that this criterion does not take into account the
size of the increase.
In contrast, we will use a utility function that gives weights to the pairs propor-
tional to the relative increment from the first to the third stages (i.e. δi) if the means
of the three phases increase, i.e., if δi > βi > 1.
We choose the utility function of Müller et al. (2006) given by
U{d, w) = c⅛wi ~ ʌ' - ⅛)wi — cD, (3.34)
2—1 2—1
where w = (wɪ, ∙ ∙ ∙ , wn) is a vector of weights, in our application we consider
wi = δiI{δi > βi> 1};
D is the number of pairs that we declare that have increasing means across the three
stages of the experiment, i.e., D = ɪɪɪ d,; c > 0 represents the cost of declaring that
a pair has increasing means; and к > 0 is such that kwi is the cost of not declaring
that the pair i has increasing means when indeed it really does.