dBUMscore is supposed to take as input a vector of p-values,
which are transformed into scores according to the fitted beta-uniform
mixture model. Also if the FDR threshold is given, it is used to make
sure that p-values below this are considered significant and thus
scored positively. Instead, those p-values above the given FDR are
considered insigificant and thus scored negatively.
dBUMscore(fit, method = c("pdf", "cdf"), fdr = NULL, scatter.bum = T)
scores: a vector of scores
The transformation from the input p-value x to the score
S(x) is based on the fitted beta-uniform mixture model with two
parameters \lambda and a: f(x|\lambda,a) = \lambda +
(1-\lambda)*a*x^{a-1}. Specifically, it considers the log-likelyhood
ratio between the signal and noise compoment of the model. The
probability density function (pdf) of the signal component and the
noise component are (1-\lambda)*a*(x^{a-1}-1) and \lambda +
(1-\lambda)*a, respectively. Accordingly, the cumulative distribution
function (cdf) of the signal component and the noise component are
\int_0^x (1-\lambda)*a*(x^{a-1}-1) \, \mathrm{d}x and
\int_0^x \lambda+(1-\lambda)*a \, \mathrm{d}x. In order to take
into account the significance of the p-value, the fdr threshold
is also used for down-weighting the score. According to how to measure
both components, there are two methods implemented for deriving the
score S(x):
S(x) =
log_2\frac{(1-\lambda)*a*(x^{a-1}-1)}{\lambda+(1-\lambda)*a} -
log_2\frac{(1-\lambda)*a*(\tau^{a-1}-1)}{\lambda+(1-\lambda)*a} =
log_2\big(\frac{x^{a-1}-1}{\tau^{a-1}-1}\big). For the purpose of
down-weighting scores, it must ensure
log_2\frac{(1-\lambda)*a*(\tau^{a-1}-1)}{\lambda+(1-\lambda)*a}
\geq 0 , that is, the constraint via \tau \leq
\big(\frac{\lambda+2*a*(1-\lambda)}{a*(1-\lambda)}\big)^{\frac{1}{a-1}}
S(x) = log_2\frac{\int_0^x
(1-\lambda)*a*(x^{a-1}-1) \, \mathrm{d}x}{\int_0^x
\lambda+(1-\lambda)*a \, \mathrm{d}x} - log_2\frac{\int_0^\tau
(1-\lambda)*a*(\tau^{a-1}-1) \, \mathrm{d}x}{\int_0^\tau
\lambda+(1-\lambda)*a \, \mathrm{d}x} =
log_2\frac{(1-\lambda)*(x^{a-1}-a)}{\lambda+(1-\lambda)*a} -
log_2\frac{(1-\lambda)*(\tau^{a-1}-a)}{\lambda+(1-\lambda)*a} =
log_2\big(\frac{x^{a-1}-a}{\tau^{a-1}-a}\big). For the purpose of
down-weighting scores, it must ensure
log_2\frac{(1-\lambda)*(\tau^{a-1}-a)}{\lambda+(1-\lambda)*a} \geq
0 , that is, the constraint via \tau \leq
\big(\frac{\lambda+2*a*(1-\lambda)}{1-\lambda}\big)^{\frac{1}{a-1}}
\tau
=\big[\frac{\lambda+(1-\lambda)*a-fdr*\lambda}{fdr*(1-\lambda)}\big]^{\frac{1}{a-1}},
i.e. the p-value corresponding to the exact fdr threshold. It can
be deduced from the definition of the false discovery rate: fdr
\doteq \frac{\int_0^\tau \lambda+(1-\lambda)*a \,
\mathrm{d}x}{\int_0^\tau \lambda+(1-\lambda)*a*x^{a-1} \,
\mathrm{d}x}. Notably, if the calculated \tau exceeds the
contraint, it will be reset to the maximum end of that constraint
# 1) generate an vector consisting of random values from beta distribution x <- rbeta(1000, shape1=0.5, shape2=1) # 2) fit a p-value distribution under beta-uniform mixture model fit <- dBUMfit(x)A total of p-values: 1000 Maximum Log-Likelihood: 319.8 Mixture parameter (lambda): 0.062 Shape parameter (a): 0.478# 3) calculate the scores according to the fitted BUM and fdr=0.01 # using "pdf" method scores <- dBUMscore(fit, method="pdf", fdr=0.01) # using "cdf" method scores <- dBUMscore(fit, method="cdf", fdr=0.01)