## Function to fit a p-value distribution under beta-uniform mixture model

### Description

dBUMfit is supposed to take as input a vector of p-values for deriving their distribution under beta-uniform mixture model (see Note below). The density distribution of input p-values is expressed as a mixture of two components: one for the null hypothesis (the noise component) and the other for the alternative hypothesis (the signal component). The noise component is the uniform density, while the signal component is the remainder of the mixture distribution. It returns an object of class "BUM".

### Usage

dBUMfit(x, ntry = 1, hist.bum = T, contour.bum = T, verbose = T)

### Arguments

x
a vector containing input p-values
ntry
an integeter specifying how many trys are used to find the optimised parameters by maximum likelihood estimation
hist.bum
logical to indicate whether the histogram graph should be drawn
contour.bum
logical to indicate whether a contour plot should be drawn to show the log likelihood as a function of two parameters (a and lambda) in the beta-uniform mixture model
verbose
logical to indicate whether the messages will be displayed in the screen. By default, it sets to true for display

### Value

an object of class "BUM", a list with following elements:

• lambda: estimated mixture parameter
• a: estimated shape parameter
• NLL: Negative log-likelihood
• pvalues: the input pvalues
• call: the call that produced this result

### Note

The probability density function of p-values under the Beta-Uniform Mixture model is formulated as: f(x|\lambda,a) = \lambda + (1-\lambda)*a*x^{a-1}. The model names after mixing two distributions:

• the uniform distribution with the density function as \frac{1}{b-a}|_{a=0}^{b=1}=1
• the beta distribution with the density function as \frac{\Gamma(a+b)}{\Gamma(a)+\Gamma(b)}*x^{a-1}*(1-x)^{b-1}|_{b=1}=a*x^{a-1}

Both are mixed via \lambda. The mixture parameter \lambda measures the contribution from the uniform distribution. Accordingly, 1-\lambda measures the contribution from the beta distribution. Notably, the probability density function of the beta distribution can be splitted into two parts (rather than the exclusitive signal):

• the constant part as noise: a*x^{a-1}|_{x=1}=a
• the rest part as signal: a*(x^{a-1}-1)

In other words, there is no signal at x=1 but all being noise. It is a conservative, upper bound estimation of the noise. Therefore, the probability density function in the model can be decomposed into signal-noise components:

• the signal component: (1-\lambda)*a*(x^{a-1}-1)
• the noise component: \lambda + (1-\lambda)*a

It is misleading to simply view \lambda as the noise component and (1-\lambda)*a*x^{a-1} as the signal component, just as wrongly do in the literatures (e.g. http://www.ncbi.nlm.nih.gov/pubmed/18586718)

### Examples

# 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: 311.1
Mixture parameter (lambda): 0.000
Shape parameter (a): 0.498
fit$lambda [1] 1e-05 fit$a

[1] 0.4979076


## Source code

dBUMfit.r

## Source man

dBUMfit.Rd dBUMfit.pdf

dBUMscore