fixed_pi0_table: Provide estimates of the functional proportion at varying...
In StoreyLab/fFDR: Functional FDR for multiple hypothesis testing

Description Usage Arguments Details Value

Estimate the functional proportion pi0(z) at varying values of lambda and tau when pi0(z) is independent of the informative variable z

fixed_pi0_table(
  p,
  z0,
  lambda = seq(0, 0.9, 0.05),
  tau = seq(0, 0.9, 0.05),
  pi0.method = NULL,
  lambda.df = 3,
  tau.df = 3
)

`p`	A vector of p-values
`z0`	A vector of observations from the informative variable, of the same length as `p`
`lambda`	Choices of the tuning parameter "lambda" (for the p-value) used to estimate pi0(z)
`tau`	Choices of the tuning parameter "tau" (for the informative variable) used to estiamte pi0(z)
`pi0.method`	The method used to estimate pi0(z), being either "smoother" (default) or "bootstrap". The method depends on the tuning parameters "lambda" and "tau"
`lambda.df`	Degrees of freedom for smoother in argument "lambda"
`tau.df`	Degrees of freedom for smoother in argument "tau"

Assume the random variable z0 may affect the power of a statistical test (that induces the p-values) or the likelihood of a true null hypothesis. The m observations z0_i, i=1,...,m of z0 are quantile transformed into z_i, i=1,...,m such that z_i = rank(z0_i) / m, where rank(z0_i) is the rank of z0_i among z0_i, i=1,...,m. Consequently, z_i, i=1,...,m are approximately uniformly distributed on the interval [0,1]. When z_i, i=1,...,m are regarded as observations from the random variable z, then z is approximately uniformly distributed on [0,1]. Namely, z0 has been quantile transformed into z, and they are equivalent. Further, z or z0 is referred to as the informative variable.

A data.frame with the following columns

`lambda`	Choices of lambda as the thresholds for p
`tau`	Choices of tau as the thresholds for z
`L`	The number of hypotheses for which p > lambda and z > tau
`pi0hat`	The estimated pi0(z) using these choices of lambda and tau
`variance`	The estimated variance of pi0hat
`se`	The estimated standard error of pi0hat
`bias`	The estimated bias of pi0hat
`MSEhat`	The estimated mean squared error (MSE) of pi0hat