Ch04-dists: The Distribution family object

In pwrFDR: FDR Power

Description

The pwrFDR package currently incorporates 3 distribution types, normal, t and F. The first two of these are strictly for statistics formed from two group comparison while the third is for statistics formed from the omnibus test of any difference among an arbitrary number of groups >=2. The structure is general and user expandable. One must specify the density, CDF and quantile function for a given distribution and its parameters under the null and under the alternative. These parameters must be expressions to be evaluated inside the kernel of the power program, functions of the arguments n.sample, groups and effect.size. This is not used directly by the user at all unless she (he) wants to add a distribution type.

Format

A data frame with 3 observations on the following 6 variables.

pars0

a list vector having components 'c(nd, p1, p2, ...)' where 'nd' is the distribution number starting with 0, and p1, p2, ..., are paramters of the distribution, which are functions of 'n.sample', 'groups' and 'effect.size' as mentioned above. These must be expressed as a call e.g. as.call(expression(c, nd, p1, p2, ...)) etc. 'pars0' are the parameters under the null.

pars1

a list vector. See directly above. Parameters under the alternative.

minv

a list vector with components given the values -Inf or 0, which will be used to decide if the two sided corrections are used or not.

ddist

a list vector with components set to functions, each one computing the probability density function corresponding to the particular distribution. A function of arguments 'x' and 'par'. See details below.

pdist

a list vector with components set to the functions, each one computing the cumulative distribution function corresponding to the particular distribution. A function of arguments 'x' and 'par'. See details below.

qdist

a list vector with components set to the functions, each one computing the quantile function (inverse cumulative distribution function) corresponding to the particular distribution. A function of arguments 'x' and 'par'. See details below.

Details

dists is a data.frame with components pars0, pars1, minv, ddist, pdist, and qdist. For the three available distribution options, "normal", "t" and "f", the components pars0 and pars1 take the following form:

1. pars0	pars1
2. c(0,ncp=0,sd=1)	c(0,ncp=.NCP.,sd=1)
3. c(1,ncp=0,ndf=.DF.)	c(1,ncp=.NCP.,ndf=.DF.)
4. c(2,ncp=0,ndf1=groups-1,ndf2=.DF.)	c(2,ncp=.NCP.^2,ndf1=groups-1,ndf2=.DF.)

The component minv gives the minumum value of the support set of the distribution. For the above named three available distribution options, minv is set to the values -Inf, -Inf and 0, respectively. The components ddist, pdist, and qdist contain functions defining the density, CDF, and quantile function, respectively. For the above named three available distribution options, ddist takes the following form:

1. ddist

2. function (x, par) dnorm(x, mean = par[2], sd = par[3])

3. function (x, par) dt(x, ncp = par[2], df = par[3])

4. function (x, par) df(x, ncp = par[2], df1 = par[3], df2 = par[4])

The components pdist and qdist are nearly identical to the component ddist, but with pnorm, pt, pf and qnorm, qt, qf replacing dnorm, dt and df, respectively.
The variables, .NCP. and .DF. named above are defined within the functions in which ddist is used based upon corresponding expressions, NCP and DF. These expressions currently contain 3 component expressions, one for each of the available test types, "paired", "balanced" and "unbalanced".
NCP is currently defined:

1. NCP

expression(n.sample^0.5*effect.size,(n.sample/groups)^0.5*effect.size,

((n.sample-1)/(1+sum((n.sample-1)/(nii.sample-1))))^0.5*effect.size)

and DF is currently defined:

1. DF

expression(n.sample - 1, groups * (n.sample - 1),

groups^2*(n.sample-1)/(1+sum((n.sample-1)/(nii.sample-1))))

Source

This isn't 'data' data, its a kind of a 'family' object.

Izmirlian G. (2020) Strong consistency and asymptotic normality for quantities related to the Benjamini-Hochberg false discovery rate procedure. Statistics and Probability Letters; <doi:10.1016/j.spl.2020.108713>

Izmirlian G. (2017) Average Power and λ-power in Multiple Testing Scenarios when the Benjamini-Hochberg False Discovery Rate Procedure is Used. <arXiv:1801.03989>

pwrFDR documentation built on May 12, 2021, 5:07 p.m.