Assuming normality of noncentrality parameters (parncpt) or a mixture of two normal distributions (parncpt2), the MLE of its standard deviation(s) (and possibly mean(s) also) is estimated from observed t-statistics
parncpt(tstat, df, zeromean = TRUE, ...)
parncpt.bfgs.0mean(tstat, df, starts, grids, approximation = "int2", ...)
parncpt.bfgs.non0mean(tstat, df, starts, grids, approximation = "int2", ...)
parncpt.momeff(tstat,n1,n2=n1,zeromean,gamma2,lower.df=6.1,upper.df=100,approx=TRUE)
parncpt2(tstat, df, common=c('mean','sd'), ...)numeric vector of t-statistics
numeric vector of degrees of freedom
logical; if TRUE, then mean of noncentrality parameters is assumed to be zero and is not estimated.
character vector. Allowed values are 'mean', 'sd', 'none'. If 'none' is present, common must be a scalar, and an unrestricted 2-component normal mixture is fit to ncp distribution. NULL is treated the same as 'none'. If mean is present, the means of the two normal components of the ncp distribution are assumed to be negative of each other. If sd is present, the sandard deviations of the two normal components of the ncp distribution are assumed to be common.
Other arguments to optim.
An optional vector of starting values. If missing, a grid search will be performed to get a good starting value.
A list of three components (lower, upper, ngrid) defining the grids to be searched in find a good starting value.
Each component is a numeric vector of the same length as the number of parameters. lower and upper give the bounds, and
ngrid specifies the number of points for each dimension.
Methods of approximating the noncentral t-density. int2 is exact for integer df, but interpolate to fractional df.
'laplace' is the laplacian approximation; 'saddlepoint' is the saddlepoint approximation; 'none' computes the (sort of) exact density using the default dt function.
Treatment 1 sample size
Treatment 2 sample size
Gamma square parameter, i.e., variance of effect sizes.
lower bound of degrees of freedom, in case of n1 is missing
upper bound of degrees of freedom, in case of n1 is missing
logical, indicating if no exact solutions are available, whether approx. solutions are returned.
Except for parncpt2, the result is a list with class attribute being c('parncpt', 'ncpest').
proportion of true nulls
mean of ncp
SD of ncp
a list of tstat and df
an object of class logLik. Call logLik.ncpest to extract. Similarly, AIC is callable.
the (effective) number of parameters in the model
estimated parameters. Call coef.ncpest to extract.
the negative loglikelihood function that is minimized
analytic gradiant at the estimate
numeric hessian at the estimate
the number of test statistics
For parncpt2, the result is a list with class attribute being c('parncpt2', 'parncpt', 'ncpest'), which is a list with the follwoing additional components:
proportion of non-nulls of which the noncentrality parameters come from the normal component with smaller mean.
the mixing proportion of the normal component of the ncp distribution with smaller mean.
the mean of the normal component of the ncp distribution with smaller mean.
the SD of the normal component of the ncp distribution with smaller mean.
the mean of the normal component of the ncp distribution with larger mean.
the SD of the normal component of the ncp distribution with larger mean.
parncpt calls either parncpt.bfgs.0mean or parncpt.bfgs.non0mean, depending whether zeromean is TRUE or FALSE.
Both parncpt.bfgs.0mean and parncpt.bfgs.non0mean use the 'L-BFGS-B' algorithm by calling optim. All gradiants are analytical, but the Hessian is only numerical approximation.
The first parmater is always pi0, i.e., the proportion of true null hypotheses; the last parameter is always the standard deviation of noncentrality parameters;
for parncpt.bfgs.non0mean the middle parameter is the mean of noncentrality parameters, whereas for parncpt.bfgs.0mean the mean is set to 0 a priori.
parncpt2 calls parncpt2.constrOptim to find the maximum likelihood estimates of parameters when the noncentrality parameter distribution is assumed to be a mixture of two normals. The parameterization being used is such that pi0 is the proportion of true nulls and pi1 is the proportion of non-nulls of which the noncentrality parameters come from the normal component with smaller mean. Therefore, for the noncentrality parameter distribution, tau=pi1/(1-pi0) is the mixing proportion for the normal component with smaller mean.
Qu L, Nettleton D, Dekkers JCM. (2012) Improved Estimation of the Noncentrality Parameter Distribution from a Large Number of $t$-statistics, with Applications to False Discovery Rate Estimation in Microarray Data Analysis. Biometrics, 68, 1178--1187.
sparncpt, nparncpt,
fitted.parncpt, plot.parncpt, summary.parncpt,
coef.ncpest, logLik.ncpest, vcov.ncpest,
AIC, dncp
# NOT RUN {
data(simulatedTstat)
(pfit=parncpt(tstat=simulatedTstat, df=8, zeromean=FALSE)); plot(pfit)
(pfit0=parncpt(tstat=simulatedTstat, df=8, zeromean=TRUE)); plot(pfit0)
(pfit2=parncpt2(tstat=simulatedTstat, df=8)); plot(pfit2)
# }
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