Package 'SetTest' reference manual

Title:	Group Testing Procedures for Signal Detection and Goodness-of-Fit
Description:	It provides cumulative distribution function (CDF), quantile, p-value, statistical power calculator and random number generator for a collection of group-testing procedures, including the Higher Criticism tests, the one-sided Kolmogorov-Smirnov tests, the one-sided Berk-Jones tests, the one-sided phi-divergence tests, etc. The input are a group of p-values. The null hypothesis is that they are i.i.d. Uniform(0,1). In the context of signal detection, the null hypothesis means no signals. In the context of the goodness-of-fit testing, which contrasts a group of i.i.d. random variables to a given continuous distribution, the input p-values can be obtained by the CDF transformation. The null hypothesis means that these random variables follow the given distribution. For reference, see [1]Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033; [2] Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379.
Authors:	Hong Zhang and Zheyang Wu
Maintainer:	Hong Zhang <[email protected]>
License:	GPL-2
Version:	0.3.0
Built:	2025-01-12 03:03:07 UTC
Source:	https://github.com/cran/SetTest

CDF of Berk-Jones statitic under the null hypothesis.

Description

CDF of Berk-Jones statitic under the null hypothesis.

Usage

pbj(q, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)
pbj(q, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

`q`	- quantile, must be a scalar.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

Value

The left-tail probability of the null distribution of B-J statistic at the given quantile.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

Examples

pval <- runif(10)
bjstat <- stat.phi(pval, s=1, k0=1, k1=10)$value
pbj(q=bjstat, M=diag(10), k0=1, k1=10)
pval <- runif(10)
bjstat <- stat.phi(pval, s=1, k0=1, k1=10)$value
pbj(q=bjstat, M=diag(10), k0=1, k1=10)

CDF of Higher Criticism statistic under the null hypothesis.

Description

CDF of Higher Criticism statistic under the null hypothesis.

Usage

phc(q, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)
phc(q, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

`q`	- quantile, must be a scalar.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

Value

The left-tail probability of the null distribution of HC statistic at the given quantile.

References

Examples

pval <- runif(10)
hcstat <- stat.phi(pval, s=2, k0=1, k1=5)$value
phc(q=hcstat, M=diag(10), k0=1, k1=10)
pval <- runif(10)
hcstat <- stat.phi(pval, s=2, k0=1, k1=5)$value
phc(q=hcstat, M=diag(10), k0=1, k1=10)

Statistical power of Berk and Jones test.

Description

Statistical power of Berk and Jones test.

Usage

power.bj(
  alpha,
  n,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)
power.bj(
  alpha,
  n,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)

Arguments

`alpha`	- type-I error rate.
`n`	- dimension parameter, i.e. the number of input statitics to construct B-J statistic.
`beta`	- search range parameter. Search range = (1, beta*n). Beta must be between 1/n and 1.
`method`	- different alternative hypothesis, including mixtures such as, "gaussian-gaussian", "gaussian-t", "t-t", "chisq-chisq", and "exp-chisq". By default, we use Gaussian mixture.
`eps`	- mixing parameter of the mixture.
`mu`	- mean of non standard Gaussian model.
`df`	- degree of freedom of t/Chisq distribution and exp distribution.
`delta`	- non-cenrality of t/Chisq distribution.

Details

We consider the following hypothesis test,

$H_0: X_i\sim F, H_a: X_i\sim G$

Specifically, $F = F_0$ and $G = (1-\epsilon)F_0+\epsilon F_1$ , where $\epsilon$ is the mixing parameter, $F_0$ and $F_1$ is speified by the "method" argument:

"gaussian-gaussian": $F_0$ is the standard normal CDF and $F = F_1$ is the CDF of normal distribution with $\mu$ defined by mu and $\sigma = 1$ .

"gaussian-t": $F_0$ is the standard normal CDF and $F = F_1$ is the CDF of t distribution with degree of freedom defined by df.

"chisq-chisq": $F_0$ is the CDF of Chisquare distribution with degree of freedom defined by df and $F = F_1$ is the CDF of non-central Chisquare distribution with degree of freedom defined by df and non-centrality defined by delta.

"exp-chisq": $F_0$ is the CDF of exponential distribution with parameter defined by df and $F = F_1$ is the CDF of non-central Chisqaure distribution with degree of freedom defined by df and non-centrality defined by delta.

Value

Power of BJ test.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

3. Jager, Leah; Wellner, Jon A. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

4. Berk, R.H. & Jones, D.H. Z. "Goodness-of-fit test statistics that dominate the Kolmogorov statistics". Wahrscheinlichkeitstheorie verw Gebiete (1979) 47: 47.

Examples

power.bj(0.05, n=10, beta=0.5, eps = 0.1, mu = 1.2)
power.bj(0.05, n=10, beta=0.5, eps = 0.1, mu = 1.2)

Statistical power of Higher Criticism test.

Description

Statistical power of Higher Criticism test.

Usage

power.hc(
  alpha,
  n,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)
power.hc(
  alpha,
  n,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)

Arguments

`alpha`	- type-I error rate.
`n`	- dimension parameter, i.e. the number of input statitics to construct Higher Criticism statistic.
`beta`	- search range parameter. Search range = (1, beta*n). Beta must be between 1/n and 1.
`method`	- different alternative hypothesis, including mixtures such as, "gaussian-gaussian", "gaussian-t", "t-t", "chisq-chisq", and "exp-chisq". By default, we use Gaussian mixture.
`eps`	- mixing parameter of the mixture.
`mu`	- mean of non standard Gaussian model.
`df`	- degree of freedom of t/Chisq distribution and exp distribution.
`delta`	- non-cenrality of t/Chisq distribution.

Details

We consider the following hypothesis test,

$H_0: X_i\sim F, H_a: X_i\sim G$

Specifically, $F = F_0$ and $G = (1-\epsilon)F_0+\epsilon F_1$ , where $\epsilon$ is the mixing parameter, $F_0$ and $F_1$ is speified by the "method" argument:

"gaussian-gaussian": $F_0$ is the standard normal CDF and $F = F_1$ is the CDF of normal distribution with $\mu$ defined by mu and $\sigma = 1$ .

"gaussian-t": $F_0$ is the standard normal CDF and $F = F_1$ is the CDF of t distribution with degree of freedom defined by df.

"t-t": $F_0$ is the CDF of t distribution with degree of freedom defined by df and $F = F_1$ is the CDF of non-central t distribution with degree of freedom defined by df and non-centrality defined by delta. "chisq-chisq": $F_0$ is the CDF of Chisquare distribution with degree of freedom defined by df and $F = F_1$ is the CDF of non-central Chisquare distribution with degree of freedom defined by df and non-centrality defined by delta.

Value

Power of HC test.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

Examples

power.hc(0.05, n=10, beta=0.5, eps = 0.1, mu = 1.2)
power.hc(0.05, n=10, beta=0.5, eps = 0.1, mu = 1.2)

Statistical power of phi-divergence test.

Description

Statistical power of phi-divergence test.

Usage

power.phi(
  alpha,
  n,
  s,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)
power.phi(
  alpha,
  n,
  s,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)

Arguments

`alpha`	- type-I error rate.
`n`	- dimension parameter, i.e. the number of input statitics to construct phi-divergence statistic.
`s`	- phi-divergence parameter. s = 2 is the higher criticism statitic.s = 1 is the Berk and Jones statistic.
`beta`	- search range parameter. Search range = (1, beta*n). Beta must be between 1/n and 1.
`method`	- different alternative hypothesis, including mixtures such as, "gaussian-gaussian", "gaussian-t", "t-t", "chisq-chisq", and "exp-chisq". By default, we use Gaussian mixture.
`eps`	- mixing parameter of the mixture.
`mu`	- mean of non standard Gaussian model.
`df`	- degree of freedom of t/Chisq distribution and exp distribution.
`delta`	- non-cenrality of t/Chisq distribution.

Details

We consider the following hypothesis test,

$H_0: X_i\sim F, H_a: X_i\sim G$

Specifically, $F = F_0$ and $G = (1-\epsilon)F_0+\epsilon F_1$ , where $\epsilon$ is the mixing parameter, $F_0$ and $F_1$ is speified by the "method" argument:

"gaussian-gaussian": $F_0$ is the standard normal CDF and $F = F_1$ is the CDF of normal distribution with $\mu$ defined by mu and $\sigma = 1$ .

"gaussian-t": $F_0$ is the standard normal CDF and $F = F_1$ is the CDF of t distribution with degree of freedom defined by df.

Value

Power of phi-divergence test.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

Examples

#If the alternative hypothesis Gaussian mixture with eps = 0.1 and mu = 1.2:#
power.phi(0.05, n=10, s=2, beta=0.5, eps = 0.1, mu = 1.2)
#If the alternative hypothesis Gaussian mixture with eps = 0.1 and mu = 1.2:#
power.phi(0.05, n=10, s=2, beta=0.5, eps = 0.1, mu = 1.2)

calculate the left-tail probability of phi-divergence under general correlation matrix.

Description

calculate the left-tail probability of phi-divergence under general correlation matrix.

Usage

pphi(q, M, k0, k1, s = 2, t = 30, onesided = FALSE, method = "ecc", ei = NULL)
pphi(q, M, k0, k1, s = 2, t = 30, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

`q`	- quantile, must be a scalar.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`s`	- the phi-divergence test parameter.
`t`	- numerical truncation parameter.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

Value

Left-tail probability of the phi-divergence statistics.

References

Examples

M = toeplitz(1/(1:10)*(-1)^(0:9)) #alternating polynomial decaying correlation matrix
pphi(q=2, M=M, k0=1, k1=5, s=2)
pphi(q=2, M=M, k0=1, k1=5, s=2, method = "ecc")
pphi(q=2, M=M, k0=1, k1=5, s=2, method = "ave")
pphi(q=2, M=diag(10), k0=1, k1=5, s=2)
M = toeplitz(1/(1:10)*(-1)^(0:9)) #alternating polynomial decaying correlation matrix
pphi(q=2, M=M, k0=1, k1=5, s=2)
pphi(q=2, M=M, k0=1, k1=5, s=2, method = "ecc")
pphi(q=2, M=M, k0=1, k1=5, s=2, method = "ave")
pphi(q=2, M=diag(10), k0=1, k1=5, s=2)

calculate the left-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Description

calculate the left-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Usage

pphi.omni(q, M, K0, K1, S, t = 30, onesided = FALSE, method = "ecc", ei = NULL)
pphi.omni(q, M, K0, K1, S, t = 30, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

`q`	- quantile, must be a scalar.
`M`	- correlation matrix of input statistics (of the input p-values).
`K0`	- vector of search range starts (from the k0th smallest p-value).
`K1`	- vector of search range ends (at the k1th smallest p-value).
`S`	- vector of the phi-divergence test parameters.
`t`	- numerical truncation parameter.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

Value

Left-tail probability of omnibus phi-divergence statistics.

References

Examples

M = matrix(0.3,10,10) + diag(1-0.3, 10)
pphi.omni(0.05, M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))
M = matrix(0.3,10,10) + diag(1-0.3, 10)
pphi.omni(0.05, M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))

Quantile of Berk-Jones statistic under the null hypothesis.

Description

Quantile of Berk-Jones statistic under the null hypothesis.

Usage

qbj(p, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL, err_thr = 1e-04)
qbj(p, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL, err_thr = 1e-04)

Arguments

`p`	- a scalar left probability that defines the quantile.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.
`err_thr`	- the error threshold. The default value is 1e-4.

Value

Quantile of BJ statistics.

References

Examples

## The 0.05 critical value of BJ statistic when n = 10:
qbj(p=.95, M=diag(10), k0=1, k1=5, onesided=FALSE)
qbj(p=1-1e-5, M=diag(10), k0=1, k1=5, onesided=FALSE, err_thr=1e-8)
## The 0.05 critical value of BJ statistic when n = 10:
qbj(p=.95, M=diag(10), k0=1, k1=5, onesided=FALSE)
qbj(p=1-1e-5, M=diag(10), k0=1, k1=5, onesided=FALSE, err_thr=1e-8)

Quantile of Higher Criticism statistics under the null hypothesis.

Description

Quantile of Higher Criticism statistics under the null hypothesis.

Usage

qhc(p, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL, err_thr = 1e-04)
qhc(p, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL, err_thr = 1e-04)

Arguments

`p`	- a scalar left probability that defines the quantile.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.
`err_thr`	- the error threshold. The default value is 1e-4.

Value

Quantile of HC statistics.

References

Examples

## The 0.05 critical value of HC statistic when n = 10:
qhc(p=.95, M=diag(10), k0=1, k1=5, onesided=FALSE)
qhc(p=1-1e-5, M=diag(10), k0=1, k1=5, onesided=FALSE, err_thr=1e-8)
## The 0.05 critical value of HC statistic when n = 10:
qhc(p=.95, M=diag(10), k0=1, k1=5, onesided=FALSE)
qhc(p=1-1e-5, M=diag(10), k0=1, k1=5, onesided=FALSE, err_thr=1e-8)

Quantile of phi-divergence statistic under the null hypothesis.

Description

Quantile of phi-divergence statistic under the null hypothesis.

Usage

qphi(
  p,
  M,
  k0,
  k1,
  s = 2,
  t = 30,
  onesided = FALSE,
  method = "ecc",
  ei = NULL,
  err_thr = 1e-04
)
qphi(
  p,
  M,
  k0,
  k1,
  s = 2,
  t = 30,
  onesided = FALSE,
  method = "ecc",
  ei = NULL,
  err_thr = 1e-04
)

Arguments

`p`	- a scalar left probability that defines the quantile.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`s`	- the phi-divergence test parameter.
`t`	- numerical truncation parameter.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.
`err_thr`	- the error threshold. The default value is 1e-4.

Value

Quantile of the phi-divergence statistics.

References

Examples

qphi(p=.95, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-3, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-3, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-6)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-6)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-8)
qphi(p=.95, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-3, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-3, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-6)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-6)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-8)

Construct Berk and Jones (BJ) statistics.

Description

Construct Berk and Jones (BJ) statistics.

Usage

stat.bj(p, k0 = 1, k1 = NA)
stat.bj(p, k0 = 1, k1 = NA)

Arguments

`p`	- vector of input p-values.
`k0`	- search range left end parameter. Default k0 = 1.
`k1`	- search range right end parameter. Default k1 = 0.5*number of input p-values.

Details

Let $p_{(i)}$ , $i = 1,...,n$ be a sequence of ordered p-values, the Berk and Jones statistic

$BJ = \sqrt{2n} \max_{1 \leq i\leq \lfloor \beta n \rfloor} (-1)^j \sqrt{i/n * \log(i/n/p_{(i)}) + (1-i/n) * \log((1-i/n)/(1-p_{(i)}))}$

and when $p_{(i)} > i/n$ , $j = 1$ , otherwise $j = 0$ .

Value

value - BJ statistic constructed from a vector of p-values.

location - the order of the p-values to obtain BJ statistic.

stat - vector of marginal BJ statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Jager, Leah; Wellner, Jon A. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

3. Berk, R.H. & Jones, D.H. Z. "Goodness-of-fit test statistics that dominate the Kolmogorov statistics". Wahrscheinlichkeitstheorie verw Gebiete (1979) 47: 47.

Examples

stat.bj(runif(10))
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.bj(p.test, k0 = 2, k1 = 20)
stat.bj(runif(10))
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.bj(p.test, k0 = 2, k1 = 20)

Construct Higher Criticism (HC) statistics.

Description

Construct Higher Criticism (HC) statistics.

Usage

stat.hc(p, k0 = 1, k1 = NA)
stat.hc(p, k0 = 1, k1 = NA)

Arguments

`p`	- vector of input p-values.
`k0`	- search range left end parameter. Default k0 = 1.
`k1`	- search range right end parameter. Default k1 = 0.5*number of input p-values.

Details

Let $p_{(i)}$ , $i = 1,...,n$ be a sequence of ordered p-values, the higher criticism statistic

$HC = \sqrt{n} \max_{1 \leq i\leq \lfloor \beta n \rfloor} [i/n - p_{(i)}] /\sqrt{p_{(i)}(1 - p_{(i)})}$

Value

value - HC statistic constructed from a vector of p-values.

location - the order of the p-values to obtain HC statistic.

stat - vector of marginal HC statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

Examples

stat.hc(runif(10))
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.hc(p.test, k0 = 1, k1 = 10)
stat.hc(runif(10))
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.hc(p.test, k0 = 1, k1 = 10)

Construct phi-divergence statistics.

Description

Construct phi-divergence statistics.

Usage

stat.phi(p, s, k0 = 1, k1 = NA)
stat.phi(p, s, k0 = 1, k1 = NA)

Arguments

`p`	- vector of input p-values.
`s`	- phi-divergence parameter. s = 2 is the higher criticism statitic.s = 1 is the Berk and Jones statistic.
`k0`	- search range left end parameter. Default k0 = 1.
`k1`	- search range right end parameter. Default k1 = 0.5*number of input p-values.

Details

Let $p_{(i)}$ , $i = 1,...,n$ be a sequence of ordered p-values, the phi-divergence statistic

$PHI = \sqrt{2n}/(s - s^2) \max_{1 \leq i\leq \lfloor \beta n \rfloor} (-1)^j \sqrt{1 - (i/n)^s (p_{(i)})^s - (1-i/n)^{(1-s)} * (1-p_{(i)})^{(1-s)}}$

and when $p_{(i)} > i/n$ , $j = 1$ , otherwise $j = 0$ .

Value

value - phi-divergence statistic constructed from a vector of p-values.

location - the order of the p-values to obtain phi-divergence statistic.

stat - vector of marginal phi-divergence statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Jager, Leah; Wellner, Jon A. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

Examples

stat.phi(runif(10), s = 2)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.phi(p.test, s = 0.5, k0 = 2, k1 = 5)
stat.phi(runif(10), s = 2)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.phi(p.test, s = 0.5, k0 = 2, k1 = 5)

calculate the omnibus phi-divergence statistics under general correlation matrix.

Description

calculate the omnibus phi-divergence statistics under general correlation matrix.

Usage

stat.phi.omni(
  p,
  M,
  K0 = rep(1, 2),
  K1 = rep(length(M[1, ]), 2),
  S = c(1, 2),
  t = 30,
  onesided = FALSE,
  method = "ecc",
  ei = NULL
)
stat.phi.omni(
  p,
  M,
  K0 = rep(1, 2),
  K1 = rep(length(M[1, ]), 2),
  S = c(1, 2),
  t = 30,
  onesided = FALSE,
  method = "ecc",
  ei = NULL
)

Arguments

`p`	- input pvalues.
`M`	- correlation matrix of input statistics (of the input p-values).
`K0`	- vector of search range starts (from the k0th smallest p-value).
`K1`	- vector of search range ends (at the k1th smallest p-value).
`S`	- vector of the phi-divergence test parameters.
`t`	- numerical truncation parameter.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

References

Examples

p.test = runif(10)
M = toeplitz(1/(1:10)*(-1)^(0:9)) #alternating polynomial decaying correlation matrix
stat.phi.omni(p.test, M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))
p.test = runif(10)
M = toeplitz(1/(1:10)*(-1)^(0:9)) #alternating polynomial decaying correlation matrix
stat.phi.omni(p.test, M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))

Multiple comparison test using Berk and Jones (BJ) statitics.

Description

Multiple comparison test using Berk and Jones (BJ) statitics.

Usage

test.bj(prob, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)
test.bj(prob, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

`prob`	- vector of input p-values.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

Value

pvalue - the p-value of the Berk-Jones test.

bjstat - the Berk-Jones statistic.

location - the order of the input p-values to obtain BJ statistic.

References

Examples

test.bj(runif(10), M=diag(10), k0=1, k1=10)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 2*(1 - pnorm(abs(stat.test)))
test.bj(p.test, M=diag(20), k0=1, k1=10)
test.bj(runif(10), M=diag(10), k0=1, k1=10)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 2*(1 - pnorm(abs(stat.test)))
test.bj(p.test, M=diag(20), k0=1, k1=10)

Multiple comparison test using Higher Criticism (HC) statitics.

Description

Multiple comparison test using Higher Criticism (HC) statitics.

Usage

test.hc(prob, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)
test.hc(prob, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

`prob`	- vector of input p-values.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

Value

pvalue - The p-value of the HC test.

hcstat - HC statistic.

location - the order of the input p-values to obtain HC statistic.

References

Examples

pval.test = runif(10)
test.hc(pval.test, M=diag(10), k0=1, k1=10)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 2*(1 - pnorm(abs(stat.test)))
test.hc(p.test, M=diag(20), k0=1, k1=10)
pval.test = runif(10)
test.hc(pval.test, M=diag(10), k0=1, k1=10)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 2*(1 - pnorm(abs(stat.test)))
test.hc(p.test, M=diag(20), k0=1, k1=10)

Multiple comparison test using phi-divergence statistics.

Description

Multiple comparison test using phi-divergence statistics.

Usage

test.phi(prob, M, k0, k1, s = 2, onesided = FALSE, method = "ecc", ei = NULL)
test.phi(prob, M, k0, k1, s = 2, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

`prob`	- vector of input p-values.
`M`	- correlation matrix of input statistics (of the input p-values).
`k0`	- search range starts from the k0th smallest p-value.
`k1`	- search range ends at the k1th smallest p-value.
`s`	- phi-divergence parameter. s = 2 is the higher criticism statitic.s = 1 is the Berk and Jones statistic.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

Value

pvalue - The p-value of the phi-divergence test.

phistat - phi-diergence statistic.

location - the order of the input p-values to obtain phi-divergence statistic.

References

Examples

stat.test = rnorm(20) # Z-scores
p.test = 2*(1 - pnorm(abs(stat.test)))
test.phi(p.test, M=diag(20), s = 0.5, k0=1, k1=10)
test.phi(p.test, M=diag(20), s = 1, k0=1, k1=10)
test.phi(p.test, M=diag(20), s = 2, k0=1, k1=10)
stat.test = rnorm(20) # Z-scores
p.test = 2*(1 - pnorm(abs(stat.test)))
test.phi(p.test, M=diag(20), s = 0.5, k0=1, k1=10)
test.phi(p.test, M=diag(20), s = 1, k0=1, k1=10)
test.phi(p.test, M=diag(20), s = 2, k0=1, k1=10)

calculate the right-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Description

calculate the right-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Usage

test.phi.omni(prob, M, K0, K1, S, onesided = FALSE, method = "ecc", ei = NULL)
test.phi.omni(prob, M, K0, K1, S, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

`prob`	- vector of input p-values.
`M`	- correlation matrix of input statistics (of the input p-values).
`K0`	- vector of search range starts (from the k0th smallest p-value).
`K1`	- vector of search range ends (at the k1th smallest p-value).
`S`	- vector of the phi-divergence test parameters.
`onesided`	- TRUE if the input p-values are one-sided.
`method`	- default = "ecc": the effective correlation coefficient method in reference 2. "ave": the average method in reference 3, which is an earlier version of reference 2. The "ecc" method is more accurate and numerically stable than "ave" method.
`ei`	- the eigenvalues of M if available.

Value

p-value of the omnibus test.

p-values of the individual phi-divergence test.

References

Examples

M = matrix(0.3,10,10) + diag(1-0.3, 10)
test.phi.omni(runif(10), M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))
M = matrix(0.3,10,10) + diag(1-0.3, 10)
test.phi.omni(runif(10), M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))

Package 'SetTest'

Help Index

CDF of Berk-Jones statitic under the null hypothesis.

Description

Usage

Arguments

Value

References

See Also

Examples

CDF of Higher Criticism statistic under the null hypothesis.

Description

Usage

Arguments

Value

References

See Also

Examples

Statistical power of Berk and Jones test.

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Statistical power of Higher Criticism test.

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Statistical power of phi-divergence test.

Description

Usage

Arguments

Details

Value

References

See Also

Examples

calculate the left-tail probability of phi-divergence under general correlation matrix.

Description

Usage

Arguments

Value

References

Examples

calculate the left-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Description

Usage

Arguments

Value

References

Examples

Quantile of Berk-Jones statistic under the null hypothesis.

Description

Usage

Arguments

Value

References

See Also

Examples

Quantile of Higher Criticism statistics under the null hypothesis.

Description

Usage

Arguments

Value

References

See Also

Examples

Quantile of phi-divergence statistic under the null hypothesis.

Description

Usage

Arguments

Value