Title:	Confidence Curves and P-Value Functions for Meta-Analysis
Version:	0.1.0
Description:	Provides tools for the combination of individual study results in meta-analyses using 'p-value' functions. Implements various combination methods including those by Fisher, Stouffer, Tippett, Edgington along with weighted generalizations. Contains functionality for the visualization and calculation of confidence curves and drapery plots to summarize evidence across studies.
License:	GPL (≥ 3)
Language:	en-US
BugReports:	https://github.com/SaveFonta/confMeta/issues
URL:	https://github.com/SaveFonta/confMeta
Encoding:	UTF-8
RoxygenNote:	7.3.3
Imports:	ggplot2, meta, metafor, patchwork, ReplicationSuccess, scales, stats, rlang
Suggests:	knitr, rmarkdown, bayesmeta, testthat (≥ 3.0.0)
Config/testthat/edition:	3
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2026-04-13 13:53:35 UTC; Menelao
Author:	Saverio Fontana [aut, cre], Felix Hofmann [aut], Leonhard Held [aut], Samuel Pawel [aut]
Maintainer:	Saverio Fontana <savefontana@gmail.com>
Repository:	CRAN
Date/Publication:	2026-04-20 12:50:08 UTC

Visualizations of confMeta objects

Description

Plots one or more confMeta objects. This function can create two types of plots: the p-value function plot (also known as drapery plot) and the forest plot. This allows for direct visual comparison of different p-value functions.

Optionally, a Bayesian meta-analysis object created with the bayesmeta::bayesmeta() function can be supplied via the bayesmeta argument. When provided, its posterior summary is displayed as an additional diamond at the bottom of the forest plot for comparison with the frequentist methods

Important Note: If supplying a Bayesian meta-analysis object via the bayesmeta argument, this function explicitly extracts the 95% credible intervals. If the Bayesian model was fit using a different credible level, the function will crash.

Usage

## S3 method for class 'confMeta'
autoplot(
  ...,
  type = c("p", "forest"),
  diamond_height = 0.5,
  v_space = 1.5,
  scale_diamonds = TRUE,
  show_studies = TRUE,
  drapery = TRUE,
  reference_methods_p = c("re"),
  reference_methods_forest = c("re", "hk", "hc"),
  xlim = NULL,
  xlim_p = NULL,
  xlim_forest = NULL,
  same_xlim = TRUE,
  xlab = NULL,
  n_breaks = 7,
  bayesmeta = NULL,
  n_points = 1000
)

Arguments

Value

An object of class ggplot containing the specified plot(s).

Creating confMeta objects

Description

Function to create objects of class confMeta. This is the main class within the package. For an overview of available methods run methods(class = "confMeta").

Usage

confMeta(
  estimates,
  SEs,
  study_names = NULL,
  conf_level = 0.95,
  fun,
  fun_name = NULL,
  w = NULL,
  MH = FALSE,
  table_2x2 = NULL,
  measure = NULL,
  method.tau = "PM",
  adhoc.hakn.ci = "IQWiG6",
  ...
)

Arguments

Value

An S3 object of class confMeta containing the following elements:

• estimates, SEs, w, study_names, conf_level: Values used to build the object.

• individual_cis: Matrix of Wald-type confidence intervals for each study.

• p_fun: The p-value function used for combined inference.

• joint_cis: Combined confidence interval(s). Calculated by finding the values where the p-value function is larger than the significant level (1 - conf_level).

• gamma: The local minima within the range of the individual effect estimates. Column x refers to the mean \mu and column y contains the corresponding p-value.

• p_max: The local maxima of the p-value function. Column x refers to the mean \mu and column y contains the corresponding p-value.

• p_0: The value of the p-value at \mu = 0.

• comparison_cis: Combined confidence intervals calculated with other methods. These can be used for comparison purposes. Currently, these other methods are fixed effect (IV or Mantel-Haenszel), random effects (IV), Hartung & Knapp, and Henmi & Copas.

• comparison_p_0: The same as in element p_0 but for the comparison methods (fixed effect, random effects, Hartung & Knapp, Henmi & Copas).

• heterogeneity: A data frame containing heterogeneity statistics (Cochran's Q, p-value for Q, I^2, \tau^2) calculated using the REML estimator.

• table_2x2: The 2x2 table data frame (if provided), otherwise NULL.

Confidence intervals

Individual confidence intervals are calculated as:

x_i \pm \Phi^{-1}(1 - \alpha/2) \cdot \sigma_i

where x_i are the estimates, \sigma_i the SEs, and \alpha = 1 - \mathrm{conf\_level}.

Combined confidence intervals are found by inverting the p-value function, identifying all \mu where the p-value exceeds the significance level (1 - conf_level).

Also, the HK method uses adhoc.hakn.ci = "IQWiG6" by default, i.e., it uses variance correction if the HK confidence interval is narrower than the CI from the classic random effects model with the DerSimonian-Laird estimator (IQWiG, 2022).

See Also

p_value_functions, autoplot.confMeta

Examples

# Simulate effect estimates and standard errors
set.seed(42)
n <- 5
estimates <- rnorm(n)
SEs <- rgamma(n, 5, 5)
conf_level <- 0.95

# Construct a simple confMeta object using p_edgington as
# the p-value function
cm <- confMeta(
  estimates = estimates,
  SEs = SEs,
  conf_level = conf_level,
  fun = p_edgington,
  fun_name = "Edgington",
  input_p = "greater"
)
    
cm2 <- confMeta(
  estimates = estimates,
  SEs = SEs,
  conf_level = conf_level,
  fun = p_edgington_w,
  w = 1/SEs,
  fun_name = "Edgington (1/SE)",
  input_p = "greater"
)

# print the objects
cm
cm2

# Plot the objects
autoplot(cm, cm2, type = "p")                   # p-value function plot
autoplot(cm, cm2, type = "forest")              # forest plot
autoplot(cm, cm2, type = c("p", "forest"))      # both

Estimation of between-study heterogeneity

Description

estimate_tau2 estimates the between-study heterogeneity \tau^{2} using an additive model. The resulting parameter \tau^2 is estimated through a call to meta::metagen with TE = estimates and seTE = SEs. Other arguments to meta::metagen can be passed via the ... argument. If no arguments are passed via ..., the following defaults are applied.

• sm = "MD"

• method.tau = "REML"

• control = list(maxiter = 1e5, stepadj = 0.25)

However, each of these defaults can be overwritten via ....

estimate_phi estimates the between-study heterogeneity \phi using a multiplicative model. The function is a modified version of Page 3 of:
Accounting for heterogeneity in meta-analysis using a multiplicative model – an empirical study, Mawdsley D. et al. 2016. doi:10.1002/jrsm.1216

Usage

estimate_tau2(estimates, SEs, ...)

estimate_phi(estimates, SEs)

Arguments

Value

For estimate_tau2: the estimated heterogeneity parameter \tau^2, a numeric vector of length 1.

For estimate_phi: the estimated heterogeneity parameter \phi, a numeric vector of length 1.

Examples

    # Example estimates and std. errors
    estimates <- c(0.21, 0.53, 0.24, 0.32, 0.19)
    SEs <- c(0.19, 0.39, 0.7, 1, 0.97)

    # Estimating heterogeneity using the additive model
    estimate_tau2(estimates = estimates, SEs = SEs)
    # Example estimates and std. errors
    estimates <- c(0.21, 0.53, 0.24, 0.32, 0.19)
    SEs <- c(0.19, 0.39, 0.7, 1, 0.97)
    
    estimate_phi(estimates = estimates, SEs = SEs)

Edgington's method

Description

Edgington’s method for combining p-values across studies. The method forms a sum of individual study p-values and evaluates it against the exact or approximate null distribution.

Under the global null hypothesis, the null distribution of the sum is given by the Irwin–Hall distribution. For a weighted generalization of this procedure, see p_edgington_w.

Usage

p_edgington(
  estimates,
  SEs,
  mu = 0,
  heterogeneity = "none",
  phi = NULL,
  tau2 = NULL,
  check_inputs = TRUE,
  input_p = "greater",
  output_p = "two.sided",
  approx = TRUE
)

Arguments

Details

The classical Edgington statistic is defined for k studies as

S = \sum_{i=1}^k p_i,

where p_i are individual study p-values. Under the global null hypothesis, each p_i is assumed to be uniformly distributed on [0, 1].

Important note on orientation: Edgington's method is orientation-invariant. The combined p-value is symmetric with respect to the direction of the one-sided p-values (controlled by the input_p argument).

Specifically, computing the Edgington combined p-value for the "greater" alternative results in 1 minus the Edgington combined p-value for the "less" alternative.

Value

A numeric vector of combined p-values corresponding to each value of mu.

Null Distribution and Approximation

The combined p-value, p_E, is the probability of observing a sum less than or equal to S under the null hypothesis. This is computed in one of two ways:

• Exact Method: The function uses the exact Irwin-Hall distribution to compute the combined p-value:

  p_E = \frac{1}{k!} \sum_{j=0}^{\lfloor S \rfloor} (-1)^j \binom{k}{j} (S - j)^k

• Normal Approximation: For a large number of studies (k \geq 12), the distribution of the sum is approximated by a Normal distribution with:

  \mathrm{E}[S] = \frac{k}{2}

  \mathrm{Var}(S) = \frac{k}{12}

Output p-value

The final output depends on the output_p and input_p arguments:

• If output_p = "two.sided" (the default) and the inputs are one-sided (input_p is "greater" or "less"), the function combines the one-sided p-values to obtain the intermediate combined p-value p_c, and returns a symmetrized, two-sided p-value: p_{2s} = 2 \min(p_c, 1 - p_c).

• If output_p = "one.sided", the function returns the inherently one-sided combined p-value p_c directly, without symmetrization.

• If input_p is "two.sided", the input p_i are already two-sided, and no further symmetrization is applied.

References

Edgington, E. S. (1972). An additive method for combining probability values from independent experiments. The Journal of Psychology, 80(2): 351-363. doi:10.1080/00223980.1972.9924813

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

See Also

Other p-value combination functions: p_edgington_w(), p_fisher(), p_hmean(), p_pearson(), p_stouffer(), p_tippett(), p_wilkinson()

Examples

# Simulating estimates and standard errors
n <- 15
estimates <- rnorm(n)
SEs <- rgamma(n, 5, 5)

# Set up a vector of means under the null hypothesis
mu <- seq(
  min(estimates) - 0.5 * max(SEs),
  max(estimates) + 0.5 * max(SEs),
  length.out = 100
)

# Using Edgington's method to calculate the combined p-value
p_edgington(
    estimates = estimates,
    SEs = SEs,
    mu = mu,
    heterogeneity = "none",
    output_p = "two.sided",
    input_p = "greater",
    approx = TRUE
)

Weighted Edgington’s method

Description

Weighted generalization of Edgington’s method for combining p-values across studies. The method forms a weighted sum of individual study p-values and evaluates it against the exact or approximate null distribution.

If all weights are equal, this reduces to the classical Edgington procedure, where the null distribution is given by the Irwin–Hall distribution.

Usage

p_edgington_w(
  estimates,
  SEs,
  mu = 0,
  heterogeneity = "none",
  phi = NULL,
  tau2 = NULL,
  check_inputs = TRUE,
  input_p = "greater",
  output_p = "two.sided",
  w = rep(1, length(estimates)),
  approx = TRUE,
  approx_rule = "neff",
  neff_cut = 12
)

Arguments

Details

The weighted Edgington statistic is defined for k studies as

S = \sum_{i=1}^k w_i p_i,

where w_i are positive study weights and p_i are individual study p-values. Under the global null hypothesis, each p_i is assumed to be uniformly distributed on [0, 1].

Value

A numeric vector of combined p-values corresponding to each value of mu.

Null Distribution and Approximation

The CDF of the test statistic S under the null, F(t) = \Pr(S \leq t), is computed in one of two ways:

• Exact Method: The function uses the exact Barrow-Smith inclusion-exclusion formula to compute the CDF.

  F(t) = \frac{1}{k! \prod_{i=1}^k w_i} \sum_{v \in \{0,1\}^k} (-1)^{\sum v_j} (t - w^T v)^k \mathbf{I}_{\{t - w^T v \ge 0\}}

  This method is computationally intensive and is infeasible for k > 18 studies, at which point the function will stop with an error if approx = FALSE is used.

• Normal Approximation: For a large number of studies or sufficiently balanced weights, S is approximated by a Normal distribution with:

  \mathrm{E}[S] = \frac{1}{2}\sum_{i=1}^k w_i

  \mathrm{Var}(S) = \frac{1}{12}\sum_{i=1}^k w_i^2

Approximation Rule

The approx = TRUE argument enables the normal approximation, but it is only used if a condition (controlled by approx_rule) is met.

• approx_rule = "n": Uses the approximation if the number of studies n > neff_cut.

• approx_rule = "neff" (default): Uses the approximation if the effective sample size n_eff > neff_cut.

The effective sample size is defined as:

n_{\mathrm{eff}} = \frac{(\sum w_i^2)^2}{\sum w_i^4} = \frac{\|w\|_2^4}{\|w\|_4^4}

The default threshold neff_cut = 12 is based on error bounds, which indicate the approximation is sufficiently accurate when this condition is met. This rule is more robust than approx_rule = "n" when weights are unbalanced.

Approximation Error

The neff_cut parameter directly controls the tolerance for the approximation error. Edgeworth Approximation: This provides an approximation of the error, which directly relates to the n_{\mathrm{eff}} criterion used in this function:

\sup_{t \in \mathbb{R}} |F_n^w(t) - \Phi(t)| \approx \frac{\|\phi^{(3)}\|_{\infty}}{20}\frac{\|w\|_{4}^{4}}{\|w\|_{2}^{4}} \approx \frac{0.028}{n_{\mathrm{eff}}}

The default neff_cut = 12 is chosen based on this, as it corresponds to an approximate maximum error of 0.028 / 12 \approx 0.0023. If you change neff_cut, you are changing this error tolerance.

Output p-value

The final output depends on the output_p and input_p arguments:

• If output_p = "two.sided" (the default) and the inputs are one-sided (input_p is "greater" or "less"), the function combines the one-sided p-values to obtain the intermediate combined p-value p_c, and returns a symmetrized, two-sided p-value: p_{2s} = 2 \min(p_c, 1 - p_c).

• If output_p = "one.sided", the function returns the inherently one-sided combined p-value p_c directly, without symmetrization.

• If input_p is "two.sided", the input p_i are already two-sided, and no further symmetrization is applied.

References

Edgington, E. S. (1972). An additive method for combining probability values from independent experiments. The Journal of Psychology, 80(2): 351-363. doi:10.1080/00223980.1972.9924813

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

Barrow, D. L., & Smith, P. W. (1979). Spline notation applied to a volume problem. The American Mathematical Monthly, 86(1): 50-51. doi:10.1080/00029890.1979.11994730

See Also

Other p-value combination functions: p_edgington(), p_fisher(), p_hmean(), p_pearson(), p_stouffer(), p_tippett(), p_wilkinson()

Examples

n <- 10
estimates <- rnorm(n)
SEs <- rgamma(n, 5, 5)
weights <- 1/SEs

p_edgington_w(
    estimates = estimates,
    SEs = SEs,
    mu = 0,
    output_p = "two.sided",
    input_p = "greater",
    w = weights,
    approx = TRUE,
    approx_rule = "neff"
)

Fisher's method

Description

Fisher's method for combining p-values across studies. The method transforms the individual study p-values using the natural logarithm and evaluates the sum against a chi-squared distribution.

Usage

p_fisher(
  estimates,
  SEs,
  mu = 0,
  heterogeneity = "none",
  phi = NULL,
  tau2 = NULL,
  check_inputs = TRUE,
  input_p = "greater",
  output_p = "two.sided"
)

Arguments

Details

The Fisher test statistic for k studies is defined as:

f = -2 \sum_{i=1}^k \log(p_i)

Under the global null hypothesis, each p_i is assumed to be uniformly distributed on [0, 1]. The test statistic f therefore follows a chi-squared distribution with 2k degrees of freedom: \chi^2_{2k}. The combined p-value, p_F, is calculated as the probability of observing a value strictly greater than f from this distribution:

p_F = \Pr(\chi^2_{2k} > f)

Important note on orientation: Unlike Edgington's method, Fisher's method is not orientation-invariant. The combined p-value depends on the direction of the one-sided p-values (controlled by the input_p argument).

Specifically, Fisher's and Pearson's methods are mirrored. Computing the Fisher combined p-value for the "greater" alternative is equal to 1 minus the Pearson combined p-value for the "less" alternative.

Value

A numeric vector of combined p-values corresponding to each value of mu.

Output p-value

The final output depends on the output_p and input_p arguments:

• If output_p = "two.sided" (the default) and the inputs are one-sided (input_p is "greater" or "less"), the function combines the one-sided p-values to obtain the intermediate combined p-value p_c, and returns a symmetrized, two-sided p-value: p_{2s} = 2 \min(p_c, 1 - p_c).

• If output_p = "one.sided", the function returns the inherently one-sided combined p-value p_c directly, without symmetrization.

• If input_p is "two.sided", the input p_i are already two-sided, and no further symmetrization is applied.

References

Fisher R.A. Statistical Methods for Research Workers. 4th ed. Oliver & Boyd; 1932. doi:10.1093/oso/9780198522294.002.0003

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

See Also

Other p-value combination functions: p_edgington(), p_edgington_w(), p_hmean(), p_pearson(), p_stouffer(), p_tippett(), p_wilkinson()

Examples

# Simulating estimates and standard errors
n <- 15
estimates <- rnorm(n)
SEs <- rgamma(n, 5, 5)

# Set up a vector of means under the null hypothesis
mu <- seq(
   min(estimates) - 0.5 * max(SEs),
   max(estimates) + 0.5 * max(SEs),
   length.out = 100
)

# Using Fisher's method to calculate the combined p-value
p_fisher(
     estimates = estimates,
     SEs = SEs,
     mu = mu,
     heterogeneity = "none",
     output_p = "two.sided",
     input_p = "greater"
)

Harmonic mean method

Description

Combines study-level results using the harmonic mean combination method

Usage

p_hmean(
  estimates,
  SEs,
  mu = 0,
  phi = NULL,
  tau2 = NULL,
  heterogeneity = "none",
  check_inputs = TRUE,
  w = rep(1, length(estimates)),
  distr = "chisq"
)

Arguments

Details

The harmonic mean statistic for k studies is defined as

x^2 = \frac{\sum_{i=1}^k \sqrt{w_i}}{\sum_{i=1}^k w_i/z_i^2},

where z_i and {w_i} are individual study z-values and weights, respectively. Under the global null hypothesis, each z_i is assumed to follow a standard normal distribution. The harmonic mean statistic then follows a chi-squared distribution with one degree of freedom. The combined p-value is calculated as the probability of observing a value equal or greater than x^2 from this distribution:

p_H = \Pr(\chi^2_{1} > x^2)

Value

A numeric vector of combined p-values corresponding to each value of mu.

References

Held, L. (2020). The harmonic mean chi-squared test to substantiate scientific findings. Journal of the Royal Statistical Society: Series C (Applied Statistics), 69:697-708. doi:10.1111/rssc.12410

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

See Also

Other p-value combination functions: p_edgington(), p_edgington_w(), p_fisher(), p_pearson(), p_stouffer(), p_tippett(), p_wilkinson()

Examples

estimates <- c(0.5, 0.8, 0.3)
SEs <- c(0.1, 0.2, 0.1)
p_hmean(estimates, SEs, mu = 0, distr = "f")

Pearson's method

Description

Pearson's method for combining p-values across studies. The method transforms the complement of the individual study p-values using the natural logarithm and evaluates the sum against a chi-squared distribution.

Usage

p_pearson(
  estimates,
  SEs,
  mu = 0,
  heterogeneity = "none",
  phi = NULL,
  tau2 = NULL,
  check_inputs = TRUE,
  input_p = "greater",
  output_p = "two.sided"
)

Arguments

Details

The Pearson test statistic for k studies is defined as:

g = -2 \sum_{i=1}^k \log(1 - p_i)

Under the global null hypothesis, each p_i is assumed to be uniformly distributed on [0, 1]. The test statistic g therefore follows a chi-squared distribution with 2k degrees of freedom: \chi^2_{2k}. The combined p-value, p_P, is calculated as the probability of observing a value less than or equal to g from this distribution:

p_P = \Pr(\chi^2_{2k} \leq g)

Important note on orientation: Unlike Edgington's method, Pearson's method is not orientation-invariant. The combined p-value depends on the direction of the one-sided p-values (controlled by the input_p argument).

Specifically, Pearson's and Fisher's methods are mirrored. Computing the Pearson combined p-value for the "greater" alternative is equal to 1 minus the Fisher combined p-value for the "less" alternative.

Value

A numeric vector of combined p-values corresponding to each value of mu.

Output p-value

The final output depends on the output_p and input_p arguments:

• If output_p = "two.sided" (the default) and the inputs are one-sided (input_p is "greater" or "less"), the function combines the one-sided p-values to obtain the intermediate combined p-value p_c, and returns a symmetrized, two-sided p-value: p_{2s} = 2 \min(p_c, 1 - p_c).

• If output_p = "one.sided", the function returns the inherently one-sided combined p-value p_c directly, without symmetrization.

• If input_p is "two.sided", the input p_i are already two-sided, and no further symmetrization is applied.

References

Pearson K. On a method of determining whether a sample of size n supposed to have been drawn from a parent population having a known probability integral has probably been drawn at random. Biometrika, 25:379-410, 1933. doi:10.2307/2332290

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

See Also

Other p-value combination functions: p_edgington(), p_edgington_w(), p_fisher(), p_hmean(), p_stouffer(), p_tippett(), p_wilkinson()

Examples

# Simulating estimates and standard errors
n <- 15
estimates <- rnorm(n)
SEs <- rgamma(n, 5, 5)

# Set up a vector of means under the null hypothesis
mu <- seq(
   min(estimates) - 0.5 * max(SEs),
   max(estimates) + 0.5 * max(SEs),
   length.out = 100
)

# Using Pearson's method to calculate the combined p-value
p_pearson(
     estimates = estimates,
     SEs = SEs,
     mu = mu,
     heterogeneity = "none",
     output_p = "two.sided",
     input_p = "greater"
)

Stouffer's method

Description

Stouffer's method for combining p-values across studies

Usage

p_stouffer(
  estimates,
  SEs,
  mu = 0,
  heterogeneity = "none",
  phi = NULL,
  tau2 = NULL,
  check_inputs = TRUE,
  w = 1/SEs,
  output_p = "two.sided"
)

Arguments

Details

Stouffer's z-statistic for k studies is defined as

z = \frac{\sum_{i=1}^k w_i z_i}{\sqrt{\sum_{i=1}^k w_i^2}},

where z_i and {w_i} are individual study z-values and weights, respectively. Under the global null hypothesis, each z_i is assumed to follow a standard normal distribution. Stouffer's z-statistic then also follows a standard normal distribution. The combined p-value is calculated as the probability of observing a value equal or greater than z from this distribution:

p_S = \Pr(Z > z)

Value

A numeric vector of combined p-values corresponding to each value of mu.

References

Stouffer SA, et al. The American Soldier. Princeton University Press, 1949. doi:10.2307/2572105

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

See Also

Other p-value combination functions: p_edgington(), p_edgington_w(), p_fisher(), p_hmean(), p_pearson(), p_tippett(), p_wilkinson()

Examples

estimates <- c(0.1, 0.2, 0.3)
SEs <- c(0.05, 0.05, 0.1)
p_stouffer(estimates, SEs, mu = 0, heterogeneity = "none")

Tippett's method

Description

Tippett's method for combining p-values across studies. This method evaluates the minimum individual study p-value.

Usage

p_tippett(
  estimates,
  SEs,
  mu = 0,
  heterogeneity = "none",
  phi = NULL,
  tau2 = NULL,
  check_inputs = TRUE,
  input_p = "greater",
  output_p = "two.sided"
)

Arguments

Details

The Tippett combined p-value, p_T, for k studies is defined directly as:

p_T = 1 - (1 - \min\{p_1, \dots, p_k\})^k

Under the global null hypothesis, each p_i is assumed to be uniformly distributed on [0, 1].

Important note on orientation: Unlike Edgington's method, Tippett's method is not orientation-invariant. The combined p-value depends on the direction of the one-sided p-values (controlled by the input_p argument).

Specifically, Tippett's and Wilkinson's methods are mirrored. Computing the Tippett combined p-value for the "greater" alternative is equal to 1 minus the Wilkinson combined p-value for the "less" alternative.

Value

A numeric vector of combined p-values corresponding to each value of mu.

Output p-value

The final output depends on the output_p and input_p arguments:

• If output_p = "two.sided" (the default) and the inputs are one-sided (input_p is "greater" or "less"), the function combines the one-sided p-values to obtain the intermediate combined p-value p_c, and returns a symmetrized, two-sided p-value: p_{2s} = 2 \min(p_c, 1 - p_c).

• If output_p = "one.sided", the function returns the inherently one-sided combined p-value p_c directly, without symmetrization.

• If input_p is "two.sided", the input p_i are already two-sided, and no further symmetrization is applied.

References

Tippett LHC. Methods of Statistics. Williams Norgate; 1931. doi:10.2307/3606890

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

See Also

Other p-value combination functions: p_edgington(), p_edgington_w(), p_fisher(), p_hmean(), p_pearson(), p_stouffer(), p_wilkinson()

Examples

# Simulating estimates and standard errors
n <- 15
estimates <- rnorm(n)
SEs <- rgamma(n, 5, 5)

# Set up a vector of means under the null hypothesis
mu <- seq(
   min(estimates) - 0.5 * max(SEs),
   max(estimates) + 0.5 * max(SEs),
   length.out = 100
)

# Using Tippett's method to calculate the combined p-value
p_tippett(
     estimates = estimates,
     SEs = SEs,
     mu = mu,
     heterogeneity = "none",
     output_p = "two.sided",
     input_p = "greater"
)

Overview of p-value combination functions

Description

The p_value_functions family is useful to combine individual study effect estimates and standard errors into a single meta-analytical p-value function. These functions are the building blocks for the confMeta function.

Details

Vectorization over mu: All of the p-value functions in this package are vectorized over the mu argument. This means that you can pass a sequence of null values (e.g., seq(-2, 2, length.out = 1000)) and the function will evaluate the combined p-value for every point. This is specifically designed to construct p-value functions and invert the tests to find confidence intervals at any desired level.

Available Methods:

• p_edgington: Classical Edgington method (Sum of p-values).

• p_edgington_w: Weighted Edgington method.

• p_fisher: Fisher's method (Log-product of p-values).

• p_pearson: Pearson's method (Log-product of complements).

• p_tippett: Tippett's method (Minimum p-value).

• p_wilkinson: Wilkinson's method (Maximum p-value).

• p_stouffer: Stouffer's method.

• p_hmean: Harmonic mean method.

References

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

See Also

confMeta

Wilkinson's method

Description

Wilkinson's method for combining p-values across studies. This method evaluates the maximum individual study p-value.

Usage

p_wilkinson(
  estimates,
  SEs,
  mu = 0,
  heterogeneity = "none",
  phi = NULL,
  tau2 = NULL,
  check_inputs = TRUE,
  input_p = "greater",
  output_p = "two.sided"
)

Arguments

Details

The Wilkinson combined p-value for k studies is defined as:

p_W = \max\{p_1, \dots, p_k\}^k

Under the global null hypothesis, each p_i is assumed to be uniformly distributed on [0, 1].

Important note on orientation: Unlike Edgington's method, Wilkinson's method is not orientation-invariant. The combined p-value depends on the direction of the one-sided p-values (controlled by the input_p argument).

Specifically, Wilkinson's and Tippett's methods are mirrored. Computing the Wilkinson combined p-value for the "greater" alternative is equal to 1 minus the Tippett combined p-value for the "less" alternative.

Value

A numeric vector of combined p-values corresponding to each value of mu.

Output p-value

The final output depends on the output_p and input_p arguments:

• If output_p = "two.sided" (the default) and the inputs are one-sided (input_p is "greater" or "less"), the function combines the one-sided p-values to obtain the intermediate combined p-value p_c, and returns a symmetrized, two-sided p-value: p_{2s} = 2 \min(p_c, 1 - p_c).

• If output_p = "one.sided", the function returns the inherently one-sided combined p-value p_c directly, without symmetrization.

• If input_p is "two.sided", the input p_i are already two-sided, and no further symmetrization is applied.

References

Wilkinson B. A statistical consideration in psychological research. Psychological Bulletin, 48(2):156-158, 1951. doi:10.1037/h0059111

Held, L, Hofmann, F, Pawel, S. (2025). A comparison of combined p-value functions for meta-analysis. Research Synthesis Methods, 16:758-785. doi:10.1017/rsm.2025.26

See Also

Other p-value combination functions: p_edgington(), p_edgington_w(), p_fisher(), p_hmean(), p_pearson(), p_stouffer(), p_tippett()

Examples

# Simulating estimates and standard errors
n <- 15
estimates <- rnorm(n)
SEs <- rgamma(n, 5, 5)

# Set up a vector of means under the null hypothesis
mu <- seq(
  min(estimates) - 0.5 * max(SEs),
  max(estimates) + 0.5 * max(SEs),
  length.out = 100
)

# Using Wilkinson's method to calculate the combined p-value
p_wilkinson(
    estimates = estimates,
    SEs = SEs,
    mu = mu,
    heterogeneity = "none",
    output_p = "two.sided",
    input_p = "greater"
)

Objects exported from other packages

Description

These objects are imported from other packages. Follow the links below to see their documentation.

ggplot2

autoplot