TFisher: Unified P-Value Combination Method

• TFisher is a unified family of p-value combination methods that generalizes classical Fisher's method and TPM using truncation and weighting schemes.
• It provides an analytic null distribution through a binomial-chi-square mixture, enabling exact p-value computations without simulations.
• Adaptive strategies like soft-thresholding and the omnibus oTFisher test optimize power across sparse and dense signal configurations.

TFisher denotes a unifying family of p-value combination statistics that incorporates general truncation and weighting schemes, subsuming classical methods such as Fisher's method and the Truncated Product Method (TPM) as special cases. TFisher provides analytic null distributions and power calculations, achieves optimality under flexible alternatives, and features adaptive methods for unknown signal configurations (Zhang et al., 2018).

1. Definition and General Formulation

TFisher combines $n$ independent one-sided p-values $P_1,\dots,P_n$ using two parameters: a truncation threshold $\tau_1 \in (0,1]$ and a scaling (weight) parameter $\tau_2 > 0$. The general TFisher statistic is

$$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$

or, equivalently,

$$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$

Special cases include:

• Fisher’s product: $\tau_1 = \tau_2 = 1$
• TPM: $\tau_1 < 1, \; \tau_2=1$
• Soft-thresholding: $\tau_1 = \tau_2 = \tau$, yielding $$W_s(\tau) = \sum_{i=1}^n [-2\log P_i + 2\log\tau]_+$$
• RTP and power-weighting: Other instances

TFisher thus unifies and extends classical p-value combination methodologies via a two-parameter family.

2. Null Distribution and Exact Calculations

Under the global null hypothesis, $P_1,\dots,P_n$0. Define $P_1,\dots,P_n$1. Conditional on $P_1,\dots,P_n$2, the sum

$P_1,\dots,P_n$3

is distributed as a shifted chi-square with $P_1,\dots,P_n$4 degrees of freedom, shifted by $P_1,\dots,P_n$5. The exact null tail probability is

$P_1,\dots,P_n$6

This closed-form allows analytic computation of p-values, eliminating the need for permutation or simulation under $P_1,\dots,P_n$7.

3. Analytical Power and Efficiency Measures

Consider a signal-detection model: $P_1,\dots,P_n$8, $P_1,\dots,P_n$9. Using $\tau_1 \in (0,1]$0, power and efficiency are expressed via one-dimensional integrals involving the discrepancy function $\tau_1 \in (0,1]$1, where $\tau_1 \in (0,1]$2 with $\tau_1 \in (0,1]$3, $\tau_1 \in (0,1]$4 cumulative distributions of $\tau_1 \in (0,1]$5 under $\tau_1 \in (0,1]$6, $\tau_1 \in (0,1]$7.

Key efficiency metrics:

• Bahadur Efficiency (BE): $\tau_1 \in (0,1]$8. $\tau_1 \in (0,1]$9 are mean and variance of $\tau_2 > 0$0 under $\tau_2 > 0$1. BE ignores alternative variance.
• Asymptotic Power Efficiency (APE): Uses both $\tau_2 > 0$2 and $\tau_2 > 0$3; power at asymptotic significance level $\tau_2 > 0$4 is $\tau_2 > 0$5.
• Asymptotic Power Rate (APR): $\tau_2 > 0$6 becomes dominant for large $\tau_2 > 0$7.

A 3-parameter skew-normal central limit theorem provides accurate power calculations under alternatives. The exact null matches Monte Carlo for $\tau_2 > 0$8 as small as 10; skew-normal CLT accurately predicts power for realistic $\tau_2 > 0$9 and $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$0.

4. Parameter Optimization and Soft-Thresholding

Optimization of $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$1 can be performed with respect to BE or APE. Under general alternatives, the BE maximizer is independent of signal proportion $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$2. For a Gaussian mixture, critical $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$3 yields $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$4; otherwise, mixtures of truncation schemes arise.

For sparse signals and moderate-to-large $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$5, $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$6 is often globally optimal for both APE and APR. The optimal cutoff $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$7 decreases as $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$8 decreases or $$T_n(\tau_1,\tau_2) = \prod_{i=1}^{n} \left(P_i / \tau_2\right)^{I\{P_i \le \tau_1\}}$$9 increases. Neither a universal fixed threshold (e.g., $$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$0) nor $$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$1 alone is optimal across all alternatives.

Soft-thresholding ($$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$2), with $$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$3, continuously down-weights moderately small $$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$4-values and strongly favors very small ones. Across a broad range of $$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$5, soft-thresholding at $$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$6 is stationary and practically optimal in theory and simulations.

5. Adaptive Omnibus Test: oTFisher

When $$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$7 are unknown, the omnibus test oTFisher adaptively aggregates across a grid of thresholds. For $$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$8, compute

$$W_n(\tau_1,\tau_2) = -2\log T_n = \sum_{i=1}^n [-2\log P_i + 2\log\tau_2] \cdot I\{P_i \le \tau_1\}.$$9

oTFisher takes

$\tau_1 = \tau_2 = 1$0

The omnibus p-value is analytically computed by evaluating the joint null distribution, which is approximately multivariate normal for large $\tau_1 = \tau_2 = 1$1, using standard MVN software. Analytic calculation of the omnibus p-value is computationally efficient ($\tau_1 = \tau_2 = 1$2milliseconds), with accurate coverage.

In power comparisons, oTFisher uniformly matches or outperforms adaptive TPM or adaptive RTP across signal settings. The selected $\tau_1 = \tau_2 = 1$3 can be interpreted to infer the relative sparsity or density of signals present.

6. Simulation Validation and Empirical Guidance

Simulation results indicate:

• The exact null distribution (binomial plus chi-square mixture) aligns with Monte Carlo simulations at $\tau_1 = \tau_2 = 1$4.
• Skew-normal CLT for alternatives yields power curves almost indistinguishable from $\tau_1 = \tau_2 = 1$5 simulation replicates; usual normal CLT underestimates accuracy for small $\tau_1 = \tau_2 = 1$6 or $\tau_1 = \tau_2 = 1$7.
• Over $\tau_1 = \tau_2 = 1$8, $\tau_1 = \tau_2 = 1$9, the global optimum $\tau_1 < 1, \; \tau_2=1$0 always achieves maximal power, with soft-thresholding virtually identical.
• Fixed soft-threshold $\tau_1 < 1, \; \tau_2=1$1 outperforms Fisher ($\tau_1 < 1, \; \tau_2=1$2) for sparse signals, inferior for dense signals; hard thresholding (TPM) never surpasses soft-thresholding.

Practical guidance:

• Use small $\tau_1 < 1, \; \tau_2=1$3 (e.g., $\tau_1 < 1, \; \tau_2=1$4–$\tau_1 < 1, \; \tau_2=1$5) for sparse signals.
• Use Fisher ($\tau_1 < 1, \; \tau_2=1$6) or larger $\tau_1 < 1, \; \tau_2=1$7 for dense signals.
• When in doubt, deploy oTFisher over a small grid. Analytic $\tau_1 < 1, \; \tau_2=1$8-value is efficient and accurate.

7. Implementation and Usage in Statistical Practice

TFisher is implemented as the R package TFisher (CRAN). Major functionalities:

Function	Description	Output
TFisher(p, tau1, tau2)	Computes $\tau_1 < 1, \; \tau_2=1$9 and its exact null p-value.	Statistic and p-value
oTFisher(p, taus)	Omnibus test over $\tau_1 = \tau_2 = \tau$0 in prescribed grid.	Minimal null-cdf and p-value

Basic usage in R:

$\tau_1 = \tau_2 = \tau$3

TFisher has been applied to, for example, exome sequencing data analysis for amyotrophic lateral sclerosis testing $\tau_1 = \tau_2 = \tau$1-values across thousands of loci (Zhang et al., 2018).

In summary, TFisher generalizes classic $\tau_1 = \tau_2 = \tau$2-value combination methods to a flexible two-parameter family, offers analytic null and power calculations, and deploys adaptive soft-thresholding and omnibus procedures to maximize performance in both sparse and dense regimes (Zhang et al., 2018).