Probability waves: adaptive cluster-based correction by convolution of p-value series from mass univariate analysis

dataset and Octave/MatLab codes/scripts for data analysis Background: Methods for p-value correction are criticized for either increasing Type II error or improperly reducing Type I error. This problem is worse when dealing with thousands or even hundreds of paired comparisons between waves or images which are performed point-to-point. This text considers patterns in probability vectors resulting from multiple point-to-point comparisons between two event-related potentials (ERP) waves (mass univariate analysis) to correct p-values, where clusters of signiticant p-values may indicate true H0 rejection. New method: We used ERP data from normal subjects and other ones with attention deficit hyperactivity disorder (ADHD) under a cued forced two-choice test to study attention. The decimal logarithm of the p-vector (p') was convolved with a Gaussian window whose length was set as the shortest lag above which autocorrelation of each ERP wave may be assumed to have vanished. To verify the reliability of the present correction method, we realized Monte-Carlo simulations (MC) to (1) evaluate confidence intervals of rejected and non-rejected areas of our data, (2) to evaluate differences between corrected and uncorrected p-vectors or simulated ones in terms of distribution of significant p-values, and (3) to empirically verify rate of type-I error (comparing 10,000 pairs of mixed samples whit control and ADHD subjects). Results: the present method reduced the range of p'-values that did not show covariance with neighbors (type I and also type-II errors). The differences between simulation or raw p-vector and corrected p-vectors were, respectively, minimal and maximal for window length set by autocorrelation in p-vector convolution. Comparison with existing methods: Our method was less conservative while FDR methods rejected basically all significant p-values for Pz and O2 channels. The MC simulations, gold-standard method for error correction, presented 2.78±4.83% of difference (all 20 channels) from p-vector after correction, while difference between raw and corrected p-vector was 5,96±5.00% (p = 0.0003). Conclusion: As a cluster-based correction, the present new method seems to be biological and statistically suitable to correct p-values in mass univariate analysis of ERP waves, which adopts adaptive parameters to set correction.

用于数据分析的数据集及Octave/MatLab代码/脚本
背景：当前p值校正方法常遭诟病，要么会增加第二类错误（Type II error），要么不当降低第一类错误（Type I error）。当涉及数千乃至数百组逐点比对的波形或图像时，该问题会更为突出。本研究针对两组事件相关电位（event-related potentials, ERP）波形开展多组逐点比对（批量单变量分析），由此得到概率向量并据此进行p值校正，其中具有统计学显著性的p值簇可用于指示真实的原假设（H0）拒绝情况。

新方法：本研究采用线索提示的强制二选任务范式下采集的ERP数据，研究注意力相关机制，数据来源包括正常受试者与注意缺陷多动障碍（attention deficit hyperactivity disorder, ADHD）患者。我们将p向量（p'）的常用对数与高斯窗进行卷积运算，高斯窗的长度设定为最短延迟时长——超过该时长后，各ERP波形的自相关可视为已消失。为验证本校正方法的可靠性，我们开展了蒙特卡洛模拟（Monte-Carlo simulations, MC），具体包括三项内容：(1) 评估数据中已拒绝与未拒绝区域的置信区间；(2) 从显著性p值分布的角度，对比校正后、未校正的p向量以及模拟得到的p向量之间的差异；(3) 通过实证分析验证第一类错误率（通过对比10000组混合样本完成，混合样本包含健康对照与ADHD受试者）。

结果：本方法缩小了与相邻节点无协方差的p'值范围，同时有效控制了第一类与第二类错误。在通过p向量自相关设定高斯窗长度的场景下，模拟得到的p向量与校正后p向量的差异最小，原始未校正p向量与校正后p向量的差异则最大。

与现有方法的对比：本方法的保守性更低，而错误发现率（False Discovery Rate, FDR）校正方法会将Pz与O2通道上几乎所有显著性p值都判定为具有统计学意义。作为误差校正的金标准方法，蒙特卡洛模拟得到的校正后p向量与原始p向量的差异为2.78±4.83%（覆盖全部20个通道）；而未校正原始p向量与校正后p向量的差异为5.96±5.00%（p=0.0003）。

结论：作为一种基于簇的校正方法，本研究提出的新方法具备生物学与统计学合理性，适用于ERP波形的批量单变量分析中的p值校正，且可通过自适应参数设定校正规则。