It’s not the p-values’ fault – reflections on the recent ASA statement (+relevant R resources)

Joint post by Yoav Benjamini and Tal Galili. The post highlights points raised by Yoav in his official response to the ASA statement (available as on page 4 in the ASA supplemental tab), as well as offers a list of relevant R resources.

Summary

The ASA statement about the misuses of the p-value singles it out. It is just as well relevant to the use of most other statistical methods: context matters, no single statistical measure suffices, specific thresholds should be avoided and reporting should not be done selectively. The latter problem is discussed mainly in relation to omitted inferences. We argue that the selective reporting of inferences problem is serious enough a problem in our current industrialized science even when no omission takes place. Many R tools are available to address it, but they are mainly used in very large problems and are grossly underused in areas where lack of replicability hits hard.

Source: xkcd

Preface – the ASA released a statement about the p-value

A few days ago the ASA released a statement titled “on p-values: context, process, and purpose”. It was a way for the ASA to address the concerns about the role of Statistics in the Reproducibility and Replicability (R&R) crisis. In the discussions about R&R the p-value has become a scapegoat, being such a widely used statistical method. The ASA statement made an effort to clarify various misinterpretations and to point at misuses of the p-value, but we fear that the result is a statement that might be read by the target readers as expressing very negative attitude towards the p-value. And indeed, just two days after the release of the ASA statement, a blog post titled “After 150 Years, the ASA Says No to p-values” was published (by Norman Matloff), even though the ASA (as far as we read it) did not say “no to P-values” anywhere in the statement. Thankfully, other online reactions to the ASA statements, such as the article in Nature, and other posts in the blogosphere (see [1], [2], [3], [4]), did not use an anti-p-value rhetoric.

Why the p-value was (and still is) valuable

In spite of its misinterpretations, the p-value served science well over the 20^th century. Why? Because in some sense the p-value offers a first defense line against being fooled by randomness, separating signal from noise. It requires simpler (or fewer) models than those needed by other statistical tool. The p-value requires only a statistical model for the behavior of a statistic under the null hypothesis to hold. Even if a model of an alternative hypothesis is used for choosing a “good” statistic (which would be used for constructing the p-value), this alternative model does not have to be correct in order for the p-value to be valid and useful (i.e.: control type I error at the desired level while offering some power to detect a real effect). In contrast, other (wonderful and useful) statistical methods such as Likelihood ratios, effect size estimation, confidence intervals, or Bayesian methods all need the assumed models to hold over a wider range of situations, not merely under the tested null.  And most importantly, the model needed for the calculation of the p-value may be guaranteed to hold under an appropriately designed and executed randomized experiment. The p-value is a very valuable tool, but it should be complemented – not replaced – by confidence intervals and effect size estimators (as is possible in the specific setting). The ends of a 95% confidence interval indicates a range of potential null hypothesis that could be rejected. An estimator of effect size (supported by an assessment of uncertainty) is crucial for interpretation and for assessing the scientific significance of the results. While useful, all these types of inferences are also affected by similar problems as the p-values do. What level of likelihood ratio in favor of the research hypothesis will be acceptable to the journal? or should scientific discoveries be based on whether posterior odds pass a specific threshold? Does either of them measure the size of the effect? Finally, 95% confidence intervals or credence intervals offer no protection against selection when only those that do not cover 0, are selected into the abstract. The properties each method has on the average for a single parameter (level, coverage or unbiased) will not necessarily hold even on the average when a selection is made.

The p-value (and other methods) in the new era of “industrialized science”

What, then, went wrong in the last decade or two? The change in the scale of the scientific work, brought about by high throughput experimentation methodologies, availability of large databases and ease of computation, a change that parallels the industrialization that production processes have already gone through. In Genomics, Proteomics, Brain Imaging and such, the number of potential discoveries scanned is enormous so the selection of the interesting ones for highlighting is a must. It has by now been recognized in these fields that merely “full reporting and transparency” (as recommended by ASA) is not enough, and methods should be used to control the effect of the unavoidable selection. Therefore, in those same areas, the p-value bright-line is not set at the traditional 5% level. Methods for adaptively setting it to directly control a variety of false discovery rates or other error rates are commonly used. Addressing the effect of selection on inference (be it when using p-value, or other methods) has been a very active research area; New strategies and sophisticated selective inference tools for testing, confidence intervals, and effect size estimation, in different setups are being offered. Much of it still remains outside the practitioners’ active toolset, even though many are already available in R, as we describe below. The appendix of this post contains a partial list of R packages that support simultaneous and selective inference. In summary, when discussing the impact of statistical practices on R&R, the p-value should not be singled out nor its usage discouraged: it’s more likely the fault of selection, and not the p-values’ fault.

Appendix – R packages for Simultaneous and Selective Inference (“SASI” R packages)

Extended support for classical and modern adjustment for Simultaneous and Selective Inference (also known as “multiple comparisons”) is available in R and in various R packages. Traditional concern in these areas has been on properties holding simultaneously for all inferences. More recent concerns are on properties holding on the average over the selected, addressed by varieties of false discovery rates, false coverage rates and conditional approaches. The following is a list of relevant R resources. If you have more, please mention them in the comments. Every R installation offers functions (from the {stats} package) for dealing with multiple comparisons, such as: 

• adjust – that gets a set of p-values as input and returns p-values adjusted using one of several methods: Bonferroni, Holm (1979), Hochberg (1988), Hommel (1988), FDR by Benjamini & Hochberg (1995), and Benjamini & Yekutieli (2001),
• t.test, pairwise.wilcox.test, and pairwise.prop.test – all rely on p.adjust and can calculate pairwise comparisons between group levels with corrections for multiple testing.
• TukeyHSD- Create a set of confidence intervals on the differences between the means of the levels of a factor with the specified family-wise probability of coverage. The intervals are based on the Studentized range statistic, Tukey’s ‘Honest Significant Difference’ method.

  Once we venture outside of the core R functions, we are introduced to a wealth of R packages and statistical procedures. What follows is a partial list (if you wish to contribute and extend this list, please leave your comment to this post):

• multcomp – Simultaneous tests and confidence intervals for general linear hypotheses in parametric models, including linear, generalized linear, linear mixed effects, and survival models. The package includes demos reproducing analyzes presented in the book “Multiple Comparisons Using R” (Bretz, Hothorn, Westfall, 2010, CRC Press).
• coin (+RcmdrPlugin.coin)- Conditional inference procedures for the general independence problem including two-sample, K-sample (non-parametric ANOVA), correlation, censored, ordered and multivariate problems.
• SimComp – Simultaneous tests and confidence intervals are provided for one-way experimental designs with one or many normally distributed, primary response variables (endpoints).
• PMCMR – Calculate Pairwise Multiple Comparisons of Mean Rank Sums
• mratios – perform (simultaneous) inferences for ratios of linear combinations of coefficients in the general linear model.
• mutoss (and accompanying mutossGUI) – are designed to ease the application and comparison of multiple hypothesis testing procedures.
• nparcomp – compute nonparametric simultaneous confidence intervals for relative contrast effects in the unbalanced one way layout. Moreover, it computes simultaneous p-values.
• ANOM – The package takes results from multiple comparisons with the grand mean (obtained with ‘multcomp’, ‘SimComp’, ‘nparcomp’, or ‘MCPAN’) or corresponding simultaneous confidence intervals as input and produces ANOM decision charts that illustrate which group means deviate significantly from the grand mean.
• gMCP – Functions and a graphical user interface for graphical described multiple test procedures.
• MCPAN – Multiple contrast tests and simultaneous confidence intervals based on normal approximation.
• mcprofile – Calculation of signed root deviance profiles for linear combinations of parameters in a generalized linear model. Multiple tests and simultaneous confidence intervals are provided.
• factorplot – Calculate, print, summarize and plot pairwise differences from GLMs, GLHT or Multinomial Logit models. Relies on stats::p.adjust
• multcompView – Convert a logical vector or a vector of p-values or a correlation, difference, or distance matrix into a display identifying the pairs for which the differences were not significantly different. Designed for use in conjunction with the output of functions like TukeyHSD, dist{stats}, simint, simtest, csimint, csimtest{multcomp}, friedmanmc, kruskalmc{pgirmess}.
• discreteMTP – Multiple testing procedures for discrete test statistics, that use the known discrete null distribution of the p-values for simultaneous inference.
• someMTP – a collection of functions for Multiplicity Correction and Multiple Testing.
• hdi – Implementation of multiple approaches to perform inference in high-dimensional models
• ERP – Significance Analysis of Event-Related Potentials Data
• TukeyC – Perform the conventional Tukey test from aov and aovlist objects
• qvalue – offers a function which takes a list of p-values resulting from the simultaneous testing of many hypotheses and estimates their q-values and local FDR values. (reading this discussion thread might be helpful)
• fdrtool – Estimates both tail area-based false discovery rates (Fdr) as well as local false discovery rates (fdr) for a variety of null models (p-values, z-scores, correlation coefficients, t-scores).
• cp4p – Functions to check whether a vector of p-values respects the assumptions of FDR (false discovery rate) control procedures and to compute adjusted p-values.
• multtest – Non-parametric bootstrap and permutation resampling-based multiple testing procedures (including empirical Bayes methods) for controlling the family-wise error rate (FWER), generalized family-wise error rate (gFWER), tail probability of the proportion of false positives (TPPFP), and false discovery rate (FDR).
• selectiveInference – New tools for post-selection inference, for use with forward stepwise regression, least angle regression, the lasso, and the many means problem.
• PoSI (site) – Valid Post-Selection Inference for Linear LS Regression
• HWBH– A shiny app for hierarchical weighted FDR testing of primary and secondary endpoints in Medical Research. By Benjamini Y & Cohen R, 2013. Top of Form
• repfdr(@github)-  estimation of Bayes and local Bayes false discovery rates for replicability analysis.  Heller R, Yekutieli D, 2014
• SelectiveCI : An R package for computing confidence intervals for selected parameters as described in Asaf Weinstein, William Fithian & Yoav Benjamini,2013 and Yoav Benjamini, Daniel Yekutieli,2005
• Rvalue– Software for FDR testing for replicability in primary and follow-up endpoints.  Heller R, Bogomolov M, Benjamini Y, 2014   “Deciding whether follow-up studies have replicated findings in a preliminary large-scale “omics’ study”, under review and available upon request from the first author. Bogomolov M, Heller R, 2013

Other than Simultaneous and Selective Inference, one should also mention that there are many R packages for reproducible research, i.e.: the connecting of data, R code, analysis output,  and interpretation – so that scholarship can be recreated, better understood and verified. As well as for analysis, i.e.: the combining of findings from independent studies in order to make a more general claim.