Multiple Testing Corrections
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Underestimation of Type I Errors in Scientific Analysis
Open Access Journal of Mathematical and Theoretical Physics Research Article Open Access Underestimation of type I errors in scientific analysis Abstract Volume 2 Issue 1 - 2019 Statistical significance is the hallmark, the main empirical thesis and the main argument Roman Anton to quantitatively verify or falsify scientific hypothesis in science. The correct use of Department of Theoretical Sciences, The University of Truth and statistics assures and increases the reproducibility of the results and findings, and Common Sense, Germany there is recently a widely wrong use of statistics and a reproducibility crisis in science. Today, most scientists can’t replicate the results of their peers and peer-review hinders Correspondence: Roman Anton, Department of Theoretical them to reveal it. Besides an estimated 80% of general misconduct in the practices, Sciences, The University of Truth and Common Sense, Germany, and above 90% of non-reproducible results, there is also a high proportion of generally Email false statistical assessment using statistical significance tests like the otherwise valuable t-test. The 5% confidence interval, the widely used significance threshold Received: October 29, 2018 | Published: January 04, 2019 of the t-test, is regularly and very frequently used without assuming the natural rate of false positives within the test system, referred to as the stochastic alpha or type I error. Due to the global need of a clear-cut clarification, this statistical research paper didactically shows for the scientific fields that are using the t-test, how to prevent the intrinsically occurring pitfalls of p-values and advises random testing of alpha, beta, SPM and data characteristics to determine the right p-level, statistics, and testing. -
©2018 Oxford University Press
©2018 PART E OxfordChallenges in Statistics University Press mot43560_ch22_201-213.indd 201 07/31/17 11:56 AM ©2018 Oxford University Press ©2018 CHAPTER 22 Multiple Comparisons Concepts If you torture your data long enough, they will tell you what- ever you want to hear. Oxford MILLS (1993) oping with multiple comparisons is one of the biggest challenges Cin data analysis. If you calculate many P values, some are likely to be small just by random chance. Therefore, it is impossible to inter- pret small P values without knowing how many comparisons were made. This chapter explains three approaches to coping with multiple compari- sons. Chapter 23 will show that the problem of multiple comparisons is pervasive, and Chapter 40 will explain special strategies for dealing with multiple comparisons after ANOVA.University THE PROBLEM OF MULTIPLE COMPARISONS The problem of multiple comparisons is easy to understand If you make two independent comparisons, what is the chance that one or both comparisons will result in a statistically significant conclusion just by chance? It is easier to answer the opposite question. Assuming both null hypotheses are true, what is the chance that both comparisons will not be statistically significant? The answer is the chance that the first comparison will not be statistically significant (0.95) times the chance that the second one will not be statistically significant (also 0.95), which equals 0.9025, or about 90%. That leaves about a 10% chance of obtaining at least one statistically significant conclusion by chance. It is easy to generalize that logic to more comparisons. -
The Problem of Multiple Testing and Its Solutions for Genom-Wide Studies
The problem of multiple testing and its solutions for genom-wide studies How to cite: Gyorffy B, Gyorffy A, Tulassay Z:. The problem of multiple testing and its solutions for genom-wide studies. Orv Hetil , 2005;146(12):559-563 ABSTRACT The problem of multiple testing and its solutions for genome-wide studies. Even if there is no real change, the traditional p = 0.05 can cause 5% of the investigated tests being reported significant. Multiple testing corrections have been developed to solve this problem. Here the authors describe the one-step (Bonferroni), multi-step (step-down and step-up) and graphical methods. However, sometimes a correction for multiple testing creates more problems, than it solves: the universal null hypothesis is of little interest, the exact number of investigations to be adjusted for can not determined and the probability of type II error increases. For these reasons the authors suggest not to perform multiple testing corrections routinely. The calculation of the false discovery rate is a new method for genome-wide studies. Here the p value is substituted by the q value, which also shows the level of significance. The q value belonging to a measurement is the proportion of false positive measurements when we accept it as significant. The authors propose using the q value instead of the p value in genome-wide studies. Free keywords: multiple testing, Bonferroni-correction, one-step, multi-step, false discovery rate, q-value List of abbreviations: FDR: false discovery rate FWER: family familywise error rate SNP: Single Nucleotide Polymorphism PFP: proportion of false positives 1 INTRODUCTION A common feature in all of the 'omics studies is the inspection of a large number of simultaneous measurements in a small number of samples. -
Simultaneous Confidence Intervals for Comparing Margins of Multivariate
Computational Statistics and Data Analysis 64 (2013) 87–98 Contents lists available at SciVerse ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Simultaneous confidence intervals for comparing margins of multivariate binary data Bernhard Klingenberg a,∗,1, Ville Satopää b a Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, United States b Department of Statistics, The Wharton School of the University of Pennsylvania, Philadelphia, PA 19104-6340, United States article info a b s t r a c t Article history: In many applications two groups are compared simultaneously on several correlated bi- Received 17 August 2012 nary variables for a more comprehensive assessment of group differences. Although the Received in revised form 12 February 2013 response is multivariate, the main interest is in comparing the marginal probabilities be- Accepted 13 February 2013 tween the groups. Estimating the size of these differences under strong error control allows Available online 6 March 2013 for a better evaluation of effects than can be provided by multiplicity adjusted P-values. Simultaneous confidence intervals for the differences in marginal probabilities are devel- Keywords: oped through inverting the maximum of correlated Wald, score or quasi-score statistics. Correlated binary responses Familywise error rate Taking advantage of the available correlation information leads to improvements in the Marginal homogeneity joint coverage probability and power compared to straightforward Bonferroni adjustments. Multiple comparisons Estimating the correlation under the null is also explored. While computationally complex even in small dimensions, it does not result in marked improvements. Based on extensive simulation results, a simple approach that uses univariate score statistics together with their estimated correlation is proposed and recommended. -
The Reproducibility of Research and the Misinterpretation of P Values
bioRxiv preprint doi: https://doi.org/10.1101/144337; this version posted August 2, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC 4.0 International license. 1 The reproducibility of research and the misinterpretation of P values David Colquhoun University College London Email: [email protected] Revision 6 Abstract We wish to answer this question If you observe a “significant” P value after doing a single unbiased experiment, what is the probability that your result is a false positive?. The weak evidence provided by P values between 0.01 and 0.05 is explored by exact calculations of false positive rates. When you observe P = 0.05, the odds in favour of there being a real effect (given by the likelihood ratio) are about 3:1. This is far weaker evidence than the odds of 19 to 1 that might, wrongly, be inferred from the P value. And if you want to limit the false positive rate to 5 %, you would have to assume that you were 87% sure that there was a real effect before the experiment was done. If you observe P = 0.001 in a well-powered experiment, it gives a likelihood ratio of almost 100:1 odds on there being a real effect. That would usually be regarded as conclusive, But the false positive rate would still be 8% if the prior probability of a real effect was only 0.1. -
P and Q Values in RNA Seq the Q-Value Is an Adjusted P-Value, Taking in to Account the False Discovery Rate (FDR)
P and q values in RNA Seq The q-value is an adjusted p-value, taking in to account the false discovery rate (FDR). Applying a FDR becomes necessary when we're measuring thousands of variables (e.g. gene expression levels) from a small sample set (e.g. a couple of individuals). A p-value of 0.05 implies that we are willing to accept that 5% of all tests will be false positives. An FDR-adjusted p-value (aka a q-value) of 0.05 implies that we are willing to accept that 5% of the tests found to be statistically significant (e.g. by p-value) will be false positives. Such an adjustment is necessary when we're making multiple tests on the same sample. See, for example, http://www.totallab.com/products/samespots/support/faq/pq-values.aspx. -HomeBrew- What are p-values? The object of differential 2D expression analysis is to find those spots which show expression difference between groups, thereby signifying that they may be involved in some biological process of interest to the researcher. Due to chance, there will always be some difference in expression between groups. However, it is the size of this difference in comparison to the variance (i.e. the range over which expression values fall) that will tell us if this expression difference is significant or not. Thus, if the difference is large but the variance is also large, then the difference may not be significant. On the other hand, a small difference coupled with a very small variance could be significant. -
Dataset Decay and the Problem of Sequential Analyses on Open Datasets
FEATURE ARTICLE META-RESEARCH Dataset decay and the problem of sequential analyses on open datasets Abstract Open data allows researchers to explore pre-existing datasets in new ways. However, if many researchers reuse the same dataset, multiple statistical testing may increase false positives. Here we demonstrate that sequential hypothesis testing on the same dataset by multiple researchers can inflate error rates. We go on to discuss a number of correction procedures that can reduce the number of false positives, and the challenges associated with these correction procedures. WILLIAM HEDLEY THOMPSON*, JESSEY WRIGHT, PATRICK G BISSETT AND RUSSELL A POLDRACK Introduction sequential procedures correct for the latest in a In recent years, there has been a push to make non-simultaneous series of tests. There are sev- the scientific datasets associated with published eral proposed solutions to address multiple papers openly available to other researchers sequential analyses, namely a-spending and a- (Nosek et al., 2015). Making data open will investing procedures (Aharoni and Rosset, allow other researchers to both reproduce pub- 2014; Foster and Stine, 2008), which strictly lished analyses and ask new questions of existing control false positive rate. Here we will also pro- datasets (Molloy, 2011; Pisani et al., 2016). pose a third, a-debt, which does not maintain a The ability to explore pre-existing datasets in constant false positive rate but allows it to grow new ways should make research more efficient controllably. and has the potential to yield new discoveries Sequential correction procedures are harder *For correspondence: william. (Weston et al., 2019). to implement than simultaneous procedures as [email protected] The availability of open datasets will increase they require keeping track of the total number over time as funders mandate and reward data Competing interests: The of tests that have been performed by others. -
From P-Value to FDR
From P-value To FDR Jie Yang, Ph.D. Associate Professor Department of Family, Population and Preventive Medicine Director Biostatistical Consulting Core In collaboration with Clinical Translational Science Center (CTSC) and the Biostatistics and Bioinformatics Shared Resource (BB-SR), Stony Brook Cancer Center (SBCC). OUTLINE P-values - What is p-value? - How important is a p-value? - Misinterpretation of p-values Multiple Testing Adjustment - Why, How, When? - Bonferroni: What and How? - FDR: What and How? STATISTICAL MODELS Statistical model is a mathematical representation of data variability, ideally catching all sources of such variability. All methods of statistical inference have assumptions about • How data were collected • How data were analyzed • How the analysis results were selected for presentation Assumptions are often simple to express mathematically, but difficult to satisfy and verify in practice. Hypothesis test is the predominant approach to statistical inference on effect sizes which describe the magnitude of a quantitative relationship between variables (such as standardized differences in means, odds ratios, correlations etc). BASIC STEPS IN HYPOTHESIS TEST 1. State null (H0) and alternative (H1) hypotheses 2. Choose a significance level, α (usually 0.05) 3. Based on the sample, calculate the test statistic and calculate p-value based on a theoretical distribution of the test statistic 4. Compare p-value with the significance level α 5. Make a decision, and state the conclusion HISTORY OF P-VALUES P-values have been in use for nearly a century. The p-value was first formally introduced by Karl Pearson, in his Pearson's chi-squared test and popularized by Ronald Fisher. -
A Partitioning Approach for the Selection of the Best Treatment
A PARTITIONING APPROACH FOR THE SELECTION OF THE BEST TREATMENT Yong Lin A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: John T. Chen, Advisor Arjun K. Gupta Wei Ning Haowen Xi, Graduate Faculty Representative ii ABSTRACT John T. Chen, Advisor To select the best treatment among several treatments is essentially a multiple compar- isons problem. Traditionally, when dealing with multiple comparisons, there is one main argument: with multiplicity adjustment or without adjustment. If multiplicity adjustment is made such as the Bonferroni method, the simultaneous inference becomes too conserva- tive. Moreover, in the conventional methods of multiple comparisons, such as the Tukey's all pairwise multiple comparisons, although the simultaneous confidence intervals could be obtained, the best treatment cannot be distinguished efficiently. Therefore, in this disser- tation, we propose several novel procedures using partitioning principle to develop more efficient simultaneous confidence sets to select the best treatment. The method of partitioning principle for efficacy and toxicity for ordered treatments can be found in Hsu and Berger (1999). In this dissertation, by integrating the Bonferroni inequality, the partition approach is applied to unordered treatments for the inference of the best one. With the introduction of multiple comparison methodologies, we mainly focus on the all pairwise multiple comparisons. This is because all the treatments should be compared when we select the best treatment. These procedures could be used in different data forms. Chapter 2 talks about how to utilize the procedure in dichotomous outcomes and the analysis of contingency tables, especially with the Fisher's Exact Test. -
Statistical Significance for Genome-Wide Experiments
Statistical Significance for Genome-Wide Experiments John D. Storey∗ and Robert Tibshirani† January 2003 Abstract: With the increase in genome-wide experiments and the sequencing of multi- ple genomes, the analysis of large data sets has become commonplace in biology. It is often the case that thousands of features in a genome-wide data set are tested against some null hypothesis, where many features are expected to be significant. Here we propose an approach to statistical significance in the analysis of genome-wide data sets, based on the concept of the false discovery rate. This approach offers a sensible balance between the number of true findings and the number of false positives that is automatically calibrated and easily interpreted. In doing so, a measure of statistical significance called the q-value is associated with each tested feature in addition to the traditional p-value. Our approach avoids a flood of false positive results, while offering a more liberal criterion than what has been used in genome scans for linkage. Keywords: false discovery rates, genomics, multiple hypothesis testing, p-values, q-values ∗Department of Statistics, University of California, Berkeley CA 94720. Email: [email protected]. †Department of Health Research & Policy and Department of Statistics, Stanford University, Stanford CA 94305. Email: [email protected]. 1 Introduction Some of the earliest genome-wide analyses involved testing for linkage at loci spanning a large portion of the genome. Since a separate statistical test is performed at each locus, traditional p- value cut-offs of 0.01 or 0.05 had to be made stricter to avoid an abundance of false positive results. -
Multiple Testing Corrections
Multiple Testing Corrections Contents at a glance I. Introduction ................................................................................................2 II. Where to find Multiple Testing Corrections? .............................................2 III. Technical details ......................................................................................3 A. Bonferroni correction...................................................................................4 B. Bonferroni Step-down (Holm) correction.....................................................4 D. Benjamini & Hochberg False Discovery Rate.............................................5 IV. Interpretation ...........................................................................................6 V. Recommendations....................................................................................7 VI. References ..............................................................................................7 VII. Most commonly asked questions............................................................8 © Silicon Genetics, 2003 [email protected] | Main 650.367.9600 | Fax 650.365.1735 1 I. Introduction When testing for a statistically significant difference between means, each gene is tested separately and a p-value is generated for each gene. If the p value is set to 0.05, there is a 5% error margin for each single gene to pass the test. If 100 genes are tested, 5 genes could be found to be significant by chance, even though they are not. If testing a group -
Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy
Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy Chapter 11 Interpreting results and drawing conclusions Patrick Bossuyt, Clare Davenport, Jon Deeks, Chris Hyde, Mariska Leeflang, Rob Scholten. Version 0.9 Released December 13th 2013. ©The Cochrane Collaboration Please cite this version as: Bossuyt P, Davenport C, Deeks J, Hyde C, Leeflang M, Scholten R. Chapter 11:Interpreting results and drawing conclusions. In: Deeks JJ, Bossuyt PM, Gatsonis C (editors), Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy Version 0.9. The Cochrane Collaboration, 2013. Available from: http://srdta.cochrane.org/. Saved date and time 13/12/2013 10:32 Jon Deeks 1 | P a g e Contents 11.1 Key points ................................................................................................................ 3 11.2 Introduction ............................................................................................................. 3 11.3 Summary of main results ......................................................................................... 4 11.4 Summarising statistical findings .............................................................................. 4 11.4.1 Paired summary statistics ..................................................................................... 5 11.4.2 Global measures of test accuracy ....................................................................... 10 11.4.3 Interpretation of summary statistics comparing index tests ............................... 12 11.4.4 Expressing