A Comparative Study Among Various Statistical Tests Using Microarray Gene Expression Data

Some Thoughts About Data Type, Distribution, and Statistical Significance

Like any other science, research in natural language processing (NLP) depends on the ability to draw correct conclusions from experiments. A key tool for this is statistical significance testing: We use it to judge whether a result provides meaningful, generalizable findings or should be taken with a pinch of salt. When comparing new methods against others, performance metrics often differ by only small amounts, so researchers turn to significance tests to show that improved models are genuinely better. Unfortunately, this reasoning often fails because we choose inappropriate significance tests or carry them out incorrectly, making their outcomes meaningless. Or, the test we use may fail to indicate a significant result when a more appropriate test would find one. NLP researchers must avoid these pitfalls to ensure that their evaluations are sound and ultimately avoid wasting time and money through incorrect conclusions.This book guides NLP researchers through the whole process of significance testing, making it easy to select the right kind of test by matching canonical NLP tasks to specific significance testing procedures. As well as being a handbook for researchers, the book provides theoretical background on significance testing, includes new methods that solve problems with significance tests in the world of deep learning and multidataset benchmarks, and describes the open research problems of significance testing for NLP.The book focuses on the task of comparing one algorithm with another. At the core of this is the p-value, the probability that a difference at least as extreme as the one we observed could occur by chance. If the p-value falls below a predetermined threshold, the result is declared significant. Leaving aside the fundamental limitation of turning the validity of results into a binary question with an arbitrary threshold, to be a valid statistical significance test, the p-value must be computed in the right way. The book describes the two crucial properties of an appropriate significance test: The test must be both valid and powerful. Validity refers to the avoidance of type 1 errors, in which the result is incorrectly declared significant. Common mistakes that lead to type 1 errors include deploying tests that make incorrect assumptions, such as independence between data points. The power of a test refers to its ability to detect a significant result and therefore to avoid type 2 errors. Here, knowledge of the data and experiment must be used to choose a test that makes the correct assumptions. There is a trade-off between validity and power, but for the most common NLP tasks (language modeling, sequence labeling, translation, etc.), there are clear choices of tests that provide a good balance.Beginning with a detailed background on significance testing, the book then shows the reader how to carry out tests for specific NLP tasks. There is a mix of styles, with the first four chapters providing reference material that will be extremely useful to both new and experienced researchers. Here, it is easy to find the material related to a given NLP task. The next two chapters discuss more recent research into the application of significance tests to deep neural networks and for testing across multiple datasets. Alongside open research questions, these later chapters provide clear guidelines on how to apply the proposed methods. It is this mix of background material and reference guidelines that I believe makes this book so compelling and nicely self-contained.The introduction in Chapter 1 motivates the need for a comprehensive textbook and outlines challenges that the later chapters address more deeply. The theoretical background material begins in Chapter 2, which introduces core concepts, including hypothesis testing, type 1 and type 2 errors, validity and power, and p-values. The reader does not need to have any prior knowledge of statistical significance tests to follow this part. However, experienced readers could still benefit from reading this chapter, as concepts such as p-values are widely misunderstood and misused (Amrhein, Greenland, and McShane 2019).The significance tests themselves are introduced in Chapter 3, categorized into parametric and nonparametric tests. The chapter begins with the intuitively simple paired z-test, then builds up to more commonly-applied techniques, showing the connections and assumptions that each test makes. Step-by-step algorithms help the reader to implement each test. Although the chapter does cite uses of tests in NLP research, the main purpose is to present the theory behind each test and point out their differences.Chapter 4 provides perhaps the most handy part of the book for reference: a correspondence between common NLP tasks and statistical tests. Each task is discussed in terms of the evaluation metrics used, then a decision tree is introduced to guide the reader toward a choice between a parametric test, bootstrap or randomization test, or sampling-free nonparametric test. Section 4.3 then links each NLP evaluation measure to a specific significance test, presenting a large table that helps readers identify which test they need for a specific task. Particular considerations for each task are also pointed out to provide more detail about the appropriate options. The final part of this chapter describes the issue of p-hacking, in which dataset sizes are increased until a significance threshold is reached without consideration for biases in the data (discussed, for example, in Hofmann [2015]). The chapter proposes a simple solution to ensure robust significance testing with large datasets.Where Chapter 4 presents well-established methods, Chapter 5 introduces the current research question of how best to apply statistical significance testing to deep learning. Non-convex loss functions, stochastic optimization, random initialization, and a multitude of hyperparameters limit the conclusions we can draw from a single test run of a deep neural network (DNN). This chapter, which is based on the authors’ ACL paper (Dror, Shlomov, and Reichart 2019), explains how the comparison process can be overhauled to provide more meaningful evaluations. Beginning by explaining the difficulties of evaluating DNNs, the chapter then introduces criteria for a comparison framework, then discusses the limitations of current methods. Reimers and Gurevych (2018) have previously tackled this problem, but their approach has limited power and does not provide a confidence score. Other works, such as Clark et al. (2011), compare DNNs using a collection of statistics, such as the mean or standard deviation of performance across runs. This book shows how such an approach violates the assumptions of the significance tests. The authors propose almost stochastic dominance as the basis for a better alternative. The chapter explains how to use the proposed method, evaluates it in an empirical case study, and finally analyzes the errors made by each testing approach.Large NLP models are often tested across a range of datasets, which presents another problem for standard significance testing. Chapter 6 discusses the challenges of assessing two questions: (1) On how many datasets does algorithm A outperform algorithm B? (2) On which datasets does A outperform B? Applying standard significance tests individually to each dataset and counting the number of significant results is likely to overestimate the total number of significant results, as this chapter explains. The authors then present a new framework for replicability analysis, based on partial conjunction testing, and discuss two variants (Bonferroni and Fisher) for when the datasets are independent or dependent. They introduce a method based on Benjamini and Heller (2008) to count the number of datasets where one method outperforms another, then show how to use the Holm procedure (Holm 1979) to identify which datasets these are. Chapter 6 provides a lot of detailed background on the proposed replicability analysis framework, and the later sections again link the process to specific NLP case studies, and step-by-step summaries help the reader to apply the methodology. Extensive empirical results illustrate the very substantial differences in outcomes between the proposed approach and standard procedures.The final two chapters present open problems and conclude, showing that the topic has many interesting research questions of its own, such as problems when performing cross-validation, and the limited statistical power of replicability analysis.Overall, I highly recommend this book to a wide range of NLP researchers, from new students to seasoned experts who wish to ensure that they compare methods effectively. The book is excellent as both an introduction to the topic of significance testing and as a reference to use when evaluating your results. For anyone with further interest in the topic, it also points the way to future work. If one could level any criticism at this book at all, it is that it does not deeply discuss the basic flaws of significance testing or what the alternatives might be. For now, though, significance testing is an integral part of NLP research and this book provides a great resource for researchers who wish to perform it correctly and painlessly.

Statistical significance tests are unnecessary even when properly done and properly interpreted: Reply to commentaries

In the absence of a sufficient amount of information for quality business decision-making, i.e. successful performance of activities without unnecessary losses in the consumption of inputs, recently the non-parametric DEA method (Data Envelopment Analysis) is most often used through the linear programming technique. In the event that company managers have enough information to make business decisions, parametric statistical tests are used that compare the company's current performance with optimal performance, i.e. those that are on the edge of efficiency. However, this situation is very rare, so before making business decisions, non-parametric and then parametric statistical tests are carried out in detail. The subject of research of this paper is primarily focused on the simultaneous application of parametric and non-parametric statistical tests in the assessment of the economic efficiency of an economic entity. After the conducted research and analysis of the obtained results, it was determined that the null hypothesis, which claims that the relative efficiency of the warehouse obtained by parametric statistical tests and the DEA method is identical and that the trends have the same direction, could not be fully accepted. Namely, it was found out that the results of one and the same economic situation using the mentioned two types of analysis differ to the extent that they are not adequate for economic decision-making, however identical results were obtained in the assessment of the trend. It can be concluded that the simultaneous application of both methods, as well as its implementation in several iterations, can provide enough quality information for effective decision-making. Stochastic processes that occur during the implementation of business decisions using the DEA technique can be minimized through the simultaneous application of statistical parametric methods and tests for evaluating the expected efficiency of DEA. The effectiveness of this method in any case depends on the size of the sample implemented in the aforementioned statistical analysis. The aforementioned statistical tests enable the measurement and detection of those input parameters that will most effectively contribute to the efficiency of business systems.

The present comment reviews practices revolving around tests of statistical significance. First, the logic of statistical significance testing is presented in an accessible manner; many people who use statistical tests might not place such a premium on the tests if these individuals understood what the tests really do, and what the tests do not do. Second, the etiology of decades of misuse of statistical tests is briefly explored; we must understand the bad implicit logic of persons who misuse statistical tests if we are to have any hope of persuading them to alter their practices. Third, three revised editorial policies that would improve conventional practice are highlighted.

Expanding the toxicologist's statistical toolbox: Using effect size estimation and dose-response modelling for holistic assessments instead of generic testing

Clinical practice involves measuring quantities for a variety of purposes, such as aiding diagnosis, predicting future patient outcomes, and serving as endpoints in studies or randomized trials. Measurements are almost always prone to various sorts of errors, which cause the measured value to differ from the true value; accordingly, studies investigating measurement error frequently appear in this and other journals. The importance of measurement error depends upon the context in which the measurements in question are to be used. For example, a certain degree of measurement error may be acceptable if measurements are to be used as an outcome in a comparative study such as a clinical trial, but the same measurement errors may be unacceptably large to make measurements usable in individual patient management, such as screening or risk prediction. In the past 20 years many papers have been published advocating how studies of measurement error should be analyzed, with a paper by Bland and Altman1 being one of the most cited and well known examples. There has been much controversy concerning the choice of parameter to be estimated and reported, and consequently confusion surrounding the meaning and interpretation of results from studies investigating measurement error. In this paper we first distinguish between the general concepts of agreement and reliability to aid researchers in considering which are relevant for their particular application. We then review the statistical methods that can be used to investigate and quantify agreement and reliability, dealing separately with the different types of measurement error study, while emphasizing the largely common techniques that should be used for data analysis. We reiterate that the judgment of whether agreement or reliability are acceptable must be related to the clinical application, and cannot be proven by a statistical test. We highlight the fact that reliability depends on the population in which measurements are made, and not just on the measurement errors of the measurement method. We discuss the advantages of method comparison studies making at least two measurements with each measurement method on each subject. A key advantage is that the cause of a correlation between paired differences and means in the so-called Bland–Altman plot can be determined, in contrast to when only a single measurement is made with each method. Throughout the paper, we try to emphasize that calculated values of agreement and reliability from measurement error studies are estimates of parameters, and as such we should report such estimates with CIs to indicate the uncertainty with which they have been estimated. We restrict our attention to measurements of a continuous quantity; alternative methods are required for categorical data2. One difficulty in the measurement error field is the number of different terms used to describe studies of measurement error. The terms 'agreement', 'reliability', 'reproducibility' and 'repeatability' are used with varying degrees of consistency in the medical literature. We first make clear the distinction between the statistical concepts of agreement and reliability3. Agreement quantifies how close two measurements made on the same subject are, and is measured on the same scale as the measurements themselves. Two measurements of the same subject may differ for a number of reasons, depending on the conditions under which the measurements were made. In a method comparison study there will be differences because of inherent variability in each of the measurement methods, as well as potentially a bias between the measurements from the methods. If the measurements are made by different observers or raters, differences may be due to biases between the observers. Agreement between measurements is a characteristic of the measurement method(s) involved, which does not depend on the population in which measurements are made, unless bias or measurement precision varies with the true value being measured. One popular way of quantifying agreement is to estimate the 95% limits of agreement, as proposed by Bland and Altman1. These limits are defined such that we expect that, in the long run, 95% of future differences between measurements made on the same subject will lie within the limits. If reliability is high, measurement errors are small in comparison to the true differences between subjects, so that subjects can be relatively well distinguished (in terms of the quantity being measured) on the basis of the error-prone measurements. Conversely, if measurement errors tend to be large compared with the true differences between subjects, reliability will be low because differences between measurements of two subjects could be due purely to error rather than to a genuine difference in their true values. The reliability parameter is also known as an intraclass correlation (ICC), as it equals the correlation between any two measurements made on the same subject. Reliability takes values between zero and one, with a value of one corresponding to zero measurement error and a value of zero meaning that all the variability in measurements is due to measurement error. As a dimensionless quantity, it is arguably quite difficult to interpret, and deciding what value constitutes sufficiently high reliability is often made in a subjective fashion. Repeatability of measurements refers to the variation in repeat measurements made on the same subject under identical conditions4. This means that measurements are made by the same instrument or method, the same observer (or rater) if human input is required, and that the measurements are made over a short period of time, over which the underlying value can be considered to be constant. Variability in measurements made on the same subject in a repeatability study can then be ascribed only to errors due to the measurement process itself. Reproducibility refers to the variation in measurements made on a subject under changing conditions4. The changing conditions may be due to different measurement methods or instruments being used, measurements being made by different observers or raters, or measurements being made over a period of time, within which the 'error-free' level of the variable could undergo non-negligible change. The first type of study we consider is a repeatability study, in which we investigate and quantify the repeatability of measurements made by a single instrument or method, and in which the conditions of measurement remain constant. The second type of study we consider is a method comparison study, in which measurements are made using two measurement methods on a sample of subjects. The use of two different methods means that this is a reproducibility study, but the term method comparison is used as it clearly communicates the changing conditions under which measurements have been made. In contrast to a repeatability study, systematic bias may exist between measurements made by two different methods, and their measurement errors may have different SDs. The last type of study we consider is one in which measurements are made by different observers or raters. Again this is a type of reproducibility study. As with method comparison studies, biases may exist between observers, and their measurement SDs may differ. As we discuss below (see Measurements with observers or raters), if interest lies in quantifying the measurement error characteristics of the particular observers in one's study, exactly the same analysis methods should be used as for a method comparison study. In order to investigate the repeatability of measurements, a repeatability study must, for an appropriately selected sample, make at least two measurements per subject under identical conditions. This means that the measurements must be made by the same measurement method, or the same observer or rater. The objective is to then quantify the agreement and reliability of measurements made by that particular method or observer. If the differences between two measurements made on a subject are approximately Normally distributed, in the long run we expect the absolute difference between two measurements on a subject to differ by no more than the repeatability coefficient on 95% of occasions. To estimate the within-subject SD, we can fit a one-way analysis of variance (ANOVA) model to the data containing the repeat measurements made on subjects. ANOVAs can be fitted in all modern statistical packages. ANOVA models partition variability in data into that which can be ascribed to differences between groups, and that remaining to within groups. For a repeatability study, the groups are defined by the subjects under measurement, and this must be specified in the statistical package used. The ANOVA model estimates how much variation in measurements can be attributed to differences in the true, or 'error-free' values of subjects, with the remainder constituting measurement error. Fitting the ANOVA model results in estimates of the between-subject and within-subject SDs (or alternatively the corresponding variances, which are the SDs squared). The estimate of the within-subject SD can be used in the above formula to give an estimate of the repeatability coefficient. We illustrate with a study by Järvelä et al., who investigated the repeatability of measurements of flow volume made by one observer using three-dimensional (3D) power Doppler ultrasonography from 22 ovaries5. The estimated repeatability coefficient was 3.97, meaning that the absolute difference between any two future measurements made by that particular observer on a particular subject/unit are estimated to be no greater than 3.97 on 95% of occasions. It is important to note that the repeatability of another observer may be different, because of differences in the training and ability of observers. Because the repeatability coefficient calculated is an estimate, it is important to calculate a CI for it to indicate how precisely it has been estimated. A CI may be given automatically by statistical software, but in the Appendix we review how a 95% CI for the within-subject SD can be calculated. Such a CI can be used to find a CI for the repeatability coefficient by multiplying the CI limits by . If the CI for the within-subject variance is given by software instead, the limits must first be square rooted to give a CI for the within-subject SD. The ANOVA model assumes that the measurement errors are statistically independent of the true 'error-free' value, and that the SD of the errors is constant throughout the range of 'error-free' values. Sometimes the SD of errors increases with the true value being measured. This should be checked by plotting paired differences between measurements against their mean, the so called Bland–Altman plot. We illustrate this later in the context of a method comparison study (see Plotting the data) and describe how such heterogeneous errors can often be dealt with by making a log transformation (see Non-uniform variability in measurement of the repeatability coefficient on the differences between measurements being approximately Normally This can be checked by a or plot of the paired differences in measurements on each subject. made in the of the CI for the within-subject SD is that the measurement errors are If this is in the may be to The reliability of a measurement method is often of interest when measurements are to be used to between subjects or groups of subjects. For example, if we have a choice of two measurement methods that could be used to an outcome in a clinical or study, using the method with reliability will give greater statistical power to differences between groups, for a given sample In this we describe how measurements from a single method can be used to estimate the reliability in a given we discuss the use of reliability to measurement methods (see Reliability in method comparison and different observers (see in The same ANOVA model as above (see can be used to estimate statistical will automatically give the estimated which is the of we can the estimates of the and within-subject SDs into the formula given above (see Agreement and to estimate the For the flow data of Järvelä et an estimated of was This means that of the variability in measurements of the flow was estimated to be due to genuine differences in flow between with the remaining being due to errors in the measurement process and the observer Because the measurements were all made by one the reliability may be to as As with the repeatability the calculated is an estimate, and a 95% CI should be statistical give a but in the Appendix we review the of a 95% CI for the For the flow data of Järvelä et the estimate of a 95% of to Because we use the same ANOVA model to estimate reliability as we describe for agreement, the same of 95% CIs for the estimate on of the true 'error-free' values. The reliability of a measurement method depends upon the of the population in which the measurements are made. the of reliability given above (see Agreement and we that the of true 'error-free' values in the measured by the between-subject SD, the value of agreement between repeat measurements is a characteristic of the method or instrument the of measurement errors is the range of true reliability depends on the of measurement errors and the true in the population in which measurements are made. The is with a a method for measuring has a within-subject SD of which does not with the underlying value being measured. we a study to estimate the reliability, and we sample from a heterogeneous population in which the between-subject SD in true is 20 This give a reliability or of . we sample our subjects from a more in which the SD of true values is the same as the variability of the within-subject error. In this the reliability is to . If studies report only an estimate of the reliability (ICC), can only make use of the estimate if the population in which the to use such measurements has We that report estimates of and within-subject SD, in to the In this can whether the measurement method will be sufficiently for their application, in which the between subjects may be we use a measurement method or in clinical we must that the measurements it are to by the measurement method used, that measurements made with the method are using the method. If the measurements from the two methods are sufficiently close and the of a patient on their basis be the the method could the method in clinical because the method is or to The as to what constitutes depends on how the measurements are to be used. If the two measurements are made on the same or we can quantify their To investigate and quantify the agreement between measurements made by two methods, we must at a a sample of subjects using the measurement method and the proposed method. The data from such a study of of measurements from each with the containing the measurements from the two measurement methods. we making two measurements on each subject using each method because the one measurement per method is the most common we by analysis. The first to such a is to plot the The plot is of measurements from the method against from the method (or If measurements were from we expect the to lie on the of the of can be used to if there is if more data are above the than this that the method on the measurements on the of fit for this plot and report a statistical for whether the of the plot from the of As by Bland and in a paper in this we expect the to have a than one if the method on the any measurement and so a of the that the true is to one is not a it the same as a plot with the of of the between the measurements from two methods is often more by plotting the difference in a measurements from the two methods against the of their measurements, as first by and this is to as the Bland–Altman and is frequently in of measurement error We illustrate the plot using an from the literature. et volume measurement in with and volume was measured using and also by We have approximately their plot of differences against means and it is as using only data from the in volume measured by three-dimensional and by against their from study by et the SD and the SD, each with 95% CI The plot the difference between calculated using and against the of two values. et not their plot or of results to indicate which measurements were from It is important to be and always the same measurement from the The the of the paired from zero an estimate of the bias between the two methods. which measurements have been from which to which method, on measurements. The indicate the estimated limits of agreement their CI which we describe The plot can be used to the differences between measurements made by the two methods. The variability of the differences between the two methods how well the methods If the variability of the paired differences is the range of measurements, and there is no between the difference and mean, we can quantify the agreement between the methods by the limits of the of the limits of agreement, we illustrate how the for limits of agreement may be and how they can be dealt on we expect the difference in volume as measured by the two methods to lie between and for 95% of future measurements. It is important to that the limits of agreement calculated in this are just and as with any type of estimate it is to quantify and report how precisely the limits are estimated of 95% The estimated limits of agreement indicate how large the between the measurements will be on 95% of occasions. The whether the methods sufficiently well must then be made depending on the context in which the measurements will be used. In contrast to the repeatability which assumes no bias between measurements, the limits of agreement method this The of the paired differences whether on one method to or measurements to the measurements of the second method, which we to as a bias between the methods. In the data in from et al., there was of a bias between the two methods, because the difference between the methods was close to at We can a statistical to whether there is of bias by a of the differences paired of the measurements from each we the that the true of the differences is corresponding to no bias between the methods. For the data in this that there was no of a bias between and made using The limits of agreement method assumes that the SD of the differences is throughout the range of measurements. Sometimes this will not be the In frequently the SD of the differences will with the In this for small values the limits of agreement will be than for values the limits will be of the plot of differences against from the study by et that the variability of the differences may be when the value being measured is The SD of the paired differences below the of the means was it was for above the This can often be by of the measurements of the two If the difference in the of the measurements results in constant we can calculate the limits of agreement and CIs for the limits in the Because the difference of two is to the of the we can all the estimates made on the log scale by Because of the limits of agreement to the of one measurements to the For the data in the of the difference in measurements was with SD The limits of agreement on the log scale are to the of with 95% limits of agreement for the of to The 95% CIs can be calculated on the log scale to the which measurements were made by which method, we cannot whether the is for measurements by to by or a plot of the of the measurements by the two methods to their mean, using a with the estimated 95% limits of agreement and their 95% This that the of a SD for the paired differences is more of measured by three-dimensional (3D) and by against of measurements, on data published by et the SD and the SD, each with 95% CI to log transformation can also be used to try to For example, et estimate limits of agreement using paired differences in measurements as a of the of the two measurements. The key is that a transformation should be used which that are approximately Normally and have approximately SD the range of It is for there to be an between the paired differences and of this is in a study by et al., who the agreement between as measured using and the calculated using in We have the data from their Bland–Altman plot of the difference between the volume and volume against the mean, and it in There to be a between the paired differences and differences between volume measured by three-dimensional and by from study by et correlation the SD and the SD, each with 95% CI We can a statistical to the for a whether the correlation coefficient between the paired differences and means from zero or by of the differences against the The is known as in the context of the SD of measurements from two methods, which for the data in is statistically for an between paired differences and how should we than models for the difference as a of the we should be given to the cause of the A between the paired differences and means when the SD of measurements from one method from the SD of measurements from the other method. There are at least two of an between paired differences and The first is that there is between the difference in measurements from the two methods and true value being the bias between methods over the range of true values. The second is that the within-subject SDs of the two methods differ. This will in the of changing bias if a method has or measurement errors than the method. This is and when a or measurement method is compared with a If the cause of the correlation is within-subject SDs and we were to plot the paired differences against the true value being there be no The between the Bland–Altman plot and a plot against true values because the paired means measurement errors, in contrast to the true values. as by and with only one measurement per method per we cannot which of two is the To we must make an that the within-subject SDs of the two methods are so that the correlation is attributed to changing or that the two within-subject SDs are and that there in no between bias and true value we may have a of For the of we with the from et by that the correlation is due to a difference in the within-subject SDs of the two methods. A correlation between differences and means will when the method with measurement error is from measurements of the method with error. For the data in the correlation is the errors than the measured by which and to the volume measured by as the true that such measurements were to no an that there is no changing we can calculate 95% limits of agreement, as by et al., because the limits of agreement method does not that the two within-subject SDs are In order to from the data whether changing bias is we must make at least two measurements with each method on each subject. This study has often been but is we describe how such studies can be to whether a correlation between paired differences and means is due to changing bias between the methods (see with two measurements from each As reliability may be a parameter with which to two different measurement methods. To estimate each reliability, we must make at least two measurements of each subject with each of the two methods. The repeat measurements from each method can then be as two repeatability studies (see estimates of each reliability, which can be advantage of using reliability to measurement methods is that it can be used to methods when their measurements are given on different or as the reliability is a dimensionless Because reliability depends on the of the true values in the population (see Reliability depends on population it is that reliability are compared only if they have been estimated from the same report a single reliability estimate from method comparison studies in which it that only a single measurement per subject per method is It appear that the estimates are using a one-way ANOVA as above (see The one-way ANOVA model assumes that there are no systematic biases between measurements within subjects, and that the within-subject SDs are for all measurements measurements are from two different methods, of may be bias may exist between the methods and their within-subject SDs may differ. If is the model is and the estimates of within and between subject SD are estimates of different The reliability estimate then has a different interpretation from the We against the estimate from the one-way ANOVA to data from two different measurement methods. As making two measurements with each method an of whether bias between the methods is constant or whether the measurement error SDs differ. the measurements from each method can be separately as two repeatability studies, using the methods estimates of the repeatability coefficient and reliability for each method. The Bland–Altman plot of paired differences against means can be by using the of a two measurements from each method in of the single As with the Bland–Altman plot on a single measurement from each method, an correlation between differences and means could be due to changing bias or a difference in measurement error between the methods. We describe a to whether any bias between measurements made by a method with the as measured by the method. We by the of the two measurements made using the method and we by the measurement made on subject by the measurement method. A plot of against can then be used to investigate whether the bias between the and methods with the as measured by the method. It is that is against rather than against because the a to the common measurement error in and It is for the same that plotting paired differences between measurements from two methods against the measurements from one method is in method comparison between values of and that the bias between the methods with the true value as measured by the method. A statistical of this can be by a for with as the The for the that the in this from zero the of against the of constant If of bias changing with true value is we may be in the at which the bias between the methods We can estimate this by the estimate from the above model by our estimate of the This quantity is an estimate of the in bias method for a in the true value as measured by the method. The choice to against is We could have against and have a different and estimate of the at which the bias This choice is because we will two different and because it means the is statistically The model underlying is a type of and as such can be fitted using software for The is to and for measurements are made by observers or raters. measurements made by two different observers are than are two measurements made by the same observer. as with two methods, measurements from two observers may differ due to bias between the observers observer and their measurement errors may also have different SDs. For example, measurements from an observer who can make more measurements will have a SD than made by a observer. If measurements in the future are to be made by different observers, we to describe and quantify the differences between such measurements in order to whether differences are genuine or may be due to measurement error. As with a method comparison study, the way to study this is for each observer to make at least two measurements of a sample of subjects. The of such a study and the type of statistical analysis that are should be by whether interest lies in a particular of observers, or whether we are in a population of observers or raters. If future measurements are to be made by a of observers, the measurement error study should each of observers making measurements of a sample of subjects, with at least two measurements per observer per subject. We can then use exactly the same methods as for method different observers as different measurement methods. If each observer two or more measurements on each we can whether there is bias or changing bias between observers. We can the bias of future measurements (in by measurements using the corresponding estimated We can also estimate repeatability and reliability for each using the one-way ANOVA model (see It may be of interest to which observers are more and if differences in reliability can be related to observer such as of or If we are to that biases between observers are we can fit a so-called model to such a for a subject and observer The estimates indicate the and for bias between observers. such models can be to the measurement error SD to differ between observers. The observers in a measurement error study can often be considered as a sample of observers from a population of observers who may be used in future studies or clinical In this we are not in the particular observers in the measurement error study, but only in the that they the population of observers. In this it is important that a number of observers is used in the measurement error study. For example, if the measurement error study only two observers, can be the population of observers, because we have a sample of just Because the observers in the measurement error study are considered a sample from the population of observers, we such a study using a model that the observer as a We use a with subject and observer Fitting a model estimates of the between-subject SD, SD and the measurement error SD to be the same for different The SD variability in measurements due to observers, or biases between the observers. such estimates one can calculate the SD of the difference in two measurements, made by the same observer or by two different observers. the SD is the will be than the This is because biases between observers to make measurements from different observers it is more difficult to distinguish between subjects on the basis of measurements made by two different observers than if the subjects been measured by the same observer. In this paper we have distinguished between the concepts of agreement and parameter is to the as they describe different characteristics of the measurement The choice of what to report in a particular study should be by how measurements are to be used in the and also by the fact that may to use a measurement method in a different We have the fact that the reliability of a method depends not only on the of the measurement errors but on the of true values in the population in which measurements are made. As measurement techniques potentially may be used in a variety of clinical and different it is to report estimates of and within-subject SDs. We have which methods we are for the analysis of repeatability studies, method comparison studies, and studies with measurements made by different observers or raters. In we not a single reliability coefficient should be used for method comparison If the reliability of two methods are to be each reliability should be estimated by making at least two measurements on each subject with each measurement method. We have how an between paired differences and means may not be by changing bias between two methods. Such an may also be by a difference in the measurement error but with only one measurement per subject per method it is not to which of is the comparison studies should make at least two measurements per subject per method. This an of the of any between paired differences and means for measurements made by the two methods, and also the repeatability and reliability of each method to be estimated. measurements an observer or measurement error studies must use an number of observers if interest lies in making a population of observers. we results for the of CIs for two in the one-way ANOVA in which two measurements are from each

Statistics serves as the foundation for various domains such as businesses, policy makers, markets and other sectors involved in activities such as data collection, planning, design, analysis, interpretation and reporting of the research ndings. Moreover, the ndings obtained from a research project carry greater signicance when subjected to analysis using statistical techniques. Various methods are employed to ensure precise data interpretation within the realm of research. Considering the data parameters, we have two methodologies – Parametric tests and Nonparametric tests. Parametric test depends on the presence of normal distributions within the data, whereas Nonparametric tests are suitable for analysing continuous data in any format. This article provides an explanation of a range of Parametric and Nonparametric tests along with R code.

Statistical tests help researchers analyze data to determine if observed differences or relationships are likely real and not just due to chance, providing a basis for making informed decisions in health research. Broadly speaking, statistical tests can be classified into two main categories: parametric tests and nonparametric tests.1 These categories are based on the assumptions and characteristics of the data being analyzed.

Different tests, different conclusions: evolutionary trees

Two main groups of statistical tests are parametric and nonparametric tests (Good, 2005). Parametric tests assume that the underlying data are normally distributed and nonparametric tests start off with the assumption that the underlying data do not have a normal distribution (Adler, 2012). Assuming that data are normally distributed is usually correct (Adler, 2012), and therefore parametric tests are more commonly applied than nonparametric ones. Nonparametric tests are also referred to as distribution-free tests, and they can be particularly useful when the data are measured on a categorical scale, i.e., when we deal with an ordered data set (Petrie and Sabin, 2005). It should be noted that nonparametric tests are less powerful in comparison to equivalent parametric tests (Vukovic, 1997).

The notion that nonparametric methods are required as a replacement of parametric statistical methods when the scale of measurement in a research study does not achieve a certain level was discussed in light of recent developments in representational measurement theory. A new approach to examining the problem via computer simulation was introduced. Some of the beliefs that have been widely held by psychologists for several decades were examined by means of a computer simulation study that mimicked measurement of an underlying empirical structure and performed two - sample Student t - tests on the resulting sample data. It was concluded that there is no need to replace parametric statistical tests by nonparametric methods when the scale of measurement is ordinal and not interval.Stevens' (1946) classic paper on the theory of scales of measurement triggered one of the longest standing debates in behavioural science methodology. The debate -- referred to as the levels of measurement controversy, or measurement - statistics debate -- is over the use of parametric and nonparametric statistics and its relation to levels of measurement. Stevens (1946; 1951; 1959; 1968), Siegel (1956), and most recently Siegel and Castellan (1988) and Conover (1980) argue that parametric statistics should be restricted to data of interval scale or higher. Furthermore, nonparametric statistics should be used on data of ordinal scale. Of course, since each scale of measurement has all of the properties of the weaker measurement, statistical methods requiring only a weaker scale may be used with the stronger scales. A detailed historical review linking Stevens' work on scales of measurement to the acceptance of psychology as a science, and a pedagogical presentation of fundamental axiomatic (i.e., representational) measurement can be found in Zumbo and Zimmerman (1991).Many modes of argumentation can be seen in the debate about levels of measurement and statistics. This paper focusses almost exclusively on an empirical form of rhetoric using experimental mathematics (Ripley, 1987). The term experimental mathematics comes from mathematical physics. It is loosely defined as the mimicking of the rules of a model of some kind via random processes. In the methodological literature this is often referred to as simulation. However, for the purpose of this paper, the terms experimental mathematics or computer simulation are preferred to because the latter is typically referred to when examining the robustness of a test in relation to particular statistical assumptions. Measurement level is not an assumption of the parametric statistical model (see Zumbo & Zimmerman, 1991 for a discussion of this issue) and to call the method used herein monte carlo would imply otherwise. The term experimental mathematics emphasizes the modelling aspect of the present approach to the debate.The purpose of this paper is to present a new paradigm using experimental mathematics to examine the claims made in the levels of measurement controversy. As Michell (1986) demonstrated, the concern over levels of measurement is inextricably tied to the differing notions of measurement and scaling. Michell further argued that fundamental axiomatic measurement or representational theory (see, for example, Narens & Luce, 1986) is the only measurement theory which implies a relation between measurement scales and statistics. Therefore, the approach advocated in this paper is linked closely to representational theory. The novelty of this approach, to the authors knowledge, is in the use of experimental mathematics to mimic representational measurement. Before describing the methodology used in this paper, we will briefly review its motivation.Admissible TransformationsRepresentational theory began in the late 1950's with Scott and Suppes (1958) and later with Suppes and Zinnes (1963), Pfanzagl (1968), and Krantz, Luce, Suppes & Tversky (1971). …

The management of large complex systems is complicated by self-regulation—the phenomenon whereby changes in one component lead to compensating changes elsewhere. One characteristic of such systems (though by no means inevitable) is the potential for chaos—an unstable dynamic that makes predictions impossible. Papadopoulos et al.1, in the JRSM, and Smethurst and Williams2, in Nature, have argued that National Health Service (NHS) waiting lists show self-regulatory behaviour and may be capable of chaotic behaviour. These workers plotted the frequency distribution of the relative changes in queue size on a double logarithmic scale. When the double logarithmic plots appeared linear, the so-called power law was taken as evidence for self-regulation. We believe this technique to be fundamentally flawed. Unregulated time series can show power laws and we have suggested alternative approaches3. Here we apply appropriate tests to data on the size of waiting lists in 10 randomly chosen hospitals over the period 1998-2001 (data taken from [www.doh.gov.ac.uk ]). To each dataset we applied three tests. First, to check for long-term trends (e.g. increases or decreases in waiting lists), we performed linear regressions of queue size on time. Second, we estimated slope of the relation between change in log queue size and log queue size to see if this was negative. This test looks for evidence that large queues tend to get smaller whilst small queues get larger, as in self-regulated systems. Statistical significance tests cannot be performed on the slopes of such regressions, since for random time series the slope of this relation is biased and is expected to be negative. Instead we were interested in whether slopes were steeper than - 1, the implication of this value being that chaos is only possible if the slope is steeper than - 1. Third, we tested for self-regulation using a test (the Pollard test) explicitly designed to detect self-regulation in time series4. The results are shown in Table 1. 6 out of 10 time series showed evidence for long-term trends, all negative, indicating significant declines in queue length. The slopes estimated from the regressions of change in log queue size on queue size were steeper than - 1 only in one case, and this was only slightly steeper (- 1.11). As noted, statistical tests on these slopes are biased. The Pollard test, which corrects for this bias, indicated that only one time series exhibited statistically significant evidence for self-regulation; moreover, in view of the number of tests (10), the marginal significance of this relation (P=0.03) should be viewed with caution. Table 1 Analysis of data on waiting lists from 10 randomly chosen hospitals for evidence of changes consistent with chaos or density dependence We do not regard the analysis we have presented as definitive; for instance, longer time series may show different behaviour. However, claims that the NHS is ‘at the edge of chaos’ are not supported by our analyses. We have shown that, for the data in question, there is no evidence of self-regulatory behaviour. Moreover, waiting lists do show evidence of clear declines, probably resulting from management strategies to reduce queue sizes.

This paper suggests a Bidirectional LSTM (ELCNN-BiLSTM) model, which is an E-Commerce Logistics CNN-based model of intelligent decision-making and timeliness prediction in the cross-border logistics processes. The data to be used in the study was gathered within a global e-commerce logistics system of a cosmetics retailer with 1,25, 000 delivery records that have 24 spatio-temporal and operational characteristics, such as the shipment weight, their origin, destination, the distance of the route, the time in customs, and the time in the warehouse. The formulated task is a binary classification task to forecast whether a delivery is on-time or delayed, with an on-time threshold parameter being delivery within +/-1 day of the expected arrival. Preprocessing was performed on the basis of alignment of the timestamps, normalisation, treatment of 3.4% missing values by interpolation, and resampling to address a 1:1.7 ratio of the class imbalance with SMOTE. The ELCNN-BiLSTM architecture proposed is a combination of spatial feature extraction with convolutional layers and long-term temporal dependencies with the bidirectional LSTM layers that allow adaptive acquisition of logistics patterns on routes and time zones. Benchmarking of the model was done against LSTM, CNN-LSTM, Att-GRU, and XGBoost baselines on measurements of Accuracy, Precision, Recall, F1-Score, PR-AUC and ROC-AUC. Findings reveal that the suggested model has a better predictive power with 95.4 % accuracy, 93.9 % precision and 96.3 % of ROC-AUC, which are significantly higher than other approaches in all the most important measures. The ability and robustness of the model to generalise and statistical significance tests were statistically proven through statistical tests and k-fold cross-validation. The study presents a powerful data-oriented model of predicting supply chain delays in real-time, which can improve the reliability of supply chains and help to make intelligent decisions in the global e-commerce business.

Within cumulus cloud fields that develop in conditionally unstable air masses, only a fraction of the cumuli may eventually develop into deep convection. Identifying which of these convective clouds is most likely to generate lightning often starts with little more than a qualitative visual satellite analysis. The goal of this study is to identify the observed satellite infrared (IR) signatures associated with growing cumulus clouds prior to the first lightning strike, or lightning initiation (LI). This study quantifies the behavior of 10 Geostationary Operational Environmental Satellite-12 (GOES-12) IR fields of interest in the 1 h in advance of LI. A total of 172 lightning-producing storms, which occurred during the 2009 convective season, are manually tracked and studied over four regions: northern Alabama, central Oklahoma, the Kennedy Space Center, and Washington, D.C. Four-dimensional and cloud-to-ground lightning array data provide a total cloud lightning picture (in-cloud, cloud-to-cloud, cloud-to-air, and cloud-to-ground) and thus precise LI points for each storm in both time and space. Statistical significance tests are conducted on observed trends for each of the 10 LI fields to determine the unique information each field provides in terms of behavior prior to LI. Eight out of 10 LI fields exhibited useful information at least 15 min in advance of LI, with 35 min being the average. Statistical tests on these eight fields are compared for separate large geographical areas. Median IR temperatures and 3.9-μm reflectance values are then determined for all 172 events as an outcome, which may be valuable when implementing a LI prediction algorithm into real-time satellite-based systems.