Fundamental Statistical Concepts Research Articles

Rolling the DICE (Design, Interpret, Compute, Estimate): Interactive Learning of Biostatistics With Simulations.

Despite the increasing relevance of statistics in health sciences, teaching styles in higher education are remarkably similar across disciplines: lectures covering the theory and methods, followed by application and computer exercises in given data sets. This often leads to challenges for students in comprehending fundamental statistical concepts essential for medical research. To address these challenges, we propose an engaging learning approach-DICE (design, interpret, compute, estimate)-aimed at enhancing the learning experience of statistics in public health and epidemiology. In introducing DICE, we guide readers through a practical example. Students will work in small groups to plan, generate, analyze, interpret, and communicate their own scientific investigation with simulations. With a focus on fundamental statistical concepts such as sampling variability, error probabilities, and the construction of statistical models, DICE offers a promising approach to learning how to combine substantive medical knowledge and statistical concepts. The materials in this paper, including the computer code, can be readily used as a hands-on tool for both teachers and students.

Age Guessing: A Game to Introduce Fundamental Statistical Concepts

We develop a spreadsheet-based game to illustrate fundamental statistical concepts in the first class of an undergraduate Statistics course to motivate students about the topics that they will learn in upcoming classes. This game has been implemented by Google Forms and Google Sheets and can be played in both online and in-person classes of small and large sizes. Statistics is one of the most anxiety-inducing courses for undergraduate students, especially if mathematics is not the focus of their program. Negative anecdotes about the course, mathematics anxiety, and not knowing what the course is exactly about and how practical it can be are among the reasons that contribute to statistics anxiety. The first class provides a good opportunity for an instructor to mitigate these negative impressions and to set a positive attitude toward the course. A pre- and post-game group discussion that we have conducted systematically for six years suggests that the game addresses the students’ negative impression about the course and helps them gain a clearer understanding of the tools and skills they will learn in Statistics. Supplementary materials for this article are available online.

Analysis on Probability Mass Function and Probability Density Function

Probability Mass Function (PMF) and Probability Density Function (PDF) are fundamental concepts in probability theory and statistics that play a crucial role in describing the probability distribution of random variables. This abstract provides a comprehensive overview of these concepts, highlighting their definitions, characteristics, and applications. The Probability Mass Function is a concept primarily associated with discrete random variables. It defines the probability of a specific outcome occurring. The PMF assigns probabilities to individual values in the sample space, providing a clear picture of the likelihood of each possible outcome. Commonly denoted as P(X=x), where X is the random variable and x is a specific value, the PMF must satisfy two essential properties: non-negativity and the sum of probabilities over all possible outcomes equals one. On the other hand, the Probability Density Function is a concept applied to continuous random variables. Unlike the PMF, which deals with specific values, the PDF deals with ranges of values. The PDF represents the probability that a continuous random variable falls within a given interval. Denoted as f(x), it is essential to note that the probability of any specific point is zero, and instead, probabilities are defined for intervals. The area under the PDF curve over a given interval corresponds to the probability of the random variable falling within that interval. Understanding the differences and similarities between PMF and PDF is crucial for statistical analysis. While PMF is discrete and deals with specific values, PDF is continuous and provides probabilities for intervals. Both functions are integral to the calculation of various statistical measures, including expected values, variance, and standard deviation. This abstract concludes with a discussion of practical applications in diverse fields, such as finance, engineering, and natural sciences, where a deep understanding of PMF and PDF is essential for making informed decisions and drawing meaningful conclusions from data. The integration of these concepts into statistical models and analyses enhances the accuracy and reliability of predictions, making PMF and PDF indispensable tools in the field of probability and statistics

Student-Teacher Level of Statistical Reasoning Ability Reviewed by Apos Theory (Action, Process, Object, Schema)

In learning statistics, there is the term statistical reasoning. Statistical reasoning arises because of the opinion that learning statistics using a traditional approach does not lead students to have statistical reasoning abilities or to think statistically. There are four statistical reasoning levels: idiosyncratic, transitional, quantitative, and analytical. Student-teachers must understand fundamental statistical concepts more deeply to develop statistical reasoning abilities. The mental construction of a statistical concept can be analyzed using APOS (Action, Process, Object, Schema) theory. Through the APOS theory, researchers can determine individual understanding of statistical concepts that cause individuals to reason or draw reasonable conclusions based on the statistical information obtained. This study aims to analyze the interrelationships and make conjectures regarding the level of statistical reasoning ability based on the APOS theory. The research subjects are. 19 informants. The research results obtained four conjectures, namely. Students at the idiosyncratic level are in the action stage; students at the transitional stage are in the action, process, or object stage; students at the quantitative stage are in the process or object stage; lastly, students at the analytical are in the schema stage.

Statistical analysis of high-dimensional biomedical data: a gentle introduction to analytical goals, common approaches and challenges

BackgroundIn high-dimensional data (HDD) settings, the number of variables associated with each observation is very large. Prominent examples of HDD in biomedical research include omics data with a large number of variables such as many measurements across the genome, proteome, or metabolome, as well as electronic health records data that have large numbers of variables recorded for each patient. The statistical analysis of such data requires knowledge and experience, sometimes of complex methods adapted to the respective research questions.MethodsAdvances in statistical methodology and machine learning methods offer new opportunities for innovative analyses of HDD, but at the same time require a deeper understanding of some fundamental statistical concepts. Topic group TG9 “High-dimensional data” of the STRATOS (STRengthening Analytical Thinking for Observational Studies) initiative provides guidance for the analysis of observational studies, addressing particular statistical challenges and opportunities for the analysis of studies involving HDD. In this overview, we discuss key aspects of HDD analysis to provide a gentle introduction for non-statisticians and for classically trained statisticians with little experience specific to HDD.ResultsThe paper is organized with respect to subtopics that are most relevant for the analysis of HDD, in particular initial data analysis, exploratory data analysis, multiple testing, and prediction. For each subtopic, main analytical goals in HDD settings are outlined. For each of these goals, basic explanations for some commonly used analysis methods are provided. Situations are identified where traditional statistical methods cannot, or should not, be used in the HDD setting, or where adequate analytic tools are still lacking. Many key references are provided.ConclusionsThis review aims to provide a solid statistical foundation for researchers, including statisticians and non-statisticians, who are new to research with HDD or simply want to better evaluate and understand the results of HDD analyses.

Three Principles for Modernizing an Undergraduate Regression Analysis Course

As data have become more prevalent in academia, industry, and daily life, it is imperative that undergraduate students are equipped with the skills needed to analyze data in the modern environment. In recent years there has been a lot of work innovating introductory statistics courses and developing introductory data science courses; however, there has been less work beyond the first course. This article describes innovations to Regression Analysis taught at Duke University, a course focused on application that serves a diverse undergraduate student population of statistics and data science majors along with nonmajors. Three principles guiding the modernization of the course are presented with details about how these principles align with the necessary skills of practice outlined in recent statistics and data science curriculum guidelines. The article includes pedagogical strategies, motivated by the innovations in introductory courses, that make it feasible to implement skills for the practice of modern statistics and data science alongside fundamental statistical concepts. The article concludes with the impact of these changes, challenges, and next steps for the course. Portions of in-class activities and assignments are included in the article, with full sample assignments and resources for finding data in the supplemental materials. Supplementary materials for this article are available online.

The Case of the Limit of Detection

Students learn early on in science courses that reliable results can only be achieved from accurate measurements. In some of the first general chemistry classes, but even more intensively in analytical chemistry classes, students are introduced to the concept of significant figures. They learn that a measurement is only as true and precise as the instrument used to carry it out, and that the level of skill of the measurer (and how they interpret and report the measurement) is also important. A classic example to introduce significant figures involves the measurement of an object’s length using a ruler. Let us say that the object is a string, and one has two rulers available to measure it. The first ruler has 1-cm markings, while the second one presents 0.1-cm markings. When performing any measurement, students learn that more accurate results will be achieved if one estimates and includes an additional digit to the measurement. Thus, as an example, let us say that the string’s length is determined to be between 3 and 4 cm using the first ruler. The measurer will then estimate the tenths decimal place and report the string’s length as 3.4 cm, for example. Although considered significant, the tenths place (i.e., 4) is an estimate and, therefore, it is uncertain. The other digit (i.e., 3) is both significant and certain. Using the second ruler, the string’s length falls between 3.4 and 3.5 cm. The measurer will again estimate an additional decimal place and report the length as 3.48 cm. In this example, the digits 3 and 4 are both significant and certain, while the hundredths place (i.e., 8) is significant but uncertain. Now, how does the concept of significant figures relate to limit of detection (LOD)? There are different interpretations for LOD, but it simply is the lowest concentration of analyte generating an instrument response that is statistically different from (and higher than) that recorded for the blank. According to the International Union of Pure and Applied Chemistry (IUPAC), the LOD is calculated as three times the standard deviation of the instrument response for the blank (Sb), divided by the calibration curve slope (m), i.e., LOD = 3Sb/m. The value Sb is calculated from repeated measurements of the blank solution, usually 10 - 20 replicates (n = 10 -20). Considering a normal distribution, 99.86% of the data is < (x ̅+3S) (i.e., < than the value corresponding to three standard deviations above the mean). Therefore, the LOD calculated at 3Sb is statistically different from, and has a 99.86% chance of being larger than the blank.4 At the LOD, the instrument response recorded for the analyte is certainly not due to the random variation of the blank signal (almost 100% certain). However, there is no certainty associated with the analyte concentration. In other words, one knows the analyte is present, but cannot quantify it with an adequate level of confidence. Considering Sb as noise, the analyte’s signal-to-noise ratio (S/N) at the LOD is 3 (i.e., the instrument response recorded for the analyte is three times higher than the noise). Because S/N is the reciprocal of the relative standard deviation (RSD), analyte signal variations higher than 33% (RSD = 1/3) are expected at the LOD. Therefore, the concentration corresponding to the LOD is, by definition, uncertain. Going back to the concept of significant figures, the LOD must be reported with a single digit, which is significant but uncertain. Many researchers fail to follow these fundamental concepts of statistics and analytical chemistry, as LODs are often reported with several significant figures (!). Perhaps, it is due to a natural difficulty to connect concepts learned at different points in one’s education. Nevertheless, “the case of the LOD” may foster an awareness that could facilitate the identification of instances in which the connection between different concepts is essential: from adequate reporting of analytical results to new ideas and discoveries.

Confidence as Likelihood

Confidence and likelihood are fundamental statistical concepts with distinct technical interpretation and usage. Confidence is a meaningful concept of uncertainty within the context of confidence-interval procedure, while likelihood has been used predominantly as a tool for statistical modelling and inference given observed data. Here we show that confidence is in fact an extended likelihood, thus giving a much closer correspondence between the two concepts. This result gives the confidence concept an external meaning outside the confidence-interval context, and vice versa, it gives the confidence interpretation to the likelihood. In addition to the obvious interpretation purposes, this connection suggests two-way transfers of technical information. For example, the extended likelihood theory gives a clear way to update or combine confidence information. On the other hand, the confidence connection means that intervals derived from the extended likelihood have the same status as confidence intervals. This gives the extended likelihood direct access to the frequentist probability, an objective certification not directly available to the classical likelihood.

Fundamental Statistical Concepts in Clinical Trials and Diagnostic Testing.

This article explores basic statistical concepts of clinical trial design and diagnostic testing, or how one starts with a question, formulates it into a hypothesis on which a clinical trial is then built, and integrates it with statistics and probability, such as determining the probability of rejecting the null hypothesis when it is actually true (type I error) and the probability of failing to reject the null hypothesis when it is false (type II error). There are a variety of tests for different types of data, and the appropriate test must be chosen for which the sample data meet the assumptions. Correcting type I error in the presence of multiple testing is needed to control the error's inflation. Within diagnostic testing, identifying false-positive and false-negative results is critical to understanding the performance of a test. These are used to determine the sensitivity and specificity of a test along with the test's negative predictive value and positive predictive value. These quantities, specifically sensitivity and specificity, are used to determine the accuracy of a diagnostic test using receiver-operating-characteristic curves. These concepts are briefly introduced to provide a basic understanding of clinical trial design and analysis, with references to allow the reader to explore various concepts at a more detailed level if desired.

A study of statistics knowledge among nurse faculty in schools with research doctorate programs

Statistics knowledge is essential for nursing faculty in both teaching and research roles. In the teaching role when discussing nursing research, nursing faculty are confronted with statistical concepts and statistical methods applications. Knowledge of fundamental statistical concepts is needed so that nursing faculty can understand and critically evaluate the literature. The purpose of this study was to assess nursing faculty knowledge of fundamental statistical concepts. A probability sample with a 7.7% response rate yielded participation from 164 nursing faculty from 26 accredited schools. Results showed that most faculty members (91.5%) read peer-reviewed health-related scientific journal articles. On average, nursing faculty answered 5.1 (SD = 1.6) out of 8 statistical knowledge questions correctly. Problematic concepts included randomization (43.3% correct), and interpreting a confidence interval (42.7%) and odds ratio (33.5%). The results of this study may be used to improve statistics education and training for future nursing faculty and strengthen scholarship for nursing faculty conducting research.

The History of Bootstrapping: Tracing the Development of Resampling with Replacement

Sampling is one of the most fundamental concepts in statistics, as the quality and accuracy of the statistical inferences made, heavily depend on the method used to obtain the sample and the sample’s ability to represent the population of inference. Despite being a simple concept, sampling presents researchers with many challenges. Generally, due to monetary and time constraints, researchers must take a smaller sample size than they would ideally use. Using statistics from these small samples, estimates for population parameters can be made, typically in the form of a confidence interval. However, the validity of these confidence intervals depends on three basic assumptions that are difficult to meet with small sample sizes. This paper traces the development of the sampling method known as bootstrapping that helps small samples to meet these assumptions. The paper touches on previous methods used before the development of bootstrapping and shows how bootstrapping has evolved over the last four decades and become widely used in the field of statistics.

Interrogating a measurement conjecture to introduce the concept of statistical association in upper elementary education

SummaryThe association or relationship between two variables, also known as covariation, has been identified as one of the fundamental statistical concepts. This article details a statistical investigation that was used to informally introduce Grade 6 students to this concept.

A cross-sectional assessment of statistical knowledge among pharmacy faculty

IntroductionThe debate about the optimal level of research methods incorporated in doctor of pharmacy curricula is ongoing. Yet relatively little has been published about the research-based knowledge of the faculty in these programs. This study seeks to assess pharmacy faculty members' knowledge of fundamental statistical concepts. MethodsA cross-sectional survey included a random sample of US-based pharmacy schools. Invitations were sent to faculty in each selected pharmacy school to participate in the online survey comprised of demographic questions and a multiple-choice statistics knowledge assessment covering eight fundamental biostatistics concepts. Each question had three answer options, with an additional fourth ‘opt-out’ option for those wishing to avoid guessing. ResultsThe random sample of selected pharmacy schools resulted in email invitations sent to 2036 faculty from 30 pharmacy schools. The final study sample consisted of 139 pharmacy faculty members with a mean of 10.2 years of experience as a faculty member. Over 95% of pharmacy faculty reported that they read peer-reviewed scientific journal articles, while nearly 99% said the understanding of statistics was either ‘somewhat important’ (35%) or ‘very important’ (64%) in their role as a researcher. Pharmacy faculty achieved a mean of 5.1 correct responses out of eight knowledge-based questions. ConclusionsPharmacy faculty provided correct responses to 64% of the items about fundamental statistical concepts. These study results are useful for making decisions about biostatistics education and curriculum content, particularly as pharmacy programs continue to incorporate a prominent role for research.

Essential Statistical Concepts for Research in Speech, Language, and Hearing Sciences.

Purpose Clinicians depend on the accuracy of research in the speech, language, and hearing sciences to improve assessment and treatment of patients with communication disorders. Although this work has contributed to great advances in clinical care, common statistical misconceptions remain, which deserve closer inspection in the field. Challenges in applying and interpreting traditional statistical methods with behavioral data from humans have led to difficulties with replication and reproducibility in other allied scientific fields, including psychology and medicine. The importance of research in our fields of study for advancing science and clinical care for our patients means that the choices of statistical methods can have far-reaching, real-world implications. Method The goal of this article is to provide an overview of fundamental statistical concepts and methods that are used in the speech, language, and hearing sciences. Results We reintroduce basic statistical terms such as the p value and effect size, as well as recommended procedures for model selection and multiple comparisons. Conclusions Research in the speech, language, and hearing sciences can have a profound positive impact on the lives of individuals with communication disorders, but the validity of scientific findings in our fields is enhanced when data are analyzed using sound statistical methods. Misunderstanding or misinterpretation of basic statistical principles may erode public trust in research findings. Recommendations for practices that can help minimize the likelihood of errors in statistical inference are provided. Supplemental Material https://doi.org/10.23641/asha.7849223.

Fuzzy soft attribute correlation coefficient and application to data of human trafficking

In this paper, we introduce fuzzy soft attribute correlation coefficient and apply it to find the correlation between vulnerability government response of various countries related to human trafficking based on six regions with the help of data from “The Global Slavery Index 2016”. Comparison of fuzzy soft attribute correlation coefficients is done with the conventional analysis of sociology by calculating Pearson’s zero-order product-moment correlations. Along with these, some fundamental concepts of mathematical statistics are developed with respect to fuzzy soft set.

Causal Graphs and Concept-Mapping Assumptions

Determining what constitutes a causal relationship between two or more concepts, and how to infer causation, are fundamental concepts in statistics and all the sciences. Causation becomes especially difficult in the social sciences where there is a myriad of different factors that are not always easily observed or measured that directly or indirectly influence the dynamic relationships between independent variables and dependent variables. This paper proposes a procedure for helping researchers explicitly understand what their underlying assumptions are, what kind of data and methodology are needed to understand a given relationship, and how to develop explicit assumptions with clear alternatives, such that researchers can then apply a process of probabilistic elimination. The procedure borrows from Pearl’s concept of “causal diagrams” and concept mapping to create a repeatable, step-by-step process for systematically researching complex relationships and, more generally, complex systems. The significance of this methodology is that it can help researchers determine what is more probably accurate and what is less probably accurate in a comprehensive fashion for complex phenomena. This can help resolve many of our current and future political and policy debates by eliminating that which has no evidence in support of it, and that which has evidence against it, from the pool of what can be permitted in research and debates. By defining and streamlining a process for inferring truth in a way that is graspable by human cognition, we can begin to have more productive and effective discussions around political and policy questions.

PRESERVICE TEACHERS COMPARING GROUPS WITH TINKERPLOTS—AN EXPLORATORY LABORATORY STUDY

Group comparisons offer students opportunities to reason about many fundamental statistical concepts like center, variation, or distribution. When doing such activities using large, real datasets, technology becomes and essential tool for exploring the data. With its large variety of features and its user-friendly handling, TinkerPlotsTM --as a software for learners and teachers--can facilitate the process of comparing distributions. In this article we focus on eight preservice teachers´ reasoning when comparing groups with TinkerPlots. We present ideas on the design of a course to develop statistical reasoning with TinkerPlots, present a framework to rate learners´ performance when comparing groups with TinkerPlots, and present results of a laboratory study about preservice teachers´ reasoning when comparing groups with TinkerPlots. Findings suggest that the TinkerPlots tool and design of the course supported these preservice teachers´ reasoning and that more learning opportunities are needed to increase their group comparison elements´ repertoire and interpretation in context. First published May 2018 at Statistics Education Research Journal Archives

Statistical Literacy as a Function of Online Versus Hybrid Course Delivery Format for an Introductory Graduate Statistics Course

ABSTRACTStatistical literacy refers to understanding fundamental statistical concepts. Assessment of statistical literacy can take the forms of tasks that require students to identify, translate, compute, read, and interpret data. In addition, statistical instruction can take many forms encompassing course delivery format such as face-to-face, hybrid, online, video capture, and flipped. In this study, we examined statistical literacy of graduate students using a validated assessment tool (the Comprehensive Assessment of Outcomes in Statistics; CAOS) across two increasingly popular delivery formats—hybrid and online. In addition, we examined condensed (six week) semesters to full (16 week) semesters to determine if course length was related to statistical literacy. Our findings suggest that, holding other factors constant, delivery format is not related to statistical literacy for graduate students. This contradicts some existing research that shows hybrid delivery outperforms online only. Our results have important implications for the teaching of statistics as well as for graduate education overall.

CORP: Minimizing the chances of false positives and false negatives.

Statistics is essential to the process of scientific discovery. An inescapable tenet of statistics, however, is the notion of uncertainty which has reared its head within the arena of reproducibility of research. The Journal of Applied Physiology's recent initiative, "Cores of Reproducibility in Physiology," is designed to improve the reproducibility of research: each article is designed to elucidate the principles and nuances of using some piece of scientific equipment or some experimental technique so that other researchers can obtain reproducible results. But other researchers can use some piece of equipment or some technique with expert skill and still fail to replicate an experimental result if they neglect to consider the fundamental concepts of statistics of hypothesis testing and estimation and their inescapable connection to the reproducibility of research. If we want to improve the reproducibility of our research, then we want to minimize the chance that we get a false positive and-at the same time-we want to minimize the chance that we get a false negative. In this review I outline strategies to accomplish each of these things. These strategies are related intimately to fundamental concepts of statistics and the inherent uncertainty embedded in them.

Statistical graph space analysis

The sample mean is one of the most fundamental concepts in statistics. Properties of the sample mean that are well-defined in Euclidean spaces become unclear in graph spaces. This paper proposes conditions under which the following properties are valid: existence, uniqueness, and consistency of means, the midpoint property, necessary conditions of optimality, and convergence results of mean algorithms. The theoretical results address common misconceptions about the graph mean in graph edit distance spaces, serve as a first step towards a statistical analysis of graph spaces, and result in a theoretically well-founded mean algorithm that outperformed six other mean algorithms with respect to solution quality on different graph datasets representing images and molecules.