A Bayesian Statistics Course for Undergraduates: Bayesian Thinking, Computing, and Research
AA Bayesian Statistics Course forUndergraduates:
Bayesian Thinking, Computing, and
Research
Jingchen Hu ∗ Mathematics and Statistics Department, Vassar College,Box 27, 124 Raymond Ave, Poughkeepsie, NY 12604,[email protected] 31, 2020
Abstract
We propose a semester-long Bayesian statistics course for undergraduate studentswith calculus and probability background. We cultivate students’ Bayesian think-ing with Bayesian methods applied to real data problems. We leverage modernBayesian computing techniques not only for implementing Bayesian methods, butalso to deepen students’ understanding of the methods. Collaborative case studiesfurther enrich students’ learning and provide experience to solve open-ended appliedproblems. The course has an emphasis on undergraduate research, where accessibleacademic journal articles are read, discussed, and critiqued in class. With increasedconfidence and familiarity, students take the challenge of reading, implementing, andsometimes extending methods in journal articles for their course projects.
Keywords:
Bayesian education, Bayesian thinking, JAGS, statistical computing, statisticseducation, undergraduate research ∗ The author gratefully acknowledges the Mathematics and Statistics Department at Vassar College fortheir support and encouragement to our experiment of Bayesian education at the undergraduate level. Theauthor thanks the Liberal Arts Collaborative for Digital Innovation (LACOL), for their generous fundingsupport to share this course across 10 campuses, resulting in wider student audience and richer pedagogyexperience for the author. The author also thanks Jim Albert, a co-author of the recently publishedundergraduate-level textbook, Probability and Bayesian Modeling ( https://monika76five.github.io/ProbBayes/ , for his experience of Bayesian education and generosity to mentor junior Bayesian educators. a r X i v : . [ s t a t . O T ] A ug Introduction
Statistics educators have been actively introducing Bayesian topics into the undergradu-ate and graduate statistics curriculum for the past few decades. At the Joint StatisticalMeetings in 1996, the Section on Statistical Education organized an invited session on theadvantages, disadvantages, rationale, and methods for teaching an introductory statisticscourse from a Bayesian perspective.
The American Statistician subsequently published aseries of papers and discussion on these topics (Berry, 1997; Albert, 1997; Moore, 1997).Moore (1997), in particular, argued that it was premature to teach the ideas and methodsof Bayesian inference in an introductory statistics course. The obstacles presented include:1) Bayesian techniques were little used, 2) Bayesians had not yet agreed on standard ap-proaches to standard problem settings, 3) the requirement of conditional probability can beconfusing to beginners, and 4) the teaching and learning of Bayesian inference might im-pede the trend toward experience with real data and a better balance among data analysis,data production, and inference.Indeed, prior to the invention and development of the Gibbs sampler and other Markovchain Monte Carlo (MCMC) algorithms in the late 1980s and early 1990s, not only theteaching in the classroom, but also the practice of Bayesian methods, had been very limited.Nevertheless, statistics educators made great effort to innovate, especially to connect to realdata problems and apply Bayesian methods to solve these problems (Franck et al., 1988).Thanks to the revolutionary computational development and the rapid spread of Bayesiantechniques used in applied problems, the teaching and learning of Bayesian methods hadtaken off, albeit mostly at the graduate level. Not only there are a number of Bayesiancourses in statistics graduate programs, the list of available textbooks keeps growing:
Bayesian Data Analysis (Gelman et al., 2013),
A First Course in Bayesian StatisticalMethods (Hoff, 2009),
Bayesian Essentials with R (Marin and Robert, 2014),
StatisticalRethinking (McElreath, 2020), and
Bayesian Statistical Methods (Reich and Ghosh, 2019),among others. Non-statistics graduate programs, including marketing and business, cogni-tive science, and experimental psychology, have seen flourishing developments of Bayesianeducation (Rossi et al., 2005; Kruschke, 2014; Lee and Wagenmakers, 2014). Some of thesetextbooks are suitable for undergraduate statistics students as well, for example,
Doing ayesian Data Analysis (Kruschke, 2014). Statisticians have also contributed to Bayesianeducation for non-statisticians (Gelman, 2008; Utts and Johnson, 2008).How about Bayesian education at the undergraduate level? Most of the educationalinnovation is taking place in the introductory statistics courses, where Bayesian inferenceis one of the many topics. Most recently, Eadie et al. (2019) designed an active-learningexercise of Bayesian inference with m&m’s, and Barcena et al. (2019) designed a websimulator to teach Bayes theorem, with application to the search for the nuclear submarine,USS Scorpion, in 1968. Educational innovation for advanced-level undergraduate statisticscourses, where Bayesian inference is typically covered as a topic in a statistical inference/ mathematical statistics course, includes Kuindersma and Blais (2007), who designed ateaching tool of Bayesian model comparison used in a physics application of a three-sidedcoin, and Rouder and Morey (2019), who proposed teaching Bayes’ theorem by looking atstrength of evidence as predictive accuracy.We are among the many Bayesian statistics educators, who believe in the huge bene-fits of introducing Bayesian methods into the undergraduate statistics curriculum, beyondbeing only a topic in introductory or statistical inference / mathematical statistics courses(Witmer, 2017; Chihara and Hesterberg, 2018). In this article, we propose a semester-longBayesian statistics course for undergraduates with a background of multivariable calculusand probability. The main learning objectives are: students are expected to 1) understandbasic concepts in Bayesian statistics, including Bayes’ rule, prior, posterior, and poste-rior predictive distributions, and 2) apply Bayesian inference approaches to scientific andreal-world problems. To achieve these learning objectives, we emphasize the cultivationof Bayesian thinking, the role of statistical computing and the use of real data. Further-more, our proposed course has three important components: case studies, discussing andcritiquing journal articles, and course projects.In Section 2, we provide a course overview, where we introduce the pre-requisites andstudents’ background, leading to a discussion of the choice of topics in the course. AsBayesian computing is an important and interwoven aspect of this course, Section 2.1describes our choices, approaches, and philosophy of computing in the course. We thenproceed to the details of the proposed course in Section 3, including the three key com-3onents of the course: case studies, discussing and critiquing journal articles, and courseprojects, where we provide recommendations based on our experience. We also discussassessments in the course in this section. The article ends with a discussion of students’experience, challenges, and future ideas with Section 4. Supplementary materials includea course schedule, sample in-class R scripts and computing labs in R Markdown, sam-ple homework, sample case studies, and a reading guide for a journal article. These areavailable in the Supplementary Materials online. Our proposed course is a popular elective for students pursuing statistics major or minorat Vassar College. We meet twice a week for 13 weeks, with a total of 26 class meetings.Each class meeting lasts 75 minutes, and some lectures are used as computing labs. Thecourse pre-requisites are multivariable calculus and probability. Among the calculus andprobability topics, we emphasize familiarity with transformation of random variables andjoint distributions (especially joint densities of conditionally independently and identically( i.i.d. ) distributed random variables), as these are key skills in expression of joint posteriordistributions. We provide a solid review of these topics at the beginning of the course.A typical student in this course might have prior statistics exposure, though havingtaken a statistics course is not required. We also do not assume prior R experience ofstudents, though most likely they have had some exposure. To ensure students are readyfor statistical computing in the course, we assign three DataCamp courses within the firstcouple of weeks of the course: Introduction to R , Intermediate R , and Introduction to theTidyverse , all of which are available through DataCamp For The Classroom . Since wedo not assume prior R experience, we have made the choice to mainly use base R insteadof tidyverse in this course, though we have seen students with tidyverse background For more information about DataCamp, visit datacamp.com . The course link: . The course link: . The course link: . For more information about the DataCamp For The Classroom, visit . tidyverse .We have two main learning objectives: students are expected to 1) understand basicconcepts in Bayesian statistics, including Bayes’ rule, prior, posterior, and posterior predic-tive distributions, and 2) apply Bayesian inference approaches to scientific and real-worldproblems. These course objectives motivate our choice of topics and their contents: • Bayesian inferences for a proportion and for a mean : Covering Bayes theorem,conjugate prior, posterior distribution, and predictive distribution, with a focus ofcomparison of the exact solutions versus the Monte Carlo simulation solutions inone-parameter Bayesian models. • Gibbs sampler and MCMC : Covering multi-parameter Bayesian models, why andhow a Gibbs sampler works, MCMC diagnostics, coding one’s own Gibbs sampler,and using Just Another Gibbs Sampler (JAGS) for MCMC estimation. • Bayesian hierarchical modeling : Covering why a hierarchical model is preferredin certain data analysis, how to specify a multi-stage (hierarchical) prior distribution,MCMC estimation, prediction, and analyzing pooling / shrinkage effects induced inhierarchical models. • Bayesian linear regression : Covering how to estimate a regression model, differentprior choices, MCMC estimation, and predictions.The above are our main topics in the course, designed to provide an adequate coverageof basic Bayesian concepts, inference methods, and computing techniques within certain ap-plied contexts. Through case studies, discussing and critiquing journal articles, and courseprojects, students are exposed to a much wider range of Bayesian methods in this course:some are innovative extensions of methodologies covered in class through case studies; oth-ers are much more advanced methods students encounter in their course projects. Thesefeatures are designed to provide ample time and space for students to dive into researchwith what they have gained to achieve what they want to do. We will provide details anddiscuss our choices of these features in Section 3.5s Bayesian computing is an important and interwoven aspect of any modern Bayesiancourse, we proceed to describe our choices, approaches, and philosophy of computing inour proposed course in Section 2.1.
At the undergraduate-level, we expect students to use computing techniques for imple-menting Bayesian methods in applied problems. Moreover, going through the computingaspect of Bayesian inference enhances students’ understanding of the methods themselves.We achieve these two computing goals by a two-stage process: • Stage 1 (first 1/3 of the course): In conjugate cases, e.g. beta-binomial, normal-normal, gamma-normal, gamma-Poisson, implement and compare exact solutionsand Monte Carlo approximation solutions to posterior and predictive inference. • Stage 2 (second 2/3 of the course): Introduce JAGS for implementing Gibbs samplers,compared to self-coded Gibbs samplers for simple cases. From then on, use JAGS forsubsequent topics, for example, Bayesian hierarchical modeling and Bayesian linearregression (not only Gibbs samplers, but also Metropolis-Hastings algorithms).
The focus of Stage 1 is to familiarize students with R programming and Monte Carlo tech-niques in simulating posterior and predictive distributions. In conjugate cases, analyticalposterior and predictive distributions are available, which are great examples for studentsto compare the exact solutions and Monte Carlo simulation solutions. Furthermore, simu-lating the predictive distributions and performing posterior predictive checks give studentsample opportunities to distill the essence of Bayesian computing. It is therefore desirablenot to introduce JAGS or any other MCMC estimation software at this stage, avoiding thetendency to use these software as a “black box”.6 .1.3 Why and How to Use JAGS
The shift from self-coding to available software such as JAGS takes place in covering theGibbs sampler and MCMC diagnostics. Simple cases, such as a two-parameter normalmodel, are used to introduce the definition and derivation of full conditional posteriordistributions, the keys to writing one’s own Gibbs sampler. Given example R scripts of aGibbs sampler (involving functions and loops), students practice writing Gibbs samplersto explore important aspects of MCMC, leading to discussions of MCMC diagnostics.Once students have a solid understanding of the mechanics of Gibbs samplers and havegained the ability to write their own Gibbs samplers, JAGS software is introduced, focusingon its descriptive nature of the specified Bayesian models and its comparison to a self-codedGibbs sampler. To show JAGS’s descriptive nature, Figure 1 presents the JAGS script,with the expressions of the sampling density and the prior distributions to its right. Tocompare JAGS output to the output of a self-coded Gibbs sampler, students are promptedto revisit previously covered aspects of MCMC, further distilling the keys to MCMC andits diagnostics.Figure 1: The JAGS script to express thesampling density and the prior distributions. - The sampling density: Y , · · · , Y n | µ, σ i.i.d. ∼ Normal( µ, σ ) . - The prior distributions: µ ∼ Normal( µ , σ ) , /σ = φ ∼ Gamma( α, β ) . Recall our two aforementioned computing goals in this course: we expect students touse computing techniques for implementing Bayesian methods in applied problems, andstudents’ understanding of the Bayesian methods should be further enhanced by going7hrough the computing aspect of these methods. Using JAGS achieves both goals: JAGS“frees up” students to explore non-conjugate priors and advanced Bayesian models, whichinvolves additional MCMC samplers, such as the Metropolis-Hastings algorithm. Moreover,it enhances students’ understanding of the Bayesian models being implemented: althoughstudents might not know the actual algorithms that JAGS performs, they need to beabsolutely clear about how to write the JAGS script to implement the Bayesian modelsthey have intended to use. We believe at the undergraduate-level, JAGS is sufficient andself-directed, that there is no need to provide an R package for the proposed Bayesianstatistics course.In addition to JAGS, Bayesian inference Using Gibbs Sampling (BUGS) is another pop-ular MCMC estimation software, whose syntax and usages are similar to those of JAGS.Stan is rapidly growing in popularity, and many available packages provide wrapper func-tions such as the stan glm() in the rstanarm package, that shares similar syntax andusages with the standard glm() R function. We have made the deliberate decision to useJAGS due to its aforementioned descriptive nature of the sampling density and prior distri-butions, its compatibility with various operating systems, and its emphasis on understand-ing the Bayesian model specification (unlike the “black box” style of Stan-based wrapperfunctions). We also note that, to the best of our knowledge, almost all undergraduate-levelBayesian textbooks use JAGS (Kruschke, 2014; Reich and Ghosh, 2019; Albert and Hu,2019). Albert and Hu (2020) provides a review of Bayesian computing in the undergraduatestatistics curriculum and their recommendations.
This section provides further details of the course. We first describe homework, computinglabs, and exams in the course. To achieve our learning objectives, we incorporate threeimportant components into our course: case studies, discussing and critiquing journal ar-ticles, and course projects. Sections 3.1 through 3.3 present and discuss these three keycomponents one-by-one and our recommendations for enhancing students’ learning expe- For more information, visit: https://mc-stan.org/
Like Allenby and Rossi (2008), we believe case studies are effective ways to illustrate appliedBayesian analysis: one constructs appropriate prior distributions, develops the likelihood,computes the posterior distributions, and finally communicates their results to address thequestions of interest. In the last 2/5 of the semester, we introduce 3 case studies in placeof homework assignments, where students are encouraged to explore extension of learnedmethods and / or create new approaches, to solve open-ended applied problems. Studentsare paired up and given one week to work on each case study. Some case studies are worthmultiple rounds of attempts and discussions, while others are more straightforward.9he open-ended nature of case studies encourages students to build upon what they havelearned and think outside of the box. Furthermore, they provide opportunities to introduceadvanced modeling techniques. For example, given a dataset consisting of two clusters ofstudents’ multiple-choice test scores, are there a knowledgable group and a random guessinggroup? If so, can we differentiate them? Furthermore, can we make inference about somegroup parameters, such as the accuracy of answering a question? Given their experience,students will usually take the hierarchical modeling approach with a pre-determined setof two groups. After a first round of case study reports and discussion, we point out aclear drawback of the approach: are the pre-determined groups are reasonable, especiallyfor those borderline scores (e.g. accuracy rates of 60% and 70%, which are higher thanthe random guessing of 50% but lower than the knowledgable 90%)? Latent class models,where no pre-determined group for each observation is assigned, are more suitable mod-eling techniques for such context. With an appropriate level of model introduction andsample R / JAGS syntax, students re-take this case study and learn by themselves thedetails of the model and estimation process. Case studies like this one could introduceimportant applied problem and new Bayesian methods simultaneously, further strengthen-ing students’ learning. Moreover, they could showcase and create discussions of limitationsof familiar approaches, and open up opportunities for more suitable, although inevitablymore advanced modeling techniques. This case study is available in the SupplementaryMaterials.In addition to designing suitable case studies, we recommend the following practicesto enrich students’ learning experience. First, assign students to case studies in pairs toencourage collaborative work. If possible, allow students to work with a different partnerin every case study. Second, ask pairs to upload their case study write-ups on the learningmanagement system (LMS, such as Moodle, Canvas etc.) before class discussion. Third,during lecture, allow students to first discuss their approaches and findings in a small group,and then to discuss and critique different approaches as an entire class.
Cobb (2015)’s five imperatives to “rethink our undergraduate curriculum from the ground10p” highly resonate with us, especially the last and the most important imperative: “teachthrough research”. We believe undergraduate students can and should be reading academicjournal articles as part of their education, provided that the articles have the right contentat the right level.Journals such as
The American Statistician are great sources of high-quality and accessi-ble journal articles for undergraduates. For our Bayesian course, Casella and George (1992)has been our favorite. Concise and nicely written, the authors gives a simple explanation ofhow and why the Gibbs sampler works, illustrates its properties with a two-by-two simplecase, designs simulation studies, and analyzes the results. Furthermore, as it was an earlypaper, some aspects of the Gibbs sampler, such as how to obtain independent parameterdraws, could be different from current practice. All these provide wonderful opportunitiesfor students to read and learn about the development of the Gibbs sampler, and to discussand critique different practices.In addition to selecting articles with the right content and at the right level, we rec-ommend the following practices to enhance students’ learning experience. First, providea reading guide containing several questions to help students navigate the article. A se-lection of questions of varied types - some about verifying mathematical expressions andothers about describing methodology in one’s own words - is a great blend. Second, askstudents to post their responses to these questions on the LMS before class discussion tofacilitate small-group and entire-class discussions in class. If possible, ask additional postsafter class discussion. Third, if the article contains simulation studies, design a computinglab to allow students to replicate simulation studies and graphical results presented in thearticle. In our case with Casella and George (1992), replicating one of their simulationstudies requires students to be very clear about how many Gibbs samplers are run and howmany iterations each run needs to take, which undoubtedly deepen their understanding ofGibbs samplers and enhance their computing skills. Our reading guide and computing labare available in the Supplementary Materials.The benefits of reading, discussing, critiquing, and replicating journal articles go beyondthe articles themselves. Through this process, students have gained confidence and skills,encouraging them to take the challenge to read, understand, implement, and sometimes11xtend methods from journal articles in their course projects, where we have frequentlywitnessed students’ growth from a focus on completing assignments to a focus on conductingresearch.
In addition to the exposure to journal articles, our next step of implementing Cobb (2015)’s“teach through research” imperative is a course project.Students in our course are encouraged to brainstorm project ideas from day one. Inaddition to several instructor-selected examples of Bayesian methods solving interestingapplied problems, our first lecture includes a few video clips of students’ projects fromprevious semesters (a 2-minute introduction video of the course project is required as partof students’ project submission). This immediately shifts the focus of interesting andexciting projects considered by us, the instructors, to those by them, the fellow students.The variety of project topics and interests not only showcases what students could achievein their projects, but also motivates them to choose what they want to explore.Depending on students’ academic background and career aspiration, there could existclusters of interests and topics. In the case of Vassar College: we have a number of doublemajors of Mathematics / Statistics and Economics, leading to groups of students workingon projects related to economics and finance; we have a Bayesian cognitive scientist facultymember in the Cognitive Science Program, leading to groups of students analyzing exper-imental data to explore learning theories. Hot topics, such as neural networks and naturallanguage processing, inevitably attract students’ attention and are reflected in their projectinterests and topic choices.Here we share additional information about practices we use to form project teams,keep every project on track, and present project outcomes. Our semester is 13 weekslong. Within the first week of the semester, students are encouraged to indicate theirproject interests through a self introduction post on the LMS. A list of project interests isextracted from their posts and shared with all students through an editable Google Doc,which students can freely browse and add more thoughts. It is at this stage that studentsstart to find shared interests with each other, and slowly project teams begin to form.12y Week 6, students settle down on their project topics, and submit a one-page projectproposal. Each project team (up to 3 students) needs to meet with the instructor beforesubmitting the project proposal, and detailed feedback of feasibility and advice is given byWeek 7, midway of a 13-week semester. From Week 8, each project team creates a weeklyschedule to complete the project. There are 2 credit-bearing check points for every team:a methodology draft by Week 10 and a project draft by Week 12.On the last day of class in Week 13, teams present their projects at a poster session. Theposter session lasts for 75 minutes. Typically we break all teams into 3 sessions, and eachsession is allocated with 15 minutes, with a 5-minute discussion break between sessions,and concluding with a 20-minute final discussion. This arrangement allows students topresent their own posters and to explore other students’ work. The 5-minute discussionbreak invites students to share their thoughts after learning about other students’ projects.In addition, each team submits a 2-minute introduction video about their project, andevery student is asked to watch the videos before the poster session. These 2-minute videoshelp presenters give a high-level pitch about their work. It also helps everyone to plan theirposter session better, for example, to spend more time on a poster which they are curiousabout based on the introduction video.Course projects naturally grow into independent studies in the following semesters. Pastand current independent study topics stemming from this Bayesian statistics course include:Bayesian estimation of future realized volatility, Bayesian inference with Python, Bayesiannonparametric models, Bayesian time series, and Bayesian variable and model selection.Students in these independent studies are engaged in almost the entire process of appliedstatistics research: literature review, collect / find datasets, implement methods, analyzethe results, and write a journal-style article / report. Some projects can be turned intoa new topic in the future iterations of this Bayesian statistics course. Moreover, workingwith students on topics of their interests exposes the instructor to new research areas andideas. 13able 1: Assessment components and their percentages of final grade.Assessment Component PercentageHomework and labs 25%Participation (DataCamp modules, case studies, and paper discussions) 10%Midterm exams 40% (20% × We present Table 1 with all course assessment components and their percentages of finalgrade. Details about when each assessment is assigned and how long students are expectedto complete can be found in the course schedule, available in the Supplementary Materials.While all other components are self-explanatory to some extent, we would like to discussour rationale of creating the participation component. It includes DataCamp modules,case studies, and paper discussions (online and in-class), with the latter two correspondingto the two unique and important features of our course, described in Sections 3.1 and3.2 respectively. They are all assessment tools of an open-ended nature. As mentionedpreviously, homework is assigned for the first 3/5 of the semester. During this period,students are gradually absorbing new knowledge and techniques, and homework consistingof a set of “close-ended” questions is a great tool to assess students’ learning. In thelast 2/5 of the semester, students would have acquired basic knowledge and techniques.Moreover, they have been exposed to a collection of Bayesian modeling strategies, whichhas prepared them to solve more open-ended questions. We therefore replace traditional“close-ended” homework with open-ended case studies, and we have deliberately made thechoice of making case studies as part of participation instead of being formally graded.In sum, assessments of “close-ended” nature, including homework, labs, and midtermexams are designed based on the learning objective of understanding basic concepts inBayesian statistics, while assessments of open-ended nature, including case studies, paperdiscussions, and projects are designed based on the learning objective of applying Bayesianinference approaches to scientific and real-world problems.14
Epilogue and Discussion
The various learning activities, tasks, and assessments in our proposed course fit wellwith many of the recommendations of the GAISE Report (Carver et al., 2016): integratereal data with a context and a purpose (case studies and projects), foster active learning(computing labs, journal articles, case studies, and projects), use technology to exploreconcepts and analyze data (R and JAGS), and use assessments to improve and evaluatestudent learning (a well-designed hybrid of “close-ended” and open-ended assessment tools).Students’ experience - through informal and formal evaluations - has been overall posi-tive. Despite a challenging and heavy workload, students recognize their knowledge buildingand skills building in this course. Many have expressed positive experience with the courseproject, and the structured weekly schedule has been highly appreciated.Since the introduction of the course in Fall 2016, we have been running it once everyacademic year at Vassar College. Our emphasis on student research has been provedsuccessful: so far we have had two 1st place winners in the intermediate statistics categoryof the Undergraduate Class Project Competition (USCLAP), organized by the Consortiumfor the Advancement of Undergraduate Statistics Education (CAUSE) and the AmericanStatistical Association (ASA) .We recognize the challenges of teaching an undergraduate-level Bayesian statisticscourse for students with limited background, and we see room for improvement. For one,our course contains nontrivial R programming and statistical methods, which can be espe-cially challenging to students with little or no R programming backgrounds. We have beensupplementing with in-class R examples, computing labs, and DataCamp courses, and wesee room for improvement on this end. With no requirement of linear algebra, we havelittle means (and little time) to cover important topics such as Bayesian model selectionand variable selection in depth. We see room for improvement, and we believe it is highlylikely for instructors at other institutions to have linear algebra as one of the pre-requisites,such that topics of model selection and variable selection can be better incorporated intotheir courses.The same goes for prior exposure to statistics, especially the classical / Frequentist For more information about the USCLAP, visit . https://github.com/monika76five/Undergrad-Bayesian-Course . Wehave a recently published textbook for undergraduate Bayesian education in the CRC Textsin Statistical Science series. Details are available at https://monika76five.github.io/ProbBayes/ . Interested readers can also refer to Hu (2019) for sharing this course througha hybrid model across several liberal arts colleges. Supplementary Materials
Please see our Supplementary Materials online for a course schedule, sample in-class Rscripts and computing labs in R Markdown, sample homework, sample case studies and areading guide for a journal article. 16 cknowledgements
We are very grateful to the editor, the associate editor, and two reviewers for their usefulcomments and suggestions, which we believe have improved our work.
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