Suggestions for Your Nonparametric Statistics Course

Abstract

Nonparametric methods form an integral part of many degree programs and concentrations in statistics. In this article a number of useful approaches are suggested to aid the instructor of a nonparametric statistics course. These include ideas for classroom presentations, projects, writing components, student-generated data, and computing. Each suggestion is discussed in the context of nonparametric methods instruction. These techniques help students develop an appreciation for the field of nonparametric statistics and the broad range of its applications in practice. Appendices include a partially annotated bibliography of textbooks and monographs from the field of nonparametric statistics and a collection of Minitab macros.

Supplemental data for this article can be accessed on the publisher's website.

• Assumptions
• Computing
• Distribution-free
• Pedagogy
• Rank-based procedures
• Projects
• Software for teaching statistics
• Student-generated data
• Writing

1. Introduction

1 Nonparametric statistics may be viewed as the collection of statistical methods that either (i) do not relate to specific parameters (the broad definition) or (ii) maintain their distributional properties irrespective of the underlying distribution of the data (distribution-free methods). Such procedures have been used for many years, although not always known by the name nonparametric. Sprent (1993) credited King Nebuchadnezzar of Babylon with an informal application of a permutation test around 600 B.C., while Cochran (1976), consistent with the Biblical account, attributed it to the prophet Daniel. Arbuthnott (1710) is frequently cited as an example of an early user of the sign test. The field of nonparametric statistics proper became established in the 1940's (see Friedman 1940, Mathisen 1943, Wilcoxon 1945, Mann and Whitney 1947, and Hoeffding 1948, for example), although isolated developments took place prior to that (e.g., Pearson 1900, Spearman 1904, Kolmogorov 1933, Friedman 1937, Kendall 1938, Smirnov 1939). The first textbook devoted to nonparametric methods (Siegel 1956) is often credited with bringing recognition to nonparametrics as a subfield of statistics, while Wolfowitz (1942) is given credit for coining the term now used to name this area (see David 1995). Today the field has grown to include such modern topics as density estimation, nonparametric regression, and semiparametrics. Applications are ubiquitous in nearly every discipline. Reflecting the increasing activity in the field, a new journal, the Journal of Nonparametric Statistics, was launched in 1991.

2 The teaching of nonparametric statistics has developed with the field itself. In the 1950's, specialized courses in nonparametric statistics were offered “in several universities” (Savage 1957). In the 1990's almost all schools offering graduate degrees in statistics have a course in nonparametrics, and many colleges and universities have undergraduate courses in the subject. A recent survey of alumni by the author's department revealed a growing interest in and demand for nonparametric methods. Instruction in nonparametric statistics has also been recommended as an important part of an undergraduate major in statistics (Pirie 1986). Even many institutions that donot offer a separate course for nonparametric methods still include material from this sub-discipline in other statistics courses.

3 Despite the prevalence of nonparametric statistics in the research, applications, and instruction of statistics, little has been published about the teaching of nonparametric statistics. Four notable exceptions are Noether (1970, 1980), Dietz (1989), Kemmerly (1990), and Katz and Tomazic (1991). Noether (1970, 1980) proposed that introductory statistics courses be taught from the nonparametric perspective; his introductory textbook (Noether 1991) does just that. Dietz (1989) gave suitable recommendations on what to include in a unit on linear regression in a nonparametric statistics course. Although Kemmerly (1990) made some useful suggestions concerning the teaching of nonparametric statistics, his paper is discipline specific, being directed to teachers of geology students. Katz and Tomazic (1991) addressed those teaching graduate service courses in nonparametric statistics. For some compelling reasons to include nonparametric statistics in graduate service courses, see Brogan (1980) and Brogan and Kutner (1986).

4 In this article we detail some suggestions for undergraduate and graduate courses in nonparametric statistics. Originality is not claimed for all the ideas set forth in this article. In fact, some have already been proposed in the broader context of statistics instruction, and many may seem obvious to experienced instructors. However, by its very nature, nonparametric statistics differs from classical statistics, and these differences call for special consideration in the teaching of the subject (details will be given in later sections). It is hoped that the collection of tools listed here and discussed in the context of the pedagogy of this important field of statistics, together with the appendices, will provide useful unified resources for instructors of courses in nonparametric statistics.

5 The ideas given in this paper have been applied in the author's experience teaching all four of the courses on nonparametric statistics offered by his department. These include one undergraduate course taken primarily by statistics majors, statistics minors, and mathematics majors; one graduate service course; one graduate course taken by master's and doctoral students in statistics; and one advanced topics course taken by doctoral students in statistics. Many of these proposals will, of course, need to be tailored to the level of the course being taught (whether it is a graduate or undergraduate course, a service course or a course for statistics majors, etc.).

6 This article is organized as follows. We discuss the class meeting or lecture in Section 2, the writing component in Section 3, the use of projects in Section 4, and computing in Section 5. There is naturally some overlap in these sections, and a suggestion made in one section may also prove to be useful in one of the other areas. Conclusions are given in Section 6. Appendix A consists of a partially annotated bibliography of textbooks and monographs in this still expanding field. Appendix B contains the templates for a collection of Minitab macros that can be retrieved electronically from the Journal of Statistics Education.

2. Class Meetings

7 One goal of a nonparametric statistics course is to convince students of the importance and usefulness of nonparametric methods. This is best accomplished by showing, rather than telling, students about the advantages of these methods over the classical procedures, although listing and discussing the advantages is certainly appropriate. Students should not be made to feel that the instructor is drawing battle lines between the classical methods and nonparametric methods, nor should they be propagandized by the instructor. Astute pupils will realize the benefits on their own as the properties of the nonparametric procedures are presented and discussed in an objective fashion.

8 An effective way to help students see the value of nonparametric methods is to demonstrate their merit through examples – preferably real ones as opposed to contrived ones (Singer and Willett 1990). This can be accomplished through handouts so that the students are provided with ‘living’ data. Previous student projects, consulting problems, and examples from colleagues in industry or other academic departments, when used with permission, can fulfill this need very effectively. Additional sources of real data sets are the Journal of Statistics Education Data Archive and the DataBank section of Teaching Statistics.

9 Ideal data sets for demonstrations of nonparametric methods are those which are clearly from non-Gaussian or asymmetric distributions (or for which symmetry and/or normality are at least suspect), those for which the population median is more relevant than the population mean, and data sets with outliers. Measured concentrations of drugs, chemicals, or toxic substances, which are nonnegative random variables that often have asymmetric distributions, provide a source of appropriate data sets. Pesticide concentrations in soil samples, for example, are rarely believed to follow a Gaussian distribution (Zacharias, Heatwole, Mostaghimi, and Dillaha 1991). Another example is comparing home prices for two residential areas. Home prices tend to come from a right-skewed distribution with some very large values pulling the mean upward, so the median home price for a particular area is usually reported instead of the mean. One would be more interested in a difference between the medians than in a difference between the means. Data sets with outliers are appropriate because nonparametric procedures tend to be inherently robust. A data set with one or more outliers can be used to illustrate differences between the sign and signed rank tests. These are only a few of many such examples. Indeed, Micceri (1989) claimed that Gaussian or symmetric light-tailed distributions occur very rarely in social and behavioral science research (thanks to a referee for pointing out this reference).

10 Presenting examples of nonparametric methods provides the instructor with an excellent opportunity to engage the students in discussion of the assumptions made by the procedures and whether or not they are satisfied. If the nonparametric procedure being presented has a classical counterpart, then the results for both procedures should be compared and/or contrasted. Many times the two approaches will lead to the same conclusion, but the assumptions may only be valid for one of the procedures. When the classical and nonparametric procedures yield different results, it is usually the nonparametric method that is considered more appropriate. It must be stressed that a comparison of the results of the two approaches is suggested here for pedagogical purposes only and not in order to determine which test is appropriate for the problem at hand. The correct procedure (parametric or nonparametric) should be chosen on the basis of the experimental design, the parameter(s) of interest, the hypotheses being tested, and the strength of the assumptions that can be made. Finally, it is important to recognize that there may not be a clear answer as to which procedure is most appropriate in a given problem (American Statistical Association 1994).

3. The Writing Component

11 The incorporation of a writing component in statistics courses has been encouraged in recent years by Radke-Sharpe (1991) and Garfield (1994). In addition to the usual merits of writing expounded by these and other authors, the inclusion of some writing has several unique benefits in a nonparametric statistics course:

1. Writing helps students to think about the assumptions behind statistical procedures, to formulate these assumptions verbally, and to critically examine the suitability of a particular procedure based on its assumptions. This is important because nonparametric methods tend to have very general assumptions. The focus on assumptions, an integral component of nonparametric statistics courses, is enhanced by student writing.

2. It facilitates student comparison of classical procedures with their nonparametric analogues, another major part of most courses in nonparametrics. Putting such comparisons into words and verbally justifying the use of a particular procedure are tools that will serve students well in their future scientific writing.

3. Writing helps students to see the “big picture” in the course. An example of this is a complete written analysis of a data set using several different techniques taught in the course (exploratory techniques, goodness of fit tests, tests for location and dispersion, etc.).

12 There are many ways to include writing as a part of a course in nonparametric statistics. Students might be asked to compare and contrast the computation or efficiencies of parametric and nonparametric procedures. Writing assignments can also be used to ensure that students understand the various meanings of the terms ‘nonparametric’ and ‘distribution-free.’ In a data analysis setting, one could ask the students to select the most appropriate procedure and justify their choices. Another possibility is to ask for a short, purely verbal description of how a particular nonparametric test works. For example, the Wilcoxon signed rank test adds the ranks of the absolute values of the positive observations rather than the ranks of the positive observations, as is sometimes incorrectly stated. In the two-sample location problem, the Hodges-Lehmann estimator of the difference in population medians based on Mood's median statistic is the difference in the sample medians, while the Hodges-Lehmann estimator based on the Mann-Whitney-Wilcoxon statistic is the median of the pairwise differences. Forcing students to put concepts such as these into words will strengthen their understanding of those concepts.

4. Projects

13 The use of projects and/or cooperative learning activities in any statistics course has been advocated by Jones (1991), Dietz (1993), Garfield (1993), and Fillebrown (1994), among others. In this section we discuss some issues related to the use of projects in nonparametric statistics courses in particular. By a project we mean, at the very least, a complete statistical analysis of a problem, including an introduction, discussion of statistical methods used, critique of assumptions, analysis of data, and conclusions. Projects may be done individually or in groups. Some of the facets of projects can also be incorporated in other aspects of a course, such as homework assignments.

14 First of all, it is recommended that the design of the experiment be included as a part of projects whenever appropriate. In this way students will learn that experimental design is not simply another topic from classical statistics, but that it is also important in nonparametric statistics. This does not mean that a course in experimental design must be a prerequisite to the nonparametric statistics course. The basic designs used in most nonparametrics courses (one and two-sample location models, one and two-way layouts and regression) are usually simple enough to be understood by students without extensive training in design.

15 Depending on the level and interest of the class, students can be asked to find a data set from a subject matter discipline. It can be very helpful for undergraduates (especially those majoring in statistics or mathematics) to see real applications of statistics in other fields of study. Often such applications yield legitimate questions about the validity of the assumptions of classical statistical procedures. This reinforces the importance of nonparametric methods to sound statistical practice. Advanced students can read journal articles and prepare statistical reports on them. In a graduate service course, the students often use data from their own research for class projects.

16 Another idea is to have students produce their own data for analysis through an experiment (Tanner 1985), a survey, or an observational study (Fillebrown 1994). The effectiveness of this approach is enhanced by requiring that a particular model, for example the one-way layout or two-way layout, be used. Students are thereby forced to think about how to design a study or an experiment so that the appropriate statistical procedures can be used. The idea of having students collect their own data has been very successful in the author's experience, with a number of students commenting favorably on this aspect of a recent course in their end-of-semester evaluations.

17 One likely question that should be anticipated by the instructor using projects in a nonparametric statistics course is the following: If, based on a critique of assumptions, the classical procedure is judged to be more appropriate than the corresponding nonparametric procedure, should I include the results for the nonparametric procedure in my report? In other words, should students analyze their data in the way they believe to be the most appropriate statistically or should the project always include the nonparametric procedures being covered in class? There are obvious advantages and disadvantages to each approach and of course some compromise between these two approaches is possible.

18 A list of some of the more interesting studies carried out by the author's undergraduate students, grouped according to the data structure for the problem, is given to provide the reader with some appreciation for the creativity demonstrated by the students, as well as some ideas for possible assignments. This list also emphasizes the point that having students collect their own data permits them to combine their personal interests with statistics, thus increasing student excitement in the course.

Simple one-sample problem (sign and Wilcoxon signed rank test)

• lengths of needles from white pine trees

• “weights of 20 random guys” (quote from a student's paper)

• ages of skydivers from one drop zone in the U.S.

• playing times of 32 classical music compact discs

Paired sample problem (sign and Wilcoxon signed rank test)

• shrinkage of material after washing and drying

• men's and women's haircut prices (matched by hair salons)

• prices of cereals at two grocery stores (matched by brand)

• computer lab usage in July and September (matched by day and hour of operation)

Two-sample problem (Mann-Whitney-Wilcoxon and Moses tests)

• burning times of an expensive and inexpensive brand of birthday candles

• heights of stairs at northwest and southeast corners of a building

• grades in the a.m. and p.m. sections of a class

One-way layout (Kruskal-Wallis test)

• toasting time for four slots of a toaster

• growth of three varieties of oak trees

Two-way layout (Friedman test)

• prices of three computer printers from 12 suppliers

• effects of cooking methods and shapes of meat on the post-cooking weight of ground beef

5. Computing

19 Both hand calculations and computer calculations should be thoroughly understood by students in nonparametric statistics courses. Because it can make for a rather dry lecture to go through hand calculations in class, a good way to demonstrate the calculations is through a handout given to students to study between classes. Questions on it can be received at the beginning of the next class meeting. Hand calculations can also be included on homework assignments or examinations, provided that the students are aware that this is expected of them.

20 Hand calculations represent one way to develop intuition for how particular procedures treat the data. In nonparametric statistics the hand calculations have a nature all their own. Evidences of this include the prevalence of integer arithmetic, counting techniques, and indicator functions. An excellent example of the type of hand calculations that should be well understood by students of nonparametric statistics (except perhaps those in service courses) is the derivation of the exact distribution, mean, and variance of the Wilcoxon signed rank statistic.

21 The computations involved in nonparametric methods are relatively simple, and yet the methods are quite powerful. They have been referred to as ‘rough and ready,’ ‘quick and dirty,’ and ‘back-of-the-envelope’ methods. The author has often reminded his students that many nonparametric procedures can be carried out by researchers while they are collecting data in the laboratory or in the field where a computer is not readily available.

22 No matter what software is used, students completing the course should know how the statistical procedures they have learned can be performed on a computer. The choice of computing package is a difficult one and no specific recommendations are made here. Several packages have been developed specifically for nonparametric methods, including NPSTAT (Lenz 1978), NPSP (Pirie 1983), NONPAR (Harley and Petruk 1989), StatXact (Mehta 1991), and XploRe (Hardle, Klinke, and Turlach 1995).

23 A set of Minitab macros developed by the author has proved helpful in assisting pupils with the computational aspects of nonparametric statistics. These programs permit students to carry out many of the nonparametric procedures that are taught in the course but are not readily available in the major statistical computing packages. Examples include the Kolmogorov-Smirnov one and two-sample tests, the Moses and jackknife two-sample dispersion tests (Moses 1963, Miller 1968), Kendall's tau (Kendall 1938), and kernel regression (Nadaraya 1964, Watson 1964). A complete list of these macros and a summary of the commands for them are given in Appendix B. Each main macro file (.MAC) is accompanied by a help file (.HLP) containing detailed information about how the program should be used. The macros can be downloaded electronically from the Journal of Statistics Education.

24 A useful way of addressing issues of computation without consuming excessive amounts of class time is to have a “Computing Corner” during the first few minutes of each class period. This name was chosen before the author was aware that the journal Teaching Statistics has had a Computing Corner in most issues published since the Spring 1989 issue. The Computing Corner could just as easily be held at the end of class or in the middle of class (in the form of a well-timed ‘commercial break’ which is especially appreciated by students in classes longer than one hour). This provides a time for discussion of issues related to computer commands, programs, and output. It gives the instructor a good opportunity to anticipate and answer students' questions and to inform students about the options for computing nonparametric procedures that are available in various statistical packages. The “Computing Corner” is often marked by high levels of interest and involvement on the part of the students. For this reason it can be difficult to bring it to a conclusion so that discussion can move on to methodology or examples. This feature of the class should therefore be restricted to a fixed time period, if possible, so that it does not detract from other parts of the course.

25 Examples of computing issues in nonparametric statistics about which students should be informed include whether the p-values reported by statistical packages are exact or approximate, whether or not a continuity correction is used, how tied values are treated, and how p-values for one-sided tests can be obtained. These questions can rarely be answered by a glance at the output of a particular package. For example, students should know that the p-value given by Minitab's STEST command is exact if the sample size is less than or equal to 50; otherwise a normal approximation with continuity correction is used. In some software packages the most common nonparametric procedures are not found under a specific procedure. Witness SAS, where the sign test (PROC UNIVARIATE), Kruskal-Wallis test (PROC NPAR1WAY), Spearman rank correlation (PROC CORR), and Friedman test (PROC FREQ) are all performed by different procedures.

6. Conclusions

26 Nonparametric methods form an integral part of an undergraduate major or concentration in statistics, as well as any graduate program in statistics. A number of useful approaches to classroom presentations and student learning and assessment have been suggested for improving the instruction of nonparametric statistics. These include comparing nonparametric methods with their classical analogues (when such analogues exist), emphasizing the importance of assumptions, demonstrating the usefulness of nonparametric techniques by real examples, having students do some writing to solidify their understanding, assigning group or individual projects, and exposing students to methods of computation for the procedures taught in class. These techniques help students develop an appreciation for the field of nonparametric statistics and the broad range of its applications in practice.

Acknowledgements

The author expresses appreciation to his students for their valuable contribution to the material presented here. Special recognition also goes to Walter R. Pirie and Jeffrey D. Vest for their assistance in developing and testing, respectively, the programs in Appendix B, and for their helpful comments on earlier drafts of this manuscript. The editor of the Journal of Statistics Education and three referees made suggestions which invigorated an earlier version of this article.

Appendix A

Partially Annotated Bibliography of Textbooks and Monographs

The purpose of this appendix is to give instructors and students of nonparametric statistics a list of textbooks and reference books in the field. Care has been taken to give adequate coverage to density estimation and nonparametric regression, the most active areas of current research in the field. The bootstrap, which is placed by some under the broad nonparametric umbrella, has been excluded from this bibliography to keep its length manageable. The intention is not to rate each book in terms of its suitability for a particular course, although comments do follow some of the entries. Instead, an extensive list of published reviews follows most of the entries. In addition, a letter appears prior to each of the 93 entries to indicate whether the entry is primarily a bibliography (B), computing manual (C), research monograph (M), proceedings (P), or textbook (T). Some of the monographs would make suitable texts for graduate courses. The format for citations of published reviews is journal (year, volume, page number(s)).

Appendix B

Summary of Minitab Macros

The Minitab macros described below were written for use with Minitab Release 10, although they should also run on Releases 9 and 11. Because of the length of the files containing the Minitab macros, only a summary of the macros is given here. Instructions for obtaining the macros and associated help files are given at the end of the article.

Macros for a Single Sample -------------------------- %KS1 C; K-S test on data in C EXPONENTIAL K K; Specify parameters for exponentiality test UNIFORM K K; Specify parameters for test of uniformity NORMAL K K. Specify mean & std. dev. for normality test %EDF1SAMP C; Plot of empirical cumulative distribution function (ECDF) & best-fitting Gaussian CDF INTERVAL K. Plot confidence bands with confidence level K %SYMMETRY C Tests for symmetry about an unknown median Macros for Two Independent Samples ---------------------------------- %ANSARIB C C; Ansari-Bradley dispersion test ALTERNATIVE K. For 1-sided tests; K = 1, 0, or −1; default = 0 %JACKNIFE C C; Miller jackknife dispersion test ALTERNATIVE K; For 1-sided tests; K = 1, 0, or −1; default = 0 NULLVALU K; H0 value of ratio of dispersion parameters; default = 1 INTERVAL K. Confidence level; default = 0.95 %MOSES C C; Moses dispersion test ALTERNATIVE K; For 1-sided tests; K = 1, 0, or −1; default = 0 NULLVALU K; H0 value of ratio of dispersion parameters; default = 1 INTERVAL K; Confidence level; default = 0.95 SUBSAMPLES K. Specify size of subsamples used %KS2 C C; K-S two-sample test on data in C C ALTERNATIVE K. For 1-sided tests; K = 1, 0, or −1; default = 0 %EDF2SAMP C C Plot of ECDFs for data in C C Macros for Two Paired Samples (including Simple Regression) ----------------------------------------------------------- %THEILSEN C C; Theil-Sen simple linear regression of C on C (Y against X) ALTERNATIVE K; For 1-sided tests; K = 1, 0, or −1; default = 0 NULLVALU K; H0 value of slope; default = 0 INTERVAL K. Confidence level for CI for slope; default = 0.95 %CORRELAT C C; Pearson, Spearman & Kendall correlations ALTERNATIVE K; For 1-sided tests; K = 1, 0, or −1; default = 0 INTERVAL K. Confidence level for CI for correlation coefficient; default = 0.95 %KERNLREG C C; Kernel regression of C on C (Y against X) KERNEL K; Specify the kernel function (default = 0) (0 = Gaussian, 1 = Epanechnikov, 2 = uniform) BANDWIDTH K; Specify a value for the bandwidth h; default is one half of the standard deviation of X data FITS C C; Save fits of estimated regression function in C C (1st column contains fits, 2nd = grid in X) DENSITY C C. Save density estimate of indep. variable in C C (1st column = density estimate, 2nd = grid in X) %LOCALREG C C; Local regression of C on C (Y against X) KERNEL K; Specify the kernel function (default = 0) (0 = Gaussian, 1 = Epanechnikov, 2 = uniform) BANDWIDTH K; Specify a value for the bandwidth h; default is one half of the standard deviation of X data DEGREE K; Specify degree of local fits (1 = linear fits, 2 = quadratic fits; default = 1) GRID K; Specify number of points in grid on X variable FITS C C. Save fits of estimated regression function in C C (1st column contains fits, 2nd = grid in X) Macro for Several Independent Samples (One Way Layout) ------------------------------------------------------ %JONCTERP C C … C Jonckheere-Terpstra test for ordered alternatives on data in C C … C Macro for Several Related Samples (Two Way Layout) ------------------------------------------------------ %PAGE K C C … C Page test for ordered alternatives on data with K blocks in C C … C