Understanding statistics using computer demonstrations.

Dunn, Peter K.

 This paper discusses programs that clarify some statistical
 ideas often discussed yet poorly understood by students. The
 programs adopt the approach of demonstrating what is happening,
 rather than using the computer to do the work for the students
 (and hide the understanding). The programs demonstrate normal
 probability plots, overfitting of models and generalized linear
 models. Although the implementation is in Matlab, any suitable
 language is appropriate.

**********

The use of technology in teaching mathematics and statistics has reached the point where it is hard to imagine teaching mathematics and statistics without using computers. The University of Southern Queensland (USQ) has chosen to standardise on using Matlab in its mathematics and statistics courses. First year students learn algebra and calculus by incorporating Matlab as a numerical and graphical tool; they learn the basic language and graphical commands. Matlab is then used in many subsequent courses, including most statistics courses.

Matlab is a computational environment for technical and numerical computing. It has a strong pedigree in robust, stable and efficient numerical algorithms, and has excellent graphical capabilities. The user-interface is typically the command-line, but Matlab has facilities for implementing a graphical user interface, as used for the programs discussed in this paper.

Cretchley, Harman, Ellerton and Fogarty (1999, 2000) analysed the influences on attitudes and learning, finding almost all students responded positively to Matlab for ease of computation and graphing. Using the software as a tool was found to strongly impact the learning strategies adopted and the students' confidence toward mathematics. In addition, "students with high computer-mathematics interaction feel that computers enhance mathematics learning by providing many examples, enable the user to focus on major ideas by reducing mechanical toil ... and find computers helpful in linking algebraic and geometric ideas" (Galbraith, Haines, and Pemberton, 1999, p. 220).

Matlab is used in numerous ways at USQ. Often it is used to actually do the work for the student, saving the student time in fiddly or prone-to-error calculations, or allowing the students to do more complex problems impossible to do "by hand." In higher-level courses, some Matlab programming is expected.

The programs discussed in this paper, however, use the computer to show what is happening, rather than to do a particular task. For example,

 It is just about impossible to illustrate Newton's method adequately
 with freehand sketches using a piece of chalk. A computer printout
 on a transparency, or even an animation using a PC, can be a great
 help. Likewise Simpson's rule, the Runge-Kutta method, and other
 numerical techniques can be best illustrated with a computer and an
 attractive display. (Krantz, 1998, p. 81)

Dunn and Harman (2000) argue that Matlab can be incorporated into learning in three main ways: Firstly, as a numerical and graphical tool. In this role, the computer enables faster illustration of the applications and better visualisation of complex ideas (such as three dimensional graphical representations). Typical in this approach is to use command line instructions in Matlab to develop the ideas while no programming skills are required; see, for example, Cretchley et al. (1999, 2000).

Secondly, Matlab can be incorporated into the material using appropriate programs then used by the student to facilitate the learning of ideas and concepts. In this approach, the programs are provided to the student for their use or possible modification. Penny and Lindfield (1995) and Borse (1997), for example, have adopted this practice; Colgan (2000) discusses a course built around this approach. A basic knowledge of programming is helpful or necessary in this approach.

Thirdly, Matlab software can be used to show the procedure or concept in a more sophisticated way. This is done by using larger, stand-alone and robust programs with well-designed graphical user interfaces. No programming skills are required in this approach. Of course, these three approaches can effectively complement one another.

In this paper, the emphasis is given to the third approach. A number of statistical and mathematical tools are presented in Matlab, though the language itself is not important. Examples from other areas of statistics can be found in Dunn (1999). Two tools for use in regression, a tool for use in teaching generalized linear models, and details on how to obtain and use the programs are presented in these findings.

TOOLS FOR USE IN REGRESSION

Regression is an important tool for any student of statistics to study and, consequently, it is an essential component of study in most University statistics programs. Two issues often discussed in relation to multiple regression include normal probability plots (also known as rankit plots, Q-Q plots, probability plots, quantile-comparison plots, and normal plots), and overfitting. This paper discusses two Matlab functions that demonstrate important points about both issues. Although Matlab is not primarily a statistics package (it does, however, have a comprehensive Statistics Toolbox), these features make it suitable for use in statistics classes at certain times. We certainly believe that our programs based on Matlab enhance the student's experience within statistics.

It has been argued that the impact of technology "should be especially strong in statistics" (Bratton, 1999, p. 666) since technology plays such an important role in the actual practice of statistics. Bratton (1999) then argues that there are three ways in which proper implementation of technology into statistics can happen: By making some topics unnecessary; by introducing new topics; and by teaching some methods better. In this third category, he especially makes mention of randomness and probability concepts. One of the programs we discuss herein addresses this issue. West and Ogden (1998) discuss a number of Java applets using statistical demonstrations that operate over the World Wide Web, though none cover the same issues as those presented here. The rationale, however, is similar to ours. Saunders (1986) discusses a number of programs used at the Open University, some of which cover similar ground, but whose implementation is far more complicated.

In the next section, the normal probability plot is discussed together with the associated Matlab program. Afterwards, the Matlab function concerning overfitting is discussed.

The Normal Probability Plot

One of the fundamental assumptions in standard linear regression is that of normal (Gaussian) errors. The assumption of normality is really only necessary for hypothesis testing in the model; for example, a t-test on one of the parameters, or an F-test of submodels.

To assess the normality of the residuals, it is often tempting to plot a histogram of the residuals and assess normality from such a graph. However, it is difficult to assess normality from such a graph: Are the tails too heavy? too light? Fox (1991, p. 44) maintains that a normal probability plot is useful for detecting departures from normality in the tails, which is where problems often occur and where they are often more serious. He also suggests that a weakness of the plots is that they do not show the overall shape, which is best done using a histogram or a stem-and-leaf plot. Our program supplies both a normal probability plot and a histogram. Marasinghe, Meeker, Cook, and Shin (1996) also include boxplots and the theoretical probability density function of the distribution.

In normal probability plots, the sample order statistics are plotted against the expected order of normal order statistics (or rankits). (Some authors prefer to use the axes in the reverse sense.) If the sample is from a normal distribution, the plot is expected to be linear. For a non-normal sample, the plot would be non-linear. Such plots are quite common in the literature. In fact, Marasinghe et al. (1996) discuss a useful Lisp-Stat program for demonstrating (among other things) the behaviour of the normal probability plot and histograms for different distributions.

Many books, such as Jennrich (1995) for example, provide a number of normal probability plots generated from actual normal distributions so students can see the types of plots generated. Jennrich (1995) also then provides numerous normal probability plots for random numbers from the double exponential distribution and the Cauchy distribution to indicate some typical data features that can be detected by such plots. Cook and Weisberg (1994) include a program enabling readers to gain experience in judging whether normal probability plots are linear. They give users a choice of plotting a normal, uniform, chi-square or t-distributions. Some authors, for example Weisberg (1985), make only cursory notes.

The program discussed here allows the user to quickly and easily generate as many plots as necessary to gain experience in recognising characteristics of the plots, and to explicitly generate plots from data that exhibit certain characteristics.

There are four main objectives in using this tool:

* To gain experience in recognising the typical characteristics of normal probability plots;

* To compare and evaluate the characteristics of normal probability plots for normal and non-normal distributions;

* To understand the difficulties in identifying normality in small samples which are actually drawn from normal distributions;

* To demonstrate the inadequacies of simply relying on a histogram to detect non-normality in a data set.

The Program

To help students understand the behaviour of the normal probability plot for normal and non-normal samples, the Matlab program nppdemo has been developed. The initial screen is shown in Figure 1. There are three regions: The left panel is where the normal probability plot is shown. The middle panel is where a histogram of the data is shown, and the controls are in the right panel.

The control panel is divided into three parts. The top section allows the user to select one of five (mutually exclusive) types of distributions to be plotted. The middle section allows the user to set parameters of the distribution, including the number of points to be randomly generated. The lower section allows the user to plot the data or to quit the program.

The top section gives the user five choices of random data to plot. So the program can operate without the use of the Statistics Toolbox (an extra-cost addition to Matlab), the random data has been generated from the two random number generators supplied with the basic Matlab program: randn for normal random variables, and rand for continuous uniform random variables.

The left panel displays the normal probability plot. To assess the normality of the errors, one has to determine if the points are approximately on a straight line. To aid the user in making this assessment, the normal probability plot is plotted with a straight line corresponding to the normal distribution with the mean and variance estimated from the data.

In the middle panel, a histogram is plotted. The histogram is plotted with bars horizontal, rather than vertical as is more common. This has been done since then the data in both graphs is plotted on the left axis. This also means the histograms mimic the stem-and-leaf plots used by Fox (1991).

The first option in the top section of the right panel generates random normal values for a given mean and variance. This plot should produce a normal probability plot that is approximately linear.

[FIGURE 1 OMITTED]

The second option generates right (positively) skewed data from an exponential distribution. (The variance cannot be supplied for an exponential distribution; it is equal to the mean.) These random numbers are easily generated from uniform random numbers (Devroye, 1986, sect. II.2.1) and are expected to produce a normal probability plot that is concave upward. A picture of the screen with 100 random points from this distribution showing the corresponding histogram and normal probability plot is shown in Figure 2.

The third option generates left (negatively) skewed data. This is generated from exponential random numbers also, and made left skewed by subtracting from the maximum value in the sample. The normal probability plot is expected to be concave down.

[FIGURE 2 OMITTED]

The fourth option generates random numbers from a distribution with long (heavy) tails. Random numbers from a t-distribution with two degrees of freedom are generated since they can be written as a function of uniform random numbers (Devroye, 1986, sect. IX.3.3). Since the variance for such a distribution is undefined, the variance cannot be entered in this case. A distribution with long tails is expected to produce a normal probability plot somewhat like a reflected 'S' shape.

The fifth option generates random numbers with short (light) tails. The numbers are generated from a continuous uniform distribution and are expected to produce a somewhat like an 'S' shape.

The middle section allows the user to set the mean and (where possible) the variance of the generated data, as well as the number of points generated. The mean can be set between - 100 and 100, and the variance can be set to a positive value less than 100. The number of random points generated can range between 5 and 1000.

The lower section contains just two buttons. Every time that Plot is pressed, a new random data set is generated according to the set parameters, and the normal probability plot and histogram updated. Quit simply closes the program.

Using the Program

There are a number of ways to use this program in the classroom in line with the objectives listed earlier. The most obvious is to demonstrate typical plots for residuals with certain characteristics. For example, if the model produces residuals skewed to the right, the typical normal probability plot can be shown by selecting the appropriate options, and pressing Plot. Students found this more engaging than simply looking at pictures in text-books.

To show the forms of such plots, it is recommended that the number of points generated be quite large, say over 200, and possibly set to the largest value permitted in the program, which is 1000. This ensures that the pattern is very clearly displayed. Then, having seen the typical patterns to watch for, smaller samples can be generated to see how easy or difficult it is to spot these patterns for smaller data sets.

The program can also be used to display how difficult it is to detect non-normality, especially for small data sets. This is probably best demonstrated by choosing a small sample size, say five or ten, and generating plots from a normal distribution. Since pressing Plot generates a new data set each time, it is quick and easy to generate a number of such plots, and it should not take too many attempts to produce a histogram or normal probability plot that doesn't look normal even though it is known to be generated from a normal distribution. Students found this enlightening, and in some respects, challenging their limited experiences. Students seemed to be very interested in the program, and were keen to experiment. In particular, they were very interested to see the shapes of the normal probability plots for small sample sizes, especially from known normal distributions. There was much discussion about how difficult it can sometimes be to spot deviations from normality unless the sample size is "large." There was much discussion over what constituted a "large" sample. In later laboratory sessions using real data sets, students often commented on the lessons learnt from the demonstration, especially when the data sets were small in size.

The program can also be used to show how difficult it is to identify a normal distribution only using a histogram. Many data sets (especially small ones) can have a normal probability plot that looks reasonably linear, but whose histogram looks non-normal.

The Overfitting Program

One dilemma that arises in multiple regression is balancing the competing criteria of having enough regressors for an adequate model, and having too many regressors that produces "artificial accuracy," or overfitting. The program overfit demonstrates this situation.

Many criteria can be used to select the "optimal" number of regressors, including the adjusted coefficient of determination (adjusted [R.sup.2]) as reported by many software packages, Mallows' [C.sub.p], and PRESS-related statistics. The adjusted [R.sup.2] penalizes the ordinary coefficient of determination when regressors are added to the model. Introducing a new, non-significant regressor will very slightly increase [R.sup.2], but will reduce the value of the adjusted [R.sup.2].

Mallows' [C.sub.p] is calculated on the basis that a good model is unbiased. This statistic, then, estimates the bias in the model which is then used as a criterion for selecting a good model (Weisberg, 1985, sect. 8.8).

The PRESS (PRediction Error Sum of Squares) residuals use a different criterion again. They are called deleted residuals by Jennrich (1995), and Atkinson (1985, chap. 3) discusses similar residuals which he calls jack-knife or cross-validatory residuals. In the case of PRESS residuals, each observation i is excluded from the analysis in turn, and the model fitted to the remaining n-1 data points. This model is then used to predict the value for the missing observation i, and the residual in predicting the missing observation is the PRESS residual for that point. (Note that alternative computational formulae exist.) The PRESS residuals can be used to calculate statistics for assessing the quality of models. The PRESS statistic is generally given as the sum of the squared PRESS residuals (also called Allen's PRESS by Jennrich (1995) after its originator). The sum of the absolute PRESS residuals can also be used. The model with the smallest PRESS statistic is chosen as the "optimal" model.

There are three major objectives for using this tool:

* To understand that adding more terms to a regression model (and hence overfitting) does not necessarily produce a better model;

* To understand that obtaining a well fitting model and obtaining an accurate predicting model (small residuals) are not necessarily the same goal;

* To realize that the criteria used for selecting an 'optimal' regression model do not necessarily agree.

The Program

To help students understand the problems of overfitting, the Matlab program overfit fits regression polynomials to a set of data. The initial screen is shown in Figure 3. There are two regions on the screen: The main figure, and controls on the right side; the main figure is where the plots are shown. The program plots data points, and then fits higher order polynomials to these data and the effects of overfitting are observed.

The panel on the right contains three sections: The upper section allows the user to specify the source of the data points used. The middle section contains information about the data points (and it depends on the source of the data selected in the region above). The lower section contains the main controls.

The top section of the right panel allows the user to specify the source of the data as either a file or randomly generated by Matlab. The data files should be text files, containing two columns of space delimited numbers; the first column contains x values and the second y. Using the computer generated data allows the student or instructor to see the effects of overfitting, but being able to load other data means that the effects can be seen on data relevant or known to the student.

The middle region of the right panel depends on the data source. If the data are sourced from a data file, the middle region gives the name of the data file used, and contains a button allowing the user to use a graphical interface to load the appropriate data file.

If the data are generated by Matlab, there are two options to be set. The bottom menu sets the type of regression line from which points are generated. The user can choose between a straight line, a quadratic or a cubic. The program will generate eleven (non-evenly spaced) points from the specified polynomial, and add an amount of Gaussian "noise" with the error variance specified in the top region of this section. The value of the error variance can be set to a positive value less than 1.

In the lower section, pressing the Plot Data button will plot (in blue) the data as specified in the top section. Two buttons under the Plot Data button have arrows, indicating fitting models of higher order (upward pointing arrow) or lower order (downward pointing arrow) than the model currently fitted (indicated in the edit window between the arrows). These arrow buttons plot polynomials from a first order (linear) up to an eighth order polynomial. These fitted polynomials are generally shown in red. The only exception is if the data is generated from Matlab, when the polynomial of the order specified (linear, quadratic or cubic) is shown in black. At any time, the user can enter any particular polynomial order (between one and eight) into the edit window between the arrows to get a plot of that particular order polynomial fitted to the data.

[FIGURE 3 OMITTED]

Using the Program

The program can easily demonstrate the folly of overfitting as listed in the objectives. Almost always (for generated data), the eighth order polynomial is very accurate in the sense of having small residuals, but the fitted curve shows it has not captured the trend of the data very well at all. See Figure 4 for an example of fitting an eighth order polynomial to data which is actually from a linear model with error variance of 0.2. The students reacted very favourably to this, even to the extent of making comments and drawing in their notes some of the plots that resulted from fitting the higher order polynomials. They seemed to understand the dangers in overfitting a model quite well after seeing the demonstration. In addition, it became clear to the students that producing small residuals and finding a good model were two different goals, a new realization for many students.

After fitting the regression models, the Matlab command window gives some statistics for each model. The order of the polynomials is given first (with the specified model flagged by stars if the data is generated using Matlab), the sum of the squared errors (SSE); the ordinary coefficient of determination [R.sup.2]; the adjusted [R.sup.2]; Mallows' [C.sub.p]; an unbiased estimate of the error variance (which can be compared to the specified error variance for generated data); and the PRESS statistic. This enables the user to see how the different criteria perform in each of the situations, and which model might be chosen by each. Importantly, the criteria do not always agree on the "optimal" model. The output for one such simulation is given below. For data loaded from a file, the full model is considered to be an eighth order polynomial for the calculation of [C.sub.p].

In this case, a quadratic model has been specified (as can be seen by the stars flagging the second order polynomial). The SSE continues to fall as the order of polynomials increase, as we would expect. Likewise, the value of [R.sup.2] continues to get closer to one. The adjusted [R.sup.2], however, seems to imply a model of sixth or eighth order. (Plotting the sixth and eighth order polynomials shows that these polynomials do not capture the trend of the data well at all.) The error variance is close to the specified 0.2 for the quadratic model, but is actually smallest for some of the higher order polynomials. The PRESS statistic implies the second, sixth or eighth order polynomials are most useful. Mallows' [C.sub.p] indicates that a quadratic model should indeed be fitted (note that [C.sub.p]=p for Matlab generated data, by the definition of [C.sub.p]). This demonstrates how difficult it is to decide on the optimal order of a polynomial for fitting to the data. Student comments, suggest they gained an appreciation of the need for different criteria and the different goals of each, and the difficulties in choosing an "optimal" model.

[FIGURE 4 OMITTED]

TOOLS FOR UNDERSTANDING GENERALIZED LINEAR MODELS

Classical linear regression is a standard and popular statistical tool, but simultaneously satisfying the necessary assumptions (normality of the data, constant variance, and independent errors) renders it inadequate in many cases. However, generalized linear models, introduced by Nelder and Wedderburn (1972) provide a wide class of statistical models that may be of use. Indeed, classical linear regression is a special case of generalized linear models.

Background

Generalized linear models were first introduced as a method of analyzing a broad class of statistical models based on exponential dispersion models (EDMs); that is, distributions having the probability density function (or mass function) in the form

f(y;[theta],[empty set]) = a(y,[empty set])exp{[y[theta] - [kappa]([theta])]/[empty set]}

where a(y,[empty set]) and [kappa]([theta]) are known functions, and the mean is [mu] = [kappa]'([theta]). For these models, Var[[UPSILON]] = [empty set][kappa]"([theta]), where V([mu]) = [kappa]"([theta]) is called the variance function, and [empty set] is a positive constant of proportionality known as the dispersion parameter. The linear predictor is generally referred to as [eta] where [eta] = X[beta]; X refers to the design matrix and [beta] is the estimated coefficients. A link function is generally chosen such that [mu] = g([eta]), where g(.) is any monotonic differentiable function. While the link function can be chosen independently of the distribution, the canonical link function deserves special mention because of its common usage; it is the link function such that [eta] = [theta] (and is therefore dependent upon the distribution). A wide variety of distributions are EDMs: continuous distributions include the normal, inverse Gaussian and gamma distributions; discrete distributions include the Poisson and binomial distributions. The theory behind such models has been well established by Nelder and Wedderburn (1972) and McCullagh and Nelder (1989). The special case of linear regression uses a normal distribution with the identity link function.

The algorithm for fitting such models is reasonably straight-forward but iterative, and requires a few tricks to use well (for example, to avoid taking logarithms of zero). The reader is referred to McCullagh and Nelder (1989).

There are five main objectives of the program:

* To understand the roles of the link function and the chosen distribution, and to realize they are separate roles;

* To understand the effect of the mean-variance relationship of an EDM;

* To recognize that the fitted values [mu] = E[Y] actually define the mean of the distribution for given X;

* To observe that a correct choice of distribution and link function can model certain types of data (such as strictly positive data) better than standard regression models;

* To understand the role of the dispersion parameter in generalized linear models.

The Program

The demonstration tool, called glmshow, shows simple generalized linear model structures and allows the components of this structure to be interactively altered. The model structure can be based on a normal, gamma or Poisson distribution, with a choice of three link functions: the identity, logarithm and reciprocal link functions. The value of the dispersion parameter can also be specified (except for the Poisson distribution, for which it is fixed at 1).

The initial screen is shown in Figure 5. In this screen, the options that can be adjusted are shown at the bottom. The distribution and link function can be changed by clicking on the current settings and making a selection from a drop-down menu. The dispersion parameter can be changed by entering a new value in the edit box.

Above the options is a slider. Moving this slider (with the mouse or arrow keys on the keyboard) changes the x-value on the graph, and the other dependent details are then updated also. The Close button closes the window.

So that the demonstration might be used in a classroom setting, the lines are thicker than standard. In addition, the font sizes are larger than standard and grow with the size of the window as it is maximized or enlarged.

Using the Program

The demonstration can be used in various ways as an aid for explaining many of the new concepts students are exposed to with generalized linear models, as listed in the objectives.

The starting point should be in familiar territory, the normal distribution-based standard regression model with the identity link function. Selecting these options (they are also the default settings) shows the screen in Figure 5. Moving the slider changes the x-value and dependent quantities. The predicted mean value is indicated by a red dot at this x-value. It is seen that a (normal) distribution of responses is predicted, and that the variance of these responses is constant for various values of x. Indeed, by specifying large values of x (and hence small values of y), the normal distribution admits negative values, which may not always be appropriate. This is shown in the demonstration by part of the normal distribution moving off the screen. Altering the dispersion parameter changes the variance of the normal distribution.

[FIGURE 5 OMITTED]

The effect of the mean-variance relationship for the gamma and Poisson distributions can be shown by choosing one of these distributions and moving the slider. The gamma distribution is shown as an appropriately scaled probability density function (see Figure 6), and the discrete Poisson distribution as an appropriately scaled probability histogram (Figure 7). It becomes clear that the variance of the predicted distribution changes with x. In particular, as x gets larger (and hence y gets smaller) the response variable can be seen to never take on values below y=0. This shows the dependence of the variance on the mean for distributions other than the normal and is an opportunity for the instructor to talk about the choice of distribution when negative values of y are inappropriate.

[FIGURE 6 OMITTED]

For any distribution, the link function can be changed independently. This is useful for discussing the separate roles of the distribution and link function. Similarly, the effect of changing the dispersion parameter can be shown for various distributions, link functions and x-values.

Students were very interested in this program; it became apparent that students had a greater understanding of the framework of generalized linear models after using the program. In particular, student comments indicated a clarification of the idea that the predicted values [mu] were means of a distribution of responses for given x.

USING AND OBTAINING THE PROGRAMS

It is almost obvious that "new technology should be layered atop a traditional foundation" (Krantz, 1998, p. 28). The programs discussed in this paper are learning aids, and cannot possibly be effective without the support of additional adequate teaching. Some notes on using the programs are presented here. In all cases, the students gain more by seeing the program initially demonstrated by an instructor, and then having time to interact with the programs themselves. It may also be useful to have guided questions for the students when using the programs to evaluate understanding, even to the extent of using the programs in assessment.

The first program (normal probability plots) can be presented during a discussion on regression diagnostics. Students should be able to gain a reasonable amount of experience with the plots in under 15 minutes. It may be useful to have the students write a short summary of their conclusions (in an assignment question, for example). Example questions include writing short answers about (i) the patterns observed in the plots for different shape distributions, (ii) what happens when sample sizes are small (say less than 10) even when they are from a normal distribution; and (iii) general guidelines for detecting non-normal residuals.

The second program (overfitting regression models) can be introduced during a discussion of the criteria for finding an "optimal" model. Since real data can be imported and used, the time spent using the demonstration can vary; simply using generated data, students should have a good understanding of the concepts within 10 minutes. The program can be used in assessment with a real (two-dimensional) data set where students are asked to select a model for their data, justifying their choice. A suggested data set is data set 384 from Hand, Daly, Lunn, McConway, and Ostrowski (1994), for which a second order polynomial seems appropriate. The impact of influential observations can also be observed; a suggested data set is the two assay experiments on hormones from Carroll and Ruppert (1998).

The third program (generalized linear models) should be introduced soon after the components of the generalized linear model have been explained. There is often confusion about the different roles of the link function and distribution. Students can manipulate the various components of the generalized linear model and see how the components can be independently chosen, yet interact to find a suitable model. In addition, the concept that the fitted values define the mean of a distribution for given x is often hard to explain and illustrate (even though this concept also belongs in the standard regression context); the program not only illustrates this but allows the value of x to vary, which serves to consolidate the concept.

[FIGURE 7 OMITTED]

The functions discussed in this paper require Matlab version 5 or 6 (for any available platform), but the Statistics Toolbox is not necessary. Because the functions use a graphical user interface, students and instructors do not need to learn Matlab to use the functions. Each function requires a single (text) file, and all functions are available from the author's Web pages at http://www.sci.usq.edu.au/staff/dunn/matlabprograms.html.

The programs must be copied to a directory or folder in the Matlab path. (This can be found by typing path at the Matlab prompt). In general, the path can be changed by typing pathtool at the Matlab prompt, but it is wise to also consult the local documentation as there are sometimes platform and version specific details to be aware of. The programs are copyright by the author and can be used freely, but cannot be exploited for commercial reasons.

Order SSE R^2 adj R^2 s^2 sum(PRESS^2) Cp p

 1 14.72 0.140 0.045 1.636 19.986 60.989 2
*2* 1.95 0.886 0.858 0.244 2.482 3.000 3
 3 1.93 0.887 0.839 0.275 2.851 3.922 4
 4 1.92 0.888 0.813 0.320 4.034 4.908 5
 5 1.30 0.924 0.848 0.261 2.862 4.343 6
 6 0.87 0.949 0.873 0.218 2.019 4.787 7
 7 0.87 0.949 0.831 0.289 3.191 6.336 8
 8 0.27 0.984 0.922 0.134 2.217 7.276 9

Acknowledgements

The author would like to thank the editor and reviewers, whose comments made substantial improvements to the article, and improved the usefulness of the programs in the classroom.

Reprinted from JCMST 22(3) because some characters in equations were inadvertently omitted. We apologize for this error.

References

Atkinson, A.C. (1985). Plots, transformations and regression: An introduction to graphical methods of diagnostic regression analysis. Oxford: Clarendon Press.

Borse, G.J. (1997). Numerical methods with Matlab: a resource for scientists and engineers. Boston: PWS Publishing.

Bratton, George N. (1999). The role of technology in introductory statistics classes. The Mathematics Teacher, 92(8), 666-669.

Carroll, R.J., & Ruppert, D. (1988). Transformation and weighting in regression. New York: Chapman and Hall.

Colgan, L. (2000). MATLAB in first-year engineering mathematics. International Journal of Mathematical Education in Science and Technology, 31(1), 15-26.

Cook, R.D., & Weisberg, S. (1994). An introduction to regression graphics. New York: Wiley.

Cretchley, P., Harman, C., Ellerton, N., & Fogarty, G. (1999). Computation, exploration, visualisation: Reaction to Matlab in first-year mathematics. In W. Spunde, R. Hubbard & P. Cretchley (Eds.), Proceedings of Delta '99: Australian Symposium on Modern Undergraduate Mathematics (pp. 81-86). Toowoomba: Delta '99 Committee.

Cretchley, P., Harman, C., Ellerton, N., & Fogarty, G. (2000). Matlab in early undergraduate mathematics: an investigation into the effects of scientific software on learning. Mathematics Education Research Journal, 12, 219-233.

Devroye, L. (1986). Non-uniform random variate generation. New York: Springer-Verlag.

Dunn, P.K. (1999). Three tools for interactively visualizing some distribution theory concepts. The American Statistician, 53(2), 137-139.

Dunn, P.K., & Harman, C. (2000). Calculus demonstrations using Matlab. International Journal of Mathematical Education in Science and Technology, 33(4), 584-595.

Fox, J. (1991). Regression diagnostics: An introduction. Sage University Paper Series on Quantitative Applications in the Social Sciences, 07-079, Newbury Park: Sage.

Galbraith, P., Haines, C. & Pemberton, M. (1999). A tale of two cities: When mathematics, computers and students meet. In J.M. Truran and K.M. Truran (Eds.), Making the difference (pp. 215-222). Sydney: Mathematics Education Research Group of Australasia Inc.

Hand, D.J., Daly, F., Lunn, A.D., McConway, K.J., & Ostrowski, E. (Eds.). (1994). A handbook of small data sets. London: Chapman and Hall.

Jennrich, R.I. (1995). An introduction to computational statistics. Englewood Cliffs, N. J.: Prentice-Hall.

Krantz, S.G. (1998). How to teach mathematics. Providence R.I.: American Mathematical Society.

Penny, J., & Lindfield, G. (1995). Numerical methods using Matlab. New York: Ellis Horwood.

Marasinghe, M.G., Meeker, W.Q., Cook, D., & Shin, T.-S. (1996). Using graphics and simulation to teach statistical concepts. The American Statistician, 50(4), 342-351.

McCullagh, P. and Nelder, J.A. (1989). Generalized linear models (2nd ed.). London: Chapman and Hall.

Nelder, J.A., & Wedderburn, R.W.M. (1972). Generalized linear models. Journal of the Royal Statistical. Society Series A, 135(3), 370-384.

Saunders, D.J. (1986). Computer graphics and animations for teaching probability and statistics. International Journal of Mathematical Education in Science and Technology, 17(5), 561-568.

Weisberg, S. (1985). Applied linear regression. New York: Wiley.

West, R.W., & Ogden, R.T. (1998). Interactive demonstrations for statistics on the World Wide Web. Journal of Statistics Education, 6(3). Retreived April 24, 2003 from JSE archive:

http://www.amstat.org/publications/jse/jse_archive.html.

PETER K. DUNN

University of Southern Queensland

Australia

dunn@usq.edu.au