Rachel Sturm-Beiss

Kingsborough Community College

(City University of New York)

Journal of Statistics Education Volume 13, Number 1 (2005), ww2.amstat.org/publications/jse/v13n1/sturm-beiss.html

Copyright © 2005 by Rachel Sturm-Beiss, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

**Key Words:** ANOVA; Java applet

The ANOVA model is the simplest linear statistical model with qualitative independent variables. ANOVA, together with simple linear regression, form a foundation for the study of general linear models. However, exercises relating ANOVA model parameters and calculated quantities are not easy to formulate. As a result, typical textbook exercises emphasize calculations, leaving the model as an abstract entity. Therefore, it is of pedagogical value to have ancillary materials that help students visualize model parameters and their relationship to sample observations. We present a tool (in the form of a java applet) that emphasizes the ANOVA probabilistic model by placing model parameters along-side observations, and giving the student the ability to manipulate values and to observe resulting effects, thus removing some of the abstraction. Taur (1999) introduced an excellent example of such a teaching tool for nonlinear regression, called “Visual Fit” and Anderson-Cook and Dorai-Raj (2003) reviewed such java applets that demonstrate the power of a test.

The ANOVA one-way approach here recommended starts with treatment group means m_{i} for i=1,2,3 and variance
(the two-way model
consists of two factor variables with three and two factor levels each). Random normal N(m_{i},
) samples are generated
within each treatment group and estimated sample means, variances and other quantities are calculated and displayed. We
display data, actual parameter values, and estimated parameter values in a scatter plot augmented with graphs, ANOVA table,
explanations, and exercises. The student can change parameter values (through standard window’s interface such as
“drag-and-drop”, text boxes, list boxes etc. ) and observe the effect on calculated quantities and model significance.
Guided exercises and explanations help the student in this process. This visual teaching tool differs from traditional
exercises in which sample observations (but not parameter values) are available. In particular, the student is able to
generate many random samples for the same set of parameters and to get a feel for statistical significance as a phenomenon
that emerges over a large number of random samples.

E(Y_{ijk}) = + +
+

where

Y_{ijk} = the k^{th} obswervation in the ij^{th} group

+ +
+ =
= the mean of the treatment group corresponding to the i^{th}
level of A and the j^{th} level of B

, i = 1, 2, 3 are factor A main effects

, j = 1, 2 are factor B main effects

, i = 1, 2, 3 and j = 1, 2 are the AB interaction effects

We assume that factor variables A and B may interact. If there is no interaction, then the interaction effects are all 0, and the model is additive: = + . The following figures and the comments below them illustrate some of the features of the ANOVA tool.

Figure 1

- A parameter treatment group mean for the population corresponding to the i = 3
^{rd}level of A and the j = 2^{nd}level of B , i.e. (*Drag and Drop point*) - The value of
- The sample treatment group mean
- Observation Y
_{12k}(the number of observations is set by the user) - A main effect of factor variable B, i.e. (
*Drag and Drop point*) - A main effect of factor variable A, i.e. (
*Drag and Drop point*) - The parameter treatment group mean (
*Drag and Drop point*) - The overall mean
- The sum of squares bar graph
- The text area for exercises and explanations
- The ANOVA table
- User controls:
- Choose one or two-way model.
- Set the value of sigma.
- View explanations of various aspects of ANOVA.
- Choose another sample.
- Choose a guided exercise.
- Choose the sample size.
- Request an explanation of the two plots.
- Display data values in the text area.

Figure 2

- A parameter treatment group mean for the population corresponding to the i = 3
^{rd}level of A, i.e. (*Drag and Drop point*) - The value of
- The sample treatment group mean
- The overall mean
- A main effect of factor variable A, i.e. (
*Drag and Drop point*) - Observation Y
_{1k}(the number of observations is set by the user) - The sum of squares bar graph
- The text area for exercises and explanations
- The ANOVA table
- User controls:
- Choose one or two-way model.
- Set the value of sigma.
- View explanations of various aspects of ANOVA.
- Choose another sample.
- Choose a guided exercise.
- Choose the sample size.
- Request an explanation of the two plots.
- Display data values in the text area .

Applets such as the one described here are tools that could easily be incorporated into lectures and assignments as accessibility through the internet is simple and not costly. We believe that the ANOVA and other related java applet visual teaching tools can help students of statistics and students from related disciplines gain a better understanding of ANOVA and other statistical techniques.

The ANOVA Visualization Tool can be viewed by clicking on Anova Applet or by going to the author's web site at www.kingsborough.edu/academicDepartmetns/math/faculty/rsturm/anova/Anova0126.html

Anderson-Cook, C. M. and Dorai-Raj, S (2003), “Making the Concepts of Power and Sample Size Relevant and Accessible to
Students in Introductory Statistics Courses using Applets,” *Journal of Statistics Education* [Online] 11 (3)
(ww2.amstat.org/publications/jse/v11n3/anderson-cook.html)

Hogg, R., and Craig, A. (1995), *Introduction to Mathematical Statistics* (5th ed.), New York: Macmillan.

Neter, J., Wasserman, W., and Kutner, M.H. (1990), *Applied Linear Statistical Models*, Chicago: Richard D. Irwin, Inc.

Taur, Y., and McCulloch, C. (1999), “A Teaching Tool for Nonlinear Regression: Visual Fit,” *Journal of Statistics Education*
[Online], 7 (2)
(ww2.amstat.org/publications/jse/secure/v7n2/taur.cfm)

Rachel Sturm-Beiss

Department of Mathematics and Computer Science

Kingsborough Community College

City University of New York

2001 Oriental Boulevard

Brooklyn, New York 11235

U.S.A.
*rsturm@kbcc.cuny.edu*

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