Bubble Plots as a Model-Free Graphical Tool for Continuous Variables

  • Keith A. Markus
  • Wen Gu
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 196)


Researchers often wish to understand the relationship between two continuous predictors and a common continuous outcome. Many options for graphing such relationships, including conditional regression lines or 3D regression surfaces, depend on an underlying model of the data. The veridicality of the graph depends upon the veridicality of the model, and poor models can result in misleading graphs. An enhanced 2D scatter plot or bubble plot that represents values of a variable using the size of the plotted circles offers a model-free alternative. The R function bp3way() implements the bubble plot with a variety of user specifiable parameters. An empirical study demonstrates the comparability of bubble plots to other model-free plots for exploring three-way continuous data.


Scatter Plot Negative Interaction Data Frame Graph Type Lyer Illusion 
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  1. 1.
    Aiken, L.S., West, S.G.: Multiple Regression: Testing and Interpreting Interactions. Sage Publications, Inc, Newbury Park (1991)Google Scholar
  2. 2.
    Baron, R.B., Kenny, D.A.: The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology 51(6), 1173–1182 (1986)CrossRefGoogle Scholar
  3. 3.
    Cleveland, W.S.: The Elements of Graphing Data. Wadsworth Advanced Books and Software, Monterey (1985)Google Scholar
  4. 4.
    Cleveland, W.S.: Visualizing Data. Hobart press, Summit, New Jersey (1993)Google Scholar
  5. 5.
    Cleveland, W.S.: The Elements of Graphing Data, revised edn. Hobart Press, Summit, NJ (1994)Google Scholar
  6. 6.
    Cleveland, W.S., Harris, C.S., McGill, R.: Experiments on quantitative judgments of graphics and maps. The Bell System Technical Journal 62(6), 1659–1674 (1982)Google Scholar
  7. 7.
    Cleveland, W.S., McGill, R.: An experiment in graphical perception. International Journal of Man-Machines Studies 25(5), 491–500 (1986)CrossRefGoogle Scholar
  8. 8.
    Cohen, J.: The cost of dichotomization. Applied Psychological Measurement 7(3), 249–253 (1983)CrossRefGoogle Scholar
  9. 9.
    Cohen, J., Cohen, P., West, S., Aiken, L.: Applied multiple regression/correlation analyses for the behavioral sciences, 3rd edn. Lawrence Erlbaum, Hillsdale, NJ (2002)Google Scholar
  10. 10.
    Fox, J.: An R and S-Plus Companion to Applied Regression. Sage Publications, Thousand Oaks (2002)Google Scholar
  11. 11.
    Fox, J.: car: Companion to Applied Regression. R package version 1.2-14 (2009). URL {}. I am grateful to Douglas Bates, David Firth, Michael Friendly, Gregor Gorjanc, Spencer Graves, Richard Heiberger, Georges Monette, Henric Nilsson, Derek Ogle, Brian Ripley, Sanford Weisberg, and Achim Zeileis for various suggestions and contributions
  12. 12.
    Hartwig, F., Dearing, B.E.: Exploratory data analysis. Sage university paper series on quantitative applications in the social sciences, series no. 07-016. Sage Publications, Beverly Hills (1979)Google Scholar
  13. 13.
    Kraemer, H.C., Kiernan, M., Essex, M., Kupfer, D.J.: How and why criteria defining moderators and mediators differ between baron & kenny and macarthur approaches. Health Psychology 27(2), S101–S108 (2008)CrossRefGoogle Scholar
  14. 14.
    MacCallum, R.C., Zhang, S., Preacher, K.J., Rucker, D.D.: On the practice of dichotomizing of quantitative variables. Psychological Methods 7, 19–40 (2002)CrossRefGoogle Scholar
  15. 15.
    MacKinnon, D.P., Fairchild, A.J., Fritz, M.S.: Mediation analysis. Annual Review of Psychology 58, 593–614 (2007)CrossRefGoogle Scholar
  16. 16.
    Maxwell, S.E., Delaney, H.D.: Bivariate median splits and spurious statistical significance. Psychological bulletin 113(1), 181–190 (1993)CrossRefGoogle Scholar
  17. 17.
    Muthen, L.K., Muthen, B.O.: Mplus user’s guide, 5th edn. Muthen and Muthen, Los Angeles, CA (2007)Google Scholar
  18. 18.
    R Development Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2009). URL {}. ISBN 3-900051-07-0
  19. 19.
    Robbins, N.B.: Creating More Effective Graphs. John Wiley & Sons, Hoboken (2005)Google Scholar
  20. 20.
    Schiffman, H.R.: Constancy and illusion. In: Sensation and perception, 5th edn., pp. 250–286. Wiley, New York (2002)Google Scholar
  21. 21.
    Schmid, C.F.: Statistical Graphics: Design Principles and Practices. Wiley, New York (1983)Google Scholar
  22. 22.
    Tufte, E.R.: The Visual Display of Quantitative Information. Graphic Press, Cheshire (2001)Google Scholar
  23. 23.
    Tukey, J.W.: Exploratory data analysis. Addison-Wesley, Reading, MA (1977)Google Scholar
  24. 24.
    Wainer, H.: Visual Revelations: graphic tales of fate and deception from Napoleon Bonaparte to Ross Perot. Copernicus, New York (1997)Google Scholar
  25. 25.
    Wright, D.B., London, K.: Modern Regression Techniques Using R. Sage, Washington DC (2009)Google Scholar

Copyright information

© Springer-Verlag New York 2010

Authors and Affiliations

  1. 1.John Jay College of Criminal Justice of The City University of New YorkNew YorkUSA

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