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Bubble Plots as a Model-Free Graphical Tool for Three Continuous Variables and a Flexible R Function to Plot Them Keith A. Markus and Wen Gu John Jay College of Criminal Justice, CUNY. Overview. Goal: Model-free graphs for 3 continuous variables. Some alternative graphs & design issues.

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Bubble Plots as a Model-Free Graphical Tool for Three Continuous Variablesand a Flexible R Function to Plot ThemKeith A. Markus and Wen GuJohn Jay College of Criminal Justice, CUNY

overview
Overview
  • Goal: Model-free graphs for 3 continuous variables.
  • Some alternative graphs & design issues.
  • The R function: bp3way().
  • An empirical study.
  • Tentative conclusions & future directions.
the goal
The Goal
  • The goal is to provide a useful graphical representation of the association between 3 continuous variables.
  • Often: 2 IVs and 1 DV.
  • Model free:
    • Exploratory data analysis.
    • Not a summary of a statistical model.
why model free
Why Model Free?
  • If the statistical model is correct: model based graphs can be very efficient.
  • If the statistical model is incorrect: model based graphs can be very misleading.
  • E.g., Multiple y~x regression lines for values of z. Misleading if...
    • y~x relationship is not linear.
    • Variance in y varies with x or z.
    • Regression lines extrapolate beyond data.
some non options
Some Non-Options
  • Scatterplot matrix.
  • y~x regression lines for fixed z values.
  • Factorial design type line plots.
  • All good plots for other applications.
  • But not good plots for present purpose.
scatterplot matrix
Scatterplot matrix
  • Does not attempt to represent 3-way distributions.
  • Same data used for all graphs (N = 100)
y x regression lines for fixed values of z
y~x regression lines for fixed values of z:
  • Model dependent: plots model not data.
  • Not clear where data leaves off.
factorial design type plots for categorized ivs
Factorial-design type plots for categorized IVs:
  • Model dependent (interpolation).
  • Arbitrary cuts (quartiles plotted here).
  • Loss of information through categorization.
some options
Some Options
  • 3D Scatterplots.
    • R Package scatterplot3d: scatterplot3()
  • Co-plots.
    • R base installation: coplot()
  • 3-way Bubbleplots.
    • Available from authors: bp3way()
3d scatterplot
3D scatterplot:
  • Natural extension of 2D scatter plot.
  • Relies on 3D illusion: some ambiguity.
co plot
Co-plot
  • Well suited to perceptual process.
  • Relies on banding of z values.
3 way bubble plot
3-Way Bubble Plot
  • 2D representation of 3D data.
  • People tend to underestimate area.
  • No literature.
some design features of the 3 way bubble plot
Some Design Features of the 3-Way Bubble Plot
  • Grid designed to make it easier to compare circle sizes across the plot surface.
  • Shading designed to accentuate bubbles.
  • Limited number of cases plotted avoids overly dense plots (in this case all 100 are plotted).
  • Margins avoid bubbles extending outside plot region.
bp3way function
bp3way() function

Usage

bp3way(x)

bp3way(x, xc=1, bc=2, yc=3, proportion=1, random=TRUE, x.margin=.1, y.margin=.1, rad.ex=1, rad.min=NULL, names=c('X', 'B', 'Y'), std=FALSE, fg='black', bg='grey90', tacit=TRUE, ...)

data parameters
Data Parameters

x is a data frame with at least 1 column.

xc, yc, and bc identify the columns used to plot the x axis, y axis, and bubbles respectively.

names is a vector of variables names used in the plot.

  • Easy to switch variables without changing the data.
  • User can use same column more than once.
  • Out of bounds values return an error.
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Parameters with data sensitive defaults:

    • rad.ex: Radius expansion rate.
    • rad.min: Minimum bubble radius.
    • proportion: % of data plotted.
    • margins and grid.
  • Other user-specified options include:
    • Plotting a random sample or first % of cases.
    • Standardization of X and Y variables.
    • labels and colors.
empirical study
Empirical Study
  • 3 Plots (Bubbleplot, 3D Scatterplot, Coplot).
    • Between subjects.
    • Within group n = 36.
  • 6 Data sets.
    • Within subjects.
  • N of subjects = 108.
  • N of observations = 108 x 6 = 648.
four dvs
Four DVs
  • Accuracy of interpretation of graphs
    • 0-3 questions answered correctly.
  • Confidence in interpretation
    • 1-5, average of 3 1-5 Likert scale items.
  • Perceived clarity
    • 1-5 Likert scale item.
  • Perceived ease of use
    • 1-5 Likert scale item.
univariate summary
Univariate Summary
  • No floor or ceiling effects, variability in DVs.
correlations between outcomes
Correlations Between Outcomes
  • Above Diagonal: N = 648 observations.
  • Below Diagonal: N = 108 participants.
multivariate model fit first
Multivariate model fit first

y* = α0 + α1'Data + α2' Data∙Graph + u1 (Level 1)

α 0 = β0 + β1'Graph + u2 (Level 2)

y = { 0 if y* ≤ τ1, 1 if τ 1 < y* ≤ τ 2, ... k if τ k-1 < y* ≤ τ k} (Threshold model)

  • Third equation not used for confidence DV.
  • Full model: Mplus
  • Confidence also fit in R using lme() function.
  • Nearly identical estimates with R or Mplus.
  • Story in interactions, not main effects.
follow up simple effects
Follow-up: Simple Effects
  • Shift focus to simple effects because we cannot usefully interpret interactions.
  • Protected Wilcox Mann Whitney Exact Tests Used for Accuracy, Clarity and Ease of Use DVs.
  • Protected t tests used for Confidence DV.
  • No one graph consistently better.
  • Mostly a story about accuracy.
tentative conclusions
Tentative Conclusions
  • Much remains to be learned about the cognition of these 3 graph types.
    • Coplot may have a slight edge over the other two.
    • But optimal plot seems data dependent.
    • Study included a limited range of data and graph conditions.
    • More detailed perceptual theory is needed to optimize graph design.
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Recommendation for exploratory analysis:

    • Use 2 or more graph types.
    • Cannot predict ahead of time which will work best.
    • Probably useful to look at data more than one way even if one graph were consistently best.
slide35

Recommendation for reporting results:

    • Use model based graphs.
      • If you understand your data well enough to fit a good model.
    • If not, try different model-free graphs and see which seems to work best.
future directions
Future Directions
  • Identify factors that impact which graph works best.
  • Identify design factors that maximize effectiveness of all 3 graph types.
  • Increase statistical power:
    • Identify individual difference covariates that account for within condition variance.
    • More sensitive outcome measures.