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Is there a proper or ideal way to annotate a bubble chart outside of what is required for a standard 2-variale scatterplot?

The bubble area is supposed to be associated with the 3rd variable, but is there something that should say so? Is there a way to properly indicate what the third variable is?

So here is my scatterplot, except that bubbles replace the points, and the bubble area is related to a third variable, not x or y.

enter image description here

How do I say what the bubble sizes are? Is there something that I need to say about the fit line being not weighted? Should I have a x=y line?

I know that I will need title, axis labels, and to clean up units on the y-axis. Is there anything else needed?

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    $\begingroup$ Are you just asking about the legend? See some examples for maps here. $\endgroup$
    – Andy W
    Commented Sep 27, 2016 at 16:25
  • $\begingroup$ @AndyW - nice reference. $\endgroup$
    – EngrStudent
    Commented Sep 28, 2016 at 18:10
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    $\begingroup$ It looks inconsistent to plot an unweighted regression line on such a bubble plot. Fitting a linear least squares model to these data seems inadvisable in the first place, because the responses appear to be constrained to 0 to 100%. Regardless, how are your readers supposed to know the bubble areas are proportional to $Z$ when nothing on the plot even mentions $Z$? $\endgroup$
    – whuber ♦
    Commented Sep 28, 2016 at 20:09
  • $\begingroup$ @whuber - I strongly agree with you. It is a horribly ugly plot. It is, however, useful. Part of the purpose of the plot is to show how bad the unweighted regression is - it is meant to be like 'The Cramer' who is "a loathsome, offensive brute, yet I can't look away". A good thrust of the question is about how should $Z$ be described properly in the plot, and also in much less ugly plots. $\endgroup$
    – EngrStudent
    Commented Sep 28, 2016 at 20:16
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    $\begingroup$ @Nick I agree generally, but there is a subtlety. One purpose of a scaled bubble plot is to reveal patterns of relationships (among the two coordinates and the bubble sizes). We are aiming for a gestalt reaction rather than an accurate decoding of each bubble area. Indeed, some cartographers have suggested scaling bubbles by a power of the variable (around 0.57 rather than 1/2) based on psychological testing(!). I have found bubble plots to be quite useful for revealing multidimensional relationships, even though I never would use, say, a pie chart if I can help it. $\endgroup$
    – whuber ♦
    Commented Apr 21, 2022 at 15:18

1 Answer 1

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Here is a tool used to argue that fault detection algorithm health should be measured using lower confidence interval given sample size instead of prevalence. It is a lower estimate, but it takes into account sample size, and protects from customer-facing falsehoods - which can be expensive.

Note the legend.

Plot:

figure with 2 subplots, upper is prevalence estimate, lower is lower-ci based estimate

Code:

# Purpose:  Show relationship between prevalence and CI for binomial

# -------------
# housekeeping
# rm(list=ls())

#libraries
require(pacman)
p_load(PropCIs)  #for Clopper-Pearson (exact)

#reproducibility
set.seed(09302016)


N_loops <- 100000  #decent sample size

#CI threshold
ci_threshold = 0.95

#stage store for loop
p_true    <- numeric(length = N_loops)
n_samp    <- numeric(length = N_loops)
prev      <- numeric(length = N_loops)
L_CI      <- numeric(length = N_loops)

store          <- data.frame(p_true, n_samp, prev, L_CI)

### main loop ###

for (i in 1:N_loops){
        
        #how many samples show up (uniform random)
        n_samp1         <- sample(x=c(1:300), size = 1, 
                                  replace = FALSE)
        
        #true prevalence (uniform random)
        p_true1         <- runif(n = 1, min = 0, max = 1)
        
        #draw samples (binomial)
        mysamples      <- rbinom(n = n_samp1, size=1, 
                                 prob = p_true1)
        
        #compute prevalence
        prev1 <- sum(mysamples)/length(mysamples)
        
        #compute the lower conf interval
        #  Agresti-Couli exact as "finish line"
        btest <- exactci(x = sum(mysamples), 
                         n = length(mysamples), 
                         conf.level=ci_threshold)
        
        L_CI1 <- btest$conf.int[1]
        
        #pack into store
        store$p_true[i]   <- p_true1
        store$n_samp[i]   <- n_samp1
        store$prev[i]     <- prev1
        store$L_CI[i]     <- L_CI1
}


#make scatterplots

#take off extreme tails
idx_kill <- which(store$p_true < 0.01 | store$p_true > 0.99)

store <- store[-idx_kill,]

#compute variance per row 
var_samp <- store$n_samp*store$p_true*(1-store$p_true)

#convert to visually intuitive form
radius <- sqrt(var_samp/pi)/3


## make plot
par(mfrow=c(2,1))

#first subplot
symbols(x = store$p_true,
        y = store$prev,
        circles = radius,inches=0.2, 
        fg="LightBlue",
        bg=rgb(red=0,green=0,blue=1,alpha=0.5),
        xlim=c(0,1),ylim=c(0,1),
        main="Prevalence based estimate", 
        xlab="True Capability", 
        ylab="Estimated Capability")

grid()
abline(a=0, b=1, col="Green", lwd=3)

#some annotations
text(0.25,0.85,"Over-estimate zone",pos = 4,bg="white")
text(0.4,0.8,"(ending up here is expensive)",bg="White")

text(0.7,0.15,"Under-estimate zone",pos = 4,bg="white")
text(0.75,0.1,"(ending up here is cheap)",bg="White")

legend(
        "topleft", 
        title = "Uncertainty scaled by 3-sigma",
        legend=c("0.3",  "1.5", "3.0","6.0","9.0"), 
        pch = 21,
        bty = "n",
        col = "black",
        pt.bg = "Blue",         
        pt.cex = c(0.1,0.5,1,2,3)
)  

#second subplot

symbols(store$p_true,store$L_CI,radius,inches=0.2,fg="yellow",
        bg=rgb(red=1,green=0,blue=0,alpha=0.6),xlim=c(0,1),ylim=c(0,1),
        main="Lower CI based estimate",
        xlab="True Capability", 
        ylab="Estimated Capability")

grid()
abline(a=0,b=1,col="Green",lwd=3)

#some annotations
text(0.25,0.85,"Over-estimate zone",pos = 4,bg="white")
text(0.4,0.8,"(ending up here is expensive)",bg="White")

text(0.7,0.15,"Under-estimate zone",pos = 4,bg="white")
text(0.75,0.1,"(ending up here is cheap)",bg="White")

legend(
        "topleft", 
        title = "Uncertainty scaled by 3-sigma",
        legend=c("0.3",  "1.5", "3.0","6.0","9.0"), 
        pch = 21,
        bty = "n",
        col = "black",
        pt.bg = "Red",         
        pt.cex = c(0.1,0.5,1,2,3)
)  

par(mfrow=c(1,1))
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