[Update 1: Sep 5, 2012: Explore the Magical Data Enhancer by IRES, using this visualization technique]
[Update 2: Sep 6, 2012: See new improved plots, and new R code!
Solomon Hsiang proposed an appealing method for visually displaying the uncertainty in regressions (see his blog , and also the discussions on the Statistical Modeling, Causal Inference, and Social Science Blog ).
I implemented the method in R (using ggplot2), and used an additional method of determining the shading (especially concerning Andrew Gelman’s comment that traditional statistical summaries (such as 95% intervals) give too much weight to the edges. In the following I will show how to produce plots like that:
I used following procedure:
- Compute smoothers from 1000 bootstrap samples of the original sample (this results in a spaghetti plot)
- Calculate a density estimate for each vertical cut through the bootstrapped smoothers. The area under the density curve always is 1, so the ink is constant for each y-slice.
- Shade the figure according to these density estimates.
Now let’s construct some plots!
The basic scatter plot:
No we show the bootstrapped smoothers (a “spaghetti plot”). Each spaghetti has a low alpha. That means that overlapping spaghettis produce a darker color and already give weight to highly populated regions.
Here is the shading according to the smoother’s density:
Now, we can overplot the median smoother estimate for each x value (the “median smoother”):
Or, a visually weighted smoother:
Finally, we can add the plain linear regression line (which obviously does not refelct the data points very well):
At the end of this post is the function that produces all of these plots. The function returns a ggplot object, so you can modify it afterwards, e.g.:
vwReg(y~x, df, shade=FALSE, spag=TRUE) + xlab("Implicit power motive") + ylab("Corrugator activity during preparation")[/cc]
Here are two plots with actual data I am working on:
The correlation of both variables is .22 (p = .003).
A) As a heat map (note: the vertical breaks at the left and right end occur due to single data points that get either sampled or not during the bootstrap):
B) As a spaghetti plot:
Finally, here's the code (sometimes the code box is collapsed - click the arrow on the top right of the box to open it). Comments and additions are welcome.