After our discussion of visual displays of regression uncertainty, I asked Solomon Hsiang and Lucas Leeman to send me their code. Both of them replied.

Solomon wrote:

The matlab and stata functions I wrote, as well as the script that replicates my figures, are all posted on my website.

Also, I just added options to the main matlab function (vwregress.m) to make it display the spaghetti plot (similar to what Lucas did, but a simple bootstrap) and the shaded CI that you suggested (see figs below). They’re good suggestions.

Personally, I [Hsiang] like the shaded CI better, since I think that all the visual activity in the spaghetti plot is a little distracting and sometimes adds visual weight in places where I wouldn’t want it. But the option is there in case people like it.

Solomon then followed up with:

I just thought of this small adjustment to your filled CI idea that seems neat. Cartographers like map projections that conserve area. We can do something similar by picking a visual-weighting scheme that conserves “ink” (which we equate with confidence). Imagine that we squirt out ink uniformly to draw the conditional mean and then smear the ink vertically so that it stretches from the lower confidence bound to the upper confidence bound. In places where the CI band is narrow, this will cause very little spreading of the ink so the CI band will be dark. But in places where the CI band is wide, the ink is smeared a lot so it gets lighter. For any vertical sliver of the CI band (think dx) the amount of ink displayed (integrated along a vertical line) will be constant. But in places where we have a lot of information, the display will have more visual weight. I think this is a somewhat more natural visual-weighting scheme for the CI band than the 1/sqrt(N) that I was using for just the mean regression. I’ve attached both for comparison.

After thinking a little more about how to visually-weight the CI bands and the spaghetti plots, I think that maybe we should be careful not to “double count” uncertainty. For example, when the estimates begin to spread out in the spaghetti plots, then the apparent coloration begins to thin out simply because there is a lower density of lines. So if we *also* fade out the lines, then we’re dimming the uncertain regions in two ways. This isn’t obviously wrong, but it does feel like we’re penalizing the graph in uncertain regions twice for the same thing.

Plotting the CI band with the “fixed ink” visual-weighting and the spaghetti plot with solid spaghetti seem like analogs to one another, since the vertically integrated quantity of ink is uniform in both plots. Commands to plot both (using the function I posted) are:

`x = randn(200,1);`

e = 4*randn(200,1).*(1+abs(x)).^1.5;

y = 2*x+x.^2+e;

figure

%"Fixed-ink" visual weighting:

vwregress(x, y, 300, .5,'CI','FILL',200,[0 0 1]);

figure

%Solid spaghetti plot without visual weighting:

`vwregress(x, y, 300, .5,'SPAG','SOLID',200,[0 0 1]);`

My reply:

I know what Solomon is saying about the double-counting; I thought about this too in my original post, which is why I’d liked the idea of the spaghetti plot with additive shading. Ultimately much depends on what these plots look like in real examples. One problem I have with many traditional statistical summaries (such as 95% intervals, whether presented graphically or as numbers in table or text) is that they emphasize the borders of the intervals. The edge of a 95% interval is not so interesting; it’s a noisy data summary.

The status quo in many fields is even worse than that, though, in that it is often standard to put little perpendicular lines at the edges of intervals to make “error bars” which emphasize the endpoints even more.

Meanwhile Lucas also responded to my request for code:

The original plot is based on a nested model with data I cannot make available. But I have here some sample code in R which will produce a similar plot:

`# Example with Probit`

library(mvtnorm)

set.seed(111)

x <- rnorm(1000) # systematic component
e <- rnorm(1000, sd=18) # random component
y.lat <- 3*x+e # latent scale
y <- rep(0,1000)
y[y.lat>0] <- 1 # observed outcome variable
mod1 <- glm(y ~ x, family = binomial(link = "probit"))
beta1 <- coef(mod1)
var1 <- vcov(mod1)
split <- 30 # spacing of x
BETA1 <- rmvnorm(200, beta1, var1) # Taking 1000 drwas of the posterior vector
xx1 <- seq(-10,10,length.out=split) # The x-range you want to use for plotting later
X1 <- cbind(1,xx1) # The Matrix of explanatory variables where
# >one variable varies from row to row

y.lat1 <- BETA1%*%t(X1) # 1000 rows (uncertainty) and 500 columns
# >(for different values of x)

#y.lat2 <- t(apply(y.lat1,1,sort)) # Sorting to cut off alpha %
y.lat2 <- y.lat1
y.pred <- pnorm(y.lat2) # translate latent to predicted probability
cib <- 0.1 # Define level for CI
lb <- round((dim(BETA1)[1] * cib)/2) # lb and ub define which predictions to be plotted
ub <- dim(BETA1)[1] - lb # >and are based on "cib"

```
``` # plot the median prediction

plot(xx1,colMeans(y.pred), type="l", col=rgb(160,32,240,255,maxColorValue=255), lwd=3, yla="Predicted Prbablity", xlab="Variable xx1")

`for (i in lb:ub){ # plot every prediction between "lb" and "ub"`

# >the denominator for the alpha (here: 18) and

# >the line thickness (lwd) are easy to change to

# >get different shades

points(xx1,jitter(y.pred[i,], factor=4), type="l", col=rgb(160,32,240,(255-abs(round(dim(y.pred)[1]/2)-i))/9,maxColorValue=255), lwd=2)

}

[…] P.S. More (including code!) here. […]

1) First chart:

I still wish that each vertical went from dark in middle to faded at edges, perhaps with a minimalist white or grey line, certainly not a dark one.2) This chart *still* has a sharp edge, and visually, there’s no shading difference between 1SDand 2SD.

2) Where in the world is it written that 95% is a magic number, and that 94% and 96% should look signifcantly different?

Meic Goodyear’s paper showed (pp.12-13) showed bars that had some useful shading, although the edges are still a little sharper than I’d wish for (but given Excel, not bad).

If such bars were drawn for every vertical slice, that would be good.

(I don’t know how to do this in R, but maybe some expert does.)

The results produced by Lucas’s R code are very sensitive to the choice of random number generator seed. See what happens when you use set.seed(115) .

Hello there is a nice R (denstrip) package along with an article in the American Statistician (2008 vol 62 340-347) that discusses this. The package is quite cool to add a shaded CI interval.

Ahuri: thanks

Google: denstrip r

Locates the relevant packages.

I’m off to conference, but that certainly sounds like the right dirt if thing.

[…] Dr. Dipl.-Psych. StartResearchTeachingPublicationsÜber mich/ About me Visually weighted regression in R (à la Solomon Hsiang)30. August, 2012 Filed in: StatisticsSolomon Hsiang proposed an appealing method for visually displaying the uncertainty in regressions (see his blog [1][2], and also the discussions on the Statistical Modeling, Causal Inference, and Social Science Blog [1][2]). […]

Hi there,

I picked up that great idea and wrote an R function for it. Concerning the comment that the CI boundaries give too much weight to the edges, I used another procedure: I computed a density estimate of the bootstrap smoothers for each vertical slice. As the area under the density always is 1, the ink is conserved for each vertical slice. So it’s something like a continuos version of the spaghetti plot.

More details, the R code, and many plots on my blog: http://www.nicebread.de/visually-weighted-regression-in-r-a-la-solomon-hsiang/

Comments and additions welcome!

Felix

Felix: I think those look interesting, nice explorations of different presentations.

I especially like the first one under

“Here are two plots with actual data I am working on:”

I really like the shading density approach as opposed to the spaghetti-graph approach, but that may be a bias on my part from having looked at spaghetti graphs where each of the strands was actually a different study, each of which ought to have it’s own shading.

It is truly a sad thing that many of the standard presentation styles still seem rooted in the 1960s or earlier, even when computing technology has long been advanced enough to do much better, assuming we can figure out better ways, try them out, and then get them encapsulated in standard software so that people can just use them.

[…] Hsiang writes: Two small follow-ups based on the discussion (the second/bigger one is to address your comment about the 95% CI […]

[…] In this case we adopted a new technology called visually weighted regression (see here, here, or here), and simply applied a new color […]