linear regression with confidence bands

[mbostock]

Inspired by https://observablehq.com/@toja/linear-regression-with-confidence-bands, using jStat for probability functions. Fixes #168.

Plot.plot({
  marks: [
    Plot.dot(cars, {x: "weight (lb)", y: "economy (mpg)", r: 2}),
    Plot.linearRegressionY(cars, {x: "weight (lb)", y: "economy (mpg)", p: 0.01})
  ]
})

TODO

• linearRegressionX
• remove sort: {channel: x}
• combine the two marks into one?

[Fil]

A few remarks after minimal testing.

• should we throw an error or at least a warning when p> 0.5? or just ignore in that case? (in any case, it should not reverse the area).
• we should not return a band if p is 0, null or undefined (the current code returns a path with infinite coordinates). By the way p:null could be the documented method to specify that the user doesn't want a band (maybe they want to draw the band below the dots, then the dots, then the line).
• options.fill is overwritten by stroke, preventing from writing "fill: 'gray'" if we want the confidence band to be gray.
• x extent: this PR draws the regression line over the whole width of the frame, but sometimes you want to limit it to the data's extent (or, I guess, the data extent +- a certain padding/inset) [this is what this older approach does, and it seems quite nice for this particular dataset]. It would be nice to have the option of limiting the line to the data extent (maybe with domain: "frame", "data", or [x1, x2]?), and another option to add (possibly negative) padding?

(I can explore variants in a PR to the PR, just raising them as comments for now)

[mbostock]

Okay, I think I’ve done everything you’ve asked! How’s it now? 😄

[Fil]

What should happen with p=0? The error message is inconsistent invalid p; not in [0, 0.5): 0 (at ca8cbf8)

[Fil]

In https://observablehq.com/@fil/plot-regression-945 I've added a quick comparison with ggplot2: p=0.05 (this default) corresponds to level=0.90 of ggplot2, and p=0.025 corresponds to level=0.95 (ggplot2 default). Suggest to change the variable name and default so that we get the same definitions as ggplot2?

It would also make it easier to document the mark, because we wouldn't have to get into details about the Student t test, cumulative distributions etc. Seems easier to write "the band in which the linear relation lay with a confidence of 95%".

[Fil]

I've added a bit of documentation, but I don't know how to describe the band simply in terms of p.

[mbostock]

I’ve replaced the p option with the more understandable ci option representing the confidence interval in [0, 1). This corresponds to ggplot2’s level option. (I think that “ci” is more self-describing than “level”.) Also I think there is a bug in Torben’s notebook, because a confidence interval of ci = 0.95 corresponds to the old p = 0.025, not 0.05. I’ve confirmed this with a visual comparison of ggplot2’s behavior using the mtcars notebook based on a blog post by Thomas Neitmann.

[teecrow]

Thanks for implementing confidence intervals - really fantastic work! As a note, to clear up any confusion (and I recognize you may have already figured this out, but in case anyone else might come across this): the confidence level C, often defaulting at 95%, corresponds to p=.05. Above the regression line, you have half your confidence interval, which is p=.025, and same below the line. I suspect that's the cause of the uncertainty about .025 being mentioned in some places. And FWIW I think using "CI" instead of "p" was a good idea.

If it's useful for your documentation, or for anyone else who might be reading, there is an excellent paper with several different recommendations for simple but accurate ways to describe the CI:
Cumming, G., & Finch, S. (2005). Inference by Eye: Confidence Intervals and How to Read Pictures of Data. American Psychologist, 60(2), 170–180. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=79e79e30c2c0f35e1add65d0375ddc38e939385d