Former Freud expert & current stats legend Andrew Gelman posted a blog (one he likely wrote in the late 1990s; he stockpiles his dispatches, so probably by the time he sees mine he'll have completely forgotten this whole thing, & even if he does respond I’ll be close to 35 yrs. old by then & will be interested in other things like drinking and playing darts) in which he said he liked one of my graphics!
Actually, he said mine was “not wonderful”—but that it kicked the ass of one that really sucked!
USA USA USA USA!
Time to get back the never-ending project of self-improvement that I’ve dedicated my life too.
The question is, How can I climb to that next rung—“enh,” the one right above “not wonderful”?
I’m going to show you a couple of graphics. They aren’t the same ones Gelman showed but they are using the same strategy to report more interesting data. Because the data are more interesting (not substantively, but from a graphic-reporting point of view), they’ll supply us with even more motivation to generate a graphic-reporting performance worthy of an “enh”—or possibly even a “meh,” if we can get really inspired here.
I say we because I want some help. I’ve actually posted the data & am inviting all of you—including former Freud expert & current stats legend Gelman (who also is a bully of WTF study producers , whose only recourse is to puff themselves up to look really big, like a scared cat would)—to show me what you’d do differently with the data.
Geez, we’ll make it into a contest, even! The “Gelman Graphic Reporting Challenge Cup,” we’ll call it, which means the winner will get—a cup, which I will endeavor get Gelman himself to sign, unless of course he wins, in which case I’ll sign it & award it to him!
Okay, then. The data, collected from a large nationally representative sample, shows the relationship between religiosity, left-right political outlooks, and climate change.
It turns out that religiosity and left-right outlooks actually interact. That is, the impact of one on the likelihood someone will report “believing in” human-caused climate change depends on the value of the other.
Wanna see?? Look!!
That’s a scatter plot with left_right, the continuous measure of political outlooks, on the x-axis, and “belief in human-caused climate change” on the right.
Belief in climate change is actually a binary variable—0 for “disbelief” and 1 for “belief.”
But in order to avoid having the observations completely clumped up on one another, I’ve “jittered” them—that is, added a tiny bit of random noise to the 0’s and 1’s (and a bit too for the left_right scores) to space the observations out and make them more visible.
Plus I’ve color-coded them based on religiosity! I’ve selected orange for people who score above the mean on the religiosity scale and light blue for those who score below the mean. That way you can see how religiosity matters at the same time that you can see that political outlook matters in determining whether someone believes in climate change.
Or at least you can sort of see that. It’s still a bit blurry, right?
So I’ve added the locally weighted regression lines to add a little resolution. Locally weighted regression is a nonmodel way to model the data. Rather than assuming the data fit some distributional form (linear, sigmoidal, whatever) and then determining the “best fitting” parameters consistent with that form, the locally weighted regression basically slices the x-axis predictor into zillions of tiny bits, with individual regressions being fit over those tiny little intervals and then stitched together.
It’s the functional equivalent of getting a running tally of the proportion of observations at many many many contiguous points along left_right (and hence my selection of the label “proportion agreeing” on the y-axis, although “probability of agreeing” would be okay too; the lowess regression can be conceptualized as estimating that).
What the lowess lines help us “see” is that in fact the impact of political outlooks is a bit more intense for subjects who are “low” in religiosity. The slope for their S-shaped curve is a bit steeper, so that those at the “top,” on the far left, are more likely to believe in human-caused climate change. Those at the “bottom,” on the right, seem comparably skeptical.
The difference in those S-shaped curves is what we can model with a logistic regression (one that assumes that the probability of “agreeing” will be S-shaped in relation to the x-axis predictor). To account for the possible difference in the slopes of the curve, the model should include a cross-product interaction term in it that indicates how differences in religiosity affect the impact of differences in political outlooks in “believing” in human-caused climate change.
That regression actually corroborates, as it were, what we “saw” in the raw data: the parameter estimates for both religiosity and political outlooks “matter” (they have values that are practically and statistically significant), and so does the parameter estimate for the cross-product interaction term.
But the output doesn’t in itself doesn’t show us what the estimated relationships look like. Indeed, precisely because it doesn’t, we might get embarrassingly carried away if we started crowing about the “statistically significant” interaction term and strutting around as if we had really figured out something important. Actually, insisting that modelers show their raw data is the most important way to deter that sort of obnoxious behavior but graphic reporting of modeling definitely helps too.
So let’s graph the regression output:
Here I’m using the model to predict how likely a person who is relatively “high” in religiosity—1 SD above the population mean—and a person who is relatively “low”—1 SD below the mean—to agree that human-caused climate change is occurring. To represent the model’s measurement precision, I’m using solid bars—25 of them evenly placed—along the x-axis.
Well, that’s a model of the raw data.
What good is it? Well, for one thing it allows us to be confident that we weren’t just seeing things. It looked like there was a little interaction between religiosity and political outlooks. Now that we see that the model basically agrees with us—the parameter that reflects the expectation of an interaction is actually getting some traction when the model is fit to the data—we can feel more confident that’s what the data really are saying (I think this is the right attitude, too, when one hypothesized the observed effect as well as when one is doing exploratory analysis). The model disciplines the inference, I’d say, that we drew from just looking at the data.
Also, with a model, we can refine, extend, and appraise the inferences we draw from the data.
You might say to me, e.g., “hey, can you tell me how much more likely a nonreligious liberal Democrat to accept human-caused climate change than a religious one?”
I’d say, well, about “12%, ± 6, based on my model.” I’d add, “But realize that even the average religious liberal Democrat is awfully likely to believe in human-caused climate change—73%, ± 5%, according to the model.”
“So there is an interaction between religiosity & political outlooks, but it's nothing to get excited about--the way somone trained only to look at the 'significance' of regression model coefficients might -- huh?” you’d say.
“Well, that’s my impression as well. But others might disagree with us. They can draw their own conclusions about how important all of this is, if they look at the data and use the model to make sense of it .”
What’s Gelman’s reservation? How come my graphic rates only “not awful” instead of “enh” or “meh”?
He says “I think all those little bars are misleading in that they make it look like it’s data that are being plotted, not merely a fitted model . . . .”
Hm. Well, I did say that the graphic was a fitted model, and that the bars were 0.95 CIs.
The 0.95 CIs *could* mislead people --if they were being generated by a model that didn't fairly convey what the actual data look like. But that's why one starts by looking at, and enabling others to see, what the raw data “look like.”
But hey--I don’t want to quibble; I just want to get better!
So does anyone have a better idea about how to report the data?
If so, speak up. Or really, much much better, show us what you think is better.
I’ve posted the data. The relevant variables are “left_right,” the continuous political outlook scale; “religiosity,” the continuous religiosity scale; and “AGW,” belief in climate human-caused-climate change =1 and disbelief = 0. I’ve also included “relig_category,” which splits the subjects at the mean on religiosity (0 = below the mean, 1 = above; see note below if you were using "relig" variable). Oh, and here's my Stata .do file, in case you want to see how I generated the analyses reported here.
So ... either link to your graphics in the comments thread for this post or send them to me by email. Either way, I’ll post them for all to see & discuss.
And remember, the winner—the person who graphically reports the data in a way that exceeds “not wonderful” by the greatest increment-- will get the Gelman Cup!
I realize now that my coding of the median split variable "relig" (used by Anoneuoid) for religiosity is counterintuitive, so I added "relig_category," which is coded "0" for below median, "1" for "above" on religiosity scale.
Not satisfied w/ his or her chances based on one entry, or perhaps trying to intimidate the billiions of other registered entrants still feverishly polishing up their graphics, Anoneuoid has now submitted a 2d graphic:
Definitely nice to have the additional information about religiosity worked in. Not only can we "see" the influence of variation in religiosity as continuous measure -- as opposed to 2 levels -- along w/ left_right at same time. In addition, we get information about correlation between religiosity & ideology--that helps to avoid inducing someone from making mistake of thinking that differences in say, "higly secular" vs. "highly religious" very conservative Strong Republican, e.g., will matter much in the real world (the former being so dominated numerically by latter).
Red & blue for belief in AGW might pose a cognitive challenge given the association with US political parties.
But forget what I think! What are the judges' views? Better than the histograms? Better than "not wonderful"? And why? For whom?
Will say that the 3d-turn is starting to remind me of the "Nukey Thompson" days... Wonder whatever happened to him?...
So during ths break in the action ...
Anoneuoid's 2d entry got me to thinking about the importance of helping people not to forget that religiosity & left_right outlooks are correlated. Pretty obvious, really, exept when it's not made obvious anymore & then people might get screwed up in their inferences, at least the practical ones they might make about the sort of religiosity/ideology interaction featured in the original analyses.
That's pretty ugly, isn't it? The scatter plot isn't adding any helpul info in my view. So I think it is fair to present just the lowess plot if one is trying to enable visualization of the "raw" data (there's no such thing, is there?, as "raw" data):
I think that would have been fine for the "raw data" reported in the post, too--although I myself think it is entertaining to see orange & blue dots; I think I "felt" the effect a bit more that way! (Hey, did you know that in Feynman's mind's eye, numbers had colors?? I think they do as well when one isn't wearing one's Gelman cup & ... you know.)
Now, here's the $1 millon question: If we were to bother modeling these data, would this be creating any sort of misimpression?
It's just an alternative to ...
I won't tell you what I think, so as not to color (as it were) your thinking. But which of those two do you prefer?!
Wow! My graphic really was "not wonderful"!
Obviously we can't know what they are thinking exactly, but the judges seemed mesmerized by this...
I can't read Anoneuoid's mind -- even when he/she reveals what's on it, he/she [maybe even it; I'm starting to think Anoneuoid might be an entry in this yr's Loebner competition, and is just sort of warming up w/ competition for the "Cup"] is pretty enigmatic.
But likely he/she it would appreciate that this graphic also conserves what Anneuoid clearly views as important information about density of observations across the political ideology/religiosity space-- but not with frequency-weighted scatter plot markers (which I, anyway, think misrepresent the frequency of observations in interior of ideology/religiosity space, likely b/c weights aren't well calibrated) but with 3-d the contours.
And speaking for myself, it's definitely cool to have the information on *how* probability varies across the 2x2 space. That's an element that is missing from Anoneuoid's last effort. It's present in my scatter plot-- but mine is inferior to both Anoneuoid's & @AdamSchwartz in loss of information associated with continuous nature of the religiosity variable.
But here is one thing... How ware we to judge the precision of the probability estimates in the @AdamSchwartz graphic? We know that certain regions are less densely populated than others and will thus involve more "error"; but how much more?
Where/when does modeling, of sort that generates parameter estimates for the religiosity & left_right measures, come in?... And what is optimal then?
I suspect that the quality of play we are seeing here is going to draw "Nukey Thompson" into the fray as surely as Paul Newman's eye-popping snooker-play flushed out Jackie Gleason in the Hustler...
Uh oh, now things are really heating up.
Provoked, apparently, by what he regards as the possibility that judges would be enchanted by the infovis magic of @Adamschwartz, and by the attempt of "Lying Ted" Anoneuod to win by stuffing the Cup ballot box, @JoeHilgard (aka "Bootstrapless Joe") has submitted 3 entries:
He supplied his R code, too.
For sure entry 2 conveys more information about relative density of observations than my massively overlplotted scatter plot (jittering didn't help much). Anoneuod's 2d entry was doing that in same say: with weighted markers. But speaking for myself (who knows what the judges will think), I think "Bootstrapless Joe's" 2d graphic is better b/c it makes it so much easier to see how changes in ideology affect the likelihood of belief in human-caused climate change & how that interacts (but in the end in a way that is likely of no practical consequence) w/ religiosity.
... Now if there were only a way to combine the nice features of "Boostrapless Joe's" 2d graphic w/ the all the extra information that Anoneuod's graphic -- by treating religiosity as one of the axes -- has on impact of religiosity at all levels & not just 2 or (as in "Bootstrapless's" 3d entry) 3... That's what I guess @Adamschwarz is aspiring to, but I did overhear some chatter that it was not "intuitive..."
Anyway, "Bootstrapless Joe" really wants that cup!
"Bootstrapless Joe" refuses to let up. Apparently, he is anxious that @Adamswartz's virtual-reality-inducing graphic might earn him the Cup, so @JoeHilgard is churning out still more entries!
Or how about yummy probability density distributions??
When I was a kid, these were the only thing I'd ever eat!
Click on it for a bigger view.
[BTW: @1RonanConnolly spotted glitches in the original rendering of the PDDs-- another reason to graph the data: to catch coding errors & like in generating predicted values! Oy!]
I used a MC simulation to generate them based on the same logistic regression model that was being graphically illustrated/reported w/ my spikey CIs and now with @Ronan's good-old CI bands
I've identified the PDD values that bound the 0.95 range for the model's predicted probability. But the whole point of PDDs is that it's ridiculous to use a "signficance" statistic. Show the precision of the model estimate in a way that enables a reflective person to form her own attitude about it.