Sunday, June 30, 2019

Bayesian estimation of severity in police use of force

In research reported in the journal Law and Human Behavior, Brad Celestin and I used Bayesian methods to measure perceived severities of police actions. For each of about two dozen actions, we had lay people rate the action's moral acceptability, appropriateness, punishability, and physical forcefulness. We regressed the ratings on the actions, simultaneously estimating latent scale values of the action severities. 

Below is a stylized graph to show the idea. The vertical axis shows the ratings, and the horizontal axis shows the underlying (latent) severity of the actions. In this graph, six actions are placed at arbitrary positions on the horizontal axis.

Below I've superimposed the regression equation. It's just linear regression, but the values of the predictors are estimated, not given.

Below is a stylized representation of the latent scale values that best fit the ratings:

Bayesian methods were especially useful for this because we obtained a complete posterior distribution on all the scale values. Bayesian methods were also very useful because the ratings were effectively censored by many respondents who pushed the response slider all the way to the top or bottom, so all we could discern from the response was that it was at least that high or low; censored dependent-variable data are handled very nicely in Bayesian analyses.

Here's the abstract from the article:
In modern societies, citizens cede the legitimate use of violence to law enforcement agents who act on their behalf. However, little is known about the extent to which lay evaluations of forceful actions align with or diverge from official use-of-force policies and heuristics that officers use to choose appropriate levels of responsive force. Moreover, it is impossible to accurately compare official policies and lay intuitions without first measuring the perceived severity of a set of representative actions. To map these psychometric scale values precisely, we presented participants with minimal vignettes describing officer and civilian actions that span the entire range of force options (from polite dialogue to lethal force), and asked them to rate physical magnitude and moral appropriateness. We used Bayesian methods to model the ratings as functions of simultaneously estimated scale values of the actions. Results indicated that the perceived severity of actions across all physical but non-lethal categories clustered tightly together, while actions at the extreme levels were relatively spread out. Moreover, less normative officer actions were perceived as especially morally severe. Broadly, our findings reveal divergence between lay perceptions of force severity and official law enforcement policies, and they imply that the groundwork for disagreement about the legitimacy of police and civilian actions may be partially rooted in the differential way that action severity is perceived by law enforcement relative to civilian observers.
A preprint of the article is here, and the published article is here. Full citation:
Celestin, B. D., & Kruschke, J. K. (2019). Lay evaluations of police and civilian use of force: Action severity scales. Law and Human Behavior, 43(3), 290-305.

Sunday, May 19, 2019

The Statistician's Error?

I just attended (and gave a talk at) the United States Conference on Teaching Statistics (USCOTS). Big thanks to Allan Rossman, who brilliantly MC-ed the conference.

• One keynote was about "moving beyond p < .05" in a talk by Ron Wasserstein and Allen Schirm, In their recent editorial in The American Statistician (with Nicole Lazar), a primary recommendation was Don't Say "Statistically Significant". Decisions with p values are about controlling error rates, but dichotomous decisions let people slip into "bright line" thinking wherein p < .05 means real and important and p > .05 means absent and unimportant.

• Another keynote, in a talk by Kari Lock Morgan, was about three possible explanations of an apparent effect of a manipulation, namely (i) genuine cause, (ii) random difference at baseline before manipulation, and (iii) random difference after manipulation.

I returned home from the conference this morning. To relax, after the intensive pre-conference preparation and during-conference insomnia, I opened a book of poetry and came across a poem by Aaron Fogel that (inadvertently) reflects upon both talks. It's a poem about how editors of printing make decisions regarding errors, and about three sources or errors, and distinguishing the sources of error. And about the role of editors (and perhaps of statisticians?).

The Printer's Error
Fellow compositors
and pressworkers!
I, Chief Printer
Frank Steinman,
having worked fifty-
seven years at my trade,
and served five years
as president
of the Holliston
Printer's Council,
being of sound mind
though near death,
leave this testimonial
concerning the nature
of printers' errors.
First: I hold that all books
and all printed
matter have
errors, obvious or no,
and that these are their
most significant moments,
not to be tampered with
by the vanity and folly
of ignorant, academic
textual editors.
Second: I hold that there are
three types of errors, in ascending
order of importance:
One: chance errors
of the printer's trembling hand
not to be corrected incautiously
by foolish professors
and other such rabble
because trembling is part
of divine creation itself.
Two: silent, cool sabotage
by the printer,
the manual laborer
whose protests
have at times taken this
historical form,
covert interferences
not to be corrected
censoriously by the hand
of the second and far
more ignorant saboteur,
the textual editor.
Three: errors
from the touch of God,
divine and often
obscure corrections
of whole books by
nearly unnoticed changes
of single letters
sometimes meaningful but
about which the less said
by preemptive commentary
the better.
Third: I hold that all three
sorts of error,
errors by chance,
errors by workers' protest,
and errors by
God's touch,
are in practice the
same and indistinguishable.
Therefore I,
Frank Steinman,
for thirty-seven years,
and cooperative Master
of the Holliston Guild
eight years,
being of sound mind and body
though near death
urge the abolition
of all editorial work
and manumission
from all textual editing
to leave what was
as it was, and
as it became,
except insofar as editing
is itself an error, and
therefore also divine.

Wednesday, March 13, 2019

Bayesian statistics at Princeton University, with a visit to the grave of John Von Neumann

I was very pleased to have had the opportunity to present a talk regarding Bayesian models of ordinal data at Princeton University, on Tuesday, March 5, 2019. An abstract of the talk is here, and the published article on which the talk was based is here.

A big thanks to Dr. Ting Qian who orchestrated the visit wonderfully. The lecture hall was packed, with people standing outside the door in the hall, in no small part due to Ting's organization of a popular statistics series. Big thanks also to the staff who made arrangements. And, of course, thanks to the many people who took time to meet with me while I was there.

I took a few hours the next morning to visit some special places in Princeton. In particular, I visited the graves of John Von Neumann and Kurt Godel. Von Neumann made contributions to Monte Carlo methods at the foundation of MCMC methods in Bayesian analysis.

Here's a snapshot of Von Neumann's grave stone:

And only a few feet away, Godel's grave stone:

After visiting the cemetery, I went to the Princeton University Art Museum and came across this painting by Georgia O'Keeffe. It's titled, "From a New Jersey Weekend II," painted in 1941.
It was an interesting coincidence that both O'Keeffe's and my visits featured two notable grave stones.

Here's a map of the Princeton Cemetery, with a marker at the approximate location of the graves of Von Neumann and Godel.

For posts about the book, DBDA2E, visiting other famous grave sites, see this post and its links to previous posts.

Thursday, February 21, 2019

A Stendhal moment on the way to Bayesian stats class

On the way to my Bayesian stats class this morning I had a few moments of Stendhal syndrome and thought I'd share it with y'all. (Aside from being on the way to Bayesian stats class, it has nothing directly relevant to Bayesian statistics.) The sun was shining through some construction fences lining a sidewalk and were unintentionally beautifying an otherwise routine Thursday morning. Here are a couple of snapshots from my office window:

It reminds me of Christo and Jeanne-Claude's Running Fence:

Sunday, October 21, 2018

Sinusoidal trend and global warming UPDATED

In a previous post from six years ago, I fit a sinusoidal trend, with auto-regressive component, to daily temperature data. (Spoiler alert: It's still getting warmer.) Recently I've received inquiries about the script for that analysis. I disinterred the ancient script, updated it, and grabbed more recent temperature data. The script and data file are linked below.

The result of the new analysis:
As you can see from the plot (above), the slope of the linear spine of the sinusoidal variation is 0.068 degrees Fahrenheit per year. The 95% HDI on the estimate spans zero, just as it did with the smaller data set in the previous post from six years ago. But I'm pretty sure that if this city were put into a big hierarchical model with lots of other cities across the globe, the high-level estimate of slope on the linear spine would be clearly greater than zero.

But evidence for global warming is not the point of this post. The point is to link the full script and data file. Here they are: R script; data. Hope this is helpful.

Wednesday, September 19, 2018

Which movie (treatment, group) is better? Opposite conclusions from different models.

Which movie is better? One way to answer is by considering the star ratings given to those movies. Just treat those 1-to-5 star ratings as numbers, throw them into a t test, and out pops your answer. Right? Not necessarily...

The analogous structure arises in many situations. Suppose, for example, we ask which group is happier, a group of poor people or a group of rich people? One way to answer is by considering subjective happiness ratings from an ordinal scale: 1 = very unhappy, 2 = mildly unhappy, 3 = neither unhappy nor happy, 4 = mildly happy, 5 = very happy. Just treat those 1-to-5 ratings as numbers, throw them into a t test, and out pops your answer. Right? Not necessarily...

Or, consider ratings of symptom intensity in different treatment groups. How bad is your headache? How depressed do you feel? Just treat the ratings as numbers and throw them into a t test, and out pops your answer. Right? Not necessarily...

Treating ordinal values as if they were numeric can lead to misinterpretations. Ordinal values do not indicate equal distances between their levels, nor equal coverage of each level. The conventional t test (and ANOVA and least-squares regression, etc.) assumes the data are metric values normally distributed around the model's predicted values. But obviously ordinal data are not normally distributed metric values.

A much better model of ordinal data is the ordered-probit model, which assumes a continuous latent dimension that is mapped to ordinal levels by slicing the latent dimension at thresholds. (The ordered-probit model is not the only good model of ordinal data, of course, but it's nicely analogous to the t test etc. because it assumes normally distributed noise on the latent dimension.)

The t test and the ordered probit model can produce opposite conclusions about the means of the groups. Here's an example involving star ratings from two movies:

The figure above shows data from two movies, labelled as Cases 5 and 6 in the first two columns. The pink histograms show the frequency distributions of the star ratings; they are the same in the upper and lower rows. The upper row shows the results from the ordered-probit model. The lower row shows the results from the metric model, that is, the t test. In particular, the right column shows the posterior difference of mu's for the two movies The differences are strongly in opposite directions for the two analyses. Each posterior distribution is marked with a dotted line at a difference of zero, and the line is annotated with the percentage of the distribution below zero and above zero. Notice the ordered-probit model fits the data much better than the metric model, as shown by the posterior predictions superimposed on the data: blue dots for the ordered probit model, and blue normal distributions for the metric model. (This is Figure 8 of the article linked below.)

Read all about it here:

Published article:

Preprint manuscript:

R code:

Friday, September 7, 2018

Analyzing ordinal data with metric models: What could possibly go wrong? (just published)

Just published:

Analyzing ordinal data with metric models: What could possibly go wrong?
We surveyed all articles in the Journal of Personality and Social Psychology(JPSP), Psychological Science (PS), and the Journal of Experimental Psychology: General (JEP:G) that mentioned the term “Likert,” and found that 100% of the articles that analyzed ordinal data did so using a metric model. We present novel evidence that analyzing ordinal data as if they were metric can systematically lead to errors. We demonstrate false alarms (i.e., detecting an effect where none exists, Type I errors) and failures to detect effects (i.e., loss of power, Type II errors). We demonstrate systematic inversions of effects, for which treating ordinal data as metric indicates the opposite ordering of means than the true ordering of means. We show the same problems — false alarms, misses, and inversions — for interactions in factorial designs and for trend analyses in regression. We demonstrate that averaging across multiple ordinal measurements does not solve or even ameliorate these problems. A central contribution is a graphical explanation of how and when the misrepresentations occur. Moreover, we point out that there is no sure-fire way to detect these problems by treating the ordinal values as metric, and instead we advocate use of ordered-probit models (or similar) because they will better describe the data. Finally, although frequentist approaches to some ordered-probit models are available, we use Bayesian methods because of their flexibility in specifying models and their richness and accuracy in providing parameter estimates. An R script is provided for running an analysis that compares ordered-probit and metric models.

Fig. 4Ordinal mean as a function of latent mean (mu) and SD (sigma). Groups marked Ⓐ and Ⓑ illustrate a false alarm (Type I error) for which the underlying means are exactly equal but the ordinal means are very different. Groups marked Ⓑ and Ⓓ illustrate a miss (Type II error) for which the underlying means are quite different but the ordinal means are exactly equal. Groups marked Ⓒ and Ⓓ illustrate an inversion for which the underlying means have μD > μC but the ordinal means incorrectly have μC > μD.

Published article:

Preprint manuscript:

R code: