## 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:
https://www.sciencedirect.com/science/article/pii/S0022103117307746

Preprint manuscript: https://osf.io/9h3et/

R code: https://osf.io/53ce9/files/