Chart/visual for negative binomial regression

Question

My wife is presenting a study at a conference poster session. She has a correlation of sorts [1] where kids' self-assessment of asthma ("I felt a lot better" or "I felt a little better or not better") matches doctors' assessments (0 to 6, low numbers are better). And we need a visual for the poster.

I'd have thought that a correlation graph would be the way to go, but the x- and y-values are both on discrete scales, so if you plot them, (1) there is no cloud of values to see and (2) there is no obvious trend.

Here are the buckets (value + count of occurrences) for people who felt a lot better:

>>> print(a_lot_better)
{0: 66, 1: 9, 2: 3, 3: 2}

And here are buckets for people who felt a little better or not better:

>>> print(a_little_better)
{0: 79, 1: 31, 2: 5, 3: 6, 6: 1}

If you sum up the numbers like this:

# python
def print_dict(d):
    total = float(sum(d.values()))
    composite = 0.0
    for k, v in sorted(d.iteritems())
        print("{0}: {1}".format(k, v / total))
        composite += (k * v) / total
    return composite

You get this:

>>> print_dict(a_lot_better)
0: 0.825
1: 0.1125
2: 0.0375
3: 0.025
0.2625

>>> print_dict(a_little_better)
0: 0.647540983607
1: 0.254098360656
2: 0.0409836065574
3: 0.0491803278689
6: 0.00819672131148
0.5327868852459017

And that is more obviously different, but I don't know how to present this visually (assuming even that this is a legitimate summary).

How should we present this data to be visually obvious that there is an interesting and significant relationship?

[1] As broadly understood by a statistics-civilian.

Answer 1

The heading "negative binomial regression" is a puzzle here as the distribution is defined only on the grades 0, ..., 6, so can't be negative binomial on that ground alone.

However, your calculation of the discrete probability distribution strikes me as exactly right and that suggests showing two histograms, which is only a step away from your probability distributions. It would be widely conventional to show bars with a little space between them to show that the scale is discrete. In fact some would go further and prefer the bars to be shrunk to spikes.

In the thread pointed to by Dimitriy Masterov, the proposals of table-like plots are essentially identical in spirit.

A Mann-Whitney test is quite encouraging here.

This isn't, in my view, a correlation problem at all. You have two distributions of ordinal (graded) variables.

Answer 2

Most important is to figure out what the message you want to communicate is and try to find a graph whose message is close to that.

Some graphs that might spur some ideas:

Comparing bar charts (aka discrete histograms) shows the different distributions but perhaps misleads since score=0 has a higher bar for "little better" than "lot better", though it's relatively less common.

A mosaic plot (aka spine plot) makes the population difference and the relative sizes clearer but it's harder to make out the distribution shape.

Overlaid smooth distribution curves emphasizes the different distribution shapes but with a loss of detail.