# [R] Diamond graphs, again.

Richard A. O'Keefe ok at cs.otago.ac.nz
Fri Sep 26 08:30:00 CEST 2003

```I have now obtained copies of all three medical papers that
the "Diamond Graphs" article based its examples on.

Figures 4 and 5: "Blood Pressure and End-Stage Renal Disease in Men".

The two predictor variables (systolic and disastolic blood pressure)
are not only continuous, they are correlated.  Recoding as some kind
of "size" (c1.diastolic + c2.systolic) and "shape" (maybe
log(systolic/diastolic) might have been interesting.

The real summary that I think anyone reading that paper would rely
on is not the 3d bar chart (figure 2) but a table (table 3) which
relates blood pressure category (optimal, normal, high-normal,
stage 1/2/3/4 hypertension) to adjusted relative risk (with 95%
confidence interval).

Reading the article, other (listed) factors also affected relative
risk, and it could have been useful to present some kind of multi-
dimensional table.

Comparing the original 3d bar plot and table with the diamond graph,
two things stand out:
(a) the higher the bar (= the bigger the hexagon), the *less* the
amount of data it is based on.  This can be seen very clearly
in the table; it cannot be seen at all in either the bar plot
or the diamond graph.  If I'm reading the article correctly
(hard, because the table and 3d bar plot don't use exactly the
same categories), the lowest bar is based on 40 times as muh
data as the highest bar (and the relative risk has a suitably
wide confidence interval).
(b) one would expect the risk to increase monotonically with
each predictor.  It doesn't.  This stands out very clearly
in the 3d bar plot.  It is very hard to see at all in the
diamond graph.  Once I saw it in the 3d plot, I could (just)
detect it in the diamond graph, but the diamond graph would
never have called my attention to it.

In fairness to both the 3d bar plot and the diamond graph, they
_could_ be made to show an equivalent of error bars.  Let the
bar (or hexagon) be coloured black from 0 to the lower end-point
of the confidence interval, then red (if colour is desired) or
grey (if it is not) from the lower end-point of the confidence
interval to the upper end-point, with a "black belt" at the nominal
value.

[Oh DRAT!  I could have patented that extension to diamond graphs!
Tsk tsk.  I'll never get rich, I'll always be 'ard up.]

Figure 7:  "Dual Effects of Weight and Weight Gain on Breast Cancer Risk".

In my previous message, I commented that I found it hard to believe
that weight *change* should be considered alone.  It wasn't.  In fact,
that's part of the point of the article.

I also commented that it seemed to me that the one categorical
predictor was probably a surrogate for a continuous variable.
Imagine me slapping my head and saying "but I _knew_ that!"

The explanation is in the editor's comment, not cited in the Diamond
Graphs paper, so here it is:
Editorial, "Weight and Risk for Breast Cancer",
Jennifer L. Kelsey & John Baron,
JAMA, November 5, 19997--Vol 278, No.17
The point is that weight and hormone treatment are *both* surrogates
for "lifetime estrogen dose profile".  In post-menopausal women,
female hormones _are_ still produced, in fat (which is an active
tissue).  I _knew_ that.  So in fact there is a _single_ explanatory
variable (some kind of weighted cumulative exposure) which both
hormone therapy and body mass index affect.  This raises the obvious
point that adding a third predictor (typical hormone levels during
years of fertility) might well be very informative.  But how would
diamond graphs cope with that?

But wait:  the abstract says "Higher [body mass index] was associated
with LOWER breast cancer incidence before menopause" but a "positive
relationship was seen among postmenopausal women who had never used
hormone replacement".  It also says "Weight gain after the age of 18
years was UNRELATED to breast cancer incidence before menopause but
was POSITIVELY associated with indicence after menopause".  The
editorial cited above makes this point also.

That is, in order to see the results of that study, you need a
display which
- shows weight
- shows weight change
- shows hormone therapy use
- distinguishes between breast cancer before menopause
and breast cancer after menopause.

The first sentence in the body of the paper is "The relation of
body weight to breast cancer is complex."

If there is an easy way to produce an "equiponderant display" with
three predictors on a two-dimensional piece of paper, I do not know
what it may be.  It's certain that diamond graphs, as described in
the TAS article, cannot do justice to the data from this study.

In contrast, the tables in the paper made the difference between
pre- and post-menopausal outcomes clear, and above all, included
confidence intervals.  Why do the confidence intervals matter?
Well, table 2 of the paper shows that the "multivariate-adjusted
relative risk" confidence intervals for premenopausal women all
contain 1 (with a fairly high p for trend), so there _might_ not,
on this evidence, be any effect at all, while the relative risk
confidence intervals for postmenopausal women all contain 1 except
for gains of 20kg or more (where the relative risk could be as low
as 1.2).  Since the study was based on 1000 premenopausal women and
1517 postmenopausal ones, while the effect is biologically plausible,
it doesn't appear to be anywhere near as strong as one might fear.

Once again, BOTH 3d bar plots AND diamond graphs are at fault for
not giving any indication of variability/noise/error bars/...,
and BOTH could be fiddled with to improve this.  In this case,
it is quite impossible to see from the diamond graphs in figure
7 of the TAS article what is quite clear from the tables in the
original source.

My background is AI, not medicine, so I came to these articles with a
"machine learning" bias.  I was expecting to see models trained on a
subset of the data and evaluated on another subset (cross-validation).
None of them did.  One of the many things to like about R that R makes
it comparatively easy to do cross-validation.

What have we seen as common themes?

1.  The so-called "categorical" variables were (in 5 out of 6 cases)
measured as continuous variables and then cut to quartiles or
quintiles or the like.

2.  More explanatory variables than 2 were considered in the sources,
and in each case more than 2 were actually important or at least useful.

3.  Presenting information without "error bars" can be seriously misleading.

How does R help?

1.  R lets us do scatter plots, smoothing, density estimation, &c.

2.  R gives us "lattice" plots, amongst others.

3.  We can construct graphs with error bars in R.

The big challenge seems to be graphical presentation of higher-
dimensional data, things like spinning plots, grand tours, &c.
And for that, there's Rgobi.

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