Some time ago I was allowed to discuss "Diamond Graphs", and whether
they would be useful in R, in this mailing list.
The August 2003 issue of The American Statistician has finally arrived
here and I have been able to read the article. A number of points of
interest arise.
1. The article is
"A Diamond-Shaped Equiponderant Graphical Display of the
Effects of Two Categorical Predictors on Continuous Outcomes"
by Xiuhong Li, Jennier M. Buechner, Patrike M. Tarwater,
and Alvaro M\~unoz.
The American Statistician, August 2003, pages 193-199.
(All of the family names are displayed in small caps except for
Xiuhong Li's. Does anyone know why?)
2. There are three examples in the paper.
a. Figures 2 and 3 display likelihood of developing AIDS
as the thing to be explained, with plasma HIV-RNA level
(measured in copies per ml) and degree of immune deficiency
(measured in count of CD4+ T-lymphocyte cells per cubic mm)
as the explanatory variables.
The explanatory variables are continuous, not categorical.
If the raw data were used, it would be possible to estimate
a 2d probability density and display that. The variables have
been made categorical by cutting to 5 levels each.
I managed to get a copy of the article this is based on. It
certainly isn't clear from the diamond graph that the viral load
categories were approximately quartiles, with the bottom
quartile split in two. So two of the viral load categories have
less data than the other three. Much the same happened with the
CD4+ levels, which is also not apparent.
Using a diamond graph instead of a density plot with rugs on the
margins hides the amount of data available for estimating the
cells (6 of the 25 cells are empty because there is missing data).
Viral load and CD4+ count were respectively the best and second
best predictor out of five predictors (seven are listed at one
point; I guess this means that CD3+ and CD8+ levels weren't useful
at all). I have skimmed the article a couple of times, and cannot
figure out why just two predictors were chosen. It would be of
interest to see graphs for one predictor, two predictors, and
three predictors. I have not yet seen any diamond graphs with
three explanatory variables...
b. Figures 4-6 display (age-adjusted rate of end-stage renal
disease due to any cause per 100,000 person-years) as the thing
to be explained, with systolic blood pressure (measured in mm Hg)
and diastolic pressure (measured in mm Hg) as the explanatory
variables.
Once again the explanatory variables are continuous, not
categorical. They are cut to 6 levels each. With the raw data,
one could perhaps get a contour plot of fitted disease rate
and a scatterplot of the explanatory variables on the same graph.
4 of the 36 cells are empty, but in this case the values in the
cells basically _are_ counts, so we are _not_ left wondering
how much data each cell is based on.
I have not yet seen the article this was based on.
c. Figure 7 has two graphs. On the left it's relative risk of
breast cancer as the thing to be explained, with adult weight
change (measured in kg; why not as a proportion of starting weight?)
and hormone use (never, past, current) as the explanatory variables.
On the right excess risk is to be explained, with the same
explanatory variables.
One of the explanatory variables is continuous. (Although it is
not obvious to me that a weight change of 10 kg in a 55kg woman
should have the same significance as a weight change of 10kg in
a 75kg woman.)
It strikes me that the other explanatory variable may well be
an approximation to a continuous predictor also (some kind of
exponentially weighted dose, perhaps).
I have not yet seen the article this was based on. I have seen
the abstract, though, which draws a conclusion somewhat at odds
with the apparent significance of these graphs. I expect that
this shows that the diamond graphs _are_ useful.
Like "a", we get no idea of how much data each cell is based
on.
In no case were there really two categorical predictors to start with.
3. I finally pinned down what these graphs remind me of: the two-way
plots described in Tukey's 1977 book "Exploratory Data
Analysis",
which is not cited in the article. Tukey's basic idea goes like this:
(1) Fit an additive model to the data (median polish, whatever).
(2) Tilt and spread the axes so that the vertical dimension is the
fitted values.
(3) Draw lines, not boxes, so that the fitted value for X=m Y=n
is the level at the intersection of the lines for X=m and Y=n.
(4) Now that you can see what the fitted values are from the
intersections, plot the residuals, either as sticks from the
intersection to the true value, or as variously sized/shaped
blobs to show the relative magnitudes of the residuals.
Once I realised this, I realised what really bothered me about these
graphs. They simply summarise the raw data (crudely). There is no
"data = fit + residuals". I found myself _itching_ for the raw
data
so that I could see what was really going on.
4. The paper compares a diamond graph (figure 5) with a trellis graph,
or rather, a pair of trellis graphs (figure 6) for the same data.
I felt much more comfortable with the trellis graph, largely because
the numbers (in the range 0..200ish) were well spread out. The
trellis graph was, however, much bigger, and is less immediately
accessible; the diamond graph conveys an impression of understanding
without needing a lot of explanation.
5. My analysis of perceptual issues was right in some respects and wrong
in others. The hexagons have been very carefully designed so that
- the area is proportional to p
- one length is proportional to p
- the difference of two other lengths is proportional to p
where p is the value in the interval [0,1] which is to be presented.
I must say that for me the visually most salient length is one which
is _not_ proportional to p (it's (1+p)/2).
6. The article neither presents nor cites any experimental data to show
that any task can be completed faster or more accurately using
diamond graphs than some other kind of display. As yet, it's a
matter of opinion.
7. It is unfortunate that the name "diamond graph" was chosen;
"diamond graph" is an established technical term in mathematics.
You could get much the same effect by plotting discs of varying sizes,
or sectors of varying width, or little thermometers, or practically
anything varying in area, on a standard horizontal & vertical table,
and then rotating the paper by hand. You won't be able to estimate
sizes accurately, but thanks to the cramped range available you aren't
going to estimate sizes accurately from a diamond graph anyway
(unless you can read the number displayed in the centre, which in my
photocopy of the article I can't).
In short, diamond graphs offer a reasonably clear way to summarise some
kinds of data, particularly for non-statisticians, but neither express
nor lead to any kind of analysis.
Since R is a statistics package rather than a "business presentation
graphics" package, perhaps Tukey-style two-way plots (are they already
available somewhere?) would be more useful than diamond graphs.
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.