This morning, my good friend Orac sent me a link to an interesting piece
of bad math. Orac is the guy who really motivated me to start blogging; I
jokingly call him my blogfather. He’s also a really smart guy, not to mention
a genuinely nice one (at least for a transparent box of blinking lights). So
when he sends me a link that he thinks is up my alley, I take a look at
the first opportunity.
Today, he sent me a link to a guy who claims to have put together
a mathematical model showing that it’s impossible to create a national
healthcare system without rationing. The argument is a great example
of what I always say about mathematical modeling: you can’t just
put together a model and then accept its results: real mathematical models
must be validated. It’s easy to put together something that looks
right, but which produces drastically wrong results.
The common way of saying it is “Garbage In, Garbage Out”. I personally
don’t like that way of describing it – because in the most convincing examples
of this, it looks like what’s going in isn’t garbage.
One way of looking at a mathematical model of something like healthcare is
as a dynamical system. Each point in the phase space of that system is
a possible situation. The model predicts how that will evolve over time. The
GIGO argument makes it sound like what matters is: which point do you choose
to describe today’s situation? If you choose the right starting point, then
run it forward, you’ll get the right end point. But the evolution function is
also important: if your evolution function doesn’t accurately model how the real
world changes over time, then the end point will be wrong, no matter how perfectly
you selected the starting point.
The selection of the evolution function is a part of the modeling process. You can
(and probably should) view it as an input to the model – in which case GIGO is
absolutely right. But most people’s intuitive understanding of GIGO doesn’t do that.
Anyway… I’m getting positively Oracian here, babbling on this long without getting to
real point. (Except, of course, that I’m not doing it nearly as well as Orac. Anyone else think
it’s unfair that a lucite box of blinking lights writes with more style and eloquence than me?).
The article is called There’s Not Enough Waste and Inefficiency in Healthcare, by a guy who writes under the name “DrRich”.
His model, as he describes it, is based on four assumptions. I’ll tell you about them,
and then I’ll try to explain both why the look right, and why they’re actually completely wrong.
I’m not going to quote him precisely – I’m going to rephrase in a way that makes the
problem more obvious – but I recommend that you look at his original article to verify
that I’m not misrepresenting his argument.
- The proportion of healthcare spending that is wasteful is currently 25%.
- The annual rate of growth of healthcare spending is constant, and is
- The annual growth rate of non-wasted healthcare spending is the same as the current
- The difference between the 10% growth rate and the non-wasted spending is
He admits that the first point is a total wild-ass guess. And that’s fine. No one is really
sure of what the correct number is, and for the sake of argument, a nice round number
like 25% is a reasonable starting point.
The second point is where things start to go awry. Health-care spending
growth isn’t constant – and that should be completely obvious. The growth
in health-care spending has been absolutely ridiculous in recent times –
but it varies enormously, depending on economic conditions. Exactly how to
predict it is very uncertain – no one is really sure exactly which variables
influence it. But we know that there are lots of factors, which push and pull
the rate in different directions. For example, in bad times, people delay
treatments – which can decrease some costs. Bad economic conditions also
increase the incidence of many stress related disorders, which can increase
some costs. Health care spending is also affected by weather, by natural disasters,
by medical innovations, by the average age of the population, and by the size of the
population. Given all that uncertainty, there’s one thing that’s very clear: the
rate of increase of health-care spending is not constant, and its
relationship to the economic environment that surrounds
it is non-linear.
The third point is also not correct. It sounds
reasonable. In fact, if you measure health care spending
per capita, then in the long term, the increase in healthcare
spending per year must eventually level out at the rate of inflation. But
that’s not what DrRich is saying. He’s saying that the total
rate of increase of healthcare spending should match inflation. And
that’s just wrong. In fact, it’s totally ridiculous. Once again,
DrRich is trying to model things using a single-variable linear model
for something that is manifestly not a single-variable linear phenomenon. And
any attempt to validate that model, by comparing it to real observations
will make it abundantly clear that the model is total rubbish.
Ignore for the moment that healthcare spending per year is highly
variable and, likely, chaotic. After all, in the long term, the chaotic
factors should damp out. (You can see it as being very similar to weather:
short-term, it’s chaotic. Long term, it can’t be.) In the long-term, what
we should expect is that on average, the rate of increase of
non-wasteful medical care would, roughly, match the rate of
economic growth in the economy, not the rate of inflation. That,
right there, is a major flaw. The population is, at the moment, constantly
increasing. That means that if health care prices stayed exactly the
same, total healthcare spending would increase every year, because
the number of people be treated increases. But by that same argument, the
economy should also grow because the number of productive workers increases
with the population. If you take a conservative model of health-care spending
increasing at a rate based on just inflation plus population growth,
but you claim that the rate of non-wasteful healthcare spending grows with
inflation, then you’ll wind up with a picture where the percentage of
healthcare spending that is going to waste/inefficiency is increasing,
without bound, every year. In fact, you’ll find that in fairly short order,
it’s pretty much 100% of cost-growth.
And – surprise! That’s exactly what DrRich’s model shows. From
that, he concludes that the rate of growth in healthcare spending
cannot be blamed on waste.
The evolution function that he chose for measuring the portion of
increase in healthcare spending that’s wasteful is totally bogus. And so,
even if the numbers that he puts into his model are absolutely, 100%
correct, his conclusions can’t be.
How much of health-care growth is really due to waste? I don’t
know. I haven’t sat down and tried to model it. But the experience
of numerous other countries, using a variety of public, private, or
hybrid universal insurance systems without experiencing the kind of
cost-growth that we see in the US is a pretty good indicator that
it’s not unmanageable. But again – I don’t know. Someone
who knows more about health care spending than I do could put together
a model that tries to show how much is wasted, and what the rate of
growth of wasteful spending really is. But it’s not an easy job. And
DrRich’s silly linear model based on the rate of inflation is clearly
not the right model.