Re: Ordinal logistic regression and the relative risk

On Thu, 30 Nov 2006 09:08:17 -0600, David Winsemius
<doe_snot@xxxxxxxxxxx> wrote:

Richard Ulrich <Rich.Ulrich@xxxxxxxxxxx> wrote in
news:2drsm214rehbh89ra7sduf0lf1ebanlr7d@xxxxxxx:

- just a few comments -

On Wed, 29 Nov 2006 21:16:05 -0600, David Winsemius
<doe_snot@xxxxxxxxxxx> wrote:
[snip, some]
RU> >
So long as the 'risks' as low, they work out like the OR.
Risks in a single period are usually very low....
DW> >
The point is that most analysts _prefer_ the risk estimate and the odds
ratio is only a marginally acceptable approximation. In the low frequency
situation the distortion induced by using odds rather than risk is small
and not a problem, but it is the odds estimate that most would consider
to be a distorted scale of risk.

I can see where the OR is 'distorted' -- I was really nonplussed
when I first heard of it, because the RR seemed so natural to
talk about. However, it did not take me very long to be sold
on it. The RR happens to be, relatively speaking, intractable
to modeling -- very inconvenient.

RU >
The main appeal of the RR is that it represents a
simple intuitive description of risk -- even though it
is relatively intractable mathematically, compared to
the OR (and it tends to be misleading when the values
are large).
DW> >>
What is so intractable about relative risks? Proportional hazards
regression of your favorite flavor will estimate coefficients that
are easily converted to risk ratios.
RU> >
Yes. For simplicity -- Do all your computations, combining
ratios across cells and computing tests, as ORs; and
then convert. But why convert?
DW> >
You don't seem to be reading for meaning. I said nothing about combining
ratios across cells.

I guess I was jumping ahead. Every 'modern' technique
combines cells, such as time periods. The OR is a natural
metric for logistic growth models, death models, etc., which gives
consistent estimates despite base-rates. In some cases, the Normal
provides a better metric (with the probit), and in those cases, the
OR gives a fine approximation in the mid-range. Relative risk is
*never* a natural metric -- it would never be expected to give
similar answers for tables with different base rates.

Either a Cox proportional hazards model or a Poisson
regression model gives beta coefficients that when exponentiated are
relative risks or hazards ratios. (The same maneuver done with betas from
logistic regression.) There is no intermediate of odds ratios. The only

The logistic unit and OR is implicit in logistic regression, so, YES,
the intermediate exists in LR. Poisson regression expects
low rates, always, in order to be 'Poisson'. The Wikipedia page
mentions that RR is a natural description for Poisson regression,
so, on considering the low rates, I must concede that there
still is a place for RR.

What I notice is that the Cox proportional hazards model will
not be especially efficient if the periods are short enough
that the hazards are large in a single interval.

You always need population base-rates to convert relative hazards
to RR. And then, the result only *applies* to populations with
that base-rate. That makes two immediate drawbacks to the RR.

reason to use odds ratios is in the situation where a case-control study
design forces you to use logistic regression. In the early days of
biostatistics, logistic regression was all you had. That did not mean
that odds ratios were the preferred effect measure, merely that they were
readily available output from a technology that was well understood at
the time. When better tools came along, people switched _because_ they
yielded estimates of risks and relative risks.

I rather thought that one of the earliest influences for
spreading the OR was the Mantel-Haenszel test for combining
2x2 tables, in 1959. The resulting statistic was *labeled* a
"Relative Risk" -- I've always assumed that they were meeting
public expectations by giving a RR, and were sneaking in the
OR because of its superiority.

- I just checked http://en.wikipedia.org/wiki/Relative_risk ,
which is pretty good. It points out 'low rates' for RR.
It points out RR is natural for describing Poisson regression.



Also, on rare occasions, one does want to
speak about the RR, mainly for making a point that
is more political than 'statistical'. That is, the RR of
90% *sounds* nearly as high as one of 95%, even though
the OR is about 2.0.

Aren't you conflating excess risks with relative risks? If the OR was
2, then the RR will be somewhat lower, but could not possibly be
below unity.

Oops. The OR in one direction is 2.1 (19/9) compared to 1.06 (19/18);
in the other direction, .47 compared to .95. The advantage of
saying "1.06" or ".95" is rhetorical, essentially *disguising*
the effect, if someone cares about the other statement.

As you can see, I still think I laid out the issues the
first time.

What issues? You started by disagreeing with the OP and then gave
incorrect information (about most analysts preferring OR as an effect

I don't have a count, but what I see reported in clincial
literature that I read is almost always the OR; occasionally,
it is mislabeled. I suppose that I should say,
"Analysts dealing with rates that are not always low
should stick with the OR as the effect measure, over RR."

measure over RR, and RR only being "good" when the risk was low) and

- I'll surely stick with that, the RR being hazardous when
the risk is not low...

finished up with a glaring error about some sort of example that you did
not describe and which still remains obscure. Even looking at your effort
at clarification, I remain unable to figure out what sort of data
situation you are positing, or what you mean by "other direction".

- I thought that 'direction' was completely obvious in a discussion
of these ratios taken abstractly. Thus: Comparing .90 vs .95 gives
relative risk of .95; comparing .95 vs .90 (opposite direction) gives
a relative risk of 1.06. And so on.

As m00es points out, if you measure the *other* condition, (1-A)
instead of (A), the RR is not constant -- .10 vs .05 is 2.0, which
*looks* very different from the same relation described as 1.06
or 0.95. Whereas, the OR is constant.

--
Rich Ulrich, wpilib@xxxxxxxx
http://www.pitt.edu/~wpilib/index.html
.

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