Re: Basis for Taguchi Loss Function 2

["Jonathan Siegel" <jmsiegel@umich.edu>]

Some further thoughts on non-Taguchi loss functions:

In recent years, I have been working as a statistician on drug trials,
primarily in oncology. Many chemotherapies are essentially poisons which
hopefully kill cancer cells faster than the rest of the body. Using them
in the body is a risky undertaking. If one overshoots one may cause
anything from nausea and diarrhea through death; if one undershoots, any
residual cancer will grow exponentially. One of the first tasks in
introducing a cancer drug into human therapy is to calculate an
appropriate dose. 

The traditional approach is to start at a very low dose and go up in
fixed intervals until one reaches a dose which causes side effects that
injure patients too severely. The modern approach involves
Bayesian-influenced  methods based on loss functions, using mathematics
somewhat similar to the pricing of stock options and derivatives. Both
the financial and the medical applications involved address a similar
concept of how high to go before one should stop.

A useful loss function in cancer research must incorporate the practical
consequences of over- and under-dosing and must generally be guided by a
theory of ethics. 

One possible candidate for an ethical approach would be Hippocrates'
dictum "first above all, do no harm." The dictum cannot be followed
strictly, however, because when therapy is risky the only way to
guarantee doing nothing harmful is to do nothing. One way one might
generalize Hippocrates's principle, however, would be to implement the
idea that doing too much is worse than doing too little. The idea
results in an asymmetric loss function, where the overshooting side has
a steeper curve than undershooting.

Hippocrates dictum, while a useful rule of thumb, does not turn out to
apply all the time, and cancer therapy can sometimes present a
counterexample. The practical consequences of undershooting and
overshooting are very different. Cancer which is not killed off grows
exponentially; side effects can sometimes be as little as mild nausea
and diarrhea, although extreme nausea and diarrhea can be deadly. 

Patients' perception of their losses varies. Some patients would rather
be treated to an inch of their lives in the hope that the damage will be
temporary and the therapy will work; others would rather deal with the
cancer than anything more than modest side effects of the drugs. It is
not necessarily for us to tell people what we think their losses should
be.

Cancer research also creates an additional problem with estimating.
People who enter the earliest phases of cancer research do so because
they have tried and survived other, unsuccessful therapies. Such people
are sometimes exceptionally tolerant of, and sometimes resistant to,
chemotherapies. For this reason, a useful loss function may need to take
into account and be somewhat insensitive to this tolerance. At the same
time, some people may be especially sensitive to the therapy, and if
this causes significant safety problems such people cannot be ignored.
Only a percentage of the people who try a therapy ever respond to it.
Therefore, one must consider the possible effect of the drug on outlying
minorities when considering a loss function. It will not do to simply
add or average everybody's. Some people's needs are more important than
others. Considerations of especially sensitive and tolerant groups will
generally result in an asymmetric loss function.

The surest approach would be an empirically-based loss function, a loss
function specific to the therapy and patient population which attempts
to assess the risks of overshooting and undershooting of the specific
therapy in accordance with the desires and ethics of the specific
population. 

We generally have to suffice with something much less than such a
method. It would take too long to gather the data needed. It is also
hard to ask people what they would do in a way that is easily
quantifiable and aggregable into a loss function. What one must do
instead is a series of judgment calls, basing the loss function on past
experience with the therapy and, especially at the beginning, therapies
one believes to be similar.

These considerations should help explain how important it is to base
loss functions on empirical criteria, evidence about practical
consequences and people's value systems. There is simply no substitute
for doing so. The fact that knowledge is imperfect, and that loss
functions are often based on guesstimations and judgment calls, does not
alter their empirical nature. Loss functions embody ones knowledge of
the practical and ethical consequences of ones actions. For a
statistician who follows the participant-observer paradigm, ones impact
is of critical importance. Thinking about ones impact means that loss
functions will often be asymmetrical. They will also often be
discontinuous. Irreversible damage is on a different plane from
reversible damage, and death is on a different plane altogether. 

I have personally seen the results of trials which had to be stopped
because the statistician devising the trial used an inappropriate loss
function, one based, not on careful observation or sound judgment about
the consequences of the experiment's actions, but on some sort of
abstract mathematical properties which either make the calculations
easier for the statistician, or were simply considered pretty. This can
result in serious damage to patients if the experiment overshoots, or
serious costs and possible cancellation of an effective product if the
drug is tried at too low doses and fails to show efficacy.

Easing the statistician's calculation burden is, when all is said and
done, simply not particularly important in the scheme of things. 

Sincerely,
 
Jonathan Siegel
jmsiegel@yahoo.com