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Bayesian Decision Theory in Biostatistics: the Utility of Utility

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Bayesian Decision Theoryin Biostatistics:the Utility of Utility
David Draper
(joint work with
Dimitris Fouskakis
and
Ioannis Ntzoufras
)
Department of Applied Mathematics and StatisticsUniversity of California, Santa Cruz, USA
draper@ams.ucsc.eduwww.ams.ucsc.edu/
∼
draper
Bayesian Biostatistics ConferenceMD Anderson Cancer Center, Houston TX
28 Jan 2009
Bayesian decision theory in biostatistics 1
What Biostatisticians Do
The practice of
statistics
in general (and
biostatistics
in particular) can beroughly divided into
four activities
:
•
Description
of
available information
(e.g., one or more
data sets
)relevant to answering a
question
of interest, without an attempt to
generalizeoutward
from the available data;
•
Inference
about aspects of the
underlying process
that gave rise to the data;
•
Prediction
of
future data values
under
interesting scenarios
; and
•
Decision-making
(choosing an
action
from among the
availablepossibilities
, in spite of the
current uncertainty
about
relevantunknowns
), e.g.,
experimental
or
sampling design
.Description is largely
non-probabilistic
and relatively
uncontroversial
.
Two probability paradigms
are in
widespread use
today in biostatistics:
•
Frequentist
probability: Restrict attention to phenomena that are
inherently repeatable
under (essentially)
identical conditions
;
Bayesian decision theory in biostatistics 2
Use of Frequentist and Bayesian Probability in Biostatistics
then, for an event
A
of interest,
P
F
(
A
) is the limiting
relative frequency
withwhich
A
occurs in the
n
(hypothetical) repetitions, as
n
→ ∞
.
•
Bayesian
probability: numerical
weight of evidence
in favor of anuncertain proposition, obeying a series of
reasonable axioms
to ensure thatBayesian probabilities are
coherent (internally logically consistent)
.
Two facts
about these paradigms:
•
With the
frequentist
approach,
inference
is
much easier
than (good)
prediction
and
decision-making
.
•
For several reasons (e.g.,
computing technology
), the
frequentistparadigm dominated work in biostatistics
in the
20th century
.An
unpleasant by-product
of these two facts is thatIn
biostatistical work
it’s a
common practice
to use
frequentistinferential tools
, such as
hypothesis testing
and
beneﬁt-onlyvariable selection methods
, for
decision-theoretic purposes
forwhich they
may not be optimal
.
Bayesian decision theory in biostatistics 3
Three Examples
In this talk I’ll give
three examples
of how
thinkingdecision-theoretically
can lead to
better results
.
•
Variable selection
in
generalized linear models
is a familiar task that isusually accomplished in what may be termed a
beneﬁt-only
manner: we try,
using inferential tools
, (e.g.) to ﬁnd a
subset
of the available predictorsthat
maximizes predictive accuracy
on
future data
.This ignores the
cost of data collection of the predictors
, which may
varyconsiderably
from one variable to another;
Bayesian decision theory
withan
appropriate utility structure
can improve on this.
References:
— Fouskakis D, Draper D (2008). Comparing stochastic optimization methods forvariable selection in binary outcome prediction, with application to health policy.
Journal of the American Statistical Association
, forthcoming.— Fouskakis D, Ntzoufras I, Draper D (2009). Bayesian variable selection usingcost-adjusted BIC, with application to cost-eﬀective measurement of quality of healthcare.
Annals of Applied Statistics
, forthcoming
Bayesian decision theory in biostatistics 4
Three Examples (continued)
— Fouskakis D, Ntzoufras I, Draper D (2009). Population-based reversible jumpMCMC for Bayesian variable selection and evaluation under cost constraints.
Journal of the Royal Statistical Society, Series C
, forthcoming.
•
When a
clinical trial
has been
adequately planned
(
“appropriately powered”
), as far as
sample size
is concerned, tobring the notions of
clinical
and
statistical signiﬁcance
into
goodagreement
with respect to its
primary objectives
, it may well still betrue that it is
“underpowered” for secondary subgroup analyses
.The use of
frequentistmultiple comparisons(inferential) methods
in such situations — e.g., to make
choices
about whether to run
newtrials
on the
promising subgroups
— is a
bad idea
that cannevertheless be seen in the
literature
(e.g., in a published trial I’m nowreanalyzing, assessing the
eﬃcacy of an HIV vaccine
).
Bayesian decision theory in biostatistics 5

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