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Many clinicians and perhaps some
statisticians are at odds regarding the correct application of Bayes
theorem in integrated risk assessments of screening programs for Down
syndrome1. Most standard textbooks show that the posterior odds = prior
odds X likelihood ratio but some publications show the use of
prior risk X likelihood ratio to calculate the posterior risk. Bayes
theorem does refer to probabilities, which is equivalent to the word
"risk". The confusion seems to arise for this reason and the
fact that for low risks the results are very similar. Some authors use
risk loosely as a generic term which might include odds. However the proof is quite trivial and we
show here clearly the correct way according to the work of Bayes. The
Royal Society published an article in 1763 entitled "An Essay towards
solving a Problem in the Doctrine of Chances" by the Reverend Thomas
Bayes (Philosophical Transactions of the Royal Society, Volume 53, pages
370-418, 1763). The article was found in Bayes" papers after his
death, and published posthumously. In it Bayes develops his famous theorem
about conditional probability: In other words, the probability of some
event A occurring given that event B has occurred is equal to the
probability of event B occurring given that event A has occurred,
multiplied by the probability of event A occurring and divided by the
probability of event B occurring. Bayes theorem states:
Let"s look at a collection of red and
white ball, some of which are large and some small. Let us say there are
more small white than large white. Then by measuring the size of the ball
we can improve our chances of correctly guessing whether the ball is red
or white. Now start with 50 red and 50 white. Without knowing how the
colors are distributed between the large and small balls we have a
probability of 0.5 picking a white ball. (Prior probability) If we are
told that all the small balls are white (the equivalent of a likelihood
ratio), then knowing the size will allow us to determine whether or not we
have picked a white or red (posterior probability). Imagine now that the
correlation between small and white is not perfect. Knowing how the size
and color is distributed in the population of balls, and the size of the
ball we have picked, will give us a better estimate of the color of the
ball. Bayes stated that the probability of a
white ball in a population of small balls is equal to the probability of
white balls in the total population multiplied by the probability of a
small ball within the white population divided by the probability of a
small in the total population.
In the context of the balls distribution shown above, Bayes theorem states:
From the table we can calculate the value of each of these probabilities as follows:
This is also called the posterior probability
i.e.
total white balls/total balls, also called prior probability
i.e.
total small balls/total balls
Using Bayes theorem to calculate the posterior probability P (w | s) from the prior probability
= (A+B)/(A+B+C+D) x A/(A+B) / (A+C)/(A+B+C+D)
This
is the correct answer for the posterior probability
Doing the calculations using ODDS
Posterior odds = prior odds x likelihood
ratio (LR) In the above example the
Using the prior ODDS, the posterior ODDS can now be calculated using the likelihood ratio:
This is the correct answer for the posterior odds Why does the confusion arise?
Thus it can be seen that to work out the posterior probability we must multiply the prior probability by
We are not aware of a name for this ratio but it is clearly not the same as the likelihood ratio.
The likelihood ratio (LR) is sensitivity / (1-specificity) and in the above illustration
How to apply the likelihood ratio in a clinical situation
While Bayes' Theorem is true for
"point probabilities", as shown above, in practice we are not
able to work with whole populations and a known distribution. We have to
infer the makeup of the population from samples of the population, and the
imprecision of the resulting estimates is represented by confidence
intervals. Let us look at a clinical problem. We want
to provide a risk assessment for Down syndrome based on maternal age. To
provide a simple example we are using age 35 or over as the cut off.
The risk of Down syndrome in the total
population is (A+B)/(A+B+C+D) this is the prior risk or prevalence of the
condition in the population under study. (In this case the whole
population.
Posterior odds = prior odds x likelihood
ratio
This result can then be converted back to probability if you wish.
The individual for whom we are trying to predict a probability has now been identified as coming from a sub-population. We can now apply a second test such as nuchal thickness.
The probability of Down syndrome in the new
population is (P+Q)/(P+Q+R+S). By applying the likelihood ratio for
"Thick NT" (using odds as explained above) we reach a new
sub-sub-population from which our individual comes and the posterior
probability for P/(P+R) can be calculated. Provided the tests are completely
independent of each other we can multiply each odds ratio by the
corresponding LR to get the new odds ratio. Posterior odds = prior odds x LR1 x LR2 x
LR3 x …….
However if there is some correlation
between any of the tests, the results using sequential likelihood ratios
will not be accurate. For example if NT increased with maternal age then
the NT test would not help to define the sub-population any better than
age had already done.
Using maternal age as described above is too crude to be helpful. We really need to have a likelihood ratio for all maternal ages to apply to the prior probability. Similarly with NT we need a continuous range of likelihood ratios for all the possible NT’s. We need to know how the likelihood ratio and the test under question are correlated. How the measurements are distributed among the normal population and how they are distributed in the Down population will determine the likelihood ratio for each measurement.
Look for more information and calculators on Dr. Hutchon web site. References 1. Hutchon DJR Absence of nasal bone and
detection of trisomy 21 (letter) The Lancet 2002;359:1343 2 Hutcon DJR Trisomy 21:91% detection
rate using second-trimester ultrasound markers.(letter) Ultrasound Obstet
Gynecol 2001;18, no 1:83 |
How to use Bayes theorem to
estimate sequential conditional risks. 