Dept of Obstetrics and Gynaecology, Memorial Hospital, Darlington, UK.

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:

P(A | B) = P(A)  x (P(B | A)) / P(B))

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:

P (w | s) = (P (w) x P (s | w)) / P(s)

From the table we can calculate the value of each of these probabilities as follows:

(1) P (w | s) =   A/(A+C)       This is also called the posterior probability

(2) P (w) =  ____(A+B) / (A+B+C+D)         i.e. total white balls/total balls - - also called prior probability

(3) P (s) =  ____(A+C) / (A+B+C+D)             i.e. total small balls/total balls

(4) P (s | w) =  __A / (A+B)

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)

=   __ (A+B) x A x (A+B+C+D)

(A+B+C+D) x (A+B) x (A+C)

=   ___A / (A+C)    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 sensitivity white for small =   __A_/ (A+B)_

specificity white for small =    __D / (C+D)

1 - specificity = 1 - __D / (C+D)

= __((C+D) - D)) / (C+D)

=     __C / (C+D)

Therefore it can be seen that the likelihood ratio = ___A / (A+B)       divided by    ___C / (C+D)

likelihood ratio = __(A x (C+D)) / (C x (A+B))

Using the prior ODDS, the posterior ODDS can now be calculated using the likelihood ratio:

Prior ODDS = _(A+B) / (C+D)

Posterior odds = Prior odds x LR =    _((A+B) x  A x (C+D)) /  ((C+D) x C x (A+B))   =   A / C

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

__(A  x (A+B+C+D)) / ((A+B) x (A+C))__

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 the

likelihood ratio =      ___(A x (C+D)) / (C x (A+B))

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.

Applying the age "test" we identify a new sub-population which has a different prevalence. The post test probability is called the posterior risk which is A/(A+C). From published work we are given the likelihood ratio (LR) for age >35. This cannot be applied directly to the prior probability. The prior risk must first be converted into odds.

This is simply odds =        __probability / (1 - probability)_

Posterior odds = prior odds x likelihood ratio

This result can then be converted back to probability if you wish.

Probability =           odds / (1 + odds)

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