# Why clinicians are natural bayesians

BMJ 2005; 330 doi: https://doi.org/10.1136/bmj.330.7499.1080 (Published 05 May 2005) Cite this as: BMJ 2005;330:1080## All rapid responses

*The BMJ*reserves the right to remove responses which are being wilfully misrepresented as published articles.

Probability estimates require an exhaustive list of possibilities. It

is this which makes us think more accurately: when you can express a thing

as a number you know something about it (Kelvin.)

For some reason articles about Bayes always seem to contain numeric

errors. Here, for example "... multiplying the negative likelihood ratio

(0.11) by the pre-test odds of 5:1.." does not "...give a posterior

probability of 0.55:1 (38%.)" It gives posterior odds of 0.55:1 and a

posterior probability of 35.4%. This isn't just nitpicking. Your sums have

to be right if you want to persuade.

Incidentally, it is difficult to glean from Bayes original paper on "The

doctrine of Chances" the ideas we now have about modification of opinion

by evidence.

Competing interests:

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**Competing interests: **
No competing interests

Clinicians are natural Bayesians when it comes to diagnosis. They

have to be. The alternative approach might be to use the methods of

classical hypothesis testing, but probably only once.

The Neyman-Pearson diagnosis of coeliac disease.

[Assume that the sensitivity and specificity of transglutaminase IgA

are both 95%]

Parent: Well doctor, have you got the result of the test yet?

Doc: Yes I have. When you brought little Johnny in with weight loss,

short stature and diarrhoea, I thought it was worth checking for coeliac

disease, and the test has come back positive.

Parent: Does that mean he has coeliac disease?

Doc: I can’t be certain, but it is likely.

Parent: Well, how likely?

Doc: I can’t actually tell you that, but given that he does not have

coeliac disease, there was a 95% probability that the test would have been

negative, and it in fact was positive.

Parent: Well, does that mean he has a 95% chance of having coeliac

disease?

Doc: No. I can’t say that.

Parent: Well what can you say?

Doc: Given that he does not have coeliac disease, there was a 95%

probability that the test would have been negative, and only a 5% chance

of obtaining this result.

Parent: We’re going around in circles. What about the fact that he

has the diarrhoea and weight loss?

Doc: Well, that’s why I did the test.

Parent: Well does it make coeliac disease more likely?

Doc: I can’t say that.

Parent: Well what do we do?

Doc: We could do the test a few more times, and if it keeps coming up

positive, it makes the diagnosis more likely.

Parent: How much more likely?

Doc: I can’t say, but more likely. In fact if I did this test on many

patients with the same signs, in the long run I wouldn’t go far wrong.

Parent: That won’t help Johnny. Are you a complete idiot or what?

Robert McCrossin

Robert_McCrossin@health.qld.gov.au

Department of Paediatrics,

Hervey Bay Hospital, Queensland 4655, Australia

Competing interests:

None declared

**Competing interests: **
No competing interests

**11 May 2005**

Dear Editor,

In this edition of the British Medical Journal Gill et al(1) describe

clinicians as natural bayesians. Such clinicians use any or all of the

features of the patient's condition: the history, the examination and the

results of special investigations, in an intuitive, probablistic and

reiterative way to arrive at a diagnosis. These reasoning processes,

though they are subjective and context-dependent, are integral to clinical

judgement and are particularly useful when diagnosing rare diseases.

A few pages on, in the same edition, Drife(2) describes the fictional

Dr House. It seems highly likely that Dr House is a bayesian. He solves

rare cases, that is, cases drawn from the smallprint of medical textbooks;

his solutions occur against a background of reiteration: the hurling back

and forth of obscure differential diagnoses; and he is intuitive: his

brilliant diagnoses are based on flashes of grim insight.

Interestingly, Dr House is likened to various fictional detectives:

Philip Marlowe, Ironside, Dirty Harry and Sherlock Holmes. This is

interesting because the detective model of medical practice has been

described previously by Downie et al(3). This model underpins clinical

judgement and has the collection of evidence (history, examination and

special investigations), and the interpretation and reinterpretation of

that evidence in a context-related way, by the clinician, at its core.

Throughout this process the evidence is continuously weighed, and accepted

or rejected, depending on the coherence it gives to the diagnostic

picture.

It seems to me that the detective model and the bayesian model are

closely related, if not the same. In the detective model probability is

expressed non-numerically, as "weight", and in the bayesian model

probability is expressed numerically. Both of these models are considered

to be important in the setting of the clinician making decisions about the

patient as an individual. Gill et al compare this to the limitations of

decision-making based on statistical inference of a frequentist type and

algorithms derived therefrom, whilst Downie et al make comparison with the

science model of medicine which, although it ensures a well validated

evidence base for practice generally, does not allow for the unique

circumstances and variations of each individual patient.

By insisting on attention to context, the bayesian approach to

practice places the patient at the centre of the clinician's

considerations. which is just where he or she should be. Whereas, the

scientific approach, with its use of frequency statistics, and its

insistence on abstraction and reductivism, can lead to the patient being

marginalised and under-valued, which is not a good thing. Long live

bayesianism!

References:

1) Gill CJ, Sabin l, Schmid CH. Why clinicians are natural bayesians.

BMJ 2005;330:1080-3.(7 May.)

2) Drife JO. House. BMJ 2005;330:1090.(7 May.)

3) Downie RS, MacNaughton J, Randall F. Clinical judgement : evidence

in practice. Oxford University Press, 2000.

Competing interests:

None declared

**Competing interests: **
No competing interests

**11 May 2005**

Editor,

The recent article on bayesian statistics brings to mind what I have

been doing in general practice for many years, that is telling people what

they are not suffering from rather than what they are suffering from, that

is to say minor rashes in children I tend to go through with the parents

classic five rashes and then say this rash is none of those but is a viral

rash and should clear up in a few days.

I am sure that many General Practitioners use this approach as it is

difficult to diagnose many things in general practice (giving the patient

the benefit of a name that they can look up on the internet or even share

in conversation with the neighbours).

Dr A M Marshall

General Practitioner.

Competing interests:

None declared

**Competing interests: **
No competing interests

**10 May 2005**

Clinical researchers found an uninspiring likelihood ratio of 2.75

for shifting dullness in ascites. This LR however is established after

referral. Cattau e.a. evaluated 21 referred patients suspected for ascites

and 6 patients appeared to have ascitis.1 The clinicians performing this

assessment had a prior-chance of 29%. The referring physicians however had

a much lower prior-chance and performed the same testing of shifting

dullness. If, say, the prior-chance in general practice is 1%, the

diagnostic procedure of the GP must have a LR of about 40 to reach a

posterior-chance of 29%. Performing the same test a second time does not

‘double’ the certainty. The problem of referral bias is described by

Knottnerus.2 The LR of shifting dullness in general practice is probably

higher than 2.75.

Reference List

1. Cattau EL, Jr., Benjamin SB, Knuff TE, Castell DO. The accuracy

of the physical examination in the diagnosis of suspected ascites. JAMA

1982;247:1164-6.

2. Knottnerus JA. The effects of disease verification and referral

on the relationship between symptoms and diseases. Med.Decis.Making

1987;7:139-48.

Competing interests:

None declared

**Competing interests: **
No competing interests

The article by Gill, Sabin and Schmid calls attention to the

importance of likelihood ratios in the process of differential

diagnosis.[1] It is important to note that to determine the sensitivity

and specificity we have to describe the results of the test necessarily

into two categories, a decision that may reduce the discriminative power

of the test, particularly for quantitative tests, such as serum ferritin

and leukocyte count. One advantage of likelihood ratios over sensitivity

and specificity is that they can be calculated for tests with more than

two possible results.

Even for test results classified into two categories, however,

likelihood ratios refine the clinical diagnosis. As mentioned by Gill et

al the only useless test is the one with likelihood ratios of 1 because

the result will not modify the probability of disease. This is exemplified

in the article by a situation where both the sensitivity and the

specificity are equal to 0.5 (i.e., 50%). In fact tests with sensitivity

and specificity near 0.5 (50%) have been described in the medical

literature. Cattau et al have studied the accuracy of the physical

examination in the diagnosis of ascites and found a sensitivity of 0.55

(55%) and a specificity of 0.5 (50%) for the puddle test.[2] Therefore,

the likelihood ratios were very close to 1. Based on this result the

conclusion is that the puddle test does not help diagnose ascites.

Likelihood ratios equal or close to one, however, could be found for tests

with sensitivity or specificity much lower or higher than 0.5 (50%).

Indeed in any situation that the sum of the sensitivity and specificity

(expressed as proportions with limits between 0 and 1) is equal to 1, both

the positive and the negative likelihood ratios will also be equal to 1.

If a test has a specificity of 0.8 (80%) but a much lower sensitivity of

0.2 (20%), for example, both the positive likelihood ratio (0.2/(1-0.8))

and the negative likelihood ratio ((1-0.2)/0.8) will be equal to 1. This

test would be useless despite its fair specificity.

Thus, to assess the diagnostic value of a dichotomous test we should

use the likelihood ratios because they take into account both the

sensitivity and the specificity.

References

1. Gill CJ, Sabin L, Schmid CH. Why clinicians are natural bayesians.

BMJ 2005;330(7499):1080-1083.

2. Cattau EL, Jr., Benjamin SB, Knuff TE, Castell DO. The accuracy of

the physical examination in the diagnosis of suspected ascites. JAMA

1982;247(8):1164-6.

Competing interests:

None declared

**Competing interests: **
No competing interests

**10 May 2005**

Thomas Bayes was not a vicar - which is a specifically Anglican job

title. He was a presbyterian minister.

Competing interests:

None declared

**Competing interests: **
No competing interests

Sir,

The authors correctly state that the pretest odds multiplied by the

likelihood ratio of a test provides the post test odds. However Bayes

work referred to probability rather than odds.

Bayes developed his famous theorem about conditional probability. He

showed that 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) devided by

P(B)

The fact that Bayes refers to probability and articles such as this

by Gillet al refers to odds, has led to confusion, with some people

thinking that it does not make any difference whether odds or probability

are used or even that it is “debatable” which is used in correct Bayesian

calculations.(ref 1 and 2)

As the authors state the pretest odds should be multiplied by the

likelihood ratio to reach the post-test odds. The likelihood ratio is a

simple calculation from the sensitivity and specificity of a test, but the

liklihood ratio as such is never used in Bayes work. As we have shown (ref

3) however it is a simple calculation to show that multiplying the pretest

odds by the likelihood ratio and deriving the post test odds is equivalent

to the proposal of Bayes.

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

3.

http://www.obgyn.net/us/us.asp?page=/us/news_articles/hutchon_bayes

Competing interests:

None declared

**Competing interests: **
No competing interests

## Why clinicians outperform algorithms, not only thanks to reverend Bayes.

To the editor,

In their interesting contribution “Why clinicians are natural

Bayesians?” Gill and collaborators ask: “How clinical judgments prove

superior to the algorithm, a diagnostic tool carefully developed over two

decades of research? Was it just a lucky guess?” They conclude that it is

the Bayesian logic that makes clinicians outperform the algorithm. We

think they do not prove this statement and that Bayesian logic is only one

of the reasons.

In the seventies a lot has been done to promote algorithms. In public

health in developing countries, they have become the main diagnostic (and

therapeutic) strategy. The algorithm has probably been developed in a

careful, but alas not in a scientific way. We could find almost no

publications about its logic, and even fewer about diagnostic accuracy and

effect on final outcome. Their development was highly influenced by

computer logic, which is by the virtue of the processor dichotomic and

serial. Human thinking follows, also by the virtue of our brain with its

neurons, a weighed and parallel approach. This counterintuitive logic of

algorithms is, we think, the main reason why they are not accepted by the

majority of clinicians. We followed nurse-practioners in Guinea Conakry,

Congo, Burkina Faso, Laos and Ecuador and we could observe that even at

that level their decisions were often fairly better than the algorithms

they were supposed to follow. Why?

Firstly because clinicians use a weighed approach. Findings are

“weighed” by the clinician, and will add up to the final probability. This

principle is part of the Bayes’ logic, but the latter adds a mathematical

formula to correctly estimate this weight. From village health workers to

university professors, we could notice that clinicians’ estimation of

weight almost constantly corresponds to the log10 of the likelihood ratio,

(an intuitive scale of 0 or “null” to 4 or “very strong” corresponds to

0;0,5;1;1,5;2 of the log10LR scale). This suggests that clinicians indeed

use the full Bayes’ theorem, but on a logarithmic scale.(1)

A second and equally important advantage of clinicians’ logic is its

parallel approach: different paths may be explored at the same time, a

feature that is not typically “Bayesian”.(2) In the algid malaria example,

a fast respiratory rate and the absence of fever may be associated both

with severe malaria and with pneumonia. One could not rule out either

without a thick film or a chest X-ray, not available at that level of

care. Following a serial approach led to a very dangerous decision.

A final but not less important feature of clinicians’ reasoning is

the notion of “threshold”, which is completely lacking in an algorithmic

approach.(3) In the example, both pneumonia and algid malaria remain

“possible”. The clinician remains with the suspicion of two severe,

potentially fatal diseases, therefore considering that a treatment

threshold for both has been reached, regardless which one he considers

“more likely”.

The promotion of the algorithm has distracted much attention from the

intuitive Bayesian logic, notwithstanding the value it had mostly in

developing countries, allowing some kind of "diagnosis", in stead of

purely symptomatic treatment. If the scientific community could produce

more reliable clinical Bayesian based decision aids, they could provide a

basis towards the rationalization of health care, and hence cost

containment.

(1) Van den Ende J, Van Gompel A, Van-den-Enden E, Van Damme W,

Janssen PAJ. Bridging the gap between clinicians and clinical

epidemiologists : Bayes theorem on an ordinal scale. Theor.Surg. 9, 195.

1994.

(2) Van Puymbroeck H, Remmen R, Denekens J, Scherpbier A, Bisoffi Z,

Van den Ende J. Teaching problem solving and decision making in

undergraduate medical education: an instructional strategy. Med Teach

2003; 25(5):547-550.

(3) Pauker SG, Kassirer JP. The threshold approach to clinical

decision making. N Engl J Med 1980; 302(20):1109-1117.

Competing interests:

None declared

Competing interests:No competing interests09 June 2005