Bayes' formula: a powerful but counterintuitive tool for medical decision-making

Learning objectives.

By reading this article you should be able to:

• •Discuss the concept of conditional probability.
• •Explain the principles underlying Bayes' formula, including the idea of prior probability and posterior probability.
• •Describe the application of Bayes' formula to disease testing, including the important effect that disease prevalence (pre-test probability) has on the positive and negative predictive values.
• •Illustrate the meaning of P-values as a conditional probability and recognise that P-values do not mean the probability that the hypothesis is false.
• •Explain the principles of Bayesian inference as an alternative to standard (frequentist) statistical testing.

Key points.

• •Conditional probability is the probability of an event occurring given another event is true; it helps to clarify terms such as positive predictive value, sensitivity, statistical power, and type I error.
• •Bayes' formula is the basis of a distinct type of statistical analysis, called Bayesian inference.
• •Bayes' formula provides a framework for working with conditional probabilities. Starting with a prior probability, Bayes' formula allows us to update the prior with ‘information’ to obtain a posterior probability.
• •Bayes' formula can help to interpret the results of disease testing. The ‘prior probability’ is the disease prevalence. The ‘information’ is the sensitivity and specificity of the test. The ‘posterior probability’ is the positive predictive value.
• •Bayes' formula can also be used to estimate the probability that a study hypothesis is false, despite a ‘positive’ statistical test result.

Introduction

Bayes' formula was first discovered more than 250 yrs ago by English clergyman Thomas Bayes (1701–1761). The formula was independently discovered and placed on a sound mathematical footing by French polymath Pierre-Simon Laplace (1749–1827). Bayes' formula is an invaluable tool for dealing with uncertainty, including the uncertainty we face as clinicians in our daily practice.

In this article, we develop Bayes' formula from the rules of joint and conditional probability, and demonstrate its usefulness for medical decision-making. First, and most importantly, we show how Bayes' formula can assist with diagnostic uncertainty. In particular, the chance that a patient who tests positive for a disease actually has the disease. Secondly, we show how Bayes' formula can provide insights into the probability that researchers have drawn the wrong conclusion when declaring a study hypothesis is true based on a statistically significant test result. Finally, we demonstrate how Bayes' formula can be used for statistical analysis in its own right.

Bayes' formula is a tool for updating the probability of an event being true (e.g. a patient has a disease) with ‘information’ (e.g. the results of a test). We start with a prior probability, then obtain information from the test and thereby update the prior probability with a posterior probability.

Joint and conditional probability

To understand Bayes' formula, it is important to understand the concept of conditional probability. Consider a man with a fever. Alarmed, he searches the internet and discovers that lymphoma is a cause of fever. He reads that 99% of patients with lymphoma have a fever. Should he be worried?

Now, consider two independent events, A and B. Let A be ‘I toss a coin and get heads’, and let B be ‘I roll a die and get a six’. The probability of both events occurring is simply

$$P\left(A\mathrm{a}\mathrm{n}\mathrm{d}B\right)=P\left(A\right)\times P\left(B\right)\text{,}$$ (1)

which, in this case, is $\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}\approx 0.083\approx 8.3\%$.

However, for two events that are not independent, we must consider conditional probabilities. Consider two events, A and B, where the probability of A occurring depends (i.e. is conditional) on the probability of B occurring, and vice versa. We have the probability of A occurring given B is true, which is written as P(A|B), and the probability of B occurring given A is true, which is written as P(B|A).

Clearly, the probabilities of having a fever and having lymphoma are related. P(fever|lymphoma) is the probability of having a fever given lymphoma, which we are told is 99%. However, what our man really wants to know is the inverse conditional probability: P(lymphoma|fever), the probability of having lymphoma given a fever. Because lymphoma is an uncommon cause of fever, P(lymphoma|fever) is low. Our febrile man should not be unduly worried.

Bayes' formula can help calculate P(lymphoma|fever). However, to do so we also need to know the prior, P(lymphoma), which is the prevalence of lymphoma. We also need to know how good a test fever is for test for lymphoma—that is, we need to know the sensitivity and specificity for fever as a test for lymphoma. In fact, we already know the sensitivity: 99%.

Bayes' formula and disease testing

This section is quite technical; it is acceptable to skip the mathematics. The rule for joint probability of two events, that holds irrespective of the events being statistically independent is

$$P\left(A\mathrm{a}\mathrm{n}\mathrm{d}B\right)=P\left(B\right)\times P(A|B)=P\left(A\right)\times P\left(B\right|A)$$ (2)

If two events are independent, then P(A) does not depend on B occurring (and vice versa). So, P(A)=P(A|B) and P(B)=P(B|A), and equation (2) simplifies to equation (1). Ignoring the left-hand term in equation (2) and dividing through by P(B) gives us Bayes' formula:

$$P\left(A|B\right)=\frac{P\left(A\right)\times P\left(B\right|A)}{P\left(B\right)}$$ (3)

When we test for a disease, there are four possible outcomes, depending on whether the test is positive (T⁺) or negative (T^–) and whether the disease is present (D⁺) or absent (D^–).

Ideally, we would only have two outcomes, a positive test when the disease is present (a true positive result) and a negative test when the disease is absent (a true negative result). The probability of testing positive given the disease is present is the sensitivity of the test, and is written as P(T⁺|D⁺). The probability testing negative given that the disease is absent is the specificity of the test and is written as P(T^–|D^–). Because no test is perfect, we also have to deal with false negatives and false positives. The conditional probabilities associated with each of these four outcomes are shown in Table 1.

Table 1.

Probability table for disease testing

	Disease present (D+)	Disease absent (D–)
Test positive (T+)	P(T+|D+) Probability of a positive test given the presence of disease Sensitivity	P(T+|D–) Probability of a positive test given the absence of disease 1–specificity
Test negative (T–)	P(T–|D+) Probability of a negative test given the presence of disease 1–sensitivity	P(T–|D) Probability of a negative test given the absence of disease specificity

When testing for disease, we are interested in P(D⁺|T⁺), the probability of having the disease given a positive test, and P(D^–|T^–), the probability of not having the disease given a negative test. P(D⁺|T⁺) is called the positive predictive value (PPV) and P(D^–|T^–) is called the negative predictive value (NPV). (Strictly speaking, P(D⁺|T⁺) and P(D^–|T^–) are post-test probabilities, whereas PPV and NPV refer to observable numbers from populations who are tested. However, for defined populations, the PPV is numerically equivalent to P(D⁺|T⁺) and the NPV is numerically equivalent to P(D^–|T^–).) Notice that sensitivity is the inverse conditional probability to PPV and specificity is the inverse conditional probability to NPV.

We can substitute into Bayes' formula to develop an equation for the PPV:

$$\text{PPV}=P\left({\text{D}}^{+}|{\text{T}}^{+}\right)=\frac{\left[P\left({\text{D}}^{+}\right)\times P\left({\text{T}}^{+}|{\text{D}}^{+}\right)\right]}{P\left({\text{T}}^{+}\right)}$$ (4)

where P(D⁺) is the prevalence of the disease and P(T⁺|D⁺) is the sensitivity of the test. To obtain a useful version of equation (4), we need further define P(T⁺), the probability of a positive test. In Table 2, we have included a term for disease prevalence to the conditional probabilities. From Table 2,

$$P\left({\text{T}}^{+}\right)=[P\left({\text{D}}^{+}\right)\times P({\text{T}}^{+}|{\text{D}}^{+})]+[P\left({\text{D}}^{-}\right)\times P({\text{T}}^{+}|{\text{D}}^{-})]$$ (5)

where P(T⁺|D^–) is the probability of a false positive test result, which is 1–specificity of the test, and P(D^–) is 1–prevalence of the disease.

Table 2.

Probability table for disease testing with prior probability (prevalence) included

	Disease present (D+)	Disease absent (D–)	Totals
Test positive (T+)	P(D+)×P(T+|D+) prevalence×sensitivity	P(D–)×P(T+|D–) (1–prevalence)×(1–specificity)	P(D+)×P(T+|D+)+P(D–)×P(T+|D–) prevalence×sensitivity+(1–prevalence)×(1–specificity)
Test negative (T–)	P(D+)×P(T–|D+) prevalence×(1–sensitivity)	P(D–)×P(T–|D) (1–prevalence)×specificity	P(D+)×P(T–|D+)+P(D–)×P(T–|D) prevalence×(1–sensitivity)+(1–prevalence)×specificity

Combining equations (4), (5), we have

$$P\left({\text{D}}^{+}|{\text{T}}^{+}\right)=\frac{\left[P\left({\text{D}}^{+}\right)\times P\left({\text{T}}^{+}|{\text{D}}^{+}\right)\right]}{\left[P\left({\text{D}}^{+}\right)\times P\left({\text{T}}^{+}|{\text{D}}^{+}\right)\right]+\left[P\left({\text{D}}^{-}\right)+P\left({\text{T}}^{+}|{\text{D}}^{-}\right)\right]}$$ (6)

Expressing equation (6) in more familiar terms, gives an equation for PPV:

$$\text{PPV}=\frac{\left[\text{prevalence×sensitivity}\right]}{\left[\text{prevalence×sensitivity}\right]+\left[(1-\text{prevalence})\text{×}(1-\text{ specificity})\right]}\text{.}$$ (7)

Using the same approach, we can develop equations for the NPV:

$$P\left({\text{D}}^{-}|{\text{T}}^{-}\right)=\frac{\left[P\left({\text{D}}^{-}\right)\times P\left({\text{T}}^{-}|{\text{D}}^{-}\right)\right]}{\left[P\left({\text{D}}^{-}\right)\times P\left({\text{T}}^{-}|{\text{D}}^{-}\right)\right]+\left[P\left({\text{D}}^{+}\right)\times P\left({\text{T}}^{-}|{\text{D}}^{+}\right)\right]}$$ (8)

$$\text{NPV}=\frac{\left[(1-\text{prevalence})\text{×specificity}\right]}{\left[(1-\text{prevalence})\text{×specificity}\right]+\left[\text{prevalence×}(1-\text{sensitivity})\right]}$$ (9)

So, knowing the prevalence of the disease along with the sensitivity and specificity of the test allows us to calculate PPV and NPV.

Sensitivity and specificity vs the positive and negative predictive values

Sensitivity and specificity are properties of the test. They tell us nothing about the patient.

In contrast, PPV and NPV do tell us about the patient. However, PPV and NPV are not fixed parameters; they vary greatly depending on the prevalence of the disease. Because lymphoma is rare and fever is non-specific for lymphoma, the PPV of fever for lymphoma is very low.

Odds ratio form of Bayes' formula

Another useful way of expressing Bayes' formula is as an odds ratio. The odds of an event is the probability of the event occurring divided by the probability of the event not occurring. So, the prior odds of having a disease are:

$$\mathrm{O}\mathrm{d}\mathrm{d}\mathrm{s}=\frac{P\left({\text{D}}^{+}\right)}{P\left({\text{D}}^{-}\right)}=\frac{P\left({\text{D}}^{+}\right)}{1-P\left({\text{D}}^{+}\right)}\text{.}$$ (10)

With this in mind, equation (6) can be reformulated as

$$\frac{P\left({\text{D}}^{+}\right|{\text{T}}^{+})}{P\left({\text{D}}^{-}\right|{\text{T}}^{+})}=\frac{P\left({\text{D}}^{+}\right)}{P\left({\text{D}}^{-}\right)}\times \frac{P\left({\text{T}}^{+}\right|{\text{D}}^{+})}{P\left({\text{T}}^{+}\right|{\text{D}}^{-})}$$ (11)

The term on the left-hand side is the post-test (posterior) odds of having the disease, which is the parameter we are interested in. The first term on the right-hand side is the pre-test (prior) odds. The second term on the right-hand side is called the likelihood ratio. The likelihood ratio is the ‘information’ used to update the prior probability to obtain a posterior probability, and is the ratio of the sensitivity to (1–specificity). The likelihood ratio tells us how much more confident we can be that a person has a disease if they test positive. In words, equation (11) can be stated as:

Post-test odds = pre-test odds × likelihood ratio (12)

Applying Bayes' formula to disease testing

Consider a rare disease with a prevalence of one in a 100,000 (i.e. P(D⁺)=0.00001) for which there is an excellent test (sensitivity 99%, specificity 99%). From equation (7), we have

$$\text{PPV}=\frac{0.00001\times 0.99}{0.00001\times 0.99+0.99999\times 0.01}=\frac{11}{11122}\approx 0.0001=0.01\%.$$

Thus, the probability of having the disease given a positive test is only 0.01%—that is, 99.99% of positive tests are false positives, even though the test is very good. Although this seems counterintuitive, it is merely a consequence of the fact that test errors are relatively common (one in 100) compared with cases of the disease (one in 100,000). Similarly, we can calculate the probability of being disease free given a negative test. From equation (9), we have

$$\text{NPV}=\frac{0.99999\times 0.99}{0.99999\times 0.99+0.00001\times 0.01}\approx 1\text{.}$$

Thus, a negative test rules out the condition.

Next, consider a subgroup where the prevalence of the same disease is one in a 100 (i.e. P(D⁺) = 0.01). Now the PPV and NPV, given the same test and the same disease, are 50% and >99%, respectively. A positive test increases the probability of having the disease from 1% (the prevalence) to 50%. A negative test still largely rules out the disease. Finally, imagine a highly selective sub-group where the prevalence of the disease is one in 10. Now, the PPV and NPV are 92% and >99%, respectively. Thus, the PPV changes dramatically depending on the prevalence of the disease.

It is rare for a test to have a sensitivity or a specificity as high as 99%. For a disease with a prevalence of 10%, consider a test with a low sensitivity (40%) and a high specificity (99%). The PPV and NPV are 82% and 94%, respectively. However, if the specificity is low (40%) but the sensitivity is high (99%), the PPV decreases to 15% and NPV remains high (>99%). Thus, a low specificity greatly reduces the PPV. This is unsurprising, because a low specificity results in a high number of false positives. A low sensitivity modestly reduces the NPV but has little effect on the PPV.

Let us now look at two real-world examples.

Mammography for breast cancer screening

Mammography is an excellent screening tool for breast cancer. The American College of Radiology recommends an annual mammogram for all women of average risk from age 40.¹ This strategy reduces mortality from breast cancer by 39.6% and averts 11.9 deaths per 1000 women.¹ These figures are in keeping with the (relatively) high sensitivity and specificity of mammography of 87% and 89%, respectively.² Assuming a prevalence of breast cancer of 0.01 (1%), we can substitute into equation (7) to obtain the PPV:

$$\text{PPV}=\frac{0.01\times 0.87}{0.01\times 0.87+(1-0.01)\times (1-0.89)}=\frac{0.0087}{0.0087+0.1089}\approx 0.07\text{.}$$

Thus, the probability that a randomly selected, asymptomatic woman who has a positive mammogram actually has breast cancer is about 7%. Thus, even for a good screening test for a common cancer, a woman who tests positive is much more likely to not have cancer than to have cancer. Similarly, we can use equation (9) to calculate the NPV:

$$\text{NPV}=\frac{(1-0.01)\times 0.89}{(1-0.01)\times 0.89+0.01\times (1-0.87)}=\frac{0.8811}{0.8811+0.0013}>0.99.$$

Thus, the NPV is more than 99%, so a negative mammogram is reassuring. However, because many more negative tests are performed than positive tests, an appreciable number of false negatives will occur, but the probability of a false negative result for an individual woman is low.

Ultrasonography is another screening test for breast cancer. In a large RCT, mammography plus breast ultrasound was compared with mammography alone as screening tools for breast cancer.³ The study demonstrated that combination screening increased sensitivity from 77.0% to 91.1%, but specificity decreased from 91.4% to 87.7%. The higher sensitivity means more cancers were detected (more true positives, fewer false negatives, higher NPV) with combination screening. However, the lower specificity means there were more false alarms (more false positives, fewer true negatives, lower PPV), which in turn would have led to further tests and anxiety for well women. So, in this case, combination screening is a trade-off: fewer missed cancers but more false alarms.

Antibody testing for heparin-induced thrombocytopenia

Heparin-induced thrombocytopenia (HIT) is a potentially fatal prothrombotic reaction to heparin caused by antibodies to platelet factor 4. Although antibody formation is common, clinical thrombocytopenia and thrombosis are less common, affecting 0.2–3% of cardiac surgical patients.⁴

There are two types of laboratory tests: immunoassays that identify antibodies to platelet factor 4 and functional assays that measure platelet activation and aggregation. Standard immunoassays for HIT have a sensitivity greater than 90% but a specificity as low as 40%.⁵ Functional assays have a high sensitivity and specificity, but are less readily available than immunoassays. Consequently, when a request is made for a HIT screen, laboratories typically perform an immunoassay.

The high sensitivity of the immunoassay results in a high NPV (>99%). Thus, a negative test is reassuring. However, the low specificity means the PPV is low. Assuming a prevalence of HIT of 1%, and a sensitivity and specificity of 90% and 40% respectively, then applying equation (7) to the immunoassay test, we have

$$\text{PPV}=\frac{0.01\times 0.90}{0.01\times 0.90+(1-0.01)\times (1-0.40)}=\frac{0.009}{0.009+0.594}\approx 0.015.$$

Thus, the PPV of the immunoassay is only 1.5%. So, a randomly selected cardiac surgical patient who has a positive test is highly unlikely to have HIT.

One way to improve the PPV for the immunoassay is to increase the pre-test probability. The 4T test is a simple clinical tool to evaluate the pre-test probability, based on the timing and severity of the thrombocytopenia and the presence of other causes of low platelets.⁴ Each parameter is scored 0, 1, or 2, with a total score of 6–8 indicating a pre-test probability of HIT of around 50%. Substituting this revised prior probability of 0.5 into equation (7), we have:

$$\text{PPV}=\frac{0.50\times 0.90}{0.50\times 0.90+(1-0.50)\times (1-0.40)}=\frac{0.45}{0.45+0.30}\approx 0.60\text{.}$$

Thus, the combination of a high 4T test score and a positive immunoassay improves the PPV from 1% to 60%. However, the 4T test alone gives a 50% PPV. Thus, the immunoassays, even when combined with the a positive 4T test, have a worryingly high rate of false positives. For this reason, guidelines recommend performing a functional assay before commencing treatment for HIT.⁴

Applying Bayes' formula to hypothesis testing

When planning a clinical trial, researchers have a key question (the study hypothesis) that they hope the trial will answer; for example, that drug A has a different (superior) effect than drug B on blood pressure. The null hypothesis (H₀) is that drug A has the same effect as drug B. The study hypothesis (H₁) is that drug A has a different effect than drug B.

When the data have been collected, hypothesis testing is performed. A test statistic is calculated which gives a P-value. If the P-value is less than or equal to a predetermined value (alpha), the null hypothesis is rejected and the study hypothesis is accepted. Thus, hypothesis testing starts from the position that the null hypothesis is true and the study hypothesis is false. The P-value is the probability that the observed (or a more extreme) outcome would occur if the null hypothesis were true.

There are two aspects of hypothesis testing that conditional probability, and Bayes' formula can help clarify. The first is interpreting P-values. It is a common error to assume that the P-value is the probability the study hypothesis is false. As hypothesis testing starts from the presumption that the study hypothesis is false, P-values cannot tell us anything about the study hypothesis. As a conditional probability, a P-value is P(data|H₀), the probability of the observed outcome (i.e. the data) given the null hypothesis is true. P-values are not P(H₀|data), the probability the null hypothesis is true (i.e. the study hypothesis is false) given the observed outcome. Mixing up these two conditional probabilities is a common reason for misinterpreting P-values. In the same way that sensitivity tells about the test and not the patient, a P-value tells us about the probability of the outcome of the study, not the study hypothesis.

The second insight relates to the probability that the wrong conclusion is drawn when a ‘statistically significant’ test result is obtained. It is a common error to assume that because alpha is 0.05, there is a 5% chance that the study hypothesis is false when a statistical test is positive. In fact, the probability the study hypothesis is false given a positive statistical test is typically much higher than 5%.

Alpha is the probability of a positive test (i.e. P≤0.05) given the null hypothesis is true, which as a conditional probability is P(T⁺|H₀). Alpha is also the probability of a type I statistical error. Alpha relates to the test, and is analogous to 1–specificity. It tells us nothing about the hypothesis. If we are interested in the probability the study hypothesis is false (i.e. the null hypothesis is true) given a positive test, we need to know P(H₀|T⁺), not alpha. P(H₀|T⁺) is called the false positive risk (FPR).⁶ The FPR is a probability concerning the hypothesis. Notice that alpha and the FPR are inverse conditional probabilities.

We can use Bayes' formula to calculate the FPR. (The derivation of equation (13) from Bayes' formula is somewhat technical. Interested readers are referred to Sidebotham for a full explanation.⁷):

$$\text{F}\text{P}\text{R}=\frac{\left[\left(1-\text{p}\text{r}\text{i}\text{o}\text{r}\right)\times \text{a}\text{l}\text{p}\text{h}\text{a}\right]}{\left[\left(1-\text{p}\text{r}\text{i}\text{o}\text{r}\right)\times \text{a}\text{l}\text{p}\text{h}\text{a}\right]+\left[\text{p}\text{r}\text{i}\text{o}\text{r}\times \text{p}\text{o}\text{w}\text{e}\text{r}\right]}$$ (13)

Looking at equation (13), we see that to calculate the FPR we need to know alpha, power, and the prior probability the study hypothesis is true (the ‘prior’). Power is P(T⁺|H₁), the probability of a positive test given the study hypothesis is true, and is analogous to sensitivity. Power is also 1–beta. Beta is P(T^–|H₁), the probability of a negative test given the study hypothesis is true, and is analogous to 1–sensitivity. Beta is also the probability of a type II statistical error.

An alpha of 0.05 and power of ≥0.8 (beta ≤0.2) are reported in virtually all medical research. If we assume the prior probability of investigating a true hypothesis is 0.5 (50%), and we use standard values for power (0.8) and alpha (0.05), then we have

$$\mathrm{F}\mathrm{P}\mathrm{R}=\frac{(1-0.5)\times 0.05}{(1-0.5)\times 0.05+0.5\times 0.8}=\frac{0.025}{0.025+0.4}\approx 0.06$$

Thus, the FPR is about 6%. However, this result assumes power is 0.8. In fact, the actual power of published studies is often much less than the reported power of 0.8, particularly for small discovery-orientated trials.⁷ Also, for a discovery-orientated trial, investigating a ‘long shot’ hypothesis, the prior might be much lower than 50%. Both of these effects (low power, low prior) dramatically increase the FPR. For instance, the prior is 0.2 and power is 0.2 (alpha 0.05), we have

$$\mathrm{F}\mathrm{P}\mathrm{R}=\frac{(1-0.2)\times 0.05}{(1–0.2)\times 0.05+0.2\times 0.2}=\frac{0.04}{0.04+0.04}=0.50$$

So, in this situation, the FPR is 50%. That is to say, there is a 50% probability that the study hypothesis is false (i.e. the null hypothesis is true) despite a statistically significant test result. Small, discovery-orientated RCTs commonly have an FPR greater than 50%, despite using an alpha value of 5%.^7,8 This type of analysis is the basis of John Ioannidis' famous paper, ‘Why most published research findings are false’.⁹

Bayesian statistical analysis

Bayesian inference can be used as an alternative to standard statistical analysis, described in the previous section. With the Bayesian approach, data (samples) are obtained and used to update a prior probability with a posterior probability. Typically, the prior probabilities of two competing hypotheses (H₁ and H₀) are compared using the odds ratio form of Bayes' formula:

$$\frac{P({\text{H}}_1\left|\text{D}\right)}{P\left({\text{H}}_0\right|\text{D})}=\frac{P\left({\text{H}}_1\right)}{P\left({\text{H}}_0\right)}\times \frac{P(\text{D}\left|{\text{H}}_1\right)}{P\left(\text{D}\right|{\text{H}}_0)}$$ (14)

where H is the hypothesis and D is the data (i.e. the findings of the study). The left-hand term in equation (14) is the posterior odds, which is the probability H₁ is true given the data divided by the probability H₀ is true given the data. The first term on the right-hand side is the prior odds of H₁ relative to H₀, which is the prior probability H₁ is true divided by the prior probability H₀ is true. The second term on the right hand-hand side is the likelihood ratio, which is the probability of the data given H₁ is true divided by the probability of the data given that H₀ is true.

Suppose a researcher decides, based on previous research, that P(H₀)=0.4 and P(H₁)=0.5, then the prior odds are:

$$\frac{0.5}{0.4}=1.25.$$

Once the study has been performed and the likelihood ratio calculated, the posterior odds might increase to, say, 4. This means that H₁ predicts the data four times better than H₀. Notice that unlike standard hypothesis testing, the Bayesian approach does not produce a binary outcome (positive or negative result). The reader decides if the posterior odds are sufficiently high to adopt or abandon the treatment. There are two main advantages of the Bayesian approach. First, compared with standard statistics, no preference is given to the null hypothesis. Second, the prior odds are updated continuously as further data are published. Thus, the Bayesian approach can be considered a dynamic process that evolves over time. The main disadvantage of the Bayesian approach is that in some circumstances the likelihood ratio is very complicated to calculate.

Bayesian interpretation of the EOLIA trial

In 2018, the ECMO to Rescue Lung Injury in Severe ARDS (EOLIA) trial was published in the New England Journal of Medicine, in which patients with acute respiratory distress syndrome (ARDS) were randomised to either extracorporeal membrane oxygenation (ECMO) or conventional treatment.¹⁰ EOLIA was a ‘negative’ trial, with 44/124 (35%) deaths in the ECMO group and 57/125 (45%) deaths in the control group (P=0.09). However, a previously published RCT¹¹ was (weakly) positive for ECMO.¹¹ Other non-randomised data also indicated a benefit for ECMO.¹² Thus, the prior probability of a benefit for ECMO at the time EOLIA was published was greater than 0.5.

In 2019, Goligher and colleagues performed a Bayesian analysis of the EOLIA data, in which they assigned various priors to a beneficial effect of ECMO.¹³ They then calculated the posterior probabilities for different outcomes. Using a ‘strongly enthusiastic prior’ the authors calculated a posterior probability of 0.95 (odds of 19) of a 4% absolute mortality reduction for EMCO and a probability of 0.65 (odds of 1.8) of a 10% absolute mortality reduction for ECMO. Thus, using a Bayesian approach, the authors found a clinically important survival advantage for ECMO, despite the fact that with standard statistical testing the trial was negative.

Conclusions

Not all the material covered here is easy to follow on first reading, and the terminology can be confusing. However, Bayes' formula provides a deep understanding the uncertainty inherent in the practice of medicine, and so we believe the effort to understand the principles—if not every equation—is worth it.

For readers interested in developing their knowledge further, we recommend two short books, one on Bayesian inference and the other on medical-decision making.^14,15 We also recommend two previous articles published in BJA Education, addressing clinical testing¹⁶ and hypothesis testing.^16,17 An analysis of hypothesis testing using Bayes' formula is the subject of a separate review by one of the authors.⁷

Declaration of interests

The authors declare that they have no conflicts of interest.

MCQs

The associated MCQs (to support CME/CPD activity) will be accessible at www.bjaed.org/cme/home by subscribers to BJA Education.

Biographies

Michael Webb BHK MSc is a senior registrar in anaesthesia in Auckland, New Zealand.

David Sidebotham FANZCA is a consultant in cardiothoracic anaesthesia and intensive care in Auckland, New Zealand.

Matrix codes: 1A03, 2A12, 3J00