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# Bayesian statistics introduction

### Bayes’ Theorem Formula

### Bayesian and Frequentist Methods: Two Statistical Philosophies

#### Bayesian Approach

#### Frequentist Approach

## Bayesian Methods: An Overview and Examples

### How Bayesian Methods Work

#### Example 1: Estimating Prevalence

#### Example 2: Using Bayes’ Theorem to Update Estimates

### Bayesian Terminology

## Prior Distributions in Bayesian Methods

### Introduction to Prior Distributions

### Common ‘Default’ Prior Distributions

### Sensitivity Analyses

### Key Points on Prior Distributions [5]

## Likelihood and Posterior Distributions in Bayesian Methods

### Likelihood

### Posterior Distribution

### Example: Combining Prior Distribution and Data

### Conjugate Distributions

## Summarizing and Presenting Bayesian Results

### Estimates

### Posterior Probabilities

### Credible Intervals (Posterior Interval)

### Significance Tests

## Using Bayesian Analyses in Medicine

### Example: GREAT – A Bayesian Re-analysis

## Using Bayesian Analyses

### Paroxetine and Suicide Attempts: A Bayesian Analysis

### Posterior Distributions

## Bayesian Checklist

### ROBUST (Reporting Of Bayes Used in Clinical STudies)

## Comparison of Bayesian and Frequentist Methods

## References

- Bayes’ Theorem Formula
- Bayesian and Frequentist Methods: Two Statistical Philosophies
- Bayesian Methods: An Overview and Examples
- Prior Distributions in Bayesian Methods
- Likelihood and Posterior Distributions in Bayesian Methods
- Summarizing and Presenting Bayesian Results
- Using Bayesian Analyses in Medicine
- Using Bayesian Analyses
- Bayesian Checklist
- Comparison of Bayesian and Frequentist Methods
- References

This material is reproduced based on the examples from: Oxford handbook of medical statistics (Janet Peacock Philip J Peacock) [1].

- Describes the Bayesian approach to statistical analysis in contrast to the frequentist approach
- Covers prior and posterior distributions, their roles, and choices
- Discusses examples, pros and cons of Bayesian methods in medicine, and how to interpret them in medical literature
- Developed by Reverend Thomas Bayes in 1763
- Statement about conditional probabilities, used in many statistical areas
- Events A and B:
- \(Pr(A \mid B)\): ‘the probability of A happening given that B has already happened’
- \(Pr(A \mid B)\) = \(\frac{Pr(B \mid A) \times Pr(A)}{Pr(B)}\)

- Bayes’ theorem forms the basis for competing statistical philosophies: Bayesian and frequentist methods
- They differ in their definitions of probability

- Probability interpreted as degree of belief that an event will occur
- Degree of belief comes from past data or experience
- Unknown quantities (e.g., means, proportions) follow probability distributions expressing certainty about true value
- Degree of belief can be updated with further information

- Probability is the long-run frequency of events (r) that occur in n trials
- Probabilities estimated directly from samples
- Unknown quantities (e.g., means, proportions) considered fixed, estimated from data with confidence intervals

- Combines anticipated values or distribution of values (prior) with new data
- Updates the distribution of the true values

- Specify anticipated value or distribution for prevalence (prior) using national data
- Collect regional data and calculate prevalence
- Combine observed area prevalence with prior distribution to update the distribution of true prevalence in the region

- Study investigated new D-dimer test for diagnosing venous thromboembolism (VTE) in patients with suspicious symptoms [3]
- Calculate updated probability that a patient has VTE given a positive D-dimer test result: \(\Pr(VTE+ \mid D+)\)

**Using Bayes’ theorem:**

- \(Pr(VTE^{+})\) = anticipated prevalence of VTE = 14% (0.14)
- \(Pr(D^{+})\) = proportion who test positive on D-dimer = 32% (0.32)
- \(Pr(D^{+} \mid VTE^{+})\) = probability of positive D-dimer test if the patient truly has VTE = 79% (0.79, sensitivity)

- Updated probability that a patient testing positive on D-dimer has VTE is approximately 35%
- Bayesian approach updates estimates, arguably providing better estimates

- Prior beliefs: Prior distribution
- New data: Likelihood
- Updated estimate: Posterior distribution

- Prior distribution: distribution of unknown quantity combined with new data to provide updated estimate
- Three categories of prior distribution [4]:
- Frequency distribution based on past data
- Objective representation of reasonable beliefs about a quantity
- Subjective measure of an individual’s beliefs

- Non-informative/reference priors: e.g., uniform distribution; used when range of values can be pre-specified but no clear opinion about which value is most likely
- Informative-sceptical prior: used to express scepticism about the estimated quantity, reduces chances of spuriously large effects
- Informative-enthusiastic prior: counterbalance of sceptical prior; used when positive effect is expected
- Informative prior based on prior beliefs, which are formally elicited: shape of distribution varies by context, Normal distribution sometimes used

- Choice of prior distribution can affect final estimate
- Good practice to test sensitivity of assumptions by using different forms
- If choice doesn’t affect updated estimate, results are more robust
- If choice matters, present range of results to demonstrate sensitivity to the prior

- Choice is based on judgement, subjectivity is unavoidable
- Use a range of options to test sensitivity of choice
- Clearly justify choice(s) of prior to make results credible to external consumers

- Summary of evidence from new data
- Combined with prior distribution to create updated posterior distribution

- Updated probability distribution for unknown quantity
- Reflects range of possible values and degree of belief associated with each value
- Less uncertainty than prior distribution; tends to be narrower than prior distribution and likelihood

- Prior distribution: evidence available before study
- Likelihood: evidence from study itself
- Posterior distribution: pools two sources of evidence by multiplying curves together [6]
- Prior distribution pulls likelihood towards null value (0), making final result less extreme

*Figure 1. Illustration of how Bayesian analysis combines a prior distribution (top graph) with the data (‘likelihood’, middle graph) to give the posterior distribution (bottom graph). Reproduced from BMJ, Spiegelhalter et al, 319, 508 1999.*

- Common for prior and posterior distributions to be related (same distribution or family of distributions)
- Makes calculations more feasible

- Summary measure (mean, median) often presented
- Other estimates (standard deviation, interquartile range) depend on distribution shape and context

- Posterior distribution is a probability distribution
- Can calculate probabilities for a specific range of values for estimated quantity
- Example: Probability that relative risk in a trial is greater than 1 (showing efficacy)

- Commonly present 95% credible intervals for posterior estimates
- Represents range within which true value lies with 95% probability
- Different from 95% confidence intervals (based on sampling distribution, not probability distribution)
- Easy to calculate for unimodal and symmetrical posterior probability distributions; more complex otherwise

**95% Credible Interval vs. 95% Confidence Interval**

- 95% probability that true value lies within 95% credible interval
- 95% probability that a 95% confidence interval contains true value
- Differences between the two are subtle and may not be practically significant for data interpretation

- No formal place in Bayesian framework
- Emphasis on distribution of estimates, not testing against a single value
- Posterior probability distribution can be used to calculate probability that true value takes specific values
- Bayesian approach provides information needed, rather than yes/no approach of significance testing
- Both Bayesian and frequentist statisticians agree that single value or test against single value is of limited usefulness

Bayesian methods are used in various medical research areas:

- Observational studies
- Design, monitoring, and analysis of trials
- Meta-analyses
- Missing data imputation
- Decision making
- Health economics

Applicable in situations similar to frequentist methods (single quantity estimation, regression analysis, multifactorial regression, multilevel models)

- Original GREAT trial examined effect of thrombolytic therapy on patients with suspected myocardial infarction
- Frequentist analysis reported significant beneficial effect on mortality
- Results challenged due to unexpected size, small trial, and more modest effect in unpublished bigger European trial
- Pocock and Spiegelhalter conducted Bayesian re-analysis, constructing prior distribution to express plausible reduction in mortality
- Bayesian analysis showed a reduction in mortality of 25% compared to 49% in frequentist analysis, suggesting over-optimistic trial results
- Demonstrates how Bayesian analysis can be used to combine prior evidence and new data for more reasonable estimates in cases of unexpected results

- Meta-analysis examined the potential link between antidepressant drugs and increased suicides in adults [10]
- Included unpublished data, corrected for duration of medication and placebo treatment
- 7 suicide attempts in patients taking the drug, 1 in patient taking placebo
- Prior distribution assumed to be gamma
- Three different prior distributions tested: pessimistic, slightly pessimistic, and slightly optimistic (Fig. 2)

*Figure 2. Three different prior distributions used in a meta-analysis of antidepressant drugs and suicide in adults. From Aursnes et al. [10]*

- Posterior distributions correspond to the three prior distributions
- Majority of each distribution is greater than 0, indicating evidence in favor of an adverse effect of paroxetine on suicide risk
- Authors reported relative risks of 2.46 (pessimistic prior), 2.20 (slightly pessimistic prior), and 2.34 (optimistic prior)
- Bayesian approach supports recent meta-analyses results, suggesting increased risk of suicidal activity in adults taking certain antidepressant drugs

*Figure 3. Three posterior distributions corresponding to the three priors used in a meta-analysis of antidepressant drugs and suicide in adults. From Aursnes et al. [10]*

Sung et. al have generated a checklist of seven items (ROBUST) that should be included when a Bayesian analysis is reported[11]. These are helpful in interpreting a Bayesian analysis.

The box below lists the items included in ROBUST. The checklist can be scored to provide a measure of the quality of reporting, but here it is given as a guide to what points to check when reading an article where Bayesian methods have been used.

- Prior distribution: specified
- Prior distribution: justified
- Prior distribution: sensitivity analysis
- Analysis: statistical model
- Analysis: analytical technique
- Results: central tendency
- Results: standard deviation or credible interval

This table is adapted from Spiegelhalter et al. and gives a helpful summary of the two approaches.

Issue |
Frequentist methods |
Bayesian methods |
---|---|---|

Prior information other than that in the study being analysed | Interpretation of the parameter of interest: Informally used when choosing a model/form of analysis, A fixed unknown value | Used formally by specifying a prior probability distribution, An unknown quantity which can have a probability distribution |

Basic statistical question | How likely are the data, given a particular value of the parameter? | How likely is the particular value of the parameter given the data? |

Presentation of results | P values, estimates, confidence intervals | Plots of posterior distribution of the parameter, calculation of specific posterior probabilities of interest, and use of the posterior distribution in formal decision analysis. Expected value and credible intervals |

Dealing with subsets in trials | Adjusted P values (e.g. Bonferroni) | Subset effects shrunk towards zero by a ‘sceptical’ prior |

*Adapted from Spiegelhalter et al. 1999.*

- Oxford handbook of medical statistics (Janet Peacock Philip J Peacock).
- Bayes T. An essay towards solving a problem in the doctrine of chances. Philos Trans Roy Soc 1763; 53:370–418.
- Kovacs MJ, Mackinnon KM, Anderson D, O’Rourke K, Keeney M, Kearon C et al. A comparison of three rapid D-dimer methods for the diagnosis of venous thromboembolism. Br J Haematol 2001; 115(1):140–4.
- Ashby D. Bayesian statistics in medicine: a 25 year review. Stat Med 2006; 25(21):3589–631.
- Spiegelhalter DJ, Abrams KR, Myles JP. Bayesian approaches to clinical trials and health-care evaluation. Chichester, West Sussex: Wiley, 2004.
- Spiegelhalter DJ, Myles JP, Jones DR, Abrams KR. Methods in health service research: An introduction to Bayesian methods in health technology assessment. BMJ 1999; 319(7208):508–12.
- Gelman A, Carlin John B, Stern Hal S, Rubin DB. Bayesian data analysis. 2nd ed. Boca Raton, FL: Chapman & Hall/CRC, 2004.
- GREAT group. Feasibility, safety, and efficacy of domiciliary thrombolysis by general practition- ers: Grampian Region Early Anistreplase Trial. GREAT Group. BMJ 1992; 305(6853):548–53.
- Pocock SJ, Spiegelhalter DJ. Domiciliary thrombolysis by general practitioners. BMJ 1992; 305(6860):1015.
- Aursnes I, Tvete IF, Gaasemyr J, Natvig B. Suicide attempts in clinical trials with paroxetine randomised against placebo. BMC Med 2005; 3:14.
- Sung L, Hayden J, Greenberg ML, Koren G, Feldman BM, Tomlinson GA. Seven items were identified for inclusion when reporting a Bayesian analysis of a clinical study. J Clin Epidemiol 2005; 58(3):261–8.