Preventive Medicine: Biostatistics 1

Bayes Theorem

• Bayes theorem is used to answer two clinical questions: what is the probability of a patient having a disease if the test results are positive, and what is the probability of a patient not having a disease if the test results are negative.
• The formula for Bayes theorem is: p(D+ | T+) = p(T+ | D+)p(D+) / [p(T+|D+)p(D+)+ p(T+|D-)p(D-)]
• The numerator of Bayes theorem describes the true-positive results (cell a in a 2x2 table), and the denominator consists of two terms: the true-positive results and the false-positive results (cell b in a 2x2 table).
• The positive predictive value (PPV) is calculated by dividing the true-positive results by the sum of true-positive and false-positive results.

Community Screening Programs

• In a population with a low prevalence of a disease, most positive test results are likely to be false positives.
• Bayes theorem can be used to predict the proportion of true-positive results among people with positive test results.
• The theorem takes into account the sensitivity and specificity of the test, as well as the prevalence of the disease in the community.

Individual Patient Care

• Bayes theorem can be used to calculate the posterior probability of a patient having a disease after a positive test result.
• The prior probability is the estimated probability of the disease before the test result, and the posterior probability is the updated probability after the test result.
• The theorem takes into account the sensitivity and specificity of the test, as well as the prior probability of the disease.

Influence of the Sequence of Testing

• The sequence of testing can affect the overall accuracy of the diagnosis.
• A sequential approach, where tests are performed one after another, can be more conservative and economical than a simultaneous approach.
• The sequence of testing should start with the most sensitive test and continue with increasingly specific tests if the previous test yields positive results.

Decision Analysis

• Decision analysis is a tool used to improve decision making under conditions of uncertainty.
• It involves identifying the problem, diagramming the options, obtaining information on each option, comparing the utility values, and performing sensitivity analysis.
• The goal of decision analysis is to help health care workers understand the types of data that must go into a clinical decision, the sequence in which decisions need to be made, and the personal values of the patient that must be considered.

Steps in Creating a Decision Tree

• Identify and set limits to the problem.
• Diagram the options.
• Obtain information on each option.
• Compare the utility values.
• Perform sensitivity analysis.

Example of a Decision Tree

• The example illustrates a decision tree for a patient with silent gallstones, where the clinician must decide whether to operate immediately or to wait.

• The decision tree shows the possible outcomes of each option, including the probability of death from operating immediately or waiting.

• The utility values of each outcome are compared to determine the best course of action.

• Sensitivity analysis is performed to see how the results of the analysis are affected by changes in the assumptions.### Decision Trees

• Decision trees are used to analyze complex issues and factors, and need to be updated as new data and assumptions arise.

• The objective is to find decisions that are clearly less satisfactory than others and prune them as they are not rational alternatives.

• Decision trees are used to choose the best branch at each decision node, working backward from the right to the left, known as folding back.

• Decision trees are applicable only in problems with non-repetitive outcomes, and not in problems with recurring outcomes, such as embolic strokes in patients with atrial fibrillation.

Applications of Decision Trees

• Decision trees are used in clinical settings, such as in the case of patients with asymptomatic gallstones.
• Decision trees are also applied to public health problems, such as strategy analysis for eliminating the problem of hepatitis B virus (HBV).

Data Synthesis

• Large data sets are essential for reaching reliable and generalizable conclusions in medical literature.
• Statistical testing of treatment effects is unnecessary when the entire population is involved.
• Large test populations are more likely to approximate truth for the population at large, providing both statistical power and external validity/generalizability.
• Data synthesis can be achieved through large, multisite intervention trials, or by aggregating findings from multiple smaller trials.

Systematic Review

• A systematic review is an aggregation of findings from multiple studies addressing a similar research question in a similar way.
• Systematic reviews have prespecified criteria for inclusion, explicit methodology, systematic search, assessment of validity, and systematic presentation and synthesis of findings.
• Systematic reviews may be purely qualitative or include quantitative data synthesis (meta-analysis).

Meta-Analysis

• Meta-analysis is a quantitative synthesis of findings from multiple studies, aggregating results to establish a composite impression or measure of the strength of a particular association.
• Meta-analysis requires a systematic review of the literature and employs strict criteria for the selection of pertinent studies.
• There are two forms of quantitative meta-analysis: analyzing reported data from the literature, or aggregating raw data from multiple studies.
• A meta-analysis begins with a systematic review, and variability in methods among studies is typically measured in a test of homogeneity.
• Forest plots are used to display the results of meta-analysis, showing the degree of statistical significance and association.

Evidence Mapping

• Evidence mapping is a method used to characterize the quantity and quality of evidence in a broad topic area, too broad to warrant a systematic review.
• It provides an overview of relevant evidence, which can guide the articulation of specific, more narrowly framed questions.
• The World Health Organization (WHO) has adopted evidence mapping as a support in healthcare policy development.

Elementary Probability Theory

• Three basic rules of probability should be kept in mind: the independence rule, the product rule, and the addition rule.
• The independence rule states that one probability is not influenced by the outcome of another probability.
• The product rule is used to determine the probability of two things being true, and the calculation depends on whether the two things are independent.
• The addition rule is used to determine the probability of one thing being true under all possible conditions, and all possible probabilities must add up to 1 (100%).

Independence Rule

• The independence rule assumes that the probability of one event is not influenced by the outcome of another event.
• If the two events are independent, the correct probabilities can be obtained by many trials of flipping an unbiased coin, repeated many times.
• The independence rule does not require that the two events have an equal probability of occurrence.

Product Rule

• The product rule is used to calculate the probability of two things being true.
• If the two events are independent, the probability of both events occurring jointly is the product of their separate probabilities.
• The product rule can be expressed as: p(H+ and W+) = p(H+) × p(W+) = 0.5 × 0.5 = 0.25

Addition Rule

• The addition rule is used to determine the probability of one thing being true under all possible conditions.
• The rule states that the sum of all possible probabilities must equal 1 (100%).
• The addition rule can be used to calculate the lifetime probability of an event occurring, taking into consideration multiple conditions.

Bayes Theorem

• Bayes theorem is a tool used in decision analysis to calculate positive predictive values and posterior probabilities.
• The numerator and denominator of Bayes theorem are based on the general product rule and the addition rule.

Decision Trees

• Decision trees are a tool used to help healthcare workers pursue a logical, step-by-step approach to exploring alternative clinical decisions.
• Decision trees can help identify the sequence of decisions that must be made, and the probabilities and utilities of each possible outcome.