🐛Biostatistics Unit 1 – Biostatistics: Intro to Probability Theory

Key Concepts and Definitions

• Probability the likelihood or chance of an event occurring, expressed as a number between 0 and 1
  • 0 indicates an impossible event, while 1 represents a certain event
• Sample space the set of all possible outcomes of an experiment or random process (rolling a die)
• Event a subset of the sample space, representing one or more outcomes of interest (rolling an even number)
• Random variable a function that assigns a numerical value to each outcome in a sample space
  • Discrete random variables have countable values (number of defective items in a batch)
  • Continuous random variables can take on any value within a range (patient's blood pressure)
• Probability distribution a function that describes the likelihood of different outcomes for a random variable
• Independence two events are independent if the occurrence of one does not affect the probability of the other

Probability Basics

• Addition rule for mutually exclusive events: $P(A \cup B) = P(A) + P(B)$
• Multiplication rule for independent events: $P(A \cap B) = P(A) \times P(B)$
• Conditional probability the probability of an event A occurring given that event B has already occurred, denoted as $P(A|B)$
  • Calculated using the formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• Bayes' theorem a method for updating probabilities based on new information or evidence
  • Formula: $P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$
• Law of total probability a way to calculate the probability of an event by considering all possible ways it can occur
• Complementary events two events that are mutually exclusive and exhaustive, meaning their probabilities sum to 1

Types of Probability

• Classical probability based on the assumption of equally likely outcomes (fair coin toss)
• Empirical probability estimated from observed data or past experiences (probability of a patient responding to a treatment based on clinical trials)
• Subjective probability based on personal belief or judgment, often used in decision-making under uncertainty (expert opinion on the likelihood of a disease outbreak)
• Axiomatic probability a formal mathematical approach that defines probability using a set of axioms
  • Non-negativity: $P(A) \geq 0$ for any event A
  • Normalization: $P(S) = 1$, where S is the sample space
  • Additivity: For mutually exclusive events A and B, $P(A \cup B) = P(A) + P(B)$
• Geometric probability involves calculating probabilities based on geometric properties (probability of a randomly thrown dart landing in a specific region of a dartboard)

Probability Distributions

• Binomial distribution models the number of successes in a fixed number of independent trials with two possible outcomes (number of patients who respond to a treatment in a clinical trial)
  • Parameters: n (number of trials) and p (probability of success in each trial)
  • Probability mass function: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
• Poisson distribution models the number of rare events occurring in a fixed interval of time or space (number of mutations in a DNA sequence)
  • Parameter: λ (average rate of events)
  • Probability mass function: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$
• Normal distribution a continuous probability distribution that is symmetric and bell-shaped, often used to model natural phenomena (distribution of heights in a population)
  • Parameters: μ (mean) and σ (standard deviation)
  • Probability density function: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
• Exponential distribution models the time between events in a Poisson process (time between patient arrivals at a hospital)
• Uniform distribution a continuous probability distribution where all outcomes within a range are equally likely (random selection of a number between 0 and 1)

Applications in Biostatistics

• Diagnostic testing calculating sensitivity, specificity, and predictive values using probability concepts
  • Sensitivity: $P(\text{positive test} | \text{disease})$
  • Specificity: $P(\text{negative test} | \text{no disease})$
  • Positive predictive value: $P(\text{disease} | \text{positive test})$
  • Negative predictive value: $P(\text{no disease} | \text{negative test})$
• Epidemiology using probability to study the distribution and determinants of health-related events in populations
  • Incidence rate: probability of developing a disease within a specified time period
  • Prevalence: probability of having a disease at a given point in time
• Clinical trials designing and analyzing studies to assess the efficacy and safety of medical interventions
  • Randomization: assigning subjects to treatment groups based on probability to minimize bias
  • Sample size calculation: determining the number of subjects needed to detect a significant treatment effect with a given probability
• Genetics applying probability concepts to study the inheritance of traits and genetic disorders
  • Mendelian inheritance: calculating probabilities of genotypes and phenotypes based on parental genotypes
  • Hardy-Weinberg equilibrium: a probability model for predicting genotype frequencies in a population

Data Analysis Techniques

• Hypothesis testing using probability to make decisions about population parameters based on sample data
  • Null hypothesis: a statement of no effect or no difference, assumed to be true unless evidence suggests otherwise
  • Alternative hypothesis: a statement that contradicts the null hypothesis, representing the research question of interest
  • P-value: the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true
  • Significance level (α): the probability threshold for rejecting the null hypothesis, typically set at 0.05
• Confidence intervals estimating a range of plausible values for a population parameter with a given level of confidence (95% confidence interval for a population mean)
• Bayesian inference updating prior probabilities based on observed data to obtain posterior probabilities
  • Prior probability: the initial probability of an event or hypothesis before considering new evidence
  • Likelihood: the probability of observing the data given a specific hypothesis
  • Posterior probability: the updated probability of a hypothesis after considering the observed data
• Markov chains a probability model for analyzing systems that transition between states over time (modeling disease progression)
• Monte Carlo simulation a technique for estimating probabilities and other quantities by generating random samples from a probability distribution (estimating the probability of a rare event)

Real-World Examples

• Medical decision-making using probability to guide diagnostic and treatment decisions (probability of a patient having a disease given their symptoms and test results)
• Insurance risk assessment calculating premiums based on the probability of events such as accidents, illnesses, or natural disasters
• Quality control monitoring manufacturing processes to ensure that the probability of defective items remains within acceptable limits
• Weather forecasting using probability to predict the likelihood of various weather events (probability of rain, hurricane landfall)
• Financial modeling estimating the probability of investment returns, loan defaults, or other economic events to inform decision-making

Common Pitfalls and Misconceptions

• Gambler's fallacy the mistaken belief that past events influence the probability of future independent events (thinking a coin is "due" for heads after a series of tails)
• Confusion of conditional probabilities misinterpreting $P(A|B)$ as $P(B|A)$ or failing to account for the base rate of an event
• Neglecting the sample space focusing only on the event of interest without considering all possible outcomes
• Misusing the law of averages believing that deviations from the expected value will be "balanced out" in the short term
• Overreliance on small sample sizes drawing conclusions based on insufficient data, leading to inaccurate probability estimates
• Misinterpreting p-values as the probability of the null hypothesis being true, rather than the probability of observing the data under the null hypothesis
• Confusing statistical significance with practical significance a result may be statistically significant but have little real-world impact
• Failing to account for multiple testing performing numerous hypothesis tests without adjusting the significance level, increasing the risk of false positives