🐛Biostatistics Unit 3 – Probability Distributions in Biology

Key Concepts in Probability

• Probability quantifies the likelihood of an event occurring ranges from 0 (impossible) to 1 (certain)
• Sample space represents all possible outcomes of an experiment or trial
• Events are subsets of the sample space can be combined using set operations (union, intersection, complement)
• Mutually exclusive events cannot occur simultaneously in a single trial (rolling a 1 and a 6 on a die)
• Independent events do not influence each other's probability (flipping a coin multiple times)
  • Probability of independent events is calculated by multiplying individual probabilities
• Conditional probability measures the likelihood of an event given that another event has occurred
  • Calculated using the formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• Bayes' theorem allows updating probabilities based on new information relates conditional probabilities
  • Formula: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

Types of Probability Distributions

• Probability distributions describe the likelihood of different outcomes in a random variable
• Discrete distributions have a countable number of possible outcomes (number of mutations in a gene)
  • Examples: Binomial, Poisson, Geometric, Hypergeometric
• Continuous distributions have an infinite number of possible outcomes within a range (height of plants)
  • Examples: Normal (Gaussian), Exponential, Gamma, Beta
• Binomial distribution models the number of successes in a fixed number of independent trials with two possible outcomes (success or failure)
  • Characterized by parameters $n$ (number of trials) and $p$ (probability of success)
• Poisson distribution models the number of rare events occurring in a fixed interval of time or space (mutations per generation)
  • Characterized by parameter $\lambda$ (average rate of events)
• Normal distribution is symmetric and bell-shaped models many biological phenomena (body weight, IQ scores)
  • Characterized by parameters $\mu$ (mean) and $\sigma$ (standard deviation)

Biological Applications of Distributions

• Hardy-Weinberg equilibrium uses binomial distribution to model genotype frequencies in a population
  • Assumes no selection, mutation, migration, or genetic drift
• Poisson distribution models rare events in biology (number of mutations, species distribution patterns)
• Exponential distribution describes waiting times between events (time between mutations, survival times)
• Normal distribution applies to many continuous biological variables (height, weight, blood pressure)
  • Central Limit Theorem states that means of large samples are approximately normally distributed
• Gamma distribution models waiting times for a specified number of events to occur (time until $k$ mutations)
• Beta distribution is used for proportions and probabilities (allele frequencies, inheritance patterns)
• Hypergeometric distribution models sampling without replacement from a finite population (genotyping individuals)

Measures of Central Tendency and Spread

• Measures of central tendency describe the typical or central value in a dataset
• Mean is the arithmetic average calculated by summing all values and dividing by the number of observations
  • Sensitive to outliers and extreme values
• Median is the middle value when data is ordered from lowest to highest
  • Robust to outliers more appropriate for skewed distributions
• Mode is the most frequently occurring value in a dataset
  • Can have multiple modes (bimodal, multimodal) or no mode
• Measures of spread describe the variability or dispersion of data points
• Variance quantifies the average squared deviation from the mean
  • Calculated as: $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}$
• Standard deviation is the square root of variance
  • Measures spread in the same units as the original data
• Range is the difference between the maximum and minimum values
  • Sensitive to outliers provides limited information about overall spread
• Interquartile range (IQR) is the difference between the first and third quartiles
  • More robust to outliers than range

Hypothesis Testing with Distributions

• Hypothesis testing uses probability distributions to make statistical inferences about populations
• Null hypothesis ($H_0$) states that there is no significant effect or difference (no difference in mean height between two plant species)
• Alternative hypothesis ($H_a$) states that there is a significant effect or difference (mean height differs between species)
• Test statistic is a value calculated from sample data used to decide between the null and alternative hypotheses
  • Examples: z-score (normal distribution), t-score (t-distribution), chi-square (chi-square distribution)
• P-value is the probability of obtaining the observed test statistic or a more extreme value, assuming the null hypothesis is true
  • Smaller p-values provide stronger evidence against the null hypothesis
• Significance level ($\alpha$) is the threshold for rejecting the null hypothesis (commonly 0.05)
  • If p-value < $\alpha$, reject $H_0$ and conclude there is a significant effect
• Type I error (false positive) occurs when rejecting a true null hypothesis
  • Probability of Type I error is equal to the significance level ($\alpha$)
• Type II error (false negative) occurs when failing to reject a false null hypothesis
  • Probability of Type II error is denoted by $\beta$ and depends on sample size and effect size

Data Visualization Techniques

• Histograms display the frequency distribution of a continuous variable
  • Divide data into bins of equal width and plot the count or frequency of observations in each bin
• Box plots (box-and-whisker plots) summarize the distribution of a continuous variable
  • Show median, quartiles, and potential outliers
• Scatter plots display the relationship between two continuous variables
  • Each point represents an observation with its x and y coordinates
• Bar plots compare frequencies or means across categories
  • Height of each bar represents the frequency or mean for that category
• Pie charts show the proportion or percentage of each category in a whole
  • Angle and area of each slice correspond to its proportion
• Heatmaps visualize patterns in two-dimensional data
  • Colors represent values in a matrix (gene expression levels)
• Violin plots combine a box plot and a kernel density plot to show the distribution shape
  • Width of the "violin" represents the density of observations at each value

Real-world Examples in Biology

• Genome-wide association studies (GWAS) use probability distributions to identify genetic variants associated with traits or diseases
  • Test for significant differences in allele frequencies between cases and controls
• Epidemiological models use Poisson and exponential distributions to predict the spread of infectious diseases
  • Estimate parameters such as transmission rate and incubation period
• Population ecology models use probability distributions to describe species abundance, dispersal, and interactions
  • Examples: species-area relationships, metapopulation dynamics
• Quantitative genetics uses normal distribution to model the inheritance of continuous traits (height, weight)
  • Estimate heritability and response to selection
• Phylogenetic inference uses probability distributions to model the evolution of DNA sequences
  • Maximum likelihood and Bayesian methods estimate tree topologies and branch lengths
• Ecological niche modeling uses probability distributions to predict species' geographic distributions based on environmental variables
  • Examples: Maxent, Generalized Linear Models (GLMs)

Common Pitfalls and Misconceptions

• Confusing probability with certainty or frequency
  • Probability is a measure of uncertainty, not a guarantee of occurrence
• Misinterpreting p-values as the probability of the null hypothesis being true
  • P-value is the probability of observing the data given that the null hypothesis is true
• Overinterpreting non-significant results as evidence of no effect
  • Failure to reject the null hypothesis does not prove it true (absence of evidence is not evidence of absence)
• Assuming that statistical significance implies practical or biological significance
  • Small effects can be statistically significant with large sample sizes but may not be biologically meaningful
• Neglecting to check assumptions of statistical tests and models
  • Violations of assumptions (normality, independence, equal variances) can lead to invalid inferences
• Overfitting models to noise in the data
  • Overly complex models may fit the sample data well but fail to generalize to new data
• Misusing summary statistics without considering the underlying distribution
  • Different distributions can have the same mean or variance but different shapes and properties
• Confusing correlation with causation
  • Correlation does not imply a causal relationship between variables (confounding factors, reverse causation)