Biostatistics
Biostatistics supports inference about populations from samples, estimation of effects with a measure of uncertainty, prediction, and the evaluation of evidence for hypotheses.
Biostatistics (also called biometry or biometrics) is the development and application of statistical methods to questions in biology, medicine, and public health. It covers the design of studies and experiments, the collection and analysis of data, and the interpretation of results under uncertainty. Biological data are variable because of differences among individuals, measurement error, and random processes, so methods are needed to separate real effects from chance. Biostatistics is used in clinical trials, epidemiology, genetics, genomics, ecology, and bioinformatics.
Biostatistics supports inference about populations from samples, estimation of effects with a measure of uncertainty, prediction, and the evaluation of evidence for hypotheses. It does not by itself establish causation. Causal conclusions depend on study design, such as randomization, and on assumptions about confounding and bias, which statistical models can help to examine but cannot replace.
Historical Evolution: How Did Biometry Shape Modern Statistical Theory?
Pioneers of Biometry and Experimental Design
Early work on the statistics of heredity and variation was done by Francis Galton, who introduced regression and correlation, and by Karl Pearson, who developed the chi-squared test, the correlation coefficient, and the journal Biometrika (1901) with W. F. R. Weldon. Ronald Fisher developed the analysis of variance, maximum likelihood, the design of experiments (including randomization), and the use of the _p_-value in the 1920s and 1930s, mainly in agriculture and genetics. William Sealy Gosset, writing as "Student", derived the _t_-distribution for small samples. Jerzy Neyman and Egon Pearson formulated the theory of hypothesis testing with error rates and confidence intervals. The first randomized clinical trials, such as the 1948 Medical Research Council trial of streptomycin, led by Austin Bradford Hill, established the use of randomization in medicine. Later developments include the Kaplan-Meier estimator (1958), the Cox proportional hazards model (1972), generalized linear models (Nelder and Wedderburn, 1972), the false discovery rate (Benjamini and Hochberg, 1995), and the wide use of computational methods such as the bootstrap and Markov chain Monte Carlo.
Probability Distributions in Biology: How Are Stochastic Biological Processes Modeled?
Parametric Distributions for Continuous and Discrete Biological Data
Statistical models describe the data by probability distributions. The choice of distribution depends on the type of variable and on how it is generated.
- Normal distribution. It is used for many continuous measurements, such as height or blood pressure, which are approximately symmetric. The central limit theorem states that sums or means of many independent contributions, under general conditions, tend toward a normal distribution. This explains why sample means are approximately normal even when the data are not, and why traits affected by many small additive genetic and environmental effects are often approximately normal. Many biological variables are skewed and are better modeled after a log transformation, such as concentrations and viral loads, which are often approximately log-normal.
- Binomial distribution. It models the number of successes k in n independent trials, each with probability p of success, with probability:
- Poisson distribution. It models counts of events that occur independently at a constant rate λ per unit of time or space:
- Negative binomial distribution. Biological counts often have a variance larger than the mean, which is called overdispersion. The reasons include heterogeneity among individuals, clustering, and variation among biological replicates. A common model is the negative binomial distribution, which can be seen as a Poisson distribution whose rate varies according to a gamma distribution. With mean μ and dispersion parameter φ, its variance is:
- Other distributions. The exponential and Weibull distributions are used for times to events, the gamma distribution for positive skewed data, the beta distribution for proportions, and the multinomial distribution for counts in more than two categories.
| Probability Distribution | Random Variable Type | Mean E(X) | Variance Var(X) | Canonical GLM Link | Primary Biological / Clinical Application |
|---|---|---|---|---|---|
| Normal (Gaussian) | Continuous (−∞, +∞) | μ | σ2 | Identity: _g_(μ) = μ | Quantitative polygenic traits, height, blood pressure |
| Binomial | Discrete count (0, 1, ..., n) | np | _np_(1 − p) | Logit: ln[p / (1 − p)] | Clinical trial response rates, Mendelian segregation |
| Poisson | Discrete count (0, 1, 2, ...) | λ | λ | Log: ln(λ) | Spontaneous mutation counts, radioactive decay, colony counts |
| Negative Binomial | Overdispersed count (0, 1, 2, ...) | μ | μ + φμ2 | Log: ln(μ) | High-throughput RNA-Seq transcript read counts (DESeq2/edgeR) |
| Exponential / Weibull | Continuous time t ∈ [0, +∞) | 1 / λ (Exponential) | 1 / λ2 (Exponential) | Log-hazard (Cox PH) | Time-to-relapse, patient survival, equipment failure times |
Study Design: How Do Experimental Architecture and Randomization Prevent Bias?
The design of a study determines which questions it can answer.
- Randomized controlled trials assign participants at random to treatments, which on average balances both known and unknown confounders. Blinding, allocation concealment, and pre-specified analysis plans reduce bias. Phase I to III trials examine safety, dose, and efficacy.
- Observational studies include cohort studies (which follow exposed and unexposed groups), case-control studies (which compare cases with controls), and cross-sectional studies. They can be affected by confounding, selection bias, and information bias.
- Experimental design principles include replication, randomization, and blocking. Fisher's principles aim to estimate error and to reduce variation by grouping similar units. Pseudoreplication, in which non-independent measurements are treated as independent replicates, is a common error.
Sample Size and Power
The sample size needed depends on the effect size, the variability, the type I error rate α, and the desired power. For comparing the means of two groups with a two-sided test, with common standard deviation σ and true difference δ, the number of participants per group is approximately:
where z denotes a quantile of the standard normal distribution. For α = 0.05 and a power of 80 percent (β = 0.2), z1−_α_/2 = 1.96 and z1−_β_ = 0.84, so (1.96 + 0.84)2 ≈ 7.85. If δ / σ = 0.5, then n ≈ 2 × 7.85 / 0.25 ≈ 63 per group. Halving the detectable difference quadruples the sample size.
Frequentist Inference: How Do Hypothesis Tests and Confidence Intervals Quantify Uncertainty?
The classical approach in biostatistics is frequentist, in which probability is the long-run frequency of outcomes in repeated sampling. A widely used procedure is null hypothesis significance testing (NHST).
Hypotheses and Error Rates
The null hypothesis H0 typically states that there is no effect or no difference. The alternative hypothesis HA states that there is one. Two errors are possible:
- A type I error is the rejection of a true null hypothesis (a false positive). Its probability is the significance level α, which is conventionally set at 0.05 in many fields. The value is a convention and not a law. Some fields use stricter values, such as 5 × 10−8 in genome-wide association studies.
- A type II error is the failure to reject a false null hypothesis (a false negative), with probability β. The power is 1 − β, the probability of detecting an effect of a given size when it exists. Underpowered studies detect real effects less often, and the significant results that they do produce tend to overestimate the effect size.
The p-Value
The _p_-value is the probability, under the assumption that the null hypothesis and all other model assumptions are true, of obtaining a test statistic at least as extreme as the one observed. It does not measure the probability that the null hypothesis is true, the probability that the result occurred by chance alone, the size of an effect, or its biological or clinical importance. A small _p_-value shows that the data are unusual under the model, and a large one does not show that the null hypothesis is true. The American Statistical Association issued a statement in 2016 that warned against the use of a threshold alone, and many journals now require effect sizes and confidence intervals. Practices that inflate false positives include testing many outcomes and reporting only those that are significant ("_p_-hacking") and flexibility in analysis choices.
Confidence Intervals
A 95 percent confidence interval is computed by a procedure that, in repeated sampling, contains the true parameter value in 95 percent of samples. It is not the probability that the true value lies in the particular interval that was computed. For a sample mean x̄ with standard deviation s and size n, an approximate interval is:
Intervals show the precision and the range of effects that are compatible with the data, so they carry more information than a _p_-value alone.
Common Tests
Common tests include the _t_-test (comparison of means), the analysis of variance (more than two groups), the chi-squared test and Fisher's exact test (counts in categories), and tests of correlation. Parametric tests assume a distribution, and their results can be misleading when the assumptions are violated, especially in small samples.
Nonparametric Methods
Nonparametric methods make fewer assumptions about the distribution. Many use the ranks of the observations and not their values. The Wilcoxon rank-sum test (equivalent to the Mann-Whitney U test) compares two independent groups, and it tests whether one group tends to have larger values (stochastic dominance). It is a test of medians only if the distributions have the same shape. The Wilcoxon signed-rank test is for paired data, and the Kruskal-Wallis test is for more than two groups. Permutation tests and the bootstrap use resampling of the data and are widely used when theory is difficult. Nonparametric tests have somewhat lower power than parametric tests when the parametric assumptions hold, and they are useful when data are skewed, have outliers, or the sample is small.
Regression and Multivariable Models: How Do Generalized Linear Models Adjust for Confounding?
Linear, Logistic, Survival, and Mixed-Effects Models
Regression models relate an outcome to one or more predictors and allow adjustment for confounders. They belong to the class of generalized linear models (GLMs), which have a distribution for the outcome, a linear predictor, and a link function.
- Linear regression. For a continuous outcome Y and predictors X1 to Xp:
- Logistic regression. For a binary outcome with probability p, the model is linear on the log-odds (logit) scale:
- Poisson and negative binomial regression are used for counts, with a log link, and for rates with an offset for person-time.
- Survival analysis. Time-to-event outcomes are often censored, which means that the event has not occurred in some participants when observation ends or they are lost to follow-up. The survival function _S_(t) = _P_(T > t) is estimated by the Kaplan-Meier estimator:
- Mixed-effects models. Observations are often correlated, for example repeated measurements on one patient, animals from one litter, or cells from one tissue sample. Treating them as independent gives standard errors that are too small and inflates false positives. Linear and generalized linear mixed models include fixed effects for the effects of interest and random effects that capture variation among clusters. Alternatives include generalized estimating equations, which estimate population-averaged effects.
Confounding and Model Building
A confounder is associated with both the exposure and the outcome and is not on the causal pathway. It can be handled by design (randomization, matching), by stratification, or by adjustment in a regression model. Adjustment cannot control for unmeasured confounders. Causal inference methods include propensity scores, instrumental variables, difference-in-differences, and Mendelian randomization, in which genetic variants serve as instruments for an exposure. Model selection and validation are important. Overfitting, in which a model describes noise in the sample, is avoided by cross-validation, penalized regression (the lasso and ridge), and external validation.
Multiple Testing and High-Dimensional Data: How Are False Positives Controlled in Genomics?
Family-Wise Error Rate versus False Discovery Rate
Genomics, proteomics, and imaging produce data in which thousands to millions of variables are tested at once. If m independent tests are done at level α when all null hypotheses are true, the expected number of false positives is mα. For m = 20,000 genes and α = 0.05, about 1,000 false positives are expected, and the probability of at least one is 1 − (1 − α)m, which is essentially 1. The multiple comparisons problem is addressed by controlling one of two error rates:
- The family-wise error rate (FWER), the probability of at least one false positive. The Bonferroni correction tests each hypothesis at α / m (for example 0.05 / 20,000 = 2.5 × 10−6). It is simple and valid under any dependence, but it is conservative, and it reduces power. The Holm procedure is uniformly more powerful with the same guarantee.
- The false discovery rate (FDR), the expected proportion of false positives among the rejected hypotheses. In the Benjamini-Hochberg procedure, the _p_-values are sorted as p(1) ≤ ... ≤ p(m), and the hypotheses with p(i) ≤ (i / m) q are rejected, where q is the target FDR. The procedure is valid for independent tests and for some forms of positive dependence. FDR control is the usual approach in transcriptomics and in other exploratory work, because it gives more power than FWER control. Adjusted values are called _q_-values.
High-dimensional data also have more variables than samples, which requires regularization, dimension reduction (such as principal component analysis), and methods that share information among genes, such as the empirical Bayes moderation in limma. Batch effects are a technical source of variation, and are controlled by design and adjusted by statistical methods. Reproducibility is improved by pre-registration, data and code sharing, and independent replication.
Bayesian Statistics: How Does Bayes' Theorem Update Prior Knowledge with Empirical Evidence?
Posterior Distributions and Markov Chain Monte Carlo Sampling
In Bayesian statistics, probability expresses a degree of belief or uncertainty about unknown quantities. Parameters are treated as random variables, and the data are fixed once observed. Bayes' theorem combines the prior distribution _p_(θ) with the likelihood _p_(y | θ) of the data to give the posterior distribution:
The prior can come from earlier studies, expert knowledge, or a weakly informative choice. From the posterior, one can compute the probability that a parameter lies in a range (a credible interval) or that a hypothesis holds, given the model and prior. Results depend on the prior, especially for small samples, so sensitivity analyses are advised. Bayesian methods are useful for hierarchical models, for combining sources of information, for adaptive clinical trials in which the design is changed by accumulating data, and for phylogenetics and population genetics, and regulators accept Bayesian designs in some settings.
A simple example is a beta prior for a binomial proportion. If the prior is Beta(a, b) and k successes are observed in n trials, the posterior is Beta(a + k, b + n − k). With a uniform prior (a = b = 1) and 7 responses among 20 patients, the posterior is Beta(8, 14), with mean 8/22 ≈ 0.36.
Posterior distributions of most realistic models cannot be found analytically, because the normalizing constant _p_(y) is an integral over a high-dimensional space. Markov chain Monte Carlo (MCMC) algorithms draw dependent samples from the posterior. They include the Metropolis-Hastings algorithm, Gibbs sampling, and Hamiltonian Monte Carlo (used in Stan). Convergence is assessed with diagnostics, such as multiple chains, the R-hat statistic, and effective sample size. Other approximations are variational inference and integrated nested Laplace approximation. Advances in computing, since about the 1990s, have made Bayesian analysis practical.
Specialized Biostatistical Domains: How Are Diagnostic Accuracy, Missing Data, and Meta-Analyses Evaluated?
Clinical Diagnostics, Imputation, and Evidence Synthesis
- Diagnostic tests. The performance of a test is described by sensitivity (the probability of a positive result in people with the disease), specificity, and the positive and negative predictive values, which depend on prevalence. By Bayes' theorem, if prevalence is 1 percent, sensitivity is 90 percent, and specificity is 95 percent, then the positive predictive value is 0.009 / (0.009 + 0.0495) ≈ 15 percent. The receiver operating characteristic curve and the area under it summarize the performance across thresholds.
- Missing data. Missing values can bias results unless handled appropriately, for example by multiple imputation. The mechanism (missing completely at random, at random, or not at random) determines which methods are valid.
- Meta-analysis. It combines estimates from several studies, weighting each by its precision, with fixed-effect or random-effects models, and it assesses heterogeneity and publication bias.
- Statistical genetics. Methods include linkage and association analysis, estimation of heritability, and polygenic scores, with correction for population structure.
- Software. Common tools include R (with Bioconductor for genomic data), Python, SAS, and Stata.
Frequently Asked Scientific Questions
Why is the negative binomial distribution preferred over the Poisson distribution for RNA-Seq data?
A Poisson distribution strictly assumes that the variance of read counts equals the mean (Var(K) = μ), which holds only for purely technical sequencing replicates. Across biological replicates, inter-individual heterogeneity and transcriptional bursting introduce substantial biological variance, causing overdispersion where Var(K) = μ + φμ2. The negative binomial distribution explicitly models this quadratic dispersion parameter (φ), preventing massive inflation of false-positive differential expression calls.
What is the difference between a p-value and the False Discovery Rate (FDR)?
A _p_-value is the probability of observing a test statistic at least as extreme as the data given that a single null hypothesis is true, controlling the false-positive rate among all truly null tests. In contrast, the False Discovery Rate (FDR, or _q_-value), controlled via the Benjamini-Hochberg procedure, estimates the expected proportion of false positives specifically among the subset of hypotheses rejected as significant.
How does right-censoring affect survival analysis in clinical trials?
Right-censoring occurs when a patient has not experienced the clinical endpoint (such as death or relapse) before the trial concludes or the patient withdraws. Discarding censored patients or treating them as event-free biases survival estimates downward. The Kaplan-Meier estimator and Cox proportional hazards model retain censored individuals in the risk set (ni) up to their exact censoring time, yielding unbiased survival probabilities under non-informative censoring.