--- module: 101-03 language: en chapter: 101 title: "Quantitative Biostatistics for Biomedical and Clinical Science" module_title: "Regression, survival, prediction, causality, and missing data" source_sha256: 9473452c9e296d3b0a9627fdb6a18492d9de58ba14730207a49faedb22396cdd --- # Regression, survival, prediction, causality, and missing data ## Regression models and linear regression ### Describe, adjust, test interaction, predict ### Adjustment does not make a coefficient causal ### Scale, form, sampling, timing define parameters ### Linear: conditional mean of a continuous outcome #### Coefficient: difference per unit, others fixed ### Check residuals: nonlinearity, variance, dependence #### Residual normality matters mainly for small samples ## Predictor form and interaction ### Categorical predictors compare with a reference ### A straight line assumes constant per-unit effect ### Splines, polynomials, transforms model curvature ### Categorising loses information and power #### Thresholds depend on chosen cut points ### Interaction: association varies by another predictor #### Changes the effect on the model's own scale #### Absent on one scale, possibly present on another ## Logistic and count models ### Logistic models the log odds of a binary outcome #### Exponentiated coefficients are odds ratios #### Common outcomes: relative change looks exaggerated ### Linearity assumed on the log-odds scale ### Poisson or negative binomial for counts ### Rates need numerator, person-time, recurrence rule ### Repeated events within a person are dependent #### First-event analysis discards later burden #### Cluster models: population-average or subject-specific ## Confounding and variable selection ### Common cause of exposure and outcome ### Adjustment blocks measured confounding paths #### Adjusting a consequence removes part of the effect #### Adjusting a common effect creates collider bias ### Causal diagrams make assumptions explicit #### Identify an adjustment set but prove nothing ### Significance-based selection is unstable ### Choose confounders from causal knowledge ### Multicollinearity: imprecise separate coefficients #### A signal of non-identifiability, not a disease ## Survival analysis ### Censoring: exact event time unknown #### Assumed conditionally independent of events #### Sicker participants leaving biases estimates ### Kaplan-Meier: event-free probability over time #### Falls at events, not at censoring times #### Median unestimable if curve stays above half ### Hazard: instantaneous rate among the event-free ### Cox model assumes a constant hazard ratio #### Hazard ratio is not a risk ratio #### Crossing hazards: restricted mean survival time ### Competing events are not always ordinary censoring #### Censoring competitors estimates a hypothetical world #### Cumulative incidence: observed-world probability ## Discrimination, calibration, and thresholds ### Discrimination: higher risk, more events ### Calibration: predicted versus observed risk #### Good discrimination yet systematic overprediction ### Utility: do threshold decisions improve outcomes ### ROC curve ignores prevalence and error costs #### Precision-recall helps with rare outcomes ### A generic index assumes equal error costs ## Overfitting and validation ### Model learns sample-specific noise ### Training performance is optimistic ### Resampling estimates optimism if whole process repeated ### External validation tests transportability #### Case mix, measurement, treatment, prevalence differ ### Recalibrate intercept or slope ### Subgroup performance when harms are unequal ## Missing data ### Mechanisms defined relative to observed information #### Completely at random: unrelated to any values #### At random: depends only on observed variables #### Not at random: depends on unseen values ### Assumptions about a process, not test results ### Complete-case analysis unbiased only narrowly #### Loses precision, may shift target population ### Single imputation understates uncertainty ### Multiple imputation combines within and between ## Sensitivity analysis and clinical deployment ### Vary untestable assumptions plausibly #### Missingness, unmeasured confounding, model form ### Robustness is not reporting only agreeing analyses ### Fragile if modest assumptions change the result ### Data leakage creates invalid performance ### Interpretability proves neither cause nor fairness ### Clinical use: validation, calibration, monitoring ### Align question, estimand, process, and model