npbuilds Biostatistics — The Clinical Lens
Statistical methods shaped by regulatory requirements, censored data, and the clinical consequences of being wrong. Every method choice here must be justifiable in a pre-specified statistical analysis plan, because the wrong decision can delay a treatment that saves lives — or approve one that doesn't.
The Estimands Framework (ICH E9 R1)
The modern organizing principle for clinical trial statistics. Before choosing any method, define the estimand — the precise clinical question your analysis answers.
An estimand has five attributes:
| Attribute | Defines | Example |
|---|---|---|
| Population | Who | Adults with moderate-to-severe plaque psoriasis |
| Treatment | What intervention | Drug X 300mg vs placebo |
| Endpoint | What outcome | PASI 90 response at Week 16 |
| Intercurrent events | What happens post-randomization that complicates interpretation | Treatment discontinuation, rescue medication, death |
| Summary measure | How treatment effects are quantified | Difference in proportions, hazard ratio, RMST difference |
Intercurrent event strategies — the key innovation:
| Strategy | Handles IC Event By | Use When |
|---|---|---|
| Treatment policy | Analyze regardless (classic ITT) | The policy question: "what happens if we prescribe this?" |
| Composite | Fold IC event into the endpoint | Death before response = non-responder |
| While-on-treatment | Only count time on assigned treatment | Interest is in direct pharmacological effect |
| Hypothetical | Estimate what would have happened without the IC event | "What if no one discontinued?" |
| Principal stratum | Focus on subgroup defined by IC event behavior | Effect in those who would comply regardless of arm |
Practical impact: Align the estimand with the clinical question BEFORE designing the trial. The estimand drives the design, data collection, and analysis — not the other way around.
Survival Analysis (Time-to-Event)
The foundational biostatistics method. Used whenever the outcome is "time until something happens" and some subjects haven't experienced the event yet (censoring).
Core Methods
Kaplan-Meier estimator: Non-parametric survival curve. The default visualization for time-to-event data. Reports median survival and survival probabilities at landmark times.
Log-rank test: Compares survival curves between groups. Optimal when hazards are proportional (constant hazard ratio over time). For non-proportional hazards (common in immunotherapy), consider:
- MaxCombo test: Combines multiple weighted log-rank tests. Better power when treatment effects are delayed or diminishing.
- RMST difference: Restricted mean survival time. Compares average survival within a clinically meaningful time horizon. Increasingly recommended as a primary or co-primary endpoint.
Cox proportional hazards model: Semi-parametric regression. Estimates hazard ratios adjusted for covariates.
- Key assumption: Proportional hazards — the hazard ratio is constant over time. Test via Schoenfeld residuals, log-log survival plots, or time-covariate interactions.
- When PH fails: Use time-varying coefficients, stratified Cox, or switch to RMST/milestone analysis.
Competing risks: When subjects can experience one of several mutually exclusive events (e.g., death from disease vs. death from other causes).
- Standard KM overestimates cumulative incidence when competing risks exist.
- Cumulative incidence function (CIF): Accounts for competing events. Fine-Gray model for regression.
- Cause-specific hazards: Alternative approach — model each event type separately.
Parametric Models
| Distribution | Hazard Shape | Use When |
|---|---|---|
| Exponential | Constant | Simple survival, memoryless process |
| Weibull | Monotone increasing/decreasing | Hazard changes steadily over time |
| Log-normal | Non-monotone (hump-shaped) | Initial risk peaks then declines |
| Log-logistic | Non-monotone | Similar to log-normal, heavier tails |
Accelerated failure time (AFT) models are the parametric alternative to Cox PH — they model how covariates accelerate or decelerate time-to-event rather than multiplicatively shifting hazards.
Clinical Trial Design
Randomization
| Method | When to Use | Key Property |
|---|---|---|
| Simple | Large trials (n > 200) | Unpredictable but may produce imbalance |
| Permuted block | Most trials | Ensures balance within blocks; block size should vary and be concealed |
| Stratified block | Important prognostic factors (site, disease severity) | Balance across strata |
| Covariate-adaptive (minimization) | Many prognostic factors | Best overall balance; slightly predictable |
| Response-adaptive | Ethical pressure to assign more patients to better arm | Complex; can introduce time trends |
Sample Size Essentials
Every trial requires a justified sample size. Key inputs:
- Significance level (alpha): Usually 0.025 one-sided (= 0.05 two-sided) for regulatory trials
- Power (1 - beta): Typically 80-90%
- Clinically meaningful difference: The minimum effect the trial is designed to detect — not the expected effect, but the smallest worthwhile effect
- Variance estimate or event rate: From prior studies, pilot data, or literature
For survival trials, additionally specify: accrual rate, accrual period, follow-up period, and expected dropout/crossover rates. The number of events (not patients) drives power.
Group Sequential Designs
Planned interim analyses that allow early stopping for efficacy or futility while controlling overall type I error.
| Boundary | Philosophy | Alpha Spent Early | Alpha Remaining for Final |
|---|---|---|---|
| O'Brien-Fleming | Conservative early, liberal late | Very little | Most |
| Pocock | Equal stopping criteria at each look | Equal portions | Less at final |
| Lan-DeMets | Flexible alpha-spending function | Customizable | Depends on function |
Critical rule: Each interim look inflates the overall type I error. You cannot simply do repeated significance tests at alpha = 0.05. The spending function controls how much alpha is "spent" at each interim and how much is reserved for the final analysis.
Adaptive Designs
Pre-specified modifications during the trial based on accumulating data:
- Sample size re-estimation: Adjust N based on observed variance or effect size (blinded or unblinded)
- Arm dropping: Remove underperforming treatment arms in multi-arm trials
- Population enrichment: Narrow the population to a responsive subgroup
- Seamless Phase 2/3: Combine learning and confirmatory phases
All adaptations must be pre-specified in the protocol and validated via simulation to demonstrate type I error control.
Diagnostic Test Evaluation
| Metric | Formula | Depends on Prevalence? | Interpretation |
|---|---|---|---|
| Sensitivity | TP / (TP + FN) | No | Probability of positive test given disease |
| Specificity | TN / (TN + FP) | No | Probability of negative test given no disease |
| PPV | TP / (TP + FP) | Yes | Probability of disease given positive test |
| NPV | TN / (TN + FN) | Yes | Probability of no disease given negative test |
| LR+ | Sensitivity / (1 - Specificity) | No | How much a positive test increases disease probability |
| LR- | (1 - Sensitivity) / Specificity | No | How much a negative test decreases disease probability |
Critical insight: PPV and NPV change dramatically with prevalence. A test with 95% sensitivity and 95% specificity has a PPV of only 16% when prevalence is 1%. Always report prevalence alongside PPV/NPV.
ROC analysis: Plot sensitivity vs. (1 - specificity) across all thresholds. AUC summarizes overall discriminative ability. Compare AUCs between tests using DeLong's test.
Epidemiological Measures
| Measure | Setting | Formula (Conceptual) | Interpretation |
|---|---|---|---|
| Relative Risk (RR) | Cohort studies, RCTs | Incidence_exposed / Incidence_unexposed | How many times more likely in exposed group |
| Odds Ratio (OR) | Case-control, logistic regression | (a/b) / (c/d) from 2x2 table | Approximates RR when outcome is rare (<10%) |
| Absolute Risk Reduction (ARR) | RCTs | Incidence_control - Incidence_treated | Absolute difference in event rates |
| Number Needed to Treat (NNT) | RCTs | 1 / ARR | Patients to treat for one additional benefit |
| Incidence rate | Cohort studies | Events / Person-time at risk | Events per unit of person-time |
| Prevalence | Cross-sectional | Cases / Population at time point | Proportion with condition |
NNT is the most clinically interpretable measure. A drug with RR = 0.70 sounds impressive, but if baseline risk is 1%, NNT = 333. If baseline risk is 30%, NNT = 10. Always report absolute measures alongside relative ones.
Meta-Analysis
Quantitatively combines results across studies to produce a pooled estimate with greater precision.
Fixed-effects vs. random-effects:
- Fixed: assumes all studies estimate the same true effect. Use only when studies are methodologically identical.
- Random: assumes true effects vary across studies. Almost always more appropriate. Produces wider confidence intervals.
Heterogeneity assessment:
- Cochran's Q: Tests whether variability exceeds sampling error. Low power with few studies.
- I-squared: Proportion of variability due to heterogeneity. 25% = low, 50% = moderate, 75% = high.
- Tau-squared: The between-study variance itself. More interpretable than I-squared for clinical meaning.
Publication bias detection: Funnel plot (visual), Egger's test (regression), trim-and-fill (sensitivity). Absence of asymmetry does not prove absence of bias.
Network meta-analysis (NMA): Compares multiple treatments simultaneously using both direct (head-to-head) and indirect (via common comparator) evidence. Requires consistency assumption — direct and indirect evidence agree. Use when no single trial compares all treatments of interest.
Multiplicity in Clinical Trials
Every additional hypothesis tested inflates the probability of a false positive. Regulatory submissions require pre-specified multiplicity strategies.
| Strategy | How It Works | Use When |
|---|---|---|
| Fixed-sequence | Test H1 → H2 → H3 in order; stop at first failure | Clear scientific ordering of hypotheses |
| Gatekeeping | Primary endpoints are "gatekeepers" for secondary | Primary/secondary hierarchy exists |
| Graphical approach (Bretz) | Hypotheses as nodes, alpha flows between them on rejection | Complex hypothesis structures |
| Hochberg step-up | Less conservative than Bonferroni | Independent or positively correlated endpoints |
| Holm step-down | Valid under any correlation structure | Default when no structure is assumed |
Rule of thumb: Pre-specify the multiplicity strategy in the protocol. Post-hoc multiplicity adjustments are viewed skeptically by regulators.
Implementation Libraries
| Task | Python | R |
|---|---|---|
| Survival analysis (KM, Cox, AFT) | lifelines, scikit-survival |
survival, survminer |
| Competing risks (Fine-Gray, CIF) | lifelines, scikit-survival |
cmprsk, tidycmprsk |
| Group sequential and adaptive designs | — (use R via rpy2) |
rpact, gsDesign |
| Clinical trial simulation | — | simtrial, rpact |
| Sample size calculation | statsmodels.stats.power |
TrialSize, pwr, rpact |
| Mixed effects / longitudinal (LMM, GEE) | statsmodels (MixedLM, GEE) |
lme4, nlme |
| Meta-analysis (fixed, random, network) | pymare, PythonMeta |
metafor, meta, netmeta |
| Epidemiological measures and causal methods | zepid |
epiR, epitools |
| ROC analysis, diagnostic test evaluation | sklearn.metrics |
pROC, DTComPair |
| Bayesian clinical models, prior elicitation | pymc + bambi, preliz |
brms, rstanarm |
| Multiple imputation (clinical-grade) | sklearn.impute.IterativeImputer |
mice |
| CONSORT flow diagram generation | — | consort |
| R package validation for regulatory use | — | valtools |
Recommended starting stack (Python): lifelines for survival analysis + statsmodels for mixed models/GEE + zepid for epidemiological measures + sklearn.metrics for diagnostics.
Recommended starting stack (R): survival + survminer for survival + rpact for trial design + metafor for meta-analysis + lme4 for mixed models.
Regulatory note: R is now fully accepted by FDA for regulatory submissions (with validated packages). Python is gaining acceptance but has fewer validated clinical packages. SAS remains common in legacy pharma workflows.
Common Mistakes
- Not accounting for multiplicity. Testing five endpoints at alpha = 0.05 gives a ~23% chance of at least one false positive. Regulators will reject uncontrolled multiplicity in pivotal trials.
- Using standard logistic regression for time-to-event data. Ignoring censoring discards information and biases estimates. Use survival methods whenever follow-up varies across subjects.
- Reporting hazard ratios when proportional hazards fails. In immunotherapy trials with delayed treatment effects, the overall HR is misleading. Report RMST or milestone survival rates instead.
- Interpreting PPV without prevalence context. A screening test with 99% sensitivity in a 0.1% prevalence population has a PPV under 10%. Always compute PPV at the relevant prevalence.
- Conflating odds ratios with relative risk. When the outcome is common (>10%), the OR substantially overstates the RR. Use log-binomial or Poisson regression with robust variance for RR estimation.
- Performing unplanned interim analyses. Looking at unblinded data mid-trial without a pre-specified spending function invalidates the type I error control. Every look at the data must be accounted for.
When This Applies
- Clinical trial design, sample size calculation, or regulatory submission
- Any analysis involving time-to-event data with censoring
- Diagnostic test evaluation or epidemiological study design
- Meta-analysis or systematic review
- Longitudinal clinical data with repeated measures
- Any analysis destined for FDA, EMA, or journal submission with CONSORT/STROBE requirements
- Studies where the estimands framework should guide the analysis plan
See references/regulatory-standards.md for a mapping of regulatory documents to statistical requirements, and references/survival-methods.md for detailed survival analysis method selection guidance.