Survival, Bayesian, Meta-Analysis & Resampling
Statistical methods for survival analysis, Bayesian inference, meta-analytic pooling, and resampling-based inference. Includes Kaplan-Meier curves, Cox proportional-hazards regression, parametric and competing-risks models, Bayesian linear/logistic/hierarchical regression with MCMC diagnostics, fixed- and random-effects meta-analysis, and bootstrap/jackknife/permutation methods.
68 procedures · 3 with documented limitations. Every result is computed by the open Python engine and is exportable to APA tables and reports.
| Procedure | What it does | Status | Key reference |
|---|---|---|---|
| Kaplan-Meier Survival Curves | Non-parametric survival estimation with Greenwood pointwise 95% CI and log-rank/multivariate log-rank test. | ✅ Full | Kaplan & Meier (1958); Greenwood (1926) |
| Cox Proportional Hazards | Semi-parametric hazard regression with Efron tie handling, PH assumption test (Grambsch-Therneau), and concordance index. | ✅ Full | Cox (1972); Efron (1977) |
| Mixture Cure Model | Fits a Berkson-Gage mixture cure model by EM, separating a logistic incidence submodel for the long-term-survivor (cured) fraction from a Weibull AFT latency submodel for the susceptible subjects who can still experience the event. | ✅ Full | Berkson & Gage (1952); Sy & Taylor (2000) |
| Piecewise-Exponential Survival | Proportional-hazards survival model with a piecewise-constant baseline hazard, fitted by the Poisson / piecewise-exponential equivalence on person-interval exposure with covariate log-hazard-ratios; reduces to the exponential model with one interval. | ✅ Full | Friedman (1982); Holford (1980) |
| Discrete-Time Hazard (cloglog) | Fits a discrete-time (grouped-duration) proportional-hazards model by expanding subjects into person-period rows and estimating a complementary-log-log binomial GLM, yielding constant-across-time log-hazard-ratios that mirror the Cox model at low per-period hazard, plus a nonparametric or polynomial baseline-hazard summary. | ✅ Full | Prentice & Gloeckler (1978); Allison (1982) |
| Actuarial Life Table | Cutler-Ederer life table with conditional survival/death probabilities, cumulative hazard rate, and Greenwood SE. | ✅ Full | Kalbfleisch & Prentice (2002); Greenwood (1926) |
| Parametric Survival Models | Univariate or AFT parametric models (Weibull, exponential, lognormal, log-logistic, Gompertz) with parameter estimates. | ✅ Full | Kalbfleisch & Prentice (2002); Akaike (1974) |
| Accelerated Failure Time | AFT regression with covariates, time ratio estimates, concordance index, and AIC. | ✅ Full | Kalbfleisch & Prentice (2002); Akaike (1974) |
| Competing Risks (Fine-Gray) | Subdistribution-hazards model with cumulative incidence functions per event type and subdistribution HR. | ✅ Full | Fine & Gray (1999); Aalen & Johansen (1978) |
| Shared Frailty Cox Model | Cluster-robust Cox with gamma frailty variance estimate and LR test vs no-frailty baseline. | ⚠️ Limited | Vaupel et al. (1979); Therneau & Grambsch (2000) |
| Schoenfeld Residuals Test | PH assumption test per covariate and globally using scaled Schoenfeld residuals with interpretation. | ✅ Full | Cox (1972); Schoenfeld (1982) |
| Nelson-Aalen Cumulative Hazard | Non-parametric cumulative hazard estimator with 95% pointwise confidence bands, optionally grouped. | ✅ Full | Nelson (1972); Aalen (1978) |
| Stratified Log-Rank Test | Log-rank test with optional stratification, multivariate for 3+ groups, and pairwise comparisons. | ✅ Full | Mantel (1966); Mantel & Haenszel (1959) |
| Restricted Mean Survival Time | RMST up to user-specified tau with SE and 95% CI; difference test and CI for 2 groups. | ✅ Full | Royston & Parmar (2013); Uno et al. (2014) |
| Time-Varying Cox Model | Cox regression with time-varying covariates (start-stop format), HR with CI and partial log-likelihood. | ✅ Full | Cox (1972); Andersen & Gill (1982) |
| Cox Residuals for Diagnostics | Five residual types (martingale, deviance, Schoenfeld, scaled Schoenfeld, score) for model diagnostics. | ✅ Full | Cox (1972); Therneau et al. (1990) |
| Bayesian Linear Regression | MCMC-based (PyMC or emcee fallback) with weakly informative priors, posterior samples, PPC, and diagnostics (R-hat, ESS). | ✅ Full | Gelman et al. (2013); Goodman & Weare (2010) |
| Bayesian Unit-Root Analysis | Closed-form Student-t posterior for the AR(1) root rho under a flat/Jeffreys prior, reporting the posterior mean/sd, a 95% credible interval, and the posterior probabilities P(rho>=1) (unit root/explosive) and P(rho<1) (stationary). | ✅ Full | Sims (1988); Sims & Uhlig (1991) |
| Bayesian VAR (Minnesota prior) | Estimates a reduced-form VAR(p) with the Minnesota (Litterman) shrinkage prior imposed by dummy observations, reporting posterior-mean coefficient matrices that center the own first lag at 1 (levels) or 0 (stationary), shrink cross-variable lags harder than own lags by 1/lag, plus the shrinkage hyperparameters, per-equation in-sample fit, and a multi-step forecast. | ✅ Full | Litterman (1986); Doan, Litterman & Sims (1984) |
| Bayesian VECM (cointegration) | Bayesian Vector Error Correction Model for cointegrated series: given a chosen cointegrating rank and lag order, places a weakly-informative Normal-inverse-Wishart prior on the short-run dynamics and adjustment coefficients and Gibbs-samples the posterior, reporting posterior means and 95% credible intervals for the speed-of-adjustment loadings (alpha), the normalised cointegrating vectors (beta), the short-run matrices, and error-correction evidence. | ✅ Full | Koop, Strachan, van Dijk & Villani (2006); Sugita (2009) |
| Bayesian CFA (one-factor) | Fits a one-factor confirmatory measurement model x_ij = lambda_j * eta_i + eps_ij by a three-block Gibbs sampler (factor scores, loadings, residual variances), returning posterior loading summaries with 95% credible intervals, residual (uniqueness) variances, implied communalities, and an SRMR-based fit summary. | ✅ Full | Lee (2007) |
| Bayesian IRT (2PL) | Bayesian two-parameter logistic item-response model fitted by a Metropolis-within-Gibbs MCMC sampler, returning posterior item discriminations and difficulties (with 95% credible intervals) plus latent person-ability summaries on the N(0,1)-identified scale. | ✅ Full | Albert (1992); Fox (2010) |
| Bayesian Meta-Analysis | Fits a random-effects meta-analysis (theta_i ~ N(mu, tau^2), y_i ~ N(theta_i, se_i^2)) by a direct Gibbs sampler with a weakly-informative half-Cauchy prior on the between-study sd, returning the posterior overall effect mu with credible interval, between-study heterogeneity tau, shrunken per-study estimates, and the posterior probability the effect is positive. | ✅ Full | DuMouchel (1990); Smith, Spiegelhalter & Thomas (1995) |
| Conjugate Bayesian Regression (Normal-Gamma) | Analytic conjugate Bayesian linear regression with a Normal-Gamma (Normal-inverse-Gamma) prior that returns closed-form multivariate-t posteriors for the coefficients and an inverse-Gamma posterior for the error variance, reducing exactly to OLS under a flat prior and shrinking slopes toward zero under a tight prior, with no MCMC. | ✅ Full | Lindley & Smith (1972); Koop (2003) |
| Bayesian Quantile Regression | Estimates the conditional tau-quantile of an outcome via the asymmetric-Laplace likelihood, sampling the posterior of the quantile-regression coefficients with the Kozumi-Kobayashi (2011) location-scale-mixture Gibbs sampler and reporting posterior means, sds and 95% credible intervals. | ✅ Full | Yu & Moyeed (2001); Kozumi & Kobayashi (2011) |
| Student-t (robust) Regression | Outlier-robust linear regression with iid Student-t(nu) errors fitted by an EM / IRLS scale-mixture ML algorithm, downweighting extreme residuals so the fit tracks the bulk of the data rather than the outliers, and reporting coefficients/SEs, the error scale, the (optionally ML-estimated) degrees of freedom, log-likelihood and per-observation robustness weights. | ✅ Full | Lange, Little & Taylor (1989, JASA); Geweke (1993) |
| Metropolis-Hastings (Bayesian regression) | Random-walk Metropolis-Hastings MCMC sampler for the joint posterior of regression coefficients and the error variance (beta, log sigma^2) in Bayesian linear regression, with a proposal auto-tuned toward the optimal ~0.234 acceptance rate, reporting posterior means/sds/95% credible intervals, the acceptance rate, and effective sample sizes. | ✅ Full | Metropolis et al. (1953); Hastings (1970) |
| Gibbs Sampler (Bayesian regression) | Conjugate Gibbs sampler for Bayesian linear regression that alternates Normal draws of the coefficients given sigma^2 and inverse-Gamma draws of sigma^2 given the coefficients, returning posterior means/sds/95% credible intervals, effective sample size and a Geweke convergence diagnostic (acceptance is identically 1), with weak-prior posteriors that coincide with OLS. | ✅ Full | Gelfand & Smith (1990, JASA); Geman & Geman (1984) |
| Laplace Approximation (Bayesian) | Approximates the posterior of a Bayesian linear or logistic regression by a Gaussian centered at the posterior mode (MAP) with inverse-Hessian covariance, reporting approximate posterior means/SDs/credible intervals, the Laplace approximation to the log marginal likelihood (model evidence), and a Tierney-Kadane fully-exponential refined posterior mean. | ✅ Full | Tierney & Kadane (1986, JASA); Rue, Martino & Chopin (2009, JRSS-B) |
| Importance Sampling (Bayesian) | Bayesian posterior inference for a linear regression by importance sampling / sampling-importance-resampling: proposals drawn from a heavy-tailed multivariate-t centred at OLS (covariance inflated) under a weakly-informative conjugate Normal-inverse-Gamma prior are reweighted to give importance-weighted posterior means, 95% credible intervals for the coefficients and error variance, the weights' effective sample size, and a resampled (SIR) posterior draw. | ✅ Full | Geweke (1989); Rubin (1987) |
| Bayesian Poisson Regression | Fits a log-linear Poisson count model by random-walk Metropolis-Hastings under a weak Normal prior, returning posterior means, standard deviations and 95% credible intervals for each coefficient, the incidence-rate ratios exp(coef), and the Metropolis acceptance rate (tuned toward the ~0.234 optimal scaling). | ✅ Full | Gelman et al. (2013, BDA 3rd ed.) |
| Bayesian Tobit | Left-censored (Tobit) regression estimated by Gibbs sampling with data augmentation of the censored latent responses, returning posterior means and 95% credible intervals for the coefficients and error standard deviation plus the censored fraction. | ✅ Full | Chib (1992) |
| Bayesian Probit (Albert-Chib) | Fits a binary probit model by Gibbs sampling with Albert-Chib truncated-normal latent-utility data augmentation under a diffuse Normal prior, returning posterior means, SDs, 95% credible intervals, effective sample sizes, and the frequentist MLE as a reference. | ✅ Full | Albert & Chib (1993, JASA) |
| Horseshoe Regression (sparse) | Bayesian linear regression with a horseshoe shrinkage prior (Carvalho-Polson-Scott 2010) sampled by the fully-conjugate Makalic-Schmidt (2016) inverse-gamma scale-mixture Gibbs sampler, reporting posterior means and 95% credible intervals per coefficient, per-coefficient shrinkage weights kappa = 1/(1 + ntau^2lambda^2), and which coefficients are effectively selected (CI excludes 0) so that on sparse designs the null coefficients are shrunk hard toward 0 while genuine signals stay near their true values. | ✅ Full | Carvalho, Polson & Scott (2010, Biometrika); Makalic & Schmidt (2016, IEEE SPL) |
| Spike-and-Slab Variable Selection | Bayesian stochastic search variable selection (George-McCulloch SSVS) that places a two-component spike-at-zero / diffuse-slab prior on every regression coefficient and Gibbs-samples Bernoulli inclusion indicators to report each predictor's posterior inclusion probability (PIP), Bayesian-model-averaged coefficients, and the most probable visited models. | ✅ Full | George & McCulloch (1993, JASA); Mitchell & Beauchamp (1988) |
| Bayesian Lasso | Bayesian lasso regression (Park & Casella) places a Laplace prior on the coefficients and samples it with a scale-mixture-of-normals Gibbs sampler, returning shrunk posterior-mean coefficients with 95% credible intervals, the posterior of the shrinkage parameter lambda, and which predictors are effectively selected (CI excludes 0). | ✅ Full | Park & Casella (2008, JASA) |
| Adaptive Lasso | Weighted lasso (Zou 2006) whose per-coefficient penalties come from an initial OLS/ridge estimate, delivering oracle variable selection: truly-zero coefficients are driven to exactly 0 while genuine signals stay nearly unbiased, with the penalty chosen by reproducible cross-validation. | ✅ Full | Zou (2006, JASA) |
| Savage-Dickey Bayes Factor | Computes the Savage-Dickey density-ratio Bayes factor (BF01/BF10) for the sharp null H0: beta=0 on one regression coefficient by taking the ratio of the marginal posterior to the prior density at zero under a conjugate Normal-prior Bayesian regression, returning posterior mean/sd, the prior sd, and a Jeffreys-scale verdict. | ✅ Full | Dickey (1971); Verdinelli & Wasserman (1995) |
| Bayesian IC (DIC/WAIC/LOO) | Bayesian predictive information criteria for a conjugate-Gibbs Bayesian linear regression, reporting DIC (Spiegelhalter), WAIC (Watanabe) and Pareto-smoothed importance-sampling LOO (Vehtari-Gelman-Gabry) on the deviance scale with their effective-parameter counts (p_DIC, p_WAIC, p_LOO) and Pareto-k diagnostics, for principled out-of-sample model comparison. | ✅ Full | Spiegelhalter et al. (2002); Watanabe (2010); Vehtari, Gelman & Gabry (2017) |
| Bayesian Model Averaging | Enumerates all 2^k linear-regression covariate subsets, scores each by BIC to obtain approximate posterior model probabilities, and reports per-predictor posterior inclusion probabilities (PIP), BMA posterior means/sds of the coefficients (averaged over models), and the top models. | ✅ Full | Raftery, Madigan & Hoeting (1997); Hoeting et al. (1999) |
| Bayesian Logistic Regression | Bernoulli likelihood with logit link, posterior odds ratios, in-sample accuracy, and MCMC diagnostics. | ✅ Full | Gelman et al. (2013); Goodman & Weare (2010) |
| Bayesian Hierarchical Model | Random-intercept multilevel model (PyMC, emcee, or OLS shrinkage fallback) with ICC, varying intercepts per group. | ⚠️ Limited | Gelman et al. (2013); Goodman & Weare (2010) |
| MCMC Diagnostics | Gelman-Rubin R-hat, ESS via autocorrelation, trace summaries with interpretations of convergence. | ✅ Full | Gelman & Rubin (1992); Gelman et al. (2013) |
| Bayes Factor | Model comparison via BIC approximation with Jeffreys-scale interpretation (decisive/strong/moderate/anecdotal). | ⚠️ Limited | Kass & Raftery (1995); Jeffreys (1961) |
| Prior vs Posterior | Visualization comparison of prior and posterior distributions per parameter. | ✅ Full | Gelman et al. (2013); Goodman & Weare (2010) |
| Posterior Predictive Check | Bayesian p-value and visual comparison of observed vs simulated data from posterior predictive. | ✅ Full | Gelman et al. (1996); Gelman et al. (2013) |
| Credible Interval | Highest density interval (HDI) and equal-tailed interval (ETI) at 94% credibility. | ✅ Full | Kruschke (2015) |
| Fixed-Effects Meta-Analysis | Inverse-variance weighting, Cochran Q, I², per-study and pooled CIs, assumes homogeneous effects. | ✅ Full | Hedges & Olkin (1985); Cochran (1954) |
| Random-Effects Meta-Analysis | DerSimonian-Laird tau² estimation, I², 95% prediction interval (Higgins-Thompson-Spiegelhalter), per-group weights. | ✅ Full | DerSimonian & Laird (1986); Cochran (1954) |
| Egger's Test for Publication Bias | Funnel-plot asymmetry test via standardized-effect intercept regression, t-test on k-2 df. | ✅ Full | Egger et al. (1997) |
| Forest Plot Data | Per-study effects with 95% CI, pooled estimate (fixed or random), I², and heterogeneity for visualization. | ✅ Full | Lewis & Clarke (2001); DerSimonian & Laird (1986) |
| Bootstrap Standard Error | Bootstrap distribution of a statistic (mean/median/std/var/quantiles) with SE, bias, and summary. | ✅ Full | Efron (1979) |
| Wild Bootstrap (OLS) | Wild (and wild-cluster) bootstrap standard errors, percentile CIs, and p-values for OLS coefficients, robust to heteroskedasticity and the standard tool for few-cluster inference. | ✅ Full | Wu (1986); Cameron, Gelbach & Miller (2008) |
| Percentile-t Bootstrap (coefficient) | Studentized (bootstrap-t) confidence interval for a single OLS regression coefficient, built by pairs-resampling the pivot t*=(b*-b)/se* and inverting it via the Hall pivot-flip formula for second-order-accurate, skewness-corrected coverage, contrasted against the normal-theory b +/- 1.96 se interval. | ✅ Full | Hall (1992); Davison & Hinkley (1997) |
| Block Bootstrap (time series) | Moving-block and circular-block bootstrap that resamples overlapping blocks of consecutive observations to deliver dependence-robust standard errors and percentile confidence intervals for the mean, median, variance, or lag-1 autocorrelation of a time series. | ✅ Full | Kunsch (1989); Politis & Romano (1992) |
| Parametric Bootstrap (OLS) | Resamples OLS errors from the fitted homoskedastic-Gaussian model (y* = X*beta_hat + N(0, sigma^2)), refits each synthetic sample, and reports per-coefficient bootstrap SEs and percentile 95% CIs that converge to the classical OLS standard errors as replications grow. | ✅ Full | Efron & Tibshirani (1993) |
| Residual Bootstrap (OLS) | Model-based (residual) bootstrap that holds the design and fitted values fixed, resamples OLS residuals with replacement to build synthetic responses, refits each, and reports per-coefficient bootstrap standard errors and percentile 95% confidence intervals without assuming any error distribution. | ✅ Full | Efron (1979); Freedman (1981) |
| Subsampling Inference | Politis-Romano-Wolf subsampling: draws subsamples of size b < n without replacement, recomputes a statistic on each, and uses the sqrt(b)-rescaled subsampling distribution to build a confidence interval and standard error that are stable as b varies. | ✅ Full | Politis & Romano (1994); Politis, Romano & Wolf (1999) |
| Bayesian Bootstrap (Rubin) | Rubin's Bayesian bootstrap for OLS coefficients: draws Dirichlet(1,...,1) weights over the observations and refits weighted least squares each replication to produce a posterior distribution (mean, sd, 95% credible interval) for every coefficient, first-order equivalent to the ordinary bootstrap. | ✅ Full | Rubin (1981, Ann. Statist.) |
| BCa Bootstrap Interval | Bias-corrected and accelerated (BCa) bootstrap 95% confidence interval for a chosen scalar statistic (mean, median, std, variance, quartiles, IQR) of one numeric variable, correcting the plain percentile interval for median bias (z0) and skewness (jackknife acceleration a). | ✅ Full | Efron (1987, JASA); Efron & Tibshirani (1993) |
| Cluster Bootstrap (OLS) | Pairs cluster bootstrap for OLS coefficient inference that resamples whole clusters with replacement and refits to produce cluster-robust bootstrap standard errors and percentile confidence intervals valid under arbitrary within-cluster error correlation. | ✅ Full | Cameron, Gelbach & Miller (2008) |
| Bootstrap Confidence Interval | Four CI methods (percentile, basic, BCa, studentized) with distribution summary and downsampled samples. | ✅ Full | Efron (1979); Efron (1987) |
| Bootstrap Regression | Row-level resampling with OLS per iteration; bootstrap SE, bias, CI, and comparison to classical estimates. | ✅ Full | Efron (1979) |
| Jackknife Estimation | Leave-one-out resampling, SE, bias correction, and bias-corrected estimate. | ✅ Full | Efron (1979); Quenouille (1956) |
| Permutation Test (Two-Sample) | Exact resampling test for mean/median/t-stat difference with null distribution and two-tailed p-value. | ✅ Full | Efron (1979); Fisher (1935) |
| Permutation Test (Correlation) | Exact test for Pearson/Spearman/Kendall correlation with null distribution and significance. | ✅ Full | Efron (1979); Fisher (1935) |
| Cross-Validation | K-fold CV (sklearn-compatible) for linear/logistic/ridge/lasso with multiple scoring metrics. | ✅ Full | Efron (1979); Stone (1974) |
| Bayesian Bootstrap | Dirichlet-weighted resampling for posterior distribution of a statistic; Bayesian alternative to classical bootstrap. | ✅ Full | Efron (1979); Rubin (1981) |
Notes & limitations
- Shared Frailty Cox Model — Gamma frailty only; variance estimated via martingale residuals.
- Bayesian Hierarchical Model — Fallback uses OLS-shrinkage approximation without full MCMC; ICC computed from variance components.
- Bayes Factor — BIC-based approximation; not exact marginal likelihood.