Time to event analysis #

The Time to event analysis module estimates how long it takes for something to happen — death, relapse, mechanical failure, customer churn — and how that timing depends on covariates. It supports five methods, three censoring patterns (right, interval, left-truncation), and competing-risks workflows when more than one kind of event is possible.

What is survival analysis? Ordinary regression treats outcomes as values you observe in full. Survival data is different: many subjects haven’t had the event yet when the study ends, so you only know they survived at least that long. That partial information is called censoring, and survival analysis is the family of methods built to handle it. The output is usually a survival curve — the probability of still being event-free at each time point — or a hazard model that shows how covariates speed up or slow down the event.

1. Pick a time variable and an event indicator
2. Assign each observed level of the event indicator a role (censored, event, or competing causes)
3. Optionally add an entry time, a grouping variable, and covariates
4. Choose an analysis method and a censoring type, then toggle method-specific options
5. Click Calculate

Variable roles #

The left column has the variable pickers; the right column has the method setup.

Time variable #

A non-negative numeric variable measuring duration from the start of follow-up to either the event or the end of observation. Days, months, and years all work — the unit only matters for interpreting the output. Negative values and missing values are dropped automatically.

Event indicator and level roles #

A categorical variable marking what happened to each subject. After picking it, a per-level table appears under Level roles: each observed value gets a dropdown where you assign one of these roles:

• Censored — the event did not occur (e.g. “alive”, “0”, “no”)
• Event — the event occurred (single-event mode)
• Event (cause 2), Event (cause 3), Event (cause 4) — distinct causes for competing-risks analysis

Defaults follow common conventions: the values 0, false, no, censored, and alive are pre-assigned to Censored; everything else to Event. Adjust as needed.

Why per-level roles? Survival data uses different conventions in different fields — sometimes 0/1, sometimes “alive”/“dead”, sometimes a multi-level cause-of-death code. Rather than asking you to recode the variable upfront, the module lets you map levels to roles in place. Two or more non-censored roles unlock the competing-risks methods automatically.

If the event variable has more than 20 distinct values, role assignment is blocked — that’s almost always a sign of picking a continuous variable by mistake. Recode it as a categorical event indicator first.

Entry time (left-truncation) #

Optional numeric variable for delayed entry: subjects join the risk set after time zero rather than from the start. Useful for age-as-time-scale analyses or when subjects enter a registry at varying ages.

What is left-truncation? Imagine studying mortality after retirement using a registry. A retiree who joined at 70 wasn’t at risk in the registry between 65 and 70 — that earlier window is left-truncated. Ignoring entry times biases survival estimates upward because you underweight people who entered the cohort late.

Left-truncation is not currently available with interval censoring or Fine-Gray regression.

Grouping and strata #

Optional categorical variable. Its role depends on the method:

• In non-parametric analysis it splits the curves and unlocks the group-comparison test
• In Cox regression it stratifies the baseline hazard — each stratum gets its own baseline, while the coefficient effects are shared across strata
• In parametric regression it adds the variable as a fixed effect (a factor on the right-hand side)
• In competing-risks regression (cause-specific only) it stratifies the baseline; Fine-Gray ignores it (add it to covariates instead)

Covariates #

Multi-select list of predictors for regression methods (Cox, parametric, competing-risks regression). Categorical covariates are encoded as factors with the first sorted level as the reference; numeric covariates enter as-is. Cumulative incidence and non-parametric analysis ignore covariates.

Censoring #

Left-truncation is handled separately via the entry time, not via the censoring dropdown. The two can combine on right-censored data.

When Interval-censored is selected an Upper bound picker appears. Pick a numeric variable holding the upper end of each interval, and use NA for right-censored rows (the time variable then carries the last-seen time).

Right vs. interval censoring: in right-censored data we know exactly when the event happened, or that it hadn’t happened by some known last-seen time. In interval-censored data we only know the event happened between two checks — typical of periodic screening. Treating an interval-censored event as if it occurred at the upper endpoint biases survival estimates downward.

Interval censoring is supported only for non-parametric and parametric methods in this checkpoint. Cox, cumulative incidence, and competing-risks regression require right-censored data.

Analysis methods #

The competing-risks methods are disabled in the dropdown until you assign at least two non-censored roles in the level-role table.

Non-parametric, semi-parametric, parametric: non-parametric methods (Kaplan-Meier, Nelson-Aalen) make no assumption about the shape of the hazard — they’re descriptive. Cox is semi-parametric: it leaves the baseline hazard unspecified but assumes covariate effects are multiplicative and constant over time. Parametric methods commit to a specific distribution (Weibull, log-normal, etc.) — riskier if the choice is wrong, but they let you extrapolate beyond the observed follow-up and produce smooth predictions.

Method-specific options #

The right column shows different cards depending on the chosen method.

Non-parametric options #

• Estimator — Kaplan-Meier (survival function) or Nelson-Aalen (cumulative hazard via Fleming-Harrington). When Interval-censored is selected the estimator field is hidden and Turnbull’s NPMLE is used automatically.
• Group comparison test — only enabled when a grouping variable is set:
  • Log-rank (ρ = 0) — the standard, weights events equally over time
  • Peto-Peto (ρ = 1) — weights early events more
  • Tarone-Ware (ρ = 0.5) — intermediate weighting
  • Skip test — produce curves only
  • With three or more groups, pairwise comparisons run with the same weight, and the p-value adjustment setting is applied (defaulting to Benjamini-Hochberg for the pairwise table)
• Survival probabilities at — space-separated time points (e.g. 12 24 36 60). Reported as S(t) with confidence bounds and risk-set size. Leave empty to skip.
• Display — toggles for the Confidence interval band and Censoring tick marks on the curve.

Choosing a log-rank weight: the standard log-rank assumes the hazard ratio between groups is roughly constant over time. If you suspect the difference is biggest early on (e.g. a treatment that helps in the first months), Peto-Peto can be more powerful. Tarone-Ware is a middle ground. Don’t pick the weight after seeing which one gave the smallest p-value — that’s p-hacking. Choose it from the substantive question.

Cox regression options #

• Ties handling — how the partial likelihood treats events that occur at exactly the same time:
  • Efron (default) — accurate and computationally cheap; the right choice for almost everyone
  • Breslow — simpler approximation; biased when ties are common
  • Exact partial likelihood — exact but slow; matches Efron in practice unless tie counts are extreme
• Robust (sandwich) variance — uses Huber-White SEs. Use this when observations may be clustered or the model is mildly misspecified.
• Diagnostics (each toggle adds an output section):
  • Proportional hazards test (Schoenfeld residuals) — cox.zph per term plus a global test, with residual scatter overlaid by a loess smoother
  • Log-log plot vs grouping variable — log(−log S(t)) curves; parallel lines support the proportional-hazards assumption
  • Functional form check (martingale residuals) — one residuals-vs-covariate panel per continuous covariate, with a loess smoother — non-flat smoothers suggest a non-linear effect
  • Influence diagnostics (dfbeta) — per-subject change in each coefficient if that subject were removed; reported as max and mean absolute dfbeta plus a per-subject scatter
  • Concordance (C-index) — a discrimination metric (0.5 = chance, 1.0 = perfect ranking)

Parametric regression options #

• Distribution family:

  Family	Type	Notes
  Weibull	AFT	Flexible monotone hazard; reduces to exponential when shape = 1
  Exponential	AFT	Constant hazard — strong assumption, useful as a baseline
  Log-normal	AFT	Hazard rises then falls; useful for “hump-shaped” risk
  Log-logistic	AFT	Similar shape to log-normal, lighter tails
  Gaussian	AFT	Linear regression in the time scale; rarely the right pick
  Gompertz (proportional hazards)	PH	Exponentially increasing hazard — common in adult-mortality models
  Compare all (AIC ranking)	—	Fits every family, reports the best-AIC fit, and shows an AIC ranking table and bar plot

  AFT models report time ratios (a coefficient × time), Gompertz reports hazard ratios.

• Overlay fitted curves on Kaplan-Meier — adds a comparison plot of the parametric survival curve against the non-parametric KM reference, split by the grouping variable when one is set.

AFT vs. PH: in a proportional-hazards model (Cox, Gompertz) a covariate multiplies the hazard. In an accelerated-failure-time model (Weibull, log-normal, log-logistic, Gaussian) it stretches or shrinks the time scale — a time ratio of 1.5 means the covariate makes events happen 1.5× later, on average. AFT effects are sometimes more intuitive to clinicians because they’re expressed in time units.

Gompertz with interval censoring: Gompertz is fit via eha::phreg, which doesn’t accept interval-censored input. The option is unavailable when interval censoring is selected, and Compare all drops Gompertz from the comparison automatically.

Competing-risks regression options #

• Approach:
  • Cause-specific hazards (Cox per cause) — fits a separate Cox model per cause, treating the other causes as censored. Answers “what predicts this cause among those still event-free?”
  • Fine-Gray subdistribution hazards — models the cumulative incidence directly via cmprsk::crr. Answers “what predicts experiencing this cause first?”

Cause-specific vs. Fine-Gray: the two answer different questions and can give different signs. Cause-specific hazards describe the etiology of each cause among current survivors. Fine-Gray sub-distribution hazards describe the prognosis — how a covariate shifts the eventual cumulative incidence, accounting for the fact that competing events block the cause of interest. For decision-making at the patient level, Fine-Gray is often more useful; for biological mechanism, cause-specific.

Sample size and EPV checks #

Before fitting any regression, the module checks the events-per-variable (EPV) ratio:

• Events < parameters — fit blocked with an error; reduce covariates or get more data
• Events < 10 × parameters — non-blocking warning; the rule of thumb is ≥ 10 events per parameter for stable estimates

Effective parameter count is the sum of covariate degrees of freedom: one per numeric covariate, (levels − 1) per factor. Parametric fits add one for the scale parameter (two for Gompertz, which has scale and shape). For competing-risks regression the check uses the smallest-event cause as the worst case, since each cause is its own model.

Why ten events per variable? With fewer events than parameters the partial likelihood is singular — there are infinitely many coefficient combinations that fit the data equally well, and you’ll see implausible standard errors. Even with a feasible fit, sub-10 EPV makes coefficients sensitive to single observations and gives unreliable hazard-ratio confidence intervals.

Reading results #

Each run produces an output card titled with the method, the time variable, and the event variable.

Non-parametric survival #

• Curve summary — N, events, censored, median survival with confidence interval, and (for Kaplan-Meier / Nelson-Aalen) restricted mean survival time (RMST) with its standard error. Median is reported as — when more than half the subjects are still at risk at the longest follow-up.
• Survival probabilities at time points (when Survival probabilities at is filled in) — S(t), confidence bounds, and number at risk for each requested time per stratum.
• Group comparison — log-rank-family chi-square, df, and p-value (omitted when the test is set to Skip test or no grouping is selected).
• Pairwise comparisons (three or more groups) — chi-square, df, raw p, and adjusted p per pair. Adjustment uses the global p-value setting.
• Survival curve plot — Kaplan-Meier, Nelson-Aalen, or Turnbull NPMLE curves with optional CI band and censoring ticks.
• Cumulative hazard plot — H(t) over time. Skipped for Turnbull (the Icens fit doesn’t expose a usable hazard).

What is RMST? Restricted mean survival time is the area under the survival curve up to a chosen cut-off — interpreted as the average event-free time over that window. Unlike the median, it’s defined even when fewer than half the subjects have had the event, and it has a clean clinical meaning (“treatment adds 4.2 months of event-free survival over 5 years”). RMST is reported with the longest common follow-up across strata.

Confidence bands on survival curves: the band shows pointwise confidence intervals on S(t). Curves whose bands separate clearly suggest a real group difference, but the formal test is the log-rank in the Group comparison table.

Cox regression #

• Model summary — N, events, ties method, robust-SE flag.
• Coefficients — one row per term (factor levels expanded). Columns:
  • HR — hazard ratio with confidence interval
  • log(HR) — the raw coefficient
  • SE — standard error
  • z with significance stars
  • p value
• Overall tests — likelihood ratio, Wald, and score (log-rank) χ², df, p. They test the joint null that all coefficients are zero.
• Concordance (C-index) (when enabled) — value and SE.
• Hazard ratio forest plot — log-scale forest with reference line at HR = 1.
• Proportional hazards test (when enabled) — per-term and global χ² from cox.zph. A significant result means the term’s effect changes over time.
• Schoenfeld residuals — scatter of scaled residuals over time, one panel per term, with a loess smoother. A non-flat smoother is the visual analogue of a significant cox.zph.
• Log-log plot (when enabled and a grouping variable is set) — log(−log S(t)) by stratum. Roughly parallel curves support PH at the strata level.
• Martingale residuals (when enabled) — one panel per continuous covariate, with the covariate on the x-axis and the residual on the y-axis. A non-flat loess smoother suggests the covariate enters non-linearly.
• Influence (dfbeta summary) + Dfbeta residuals per subject (when enabled) — max and mean absolute dfbeta per term, plus a per-subject scatter for spotting individual outliers.
• Baseline survival — survfit.coxph reference curve(s); numeric covariates are evaluated at their sample mean and factor covariates at their reference level (not at zero), so the curve is the survival of an “average” subject.

Reading hazard ratios: HR = 1 means the predictor has no effect on the hazard. HR = 2 means the hazard is doubled per one-unit increase. HR = 0.5 means it’s halved. The CI is the part you check first — if it crosses 1, the effect isn’t significant. For a binary 0/1 covariate the HR is the ratio of hazards between the two groups, controlling for the other terms.

What does a violated PH assumption mean? A significant cox.zph says the hazard ratio for that term isn’t constant over time — the covariate’s effect grows or shrinks as follow-up accumulates. The fitted HR is then a kind of weighted average across the follow-up window. Common fixes: stratify on the offending variable, fit a time-varying coefficient (not yet supported), or switch to a parametric model where the time-shape can be inspected directly.

Parametric regression #

• Model summary — distribution, N, events, AIC, log-likelihood, scale parameter (where applicable).
• AIC ranking (Compare all mode) — every family’s AIC, ΔAIC, log-likelihood, and convergence flag, sorted best-first. Followed by an AIC by distribution bar plot.
• Coefficients — same shape as Cox, but the exponentiated column is Time ratio for AFT families and HR for Gompertz. Distribution parameters (Log(scale), log(shape)) appear as separate rows at the bottom — these are not covariate effects.
• Kaplan-Meier vs parametric fit (when Overlay fitted curves on Kaplan-Meier is on) — KM reference (with CI band) and the smooth parametric curve. Curves split by the grouping variable when one is set. The smooth curve uses numeric covariates at their mean and factor covariates at the reference level.

Reading time ratios: in an AFT model, time ratio = exp(coefficient). A time ratio of 1.4 for a binary covariate means events occur 1.4× later (40% later) for that group on average. Smaller time ratios mean earlier events.

AIC ranking caveat: AIC compares fits on the same data, so the ranking is meaningful, but lower AIC isn’t a guarantee that the winning family is “right”. Always sanity-check the overlay plot — a parametric curve that systematically misses the KM in the tail is a red flag even if the AIC is lowest.

Cumulative incidence (competing risks) #

• Cumulative incidence summary — N and events per group × cause.
• Gray’s K-sample test (when a grouping variable is set) — χ², df, p per cause. Tests whether the cumulative incidence functions differ across groups for that cause.
• Cumulative incidence curves — F(t) per cause × group. Confidence bounds use a complementary log-log transform on F.

Cumulative incidence vs. 1 − KM: treating non-cause-of-interest events as censored and reporting 1 − KM systematically over-estimates the cause-specific incidence — it pretends competing events would eventually convert into the cause of interest if you waited long enough. The CIF (Aalen-Johansen / cmprsk::cuminc) divides the hazard so that all causes’ incidences sum to 1 minus event-free survival.

Competing-risks regression #

• Fit summary — N, events, and convergence flag per cause.
• Cause-specific hazards: {cause} or Sub-distribution hazards: {cause} — coefficient table per cause. Sub-distribution columns use sHR (sub-distribution hazard ratio) and log(sHR) instead of HR / log(HR).
• Overall tests: {cause} (cause-specific only) — LR / Wald / score per fit, same shape as Cox.
• HR forest plot (cause-specific) or sHR forest plot (Fine-Gray) — combined forest with one row per (cause, term).

If a cause’s fit fails to converge (small subgroup, separation, or numerical issues) it appears as a warning row instead of a coefficient table.

Dropped rows and missing data #

Each output card ends with a “rows excluded” note when any data was dropped. The module silently excludes rows with:

• A non-finite or negative time
• A blank event level, or an event level that wasn’t assigned a role
• For interval censoring: an upper bound that’s missing (when the event occurred) or smaller than the lower bound
• A missing grouping or entry-time value
• A missing covariate value (for regression methods)

This is method-local listwise deletion — the global missing-data setting does not apply here, since survival analysis needs exact follow-up times to be meaningful.

Reporting checklist #

Method:

• Time variable, event indicator, and how each event level was assigned a role
• Censoring scheme (right vs. interval; left-truncation if used)
• Method (Kaplan-Meier, Cox, parametric Weibull, etc.) and any assumptions checked
• For Cox: ties handling, robust SE if used, diagnostics performed
• For parametric: family chosen and (when comparing) selection criterion
• For competing risks: cause-specific or Fine-Gray, definition of each cause
• Covariates and how factor reference levels were chosen
• N before and after exclusions; events per cause where applicable

Results:

• Median survival (or RMST) per group with confidence intervals
• Survival probabilities at the clinically relevant time points
• For comparison tests: test (log-rank / Peto-Peto / Tarone-Ware / Gray), χ², df, p; pairwise adjusted p where applicable
• For Cox: HRs with CIs, overall test χ²/p, and an explicit statement on the proportional-hazards assumption
• For parametric: time ratios (or HRs for Gompertz) with CIs, AIC, and the comparison table where relevant
• For competing-risks regression: per-cause coefficient table with the appropriate hazard label (HR or sHR)

Reproducibility #

Every analysis prints the underlying R code to the R console — you can inspect, copy, or re-run the exact commands. The module uses survival (Kaplan-Meier, Nelson-Aalen, log-rank, Cox, parametric AFT), eha (Gompertz proportional-hazards fit), Icens (Turnbull NPMLE for interval censoring), and cmprsk (cumulative incidence and Fine-Gray subdistribution regression). Confidence levels follow the global confidence setting, and the pairwise non-parametric comparisons honour the p-value adjustment setting. Citations for R packages used in your analysis appear automatically at the top of the output card.

Common pitfalls #

Treating censored as event-free forever. Censoring means the event hadn’t occurred as of last contact, not that it never will. Reports like “75% survived” should always cite a follow-up window — KM medians are undefined when more than half the subjects are still at risk.

Ignoring competing risks. If subjects can experience more than one kind of event and you analyse one cause as if the other simply censored those subjects, 1 − KM over-estimates the cause-of-interest incidence. Use cumulative incidence (or Fine-Gray) when competing events are non-trivial.

Checking proportional hazards only globally. A non-significant global cox.zph doesn’t mean every term is fine — a single term can violate PH while the global test stays clean. Always look at the per-term test and the Schoenfeld residual plots.

Choosing a parametric distribution by AIC alone. AIC ranks fits within your data but doesn’t guarantee any of them describes the true hazard well. Compare the parametric overlay against the KM curve before reporting an AFT effect.

Informative censoring. Survival methods assume that subjects who are censored are no different in risk from those still at risk. If censoring is driven by worsening prognosis (sick patients drop out), the KM and Cox results are biased — and there’s no built-in fix. Document the censoring mechanism in your write-up.

Immortal time bias. Defining group membership using something that happens during follow-up (e.g. “patients who eventually got drug X”) gives those subjects a guaranteed event-free window before they could possibly be classified, biasing the comparison. Define groups using only baseline information, or use a time-dependent covariate (not yet supported in this module — pre-process the data via the R console for now).