Cox Regression with Time-Dependent Covariates

A time-dependent (time-varying) Cox model generalizes the standard proportional-hazards model by letting a covariate `x(t)` change value during follow-up. Mechanically, each subject's follow-up is split into multiple `(start, stop]` intervals — the counting-process (Andersen–Gill) data layout — and at every event time the partial likelihood compares the covariate value of the person who failed against the current covariate values of everyone still at risk. The hazard is `h(t | x(t)) = h0(t) * exp(beta' x(t))`, and `exp(beta)` is the hazard ratio per unit of the covariate as it is at time t.

This is the correct way to model a status that switches on during follow-up (started a drug, was hospitalized, crossed a lab threshold, accrued cumulative dose) without committing the cardinal sin of using a future value to classify present person-time.

Core conceptual distinction

. The decision that does the work is when person-time is allowed to count toward an exposure group. A naive time-fixed analysis assigns a subject to the "exposed" group for their entire follow-up because they were exposed at some point, which credits the exposed group with person-time that was lived before exposure occurred — immortal time / guarantee-time bias, because to become exposed the subject had to survive event-free until the exposure happened. A time-dependent covariate fixes this: the subject contributes unexposed person-time from time zero until the moment of exposure, then exposed person-time thereafter, and the exposure indicator is 0 then 1 across the split intervals.

The estimand is a hazard ratio comparing the instantaneous hazard under the covariate value held at each instant — an internal, association-style contrast. Critically, this is NOT a causal contrast of treatment strategies: when the time-varying exposure is affected by, and itself affects, time-varying confounders (e.g., disease severity drives both treatment switching and the outcome), standard time-dependent Cox is biased no matter how you condition, because conditioning on a post-baseline confounder that is also a mediator opens a collider path.

The fix for that is a marginal structural model (IPTW-weighted pooled logistic / Cox) or g-estimation — not a richer time-dependent Cox.

Pros, cons, and trade-offs

• vs naive baseline-only ("ever exposed") Cox: time-dependent coding removes immortal-time bias and correctly aligns exposed person-time with exposed status. Cost: more programming, a counting-process dataset, and a model whose HR is harder to communicate (it is per-instant, not "over the study"). Prefer time-dependent whenever exposure status, dose, or a covariate genuinely changes after time zero and you are not estimating a strategy contrast.
• vs landmark analysis: landmarking sidesteps immortal time by fixing exposure status at a pre-specified landmark time and analyzing only survivors to that point — simpler, robust, and avoids the post-baseline-confounding trap, but discards early events and answers a conditional question (survivors to the landmark). Prefer landmarking for a single one-way status change and a clean clinical message; prefer time-dependent Cox for repeatedly changing exposures, cumulative dose, or when you cannot afford to drop pre-landmark events.
• vs marginal structural models / g-methods: when the time-varying exposure has time-varying confounders that are also on the causal pathway (confounder affected by prior treatment), time-dependent Cox is structurally biased and an MSM with inverse-probability-of-treatment weights (or g-estimation) is required. Prefer plain time-dependent Cox only when later covariates are not affected by earlier exposure; otherwise it is actively misleading. See marginal-structural-models-g-methods.
• vs Fine–Gray / cause-specific competing-risks models: orthogonal concern — if a competing event (e.g., non-outcome death) precludes the event of interest, the time-dependent Cox should be specified as cause-specific, and absolute-risk statements need a cumulative-incidence (Fine–Gray) companion. See competing-risks-cause-specific-fine-gray-rwe.

When NOT to use — and when it is actively misleading or dangerous

.

• Time-varying confounding affected by prior treatment. This is the headline failure. If a confounder (CD4, eGFR, disease activity) responds to earlier exposure and also predicts later exposure and the outcome, conditioning on it in a time-dependent Cox biases the effect (collider/over-adjustment) while omitting it leaves residual confounding — there is no safe option. Use an MSM/g-methods. Mistaking this for "I just need more covariates" is the dangerous error.
• Using a covariate value not yet observed at time t. Carrying a final or peak value backward, or coding exposure as 1 for person-time before the exposure date, silently re-introduces immortal-time/guarantee-time bias — the very bias the method exists to prevent. Every interval's covariate must be knowable from data available strictly at `start`.
• A defined treatment strategy is the question (e.g., "initiate and stay on drug A vs B"). That is a sustained-strategy estimand for clone-censor-weight or an MSM, not an instantaneous time-dependent HR.
• Reverse causation / the covariate is part of the outcome process. A lab that moves because the event is imminent (e.g., falling albumin in the days before death) will absorb the effect; the HR is then descriptive, not causal.
• Proportional hazards is violated for the time-dependent term (its effect changes over time); then add an explicit covariate-by-time interaction or move to RMST/flexible models rather than reporting a single misleading HR.

Data-source operational depth

.

• Claims (FFS vs MA): the time-varying exposure is built from pharmacy `fill_date` + `days_supply` (on-treatment episodes with stockpiling and grace rules) or from inpatient/procedure dates. The interval split must use the claim/service date, never the paid/adjudication date, or exposure onset is shifted by months of lag. Failure mode: Medicare Advantage enrollees lack fee-for-service claims, so on/off-drug and hospitalization intervals are unobservable — MA-only person-time masquerades as "unexposed," biasing the time-dependent indicator; restrict to A/B/D FFS (or commercial with pharmacy benefit) and exclude MA-only spans rather than treating them as exposure-free. Same-day duplicate/reversed claims and bundled services create spurious zero-length or overlapping intervals — collapse them before splitting. Plan switching fragments enrollment; left-truncated person-time (enrollment starts mid-history) must enter via `start` > 0, not at 0.
• EHR: time-updated labs/vitals arrive at irregular, encounter-driven times; between measurements you must choose a carry-forward (last-observation-carried-forward) window and cap it, because a value carried for 18 months is fiction. Visit-driven capture means sicker patients are measured more often — informative observation that can bias a measurement-triggered exposure. External care leakage (treatment received outside the system) makes the on/off-exposure indicator wrong; prefer linkage to dispensing.
• Registry: strong for adjudicated time-stamped clinical events (stage change, transplant date) but typically sparse between scheduled visits; reconcile registry event dates with claims/EHR before splitting, and treat between-visit gaps as genuinely unobserved.
• Linked claims–EHR–vital records: the ideal substrate (EHR labs for the time-varying covariate, claims for complete drug/hospitalization episodes, death index for the outcome), but order/fill/service-date discrepancies across sources must be reconciled so each interval's covariate is anchored to a single, defensible clock — otherwise overlapping intervals from different sources double-count person-time.

Worked claims example

Question: does current statin exposure (time-varying on/off) change the hazard of incident myocardial infarction among adults with type 2 diabetes in a commercial + Medicare FFS database?

1. Cohort & time zero: age ≥18, ≥2 diabetes diagnoses, ≥365 days continuous A/B/D (or commercial medical+pharmacy) enrollment before time zero; time zero = first diabetes-qualifying date after enrollment; exclude MA-only person-time.
2. Outcome: first validated MI (e.g., 1 inpatient primary-position MI claim), date = service date.
3. Build statin episodes from `fill_date` + `days_supply`, stitching overlapping fills (stockpiling, cap at, say, 90 days) and bridging gaps ≤30 days (grace period); an MI's risk at a given day is classified by whether that day falls inside an on-statin episode.
4. Split each person's follow-up at every episode start/stop into `(start, stop]` intervals, set `statin_on` = 0/1 per interval, and set the event flag = 1 only on the interval whose `stop` is the MI date.
5. Right-censor at disenrollment, death, end of data; left-truncate any person-time before continuous enrollment via positive `start`.
6. Fit a Cox model on the counting-process data with `statin_on` as a time-dependent covariate plus baseline confounders (and time-updated confounders only if they are NOT affected by prior statin use — otherwise switch to an MSM).
7. Check PH for `statin_on` via scaled Schoenfeld residuals; if violated, add a `statin_on * log(t)` interaction. The contrast is the hazard of MI while currently on statin vs currently off, with each day of person-time credited to the status truly held that day — no immortal time.

Interpreting the output

A time-dependent Cox model of statin use and myocardial infarction returns: HR = 0.82 (95% CI 0.68–0.98) for current statin use vs current non-use, counting-process layout.

Formal interpretation. The estimated HR of 0.82 means that, at each instant of follow-up, a patient currently on statin has, on average, 82% of the instantaneous MI rate of an otherwise similar patient currently not on statin, among those still event-free.

Because person-time is classified by actual statin status on each day — using a (start, stop, event) counting-process layout — days before the first fill contribute to unexposed person-time even for patients who later initiate, eliminating the immortal-time that inflates apparent benefit in naive time-fixed analyses. The PH assumption implies this ratio does not change with time; if the effect is plausibly early-versus-late, a time-split or log(t) interaction should be checked.

Practical interpretation. The HR captures "currently on" vs "currently off" treatment, not "ever-treated" vs "never-treated." This distinction matters when adherence is variable: partial users contribute both on- and off-treatment person-time, diluting the contrast toward the null. Report both the HR and a cumulative incidence difference at a fixed horizon to convey absolute benefit alongside the relative rate estimate.

Decision diagram

flowchart TD
  T0[Time zero: continuous enrollment + cohort entry] --> Epi[Derive exposure/covariate episodes<br/>service dates, days_supply + grace]
  Epi --> Split[Split follow-up at every change into start,stop intervals]
  Split --> Code[Per interval: covariate = value known at start; event = 1 only on outcome interval]
  Code --> Trunc[Left-truncate pre-enrollment time via start &gt; 0; censor at disenroll/death/data end]
  Trunc --> Fit[Cox partial likelihood on counting-process data<br/>HR = effect of CURRENT covariate value]
  Fit --> PH[Check PH on the time-dependent term<br/>Schoenfeld residuals]
  PH -->|PH holds| HR[Report current-exposure hazard ratio]
  PH -->|PH violated| Inter[Add covariate*log t or step interaction / epoch-specific HRs]

flowchart LR
  A[Subject exposed at some point] --> B{How is exposed person-time coded?}
  B -->|Naive: exposed for ALL follow-up| C[Pre-exposure event-free time credited to exposed group]
  C --> D[Immortal / guarantee-time bias: exposure looks protective]
  B -->|Time-dependent: 0 before exposure, 1 after| E[Person-time credited to the status truly held]
  E --> F[Unbiased current-exposure hazard ratio<br/>if no exposure-confounder feedback]
  F --> G{Time-varying confounder affected by prior exposure?}
  G -->|Yes| H[Time-dependent Cox is biased -&gt; use MSM / g-methods]
  G -->|No| I[Time-dependent Cox is appropriate]

Worked example

Scenario

A researcher wants to know whether being currently on a statin reduces the rate of heart attack among adults with type 2 diabetes. Patient 1001 enters the study on 2023-01-01 without a statin prescription. She fills her first statin on day 60 (2023-03-01). She has a heart attack on day 180 (2023-06-29). The key question for the data analyst is: how should her 180 days of follow-up be divided between unexposed time (no statin) and exposed time (on statin)?

Dataset

Counting-process table for patient 1001 after splitting her follow-up at her statin start date. Each row is one interval; the exposure indicator reflects her actual status during that stretch of time.

FIG. 1 — DESIGN TIMELINE

Steps

Result

Unexposed person-time = 59 days (statin_on = 0); exposed person-time = 121 days (statin_on = 1); event attributed to the exposed interval. The hazard ratio for statin_on produced by the Cox model reflects the instantaneous rate of heart attack while currently on statin versus currently off statin, with no immortal time in the calculation.

Trade-offs

Runnable example

import pandas as pd
from lifelines import CoxTimeVaryingFitter
from lifelines.statistics import proportional_hazard_test

# df is the counting-process table described in the header (one row per person-interval).
# Sanity check: intervals are well-formed and non-overlapping within person.
assert (df["t_stop"] > df["t_start"]).all(), "each interval needs t_stop > t_start"

ctv = CoxTimeVaryingFitter(penalizer=0.0)
ctv.fit(
    df,
    id_col="person_id",
    start_col="t_start",
    stop_col="t_stop",
    event_col="event",
    # statin_on enters at the value it holds in each interval -> time-dependent HR
    formula="statin_on + age + female + prior_cvd",
)
ctv.print_summary()  # exp(coef) for statin_on = HR of MI while currently on vs off statin

# Proportional-hazards check for the time-dependent term via scaled Schoenfeld residuals.
# If statin_on violates PH, refit with a step (Heaviside) or statin_on:log(t) interaction.
ph = proportional_hazard_test(ctv, df, time_transform="log")
print(ph.summary)

library(survival)

# Follow-up time in days from time zero; left truncation handled by tstart of the first interval.
base$fu_days   <- as.numeric(base$exit_date - base$time_zero)
statin$start_d <- as.numeric(statin$episode_start - base$time_zero[match(statin$person_id, base$person_id)])
statin$stop_d  <- as.numeric(statin$episode_end   - base$time_zero[match(statin$person_id, base$person_id)])

# Build the counting-process dataset: split at episode boundaries, carry the event to the final interval.
cp <- tmerge(base, base, id = person_id,
             mi = event(fu_days, event))
cp <- tmerge(cp, statin, id = person_id,
             statin_on = tdc(start_d))        # 1 from each episode start onward within the episode span

fit <- coxph(Surv(tstart, tstop, mi) ~ statin_on + age + female + prior_cvd,
             data = cp, id = person_id, ties = "efron")
summary(fit)                                  # exp(coef) for statin_on = current-exposure HR

# Proportional-hazards diagnostic on the time-dependent term (scaled Schoenfeld residuals).
cox.zph(fit)

/* (A) Programming-statement time-dependent covariate: statin_on is re-evaluated at each risk set. */
/*     stat_start = days from time zero to statin initiation (missing if never treated).            */
proc phreg data=work.cohort;
  class female prior_cvd / param=ref;
  model fu_days*event(0) = statin_on age female prior_cvd / rl ties=efron;
  /* At each event time, a subject is "on statin" iff treatment started at or before that time. */
  if stat_start ne . and stat_start <= fu_days then statin_on = 1; else statin_on = 0;
  assess var=(age) ph / resample;   /* proportional-hazards check */
run;

/* (B) Counting-process (start, stop] form with a pre-split dataset and left truncation via tstart. */
proc phreg data=work.cp;
  class female prior_cvd / param=ref;
  model (tstart, tstop)*event(0) = statin_on age female prior_cvd / rl ties=efron;
  /* HR for statin_on = hazard while currently exposed vs currently unexposed. */
run;

Citations

1. [1]Fisher LD, Lin DY. Time-dependent covariates in the Cox proportional-hazards regression model. Annual Review of Public Health. 1999;20:145-157.
2. [2]Andersen PK, Gill RD. Cox's regression model for counting processes: a large sample study. The Annals of Statistics. 1982;10(4):1100-1120.
3. [3]Cox DR. Regression models and life-tables. Journal of the Royal Statistical Society Series B. 1972;34(2):187-220.

1. [4]Giobbie-Hurder A, Gelber RD, Regan MM. Challenges of guarantee-time bias. Journal of Clinical Oncology. 2013;31(23):2963-2969.

1. [5]Suissa S. Immortal time bias in pharmacoepidemiology. American Journal of Epidemiology. 2008;167(4):492-499.