Why Last Observation Carried Forward Fails in Survival Data

Why a Carried-Forward Value Can Misrepresent Risk

In survival analysis, a time-varying covariate describes a person’s changing state during follow-up. Examples include a biomarker, a treatment status, or a measure of disease severity. If a value is missing at a particular time, a tempting shortcut is to substitute the most recently observed value and use it until another value is available. This is called last observation carried forward (LOCF).

LOCF may look plausible in a table, but it makes a substantive claim: the covariate stayed at its last observed value throughout the gap. When the covariate actually changed, the resulting survival record assigns the person to the wrong exposure state for some period of time. That can change the modeled hazard, the cumulative hazard, and the estimated relationship between the covariate and the event. The direction and size of the effect depend on the pattern of change and the analysis; they are not guaranteed to be the same in every data set.

Definition: For a time-varying covariate \(X_i(t)\) on person \(i\), LOCF replaces a missing value at time \(t\) with the most recent observed value before \(t\). It then treats that substituted value as applying over the relevant interval until a later value is observed. This is a deterministic substitution, not a draw from a model for the missing covariate.

In start-stop survival records, a covariate value is associated with an interval during which the person is at risk. The key question is not simply whether a carried value is numerically close to a later measurement. It is whether that value represents the person’s state over the interval in which events could occur. For the survival-record structure and risk sets used here, the timing of a covariate value matters because each person’s hazard can depend on the covariate history up to that time.

How the Hazard History Enters

Suppose the hazard for person \(i\) at time \(t\), conditional on the covariate history available up to that time, follows a proportional-hazards model:

\lambda_i(t\mid X_i(t))=\lambda_0(t)\exp\{\beta X_i(t)\}.

Here \(\lambda_0(t)\) is the baseline hazard, and \(\beta\) describes the log hazard ratio for a one-unit difference in the covariate, under the model. This interpretation relies on the proportional-hazards specification, correct alignment of covariate values with risk intervals, and a covariate value that represents the state at that time. In an analysis, valid inference also requires suitable handling of censoring and the other modeling assumptions. LOCF does not make those assumptions true.

The cumulative hazard through time \(t\) is the integral of the instantaneous hazard. On intervals where the covariate is constant, it is a sum of interval contributions:

H_i(t)=\int_0^t \lambda_0(u)\exp\{\beta X_i(u)\}\,du =\sum_j \Delta\Lambda_{0j}\exp\{\beta x_{ij}\},

where interval \(j\) has covariate value \(x_{ij}\), and \(\Delta\Lambda_{0j}\) is the baseline cumulative-hazard increment over that interval. If LOCF substitutes \(\widetilde{x}_{ij}\), the difference in total cumulative hazard is

\widetilde H_i(t)-H_i(t) =\sum_j \Delta\Lambda_{0j} \left[\exp\{\beta\widetilde{x}_{ij}\}-\exp\{\beta x_{ij}\}\right].

A discrepancy in a particular interval changes that interval’s contribution when the baseline hazard increment is positive and \(\beta\ne 0\). But the total discrepancy is the sum across intervals: positive and negative differences can offset one another. Therefore, different covariate histories do not necessarily imply different cumulative hazards at the end of follow-up. Even when the totals agree, the timing of hazard contributions can still differ.

Worked Example: A Missed Change Alters Cumulative Hazard

Consider a hypothetical person followed for 6 months. Let the baseline hazard be constant at \(0.10\) per month, and let \(X\) be binary, with \(\beta=\log 2\). The hazard is thus \(0.10\) per month when \(X=0\) and \(0.20\) per month when \(X=1\). Suppose the true state is \(X=0\) from month 0 to 2, \(X=1\) from month 2 to 3, and \(X=0\) from month 3 to 6. The value observed at month 2 is carried forward through month 5, so the LOCF record incorrectly uses \(X=1\) from month 2 to 5. At month 5, \(X=0\) is observed again.

For the true history, the cumulative hazard is \(H=0.10(2)\exp(0)+0.10(1)\exp(\log 2)+0.10(3)\exp(0)\). Since \(\exp(0)=1\) and \(\exp(\log 2)=2\), this gives \(H=0.20+0.20+0.30=0.70\).

For the carried-forward history, it is \(\widetilde H=0.10(2)(1)+0.10(3)(2)+0.10(1)(1)=0.20+0.60+0.10=0.90\).

Under this specified hazard model, the corresponding survival probabilities through month 6 are \(S(6)=\exp(-0.70)=0.4966\) and \(\widetilde S(6)=\exp(-0.90)=0.4066\). The LOCF history therefore implies a larger cumulative hazard and lower survival in this hypothetical calculation. These values illustrate consequences of the assumed histories and model; in real data, the unobserved true change is usually unknown.

Why the Error Has No Guaranteed Direction

A frequent but incorrect conclusion is that LOCF always overestimates risk because it keeps an old, perhaps higher, exposure value in place. If the old value is lower than the person’s actual value, it can instead understate the interval’s hazard. If the covariate rises and falls between measurements, the errors can change sign across intervals. Their combined effect depends on the durations, baseline-hazard increments, and covariate effects in those intervals.

Worked Example: Different Histories With the Same Total

Suppose there are two equal-length intervals, each with baseline cumulative-hazard increment 1, and \(\beta=\log 2\). Compare an actual history with \(X=1\) in the first interval and \(X=0\) in the second to a carried-forward history with \(X=0\) first and \(X=1\) second.

The actual interval contributions are \(1\exp(\log 2)=2\) and \(1\exp(0)=1\), so \(H=2+1=3\). The carried-forward contributions are \(1\exp(0)=1\) and \(1\exp(\log 2)=2\), so \(\widetilde H=1+2=3\). The histories differ in both intervals, and each interval’s contribution differs, yet the total cumulative hazards are equal because the differences cancel.

Thus, at the end of the second interval, both histories imply survival \(\exp(-3)=0.04979\) under this model. That equality does not make the histories equivalent: the hazard contribution is larger in the first interval for the actual history and in the second for the carried-forward history. A different event time or risk-set comparison can therefore be affected even though these endpoint cumulative hazards match.

This calculation also clarifies the conditions for a contribution to change. If \(X\) differs on an interval with positive baseline cumulative-hazard increment and \(\beta\ne0\), that interval’s contribution differs. When \(\beta=0\), the covariate does not affect the modeled hazard; when the baseline increment is zero, the interval contributes no baseline hazard. Across multiple intervals, however, the total differences can cancel.

How Exposure Misclassification Can Distort an Association

LOCF can also misclassify person-time as exposed or unexposed. This can alter comparisons of event rates and can influence a model’s estimated covariate association. The following deliberately simplified calculation illustrates attenuation; it is not a formula for the bias of every survival model.

Worked Example: A Crude Rate Contrast Is Attenuated

Imagine hypothetical person-time with true event rates of 20 events per 1,000 person-months while exposed and 10 per 1,000 person-months while unexposed. In a simplified mixture, person-time labeled exposed by LOCF consists of 500 exposed person-months and 500 unexposed person-months. The expected event count is \(500(20/1000)+500(10/1000)=10+5=15\), giving a labeled-exposed rate of \(15/1000=0.015\) per person-month.

Person-time labeled unexposed consists of 500 exposed person-months and 1,500 unexposed person-months. Its expected event count is \(500(20/1000)+1{,}500(10/1000)=10+15=25\), giving a labeled-unexposed rate of \(25/2000=0.0125\) per person-month. The crude rate ratio based on the LOCF labels is \(0.015/0.0125=1.20\), whereas the true-state rate ratio in this construction is \(0.020/0.010=2.00\).

The example shows how mixing states can weaken a contrast. It does not establish that LOCF always attenuates a hazard ratio: the mixture, timing of events, risk-set composition, confounding, and model specification all matter. Nor does either rate ratio by itself establish a causal effect.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to compare each covariate’s observation times with the start-stop intervals and event times. Long gaps, frequent changes at observed visits, and large differences between successive values make a constant-between-visits assumption less credible. Where possible, compare results under plausible alternative assumptions about when changes occurred. Such checks reveal sensitivity; they cannot recover a change that was never observed without additional information or assumptions.

LOCF is especially problematic when it is used as if it were a general missing-data method. It creates one completed value with no allowance for uncertainty about what the missing value could have been. It also assumes, rather than estimates, that the covariate stayed unchanged. If the covariate predicts the event and changes during follow-up, those choices can distort both the modeled history and the estimated relationship. Adding more imputed data sets does not solve this if every data set is produced by the same deterministic carry-forward rule.

The main alternatives depend on what is known and why values are missing. If repeated covariate measurements are available, represent their timing and observed changes in the survival records rather than replacing those changes with a single earlier value. If values are missing, a multiple-imputation model can draw plausible values using relevant observed information and survival-related predictors, while respecting the time ordering and structure of the records. The earlier material on imputation of intermittently missing time-varying covariates gives the basic setting; here the essential contrast is that imputation represents uncertainty, whereas LOCF fixes a value by rule.

LOCF can be defensible as a data representation when the scientific variable is genuinely defined to remain constant until a new recorded update, and that update process matches the real exposure process. It may also be a reasonable approximation over a short gap if stability is substantively plausible. Those are assumptions to justify, not properties guaranteed by the method. No imputation strategy can compensate for an ill-defined covariate, poor measurement timing, or a survival model that does not represent the relevant risk structure.

Common pitfall: “The last measured value is the best available value, so it is safe to use until the next visit.” Availability is not evidence of stability. The correction is to ask whether the covariate could have changed during the interval, whether that change affects the hazard, and how uncertainty about the missing history is represented. Also, a changed history can have a different contribution in each interval while still producing the same total cumulative hazard through cancellation.

A Practical Review

Define the covariate in time.
State what \(X_i(t)\) represents and when each measurement applies to the risk interval.

Identify what LOCF assumes.
For every gap, it assumes the most recently observed value remained applicable throughout that gap.

Check interval contributions.
Use the hazard model to determine how substituting a value changes each interval’s contribution; then check whether differences can cancel in the total.

Compare with a suitable alternative.
Use observed changes where available, and represent uncertainty in missing values rather than treating a deterministic substitution as known truth.

Check Your Understanding

Answer each question, then compare with the brief solutions.

If \(\beta=0\), does changing a covariate value alter its modeled hazard contribution? Why?
With \(\beta=\log 2\) and a baseline cumulative-hazard increment of \(0.4\), calculate the interval contribution for \(X=0\) and \(X=1\).
Can two different covariate histories have the same total cumulative hazard? Give the condition that allows this.
Why is a deterministic LOCF substitution not equivalent to multiple imputation?

Solutions. (1) No. Since \(\exp\{\beta X\}=\exp(0)=1\) for any \(X\), the contribution is the baseline increment. (2) For \(X=0\), the contribution is \(0.4\exp(0)=0.4\). For \(X=1\), it is \(0.4\exp(\log 2)=0.8\). (3) Yes. Interval-specific differences can have opposite signs and sum to zero, as in the two-interval example. (4) LOCF fixes each missing value to a single prescribed value and does not represent uncertainty by drawing plausible values from a model.