Why Entry Time Changes the Likelihood
In many PK/PD studies, a participant is not observed from the time at which they first become at risk. A patient may begin treatment or enter a risk period at time zero but enroll in a study later. If the event of interest must not have occurred before enrollment, the observed sample includes only people who remained event-free long enough to enter. This is left truncation, also called delayed entry.
The key consequence is that an observed participant’s event-time distribution is conditional on surviving event-free to their entry time. The likelihood must therefore compare survival after entry with survival to entry, rather than treating the participant as if their event-free time before observation had also been observed. This matters especially when the hazard depends on time since treatment, calendar time, or a changing drug concentration.
Suppose the hazard for subject \(i\) at time \(t\), given their modeled covariate and exposure history, is \(h_i(t)\). Define cumulative hazard from the chosen time origin as \(H_i(t)=\int_0^t h_i(u)\,du\). Then survival from that origin is \(S_i(t)=\exp[-H_i(t)]\). These definitions apply whether the hazard is constant or changes with time and exposure.
Deriving the Conditional Likelihood
For an exact event at time \(t>L_i\), the ordinary event-time density is \(f_i(t)=h_i(t)S_i(t)\). But the subject is in the observed sample only given survival to entry, \(T_i>L_i\). Divide the density by the probability of that selection event:
For a subject censored at \(c\ge L_i\), the conditional contribution is the probability of remaining event-free from entry through censoring:
Together, the contributions for exact events and right censoring can be written in one expression. The cumulative hazard increment is from entry to the observed time, not from time zero.
This likelihood assumes that the event-time model is correctly specified over the interval relevant to the calculation, that event and censoring times are recorded consistently, and that censoring is independent of the event process conditional on modeled information. It also assumes that selection into observation is adequately represented by conditioning on event-free survival to entry. If entry depends on unmodeled prognostic factors, or on event risk in ways not captured by the model, this conditional likelihood need not remove selection bias.
A time-varying exposure requires a further practical condition: the exposure history used to calculate \(h_i(t)\) must be available or modeled over the interval being integrated. If risk depends on concentration before entry, that prior history may be needed to specify the model and interpret predictions. The likelihood contribution after entry uses \(H_i(Y_i)-H_i(L_i)\); it does not make unknown pre-entry exposure history known.
Worked Likelihoods and Risk Calculations
Worked Example: Constant Hazard With Delayed Entry
Consider four hypothetical participants, all entering at day 5. Two experience the event at days 6 and 7; the other two are censored at day 10. Assume a common constant hazard \(\lambda\) per day, independent censoring, and no covariates. These assumptions make the hazard \(h(t)=\lambda\) and the cumulative hazard \(H(t)=\lambda t\).
The observed time at risk after entry is \(1, 2, 5,\) and \(5\) days, respectively, for a total of 13 days. There are two events. The conditional likelihood is therefore
Differentiate and set the score to zero: \(2/\lambda-13=0\). Thus the maximum-likelihood estimate is \(\hat{\lambda}=2/13=0.1538\) per day. The estimate describes the constant hazard under the assumed model; the small hypothetical data set is only for illustrating the calculation.
If the entry times are ignored and all four participants are incorrectly treated as at risk from day 0, the total time used becomes \(6+7+10+10=33\) days. The resulting estimate is \(2/33=0.06061\) per day, substantially lower because it counts 20 unobserved, event-free days as though they had been observed at risk. Under the correctly conditioned estimate, the fitted probability of an event in the five days after entry is \(1-\exp[-(0.1538)(5)]=0.5365\). Under the incorrect estimate it is \(1-\exp[-(0.06061)(5)]=0.2614\). These are model-based fitted probabilities, not observed proportions.
The fitted absolute risk from day 0 to day 10 under the correctly conditioned estimate would be \(1-\exp[-(2/13)(10)]\approx0.7853\). That is an extrapolation under the constant-hazard model: the study did not observe these participants from day 0, so the data alone do not directly establish their pre-entry risk.
Worked Example: A Changing Concentration After Entry
For a hypothetical subject, suppose the hazard is \(h(t)=0.0200\exp[0.800C(t)]\) per day, where concentration \(C(t)\) is in mg/L. The subject enters at day 4 and has an event at day 7. Concentration is \(0.500\) mg/L from day 4 through day 6 and \(1.00\) mg/L from day 6 through day 7. Assume this hazard model and the specified concentration history are correct.
The hazard is \(0.0200\exp(0.800\times0.500)=0.02984\) per day during days 4–6, and \(0.0200\exp(0.800\times1.00)=0.04451\) per day during days 6–7. The cumulative hazard increment from entry to the event is
Since the event occurs when concentration is \(1.00\) mg/L, the conditional likelihood contribution is
This contribution combines the event hazard at the event time with survival only over the observed interval after entry. Treating survival from day 0 to day 4 as observed would add an unjustified factor. If, in addition, we assume concentration was \(0.500\) mg/L from day 0 through day 4, then \(H(4)=4(0.02984)\approx0.1194\), so that erroneous factor would be \(\exp(-0.1194)\approx0.8875\). In a parameter-estimation problem, including such a factor can change the likelihood’s preference among parameter values.
Worked Example: Conditional Risk Under a Piecewise Hazard
Suppose a hypothetical drug-related event has hazard \(0.0400\) per day from day 0 to day 5 and \(0.100\) per day after day 5. A participant is known to be event-free at entry on day 3. Calculate the probability of an event by day 8 conditional on being event-free at entry. Assume the piecewise hazard is correctly specified and there is no competing event.
The cumulative hazard from day 3 through day 8 is \((5-3)(0.0400)+(8-5)(0.100)=0.380\). Thus conditional survival through day 8 is \(\exp(-0.380)=0.6839\), and the conditional event probability is \(1-0.6839=0.3161\), or 31.61%.
For comparison, cumulative hazard from day 0 through day 8 is \((5)(0.0400)+(3)(0.100)=0.500\), giving unconditional event risk by day 8 of \(1-\exp(-0.500)=0.3935\), or 39.35%. This is not the appropriate risk for someone selected because they were event-free at day 3. Conditioning removes the risk accumulated before entry from the post-entry probability.
Risk Sets, Diagnostics, and Common Alternatives
In a risk-set analysis, a subject contributes to the risk set only from their entry time until their event or censoring time. At time \(t\), an eligible subject must satisfy \(L_i\le t\le Y_i\), with endpoint conventions set consistently for events and censoring. A subject who has not yet entered cannot contribute person-time or serve as an event-free comparison subject at that time. This is the risk-set counterpart of subtracting \(H_i(L_i)\) in the conditional likelihood.
Useful checks include confirming that every observed endpoint is at or after entry, verifying that exposure and follow-up units agree, and checking that each subject’s accumulated hazard uses only the correct interval. Compare the number entering and the number at risk over time; an unexpectedly large risk set before enrollment or a contribution from a subject before \(L_i\) signals a data or implementation error. Predictions should also be checked on the scale actually required: event probability after entry is conditional on being event-free at entry, whereas risk from the original time origin additionally relies on assumptions about the pre-entry process.
The usual alternatives answer different questions. A model that resets time to zero at each subject’s entry can be reasonable when the scientific time scale is time since entry and the hazard is appropriately modeled on that scale. It does not automatically represent a model whose baseline hazard depends on time since treatment began or calendar time. In a proportional-hazards analysis, delayed entry can instead be implemented by placing each subject in risk sets only after entry; this preserves the relevant time origin while respecting delayed observation. Neither approach repairs informative entry or an incorrectly specified exposure history.
Left truncation also limits what the data identify. Among entrants alone, the post-entry conditional event distribution is estimable under suitable assumptions, but the pre-entry survival distribution generally cannot be recovered without additional information or assumptions about the hazard and the entry process. A parametric model may extrapolate backward, but that extrapolation is model-dependent, not a direct observation. If entry depends on unmeasured risk, conditioning on observed survival to entry is not sufficient to make the sample representative of the original at-risk population.
Check Your Understanding
Try each question, then compare your reasoning with the solutions below.
- A subject enters at day 2, has an exact event at day 5, and has a constant hazard of \(0.120\) per day. Write and calculate the conditional likelihood contribution.
- Why does a right-censored subject entering at \(L\) and censored at \(c\) contribute \(\exp[-\{H(c)-H(L)\}]\), rather than \(\exp[-H(c)]\)?
- For the piecewise hazard in the third worked example, what is the conditional probability of remaining event-free through day 8 for a subject entering at day 3?
- Name two conditions or checks that matter when using delayed-entry event data with time-varying PK exposure.
Solutions. (1) The contribution is \(h(5)\exp[-\{H(5)-H(2)\}]=0.120\exp[-(0.120)(3)]=0.120\exp(-0.360)\approx0.08372\) per day. (2) The subject is observed conditional on being event-free at \(L\), so survival through \(c\) must be divided by survival through \(L\); the ratio is \(\exp[-H(c)]/\exp[-H(L)]\). (3) The conditional survival is \(\exp(-0.380)=0.6839\), or 68.39%. (4) Examples include verifying that the exposure trajectory is available or modeled over the interval used to calculate hazard, checking that no subject contributes before entry, and assessing whether entry or censoring depends on unmodeled event risk.