A Structural Description of Biases That Generate Immortal Time

Abstract

Immortal time may arise in survival analyses when individuals are assigned to treatment strategies based on post-eligibility information or selected based on post-assignment eligibility criteria. Selection based on eligibility criteria applied after treatment assignment results in immortal time when the analysis starts the follow-up at assignment. Misclassification of assignment to treatment strategies based on treatment received after eligibility results in immortal time when the treatment strategies are not distinguishable at the start of follow-up. Target trial emulation prevents the introduction of immortal time by explicitly specifying eligibility and assignment to the treatment strategies, and by synchronizing them at the start of follow-up. We summarize analytic approaches that prevent immortal time when longitudinal data are available to emulate the target trial from the time of treatment assignment. The term "immortal time bias" suggests that the source of the bias is the immortal time, but it is selection or misclassification that generates the immortal time, leading to bias.

Figure 1.

Survival curves from a follow-up study based on all data (A), on data restricted to those who survive at least 3 months with follow-up starting at assignment (B), on data restricted to those who survive at least 3 months with follow-up starting at 3 months (C).

Figure 2.

Causal directed acyclic graph of a follow-up study with treatment A₀ at time 0 and outcome Y₁₂ at time 12, and with selection represented by the conditioning on Y₃=0 at time 3, a post-assignment eligibility criterion. U represents unmeasured susceptibility factors. Immortal time arises when the start of follow-up in the analysis remains at time 0.

Figure 3.

Schematic representation of a study with 16 individuals (horizontal lines) assigned to one of two strategies indistinguishable at time zero: (A) original assignment Z, (B) reconstructed assignment Z* with immortal time, (C) reconfiguration of the data to emulate a study in which individuals are assigned to strategies that are distinguishable at time zero (no immortal time), and (D) reconfiguration of the data to allocate person-time to “exposed” and “unexposed” categories (no immortal time).

Figure 3.

Schematic representation of a study with 16 individuals (horizontal lines) assigned to one of two strategies indistinguishable at time zero: (A) original assignment Z, (B) reconstructed assignment Z* with immortal time, (C) reconfiguration of the data to emulate a study in which individuals are assigned to strategies that are distinguishable at time zero (no immortal time), and (D) reconfiguration of the data to allocate person-time to “exposed” and “unexposed” categories (no immortal time).

Figure 4.

Causal directed acyclic graph of a follow-up study with treatment A₀, A_1, and A₂ (jointly represented as A_0-2) at times 0, 1, and 2, respectively, and outcome Y_1-2 at times 1 and 2, and Y₁₂ at time 12. Z* represents post-eligibility assignment (to a treatment strategy) that is constructed as a deterministic function of A_0-2 and Y_1-2. U represents unmeasured susceptibility factors. Immortal time arises for some treatment strategies because individuals can only be assigned to them if they survived the early follow-up period (Y_1-2=0).