Sensitivity Analysis of Life History Data with Loss to Follow-up: Extending Multistate Models Using Multiple Imputation

Abstract

Life history data describes a process that progresses through various stages before reaching a terminal event, providing detailed insights into the entire disease trajectory. It is increasingly used in health research to evaluate treatment effects, risk factors, and policy implementations. However, handling loss to follow-up (LTF) remains a major challenge since conventional multistate models often assume that LTF-induced censoring is independent of the life history process, an assumption that may not hold in practice. This paper addresses this issue through sensitivity analysis, which characterizes deviations from independent censoring and evaluates the impact on treatment effect estimation. We extend the classical multistate model to include separate pre- and post-LTF transition intensities. Using a delta-adjusted (DA) sensitivity assumption, we apply multiple imputation (MI) to generate potential outcomes beyond LTF. The proposed framework is illustrated using real-world data to assess the impact of the World Health Organization’s Treat All policy on HIV disease progression.

1. Introduction

In health research, interest often lies in modeling a life history process where individuals may transition through various intermediate stages before reaching a terminal event such as death. Classical survival analysis, which typically models the duration from a time origin to a single terminal event, is limited in its ability to capture the complexities of these intermediate events [1]. The multistate modeling (MSM) framework naturally extends survival models by characterizing transitions between multiple disease states, thereby providing a more comprehensive view of disease progression [2]. This approach is particularly useful for understanding the effects of treatment, policy interventions, and risk factor exposures on different stages of the disease process.

However, life history data are often incomplete due to individuals becoming lost to follow-up (LTF). Conventional MSM analyses assume that LTF-induced censoring is independent of the underlying disease process (i.e., non-informative censoring). This assumption may not be realistic in many clinical and observational settings, where LTF is likely related to a patient’s disease severity or other unobserved factors [3]. As a result, failure to account for dependent LTF can bias estimates of transition intensities and treatment effects.

To address this challenge, we propose a sensitivity analysis framework using multiple imputation (MI) to handle dependent LTF in life history data. Our approach is motivated by data from a quasi-experiment evaluating the World Health Organization’s Treat All policy for people living with HIV. We extend the conventional MSM framework by introducing separate transition intensities for pre- and post-LTF states using a delta-adjustment (DA) sensitivity assumption [4–8]. The DA assumption scales the pre-LTF hazard by a factor $\delta$, allowing for systematic deviations in post-LTF behavior. This extension provides a flexible way to explore different scenarios and assess the robustness of treatment effect estimates when LTF is potentially informative.

The remainder of this paper is organized as follows. In Section 2, we describe the motivating example and the conventional MSM model. Section 3 presents our proposed framework, including the sensitivity assumptions and MI procedure. Section 4 applies the method to real-world data, and Section 5 concludes with a discussion.

2. Background

2.1. Example Study

In September 2015, the World Health Organization (WHO) recommended immediate antiretroviral therapy (ART) initiation for people living with HIV, regardless of their CD4 cell count or disease stage [9]. This policy, commonly referred to as Treat All, was adopted by 84% of low- and middle-income countries by 2018 [10]. While the impact of the Treat All policy has been evaluated in terms of improving timely ART initiation, viral suppression, and reducing mortality [11–13], its effect on HIV disease progression through intermediate disease stages remains underexplored. Understanding the effect of the Treat All policy on the entire disease process is crucial for assessing long-term outcomes and optimizing HIV care.

In resource-limited settings where CD4 and HIV viral load testing are less accessible, the WHO HIV/AIDS Clinical Staging System serves as a primary tool for monitoring patient status [14]. This system categorizes patients into four disease stages: Stage 1 (asymptomatic), Stage 2 (mild symptoms), Stage 3 (advanced symptoms), and Stage 4 (severe symptoms, i.e., AIDS). Our study uses data from the Central Africa cohort of the International Epidemiology Databases to Evaluate AIDS (CA-IeDEA) consortium, spanning from 2013 to 2019, which includes patients from Burundi, Cameroon, the Democratic Republic of Congo, the Republic of Congo, and Rwanda.

The primary goal of our analysis is to assess the impact of the Treat All policy on HIV disease progression using WHO clinical staging data. We define two mutually exclusive cohorts: (1) the Treat All Guideline (TAG) cohort, which includes patients enrolled after each country adopted the policy, and (2) the Non-Universal Guideline (NUG) cohort, which includes patients enrolled before the Treat All adoption date which is chosen to ensure equal observation windows across the two cohorts. The data consists of 9,046 patients after excluding those without baseline staging information, with 4,607 in the TAG cohort and 4,439 in the NUG cohort. Table 1 presents the baseline characteristics of the two cohorts, while Table 2 summarizes the observed transitions among disease stages.

Table 1.

Summary of enrollment by country and patient baseline characteristics

		TAG (n=4607)	NUG (n=4439)
WHO stage	1 (asymptomatic)	2919 (63.4%)	2622 (59.1%)
	2 (mild)	814 (17.7%)	813 (18.3%)
	3 (advanced)	720 (15.6%)	843 (19.0%)
	4 (severe)	154 (3.3%)	161 (3.6%)
	n-miss	0	0
Age	Mean (SD)	33.5 (12.9)	32.0 (13.4)
	n-miss	21	11
Sex	Male	1878 (40.8%)	1676 (37.8%)
	Female	2728 (59.2%)	2762 (62.2%)
	n-miss	1	1
Season	Dry	2229 (48.4%)	2356 (53.1%)
	Rainy	2378 (51.6%)	2083 (46.9%)
	n-miss	0	0
Country	C1 (Burundi)	1241 (26.9%)	1315 (29.6%)
	C2 (Cameroon)	822 (17.8%)	264 (5.9%)
	C3 (Democratic Republic of Congo)	255 (5.5%)	315 (7.1%)
	C4 (Republic of Congo)	82 (1.8%)	175 (3.9%)
	C5 (Rwanda)	2207 (47.9%)	2370 (53.4%)
	n-miss	0	0
Marital Status	M1 (Single/never married)	1358 (35.5%)	1186 (31.7%)
	M2 (Divorce/separated/widow)	716 (18.7%)	768 (20.5%)
	M3 (Married/living with a partner/engagement)	1678 (43.8%)	1727 (46.2%)
	M4 (Other)	76 (2.0%)	59 (1.6%)
	n-miss	779	699
CD4 value at baseline	Median (IQR)	378 (207, 560)	399 (239, 605)
	n-miss	3710	3001

Table 2.

Observed state-to-state transitions in CA-IeDEA.

	To
	Treatment						Control
From	1	2	3	4	5	9	1	2	3	4	5	9
1	4667	24	21	7	9	221	3459	24	36	2	24	268
2	0	2024	23	3	8	45	0	1784	40	9	14	57
3	0	0	3578	14	27	27	0	0	4736	16	35	43
4	0	0	0	676	14	7	0	0	0	927	17	4

1: asymptomatic; 2: mild; 3: advanced; 4: severe (i.e., AIDS); 5: death; 9: LTF

The main methodological challenge is handling the high frequency of LTF, which can occur from any of the transient states (Stages 1, 2, 3, or 4) and may be dependent on disease stage at the time of LTF. Without appropriately accounting for LTF, conventional analyses may produce biased estimates of the policy effect on disease progression.

2.2. Conventional MSM Model

A continuous-time multistate model allows for transitions between disease states at unknown times, accommodating multiple state-to-state transitions over the observation period. Let ${\mathit{y}}_{\mathit{i}\mathit{t}}$ denote the disease stage of individual “$i$” at time $t\ge 0$. For a transition from state $r$ to $s$, the transition intensity (or hazard) is defined as:

$${\mathit{\lambda }}_{\mathit{i},\mathit{r}\mathit{s}}(\mathit{t})=\underset{\Delta t\downarrow 0}{\text{lim}}\frac{P\left({\mathit{y}}_{\mathit{i}\mathit{t}+\Delta \mathit{t}}=s\mid {\mathit{y}}_{\mathit{i}\mathit{t}}=r\right)}{\Delta t},$$ (1)

which is the instantaneous hazard of entering state $s$, given that the previous state occupied was a state $r$. The hazard can be modeled as

$${\mathit{\lambda }}_{\mathit{i},\mathit{r}\mathit{s}}\left(\mathit{t}\right)={\mathit{\lambda }}_{\mathit{r}\mathit{s}}^{\left(\mathbf{0}\right)}\left(\mathit{t}\right)\text{exp}\left({\mathit{\beta }}_{rs}^T{\mathit{Z}}_{\mathit{i}}\right),$$ (2)

where ${\mathit{\lambda }}_{\mathit{r}\mathit{s}}^{\left(\mathbf{0}\right)}(\mathit{t})$ is the baseline hazard function, ${\beta }_{rs}$ is the transition-specific regression coefficient, and $Z_i$ is the vector of independent variables and covariates (e.g., treatment assignment, age, gender, etc.). For simplicity, we assume an Exponential distribution for the baseline hazard function, given its memoryless property, which suits the HIV disease progression process where left censoring is present.

While the multistate model provides a more nuanced understanding of disease progression compared to traditional survival models, conventional implementations typically assume that LTF is independent of the disease process. This assumption, known as noninformative censoring, implies that the hazard of becoming LTF is unrelated to a patient’s future transitions, conditional on observed covariates. When this assumption is violated, LTF may act as an informative censoring mechanism, potentially biasing standard MSM estimates.

To address this, we propose a sensitivity analysis framework that extends the conventional MSM by allowing for dependent censoring mechanisms. Specifically, we introduce separate pre- and post-LTF transition intensities to capture deviations from the noninformative assumption. In the next section, we describe our proposed framework and demonstrate how to conduct sensitivity analysis using multiple imputation under different sensitivity assumptions.

3. Sensitivity Analysis

3.1. Extended MSM and Sensitivity Assumptions

We propose an extended MSM as illustrated in Figure 1 where the superscript ‘$p$’ denotes the post-LTF state and corresponding hazard. The functions ${\alpha }_1\left(t\right),{\alpha }_2\left(t\right),{\alpha }_3\left(t\right)$, and ${\alpha }_4\left(t\right)$ represent the censoring intensities when LTF occurs from state 1, 2, 3, and 4, respectively. This model extends conventional multi-state models to distinguish transitions occurring before and after LTF, building upon the framework established by Lawless and Cook [15]. By introducing post-LTF intensities, denoted as ${\mathit{\lambda }}_{\mathit{r}\mathit{s}}^{\mathit{p}}\left(\mathit{t}\right)$, we can flexibly model different censoring mechanisms and assess their impact on disease progression.

Figure 1:

Extended MSM model for HIV disease evolution with lost to follow-up where ${\mathit{k}}^{\mathit{p}}$ represents the state k after LTF.

Independent Censoring Assumption (IC)

Under the independent censoring (IC) assumption, the hazard of becoming LTF is unrelated to the future disease trajectory. In other words, the probability that a subject is censored does not depend on unobserved future transitions:

$$P\left(T_{i,rs}>t\mid C_i>t,Z_i\right)=P\left(T_{i,rs}>t\mid Z_i\right),\forall t.$$

This implied that, conditional on the covariates $Z_i$, the transition time $T_{i,rs}$ and the censoring time $C_i$ are independent. Consequently, for a subject censored at $C_i$, the transition intensity post-LTF remains the same as the pre-LTF intensity:

$${\mathit{\lambda }}_{\mathit{i},\mathit{r}\mathit{s}}^{\mathit{p}}(\mathit{t})={\mathit{\lambda }}_{\mathit{i},\mathit{r}\mathit{s}}(\mathit{t}),\mathit{t}>C_i$$ (3)

In Figure 1, this assumption is depicted by the equality of pre- and post-LTF hazards. Because the post-LTF transitions mirror the pre-LTF transitions, standard MSM models yield unbiased estimates under this assumption, provided that the LTF mechanism is non-informative.

Delta-adjustment Assumption (DA)

The DA assumption introduces a sensitivity parameter ${\delta }_r$ that scales the pre-LTF transition intensity:

$${\mathit{\lambda }}_{\mathit{i},\mathit{r}\mathit{s}}^{\mathit{p}}\left(\mathit{t}\right)={\delta }_r{\mathit{\lambda }}_{\mathit{i},\mathit{r}\mathit{s}}\left(\mathit{t}\right),\mathit{t}>C_i$$ (4)

This scaling allows the hazard to increase or decrease post-LTF depending on ${\delta }_r$. For instance, ${\delta }_r>1$ indicates an increased hazard (e.g., worsening disease progression) for transition from $r$ to $s$, while ${\delta }_r<1$ indicates a reduced hazard. By varying ${\delta }_r$, one can conduct a sensitivity analysis to explore different post-LTF behaviors. Note that independent censoring (IC) can be viewed as a special case with ${\delta }_r=1$. We describe it separately for the conceptual understanding.

3.2. Multiple Imputation Analysis

A comprehensive discussion of the multiple imputation (MI) method was provided by Rubin [16]. MI offers a general framework to handle missing or incomplete data. In classical survival analysis, censored event times can be treated as missing data [17], making MI an appropriate tool for imputing these unobserved values—a strategy explored by various researchers [18,19]. For illness-death models, which represent the simplest form of life history data, Yu et al. [20] proposed an MI-based method using two approaches: Poor Man’s Data Augmentation (PMDA) and Asymptotic Normal Data Augmentation (ANDA) for interval-censored data. Their findings suggest that PMDA may underestimate the true variability when the proportion of missing data is high, making ANDA a more reliable alternative in such cases.

In our problem, since censoring times are typically unobserved for individuals who become lost to follow-up, the analysis can be framed as a missing data problem, where the censored transition times are treated as missing values. We therefore adopt the ANDA approach.

We begin by estimating the baseline hazard parameters ${\mathit{\lambda }}_{\mathbf{0},\mathit{r}\mathit{s}}$ and the coefficients ${\mathit{\beta }}_{\mathit{r}\mathit{s}}$ governing the transition intensities between states, along with their covariance matrix $\mathit{\Sigma }$, using the observed data and a conventional MSM model. Denote ${\mathit{\theta }}_{\mathit{r}\mathit{s}}=\left(\mathbf{l}\mathbf{o}\mathbf{g}\left({\mathit{\lambda }}_{\mathbf{0},\mathit{r}\mathit{s}}\right),{\mathit{\beta }}_{\mathit{r}\mathit{s}}\right)$. For each imputation iteration, we draw ${\tilde{\mathit{\theta }}}_{\mathit{r}\mathit{s}}~N\left({\mathit{\theta }}_{\mathit{r}\mathit{s}},\Sigma \right)$ to compute the transition intensities ${\mathit{\lambda }}_{\mathit{r}\mathit{s}}\left(\mathit{t}\right)$ and the state-specific censoring hazard function ${\mathit{\alpha }}_{\mathit{k}}\left(\mathit{t}\right)$ for each treatment group (e.g., intervention or placebo).

To impute the latent states after LTF for censored individuals, we follow these steps:

1. Start at the last observed visit time $\left({\mathit{T}}_1\right)$ and consider the subject’s current state (e.g., state r).

2. Simulate the next transition time $\left({\mathit{T}}_2\right)$ from state r to state ${\mathit{r}}^{\mathit{p}}$ (a post-LTF state), using censoring hazard ${\mathit{\alpha }}_{\mathit{r}}\left(\mathit{t}\right)$.

   • Draw from a uniformly distributed $u$ to determine the time until the next event, calculated as:

     $${\mathit{T}}_{\mathbf{2}}={\mathit{T}}_{\mathbf{1}}-\mathbf{l}\mathbf{o}\mathbf{g}(\mathbf{1}-\mathit{u})/{\mathit{\alpha }}_{\mathit{r}}\left(\mathit{t}\right)$$
   • If ${\mathit{T}}_{\mathbf{2}}$ exceeds the end of study time, set ${\mathit{T}}_{\mathbf{2}}=\text{END}$ and stop.
   • If ${\mathit{T}}_{\mathbf{2}}$ occurs before the end of study, assign the transition to state ${\mathit{r}}^{\mathit{p}}$
3. Continue for Subsequent States:

   • For each post-LTF state ${\mathit{r}}^{\mathit{p}}$, simulate the next transition time based on competing risks [21]. The transition to state ${\mathit{s}}^{\mathit{p}}$ would occur with probability proportional to the transition intensity ${\mathit{\lambda }}_{{\mathit{r}}^{\mathit{p}}{\mathit{s}}^{\mathit{p}}}\left(\mathit{t}\right)$

     $$P\left({\mathit{r}}^{\mathit{p}}\mathbf{\to }{\mathit{s}}^{\mathit{p}}\right)=\frac{{\mathit{\lambda }}_{{\mathit{r}}^{\mathit{p}}{\mathit{s}}^{\mathit{p}}}\left(\mathit{t}\right)}{{\mathbf{\sum }}_{{\mathit{s}}^{\mathit{p}}}{\mathit{\lambda }}_{{\mathit{r}}^{\mathit{p}}{\mathit{s}}^{\mathit{p}}}\left(\mathit{t}\right)}$$
   • Continue this process until reaching absorbing state (state 5) or END.

We use above procedure M times to generate M complete datasets. We then analyze the M complete datasets using standard MSM analysis. Finally, we combine the results from the M complete datasets to produce inferential results using Rubin’s rule.

In this procedure, we implicitly assume a Markov process, meaning that the future transitions depend only on the current state and not on the history of prior transitions. This assumption simplifies the modeling of transitions by allowing the transition intensities ${\mathit{\lambda }}_{\mathit{r}\mathit{s}}(\mathit{t}$) and the censoring hazard ${\mathit{\alpha }}_{\mathit{r}}\left(\mathit{t}\right)$ to depend solely on the current state $r$.

However, this approach can be extended to non-Markov or semi-Markov models. In a semi-Markov model, for example, transition intensities may depend on both the current state and the time spent in that state. Extending the imputation procedure to a non-Markovian framework would involve adjusting the transition probabilities to account for the individual’s history or the duration spent in each state, adding complexity but potentially improving the accuracy of the model in cases where the Markov assumption does not hold.

4. Application

4.1. Choosing M for the MI Approach under IC Assumption

In this section, we evaluate the performance of the MI method under the independent censoring (IC) assumption, which corresponds to setting ${\delta }_r=1$ in the DA framework described earlier. With ${\delta }_r=1$, the imputed data are generated from the same conditional failure time distributions as estimated by the conventional MSM approach, assuming that the censoring mechanism is independent and analogous to the MAR (Missing at Random) assumption [22].

To determine the optimal number of imputations (M), we follow the guidelines outlined in [23], which recommend assessing the stability of the estimator across different values of M. For this purpose, we conduct multiple imputations separately for each treatment group, setting the hazard ratio to 1, and generate 100 replicate datasets for each of the following values of M: 3, 5, 10, 20, 30, 40, 50, 70, and 100. This yields nine different sets of imputations for comparison.

The variability of the MI estimates for the transition-specific hazard ratios is summarized in boxplots (Figure 2), using the conventional MSM estimates as the reference (which treat follow-up times of LTF patients as censored). The results indicate that, for small numbers of imputations (M ≤ 30), the MI estimates are less stable, with increased variability across replicates. As M increases, the variability of the estimates decreases and stabilizes around M=50 for both treatment groups.

Figure 2:

Transition-specific hazard ratio for 100 replications of different numbers of imputations. The standard MSM estimates are indicated with the horizontal line.

Thus, we recommend using M=50 imputations for this analysis, as it provides a reasonable balance between computational efficiency and estimate stability, ensuring robust inference for the given dataset.

4.2. Sensitivity Analysis under DA

Under the DA assumption, we conduct sensitivity analyses by varying the sensitivity parameter ${\delta }_r$ over a plausible range which is chosen to explore how the estimated treatment effect changes for different post-LTF scenarios, reflecting varying degrees of deviation from the independent censoring assumption. Specifically, we set ${\delta }_r=\delta =3.0$ as the upper bound, based on a value of 1/0.33, where 0.33 represents a reasonably large effect size derived from preliminary data analysis for a clearly effective Treat All policy. This selection indicates that patients lost to follow-up are assumed to experience events three times more rapidly compared to those who remain in the study.

Such a specification reflects a worst-case scenario, where premature discontinuation significantly increases the hazard of adverse events.

Even though it is plausible that individuals who lost to follow-up in the pre-Treat-All cohort may also have a higher hazard of progression to worse disease states, we assume, for the purpose of this analysis, that their disease progression continues at a similar rate as observed before LTF. Therefore, this analysis serves as a benchmark to understand the sensitivity of our conclusions under varying assumptions about the impact of LTF on disease progression.

We vary $\delta$ in increments of 0.25 from 1 to 3.0, resulting in nine different assessments of the treatment effect. The results are visualized using contour plots (Figure 3) and summarized in Table 3, showing the hazard ratio (HR) estimates and corresponding p-values for each transition type. For transitions from state 1 to 5, 2 to 5, and 3 to 5, the estimated treatment effects remain close to the null value (HR = 1) as $\delta$ increases. Interestingly, the p-values for transitions from state 1 to 5 remain significant across all values of $\delta$, suggesting the robustness of the Treat All policy’s beneficial effect on these transitions.

Figure 3:

Sensitivity analysis with MI with the transition-specific hazard ratios (HRs) under delta adjustment (DA) assumption.

Table 3.

Sensitivity analysis with MI with the transition-specific hazard ratios and p-values under delta-adjustment (DA) assumption.

$\delta$	1→2		1→5		2→3		2→5		3→4		3→5		4→5
	HR	PV	HR	PV	HR	PV	HR	PV	HR	PV	HR	PV	HR	PV
1.0	0.7	0.08	0.3	0.01	0.6	0.01	0.4	0.07	1.0	0.87	0.7	0.21	0.7	0.36
1.3	0.8	0.23	0.3	0.01	0.7	0.02	0.4	0.06	1.2	0.58	0.8	0.28	0.7	0.39
1.5	0.9	0.71	0.3	0.01	0.7	0.06	0.5	0.11	1.2	0.39	0.8	0.36	0.8	0.49
1.8	1.0	0.80	0.4	0.01	0.8	0.22	0.5	0.11	1.4	0.26	0.8	0.45	0.8	0.56
2.0	1.1	0.41	0.4	0.01	0.9	0.59	0.5	0.15	1.5	0.15	0.8	0.48	0.9	0.63
2.3	1.2	0.15	0.4	0.02	0.9	0.93	0.5	0.14	1.5	0.09	0.9	0.58	0.9	0.84
2.5	1.3	0.12	0.4	0.02	1.0	0.76	0.5	0.16	1.6	0.06	0.9	0.65	0.9	0.94
2.8	1.4	0.02	0.4	0.03	1.1	0.59	0.5	0.16	1.8	0.01	0.9	0.71	1.0	0.98
3.0	1.5	0.01	0.4	0.03	1.1	0.40	0.6	0.19	1.9	0.01	0.9	0.73	1.1	0.83

However, for transitions from state 1 to 2, 2 to 3, and 4 to 5, the beneficial effect of the Treat-All policy diminishes as $\delta$ increases. Specifically, the hazard ratio for the transition from state 1 to 2 reverses direction when $\delta$ reaches approximately 1.75, while for the transition from state 4 to 5, the reversal occurs at a larger value of $\delta =2.75$. Additionally, the transition from state 3 to 4 shows an adverse effect of the Treat-All policy even under the conventional analysis $(\delta =1)$, with the magnitude of this effect intensifying as $\delta$ increases (HR ranging from 1.04 to 1.96). This adverse effect becomes statistically significant when $\delta$ reaches 2.75.

5. Discussion and Conclusion

This study proposed an extended MSM framework to handle life history data with LTF under varying sensitivity assumptions. The conventional MSM approach often assumes that censoring, including LTF, is independent of the underlying disease process—a strong assumption that is seldom valid in real-world applications. This work extends previous efforts by integrating a flexible set of post-LTF transition intensities, denoted as ${\mathit{\lambda }}_{\mathit{r}\mathit{s}}^{\mathit{p}}\left(\mathit{t}\right)$, to account for different scenarios where the LTF mechanism is dependent on unobserved factors that may influence disease progression.

We developed a new approach by extending the conventional MSM model to incorporate separate pre- and post-LTF transition intensities. This model allows us to specify the hazard functions before and after LTF separately, using sensitivity assumptions such as the reference-based (RB) and delta-adjusted (DA) models, which have been well-established in longitudinal data settings but rarely applied in life history data. The inclusion of the LTF hazard ${\mathit{\alpha }}_{\mathit{r}}\left(\mathit{t}\right)$ in the model allows a precise representation of the LTF process, making it possible to estimate the effect of LTF on the disease progression.

We adapted the MI procedure to perform sensitivity analysis for life history data. The proposed MI procedure involves sampling from the posterior distribution of the transition parameters, while also incorporating the censoring hazard ${\mathit{\alpha }}_{\mathit{r}}\left(\mathit{t}\right)$. This approach accommodates uncertainty in parameter estimates, resulting in more robust inferences. Our application of MI extends beyond traditional survival data by considering multiple transitions and competing risks in the life history framework.

Using data from the Central Africa cohort of the International Epidemiology Databases to Evaluate AIDS (CA-IeDEA), we demonstrated the utility of the proposed sensitivity analysis in evaluating the WHO’s Treat All policy. The results showed that the impact of LTF on disease progression can significantly alter the estimated treatment effects. For instance, our analysis under the DA assumption indicated that the effectiveness of the Treat All policy diminishes as the sensitivity parameter $\delta$ increases, suggesting that LTF may be masking adverse outcomes in the conventional analyses.

This work has both theoretical and practical implications for researchers handling life history data. Theoretically, the integration of separate post-LTF transition intensities provides a new way to handle nonignorable censoring mechanisms, bridging the gap between conventional MSMs and more complex settings where LTF is a key concern. Practically, the MI-based sensitivity analysis is straightforward to implement using standard software, such as the msm package in R [24], making it accessible for applied researchers.

The findings highlight the importance of conducting sensitivity analysis in settings with potentially informative censoring. While standard methods may yield biased estimates under noninformative censoring, our approach offers a way to quantify the potential impact of LTF, thereby improving the robustness and interpretability of the results.

Several limitations must be acknowledged. First, in the current sensitivity analysis, we assume that the sensitivity parameter ${\delta }_r$ is constant across different starting states $r$. This simplification may not capture potential heterogeneity in post-LTF behavior across different disease stages. For example, patients who become LTF at more advanced stages of disease progression may experience a different post-LTF hazard compared to those lost at earlier stages. A natural extension of this work would be to allow ${\delta }_r$ to vary depending on the starting state r, thereby incorporating more flexible and realistic assumptions. The assumption of a constant ${\delta }_r$ can be relaxed when auxiliary information, such as tracing data or vital status updates, is available. In such cases, a varying ${\delta }_r$ model could be used to reflect different post-LTF intensities specific to certain subgroups. This would enable us to leverage tracing data to better estimate post-LTF behaviors and improve the precision of the sensitivity analysis. Implementing state-specific ${\delta }_r$ values would enhance the flexibility of the model, allowing for more tailored inferences that better capture the complexities of disease progression in the presence of LTF.

Second, our imputation procedure relies on a Markov assumption where transitions depend only on the current state and not on the entire history. Extending this approach to semi-Markov models would be a promising direction for future work, particularly for chronic diseases with long latency periods. Lastly, our analysis focused on a single example in HIV care. Extending the approach to other disease areas such as cancer or cardiovascular disease would provide further validation and highlight the generalizability of the method.

The proposed sensitivity analysis framework using extended MSM and MI provides a novel approach to handling LTF in life history data. By specifying post-LTF hazards under different sensitivity assumptions, we can evaluate how deviations from the independent censoring assumption affect the results. Our application to the WHO’s Treat All policy demonstrates the practical utility of the method and underscores the importance of accounting for LTF in policy evaluation. Future research should focus on extending this framework to non-Markovian settings and exploring additional sensitivity assumptions to broaden the applicability of the method.