Survival analysis of left-truncated and right-censored HIV data: comparison of Cox regression and alternative models

Abstract

Background

This study examined the impact of delayed entry in HIV cohorts by analyzing left-truncated and right-censored (LTRC) survival data from 69 HIV-positive male patients (24 deaths) followed at a tertiary infectious diseases center. The primary objective was to empirically compare commonly used survival models in estimating survival and identifying prognostic factors under LTRC conditions.

Methods

Three LTRC-adapted modeling frameworks were applied with identical covariates: the semiparametric Cox proportional hazards model, accelerated failure time (AFT) models, and parametric proportional hazards models. Median survival time was estimated using Kaplan–Meier methods under both right-censoring-only and LTRC specifications to assess truncation-related differences. Hazard ratios (HRs) and 95% confidence intervals (CIs) were obtained from the LTRC-adjusted Cox model. Model performance was evaluated using information criteria (AIC, BIC, HQIC).

Results

Ignoring left truncation substantially inflated median survival estimates (3,885 vs. 2,626 days). In the LTRC-adjusted Cox model, age (HR = 1.049, 95% CI: 1.011–1.089) and log-transformed HIV RNA (HR = 1.214, 95% CI: 1.055–1.400) were significant predictors, whereas CD4 count and comorbidity status were not. Among the evaluated models, the Cox model showed the lowest information criterion values within this dataset.

Conclusions

Appropriate risk-set specification under left truncation is essential for reliable survival estimation in delayed-entry HIV cohorts. Within this empirical dataset, the LTRC-adapted Cox model showed favorable performance relative to AFT and parametric PH alternatives.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12874-026-02874-5.

Introduction

Survival analysis comprises a set of statistical methodologies used to model the time until the occurrence of an event, such as death, disease relapse, or treatment failure. These methods are widely applied across disciplines—including medicine, engineering, economics, and social sciences—and are especially important in clinical research for evaluating patient outcomes [1]. A major methodological challenge in survival data arises from censoring and truncation, which occur when event times cannot be fully observed.

Right-censoring occurs when individuals leave the study before experiencing the event of interest or when the study ends while they remain event-free; thus, the exact event time is only known to exceed a certain point [2]. In contrast, left-truncation arises when subjects enter the study after becoming at risk. As a result, only individuals who survive beyond the truncation time are included in the sample [3, 4]. If not properly accounted for, this selective inclusion mechanism leads to sampling bias and distorted survival estimates.

Several studies have demonstrated that ignoring left-truncation results in inflated and biased survival estimates. For example, Wolfson et al. [5] reported that the median survival of dementia patients was 6.6 years when truncation was ignored but only 3.3 years when truncation was properly accounted for. Asgharian and Wolfson [6] further showed that disregarding truncation can bias both survival time estimation and regression coefficients. Consequently, appropriate modeling of truncation and censoring is essential, particularly in epidemiological settings characterized by delayed entry [7, 8].

Human immunodeficiency virus (HIV) infection weakens the immune system, increasing susceptibility to opportunistic infections and malignancies. Although antiretroviral therapy (ART) has transformed HIV into a chronic, manageable condition, identifying prognostic factors associated with survival remains a critical research focus. Covariates such as age, CD4 cell count, viral load (HIV RNA), and coinfections (e.g., tuberculosis, PCP) have been widely studied for their prognostic relevance [9–11]. HIV cohort studies frequently feature both right-censoring and left-truncation due to late diagnosis or delayed initiation of clinical follow-up [12].

Commonly employed survival modeling approaches include the semiparametric Cox proportional hazards (PH) model [13], accelerated failure time (AFT) model, and fully parametric PH model. The Cox model is widely used for its flexibility and robustness when the proportional hazards assumption holds, although this assumption must be assessed carefully [14]. AFT models serve as valuable alternatives when the logarithm of survival time follows distributions such as Weibull, log-normal, or log-logistic distributions [15, 16]. Parametric PH models can also provide efficient inference when the underlying time-to-event distribution is known or can be reasonably specified [17].

The primary objective of this study is to compare Cox, AFT, and parametric PH models using left-truncated and right-censored HIV patient data to determine the most suitable modeling framework for this data structure. Additionally, the study evaluated the effects of key clinical covariates—age, CD4 + T cell count, HIV RNA level, and comorbidities—on survival time. By quantifying the bias introduced when left-truncation is ignored, the study underscores the methodological importance of appropriately handling delayed entry in survival analysis.

This study is intended as an applied empirical evaluation of survival modeling strategies under LTRC conditions rather than a formal methodological development or simulation-based investigation.

Specifically, this study makes three key contributions to the survival analysis literature.

Structured model comparison

It provides a rigorous comparison of Cox, AFT, and parametric PH models under an explicitly defined left-truncated and right-censored (LTRC) structure using identical covariates, allowing direct evaluation of model behavior under delayed-entry conditions.

Bias and inference evaluation

Beyond quantifying survival inflation when left-truncation is ignored, the study systematically examines how proper risk-set specification affects covariate significance, effect size interpretation, and overall model fit.

Practical modeling implications

The findings demonstrate that model choice and distributional assumptions can materially influence statistical inference and clinical interpretation in delayed-entry survival data. By highlighting how improper risk-set definition may alter substantive conclusions, the study underscores the applied methodological consequences of LTRC specification.

Methods

Study population and data description

This retrospective study utilized clinical follow-up data from 69 HIV-positive male patients who where diagnosed and monitored at the Department of Infectious Diseases, Ondokuz Mayıs University Faculty of Medicine (Samsun, Türkiye). The time of origin for survival analysis was defined as the date of HIV diagnosis (t = 0). The event of interest was all-cause mortality (n = 24). Patients who were alive at the end of the study or lost to follow-up (n = 7) were treated as right-censored, yielding a total of 45 censored observations.

The observation period spanned from January 1, 2006 to January 1, 2020, which served as the common right-censoring time . Although some HIV-positive patients had been diagnosed prior to January 1, 2006, these individuals were not included in the analytical cohort because they were not under active clinical follow-up during the observation window. To account for early clinical variability in laboratory markers, a left-truncation time of 90 days post diagnosis (L = 90) was imposed. This threshold reflects the period required for stabilization of CD4 + T cell counts and HIV RNA viral load levels, reducing bias from early, highly volatile measurements.

Figure 1 summarizes the patient selection process, including initial identification, exclusions due to incomplete baseline data or insufficient follow-up, and application of the 90-day left-truncation criterion. Figure 2 illustrates individual follow-up timelines under left-truncation and right-censoring, demonstrating how delayed entry defines the risk set within the LTRC framework.

Fig. 1.

Flow diagram of patient selection and application of the 90-day left-truncation criterion. Flowchart illustrating the identification, exclusions, and application of the 90-day left-truncation procedure used to derive the final analytical cohort of 69 HIV-positive patients from the initial clinical database (2006–2020)

Fig. 2.

Individual follow-up timelines showing left-truncation and right-censoring. Visualization of left-truncation and right-censoring for HIV-positive patients (n = 69). The solid lines represent each patient’s observation interval, the dashed lines indicate left-truncation times, and the red markers denote censored observations. Dark segments indicate patients who experienced the event (death)

Only patients who survived beyond 90 days after diagnosis were included in the final analytical cohort. Individuals who died or were censored within the first 90 days were excluded to ensure consistency with the imposed left-truncation time (L = 90). Therefore, all n = 69 patients satisfy the left-truncated and right-censored (LTRC) condition.

All analyses were conducted on individuals satisfying the left-truncated, right-censored (LTRC) condition:

The primary clinical covariates included the following:

• Age (years)

• CD4 + T- cell count (cells/mm³)

• HIV RNA viral load (copies/mL)

• Comorbidities (binary indicators for opportunistic infections such as tuberculosis or PCP)

All covariates were defined at baseline. Age corresponded to the age at HIV diagnosis. CD4 cell count and HIV RNA viral load values were obtained from the initial clinical assessment following diagnosis and prior to the application of the 90-day left-truncation threshold. Comorbidity status was recorded at baseline based on clinical documentation. All patients received antiretroviral therapy (ART) during follow-up according to routine clinical care; however, because ART exposure was universal in this cohort, it was not included as a covariate in the regression models.

To improve model interpretability and meet statistical assumptions, HIV RNA values were log-transformed using base 10, and the CD4 count was evaluated both continuously and categorically, with a clinically established cutoff of ≤ 200 cells/mm³ to reflect severe immunosuppression.

Survival models

To evaluate survival under left-truncated and right-censored (LTRC) conditions, both nonparametric and model-based approaches were considered. Specifically, the Kaplan–Meier estimator was used as a nonparametric method, alongside three modeling frameworks: the semiparametric Cox proportional hazards (PH) model, accelerated failure time (AFT) models, and fully parametric PH models.

For each approach, appropriate methods were applied to account for delayed entry due to left-truncation. In addition, analyses based on standard right-censored assumptions were also considered to enable comparison with models that properly incorporate left-truncation.

All models were fitted using identical covariates to ensure consistent comparison of effect estimates and model fit across different methodological approaches.

Kaplan–Meier estimator

The Kaplan–Meier (KM) estimator is a nonparametric method used to estimate the survival function . In the presence of both right-censoring and left-truncation (delayed entry), the standard KM estimator must be modified to account for the fact that individuals are not at risk from time , but only after their respective truncation times .

The modified risk set at time is defined as:

where is the entry time and is the time of event or censoring for individual .

Difference between ignoring and accounting for left-truncation:

• Ignoring left-truncation: If is ignored (i.e., assuming for all ), individuals are incorrectly assumed to be at risk during the interval . This results in an artificially larger risk set (), which typically leads to an overestimation of the survival probability and potentially masks early-term hazards.

• Accounting for left-truncation: By conditioning on , the estimator only includes individuals in the risk set who have actually entered the study and have not yet experienced the event or been censored. This provides an unbiased estimate of the conditional survival function .

Cox proportional hazards model

The Cox proportional hazards (PH) model [13] is one of the most widely used semiparametric regression approaches in survival analysis. It relates the hazard function for subject at time , denoted , to a covariate vector through a log-linear form:

where is the baseline hazard function and where is the vector of regression coefficients. Because the baseline hazard is left unspecified, the model is regarded as semiparametric.

Estimation is performed through the partial likelihood, which depends only on the ordering of event times rather than the full distribution of survival times.

In the context of the present clinical follow-up data, for each individual , let denote the true event time measured from the date of HIV diagnosis, denote the left-truncation time (90 days post-diagnosis), and denote the right-censoring time (death or administrative censoring at January 1, 2020). The observed data therefore satisfy , and represents the vector of baseline covariates (age at diagnosis, CD4 cell count, log-transformed HIV RNA level, and comorbidity status).

For left-truncated and right-censored (LTRC) data, the partial likelihood is appropriately modified to account for delayed entry. Specifically, the risk set at event time t includes only those individuals j satisfying:

meaning individuals must have already entered observation and not yet failed or been censored. This adjustment ensures unbiased and consistent estimator properties under left-truncation.

A key assumption of the Cox model is proportional hazards, which state that hazard ratios between individuals remain constant over time. This assumption was assessed via Schoenfeld residuals and log–log survival plots [18], and no violations were detected.

In this study, the Cox model estimated using the LTRC-adapted partial likelihood served as the primary semiparametric framework for identifying prognostic factors among HIV patients and provided a baseline for comparison against AFT and fully parametric PH models.

Accelerated Failure Time (AFT) models

The Accelerated failure time (AFT) model provides an important alternative to the Cox proportional hazards model, particularly when the proportional hazards (PH) assumption is violated. While there were originally popular in engineering and industrial reliability analysis, AFT models have become increasingly relevant in biomedical survival studies [19, 20].

Unlike the Cox model, which assumes a multiplicative effect of covariates on the hazard function, the AFT model assumes that covariates act multiplicatively on the survival time itself. The model can be expressed in a log-linear form:

In the context of the present HIV cohort, denotes the time from HIV diagnosis to death for individual . The observed data satisfy the left-truncated and right-censored condition where represents the prespecified truncation time of 90 days post-diagnosis and denotes the administrative censoring time (January 1, 2020) or loss-to-follow-up. The covariate vector includes baseline clinical characteristics: age at diagnosis, CD4 cell count, log-transformed HIV RNA level, and comorbidity status.

where denotes the survival time, represents the covariates, and represents the error term following a specific distribution (e.g., extreme value, normal, or logistic). The exponent of the regression coefficient, , is known as the acceleration factor, which represents how a covariate accelerates or decelerates the time to event. An acceleration factor greater than 1 implies longer survival, whereas a value less than 1 indicates shorter survival [21, 22].

AFT model parameters were estimated via maximum likelihood. For left-truncated and right-censored (LTRC) data, the likelihood was conditioned on survival beyond each individual’s truncation time to account for delayed entry.

LTRC-adapted AFT models were fitted under several distributions (Weibull, log-normal, log-logistic, Gompertz, and extreme value) and compared using information criteria (AIC, BIC, HQIC, CAICF) to evaluate relative model fit.

Parametric proportional hazards models

Parametric proportional hazards (PH) models assume that the baseline hazard function follows a specific distribution. When this assumption is approximately valid, parametric PH models typically provide more efficient and precise estimates than semiparametric methods such as the Cox model [23]. These models offer direct expressions for the hazard function, survival probability, and median survival time, often with reduced standard errors.

Let denote the density function and denote the survival function of the event time .

In the present HIV cohort, represents the time from HIV diagnosis to death or censoring for individual , observed under the left-truncated and right-censored condition , where denotes the 90-day post-diagnosis truncation time and denotes the administrative censoring time (January 1, 2020) or loss-to-follow-up. Baseline covariates include age at diagnosis, CD4 cell count, log-transformed HIV RNA level, and comorbidity status.

The hazard function is defined as:

Several commonly used parametric distributions were considered in this study:

• Exponential model: assumes a constant hazard , yielding .

• Weibull model: extends the exponential form with , allowing hazards that increase or decrease depending on the shape parameter .

• Gompertz model: defined by , representing exponentially increasing or decreasing hazards.

The Weibull and Gompertz models were considered due to their flexibility in modeling monotonic hazard patterns. For left-truncated and right-censored (LTRC) data, estimation was based on likelihood functions conditioned on survival beyond each individual’s truncation time.

Parametric PH models were fitted alongside Cox and AFT models using identical covariates and compared using information criteria (AIC, BIC, HQIC, and CAICF).

All statistical analyses were performed using R statistical software [24]. The main packages used were “survival”, “truncreg”, and “eha”.

Results

Descriptive statistics

The study utilized clinical follow-up data from HIV-positive male patients treated at the Department of Infectious Diseases, Ondokuz Mayıs University Medical Faculty Hospital (Samsun, Türkiye). Female patients were excluded due to their small representation in the dataset. The recorded variables included age (years), CD4 cell count (cells/mm³), viral load (HIV RNA, copies/mL), comorbidity status (e.g., tuberculosis, PCP), and diagnosis date.

The CD4 count reflects the number of immune cells targeted by HIV, with lower values indicating greater immunosuppression. Viral load (HIV RNA) quantifies the amount of circulating virus and serves as a key indicator of treatment response. In this study, a truncation time of 90 days post-diagnosis was defined to allow for initial stabilization of CD4 and viral load measurements. The observation period covered 2006–2020, with data right-censored on January 1, 2020 (pre-pandemic).

Descriptive statistics for continuous variables are presented in Table 1, and categorical distributions are presented in Table 2. The final cohort included in the analysis consisted of 69 patients; this sample size was obtained after excluding patients who died or were censored before the truncation time (90 days), satisfying the left-truncated right-censored (LTRC) condition . Among these patients, 45 (65%) were censored, and 24 (35%) experienced the event of interest (death).

Table 1.

Descriptive statistics of continuous variables (n = 69)

Variable	Min	Max	Mean	SD	Median
Age (years)	22	67	42.23	11.50	41
CD4 (cells/mm3)	2	1460	335.71	330.65	267
HIV RNA (copies/mL)	20	23,400,000	193,876.02	339,958.46	54,300

SD Standard deviation

Table 2.

Frequency distribution of categorical variables

Variable	Category	n (%)
Age (years)	< 30	11 (15.9)
	30–39	15 (21.7)
	40–49	23 (33.3)
	≥ 50	20 (29.0)
CD4 (cells/mm³)	15–199	31 (44.9)
	200–349	12 (17.4)
	350–499	13 (18.8)
	≥ 500	13 (18.8)
HIV RNA (copies/mL)	< 10,000	26 (37.7)
	10,000–99,999	9 (13.0)
	≥ 100,000	34 (49.3)

Percentages are calculated within each variable category

In subsequent analyses, this dataset was evaluated under both standard right-censored assumptions and left-truncated and right-censored (LTRC) conditions, reflecting different modeling approaches rather than different datasets.

The mean age was 42.23 years (SD = 11.50), the mean CD4 count was 335.71 cells/mm³ (SD = 330.65), and mean viral load was 193,876 copies/mL (SD = 339,958). Most patients (33.3%) were aged between 40 and 49 years. More than half (55%) had CD4 counts above 200 cells/mm³, and 50.7% had viral loads below 100,000 copies/mL. When the comorbidity status was examined, 20.3% (n = 14) of the patients had at least one comorbid disease such as tuberculosis or PCP.

Kaplan–Meier survival estimates under right-censoring and LTRC assumptions

In this section, the same dataset is analyzed under two different assumptions: (i) standard right-censoring, where delayed entry is ignored, and (ii) left-truncated and right-censored (LTRC) conditions, where delayed entry is explicitly accounted for. This comparison allows us to evaluate the impact of left-truncation on survival estimates.

The Kaplan–Meier (KM) estimator is widely applied in survival analysis; however, many studies fail to incorporate left-truncation, which can lead to biased and overly optimistic survival estimates. Table 3 summarizes the mean and median survival times obtained using (i) right-censored data only and (ii) left-truncated and right-censored (LTRC) data.

Table 3.

K–M survival estimates under different data assumptions

Survival time	Right-censored	Left-truncated right-censored
Mean (days)	2,602.47	2,902.57
Median (days)	3,885	2,626

The inclusion of left-truncation results in shorter estimated median survival, indicating that ignoring truncation may inflate survival estimates

As shown in Table 3, incorporating left-truncation yields a substantially shorter median survival time, indicating that analyses ignoring delayed entry may lead to overestimation of survival duration.

The apparent discrepancy between mean and median survival estimates reflects their different statistical sensitivities. The median survival time is determined by the point at which the survival probability reaches 0.5 and is therefore primarily influenced by the early portion of the survival curve and risk-set redefinition under left-truncation. In contrast, the mean survival time corresponds to the area under the Kaplan–Meier curve and is affected by the entire follow-up period, including the tail of the distribution. Consequently, modifications in the early risk set may reduce the median survival estimate while simultaneously increasing the estimated mean survival.

It should be noted that the time origin for all analyses was defined as the date of HIV diagnosis (t = 0). The 90-day truncation does not imply that survival times were shifted or that 90 days were added to the estimated mean or median survival. Instead, left-truncation affects only the definition of the risk set, such that individuals contribute to the analysis only after surviving beyond the truncation time.

Figure 3 presents the Kaplan–Meier curve based on the LTRC dataset. The blue markers indicate uncensored (event) observations, whereas the red markers represent censored cases.

Fig. 3.

Kaplan–Meier survival curve for left-truncated and right-censored HIV data

Cox model results under right-censoring and LTRC assumptions

In this section, the same dataset is analyzed under two different modeling assumptions: (i) standard right-censoring, where left-truncation is ignored, and (ii) left-truncated and right-censored (LTRC) conditions, where delayed entry is explicitly incorporated. This comparison allows us to assess how ignoring versus accounting for left-truncation affects hazard ratio estimates and statistical inference.

The Cox proportional hazards (PH) model was first applied using right-censored data only ignoring potential bias from left-truncation. This preliminary analysis serves as a benchmark to assess how delayed entry may distort hazard estimates.

Bias assessment via right-censored analysis.

Table 4 presents the estimated hazard ratios from the standard right-censored Cox PH model. CD4 count was evaluated as a continuous variable to ensure consistency with the LTRC-adjusted model and to facilitate direct comparison across model specifications.

Table 4.

Right-censored Cox proportional hazards model results

Variable		p-value
Comorbidity	1.077 (1.036–1.120)	0.876
CD4	0.999 (0.998–1.002)	0.024
Log(HIV RNA)	1.000 (0.999–1.001)	0.429
Age	1.046 (1.006–1.088)	0.023

HR Hazard Ratio, 95% CI 95% Confidence Interval

CD4 count was treated as a continuous variable in the right-censored analysis. Higher CD4 levels were associated with a lower risk of mortality; however, this effect should be interpreted cautiously, as the analysis does not account for delayed entry and may therefore be subject to bias.

LTRC-adjusted Cox model

The full left-truncated and right-censored (LTRC) dataset was subsequently analyzed using the partial likelihood function modified to incorporate delayed entry. Continuous covariates were used in this definitive analysis, and HIV RNA was transformed using log base 10 to improve model performance (Table 5).

Table 5.

Left-truncated right-censored Cox proportional hazards model results

Variable	HR (95% CI)	p-value
Comorbidity	0.632 (0.241–1.659)	0.352
CD4	0.999 (0.997–1.001)	0.927
Log(HIV RNA)	1.214 (1.055-1.400)	0.007**
Age	1.049 (1.011–1.089)	0.013*

HR Hazard Ratio, 95% CI 95% Confidence Interval

*p < 0.05, **p < 0.01

In the LTRC-adjusted model, age and log(HIV RNA) remained statistically significant predictors of mortality. Each additional year of age increased the risk of death by approximately 5% (HR = 1.049, 95% CI: 1.011–1.089), and each one-unit increase in log(HIV RNA) increased the hazard by 21% (HR = 1.214, 95% CI: 1.055–1.400).

Importantly, the CD4 count which appeared significant in the non-LTRC model became nonsignificant after properly accounting for delayed entry (HR = 0.999, 95% CI: 0.997–1.001; p = 0.927). This shift illustrates the bias introduced when left-truncation is ignored and highlights the necessity of LTRC methods for reliable inference.

Proportional hazards assumption

The proportional hazards assumption was assessed using Schoenfeld residuals. As shown in Table 6, all covariates had p-values greater than 0.05, and the global test yielded p = 0.243, indicating no violation of the assumption.

Table 6.

Proportional hazards assumption test (Schoenfeld residuals)

Variable		df	p-value
Comorbidity	3.053	1	0.081
CD4	3.261	1	0.071
Log(HIV RNA)	0.006	1	0.936
Age	0.013	1	0.909
Global	5.465	4	0.243

Figure 4 presents the scaled Schoenfeld residuals. The residuals fluctuate randomly around zero and remain within ± 2 standard error bounds, supporting the validity of the PH assumption for the LTRC Cox model. It should be noted that, under left-truncation, Schoenfeld residuals are computed conditional on the delayed-entry risk sets; therefore, the proportional hazards assessment reflects the hazard structure among individuals who have survived beyond the truncation time.

Fig. 4.

Scaled Schoenfeld residuals for covariates in the Cox PH model

Model comparison and selection

The semiparametric Cox proportional hazards (PH) model, accelerated failure time (AFT) model, and fully parametric PH model were compared to identify the most appropriate method for analyzing left-truncated and right-censored (LTRC) HIV survival data. All the models were fitted with the same covariates (age, log(HIV RNA), CD4, and comorbidity status) to ensure a fair comparison.

For the Cox model, the proportional hazards assumption was verified using Schoenfeld residuals and global chi-square tests (p > 0.05), confirming that the model satisfied proportionality requirements.

To evaluate the relative model performance, we calculate several information criteria (AIC, BIC, AICc, CAICF and HQIC). Lower values indicate a more parsimonious, better-fitting model. The results are summarized in Table 7.

Table 7.

Model comparison on the basis of information criteria

Model	Log-likelihood	AIC	BIC	AICc	CAICF	HQIC
Cox PH	−67.887	143.774	152.710	145.879	185.355	147.319
AFT (Weibull)	−212.953	437.905	451.309	442.846	482.034	443.223
AFT (Log-normal)	−213.041	438.082	451.487	443.023	481.285	443.400
AFT (Log-logistic)	−213.610	439.219	452.624	444.161	485.386	444.537
AFT (Gompertz)	−213.812	439.624	453.029	444.566	482.156	444.943
AFT (Extreme value)	−212.948	437.895	451.300	442.836	483.550	443.213
Parametric PH (Weibull)	−212.953	437.905	451.309	442.846	488.335	443.223
Parametric PH (Extreme value)	−213.167	438.334	451.739	443.275	488.462	443.652
Parametric PH (Gompertz)	−213.812	439.624	453.029	444.565	479.806	444.942

Interpretation and quantification of bias

Across all the models, the Cox proportional hazards model demonstrated the lowest information criterion values (AIC = 143.774; BIC = 152.710), indicating the best overall fit and parsimony for the LTRC HIV dataset. All the parametric PH and AFT models presented substantially higher AIC/BIC values—for example, the best-performing AFT model had an AIC of 437.895—suggesting inadequate representation of the underlying hazard structure. These findings identify the Cox PH model as the most robust and appropriate approach for this dataset.

A key methodological finding was the quantification of survival bias arising from ignoring left-truncation. When the Cox model was fit to right-censored data only, the estimated median survival was 3885 days. In contrast, the LTRC-adjusted Cox model produced a median survival of 2626 days, representing an artificial inflation of approximately 47%. This substantial discrepancy highlights the severe sampling bias introduced by delayed entry and underscores the necessity of employing LTRC-appropriate modeling techniques in studies of HIV survival.

Interpretation of significant covariates

In the LTRC-adjusted Cox proportional hazards model, age and log-transformed HIV RNA were identified as statistically significant predictors of mortality among HIV-positive patients (p < 0.05).

The estimated hazard ratio for age (HR = 1.05) indicates that each additional year of age increases the risk of death by approximately 5%. This finding suggests that older individuals may experience faster disease progression or diminished response to antiretroviral therapy (ART). This is consistent with previous studies reporting reduced immune recovery and increased comorbidity burden among older HIV-infected adults [25, 26].

Similarly, the hazard ratio associated with log(HIV RNA) (HR = 1.21) demonstrated that a higher viral load substantially elevates the risk of mortality. Increased HIV RNA levels reflect active viral replication and impaired immune function, which increase vulnerability to opportunistic infections. These findings reinforce the central role of achieving and maintaining viral suppression through ART to improve long-term survival outcomes.

In contrast, the CD4 cell count and comorbidity status were not statistically significant predictors in according to LTRC Cox model. This lack of significance may be attributed to the relatively small sample size or to the stabilizing effects of modern ART, which can reduce the variability of CD4 levels and mitigate the short-term impact of opportunistic infections.

Overall, the results highlight age and viral load as the two most robust prognostic factors in this cohort, which is consistent with the model selection results indicating that HIV RNA and age are the most stable predictors across all survival modeling approaches.

Discussion

This study evaluated left-truncated and right-censored survival data from 69 HIV-positive male patients followed at the Infectious Diseases Department of Ondokuz Mayıs University Medical Faculty Hospital. The primary aim was to compare the performance of Cox proportional hazards (PH), parametric PH, and accelerated failure time (AFT) models in estimating survival and identifying prognostic factors under delayed entry conditions a common feature in HIV cohorts due to late diagnosis, irregular follow-up patterns, and the time required for initial clinical stabilization.

Consistent with prior studies, younger individuals tend to present with higher baseline HIV RNA levels and delayed viral suppression [27]. In our LTRC-adjusted analyses, age emerged as a significant predictor, with each additional year increasing mortality risk by approximately 5%. This aligns with earlier studies reporting impaired immune recovery and a greater burden of comorbidities among older HIV-infected adults. Similarly, higher viral load measured using log-transformed HIV RNA—was strongly associated with increased mortality risk, reflecting its direct relationship with ongoing viral replication, immune decline, and vulnerability to opportunistic infections.

K–M analyses demonstrated the critical impact of accounting for left-truncation: median survival was overestimated (3,885 vs. 2,626 days) when truncation was ignored. Importantly, the naïve Kaplan–Meier estimator that ignores left-truncation is not targeting the same estimand as the LTRC-adjusted estimator. While the LTRC approach estimates survival conditional on individuals having survived beyond the truncation time, the naïve estimator implicitly assumes full risk-set observability from time zero, leading to a fundamentally different target quantity. This finding supports earlier research [5], which documented substantial upward bias in survival estimates when delayed entry was not modeled appropriately.

Given that the truncation specification plays a central role in defining the risk set, the choice of the truncation threshold warrants clarification. The 90-day left-truncation threshold was prespecified based on routine clinical follow-up intervals in HIV care, where CD4 and viral load assessments are typically conducted at three-month intervals during early treatment. This period is characterized by substantial early immunological and virological variability following diagnosis and treatment initiation. While alternative truncation points could be evaluated in sensitivity analyses, the modest sample size of the present cohort would substantially alter the effective risk set and event counts. Future studies with larger populations may benefit from systematically assessing robustness across different truncation thresholds.

Initial Cox regression using right-censored data suggested that CD4 categories were significant predictors. However, after left-truncation was incorporated, the CD4 count and comorbidity status were no longer significant, whereas age and log(HIV RNA) remained robust predictors. This shift illustrates the methodological importance of correcting the risk set definition; failing to account for delayed entry can lead to misleading clinical interpretations and biased covariate effects.

The relatively small sample size (69 patients, 24 events) limits statistical power, particularly for detecting moderate associations for CD4 count and comorbidity status. Therefore, the absence of statistical significance for certain covariates should not be interpreted as definitive evidence of no effect. Importantly, the methodological conclusions regarding the behavior of survival models under left-truncation are conceptually independent of cohort size, whereas the clinical predictor findings are specific to this study population and should be interpreted with caution.

AFT models were explored as alternatives, particularly for scenarios where the proportional hazards assumption might be violated. Although several AFT distributions (e.g., log-logistic, Gompertz) yielded significant covariates, none demonstrated superiority to the Cox model on the basis of information criteria. The extreme value distribution performed best among the AFT models, yet its goodness-of-fit remained considerably lower than the Cox PH model. Similarly, among parametric PH models, the Weibull model provided the best fit, but remained inferior overall.

Collectively, model comparison using AIC, AICc, BIC, CAICF, and HQIC confirmed that the semiparametric Cox model was the most parsimonious and best-fitting approach for this dataset. It should be noted that information criteria derived from partial likelihood (as in the Cox proportional hazards model) and full likelihood (as in fully parametric survival models) are not strictly equivalent from a theoretical standpoint. The Cox model leaves the baseline hazard unspecified and is estimated using partial likelihood rather than a complete parametric likelihood formulation [18]. In contrast, AFT and parametric PH models rely on fully specified likelihood functions. Because likelihood-based information criteria are formally grounded in full likelihood theory [28], the model comparison presented in this study should be interpreted pragmatically as an empirical assessment of relative fit within the observed dataset, rather than as a definitive inferential ranking across fundamentally different likelihood structures. These findings highlight the flexibility and robustness of the Cox model particularly when appropriately adapted to left-truncated and right-censored survival data structures.

Importantly, these methodological considerations have practical implications for public health. Accurate survival estimation is essential for monitoring progress toward the World Health Organization’s 2030 goals, which include eliminating new HIV infections and reducing HIV-related mortality. As Turkey is still in an early phase of its HIV epidemic, rigorous survival modeling using appropriately specified risk sets is vital for guiding clinical decision-making and policy development.

In addition, this study may be subject to length-biased sampling due to delayed entry. Although individuals with survival times shorter than 90 days were excluded to reduce early mortality bias, patients who survived longer were more likely to enter the cohort, while individuals with shorter survival times may have died before observation. Therefore, the observed survival estimates may be higher than the true survival experience in the target population.

Future research could benefit from expanding the sample size and incorporating additional covariates such as sex, substance use, ART adherence, and comorbidity burden. Such enhancements would strengthen predictive accuracy and improve generalizability across more diverse HIV populations.

Conclusion

This study demonstrated that properly accounting for left-truncation is essential for obtaining unbiased survival estimates in HIV cohorts with delayed entry. When left-truncation was incorporated, estimated survival times decreased substantially and the set of significant predictors changed, underscoring the methodological importance of accurate risk-set specification.

Among the evaluated models, the Cox proportional hazards model showed the most favorable information criterion values within this dataset relative to the AFT and parametric PH alternatives.

Age and log(HIV RNA) count emerged as the strongest prognostic factors, reinforcing the clinical importance of early diagnosis and effective viral suppression strategies. These findings support the continued need for targeted interventions aimed at reducing the viral load and improving long-term outcomes in HIV-infected individuals.

Importantly, this analysis reinforces that ignoring delayed entry can introduce significant upward bias in survival estimation and distort covariate effects. The present findings underscore the necessity of incorporating left-truncation mechanisms in future HIV survival analyses to avoid biased conclusions and misleading clinical recommendations.

Although the present study focuses on empirical evaluation within a real-world HIV cohort, future research may extend these findings through simulation-based investigations to systematically examine estimator performance and truncation-induced bias under varying data-generating scenarios.

Future studies with larger, more diverse cohorts—and with expanded clinical and behavioral variables—are recommended to further improve the prediction accuracy and support health policy planning in alignment with the global 2030 HIV targets.

Authors’ contributions

TÇ: Study design, data curation, statistical modelling, results interpretation, manuscript preparation. YT: Methodological oversight, statistical guidance, interpretation review, manuscript revision. Both authors approved the final manuscript.

Funding

No extramural funding.

Data availability

The datasets generated and/or analyzed during the current study were collected under institutional ethics committee approval and include confidential patient-level clinical data. In accordance with ethical regulations and institutional data protection policies, these data are not publicly available and cannot be shared.

Declarations

Ethics approval and consent to participate

This study was approved by the Ethics Committee of Ondokuz Mayıs University (Approval No: B30.2.ODM.0.20.08/523, dated 24 July 2020). As this was a retrospective study using anonymized clinical records, the requirement for informed consent was waived by the ethics committee.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.