2.1 Simulation results

The detailed description of the simulation study is shown in the Method section. The variable selection performances of different regularization methods, showing the proportion of the first 30 covariates selected for two GPS models in 100 replications, are visualized in Figs 1–2 and Figs A-B in S2 Appendix. The direct and indirect mediation effects estimated by the proposed method and other comparing methods based on 100 replications are presented in Tables 1–3 and Table 3 Tables A-C in S1 Appendix, comprehensively showing the estimation performance under different settings, with respect to the association strengths of covariate with outcome or treatment, the kernel procedures by standard or undersmoothing bandwidths, covariate correlation of ρ = 0 or 0.3, and sample sizes and covariate dimensions (n, p) = (2000, 100) or (5000, 200). The estimation performance under typical scenarios is further visualized to comprehensively illustrate the estimation accuracy and stability across the exposure spectrum by different methods (Figs 3–4).

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Table 1. Mediation effects estimated by weighting with a parametric generalized propensity score based on different covariate sets under Scenario 1 (SoSt) using an undersmoothing kernel bandwidth.

https://doi.org/10.1371/journal.pcbi.1014436.t001

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Table 2. Mediation effects estimated by weighting with a parametric generalized propensity score based on different covariate sets under Scenario 2 (SoWt) using an undersmoothing kernel bandwidth.

https://doi.org/10.1371/journal.pcbi.1014436.t002

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Table 3. Mediation effects estimated by weighting with a parametric generalized propensity score based on different covariate sets under Scenario 3 (WoSt) using an undersmoothing kernel bandwidth.

https://doi.org/10.1371/journal.pcbi.1014436.t003

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Fig 1. Proportion of the top 30 covariates selected using GOAL, AdaLASSO, and LASSO-based methods in the treatment generalized propensity score model given covariates under Scenarios 1–3 with (n, p) = (2000, 100).

Notes: The horizontal axis represents the index of the simulated covariates, and the vertical axis denotes the proportion of covariates repeatedly selected across 100 simulations. Abbreviations: LASSO, the least absolute shrinkage and selection operator; GOAL, generalized outcome-adaptive LASSO; AdaLASSO, adaptive LASSO.

https://doi.org/10.1371/journal.pcbi.1014436.g001

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Fig 2. Proportion of the top 30 covariates selected using GOAL, AdaLASSO, and LASSO-based methods in the treatment generalized propensity score model given both covariates and the mediator under Scenarios 1–3 with (n, p) = (2000, 100).

Notes: The horizontal axis represents the index of the simulated covariates, and the vertical axis denotes the proportion of covariates repeatedly selected across 100 simulations. Abbreviations: LASSO, the least absolute shrinkage and selection operator; GOAL, generalized outcome-adaptive LASSO; AdaLASSO, adaptive LASSO.

https://doi.org/10.1371/journal.pcbi.1014436.g002

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Fig 3. Comparison of absolute bias in causal mediation effect estimation across different scenarios with a sample size of 2000.

Note: The figure illustrates the distribution of absolute bias (calculated by the absolute value of the difference between the estimates and the true values across the exposure range) for the proposed GOAL-based method and competing approaches. The dotted line indicates the state with zero absolute bias. The results are based on the setting of sample size n = 2000 and covariate correlation (ρ) of 0. The NDE1 and NDE0 separately represent natural direct effects (NDE) under treatment and non-treatment, and the NIE1 and NIE0 separately represent natural indirect effects (NIE) under treatment and non-treatment. Abbreviation: LASSO, the least absolute shrinkage and selection operator; GOAL, generalized outcome-adaptive LASSO; AdaLASSO, adaptive LASSO; DoubleML, double machine learning.

https://doi.org/10.1371/journal.pcbi.1014436.g003

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Fig 4. Comparison of absolute bias in causal mediation effect estimation across different scenarios with a sample size of 5000.

Note: The figure illustrates the distribution of absolute bias (calculated by the absolute value of the difference between the estimates and the true values across the exposure range) for the proposed GOAL-based method and competing approaches. The dotted line indicates the state with zero absolute bias. The results are based on the setting of sample size n = 5000 and covariate correlation (ρ) of 0. The NDE1 and NDE0 separately represent natural direct effects (NDE) under treatment and non-treatment, and the NIE1 and NIE0 separately represent natural indirect effects (NIE) under treatment and non-treatment. Abbreviation: LASSO, the least absolute shrinkage and selection operator; GOAL, generalized outcome-adaptive LASSO; AdaLASSO, adaptive LASSO; DoubleML, double machine learning.

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As shown in Figs 1–2 and Figs A-B in S2 Appendix, the GOAL method excels in selecting the prognostic variables (X₁₁ and X₁₆) with a proportion approaching 100% in most cases, while the AdaLASSO and LASSO methods, in contrast, have a low proportion. The GOAL method also outperforms other methods in selecting the true confounders (X₁ and X₆) in most settings, especially when the confounders are more strongly associated with the outcome than the treatment. Moreover, AdaLASSO and LASSO methods always choose instrumental variables X₂₁ and X₂₆ in the GPS models of treatment and have a higher possibility of including spurious variables, while the GOAL method instead presents a lower percentage of including instrumental variables and spurious variables, especially when there’s no correlation between covariates ( = 0).

In the effect estimation stage, the bandwidth diagnostic results for different candidate bandwidths suggest that the h_ROT bandwidth fails to achieve localized smoothing due to its excessively large effective window half-width, which approaches the entire exposure span, consequently precluding its utilization in the simulations. Meanwhile, both h_wp and h_{wp us} maintain sufficient local effective information and favorable covariate balance and weight stability, characterized by non-extreme effective sample size (ESS), maximum standardized mean difference (SMD) all below 0.25, and maximum weight coefficient of variation (CV) all lower than 1 (Table D in S1 Appendix). Therefore, both bandwidths are adopted in the simulations, with trade-offs made based on the estimation results.

The results in Tables 1–3 and Tables A-C in S1 Appendix separately report the average of the relative absolute bias (Bias%), standard deviation (SD), root mean squared error (RMSE), and coverage probability (CP) for each causal mediation effect over the range of a ∈ {-1.0, -0.9,  ..., -0.1, 0.1,  ..., 0.9, 1.0}. The Stability check of variance estimates based on bootstrap resampling indicates that the estimates remain consistent across different levels of replication, supporting the robustness and reliability of our bootstrap-based inference (Table E in S1 Appendix). Due to no interaction effect considered between treatment and mediator, the estimation of causal mediation effects shows an overall consistency under treatment and non-treatment for either NDE or NIE. Notably, we observe slight differences between the estimation bias of NIE under treatment and non-treatment. These discrepancies may be related to random fluctuations arising from different estimation mechanisms and algorithms in NIE estimators under treatment and non-treatment, which have also been reported in previous related literature [5,9]. Moreover, the results of the robustness test on non-linearity and interaction show that the GOAL-based estimator yields similar bias and SD across all the scenarios under data generated by interactive and nonlinear outcome models compared to the original simplified model (Table F in S1 Appendix). This suggests that the proposed method may be relatively robust to moderate violations of the linearity and no-interaction assumptions.

It is notable that estimation based on undersmoothing kernel bandwidth generally maintains a CP of above 90% and closely distributed around the 95% level, while the standard bandwidth cases suffer from unstable coverage of true values due to relatively large bias and narrow confidence intervals. Its superiority is particularly highlighted in covariate correlation settings (ρ = 0.3), and remains consistent across different sample sizes and multiple scenarios compared to the standard bandwidth case. The advantage of undersmoothing bandwidth may stem from its strictly faster rate of bias vanishing, which effectively prioritizes bias reduction over variance minimization [31], making this bandwidth option a more preferable strategy to derive more stable and valid inferences. We further examine the comprehensive results of CP performance across the entire exposure spectrum. The results reveal that the minimum coverage probability of the h_{wp us} bandwidth safely avoids severe under-coverage, and its CP distributions across the exposure range maintain relatively stable around the average with low variation (Table G in S1 Appendix). These findings demonstrate the stability and consistency of our inference framework under appropriate bandwidth specification across the exposure evaluation range, and also support the reliability of the mean-based CP metrics.

From Tables 1–3 and Tables A-C in S1 Appendix, we can observe that both NDE and NIE estimates of the GOAL-based estimators outperform other regularization method-based methods under most scenarios, yielding the smaller relative absolute bias, SD, and RMSE. This result is in line with common perspectives in causal inference literature that incorporating outcome information by including prognostic variables in the propensity score model can improve the efficiency of the estimation [18], while additionally adding non-outcome related variables may increase SD and RMSE [32]. It also aligns with the results of the referenced outcome benchmark GPS models that only include confounders and prognostic variables. In NIE estimation, although the SD and RMSE of our proposed GOAL-based method are generally smaller or close to those of other competing methods, its relative absolute bias remains consistently lower across different scenarios. By contrast, other regularization-based methods experience significantly inflated estimation bias and lower CP of true values with correlated covariates (ρ = 0.3) or with a smaller sample size. Similar results can also be observed in the benchmark GPS models that include instrumental variables and exclude outcome-related variables. Notably, the double machine learning method generally exhibits superior performance characterized by higher efficiency and robust estimation in NDE estimation compared to our proposed method. Nevertheless, its reduction in variance is offset by pronounced inflation in estimation bias for NIE estimation, leading to suboptimal coverage of the true values and a substantial decline in inferential validity. Whereas, our method remains consistent in such cases, demonstrating an overall comparative advantage in estimating both NDE and NIE effects, achieving robust estimation without substantially compromising variance. Furthermore, as the sample size increases, our method closely resembles the machine learning approach and similarly demonstrates comparable advantages. Therefore, when compared to the double machine learning method, our method offers a relatively more stable efficiency-bias balance, while also enabling informative covariate identification and noise filtering, thereby enhancing model interpretability and facilitating deeper insights into the underlying mechanisms.

The estimates generally present significantly lower relative bias, SD, and RMSE in the condition of independent covariates than in covariate-correlated settings in most settings for all the regularization-based methods. Both the bias and variance are observed to be significantly reduced when the sample size increases from 2,000–5,000, especially for the NDE estimates. Moreover, as the correlation strength between covariates and the outcome decreases and the correlation with the treatment enhances, i.e., from Scenario 1–3, the proposed method exhibits a relatively increased bias. However, the bias remains smaller than that of competing methods, whilst maintaining high estimation efficiency and valid inference.

The results of the robustness test regarding the method sparsity demonstrate that our proposed method maintains high selection accuracy approaching 100% and yields stable estimations as the proportion of true signals increases (Table H in S1 Appendix and Fig C in S2 Appendix). It justifies the consistency and reliability of our framework in scenarios characterized by weaker sparsity or denser signal structures. Furthermore, the sensitivity analysis exploring a continuous spectrum of covariate-to-sample size (p/n) ratios shows that the proposed method maintains overall robust performance across low-to-moderate ratios (≤ 0.6). As the ratio escalates to 0.9, a noticeable decline in estimation efficiency emerges, primarily characterized by moderately fluctuating increases in SD and RMSE (Table I in S1 Appendix). This precision loss may stem from the inclusion of spurious noise variables induced by selection degradation in heavily congested covariate space. Nevertheless, by virtue of the sparse true model structure and the identifiability of the initial ordinary least squares regression in adaptive penalty weights construction, no severe breakdown is observed within the p < n regime. This result suggests that the proposed approach is capable of accommodating moderately large covariate spaces, while also reflecting the potential impact of higher dimensionality on estimation accuracy. In addition, our method demonstrates consistent estimation performance both in scenarios with normally distributed errors and with binary outcomes, generally exhibiting low estimation bias and valid statistical inference (Tables J-K in S1 Appendix), which highlights the robustness of our approach to distributional assumptions and different outcome settings. Despite the misspecified GPS models of treatment A given X and given (X, M), the GOAL-based method, in most cases, manifests a better property in balancing bias and efficiency compared to other methods. Its superiority appears more distinct in undersmoothing bandwidth-based estimation, which implies that misspecification of the GPS models may not entail important biases under a sufficiently small kernel bandwidth, as interpreted in the prior work of Huber et al. [5].

To sum up, the simulation results illustrate the superiority of the proposed GOAL-based method in accurately identifying outcome-related covariates and excluding instrumental variables and spurious variables, and in obtaining accurate and valid estimation while improving estimation efficiency. The results imply that a more promising performance of the estimation can be obtained under an appropriate bandwidth strategy based on sample sizes with several thousand observations, which is easily met in practical biomedical studies.

2.2 Real data application results

The detailed information of the real-world analysis that explores the mediating role of apolipoprotein (ApoB) levels in the association between Finnish Diabetes Risk Score (FINDRISC)-represented potential diabetes risk and overall cancer incidence is presented in the Method Section and Tables L-O in S1 Appendix.

Our proposed GOAL-based method selects 45 key covariates in both the treatment GPS model given covariates, and the model given the combination of covariates and the mediator. In comparison, the AdaLASSO and LASSO-based methods select 52 and 92 variables, respectively, across both of the aforementioned GPS models. The factors identified by the GOAL method cover a wide range of biologically plausible risk factors, including the demographic and socioeconomic characteristics (e.g., sex, education attainment, income level, etc.), lifestyle, dietary and behavior patterns (e.g., sedentary behavior, tea and coffee intake, etc.), genetic risk factors (standard polygenic risk score for breast cancer, cardiovascular disease, and type 1 diabetes, etc.), mental health problems and stress-related events (physical and psychological abuse in the childhood, etc.), and biochemical and metabolic measures (systolic blood pressure, alanine aminotransferase, triglycerides, etc.). Notably, many of the identified covariates are in concordance with findings from prior research [33–44]. For instance, previous research indicates that certain genes associated with breast cancer risk also participate in diabetes-related symptoms, lipid metabolism-related molecules, and the occurrence of multiple other cancers [45–47]. It provides consistent evidence for our selected variable of genetic risk of breast cancer in the diabetes-cancer pathway. Also, the number of treatments/medications taken is supported by established evidence, especially implying that polypharmacy can both have an impact on diabetes risk and cancer incidence [48,49]. Moreover, some identified urinary-related indicators (e.g., sodium, potassium, and creatinine in urine) represent key physiological states such as blood pressure related to dietary habits, muscle mass, and renal function, and their confounding effects have also been consistently reported in prior findings [50–55]. This consistent evidence confirms the biological plausibility of our variable selection results.

The bandwidth diagnostic checks reveal that the effective window half-widths of the h_wp and h_{wp us} bandwidths are smaller than the minimum exposure spacing of 1, failing to secure adequate local empirical support for valid estimation (Table P in S1 Appendix). On the other hand, the 1.5h_ROT and 2h_ROT specifications are similarly excluded, as their excessively large effective half-widths approach or even surpass the full exposure spectrum, fundamentally defeating the purpose of localized smoothing. Consequently, diagnostic results suggest that the ideal bandwidth approaches or falls within the range between 0.75h_ROT and 1.25h_ROT. Supplementary evaluations across this spectrum further demonstrate generally consistent estimation performance across the exposure levels (Figs D-E in S2 Appendix). Therefore, we adopt the h_ROT bandwidth as a representative specification to obtain robust estimation. As shown in Table 4, our proposed GOAL method achieves relatively high estimation efficiency with lower SD for both direct and indirect effects estimation compared to other regularization-based methods. Also, compared to traditional models including all available covariates, which are more likely to introduce noise from irrelevant factors, our method selection-integrated approach enhances the reliability and precision of clinical inferences by focusing on the most relevant biological predictors. The variability plots (Figs F-G in S2 Appendix) illustrate that the GOAL method applied to our real data application has a solid capacity to identify important and high-frequency variables with a selection proportion ≥ 60% in 100 bootstrap resampling. This result demonstrates the stability and reliability of the variable selection results of the GOAL method in the current empirical analysis.

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Table 4. The average standard deviations of the estimation based on different variable selection procedures in the real data analysis.

https://doi.org/10.1371/journal.pcbi.1014436.t004

Figs 5 and 6 illustrate the dose-response causal mediation effects of varying levels of potential diabetes risk (from risk score of 3–13) on the overall cancer incidence through the ApoB pathway under different identified factors in both treatment and nontreatment groups. From Fig 5, we can find that the NDE estimates by the GOAL method-based approach show an overall upward trend as the treatment value increases, revealing that the direct effect of diabetes risk on cancer incidence markedly intensifies as the former risk level rises. Also, it exhibits a significant sign in the early stages of the diabetes risk window. This result aligns with previous epidemiological findings that patients with moderate- and high-diabetes risk might suffer an increased risk of overall cancer incidence compared to those with low risk or healthy populations [56,57]. Furthermore, the superiority of the GOAL-based method gets more pronounced at higher diabetes risk levels, where the sample size is smaller, demonstrating its inferential stability and reliability across the entire exposure spectrum.

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Fig 5. Direct effects and estimation using the GOAL-based method and the competing methods.

Notes: The treatment (i.e., FINDRISC) value a ∈ {3, 4,  ..., 13}. The horizontal axis represents the continuous treatment value in increments of 2, and the vertical axis denotes the magnitude of the natural direct effect (NDE) under treatment or non-treatment. The solid black line with hollow dots depicts the point estimates of the NDE at varying treatment levels, and the grey dashed lines surrounding the solid line represent the 95% pointwise confidence intervals derived from 500 bootstrap resampling. The horizontal dashed line at zero represents no causal effect observed on the risk difference scale through the direct pathway between the treatment and the outcome. Estimated values above zero suggest a risk effect, while values below zero indicate a protective effect. Abbreviations: LASSO, the least absolute shrinkage and selection operator; GOAL, generalized outcome-adaptive LASSO; AdaLASSO, adaptive LASSO; FINDRISC, Finnish Diabetes Risk Score.

https://doi.org/10.1371/journal.pcbi.1014436.g005

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Fig 6. Indirect effects and estimation using the GOAL-based method and the competing methods.

Notes: The treatment (i.e., FINDRISC) value a ∈ {3, 4,  ..., 13}. The horizontal axis represents the continuous treatment value in increments of 2, and the vertical axis denotes the magnitude of the natural indirect effect (NIE) under treatment or non-treatment. The solid black line with hollow dots depicts the point estimates of the NIE at varying treatment levels, and the grey dashed lines surrounding the solid line represent the 95% pointwise confidence intervals derived from 500 bootstrap resampling. The horizontal dashed line at zero represents no causal effect observed on the risk difference scale through the indirect pathway between the treatment and the outcome. Estimated values above zero suggest a risk effect, while values below zero indicate a protective effect. Abbreviations: LASSO, the least absolute shrinkage and selection operator; GOAL, generalized outcome-adaptive LASSO; AdaLASSO, adaptive LASSO; FINDRISC, Finnish Diabetes Risk Score.

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Fig 6 presents the indirect effect mediated by the level of ApoB. Similar to the estimation of NDE, our method provides more precise effect estimates than alternative models, especially at extreme risk values. The findings indicate that the indirect impact of ApoB-related biological pathways in the diabetes-cancer link intensifies as diabetes risk rises. This supports the biological plausibility of ApoB as a key mediator, consistent with the established evidence that dysregulation of lipid metabolism involving ApoB may partly link potential diabetes risks to oncogenic processes [58–60]. It also provides a novel perspective on clinical management that targeting ApoB-related lipid metabolism pathways could serve as an effective preventive strategy to prevent cancer development in high-risk diabetic populations.

In the sensitivity analysis to examine the stability of the model through 100 bootstrap resampling, we find that the average outcome incidence is 14.9% (14.0%, 15.8%) and the average SDs for the GOAL-based estimates of causal mediation effects are also similar to those of the original data (Table Q in S1 Appendix). These results indicate a stable outcome distribution under bootstrapping and support the estimation performance consistency and model stability of our method. Additionally, the estimation results of the sensitivity analysis adopting different bandwidths show that the proposed GOAL-based method consistently produces the smallest average SD among other alternative methods, demonstrating the highest estimation efficiency (Table R in S1 Appendix). Furthermore, the estimated direct and indirect effects remained stable across these bandwidth settings, with no substantial changes observed. (Figs H-I in S2 Appendix) These results suggest that our kernel-based estimation procedure is relatively robust to moderate bandwidth variation.

In summary, the mediation analysis results indicate that our proposed GOAL-based method is able to identify risk factors with evidence-based biological plausibility from large-dimensional covariates with higher efficiency of estimation. Moreover, our investigation provides a novel perspective that the elevated underlying diabetes risk may not only have a direct effect on the increase of the risk of overall cancer incidence, but also exerts a significant indirect effect through ApoB-related metabolic pathways. Clinical strategies and health management based on ApoB-related metabolic pathways may be considered as indirect intervention targets between potential diabetes risk and cancer occurrence in the future. Further exploration into more potential mediating pathways and more complicated modeling forms can be undertaken in the future to provide a comprehensive understanding of the mechanisms involved.

4.1 Ethics statement

The real-world data used in empirical analysis were obtained from the UK Biobank, which has received ethical approval from the North West Multi-Centre Research Ethics Committee to function as a Research Tissue Bank (RTB). During the recruitment stage, written informed consent has been obtained from all participants. The RTB approval allows researchers to operate under this approval with no need for further ethical approval, other than exceptions such as re-contact applications. The approval was granted initially in 2011 and is renewed on a five-yearly cycle, with the most recent renewal approved in 2021 (https://www.ukbiobank.ac.uk/wp-content/uploads/2025/01/Ethics-approval-renewal-2021.pdf).

4.2 Notation and reviews of the semiparametric estimation approach

Our goal is to disentangle the average treatment effect (ATE) of a continuous treatment variable A on an outcome variable Y into a direct effect and an indirect effect through a mediator M, which can be either discrete or continuous (hereafter, we use the continuous mediator to elaborate the methodology). Suppose the observed data (A_i, M_i, Y_i, X_i), i = 1, 2,  ..., n, are identically distributed and independent samples drawn from a common joint distribution f(A, M, Y, X), where represent the p-dimensional pre-treatment covariates. For generic random variables A and M, and denote the continuous support of A and M, respectively. For the definition of causal mediation effects, we apply the potential outcome framework [73] in the context of mediation analysis as described in previous literature [74–76]. Under treatment values , M(a) is defined as a function of the treatment, denoting the potential mediator state that would have been observed when the treatment level is set to be a; and Y(a, M(a’)) is defined as a function of the treatment and the potential mediator, denoting the potential outcome that would have been observed with the treatment set to a and the mediator set to the value it would attain under treatment a’. The mean potential outcomes are then defined as and with a ≠ a’. has also been stated as the average dose-response function for a continuous treatment in the literature [13,14].

Under this notation, the ATE of treatment value a versus a’, denoted by , can be written as

(1)

The ATE can be interpreted as the total average effects of A on Y from both the direct pathway and the indirect pathway via potential mediators M(a) and M(a’). The average natural direct effect (NDE) is defined as the average outcome difference between a and a’ when keeping the potential mediator fixed at the level of M(a) or M(a’):

(2)

Analogously, the average natural indirect effect (NIE) is the average outcome difference between M(a) and M(a’) when keeping the treatment fixed at a or a’:

(3)

By simply performing addition or subtraction between Equations 2 and 3, it can be readily drawn that ATE equals the sum of NDE and NIE, i.e., . In fact, and can be respectively different from and if the NDE and NIE are heterogeneous in A and M, which indicates the presence of interactions between A and M. The identification of the direct and indirect mediation effects is based on the assumptions of consistency, positivity, and conditional independence (Assumptions 1–3, Text A in S3 Appendix).

In this paper, we will follow the GPS-weighted semiparametric estimation approach proposed by Huber et al. to estimate the causal mediation effects for continuous treatment and mediator [5]. This method is built upon the weights constructed by the inverse of two versions of the GPSs, separately corresponding to the conditional treatment densities given observed covariates and given covariates and the mediator. Due to the probability of a specific value a being equal to zero for the continuous treatment A, we use kernel density instead of indicator function 1(A = a) (adopted in the binary treatment context) [4] to modify the weighting function of the estimator, which is defined as , where K is a symmetric second-order kernel function and h is a bandwidth. The weighting function depends on the distance between A and the reference value a and a nonnegative tuning parameter h, where larger discrepancies between A and a gain less weight as h approaches zero.

Let and be the GPS models separately corresponding to and . Under the assumption that and are continuous in a, the inverse-GPS weighted mean potential outcomes and with a ≠ a’ can be written as

(5)

(6)

as h goes to zero.

For an s-dimensional vector u = u (u₍₁₎,  ..., u_(s))’, let be a product kernel with a generic kernel function and bandwidth h for estimating the mean potential outcomes. To parametrically specify the GPS models and in the first step of the weighted semiparametric estimation of and , we also invoked Assumption 4 (Text A in S3 Appendix) for parametric generalized propensity scores.

The sufficient conditions for Assumption 4 have been illustrated by Huber et al.[5] The estimator of the GPS model , , satisfies , where such that for all . Similarly, the estimator of the GPS model , , satisfies , where such that for all and . The estimator and are the root-n consistent estimators for and (typically based on maximum likelihood), respectively. Let , , be the marginal density function and be the joint density function. Then, and can be consistently estimated by and , respectively. Then, semiparametric estimators for and can be written as

(7)

Assumption 5 invokes several regularity conditions required for the consistency and asymptotic normality of the proposed estimator (Text A in S3 Appendix).

We employ the bootstrap method to non-parametrically estimate variance [77], which is achieved by replacing the random sample with the bootstrap sample and substituting the population distribution with the empirical distribution [5]. The detailed explanation is provided in the next subsection ‘Generalized outcome-adaptive LASSO for mediation analyses’.

The algorithmic overview of the semi-parametric GPS-weighted mediation effect estimation is presented as Algorithm 1:

Algorithm 1 Semi-parametric GPS-weighted mediation effect estimation

1. Input: The outcome value y, treatment value a, mediator value m, and the specified covariates x_ax and x_mx for two GPS models;

2. Fit assumed working models of treatment GPS , where and , where , and obtained the treatment density and ;

3. For each individual i, compute kernel weights using an Epanechnikov kernel centered at a or a’ with bandwidth h;

4. Compute the kernel-based inverse-GPS weighted and (Equation 7);

5. Calculate the causal mediation effects ;

6. Use the bootstrap method to obtain the estimated standard error (SE) and p-value;

7. Output: estimates, SE and p-value.

4.3 Generalized outcome-adaptive LASSO for mediation analyses

To precisely specify the covariate sets for the GPS models and of the weighted semiparametric estimation for continuous treatment, we utilized the GOAL method [22] to select important variables to eliminate the confounding effect and improve estimation efficiency, which shrinks the coefficients of the outcome-unrelated covariates to zero.

Let X_C denote confounders that have an impact on treatment, outcome, and mediator. Let X_P denote prognostic covariates that only predict the outcome and are unrelated to the mediator and treatment. Let X_I denote instrumental covariates that only predict the treatment but have no association with the mediator and outcome. Let X_S denote spurious covariates that are unrelated to treatment, outcome, or mediator. To avoid the confounding bias and increase the statistical efficiency, we shall include all X_C and X_P and exclude X_I and X_S in the GPS models.

The corresponding GOAL estimates and are defined as

(8)

and

(9)

where λ_x and λ_mx (uniformly denoted by λ) separately represent the nonnegative tuning parameters in the GOAL estimates for two GPS models, with the power parameter τ > 1, is the unpenalized coefficient of the jth covariate in the ‘full’ generalized linear outcome regression model and can be estimated by ordinary least squares: . represents the computation of the squared sum. Under the conditions that / and for τ > 1, GOAL imposes heavier weights on covariates that are less associated with the outcome given treatment and mediator (i.e., X_I and X_S), such that these covariates tend to be excluded in the GPS models [21]. Notably, for binary outcomes, the unpenalized coefficients can be obtained via maximum likelihood estimation under a logistic regression model. The corresponding algorithmic details are provided in Text B in S3 Appendix.

Given the complexity of the final multi-stage effect estimator involving large-scale variable selection, statistical inference based on asymptotic linear properties (e.g., influence function-based inference) may become invalid [66,67,78]. We therefore adopt the nonparametric bootstrap method to approximate the sampling distribution and estimate variance.

We summarize the above estimation procedure in Algorithm 2:

Algorithm 2 Generalized outcome-adaptive LASSO estimation

1. Input: The sample size n, the target data with (Y, A, M, X), and the pre-specified set of tuning parameter λ;

2. Fit the full generalized linear outcome model g(Y)~A + M + X, and obtain the coefficients β of the covariates X;

3. Choose the optimized λ_x and λ_mx that minimize the deviance with 10-fold cross-validation for two GPS models;

4. Obtain the LASSO estimates and (Equations 8 and 9) using coefficients β-weighted penalty under optimized λ_x and λ_mx.

5. Select covariates with non-zero coefficients in two GPS models for sequential estimation.

6. Output: optimized λ_x and λ_mx and selected covariates for two GPS models.

4.4 Simulation setup

We conduct a simulation study to estimate the finite sample properties of the proposed method. To ensure and hold in Equations 2 and 3, we assume that there is no interaction between the treatment and mediator and that the associations between outcome and treatment, mediator, or covariates are linear. Referring to Huber et al.[5], we adopt the Gaussian assumption in the algorithm as a quasi-likelihood framework to parametrically estimate two treatment GPS models conditioned on covariates and/or the mediator. To define the NDE and NIE, we set a’ = 0 and let a take a series of grid points between (and including) -1 and 1 with a step size of 0.1 but excluding 0, defined by , where is a set of the grid points. Considering the optimal efficiency in symmetric second-order kernels [79,80] and the requirements for valid semiparametric inference, we utilize second-order Epanechnikov kernels and adopt an undersmoothing bandwidth strategy that can efficiently suppress asymptotic bias and ensure the root-n consistency and inferential validity of our estimator. Specifically, we compare the estimation under two bandwidth h settings: a small bandwidth (h_wp) C·n^-0.25 with C = 2.34, and a further undersmoothing version (h_{wp us}) set at half of h_wp bandwidth, following the empirical recommendations of Huber et al [5]. In addition, we supplement relevant theories regarding bandwidth specifications in Text C in S3 Appendix, and conduct a series of diagnostics and examinations on three candidate bandwidths: h_wp, h_{wp us}, and the rule-of-thumb (ROT) bandwidth h_ROT that is set to C·sd(A)·n^-1/5.

We repeat 100 times of simulations for each setting and consider two combinations (n, p) of sample size (n) and covariate dimension (p): (2000, 100) and (5000, 200). Within each simulation run, the standard error of the proposed estimator is estimated using a non-parametric bootstrap procedure based on 500 replicates. Average relative absolute bias (Bias%), standard error (SD), and root mean squared error (RMSE) are measured to evaluate the performance of the estimators across all the treatment values . Specifically, the average relative absolute bias was calculated as , and the average RMSE = , where denotes the number of elements in the set , denotes the estimate of the function for of the ith simulation, is the true effect at level , and denotes the number of simulation iterations. Also, as the bootstrap approach does not account for the impact of systematic estimation bias, we therefore evaluate the coverage probability of the bootstrap confidence intervals to assess the inference validity. To further validate the stability of the bootstrap estimates of variance in our simulations, we additionally conduct a sensitivity analysis with a representative simulated dataset using the bootstrap replicates of 200 and 1000 for comparison.

Of all covariates, X₁ and X₆ are true confounders (X_C), X₁₁ and X₁₆ are prognostic variables (X_P), X₂₁ and X₂₆ are instrumental variables (X_I), and all the others are deemed as spurious covariates (X_S). We denote the covariance of covariates as and and generate pre-treatment covariates X from a uniform distribution (-1, 1) with two covariate correlation patterns, where ρ = 0 indicates an independent covariate pattern, and ρ = 0.3 indicates all the covariates are correlated with a coefficient of 0.3. Treatment A is a linear function of the observed variables X and an unobserved W, mediator M is a linear function of the A, X and an unobserved V, and outcome Y is a linear function of A, M, X, and an unobserved U. The unobserved variables U, W, V are independent of each other and all follow uniform distributions (-2, 2). Here, we focus our simulation on continuous-outcome settings, which allows for leveraging richer information compared to binary cases and providing a more rigorous assessment of method performance. However, it is worth noting that the variable selection mechanism of our proposed method consistently operates under the general generalized linear model framework (including both linear and logistic models) [20].

Then the data-generating processes for A, M, and Y are as follows:

In our simulation design, the direct effects are defined as , and the indirect effects are defined as .

Three scenarios are then framed to assess the estimation performance of different strengths of confounder-outcome and confounder-treatment association: 1) confounders are strongly associated with both outcome and treatment (SoSt); 2) confounders are more strongly associated with outcome than with treatment (SoWt); 3) confounders are more strongly associated with treatment than with outcome (WoSt).

1. Scenario 1 (SoSt)

In the case that X_C (X₁ and X₆) are strongly associated with both and , the coefficients of in the aforementioned function of A and Y, that is, γ and β, are as follows:

1. Scenario 2 (SoWt)

In Scenario 2, X_C are more strongly associated with Y than with A. The coefficients γ and β are

1. Scenario 3 (WoSt)

The consistency of variable selection can be influenced by two nuisance parameters λ and τ. In each scenario, we consider λ from a set of eight values, the same as those used by Shortreed et al.[21]: {} and determine the optimal λ by minimizing deviance with 10-fold cross-validation. Also, following the recommendation by Gao et al.[22], we set , where the power parameter of n (i.e., the value of 2) is called the convergence factor.

To examine the sensitivity of the proposed method to the distributional assumption of the error terms, we introduce an additional simulation scenario in which the error terms for the exposure, mediator, and outcome models were generated from normal distributions (Text D in S3 Appendix). In addition, we perform a robustness check to examine the estimation performance of the proposed GOAL-based method under interactive and nonlinear outcome model settings. In this sensitivity analysis, we separately assess the estimation performance of our method under an outcome generation model incorporated with a treatment–mediator interaction term and under a nonlinear outcome model with the interaction term and the quadratic terms in both the treatment and the mediator (Text E in S3 Appendix). Moreover, to validate the consistency of our method in different outcome settings, we repeat our simulations under a binary outcome scenario (Text B in S3 Appendix). Also, to evaluate the robustness of our proposed GOAL-based method regarding the sparsity structure, we conduct a sensitivity analysis by incrementally increasing the number of true active signals in the data generation process (Text F in S3 Appendix). We additionally evaluate the robustness of our method using increased covariate-to-sample size ratios (p/n) from 0.1 to 0.9 to investigate its performance under varying dimensionality levels (Text G in S3 Appendix).

Further, we compare the performance of the proposed GOAL-integrated estimator with the full-covariate GPS-weighted semiparametric estimator and other regularization method-based methods, including AdaLASSO and LASSO. For AdaLASSO, we fit two linear regression models corresponding to two GPS models in our semiparametric estimators: one model regresses A on X, and the other model regresses A on M and X. The regression coefficients are then used to compute the corresponding penalty weights for two GPS models. The estimates of AdaLASSO and LASSO are presented in Text H in S3 Appendix. The tuning parameter λ is also chosen by minimizing deviance with 10-fold cross-validation. Additionally, to compare the performance of our method with other non-regularization approaches, such as modern machine learning methods, we follow the work of Yang et al. [26] to implement a double machine learning framework-based mediation analysis under high-dimensional confounding. Specifically, this method utilizes a partially linear mediation structure to isolate causal parameters and employs the machine learning algorithms (here we adopt the LASSO algorithm in our replication) to estimate nuisance parameters within the high-dimensional covariate space. Building on this framework, the mediation effects are subsequently estimated using traditional structural equation modeling, where the indirect effect is derived from the product of coefficients. Notably, under a linear model specification without exposure-mediator interaction or nonlinear effects, the direct and indirect effect estimates yielded from this approach are approximately comparable to the causal mediation effects defined in our potential outcome-based framework. Furthermore, we fit four benchmark models based on different subsets of covariates for comparison:

1. (i). Benchmark model (true): GPS models are fitted using X_C and X_I, indicating “true” treatment models.
2. (ii). Benchmark model (outcome): GPS models are fitted using X_C and X_P, which involve outcome information; the target model of our proposed method.
3. (iii). Benchmark model (true + outcome): GPS models are fitted using X_C, X_I, and X_P.
4. (iv). Benchmark model (full): GPS models are fitted using X_C, X_I, X_P, and 10 randomly selected X_S, representing the natural GPS models without variable selection. Of note, we select 10 spurious covariates considering both the computational efficiency and the sufficiency of representativeness of the influence of spurious covariate variables.

4.5 Real data application

The established link between diabetes risk and cancer incidence [81–83] is increasingly attributed to metabolic dysregulation, particularly dyslipidemia [58–60]. Apolipoprotein B (ApoB), an essential structural and functional component of lipoprotein particles [84–86], has emerged as a key mechanistic link. Specifically, abnormal changes in ApoB levels have been widely suggested as potential downstream consequences of diabetic conditions [87–91]. In turn, abnormal ApoB-related disturbances are associated with carcinogenesis through mechanisms, including transduction, oxidative stress, and tumor proliferation [61,62,92–97]. These findings position ApoB as an underlying plausible mediator in the diabetes-cancer pathway. Furthermore, as large-scale databases with a vast quantity of variables become prevalent, traditional observational studies struggle to fully account for potential confounders based on established knowledge, making it difficult to deliver causal estimates. Therefore, we apply our method to investigate the mediating effect of ApoB levels on the association between the Finnish Diabetes Risk Score (FINDRISC)-represented potential diabetes risk and the overall incidence of 24 site-specific cancers with large-scale biomedical data obtained from the UK Biobank.

UK Biobank is a large population-based cohort study conducted in England from 2006 to 2010, comprising approximately 500,000 participants aged 40–69 years [98]. Based on the UK Biobank, the treatment variable A is defined as a continuous composite indicator of FINDRISC constructed by age, body mass index (BMI), waist circumference, activity, vegetable and fruit intake, history of hypertension medication, history of hyperglycemia, and family history of diabetes, ranging from 0 to 25 [99,100] (Table L in S1 Appendix). The outcome variable Y is a binary indicator of overall cancer occurrence defined by the first record of hospitalization for cancer, cancer registry, or self-reported data according to the International Classification of Diseases (ICD)-9, ICD-10, and self-reported codes (Table M in S1 Appendix). The mediator variable M is defined as a continuous variable of the ApoB levels. We include a total of 98 variables in the covariate space, covering a wide range of demographic characteristics, socioeconomic status, lifestyle and environmental factors, genetic risk indicators, physical and mental health-related factors and measures, and blood biochemical and metabolic biomarkers (Tables N-O in S1 Appendix). The data on FINDRISC and ApoB are collected or measured at baseline, and the information on cancer occurrence is obtained by linking data from the cancer and death registry. By integrating the GOAL method into the GPS models of the semiparametric estimation, we expect to select important variables from the large covariate space.

We sequentially exclude the pregnant individuals, withdrawals, participants with missing information on FINDRISC, ApoB and cancer, and participants who reported cancer occurrence at baseline. These procedures restrict the analysis to individuals with a positive conditional treatment density and ensure the temporal sequence of the treatment and mediator preceding the outcome. We implement a complete case analysis where participants who had missingness on any of the 98 covariates are excluded. Ultimately, the analysis sample contains 7,054 participants, of which 1,057 developed cancer (Y = 1) and the remaining 5,997 did not (Y = 0), with a cancer incidence of 14.98%. The density distribution of FINDRISC is presented in Fig 7. The mean value of FINDRISC (A) of the 7,054 observations is 7.67, and the median value is 7. And the mean and median values of the level of ApoB (M) are 0.99 g/L and 0.98 g/L, respectively, with the density distribution plot in Fig J in S2 Appendix.

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Fig 7. Density histogram of FINDRISC (treatment).

Notes: The horizontal axis displays the treatment values, i.e., Finnish Diabetes Risk Score (FINDRISC), in increments of 2, and the vertical axis indicates the number of participants within each treatment interval. Abbreviations: FINDRISC, the Finnish Diabetes Risk Score.

https://doi.org/10.1371/journal.pcbi.1014436.g007

We examine the key assumptions underlying the application of this method to real-world data (Text I in S3 Appendix). We conduct an exploratory data analysis by using the test.TMint function in the R ‘mediation’ package [101] to preliminarily examine the interaction effect between A and M. The results all show no significant sign of an interaction effect between A and M (P = 0.416), implying the plausibility of omitting the interaction term in the subsequent analyses. In addition, we fit an outcome model with an A-M interaction term and the quadratic terms of A and M to pre-test the linearity of the outcome model. The results also indicate non-significant effects for the interaction term and quadratic terms (, and ), providing supportive evidence for the appropriateness of the model linearity. Furthermore, we fit a generalized additive outcome model with A and the penalized spline term of the M using the R ‘mgcv’ package [102,103] to extensively capture the potential nonlinear effect of the mediator. The effective degrees of freedom (EDF) for the smooth term are close to 1 and not statistically significant (P = 0.324), suggesting no evidence of nonlinearity between the mediator and the outcome.

The mediation effects are evaluated using the GOAL-embedded GPS-weighted semiparametric estimation approach, where the weight for each observation is constructed by parametric GPSs of treatment density conditioned on the selected covariates or the covariates and the mediator (Equation 7). Also, analyses based on the GPS models with full covariates are performed for comparison. The convergence factor takes the value of 2 in all the variable selection procedures. Under the condition of positive treatment intensity, we estimate the direct and indirect effects under both treatment and non-treatment status in the treatment range of 3–13 in a step of 1 versus the reference value of 2, that is, , , and for each of a ∈ {3, 4,  ..., 13} and a’ = 2. We employ the second-order Epanechnikov kernels and utilize the rule-of-thumb bandwidth h for kernel methods with h = C·sd(A)·n^-1/5, where sd(A) is the standard deviation of A and C = 2.34. It is more pragmatic and widely adopted in the empirical studies with an unknown data-generating process to balance bias and variance, which was also consistently applied and described in Huber et al [5]. We also provide a set of diagnostic checks for a series of candidates covering simulation bandwidths and a set of ROT-based candidates that follows the conventional practice in kernel-based sensitivity analyses: (Table P, Figs D-E, and Text C in S1–S3 Appendices). Further, to examine the sensitivity of kernel-based estimation to bandwidth specification, we repeat the estimation using a range of bandwidth constants C = 1.7, 2.0, 2.5 under the rule-of-thumb formula h = C·sd(A)·n^-1/5, where C = 2.34 is compared as a reference (Table R in S1 Appendix and Figs H-I in S2 Appendix).

We derive the variance estimation by bootstrapping 500 times and measure the average standard deviation of the effects across all treatment values. We also conduct a sensitivity analysis with 100 bootstrap resamples for the GOAL-based estimation to visualize and examine the stability of variable selection under bootstrapping and regularization (Table Q in S1 Appendix and Figs F-G in S2 Appendix).