Derivation of a multivariate longitudinal causal effects model

Abstract

This paper presents a causal inference estimation method for longitudinal observational studies with multiple outcomes. The method uses marginal structural models with inverse probability treatment weights (MSM-IPTWs). In developing the proposed method, we re-define the weights as a product of inverse weights at each time point, accounting for time-varying confounders and treatment exposures and possible correlation between and within (serial) the multiple outcomes. The proposed method is evaluated by simulation studies and with an application to estimate the effect of HIV positivity awareness on condom use and multiple sexual partners using the Malawi Longitudinal Study of Families and Health (MLSFH) data. The simulation study shows that the joint MSM-IPTW performs well with coverage within the expected 95% level for a large sample size (n = 1000) and moderate to strong between and within outcome correlation strength ( ${\rho }_j=0.3$, 0.75, ${\rho }_k=0.4$, 0.8) when the effects are similar. The joint MSM-IPTW performed relatively the same as the adjusted standard joint model when the treatment effect estimate was the same for the outcomes. In the application, HIV positivity awareness increased the usage of condoms and did not affect the number of sexual partners. We recommend using the proposed MSM-IPTWs to correctly control for time-varying treatment and confounders when estimating causal effects for longitudinal observational studies with multiple outcomes.

1. Introduction

Longitudinal studies are well suited for assessing causal effects because data is collected over time. This ensures that treatment exposure occurs before the outcomes are measured, creating a clear temporal relationship between the treatment and outcome. In health research utilizing longitudinal data, researchers often face the challenge posed as a result of collecting multiple outcomes from study participants. For instance, in HIV research, being aware of one's positive HIV status is crucial for the timely initiation of treatment, as outlined in the UN's 95-95-95 strategy [38]. However, awareness of being HIV-positive can significantly influence individuals' behavior changes. Understanding how awareness of one's HIV-positive status affects risky sexual behaviors, such as inconsistent condom use and having multiple sexual partners over time, is crucial for developing effective public health interventions. Condom use and the number of sexual partners an individual has are interrelated and can change over time [9,10,28,37]. To accurately estimate the impact of awareness of HIV positivity, it is essential to consider this dependency and control for other confounding factors that can influence the outcomes and awareness itself. Figure 1 illustrates a potential mechanism that contributes to the complex interactions among condom use, the number of sexual partners, awareness of HIV status, and time-dependent confounders, such as education level (measured in years). For example, individuals with higher educational attainment are more likely to be aware of their HIV status, to have used condoms during their last sexual encounter, or to have had only one sexual partner. Additionally, HIV status awareness at baseline can affect future behaviors related to multiple sexual partners and condom use, as well as ongoing awareness [9,28,29,39]. Furthermore, years of education and awareness of HIV status can influence future condom use and the number of sexual partners an individual may have [37]. Past behaviors, such as condom use and the number of sexual partners, may also affect an individual's future awareness of their HIV status. A proper statistical analysis of this issue must account for the correlation of measurements within individual subjects and between the various outcomes, while also controlling for time-varying confounders.

Figure 1.

DAG for the complex relationship between the correlated outcomes, the time-varying and baseline confounder for three time points. Ed, education level; HIV, HIV awareness; Con, condom use; Mult, multiple sexual partners.

Several statistical methods have been developed to analyze multiple outcomes measured repeatedly over time, addressing the dependency between these measurements. Extensions of the random effects model have led to mixed models that impose a joint multivariate distribution on random effects [14]. Reinsel [30,31] has expanded this model for balanced multivariate longitudinal data. Random effects models can be computationally intensive and may become unstable with increased parameters [11]. Thus, pairwise regression models have been developed to simplify the analysis of high-dimensional data by focusing on bivariate associations [12]. Generalised estimating equations (GEE) have been used for multivariate data as this estimation does not require specifying the entire distribution. Asar and Ik [4] applied GEE for multivariate marginal models, and Cho [7] used the quadratic inference functions (QIF) to analyze multivariate longitudinal data efficiently. Liang [23] proposed an M-mix-GEE for working correlation matrices, and Inan [19] extended prior studies [13] to a general class of joint GEE models. Despite these advancements, many methods do not adequately control for time-varying confounders, limiting causal inference. Estimating causal effects in longitudinal observational studies can be challenging, especially with time-varying treatment exposures and confounders [16,17,32]. The Marginal Structural Models with inverse probability treatment weights (MSM-IPTWs) estimate treatment effects while accounting for time-varying confounders [16–18,24,34]. While Structural Nested Models (SNMs) can also handle such confounders, MSMs are more commonly used due to their simplicity and compatibility with standard software.

Research on causal models for longitudinal data with multiple outcomes is still developing. Some studies, like those by Lupparelli et al. [25] and Mattei et al. [26], have explored causal inference for multiple outcomes simultaneously at single time-points. More recent work has focused on longitudinal data; Vanderweele et al. [39], provided examples of causal effect estimation for longitudinal data with multiple outcomes, and Moodie et al. [27], used marginal structural models for analyzing multiple-outcome survival data. Agogo et al. [1], employed a shared random effects model for concurrent binary outcomes but did not address time-varying confounding.

This paper proposes methods for causal inference in longitudinal observational studies that involve multiple outcomes. Our approach is centered on the Marginal Structural Model, which uses inverse probability treatment weights to account for time-varying treatment exposures and confounders. Specifically, we develop a novel technique that considers the relationships between the outcomes while addressing potential confounding factors. To illustrate the applicability of our proposed methods, we apply them to the Malawi Longitudinal Study of Families and Health (MLSFH) to estimate the impact of HIV positivity awareness on condom usage and the number of sexual partners. The subsequent sections outline the estimation of causal effects for univariate outcomes in longitudinal studies. Section 3 introduces the methodology for estimating causal effects on multiple outcomes in such studies, and Section 4 details the model estimation process. In Section 5, we present simulation studies conducted to evaluate our methods. Section 6 reports the results from analyzing the effect of HIV positivity awareness on condom use and multiple sexual partners using the MLSFH data. Finally, we discuss our methods and findings in Section 7.

2. Marginal structural models for univariate longitudinal outcomes

Consider a set of subjects indexed by i, where $i=1,\dots ,N$, with observed q outcomes in vector $Y_{\mathrm{it}}$ at each subject's tth visit. At each time point t, we also observe a time-dependent exposure $A_{\mathrm{it}}$ (either 0 or 1) and a set of time-dependent confounders $L_{\mathrm{it}}$, which may be influenced by previous exposures, alongside baseline confounders $X_0$. For each subject i, we denote the exposure and covariate histories as ${\overline{A}}_{\mathrm{it}}=\left\{A\right(k);k=1,2,\dots ,t\}$ and ${\overline{L}}_{\mathrm{it}}=\left\{L\right(k);k=1,2,\dots ,t\}$. Assuming that the data for each subject are drawn independently from a common distribution, we can write the outcome as $Y_t$.

The causal estimate of interest is the difference in means of potential outcome $E\left[Y_t\right(\overline{a}\left)\right]$ between different exposure groups, expressed as:

$$E\left[Y_t\right(\overline{a}\left)\right]-E\left[Y_t\right(\overline{0}\left)\right]$$ (1)

A marginal structural model is proposed to estimate this causal effect [17,36]:

$$E\left[Y_t\right(\overline{a}\left)\right]=g\left(h\right({\overline{a}}_{t-1});\mathrm{\text{β}})$$ (2)

Where $h\left({\overline{a}}_{t-1}\right)$ is the function of the exposure history. Here, $g(\overline{a};\mathrm{\text{β}})$ can take various forms, such as additive or multiplicative functions that relate exposure history to the mean potential outcome. If focusing on the cumulative effect, we rewrite it as:

$$E\left[Y_t\right(\overline{a}\left)\right]={\mathrm{\text{β}}}_1\mathrm{cum}\left({\overline{a}}_t\right)$$ (3)

where $\mathrm{cum}\left({\overline{a}}_t\right)=\sum _{k=1}^t{a}_t$ and ${\mathrm{\text{β}}}_1$ measures the impact of the cumulative exposure on the mean outcome. To obtain estimates, we calculate $E[Y_t|{\overline{A}}_t={\overline{a}}_t,X_0,L_t]$. However, the observed expectation does not equal the true expectation due to confounding by ${\overline{L}}_t$. To address confounding, inverse probability treatment weights (IPTW) are utilized, defined as the inverse of the conditional probability of treatment given past exposure and confounders:

$$W_k=\prod _{k=1}^t\frac{1}{P(A_k=1|{\overline{A}}_{k-1},X_0,{\overline{L}}_k)}$$ (4)

IPTW can be subject to extreme values, so stabilized weights and weight truncation methods are often employed. Additionally, missing data due to loss to follow-up can be managed by extending IPTW to account for censoring.

3. Marginal structural model for multiple outcomes

We extend the notation from Section 2 to multiple outcomes. For each subject i (where $i=1,\dots ,n$), we observe J outcomes measured over time t. Each subject has a JT vector of outcomes, represented as $\mathbf{\text{Y}}=(Y_{\mathrm{ijt}}{)}^{\mathrm{T}}$, where $Y_{\mathrm{ijt}}$ denotes the measurement of the jth outcome for subject i at time t.

The potential outcome framework is used in a longitudinal setting, where ${\mathbf{\text{Y}}}_i\left({\overline{a}}_t\right)=\left(Y_{\mathrm{ijt}}\right({\overline{a}}_t){)}^{\mathrm{T}}$ represents the potential outcome for subject i based on treatment history ${\overline{a}}_t$. The average causal effect for subject i is defined as:

$${\mathbf{\text{Y}}}_{\mathrm{ij}}\left(\overline{a}\right)=\frac{1}{t}\sum _{k=1}^t{Y}_{\mathrm{ijt}}\left(\overline{a}\right).$$

We extend the definition of the Average Treatment Effect (ATE) from univariate longitudinal data to a multivariate context. The average response vector for the entire population of subjects who received treatment a = 1 over time is represented as $E\left[Y_{\mathrm{ijt}}\right(\overline{a}=1\left)\right]$, while the average response vector for those who received treatment a = 0 is represented as $E\left[Y_{\mathrm{ijt}}\right(\overline{a}=0\left)\right]$. The ATE for outcome j is given by:

$$E\left[Y_{\mathrm{ijt}}\right({\overline{a}}_t=1\left)\right]-E\left[Y_{\mathrm{ijt}}\right({\overline{a}}_t=0\left)\right].$$

In vector notation:

$$E\left[\mathbf{\text{Y}}\right({\overline{a}}_t=1\left)\right]-E\left[\mathbf{\text{Y}}\right({\overline{a}}_t=0\left)\right].$$

We extend the marginal structural model for longitudinal outcomes, modeling the effect of exposure history on the vector of marginal means:

$$E\left[\mathbf{\text{Y}}\right({\overline{a}}_t\left)\right]=g\left(h\right({\overline{a}}_t);{\mathrm{\text{β}}}_j).$$

To assess how HIV status awareness affects risky sexual behaviors, we generalize as follows:

$${\mathbf{\text{Y}}}_{\mathrm{ij}}={\mathrm{\text{β}}}_0+{\mathrm{\text{β}}}_{j1}\mathrm{cum}\left({\overline{a}}_t\right),$$ (5)

where $\mathrm{cum}\left({\overline{a}}_t\right)=\sum _{k=1}^t{a}_k$, and ${\mathrm{\text{β}}}_{j1}$ is the causal effect on the jth mean potential outcome $E\left[Y_{\mathrm{ij}}\right({\overline{a}}_t\left)\right]$ for subject i's cumulative exposure.

3.1. Assumptions to identify ATE for multiple longitudinal outcomes

To identify the causal effect ${\mathrm{\text{β}}}_{j1}$ for the jth outcome measured at times $s=1,\dots ,t$, we adapt the assumptions by Rosenbaum and Rubin [35] to account for longitudinal data with multiple outcomes. The assumptions are as follows:

1. Consistency: For any subject i, the observed outcome $Y_{\mathrm{ijt}}$ under the actual exposure $A_i$ equals the potential outcome $Y_{\mathrm{ijt}}\left({\overline{a}}_i\right)$:

   $${\mathbf{\text{Y}}}_{\mathrm{ij}}={\mathbf{\text{Y}}}_{\mathrm{ij}}\left({\overline{a}}_t\right).$$
2. Positivity: For all combinations of covariates and time-varying covariates, the probability of receiving each possible treatment history $\overline{a}$ must be non-zero, and there is no discontinuation of the treatment:

   $$P(A_t=a_t|{\overline{A}}_{t-1},{\mathbf{\text{L}}}_{\mathrm{it}},{\mathbf{\text{X}}}_0,{\mathbf{\text{Y}}}_{i,t-1})>0.$$
3. Sequential Ignorability: Given the past treatment history and covariates, including past outcomes, the treatment assignment at each time point is independent of the potential outcomes:

   $${\overline{A}}_{\mathrm{it}}⨿\left\{{\mathbf{\text{Y}}}_{1j}\right(a_1),{\mathbf{\text{Y}}}_{2j}(a_2),\dots ,{\mathbf{\text{Y}}}_{\mathrm{tj}}(a_t\left)\right\}\mid {\overline{A}}_{t-1},{\mathbf{\text{L}}}_{i,t-1},{\mathbf{\text{X}}}_0,{\mathbf{\text{Y}}}_{i,t-1}.$$

   Furthermore, since the outcomes are correlated, they are assumed to be influenced by the same covariates, with no outcome-specific confounders.
4. Stable Unit Treatment Value Assumption (SUTVA): The potential outcomes for each individual are unaffected by the treatment assignments of others:

   $$\left\{{\mathbf{\text{Y}}}_{1j}\right(a_1),{\mathbf{\text{Y}}}_{2j}(a_2),\dots ,{\mathbf{\text{Y}}}_{\mathrm{tj}}(a_t\left)\right\}⨿{\overline{A}}_t.$$

3.2. Inverse probability treatment weighting for multivariate longitudinal data

We extend the inverse probability treatment weights from equation (4) to include multiple outcomes (j). By incorporating the history of past outcomes, we control for confounding relationships between outcomes and exposure. This modification allows us to evaluate the treatment's average effect on multiple outcomes without the influence of past exposure, baseline factors, or time-varying confounders.

The unstabilized weight for subject i's jth outcome at time t is given by:

$$W_k=\prod _{k=1}^t\prod _{j=1}^J\frac{1}{P(A_k|{\overline{A}}_{k-1},{\overline{\mathbf{\text{L}}}}_k,{\mathbf{\text{X}}}_0,{\overline{\mathbf{\text{Y}}}}_{j,k-1})}$$

The estimator can be consistently estimated using the modified weights, with the stabilized weight defined as:

$$S{W}_k=\prod _{k=1}^t\prod _{j=1}^J\frac{P(A_k|{\overline{A}}_{k-1},{\mathbf{\text{X}}}_0)}{P(A_k|{\overline{A}}_{k-1},{\overline{\mathbf{\text{L}}}}_k,{\mathbf{\text{X}}}_0,{\overline{\mathbf{\text{Y}}}}_{j,k-1})}$$

This approach creates a pseudo-population where the distribution of confounders remains consistent across multiple outcomes.

4. Estimation of cause-effect relationships in a multivariate longitudinal study

We employ the Generalized Estimating Equation (GEE) approach to estimate our proposed model, as it effectively provides consistent estimates of treatment exposure effects using Inverse Probability Treatment Weights (IPTWs) [15,17]. Our method includes a working correlation matrix that addresses both serial correlation and correlations among outcomes.

To obtain a consistent estimator from model 5, we solve the following estimating equations:

$$\sum _{i=1}^N\sum _{j=1}^J\sum _{k=1}^t{D}_i\left(\mathrm{\text{β}}\right)V^{-1}{W}_k\left(Y_{\mathrm{ijt}}\right({\overline{a}}_t)-{\mu }_i({\mathrm{\text{β}}}_j\left)\right)=0$$ (6)

Here, ${\overline{a}}_t$ represents the counterfactual exposure history, and $D_i\left(\mathrm{\text{β}}\right)$ is the derivative of the mean vector, with $W_k$ denoting the inverse probability treatment weights. The marginal covariance matrix ${\mathbf{\text{V}}}_i$ is constructed as:

$${\mathbf{\text{V}}}_i=\varphi {\mathbf{\Sigma }}_i$$

where ${\mathbf{\Sigma }}_i=\varphi {\mathbf{\text{A}}}^{1/2}\left({\mathbf{\text{R}}}_y\right(\mathit{\tau })\otimes {\mathbf{\text{R}}}_t(\mathit{\alpha }\left)\right){\mathbf{\text{A}}}^{1/2}$ incorporates the variance function and includes the $\mathrm{JT}\times \mathrm{JT}$ working covariance matrices for the serial ( $\mathit{\tau }$) and between outcome ( $\mathit{\alpha }$) correlations. We also propose a Kronecker structure for the working correlation matrix to reduce the complexity of parameter estimation [20].

The correlation structures for repeated measures can be modeled as exchangeable, first-order autoregressive (AR(1)), or unstructured, depending on the relationships we want to capture. For instance, we can define the correlation for two outcomes as:

$$\mathrm{Corr}(Y_{\mathrm{ijt}},Y_{i{j}^{\mathrm{\prime }}{t}^{\mathrm{\prime }}})=\tau {\alpha }^{|t-t^{\mathrm{\prime }}|},\mathrm{\forall t}\ne t^{\mathrm{\prime }},j\ne j^{\mathrm{\prime }}$$

To estimate causal parameters, we employ the observed data under the consistency and strong ignorability assumptions, leading to the following estimating equations:

$$\sum _{i=1}^n\sum _{j=1}^J{\mathbf{\text{D}}}_i\left(\mathrm{\text{β}}\right).{\mathbf{\text{V}}}_i^{-1}\hat{W}({\mathbf{\text{Y}}}_{\mathrm{ij}}-\mathit{\mu }(\mathrm{\text{β}}\left)\right)=\mathbf{\text{0}}$$

The GEEs can be solved using the Fisher scoring algorithm, resulting in:

$$\widehat{\mathrm{\text{β}}}=\widetilde{\mathrm{\text{β}}}+{\left(\sum _{i=1}^n{\widetilde{\mathbf{\text{D}}}}_i\right(\mathrm{\text{β}}\left){\widehat{\mathbf{\text{V}}}}_i^{-1}\hat{W}{\widetilde{\mathbf{\text{D}}}}_i\right(\mathrm{\text{β}}\left)\right)}^{-1}\left(\sum _{i=1}^n{\widetilde{\mathbf{\text{D}}}}_i\right(\mathrm{\text{β}}\left){\widehat{\mathbf{\text{V}}}}_i^{-1}{\hat{W}}_{\mathrm{it}}\right({\mathbf{\text{Y}}}_i-{\tilde{\mathit{\mu }}}_i\left)\right)$$

The asymptotic covariance matrix for the GEE estimators is estimated using:

$$\mathrm{Var}\left(\widehat{\mathrm{\text{β}}}\right)={\widehat{\mathbf{\text{A}}}}^{-1}\widehat{\mathbf{\text{B}}}{\widehat{\mathbf{\text{A}}}}^{-1}$$

This framework offers a robust method for estimating causal effects in multivariate longitudinal studies while accounting for data correlations.

5. Simulation study

In this section, we evaluate performance of the proposed estimation procedure through simulation studies and in the next section we apply to a real life data.

Following Asar's [4] and Inan's [20] data-generating approach for simulating multivariate longitudinal data, we implemented a longitudinal design with three time points (t = 1, 2, 3) per subject. We assumed two jointly continuous random variables observed for N subjects [7], with each variable measured at all three time points. We generate the baseline confounder $X_0$ representing a hypothetical gender of the subject as a time-invariant covariate from a binomial distribution as $X_0\sim \mathrm{Binomial}(N,0.5)$. We simulated each subject's hypothetical age as a time-varying covariate.

The exposure variable, awareness of one's HIV status, was simulated as a binary response influenced by these covariates using a logistic regression model. We generate initial values of $A_1,L_1$ at baseline and iterate for the remaining time points t = 2, 3. We generate $L_1$ from a normal distribution $N(0,0.5)$ and $A_1$ from the following logistic model $P(A_1=1|L_1,X_0)={\mathrm{logit}}^{-1}({\psi }_0+{\psi }_1{L}_1+{\psi }_2{X}_0$), where $\mathit{\psi }=(1,0.5,0.7)$. $L_t$ was generated using the following model

$$L_t={\delta }_0+{\delta }_1{L}_{t-1}+{\delta }_2{A}_{t-1}+{ϵ}_L$$

where ${ϵ}_L\sim N(0,1)$. We simulated each subjects's treatment as a function of the time-varying and baseline confounder using a logistic regression. The logistic regression model was given as follows;

$$P(A_t=1|A_{t-1},L_{t-1},X_0)={\mathrm{logit}}^{-1}({\theta }_0+{\theta }_1{A}_{t-1}+{\theta }_2{L}_{t-1}+{\theta }_3{X}_0)$$

Where $\mathit{\theta }=(1,0.5,0.2,0.2)$.

The correlated multivariate outcomes ( $Y_{i1t},Y_{i2t}$) at time t were generated using the following model

$${\mathbf{\text{Y}}}_{\mathrm{ij}}={\mathrm{\text{β}}}_{0j}+{\mathrm{\text{β}}}_{1j}{A}_t+{\mathrm{\text{β}}}_{2j}{L}_t+{\mathit{ϵ}}_i$$

The covariance-variance structure for the two outcomes is determined by the independent error terms ${\mathit{ϵ}}_i$, where $\mathit{ϵ}$ has a multivariate normal distribution with mean 0, marginal variance 1, ${\mathit{ϵ}}_i\sim \mathcal{N}(\mathbf{\text{0}},\mathrm{\Sigma })$ and an exchangeable correlation structure with a correlation coefficient ${\rho }_{j{j}^{\mathrm{\prime }}}$.

$${\mathit{ϵ}}_i=({ϵ}_{i1},{ϵ}_{i2}{)}^{\mathrm{T}}\sim N(\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}1& {\rho }_{12}\\ {\rho }_{21}& 1\end{array}\right))$$

The joint outcomes were generated under two scenario's.

1. We assume that the treatment effect is the same were we first simulate for ${\mathrm{\text{β}}}_{1j}=0.3$ and when ${\mathrm{\text{β}}}_{1j}=0.8$.

2. We assume each outcome has a different treatment effect ${\mathrm{\text{β}}}_{1j}=(0.8,0.3{)}^{\mathrm{T}}$.

For both scenarios, we fixed ${\mathrm{\text{β}}}_{2j}=(0.5,0.5{)}^{\mathrm{T}}$. The simulated treatment effects for condom use and multiple sexual partners were informed by findings from prior studies [9,29], which fall within the range of observed effects in the literature. The correlated outcomes were assumed to follow an exchangeable correlation structure with varying correlation strengths. While the repeated outcomes ${\mathbf{\text{Y}}}_{\mathrm{ij}.}=(Y_{\mathrm{ij}1},Y_{\mathrm{ij}2},Y_{\mathrm{ij}3})$ were assumed to follow a serial correlation. The simulation run into three values of the serial correlation ( ${\rho }_k=0.1$, 0.4, 0.8).

We examined varying strengths of between-outcome correlations ( ${\rho }_j=0$, 0.3, 0.75) and sample sizes of 50, 200, 500 and 1000. For each scenario, we generated 1000 replicated datasets and fitted two models to each dataset.

1. Non-weighted joint model

2. IPTW weighted joint model (MSM)

The accuracy of each of the estimated effect ${\mathrm{\text{β}}}_{1j}$, j = 1, 2 was assessed with the following measures:

1. The Bias in ${\widehat{\mathrm{\text{β}}}}_{1j}:\mathrm{Bias}\left({\widehat{\mathrm{\text{β}}}}_{1j}\right)=({\widehat{\mathrm{\text{β}}}}_{1j}-{\mathrm{\text{β}}}_{1j})$

2. The Root Mean Square Error

   $$\mathrm{RMSE}=\sqrt{\frac{{\widehat{\mathrm{\text{β}}}}_{1j}-{\mathrm{\text{β}}}_{1j}}{N}}$$

3. Coverage which is defined as the proportion of times the 95% confidence interval of ${\widehat{\mathrm{\text{β}}}}_{1j}$ contains the true ${\mathrm{\text{β}}}_{1j}$

The weights were obtained using the ipw package in R and the weighted joint MSM was fit using the geepack and JGEE packages. The simulation study and statistical analyzes were performed using R version 4.4.1.

5.1. Simulation results

Table 1 presents the mean effects, bias, root mean square error (RMSE), and coverage for estimating a treatment effect of 0.3. Both non-weighted and weighted models showed reduced bias and RMSE as sample size increased, with coverage remaining consistent at approximately 0.93–0.95 for non-weighted models and stabilizing around 0.94–0.96 for weighted models. Non-weighted models performed well across all sample sizes when the between-outcome correlation ( ${\rho }_j$) was low ( ${\rho }_j=0$), producing low bias and high coverage. In scenarios with moderate or high between-outcome correlation ( ${\rho }_j=0.3$, 0.75), weighted models demonstrated slightly better performance, particularly at larger sample sizes, with lower RMSE and higher coverage, especially for ${\rho }_j=0.3$, ${\rho }_k=0.8$ and ${\rho }_j=0.75$, ${\rho }_k=0.4$. High serial correlation ( ${\rho }_k=0.8$) increased variability in smaller samples (N = 50), leading to higher RMSE, but both models stabilized as the sample size grew. Weighted models generally outperformed non-weighted models in terms of RMSE and coverage when either ${\rho }_j$ or ${\rho }_k$ was high, demonstrating their robustness in complex dependency structures. However, they were more sensitive to small sample sizes, showing increased bias and RMSE under these conditions.

Table 1.

Treatment effect mean, bias, root mean square error, and coverage probability of mean estimate for effects of treatment on the joint outcomes for varying sample size and correlation when the true treatment effect is 0.3.

			Standard bivariate				Joint MSM-IPTW
${\rho }_j$	${\rho }_k$	Size	$\widehat{\mathrm{\text{β}}}$	Bias	RMSE	Coverage	$\widehat{\mathrm{\text{β}}}$	Bias	RMSE	Coverage
0	0.1	50	0.3065	0.0065	0.4089	0.94	0.052	0.244	0.8376	0.9
		200	0.3105	0.0105	0.1315	0.95	0.295	0.0048	0.272	0.95
		500	0.3134	0.0134	0.3817	0.95	0.3221	0.0221	0.1651	0.96
		1000	0.3124	0.0124	0.0606	0.95	0.3324	0.0324	0.1066	0.96
	0.4	50	0.304	0.004	0.2925	0.93	0.2957	0.0043	0.3557	0.9
		200	0.3196	0.0196	0.2241	0.94	0.3079	0.0079	0.177	0.94
		500	0.3037	0.0037	0.0958	0.95	0.3079	0.0079	0.1089	0.96
		1000	0.3027	0.0027	0.068	0.97	0.3048	0.0048	0.0784	0.96
	0.8	50	0.3729	0.0729	0.8277	0.95	0.1943	0.1057	1.7425	0.95
		200	0.2622	0.0378	0.4087	0.95	0.2678	0.0322	0.344	0.96
		500	0.2621	0.0379	0.2503	0.95	0.2779	0.0221	0.1917	0.96
		1000	0.251	0.0486	0.1215	0.94	0.2599	0.0401	0.1408	0.94
0.3	0.1	50	0.3071	0.0071	0.2627	0.93	0.3719	0.0719	0.5026	0.91
		200	0.3158	0.0158	0.1311	0.94	0.3119	0.0119	0.1931	0.94
		500	0.3143	0.0143	0.0848	0.94	0.3177	0.0177	0.1002	0.94
		1000	0.314	0.014	0.0614	0.94	0.3152	0.0211	0.0716	0.94
	0.4	50	0.3049	0.0049	0.2739	0.93	0.3029	0.0029	0.3599	0.92
		200	0.3123	0.0123	0.141	0.94	0.313	0.013	0.167	0.93
		500	0.3107	0.0107	0.0909	0.94	0.3157	0.0157	0.1038	0.94
		1000	0.3095	0.0095	0.0637	0.94	0.3115	0.0115	0.075	0.94
	0.8	50	0.286	0.014	0.4137	0.95	0.2889	0.011	0.7601	0.94
		200	0.29	0.01	0.2076	0.94	0.2976	0.0024	0.3502	0.96
		500	0.2876	0.0124	0.1259	0.95	0.302	0.002	0.2303	0.96
		1000	0.2854	0.0146	0.1371	0.94	0.2939	0.0061	0.1037	0.95
0.75	0.1	50	0.3068	0.0068	0.2701	0.93	0.3161	0.0161	0.4797	0.89
		200	0.3211	0.0211	0.1297	0.94	0.3208	0.0208	0.1542	0.93
		500	0.3152	0.0152	0.0853	0.94	0.3189	0.0189	0.1006	0.93
		1000	0.3159	0.0159	0.0617	0.94	0.3175	0.0175	0.0706	0.94
	0.4	50	0.3048	0.0048	0.2681	0.92	0.3913	0.0913	0.5744	0.93
		200	0.3294	0.0207	0.1315	0.94	0.234	0.066	0.3914	0.96
		500	0.316	0.016	0.0861	0.93	0.2007	0.0993	0.3288	0.95
		1000	0.3158	0.0158	0.0616	0.94	0.3699	0.0699	0.1997	0.95
	0.8	50	0.2983	0.0017	0.2887	0.94	0.321	0.021	0.467	0.91
		200	0.3157	0.0157	0.1474	0.93	0.3175	0.0175	0.1658	0.94
		500	0.3138	0.0138	0.0945	0.94	0.319	0.019	0.1109	0.94
		1000	0.3123	0.0123	0.0656	0.93	0.3145	0.0145	0.0759	0.95

From the results in Table 2, both models produce reduced bias and RMSE as sample size increases. Coverage remains stable across sample sizes, consistently near 0.93–0.96. For smaller samples (N = 50), RMSE is higher due to greater variability, with weighted models showing slightly higher RMSE and bias. When there is no between-outcome correlation ( ${\rho }_j=0$), both models perform well, with minimal bias, consistent coverage, and low RMSE, particularly in larger samples and for ${\rho }_k=0.8$. Under high between-outcome correlation ( ${\rho }_j=0.75$), non-weighted models maintain low bias and consistent coverage but experience slightly higher RMSE in smaller samples, while weighted models stabilize only with larger sample sizes. For scenarios with increased serial correlation ( ${\rho }_k=0.8$ for ${\rho }_j=0.3$ and ${\rho }_k=0.4$ for ${\rho }_j=0.75$), both models remain robust at larger sample sizes ( $N\ge 200$), but RMSE is higher for smaller samples (N = 50). Weighted models outperform non-weighted models in bias reduction and RMSE under moderate to high correlations ( ${\rho }_j$ or ${\rho }_k$) when sample sizes are sufficient ( $N\ge 500$). However, non-weighted models are more stable in smaller samples. Across all conditions, coverage remains high (approximately 0.94–0.96), indicating confidence intervals effectively capture the true effect size.

Table 2.

Treatment effect mean, bias, root mean square error, and coverage probability of mean estimate for effects of treatment on the joint outcomes for varying sample size and correlation when the true treatment effect is 0.8.

			Standard bivariate				Joint MSM-IPTW
${\rho }_j$	${\rho }_k$	Size	$\widehat{\mathrm{\text{β}}}$	Bias	RMSE	Coverage	$\widehat{\mathrm{\text{β}}}$	Bias	RMSE	Coverage
0	0.1	50	0.8089	0.0089	0.2578	0.94	0.552	0.248	0.7242	0.94
		200	0.8105	0.0105	0.1315	0.95	0.7969	0.0031	0.3301	0.95
		500	0.8134	0.0134	0.0854	0.94	0.8262	0.0262	0.1651	0.96
		1000	0.8124	0.0124	0.0606	0.95	0.8324	0.0324	0.1066	0.96
	0.4	50	0.804	0.004	0.2925	0.94	0.7957	0.0043	0.3557	0.92
		200	0.8028	0.0028	0.1522	0.94	0.8079	0.0079	0.177	0.94
		500	0.8037	0.0037	0.0958	0.95	0.8079	0.0171	0.1089	0.96
		1000	0.8027	0.0027	0.068	0.95	0.8048	0.0048	0.0784	0.95
	0.8	50	0.8729	0.0729	0.8277	0.97	0.6943	0.1057	1.7425	0.96
		200	0.7555	0.0445	0.2796	0.95	0.7678	0.0322	0.344	0.96
		500	0.7518	0.0482	0.1672	0.94	0.7621	0.0379	0.1917	0.96
		1000	0.7514	0.0486	0.1215	0.94	0.7599	0.0401	0.1408	0.94
0.3	0.1	50	0.8071	0.0071	0.2627	0.93	0.8904	0.0904	0.5026	0.92
		200	0.8158	0.0158	0.1311	0.94	0.8119	0.0119	0.1931	0.94
		500	0.8143	0.0143	0.0848	0.94	0.8177	0.0177	0.1002	0.94
		1000	0.814	0.014	0.0614	0.94	0.8152	0.0152	0.016	0.94
	0.4	50	0.8049	0.0049	0.2739	0.93	0.8029	0.0029	0.3599	0.92
		200	0.8123	0.0123	0.141	0.94	0.813	0.013	0.1672	0.93
		500	0.8107	0.0107	0.0909	0.94	0.8157	0.0157	0.1038	0.94
		1000	0.8095	0.0095	0.0637	0.94	0.8115	0.0115	0.075	0.94
	0.8	50	0.786	0.014	0.4137	0.95	0.7889	0.0111	0.505	0.94
		200	0.79	0.01	0.2076	0.94	0.7976	0.0024	0.3502	0.96
		500	0.7876	0.0124	0.1259	0.95	0.7958	0.0042	0.1407	0.96
		1000	0.7854	0.0146	0.0902	0.94	0.7939	0.0061	0.1564	0.95
0.75	0.1	50	0.8068	0.0068	0.2701	0.93	0.8251	0.0251	0.3341	0.89
		200	0.8211	0.0211	0.1297	0.94	0.8208	0.0208	0.1542	0.93
		500	0.8152	0.0152	0.0853	0.94	0.8189	0.0189	0.1006	0.93
		1000	0.8159	0.0159	0.0617	0.94	0.8175	0.0175	0.0706	0.94
	0.4	50	0.8048	0.0048	0.2681	0.92	0.8913	0.0913	0.4071	0.93
		200	0.8207	0.0207	0.1315	0.94	0.734	0.066	0.3914	0.96
		500	0.816	0.016	0.0861	0.93	0.7007	0.0993	0.3288	0.95
		1000	0.8158	0.019	0.0616	0.94	0.8699	0.0777	0.1997	0.95
	0.8	50	0.7983	0.0017	0.2887	0.94	0.821	0.021	0.467	0.91
		200	0.8157	0.0157	0.1474	0.93	0.8175	0.0175	0.1658	0.92
		500	0.8138	0.0138	0.0945	0.94	0.819	0.0208	0.1673	0.94
		1000	0.8123	0.0123	0.0656	0.93	0.8145	0.0145	0.0759	0.95

The simulation results from Table 3 indicate that both standard and bivariate models have reduced bias consistently for both ${\mathrm{\text{β}}}_1$ and ${\mathrm{\text{β}}}_2$, with smaller biases observed at higher sample sizes (N = 500 and 1000). RMSE also declines as sample size grows, reflecting improved precision, particularly for higher effect sizes ( $\mathrm{\text{β}}=0.8$) and lower correlations. For scenarios with no correlation ( ${\rho }_j=0$), both models maintain minimal bias and high coverage (approximately 0.93–0.96). With moderate or high between-outcome correlation ( ${\rho }_j=0.75$), bivariate models show slight increases in RMSE and bias, especially in smaller samples (N = 50 or N = 200), but remain robust in larger samples (N = 500 or N = 1000). Similarly, high serial correlation ( ${\rho }_k=0.8$) leads to increased variability in smaller samples but is well-handled by both models with sufficient sample sizes. Coverage may decrease slightly under high ${\rho }_j$, particularly for bivariate models with smaller sample sizes. When serial correlation ( ${\rho }_k$) increases (from 0 to 0.4), bias and RMSE remain stable under standard models, but bivariate models sometimes provide higher RMSE and reduced coverage, particularly in smaller samples. For effect sizes, ${\mathrm{\text{β}}}_1=0.3$ shows consistently low bias and significant reductions in RMSE as sample size increases, with coverage remaining well-maintained, though slightly reduced for bivariate models under smaller sample sizes or higher ${\rho }_j$. For ${\mathrm{\text{β}}}_2=0.8$, similar trends are observed, though RMSE is generally higher, especially for smaller sample sizes or greater ${\rho }_j$ and ${\rho }_k$.

Table 3.

Treatment effect mean, bias, root mean square error, and coverage probability of mean estimate for effects of treatment on the joint outcomes for varying sample size and correlation (True ${\mathrm{\text{β}}}_{11}=0.8$, ${\mathrm{\text{β}}}_{12}=0.3$).

Size	Method	${\rho }_k$	${\rho }_j$	${\widehat{\mathrm{\text{β}}}}_1$	Bias1	RMSE1	Cov1	${\widehat{\mathrm{\text{β}}}}_2$	Bias2	RMSE2	Cov2
50	Standard	0.1	0	0.3089	0.0089	0.2578	0.94	0.8065	0.0065	0.4089	0.93
			0.3	0.3071	0.0071	0.2627	0.93	0.8081	0.0081	0.4069	0.94
			0.75	0.3068	0.0068	0.2701	0.93	0.8083	0.0083	0.4048	0.94
	Joint		0	0.052	0.248	0.7242	0.94	0.5556	0.244	0.8376	0.93
			0.3	0.3904	0.0904	0.5026	0.92	0.8719	0.0719	0.6333	0.91
			0.75	0.3251	0.0251	0.3341	0.89	0.8161	0.0161	0.4797	0.89
200	Standard	0.1	0	0.3105	0.0105	0.1315	0.95	0.8289	0.0289	0.196	0.94
			0.3	0.3158	0.0158	0.1311	0.94	0.8299	0.0299	0.197	0.94
			0.75	0.3211	0.0211	0.1297	0.94	0.8301	0.0301	0.1968	0.94
	Joint		0	0.2952	0.0048	0.272	0.95	0.7969	0.0031	0.3301	0.95
			0.3	0.3119	0.0119	0.1931	0.94	0.8219	0.0219	0.2657	0.94
			0.75	0.3208	0.0208	0.1542	0.93	0.8253	0.0253	0.2246	0.93
500	Standard	0.1	0	0.3134	0.0134	0.0854	0.94	0.8163	0.0163	0.127	0.94
			0.3	0.3134	0.0143	0.0848	0.94	0.8167	0.0167	0.1263	0.93
			0.75	0.3152	0.0152	0.0853	0.94	0.8166	0.0166	0.126	0.93
	Joint		0	0.3262	0.0262	0.1651	0.96	0.8221	0.0221	0.2018	0.95
			0.3	0.3177	0.0177	0.1002	0.94	0.8224	0.0224	0.1483	0.93
			0.75	0.3189	0.0189	0.1006	0.93	0.8215	0.0215	0.1467	0.93
1000	Standard	0.1	0	0.3124	0.0124	0.0606	0.95	0.8177	0.0177	0.089	0.94
			0.3	0.314	0.014	0.0614	0.94	0.8185	0.0185	0.0876	0.94
			0.75	0.3159	0.0159	0.0617	0.94	0.8188	0.0188	0.0874	0.94
	Joint		0	0.3324	0.0324	0.1066	0.96	0.8404	0.0404	0.1368	0.95
			0.3	0.3152	0.0152	0.0716	0.94	0.8211	0.0211	0.1017	0.94
			0.75	0.3175	0.0175	0.0706	0.93	0.8212	0.0212	0.1012	0.94
50	Standard	0.4	0	0.304	0.004	0.2925	0.94	0.7979	0.0021	0.4723	0.93
			0.3	0.3049	0.0049	0.2739	0.93	0.8017	0.0017	0.4363	0.93
			0.75	0.3048	0.0048	0.2681	0.92	0.8045	0.0045	0.4071	0.93
	Joint		0	0.3155	0.0155	0.3557	0.92	0.7957	0.0043	0.52	0.9
			0.3	0.3029	0.0029	0.3599	0.92	0.7865	0.0135	0.5177	0.91
			0.75	0.3913	0.0913	0.5744	0.93	0.8637	0.0637	0.6785	0.92
200	Standard	0.4	0	0.3028	0.0028	0.1522	0.94	0.8196	0.0196	0.2241	0.95
			0.3	0.3123	0.0123	0.141	0.94	0.8249	0.0249	0.2083	0.94
			0.75	0.3207	0.0207	0.1315	0.94	0.8294	0.0294	0.1972	0.94
	Joint		0	0.3079	0.0079	0.177	0.92	0.8142	0.0142	0.2657	0.94
			0.3	0.313	0.013	0.1672	0.93	0.8188	0.0188	0.2402	0.94
			0.75	0.234	0.066	0.3914	0.96	0.7219	0.0781	0.4272	0.95
500	Standard	0.4	0	0.3037	0.0037	0.0958	0.95	0.8103	0.0103	0.1456	0.95
			0.3	0.3107	0.0107	0.0909	0.94	0.8148	0.0148	0.1344	0.94
			0.75	0.316	0.016	0.0861	0.93	0.8178	0.0178	0.1264	0.94
	Joint		0	0.3079	0.0079	0.1089	0.96	0.8171	0.0171	0.1688	0.93
			0.3	0.3157	0.0157	0.1344	0.94	0.8205	0.0205	0.1038	0.93
			0.75	0.188	0.112	0.2981	0.95	0.7007	0.0993	0.3288	0.95
1000	Standard	0.4	0	0.3027	0.0027	0.068	0.95	0.8087	0.0087	0.0998	0.94
			0.3	0.3095	0.0095	0.0637	0.94	0.8144	0.0144	0.0935	0.94
			0.75	0.3158	0.0158	0.0616	0.94	0.819	0.019	0.088	0.94
	Joint		0	0.3048	0.0048	0.0784	0.95	0.8121	0.0121	0.1164	0.94
			0.3	0.3115	0.0115	0.075	0.94	0.8171	0.0171	0.108	0.94
			0.75	0.3699	0.0699	0.1997	0.95	0.8777	0.0777	0.2109	0.95
50	Standard	0.8	0	0.3729	0.0729	0.8277	0.97	0.784	0.016	1.3274	0.97
			0.3	0.286	0.014	0.4137	0.95	0.7707	0.0293	0.6596	0.95
			0.75	0.2983	0.0017	0.2887	0.94	0.7957	0.0043	0.4366	0.93
	Joint		0	0.1943	0.1057	1.7425	0.94	1.1746	0.3746	1.9613	0.94
			0.3	0.2889	0.0111	0.505	0.96	0.7697	0.0303	0.7601	0.96
			0.75	0.321	0.021	0.467	0.91	0.8105	0.0105	0.6312	0.91
200	Standard	0.8	0	0.2555	0.0445	0.2796	0.95	0.7622	0.0378	0.4087	0.96
			0.3	0.29	0.01	0.2076	0.94	0.7951	0.0049	0.3082	0.96
			0.75	0.3157	0.0157	0.1474	0.93	0.822	0.022	0.2204	0.95
	Joint		0	0.2678	0.0322	0.344	0.93	0.7645	0.0355	0.4844	0.96
			0.3	0.2962	0.0038	0.2419	0.93	0.7976	0.0024	0.3502	0.96
			0.75	0.3175	0.0175	0.1658	0.92	0.822	0.022	0.2472	0.94
500	Standard	0.8	0	0.2518	0.0482	0.1672	0.94	0.7621	0.0379	0.2503	0.95
			0.3	0.2876	0.0124	0.1259	0.96	0.7929	0.0071	0.1902	0.95
			0.75	0.3138	0.0138	0.0945	0.94	0.8156	0.0156	0.1385	0.95
	Joint		0	0.2621	0.0379	0.1917	0.96	0.7779	0.0221	0.3103	0.95
			0.3	0.2958	0.0041	0.1407	0.95	0.802	0.002	0.2303	0.95
			0.75	0.319	0.019	0.1109	0.94	0.8208	0.0208	0.1673	0.93
1000	Standard	0.8	0	0.2514	0.0486	0.1215	0.94	0.7567	0.0433	0.1823	0.94
			0.3	0.2854	0.0146	0.0902	0.94	0.7893	0.0107	0.1371	0.94
			0.75	0.3123	0.0123	0.0656	0.93	0.8151	0.0151	0.0986	0.95
	Joint		0	0.2599	0.0401	0.1408	0.94	0.765	0.035	0.2122	0.94
			0.3	0.2905	0.0095	0.1037	0.94	0.7939	0.0061	0.1564	0.95
			0.75	0.3145	0.0145	0.0759	0.93	0.8167	0.0167	0.1124	0.94

Cov1, coverage ( ${\widehat{\mathrm{\text{β}}}}_1$); Cov2, coverage ( ${\widehat{\mathrm{\text{β}}}}_2$),

6. Application: Malawi longitudinal study of families and health

The Malawi Longitudinal Study of Families and Health (MLSFH) is an ongoing panel survey that has been conducted bi-annually since 1998 in 120 villages in rural areas of three districts; Rumphi, Mchinji, and Balaka in Malawi [5,21,22]. MLSFH has collected socio-economic, fertility, and HIV data for up to 4000 couples in the 120 villages across the three districts. The design and methods of the MLSFH have been described in detail elsewhere [21,22].

We use the MLSFH to estimate the effect of HIV positivity awareness on condom use and multiple sexual partners (j = 1, 2) across the years 2004, 2006, 2008, 2010, and 2012, 2013, 2017, 2018 and 2019 ( $k=0,1,2,3,\dots ,8$). The proposed MSM-IPTWs was fit using a logit link function for the two binary outcomes as a function of HIV positivity awareness and time under the strongly ignorable assumption as follows:

$$\mathrm{logit}\left(\mathrm{Pr}\right(Y_{\mathrm{ijk}}=1|\text{HIV awareness},k\left)\right)={\mathrm{\text{β}}}_{1j}{\text{HIV awareness}}_i+{\mathrm{\text{β}}}_{2j}k$$ (7)

where $Y_{\mathrm{ijk}}$ is the the jth outcome, j = 1 for condom use and j = 2 for two or more partners, of the ith individual at time k. We assumed two Kronecker correlation structures and denoted the serial correlation structure as $R_k\left(\mathit{\alpha }\right)$ and the between-outcome correlation structure as $R_j\left(\mathit{\tau }\right)$. The causal parameters from the marginal model in equation 7 were estimated under the following assumed working correlation structures combination;exchangeable between-outcome and AR(1) serial correlation structures, see Table 7. We fitted the multivariate model considering the inverse propensity treatment weights to control for confounding. We estimated the stabilized weights by first fitting a multivariable logistic regression model for exposure HIV status awareness as a function of all baseline, time-varying, and past outcome confounders for each outcome using the IPW package in R.

Table 7.

HIV status awareness effects on condom use and multiple sexual partners using the joint MSM with IPTW.

	Odds ratio (SE)	95% Conf. Int
Condom use
HIV negativity	1.00
HIV positivity	1.90 (0.09)	(1.59, 2.27)
Multiple partners
HIV negativity	1.00
HIV positivity	0.95 (0.10)	(0.78, 1.16)

Conf. Int., confidence interval.

We assessed the statistical approach in terms of the association with and without stabilized weights for comparison purposes, where the dependence between outcomes is modeled by specifying a working correlation structure. The causal parameter for the multivariate marginal model using the joint generalized estimating equation solver package JGEE in R where both working correlation structures are specified [20]. We modified the JGEE source code to incorporate the stabilized weights in the estimation as initially, this was undefined.

6.1. Standard bivariate logistic and univariate MSM models

Table 4 presents the results of fitting a standard bivariate logistic model to assess the association between HIV positivity awareness and the two outcomes: condom use and multiple sexual partners, without adjusting for time-varying confounding. Table 5 displays the findings from a marginal structural model that estimates the effects of HIV positivity awareness on the separate outcomes. A detailed explanation of the effects in Table 5 can be found in a previous study [37]. In this paper, we compare the results of the univariate marginal structural model with our proposed model.

Table 4.

Estimates for a bivariate logistic regression model without IPTW.

	Odds ratio	95% conf. int	Std. error
Outcome: Condom use
Intercept	0.19	(0.18, 0.21)	0.03
HIV aware (HIV negativity vs HIV positivity)	2.53**	(2.05, 3.11)	0.11
Gender (women vs men)	1.55**	(1.37, 1.76)	0.07
Age (in years)	0.95**	(0.94, 0.96)	0.003
Wealth (poor vs medium/rich)	1.03	(0.91, 1.17)	0.07
Education level (years)	1.32**	(1.09, 1.59)	0.10
Religion (Muslim or other vs Christian)	1.04	(0.91, 1.19)	0.07
Region (North vs Central)	0.67	(0.44, 1.02)	0.22
Region (North vs South)	1.21	(0.80, 1.85)	0.23
Wave	0.87	(0.85, 0.89)	0.01
Outcome: Multiple sexual partners
Intercept	0.25	(0.23, 0.27)	0.04
HIV aware (HIV negativity vs positivity)	1.42**	(1.13, 1.77)	0.12
Gender (women vs men)	4.74**	(4.17, 5.39)	0.07
Age (in years)	0.97**	(0.96, 0.98)	0.002
Wealth (poor vs medium/rich)	0.79**	(0.71, 0.88)	0.05
Education level (years)	0.72**	(0.62, 0.83)	0.08
Religion (Muslim or other vs Christian)	0.85**	(0.76, 0.96)	0.06
Region (North vs Central)	1.31**	(1.09, 1.56)	0.09
Region (North vs South)	1.05	(0.87, 1.26)	0.10
Wave	0.68	(0.66, 0.70)	0.02

Conf. int., confidence interval; Std. error, standard error.** p-value <0.001.

Table 5.

Effect Estimates of awareness of HIV positivity on risky sexual behavior using stabilized weights.

	Odds ratio	95% CI	Std. error
(a) Condom use
HIV negativity	1.00
HIV positivity	2.22	(1.79, 2.75)	0.12
(b) Multiple sexual partners
HIV negativity	1.00
HIV positivity	0.96	(0.83, 1.35)	0.14

i. We controlled for the following confounders:- gender, wealth, age, education, time (wave), religion, comprehensive HIV knowledge, distance to VCT, received an incentive, perceived partners HIV status, likelihood has HIV, region.ii. 95% CI, 95% Confidence Interval.

6.2. Generation of IPTWs for multiple outcomes

The stabilized weights for each outcome j were generated as follows;

$$S{W}_{\mathrm{ijt}}=\prod _{t=0}^8\prod _{j=1}^2\frac{P(A_t=1|{\overline{A}}_{t-1},{\mathbf{\text{X}}}_0)}{P(A_t=1|{\overline{A}}_{t-1},{\overline{\mathbf{\text{X}}}}_t,{\mathbf{\text{X}}}_0,{\mathbf{\text{Y}}}_{j,t-1})}$$

To account for drop-out (refused to participate, dead, moved out or not eligible) we generated the dropout weights

$$C{W}_{\mathrm{it}}=\prod _{t=0}^8\frac{P({\mathrm{Censored}}_t=0|{\overline{\mathrm{Censored}}}_{t-1}=0,{\overline{A}}_t,{\mathbf{\text{X}}}_0)}{P({\mathrm{Censored}}_t=0|{\overline{\mathrm{Censored}}}_{t-1}=0,{\overline{A}}_t,{\overline{\mathbf{\text{X}}}}_t,{\mathbf{\text{X}}}_0}$$

The final weight $S{W}_i=S{W}_{\mathrm{ijt}}\times C{W}_{\mathrm{it}}$

From Table 6, we observe that the final weights have mean of 1 and are less variable, implying stability. These weights were used to fit the proposed bivariate MSM-IPTW.

Table 6.

Distribution of inverse probability treatment weights.

	Mean (SD)	Median	Interquartile range	Min	Max
Stabilized	1.01 (0.13)	1.01	(0.99, 1.01)	0.1	2.60
LTFU weights	0.99 (0.07)	1.00	(0.99, 1.01)	0.33	1.85
Final weights	1.00 (0.14)	1.00	(0.99, 1.01)	0.1	2.53

6.3. Estimation of HIV-awareness effects using the proposed joint MSM-IPTW

Table 7 presents estimates for the proposed MSM with inverse probability treatment weights to control for confounding. We observe from Table 7 a positive effect of HIV positivity awareness on condom use. Individuals aware of their HIV positivity were more likely to use condoms compared to those aware of their HIV negativity, similar to findings from other studies [10,28,37]. There was no significant causal association between awareness of HIV positivity and multiple sexual partners.

The HIV-positivity awareness effects obtained using the proposed bivariate MSM-IPTW differed significantly from those obtained from the standard bivariate model. The bivariate MSM-IPTW model produced smaller standard errors, although the estimates themselves were relatively similar, particularly for the HIV-positivity awareness effect on multiple sexual partners. Furthermore, comparing the causal estimates from univariate methods (see Table 4) to the proposed bivariate methods revealed that the proposed method achieved smaller standard errors, resulting in a more significant causal association compared to the univariate approaches [37].

6.4. Sensitivity analysis

The causal associations for the proposed methods after considering the stabilized weights presented in Table 7 have been obtained under the assumption of no unmeasured confounders. Vanderweele et al. [39] suggested the use of the E-value approach when dealing with multiple outcomes that are repeatedly measured. The E-value is the minimum strength of association that an unobserved confounder would have with both the exposure and outcome conditional on observed measured confounders that would minimize the significance of the observed causal effect. Thus, we assessed the robustness of the estimated causal associations from unmeasured confounding using the E-value statistics for an estimated odds ratio. Table 8 presented the E-value statistics for the point estimates of the causal associations when the within-subject correlation was assumed to be AR(1), and the between-outcome correlation was assumed to be exchangeable as shown in Table 7. We observed that the estimated causal effect for condom use was more robust to unmeasured confounding than the estimates for multiple sexual partners.

Table 8.

E-value to assess robustness to unmeasured confounding for the causal effect of HIV status awareness on condom use and multiple partners.

Outcome	E-value for point estimate	E-value for LL of CI
Condom use
HIV positivity	2.16	1.85
Multiple partners
HIV positivity	1.48	1.00

LL, lower limit; CI, confidence interval. E-value parameters calculated as a function of the adjusted odds ratio (OR) for the effect of HIV positivity awareness on the outcomes.

7. Discussion

This study aimed to develop causal inference methods for longitudinal data with multiple outcomes, using marginal structural models with inverse probability of treatment weights. We evaluated these methods through a simulation study that varied sample sizes, correlation strengths, and effect sizes. Additionally, we applied the methods to data from the Malawi Longitudinal Study of Families and Health (MLSFH) to estimate the impact of HIV positivity awareness on condom use and the number of sexual partners.

The simulation study showed that our proposed model estimated treatment effects more accurately than a joint covariate-adjusted model, particularly for larger sample sizes and strong correlations. When outcomes share a common treatment effect, our method proved effective, maintaining high coverage for confidence intervals. These findings align with existing literature, which indicates that treatment effect estimation is more efficient with appropriate joint models that control for confounders [1,27]. In the MLSFH data, awareness of HIV positivity was associated with increased condom use and fewer sexual partners [2,3,10,37]. The estimated effects were more precise compared to those derived from standard bivariate and univariate models, suggesting that our methods better adjusted for confounding and outcome dependencies.

The differences between the bivariate logistic model and our proposed methods highlight the importance of correctly adjusting for confounders and dependencies between outcomes. Standard covariate adjustments in longitudinal data may overestimate treatment effects by blocking indirect effects of time-varying exposures [18,32,33,39]. While marginal structural models (MSMs) for univariate data often do not account for correlations between multiple outcomes, our approach incorporates the dependency structure into the MSM-IPTW framework to jointly analyze multiple outcomes.

However, limitations include the assumption that correlated outcomes are influenced by the same confounders and that the simulation study considered only two continuous outcomes, leaving out other distributional forms like binary, count or ordinal data. In our simulations, there is a possibility that controlling for a few confounders may have resulted in extreme weights in the inverse probability weighting (IPW) model. These extreme weights can reduce efficiency and stability, inflating variance and confidence intervals, especially in smaller samples [6,8]. While the standard bivariate model adjusts directly for covariates, reducing variability from weights and providing greater stability in such scenarios. However, the multivariate IPW approach is advantageous in managing complex exposure-confounder relationships and producing less biased estimates in cases of time-varying confounding. We recommend using marginal structural models that account for dependency structures and time-varying treatments to estimate causal effects in longitudinal studies with multiple outcomes. Future studies could evaluate the stability of weights, consider trimming extreme weights, and ensure the correct specification of the treatment model during simulations to enhance the validity of the proposed estimator. Additionally, further research should address missing data and explore alternative balancing methods, such as G-estimation techniques and instrumental variable approaches, to improve confounding control.

Funding Statement

This work was supported through the DELTAS Africa Initiative a scheme of the African Academy of Sciences (AAS), Alliance for Accelerating Excellence in Science in Africa (AESA) and supported by the New Partnership for Africa's Development Planning and Coordinating Agency (NEPAD Agency), Grant No. 107754/Z/15/Z-DELTAS Africa SSACAB, with funding from the Wellcome Trust and the UK government. This work was partly funded by the L'Oreal-UNESCO for Women in Science Sub-Saharan Africa Young Talents Programme through the Young Talents award provided to Ms. Twabi. Part funding was obtained from the South African Medical Research Council (SAMRC) supporting Ms. Twabi's research visits to SAMRC and Samuel Manda's time. The views expressed in this publication are those of the author(s) and not those of the funders (AAS, NEPAD Agency, Wellcome Trust or the UK government, L'Oreal, SAMRC).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Ethical approval and consent to participate

The data collections of the MLSFH have been approved by the IRB Board at the University of Pennsylvania (IRB Protocols no. 815 016 and no. 826828), and in Malawi, the MLSFH research has been approved by the Ethics Committee of the College of Medicine, Malawi (COMREC, Protocols no. P01/12/1165 and no. P.04/17/2160) and the National Health Sciences Research Committee (NHSRC, Protocol no. 19/01/2214).

Author contributions

HST read the relevant literature, formulated the proposed methods, conducted the analysis and wrote the drafts of the manuscript. SOMM conceptualized and suggested the direction for this paper, reviewed the statistical analysis and assisted with the revision of the manuscript. DSS helped with the concept, direction and revisions. HPK helped with providing the substantive concept of the paper and the revisions of the manuscript. All authors have read and approved the final manuscript.

Data availability statement

The data is available upon request from the Principal investigator of the MLSFH that can be done through email: hpkohler@pop.upenn.edu.