Abstract
Survival analysis has seen in recent times more of joint modelling of longitudinal and survival data using approximate hierarchical Bayesian method. This study modelled jointly multivariate and cure proportion with spatial variation using latent Gaussian models (LGMs). Spline specification was used to capture nonlinear trajectories of the multivariate longitudinal biomarkers in the LGM paradigm. Spatial dependence was assumed for neighbouring locations with an improper multivariate conditionally autoregressive prior on the spatial random effects, while inverse-Wishart prior was assumed for the covariance matrix of the random effects and Gaussian priors for the fixed effects. The penalised complexity prior was assumed for the Weibull shape parameters of the baseline hazard function for the event-time and binomial distribution for latent cure classification variable. Posterior distributions were evaluated using INLA. The study was applied to longitudinal and survival HIV/AIDS patients datasets. The random spatial variability was seen to be significant for both biomarkers, random intercept for BMI and cure probabilities. The full conditional distribution of latent variable gave a predicted cure proportion of 61%. The susceptible group on the hand, had a full conditional distribution of latent incidence variable predicting susceptible proportion of 39%. Flexibility of LGM for Bayesian hierarchical was demonstrated easily for multivariate joint model with spatial cure proportion.
Keywords: Spatial, cure proportion; Laplace approximation; Multivariate; Shared random effect; Nonlinear trajectory
Subject terms: Computational biology and bioinformatics, Diseases, Mathematics and computing, Medical research, Risk factors
Introduction
Classical survival analysis models such as Cox proportional hazard (PH) model1 and accelerated failure time (AFT)2 model are based on the assumption that given enough follow-up time, every individual will eventually encounter the event otherwise be censored within experiment or follow-up period. Usually in most real settings, it is possible that some individuals may not encounter the event, even with a follow-up time of several decades. For example, in HIV/AIDS trials, availability of effective treatments, has made it possible that patients can live long, healthy lives and eventually die as a result of old age or other age-related ailments. Based on this, one cannot say the patients have experience the event from HIV/AIDS, hence it can be said that such patients are “cured”. In the case of possibility of cured patients, the population is assumed to be made up of two groups: the susceptible group who will one day encounter the event, and the cured group who may not encounter the event in a long time. Cancer trials are also cases where there is possibility of cured patients when cancer is removed as a result of treatment, especially if the cancer was detected at early stages3. Identifiability has been noted to be one of the problems with cure models as a result of lack of censoring before the close of the follow-up period. So the influence of covariates on failure time is confounded by that on mixture probability, resulting in unstable estimates. A plausible remedy is including a time-dependent information that may have some association with the survival process4. This association may be derived from longitudinal measurements taken simultaneously as the survival measurements.
Joint modeling has emerged as a widely used method for data analysis in many applicable contexts where time-to-event and longitudinal outcomes are simultaneously observed. A thorough assessment of the literature on the application of joint models including multiple event times per subject was provided by Hickey et al.5. A survey of advancements in Bayesian joint models, encompassing publications published to July 2019, was presented by Alsefri et al.6. Even with a frequentist approach, joint modeling with spatial components is uncommon, according to the literature. According to the literature, Martins et al.7,8 and Rappl et al.9 dealt with Bayesian joint modeling with a spatial component. Martins et al.8 also included a survival model with spatial long-term survivors, and they evaluated the posterior distribution using Markov chain Monte Carlo (MCMC). The piecewise additive strategy employed by Rappl et al.9 had the drawback of requiring data augmentation, which is a step in the time-to-event process. The dataset is artificially enlarged by this augmentation, which results in lengthier or impractical runtimes when the original data is huge. Due to the huge number of parameters that required to be estimated, the delayed convergence of Markov chain Monte Carlo (MCMC) methods was a limitation of the Bayesian strategy utilized by Martins et al.7,8. Martins et al. 8 came to the conclusion that their modeling was computationally intensive and time-consuming, which suggested that fewer models could be easily fitted, hence decreasing the statistical analysis’s flexibility.
Notwithstanding the computational burden, a large number of publications on Bayesian joint modeling use MCMC sampling-based inference, which has been demonstrated to be restricted to relatively small samples and model specifications for joint modeling10. Rue et al.11 introduced the approximate Bayesian approach, the integrated nested Laplace approximation (INLA), which is gradually becoming more popular as an alternative to MCMC for joint modeling; see10,12–19. In contrast to the joint posterior distribution of the model parameters as assessed in MCMC, the INLA method concentrates on estimating the posterior marginals of the model parameters20. Among INLA’s advantages are its simplicity in implementing hierarchical models and its quick computation times for fitting various models21,22. This paper expanded on the joint model in Ekong et al.18 to take into consideration spatial variation and nonlinear multivariate longitudinal trajectories.
Materials and methods
Longitudinal and survival HIV/AIDS patients’ dataset
Given that the i-th patient (i = 1, …, N) from the k-th geographic location having nk patients, has sample observations from the p-th longitudinal variable as yipk (p = 1 for CD4 count and 2 for BMI for the motivating dataset). Let Tik be the observed event time which might be right censored. At the end of the follow-up, let Zik be the latent variable that determines whether the patient had been cured or not. The event indicator is presented as δik = 1 if the event is witnessed and δik = 0 if it is censored. We note that any patient whose survival time is monitored at a specific moment is included in the group of patients who have not been cured.
The observed data for the i-th patient without any covariate is Dik = { yipk, Tik, δik, Zik }. The Dik ‘s are assumed to be independent across patients, i.e., the disease process evolves independently for each patient. We also assume that Tik and yipk are conditionally independent given the independent variables and a set of unobserved patient-level and location-level random effects.
The dataset used in this study was a secondary, retrospective dataset from a PhD study by investigator, Ogunmiluyi23, on the effect of ready-to-eat-therapeutic-snack on the nutritional status of children living with HIV/AIDS, carried out in 2023 in Abeokuta, Ogun state, Nigeria. The study obtained informed consent from parents and/or guardians of participating children and interventions, with data collection followed relevant guidelines and regulations. Protocol approval was obtained from the licensing committee - the Health Research Ethical Committee (HREC) of Federal Medical Centre Abeokuta, Ogun state, Nigeria. The longitudinal biomarkers used for this paper included, CD4 counts and body mass index (BMI), including other covariates such as gender, age and local government area (LGA) of residence.
Longitudinal submodel
Given pth longitudinal variable observation yip assumed to be are conditionally independent and follow a well-defined distribution, G, with some density function 
, linear predictor ηL and hyperparameters θL, hence a structured additive model for the multivariate longitudinal component is given as19
 |
1 |
where the longitudinal trajectories are defined by spline functions in order to account for potential nonlinear routes and are given by the p-th latent random effect 
of covariate 
for i-th patients. 
is the total number of observations on the p-th biomarker and 
is the number of biomakers. The unstructured random effects for the i-th patient with p-th biomarker is denoted by 
, and these captures the independent Gaussian measurement errors for the each p biomarkers. It is assumed that each of 
for all p, is independently identically distributed normally with mean zero and inverse variance (precision) 
. The fixed effects of the covariates X are represented by β. Assuming that the biomarkers are multivariate normally distributed, 
is the vector of random effects of intercept measuring the correlation of the biomarkers10 within-patient, i.e.,
 |
Accounting for the dependence between the different longitudinal biomarkers using 
, leads us to take the covariance between different biomarkers to be equal to 
for 
. This latent process subsequently links the separate longitudinal biomarker models with the survival submodel through association parameters.
Due to the hierarchical nature of the data, the model was also to account for a fixed effect and random intercept component for each patient, taking into consideration the association between the longitudinal biomarkers. This is as a result of the fact that each patient has their own baseline biomarker measurement. Hence, the overall fixed effect and the variance in the baseline longitudinal measurements across patients will be estimated.
Cure proportion submodel with Spatial variation
The cure random variable, Zik = 0 if the patient is prone to or will eventually experience the event of interest and Zik = 1 if the patient is cured or will not experience the event in the farthest distant time, given the observed event time Tik. The cure and uncured probability are P(Zik = 1) = πk and P(Zik = 0) = 1 - πk, respectively, because we additionally assume a location-specific cure proportion, as stated by Martins et al.8. In addition to assuming that πk = 1, …, K are mutually independent, the assumed common cure fraction for the patients residing in the k-th location is 1 - πk8. The survival functions for patients in the cured and uncured population, Sc(t) and Su(t), t > 0, respectively, are
 |
2 |
A combination of cured and uncured populations can be used to represent the general survival function for 
as the sum of the cure probability and the product of the uncured probability with uncured population survival function:
 |
3 |
The incidence model is another name for the location-specific cure fraction πk, and the latency model is another name for the event time Tik
in the uncured population3.
Covariates in the incidence model.
The spatial effects are assumed to follow an intrinsic conditionally autoregressive (ICAR) prior24, 
, with patients in the same geographic location sharing a common spatial effect 
, which is assumed to have a spatial dependence at neighbouring locations25. The spatially autocorrelated random effects in event time data across nearby locations are taken into consideration by this prior.
The effect of a baseline covariate vector x1 on the cure proportion is usually modeled using a logistic link function given below. For a patient who has experienced the event (i.e., δik = 1), we know Zik = 1, but for a censored patient (i.e., δik = 0), we do not observe Zik.
 |
4 |
where 
is the vector of regression coefficients associated to x1 and π is the cure proportion.
We assume as in Martins et al.8 a multivariate normal for the ICAR parameter
 |
where 
is the set of labels of the neighbours of the kth region and 
is its total number of neighbours; 
the vector of all spatial frailties except 
. Also, 
is assumed to circumvent identifiability issues. The assumption of spatial frailties across location is made, however, the variance, 
, was of interest to assess the location differences in the survival of patients.
Covariates in the latency
It is assumed that the time to incident for patients with Zik = 1 follows a parametric distribution. The hazard function for the event time is typically formulated using the Cox proportional hazards model. The association parameter 
links the longitudinal component to the survival model’s latency portion as
 |
5 |
The joint model is mainly formulated based on the shared effects from both submodels, hence the linear predictor for the of the joint model from the longitudinal and survival submodels becomes,
 |
6 |
The Cox proportional hazard for the survival time follows a Weibull baseline hazard function 
with 
and 
as the scale and shape parameters respectively. 
is the association parameter estimating the strength of association between the survival and the p-th longitudinal component.
Likelihood function of joint model
Typically, the likelihood function for a joint model is a product of the likelihoods of the individual longitudinal and survival submodels, conditional on shared random effects. Multivariate integral in the likelihood function of joint models are computationally intensive, and this study employs the integrated nested Laplace approximation for the computations, where the joint model is formulated as a Latent Gaussian Model (LGM), where both fixed and random effects are treated as a large latent Gaussian field (LGF). The first likelihood of the multivariate longitudinal outcomes given the parameters 
can be given as
 |
7 |
Given survival observations 
and parameter vector 
, the likelihood for the mixture cure survival also is given as
 |
8 |
where 
defines the effects on the cure proportion including the spatial effects and the effect on the latency model including the association effects on the longitudinal biomarkers. These components incorporates the joint effects in Eq. (6) by means of the association effects terms in the likelihood of the mixture cure survival.
The complete likelihood becomes
 |
9 |
where we define the latent field 
, a vector of hyperparameters 
and D is the complete data of the survival and longitudinal measurement.
The latent field χ which is assumed to have a multivariate Gaussian prior with zero mean, such that it forms a Gaussian Markov random field with sparse precision matrix matrix 
, i.e. 
, this is given as
 |
Then at the final level of the hierarchy, a prior on the hyperparameter vector 
can then be formulated for the set of hyperparameters 
, which could be non-normal. This enables us to assume normal prior for the vector of fixed effects for the pth longitudinal variable as 
, where 
and 
. To complete the model specification, we assume the inverse-Wishart prior distribution for the covariance matrix of the random effects, while the penalised complexity prior 
is assumed for the Weibull shape parameters of the baseline hazard function.
Posterior estimation using integrated laplace approximation
From this hierarchical Bayesian formulation as an LGM, the joint posterior distribution is then given as:
 |
 |
10 |
Within this framework the joint posterior density (10) and subsequently the marginal posterior densities, 
; i = 1, …, n and 
can be efficiently and accurately calculated using the integrated Laplace approximation (INLA) methodology developed by Rue et al.7. INLA approximates marginal posterior distributions of θ and 
, instead of their joint posterior distributions, for which θ is required to obtain 
7. The marginal posterior densities of 
becomes
 |
11 |
and the marginal posterior of the j-th hyperparameter is
 |
12 |
First, 
from (12) is evaluated using Laplace approximation as
 |
13 |
based on the second-order Taylor expansion of log 
centered at the maximum a posteriori estimate. 
is given as
 |
14 |
where 
. The approximation locates the posterior mode 
, taking it as the mean of the Gaussian approximation26. The Gaussian approximation of 
, given as 
is obtained by matching the mode and the curvature at the mode of 
. The mode is computed iteratively by using a Newton Raphson method.
The Gaussian approximation of 
, given as 
is obtained by matching the mode and the curvature at the mode of 
. The mode is computed iteratively by using a Newton Raphson method. Let 
be the initial value of the mode, and expand 
around 
to the second order Taylor expansion,
 |
15 |
where 
and 
depend on 
Substituting Eq. (15) into Eq. (14) yields
 |
16 |
 |
The Gaussian approximation of 
is obtained, with the precision matrix 
and mode 
, which is the solution of 
. The process can then be iterated, with 
as the new starting value, until it converges to a Gaussian distribution with, say, mean 
and precision matrix 
where an appropriate convergence criterion must be used.
If the mean of 
is 
, the density of 
is
 |
17 |
The sparse matrix 
is factorised as Cholesky triangle product 
, and only non-zero terms are computed due to the Markov property and 
. Let 
where 
, then we have that 
for 
. Multiplying each side with 
and taking the expectation yields the recursion
 |
where 
is the covariance matrix and 
if 
and 
otherwise. These recursion results in Gaussian approximations 
with mean 
and marginal variance 
.
The resulting approximation will then be27:
 |
18 |
where c is the second-order term in the Taylor expansion of 
at modal value 
.
The Gaussian approximation is generally not best if the true density of 
is not symmetric, the full Laplace approximation is a correction of Gaussian approximation and accurate but at a very expensive computational cost, the simplified Laplace approximation which is based on the Taylor series expansion of the full Laplace approximation is sufficiently accurate for most applications28.
Laplace approximation
Laplace approximation is obtained from Gaussian approximation to 
, 
. Laplace approximation is given as
 |
19 |
Rue et al.11 noted that 
is recomputed for each value of 
since its precision matrix depends on 
, and they proposed two modifications. The first avoids the optimisation step in computing 
by approximating the modal configuration,
 |
The second modification uses, for some 
when 
,
 |
The idea is to compute a set around 
, 
, where only 
s for which the dependence between 
and 
decay as the distance between nodes 
and 
increases determine the marginal of 
, hence 
.
The density 
is then represented by
 |
20 |
The cubic spline is fitted to the difference of the log-density of 
and 
at the selected abscissa points, and then the density is normalized by using quadrature integration11.
Equation (13) becomes
 |
21 |
with 
being the Laplace approximation of 
, 
is the Gaussian approximation to the full conditional distribution 
with posterior mode 
, called maximum a-posteriori estimator (MAPE) which is obtained by taking 
proportional to the likelihood function 
, weighted by the Gaussian prior distribution 
and the hyperprior distribution
using Bayes’ theorem26.
Next is the evaluation of (11), the marginal conditional posterior distribution, 
, needed to compute the marginal posterior of 
, 
, is sought using simplified full Laplace approximation11. It involves initially rewriting the latent field parameter vector as 
, 
) and uses Gaussian approximation to obtain 
. Using the conditional probability rule, 
can be expressed as
 |
22 |
from which the following the following can be obtained
 |
 |
 |
 |
23 |
where 
is the Gaussian approximation of 
evaluated at its mode 
.
Having obtained 
and 
, the marginal posterior distribution 
in Eq. (11) is approximated by
 |
24 |
with the integral numerically given via a finite weighted sum as
 |
25 |
for a set of integration points 
with a corresponding weight 
, with 
denoting the 
th grid and 
denoting the number of evaluation grid points11. The marginal posterior densities of 
, the vector of parameters of interest, elucidates approximate summary statistics, posterior mean, posterior standard deviation and posterior credible intervals26.
Under the Bayesian approach, the parameter of interest is a random variable with probability distribution allowing for probabilistic statements28. The credible interval for the parameter of interest is given as the quantiles 
and 
such that the parameter lies in the interval with 100(1 – α)% posterior probability and is such that
 |
The quantiles 
and 
can be obtained through quantile function of the posterior distribution
 |
 |
where 
. The quantile function 
is an inverse of the cumulative density function 
of the posterior distribution.
Application to HIV/AIDS patients’ dataset
We apply the model with longitudinal biomarkers are CD4 count and BMI, we used the square root transformation on the CD4 count for the modelling. As earlier stated, due to the hierarchical nature of the data, the model included a random intercept term to account for fixed effects and a random intercept for each patient. Hence, the terms 
and 
for CD4 count and BMI were modelled in the additive structure of Eq. (1). The random intercepts estimates the degree of association between the CD4 count and BMI, via the covariance. The models below have rut (snacks treatment with two levels, TS, those who received the treatment and No TS, those who did not receive the treatment), sex (male and female) and age (2.0-4.0yrs, 4.1-6.0yrs, 6.1-8.0yrs, 8.1-10yrs) as fixed effect factors, time is the random effect variable.
Longitudinal component with spline trajectories:
 |
 |
Latency model component:
 |
Incidence model component:
 |
where 
are independent Gaussian measurement errors for the longitudinal outcomes, 
and 
are natural cubic spline basis functions with internal knots at 2 points year 1 and 4. For priors specification, as noted earlier, we assume a multivariate Gaussian prior for the latent field 
with precision matrix 
conditioned on 
, which we also assume prior distributions 
for which all the regression coefficients and the Weibull log(λ) scale parameters follow a vague normal distribution centred at zero (
) while the shape parameter, α, is assumed to follow the penalised complexity prior PC(5), α~ PC(λ=5), 
, 
and the inverse-Wishart prior distribution is assumed for the covariance matrix of the random effects 
and Gaussian priors for all the fixed effects. Analyses were carried out the R programming language29, version 4.3.3 (2024-02-29 ucrt). Posterior statistics were estimated using R packages INLA11,21, INLAMSM20 and INLAjoint10, while data wrangling was done with tidyverse30 package.
Results
From Table 1, the posterior mean, standard deviation, and 95% credible interval (CI) for the multivariate spatial joint model on the motivating dataset are displayed in Tables 1 and 2. Age and sex were found to be important factor influencing the incidence model binary variable Z. Compared to male patients, female patients had a significantly higher chance of being cured (exp(1.419) = 4.133, with a 95% CI of (exp(-0.762) = 0.467, exp(3.618) = 37.263). Compared to other age groups, patients aged 2 to 4 had a considerably higher chance of being cured (exp(1.218) = 3.380, with a 95% confidence interval of (exp(-2.888) = 0.056, exp(1.889) = 6.606). Because each age group level was significant and the probability of the failure event was higher for the higher age group levels due to the positive sign in the coefficients, the latency model demonstrated that the failure event is dependent on the age group level. In contrast, the probability of the failure event was lower for the lower age group levels.
Table 1.
Fixed effects posterior mean and standard deviation of therapeutic snack consumption data.
| Mean | sd | Lower 95% CI | Upper 95% CI | Mean | sd | Lower 95% CI | Upper 95% CI | ||
|---|---|---|---|---|---|---|---|---|---|
| Fixed effects (CD4 biomarker) | Latency outcome | ||||||||
| β10 | 16.9058 | 1.0168 | 14.9129 | 18.8987 | α | 1.0384 | 0.0254 | 0.9900 | 1.0902 |
| β1time | 0.3857 | 2.0160 | -3.5655 | 4.3369 | λ | 0.0049 | 0.0013 | 0.0028 | 0.0078 |
| β1ruts | -0.2374 | 0.2525 | -0.5423 | 0.4475 | β3ruts | -1.8278 | 0.4227 | -2.6563 | -0.9994 |
| β1sex | 0.7076 | 0.2554 | 0.2070 | 1.2082 | β3sex | -0.5881 | 0.3243 | -1.2238 | 0.0477 |
| β1age2 | 0.6425 | 0.2858 | 0.0824 | 1.2025 | β3age2 | -0.2046 | 0.3256 | -0.8428 | 0.4336 |
| β1age3 | -1.0760 | 0.4595 | -1.9767 | -0.1753 | β3age3 | 0.5562 | 0.4467 | -0.3192 | 1.4317 |
| β1age4 | -0.1723 | 0.9486 | -2.0315 | 1.6869 | β3age4 | 0.5842 | 0.7347 | -0.8558 | 2.0241 |
| σe1 | 29.5569 | 0.8186 | 27.9586 | 31.1748 | Incidence outcome | ||||
| Fixed effects (BMI biomarker) | β40 | -0.4490 | 1.1240 | -2.6600 | 1.7500 | ||||
| β20 | 13.9007 | 0.8808 | 12.1744 | 15.6270 | β4ruts | 0.0580 | 1.0430 | -1.9960 | 2.0960 |
| β2time | 6.2152 | 1.7598 | 2.7662 | 9.6643 | β4sex | 1.4190 | 1.1160 | -0.7620 | 3.6180 |
| β2ruts | 0.1776 | 0.1609 | -0.1377 | 0.4929 | β4age2 | -0.5080 | 1.2180 | -2.8890 | 1.8880 |
| β2sex | -0.2186 | 0.1616 | -0.5354 | 0.0982 | β4age3 | 0.0000 | 31.6230 | -62.0090 | 62.0090 |
| β2age2 | -0.3105 | 0.1877 | -0.6783 | 0.0574 | β4age4 | 0.0000 | 31.6230 | -62.0090 | 62.0090 |
| β2age3 | -0.4708 | 0.2929 | -1.0449 | 0.1033 | Association parameters | ||||
| β2age4 | 1.2153 | 0.6677 | -0.0933 | 2.5239 | γcd4_S | -0.2856 | 4.8630 | -9.5379 | 9.6096 |
| σe2 | 2.5119 | 0.0780 | 2.3618 | 2.6682 | γbmi_S | 0.0872 | 4.8646 | -9.4960 | 9.6578 |
Table 2.
Random effects posterior mean and standard deviation of therapeutic snack consumption data.
| Mean | sd | Lower 95% CI | Upper 95% CI | |
|---|---|---|---|---|
| Random effects | ||||
| σ2b1 | 0.0275 | 0.0039 | 0.0224 | 0.0371 |
| σ2τ1 | 0.1829 | 0.1312 | 0.0550 | 0.5035 |
| σ2b2 | 0.1776 | 0.1203 | 0.0564 | 0.4947 |
| σ2τ2 | 0.1653 | 0.0995 | 0.0595 | 0.4202 |
| covb1,τ1 | 0.0000 | 0.0181 | -0.0367 | 0.0392 |
| covb1,b2 | 0.0004 | 0.0172 | -0.0337 | 0.0359 |
| covb1,τ2 | 0.0002 | 0.0168 | -0.0357 | 0.0359 |
| covτ1,b2 | 0.0022 | 0.0840 | -0.1698 | 0.1882 |
| covτ1,τ2 | -0.0008 | 0.0832 | -0.1688 | 0.1721 |
| covb2,τ2 | 0.0006 | 0.0847 | -0.1744 | 0.1754 |
| Spatial effect | ||||
|
340.23 | 54.12 | 10.271 | 522.18 |
For the fixed effects of the biomarkers, the longitudinal trajectory was significant in the measurement of BMI than CD4 counts. The treatment effects was significant for both, however the negative sign in the posterior mean for CD4 count showed that patients in the intervention group had higher CD4 count. While the reverse was the case for patients’ BMI. The random effects of intercept measuring the within-patient correlation of the BMI was significant and that of Cd4 was not significant, as shown in Table 2. Also, the associations between the mixture survival component and CD4 profile was negative (-0.2856) and significant, but positive for BMI (0.0872) and not significant. The random intercept covariance of CD4 and BMI was very low, indicating a lack of correlation between these two biomarkers according to the dataset used. However, CD4 longitudinal measurements had the stronger association with the survival process than the BMI measurements as earlier noted.
Whereas Z, that defines the binary outcome of a patient falling into either cured or susceptible group, is a latent variable, this study was interested in the fitted posterior cure proportion of the full conditional distribution of the latent variable Z. The cure proportion evaluated by the joint modelling approach using integrated Laplace approximation, were obtained for the different factor-level combinations of the factors in the application data, and the posterior mean and standard deviations were presented in Tables 3 and 4, respectively, for the fitted cured and susceptible groups. Regardless of the treatment group they were in, female patients between the ages of 2 and 8 had higher cure probability for the cure group in Table 3. Older patients (those aged 8 and beyond) had significantly lower cure probabilities than male patients, while male patients had lower cure probabilities than female patients. Therefore, whether or not patients received the therapeutic snack (TS), their age and sex were the most important covariates to predict cure probabilities. Patients 6years of age and younger had the highest number of male patients with better cure probabilities. There were instances of female patients between the ages of 8 and 10 who had mean cure probabilities greater than 60%. In summary, from Tables 3 and 4, younger female patients had higher predicted cure proportions compared to male patients and older. Also, the cure proportion were lower among the control group, who did not receive the therapeutic snacks, but was higher for the intervention groups. The whole conditional distribution of latent incidence variable Z yielded a predicted cure proportion of 61%.
Table 3.
Posterior means of predicted cure probability of fitted cure group for covariates’ levels combinations.
| Ruts | Sex | Age | Cure prob. | SD |
|---|---|---|---|---|
| No TS | Male | 6.1-8.0yrs | 0.377 | 0.000742 |
| TS | Male | 6.1-8.0yrs | 0.392 | 0.00128 |
| No TS | Male | 4.1-6.0yrs | 0.502 | 0.000688 |
| TS | Male | 4.1-6.0yrs | 0.517 | 0.00472 |
| No TS | Female | 8.1-10yrs | 0.600 | 0.000709 |
| TS | Female | 8.1-10yrs | 0.614 | 0.000113 |
| No TS | Male | 2.0-4.0yrs | 0.627 | 0.00621 |
| TS | Male | 2.0-4.0yrs | 0.641 | 0.001 |
| No TS | Female | 6.1-8.0yrs | 0.714 | 0.00208 |
| TS | Female | 6.1-8.0yrs | 0.726 | 0.00036 |
| No TS | Female | 4.1-6.0yrs | 0.806 | 0.00255 |
| TS | Female | 4.1-6.0yrs | 0.815 | 0.00039 |
| No TS | Female | 2.0-4.0yrs | 0.874 | 0.00356 |
| TS | Female | 2.0-4.0yrs | 0.880 | 0.000518 |
Table 4.
Posterior means of predicted cure probability of fitted susceptible group for covariates’ levels combinations.
| Ruts | Sex | Age | Cure prob. | SD |
|---|---|---|---|---|
| No TS | Male | 8.1-10yrs | 0.267 | 0.00192 |
| No TS | Male | 6.1-8.0yrs | 0.377 | 0.00419 |
| TS | Male | 6.1-8.0yrs | 0.392 | 0.000738 |
| No TS | Male | 4.1-6.0yrs | 0.502 | 0.00455 |
| TS | Male | 4.1-6.0yrs | 0.517 | 0.000675 |
| No TS | Male | 2.0-4.0yrs | 0.627 | 0.0056 |
| TS | Male | 2.0-4.0yrs | 0.641 | 0.00528 |
| No TS | Female | 6.1-8.0yrs | 0.714 | 0.00117 |
| TS | Female | 6.1-8.0yrs | 0.726 | 0.00197 |
| No TS | Female | 4.1-6.0yrs | 0.806 | 0.00287 |
| TS | Female | 4.1-6.0yrs | 0.815 | 0.000332 |
| No TS | Female | 2.0-4.0yrs | 0.874 | 0.00375 |
| TS | Female | 2.0-4.0yrs | 0.880 | 0.000433 |
Figure 1 displays the spatial distribution of the posterior fitted incidence variable, Z values from the IMCAR prior throughout Ogun State. Maps in Figs. 1 and 2 were generated using R programming language mapping packages, including rgdal31, rgeos32, tmap33, sf34,35 and ggplot236. In both the northern and western parts of the state, it revealed higher incidence proportions of cure. Ipokia had the highest proportion, followed by Sagamu, Odogbolu, and Imeko Afon LGAs, while Ijebu East LGA had the lowest proportion. As shown in Fig. 2, we can also examine the weighted mean of the cure probability for the entire state and for the two surviving groups of susceptible and cure patients.
Fig. 1.
Posterior means of fitted values with IMCAR prior.
Fig. 2.
Spatial distribution of weighted Posterior mean of cure probabilities across the state, cure group and susceptible group.
According to Fig. 2, overall across the state, Remo LGA has the highest weighted mean of cure probability, followed by Odeda and Yewa South LGAs, while Ewekoro LGA has the lowest values fitted for patients. Regarding the cure group, patients living in Remo, followed by other locations around the state, had the highest weighted mean of cure probability, while patients living in Ewekoro and Odogbolu LGAs had the lowest, in that order. Patients who lived in Ipokia LGA had the lowest weighted mean for cure probability, according to an analysis of the susceptible group. Residents of Odeda LGA had the highest values, followed by residents of Odogbolu LGAs, in that order.
Discussion
The intervention of TS seemed not to have significant effect on the BMI of patients and the incidence model, however it was significant for CD4 count and latency model. The associations between the mixture survival component and CD4 profile was negative and significant, indicating that increased CD4 count resulted in less risk in loghazard of survival event. The longitudinal trajectories for both biomarkers as modelled using spline specification showed a significant posterior mean effect, indicating that spline specification does fit the biomarkers trajectories appreciably.
From the model coefficients for the survival components, there seemed to be significant variation of the biomarkers and mixture survival effect on the various covariates considered in the study. The incidence model coefficients revealed that sex was a major covariate determine the incidence of cure or susceptible for the dataset, with more of female patients in the cure group, whereas for the age groups, the lowest level 2–4 years were more in the cure group compared to other age groups. The latency model showed that the failure event was depended on the age group as each level was significant and whereas the probability of the failure event was higher for the higher age group levels, as a result of the positive sign in the coefficients. The probability of the event was lower in the lower age group levels.
The sex of patients and age were the most significant covariates to predicating cure probabilities, whether or not they received the therapeutic snack (TS). For the male patients, the number of patients with higher cure probabilities was highest in patients of ages 4 years and below. For the female patients there were cases of patients in the age ranges of 8 to 10 years with mean cure probabilities above 60%. The susceptible group on the hand, had a full conditional distribution of latent incidence predicting proportion of 39%. Female patients had higher cure probabilities than male patients and the cure probabilities also were highest in younger patients as expected for both male and female patients. It was also seen that the intervention of TS had no significant effect on these probabilities.
The cure probabilities was highest in Remo LGA, followed by those in Odeda and Yewa South LGAs, while the least values were predicted for patients in Ewekoro LGA. Considering the cure group, the highest weighted mean of the cure probabilities were found for patients residing in Remo, and then scattered across the state, while the least were found for patients residing in Ewekoro and Odogbolu LGAs in that order. Examination of the susceptible group revealed that the patients residing in Ipokia LGA had the least weighted mean for cure probabilities. The patients with the highest values were residents of Odeda LGA, followed by those resident in Odogbolu LGAs in that order.
Conclusion
As an extension of the univariate approach described in Ekong et al.18 this paper introduced the spatial joint modeling of multivariate longitudinal biomarkers and mixture cure survival under shared random effect using latent Gaussian modeling approach. The posterior distribution of the resulting Bayesian modeling was evaluated using deterministic approximate Bayesian inference of Laplace approximation11. A spline specification was used for the longitudinal biomarkers to capture the nonlinear trajectories, and Lázaro et al.13 used specifications for the survival cure component, which used latent indicators in the inferential process to classify patients into susceptible and cured groups based on a latent indicator variable. We assumed spatial dependence between neighboring places in order to represent the spatial variability using a shared unknown random effect across patients residing in the same geographic region. It is believed that an intrinsic conditionally autoregressive (IMCAR) prior governs the spatial effects24. In the spatial multivariate joint cure modeling approach, the joint model was expressed as a hierarchical structure that fit the structure of latent Gaussian models under the Bayesian paradigm. The method of this study is limited to settings where models can be expressed in the hierarchical form of LGMs. The method was applied to a dataset comprising patients of children ages, this limits the conclusions of the results to this demography. However, the method can be applied to datasets comprising wider range of ages. While the survival event in this study was a single outcome, other extension for the survival part could be competing risk and semi-competing risks. Although MCMC is computational intensive with unguaranteed convergence and has high correlation among simulated values especially for large datasets, INLA is less accurate than MCMC in small sample sizes26.
Author contributions
E.A. wrote the main manuscript text and O.O., D. G. and O.I. supervised.
Data availability
The datasets analysed during the current study are not publicly available due to restriction from the original investigator’s institution, however, a cross-section of the data is available from the corresponding author on reasonable request with permission from the investigator.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The datasets analysed during the current study are not publicly available due to restriction from the original investigator’s institution, however, a cross-section of the data is available from the corresponding author on reasonable request with permission from the investigator.