Bayesian Latent Class Model for Predicting Gestational Age in Health Administrative Data

Summary

Health administrative data are oftentimes of limited use in obstetric research due to lacking accurate estimation of gestational age at birth (GAB). Although several studies have proposed algorithms to estimate GAB using claims database, failing to incorporate the unique distributional shape of GAB, can introduce bias in estimates and subsequent modeling. To address this gap, we develop a Bayesian Latent class model to predict GAB. We propose a mixture of Gaussian distributions and jointly fit a linear model within each class. Our Bayesian approach allows modeling heterogeneity in the population by identifying latent subgroups and estimation of class-specific regression coefficients. Posterior computation is conducted using Markov Chain Monte Carlo methods with a Gibbs sampler structure. We use the Deviance Information Criterion and the Watanabe - Akaike Information Criterion to select the optimal number of latent classes. The method is illustrated with a dataset of 10,043 Rhode Island Medicaid women. We found that the 3-class and 6-class mixture specifications maximize prediction accuracy. Based on our results, Medicaid women were partitioned into three classes, featured by extreme preterm or preterm birth, preterm or “early” term birth, and “late” term birth. Obstetrical complications appeared to pose more significant influence on class-membership allocation than other patient-level characteristics. Altogether, compared to traditional linear models our approach shows an advantage in predictive accuracy, because of superior flexibility in modeling a skewed response and population heterogeneity.

1 |. INTRODUCTION

Gestational age at birth (GAB) is a crucial variable useful in determining the onset and duration of pregnancy, assessing when exposure occurred in pregnancy, and evaluating obstetric and neonatal outcomes. Two conventional GAB estimation methods in a clinical setting include using self-reported Last Menstrual Period (LMP) and relying on early ultrasound measures, and the clinically estimated GAB has begun to be documented in the US birth certificates since 2003.¹ Inaccurate estimation of the gestational age is detrimental to pregnancy research because it might lead to both exposure misclassification (i.e., incorrect ascertainment of timing of an exposure) and outcome misclassification (e.g., misclassified premature birth).² Considering that pregnant women are often excluded from clinical trials and drug use is infrequent during pregnancy, administrative health utilization data, from either state Medicaid or commercial health plans, has been widely used as a rich data source (for examples, see Patorno et al. 2017 and Cohen et al. 2017).^3,4 Despite the appealing features of longitudinal data collection and large sample size, the start or length of pregnancy is often absent from healthcare utilization data, thus substantially impeding its use in obstetric research.

There has been a growing focus on estimating gestational age in automated databases. For simplicity and easiness of interpretation, methods that assign a duration of pregnancy depending on proxy variables (e.g., indications of preterm delivery, birth outcomes, birth weight, or the timing of prenatal care visits) have been reported in the literature.^2,5,6,7 Of note, assuming a uniform duration for both preterm and post-term deliveries is inappropriate due to the potential for misclassification aforementioned. An alternative option is to employ traditional multivariable regression models in which individual characteristics are incorporated.² Eworuke et al. developed and validated an algorithm to identify prematurity in administrative claims data by multivariate logistic regression models.⁸ Despite considerably high validity, the application of this multivariate regression algorithm is confined to only identify infants born before 34 complete weeks from near preterm infants (34-36 gestational weeks) or term infants (≥37 gestational weeks). Standard linear regression models have also been used to estimate the gestational age as a continuous variable.⁹ However, modeling for gestational age data is complicated by its skewed distribution with a long left tail for extreme preterm delivery. Also, it is characterized by an almost perfect truncation on the right tail, given that gestation does not exceed 45 weeks due to medical induced labor. Consequently, in this case, the normality assumption is unrealistic and models based on such assumption are subject to bias. Although transformations for gestational age data have been attempted, Sauzet et al. suggested that no satisfactory transformation was thought to be reliable.¹⁰

Finite Gaussian mixtures can be successfully used to approximate a wide class of continuous distributions, and to account for skewness, excess of kurtosis and multimodality. This framework allows us to propose a Bayesian latent class model (LCM) to predict gestational age at birth using maternal risk factors identified in an administrative database. Moreover, LCM is advantageous over traditional models as it is capable of identifying heterogeneous latent subgroups in that we can further investigate between-class difference, class-specific regression parameters, and patient-level characteristics associated with the class allocation.¹¹ Although our response variable is in whole weeks of gestational age, we preferred to avoid a mixture of discrete distributions in order to be able to predict gestational age in a continuous time grid.

Over the past decades, the applications of LCMs have increased considerably in medical research. Surrounding the “paradox” of birthweight and gestational age, Schwartz et al. (2009) have applied a joint Bayesian analysis of birth weight and gestational age using finite mixture models to investigate neonatal outcomes such as small for gestational age.¹² Similarly, Tentoni et al. (2004) have used two-component Gaussian mixture models to describe the distribution of birth weight stratified by gestational age, and thus correcting gestational age misclassification and weight-for-gestational-age percentile curves.¹³ LCM has also been applied to investigating the association between risk factors and health outcomes. Slaughter et al. used a Bayesian latent variable mixture model for identifying intrauterine growth restriction.¹⁴ Within the Bayesian framework, Neelon et al. (2011) proposed a growth mixture model (GMM) to examine maternal hypertension during pregnancy and birth outcomes.¹⁵ In addition, Bayesian LCMs are becoming a popular approach to exploring the performance of medical diagnostic tests. Pan-ngum et al. (2013) carried out Bayesian LCMs to assess the accuracy of rapid diagnostic tests for dengue infection.¹⁶ Bayesian LCMs were also used to estimate the prevalence of malaria infection, along with sensitivity, specificity, and predictive value of diagnostic tests by Gonçalves et al. (2012).¹⁷ Bayesian LCMs have demonstrated the flexibility and advantages of analyzing complex data and making inferences in clinical settings. However, predicting gestational age in the absence of birth weight has not been explored in a Bayesian framework.

Altogether, the objective of our analysis is to develop a LCM to predict GAB in the absence of birth weight and to identify latent heterogeneous subgroups. The study is outlined as follows: Section 2 describes an overview of the data source and study samples. Section 3 presents the proposed LCM, posterior computation, MCMC algorithms, model selection, and assessment of predictive accuracy. In section 4, we apply the LCMs to Medicaid women with infant data from birth certificates in the state of Rhode Island in the US. Lastly, we conclude the study findings and discuss the future research plan in section 5.

2 |. DATA SOURCE AND STUDY COHORT

The study data were obtained from Rhode Island Medicaid program linked with vital statistics from the Department of Health (DOH) throughout 2008 to 2015. The linked dataset contains healthcare utilization information including pharmacy claims and medical claims, sociodemographic characteristics as well as infant birth outcomes. Inpatient and outpatient medical claims comprise fields of diagnoses and procedures (coded by International Classification of Diseases, 9th Revisions, Clinical Modification [ICD-9-CM], and Current Procedural Terminology [CPT], Fourth Edition), dates of service, cost of services and others. In addition, date of delivery, LMP, and the ultrasound-based estimation of gestational age at birth were registered in vital statistics. Two contradictory methods have been proposed to estimate gestational age in a clinical setting: menstrual-based dating and ultrasound-based dating.^18,19 Due to concerns about the proportion of missing LMP values (36.3%) and the potential for recall bias with menstrual-based dating, we considered the ultrasound-based estimation of gestational age in complete weeks as the “gold standard” gestational age in the analyses.

We included women aged 12-55 years with a live birth between January 1, 2008 and September 30, 2015. Women were required to have continuous enrollment since 6 months preceding the conception date, throughout pregnancy, and 6 weeks after delivery. The date of conception was identified by subtracting gestational ages in days from the date of delivery. Women with invalid gestational age (i.e., missing values and outside the range of 20-44 complete weeks) were excluded.⁹

Risk factors were selected based upon a discreet review of published articles in the literature. Obstetrical risks including severe pre-eclampsia, cervical incompetence, intrauterine infections, oligohydramnios, polyhydramnios, isoimmunization and vaginal bleeding have been found to contribute a high likelihood of potential premature delivery.²⁰ Maternal co-morbid conditions prior to or during pregnancy (including diabetes, hypertension, chronic renal disease, anemia particularly during pregnancy, and epilepsy)^21,22 and mental illness or stress such as depression, anxiety, post-traumatic stress disorder (PTSD), and psychosis have been linked with adverse birth outcomes such as low birth weights and low gestational age.^23,24,25 A proxy variable indicative of disorders relating to short gestation or low birth weight or early or threatened labor proposed by Margulis et al. was also considered.⁹ Maternal risk factors were assessed by ICD-9-CM diagnosis and procedure codes at baseline and during pregnancy. To ensure the proposed mixture models replicable, we only adopted risk factors that can be identified in an automated database via medical claims. This project was approved by the Institutional Review Boards of Rhode Island Department of Health and the University of Rhode Island.

3 |. PROPOSED LATENT CLASS MODEL

To fit the gestational age (completed weeks) for Medicaid women, we propose a finite mixture of Gaussian distributions to relax the marginal normality assumption. In addition, women could be classified into latent groups depending on their demographic and clinical characteristics. The proposed LCM is expressed as

$$\begin{gathered} f\left(y_i|{\pi }_k,{\beta }_k,{\sigma }_k^2;x_i\right)=\sum _{k=1}^K{\pi }_k\mathcal{N}\left(y_i;{\eta }_{ik},{\sigma }_k^2\right) \\ {\eta }_{ik}=x_i^T{\beta }_k,i=1,\dots ,n,k=1,\dots ,K, \\ {\pi }_k=\mathrm{Pr}\left(C_i=k\right) \end{gathered}$$ (1)

where y_i denotes the gestational age for subject i; $\mathcal{N}\left(·|\mu ,{\sigma }^2\right)$ denotes a normal distribution with mean μ and variance σ²; we introduce latent class allocation variables C_i ∈ {1, …,K} that have a multinomial distribution $ℳ\left({\rho }_{i1},\dots ,{\rho }_{iK}\right);{\pi }_k$ denotes the probability of membership of the kth component $\left({\sum }_{k=1}^K{\pi }_k=1\right),{\beta }_k$ is a p×1 vector denoting class-specific regression coefficients corresponding to p covariates.

3.1 |. Prior specification

We assume weakly informative prior distributions to all parameters $\left\{{\pi }_k,{\beta }_k,{\sigma }_k^2\right\}$. For class-membership probabilities, we assign a conjugate Dirichlet prior $𝒟\left(a_1,\dots ,a_K\right)$ where K is the number of mixture components. For class-specific regression parameters β_k, we assign a conditionally conjugate multivariate normal prior ${\mathbf{\text{N}}}_p\left({\mu }_{\beta },{\sigma }_{\beta }^2{\mathit{I}}_p\right)$. We assume a conjugate inverse gamma prior for the variance of the class specific regression, that is, ${\sigma }_k^2~ℐ\mathcal{G}(a,b)$. The prior hyperparameters are assumed to remain the same across all latent classes.

3.2 |. Posterior computation

According to the Bayes’ Theorem and assuming independence of priors, the joint posterior distribution is given by:

$$\pi \left({\beta }_k,{\sigma }_k^2,{\pi }_k|{\mathit{Y}}_{\mathit{i}},{\mathit{X}}_{\mathit{i}}\right)\propto \pi \left({\beta }_k\right)\pi \left({\sigma }_k^2\right)\pi \left({\pi }_k\right)\prod _{k=1}^K{[{\pi }_k\prod _{i=1}^{N}f\left(y_i|{\eta }_{ik},{\sigma }_k^2\right)]}^{I_{\left(C_i=k\right)}}$$

where $f\left(y_i|{\eta }_{ik},{\sigma }_k^2\right)$ is given in equation (1).

Given the complexity of the joint posterior distribution, we propose a Gibbs sampler structure to draw samples from the full conditional distribution of each parameter. The full conditional forms are given by

$$\begin{gathered} {\pi }_k|·\propto \prod _{i=1}^N\mathrm{Pr}{\left(C_i=k|{\pi }_k\right)}^{I\left(C_i=k\right)}\pi \left({\pi }_k\right) \\ =\mathcal{D}\left(a_1+\sum I\left(C_i=1\right),\dots ,a_K+\sum I\left(C_i=K\right)\right) \\ {C}_i|·~ℳ\left({\rho }_{i1},\dots ,{\rho }_{iK}\right)=\mathrm{Pr}\left(Y_i|C_i,{\beta }_k,{\sigma }_k^2;{\mathit{x}}_{\mathit{i}}\right)\mathrm{Pr}\left(C_i|{\pi }_k\right) \\ {\beta }_k|·\propto \mathrm{Pr}\left(Y_k|C_i=k,{\sigma }_k^2;{\mathit{x}}_{\mathit{k}}\right)\pi \left({\beta }_k\right) \\ {\sigma }_k^2|·\propto \mathrm{Pr}\left(Y_k|C_i=k,{\beta }_k;{\mathit{x}}_{\mathit{k}}\right)\pi \left({\sigma }_k^2\right) \end{gathered}$$

where Y_k and x_k denote the response variable and the regression covariates for subjects allocated to class k, respectively. With regard to sampling the component indicator variable C_i, the probability of subject i belonging to class k is as follows

$${\rho }_{ik}=\frac{{\pi }_{k}f\left(y_i|{\beta }_k,{\sigma }_k^2;{\mathit{x}}_{\mathit{i}}\right)}{{\sum }_{l=1}^K{\pi }_{l}f\left(y_i|{\beta }_l,{\sigma }_l^2;{\mathit{x}}_{\mathit{i}}\right)}$$

Since closed forms are available for the conditional posterior distributions of all parameters π, β, σ², we simulate the posterior distribution using the Gibbs sampler based on their full conditional distributions given above:

1. Assign initial values to ${\beta }_k,{\sigma }_k^2$ for k = 1, …,K, and C_i for i = 1, …,N;

2. For k = 1, …,K, sample the vector π_k from its full conditional distribution using a Gibbs sampler;

3. Sample the latent class indicators C_i from a discrete categorical distribution with the probability vector ρ_ik(i = 1, …, n);

4. Given C_i = k, update β_k from a full conditional distribution using a Gibbs sampler;

5. Given C_i = k, update ${\sigma }_k^2$ from a full conditional distribution using a Gibbs sampler.

To ensure the mixing of MCMC and to check label switching, we examine the trace plots of each parameter. Label switching is a common, yet challenging issue encountered in the context of mixture models. It occurs when the posterior distribution is invariant to relabeling of the mixture components, thus leading to inappropriate inference on their class-specific parameters^26,27. To address this issue we adopt a commonly used approach by imposing an identifiability constraint on the parameter space.²⁸ To evaluate the speed of mixing of the Gibbs sampler, we check the sample autocorrelation function and the effective sample size for each parameter. We thin out the posterior MCMC samples to reduce autocorrelation issues.

3.3 |. Model selection and predictive accuracy

To determine the optimal number of latent classes, we employ several information criteria approaches, namely, the Deviance Information Criterion (DIC),²⁹ an adapted version of DIC (DIC3) proposed by Celeux et al.³⁰ which was found to be successful in the context of mixtures, and the Watanabe-Akaike Information Criterion (WAIC).³¹ DIC is defined as $-2\log p\left(y|{\widehat{\theta }}_{Bayes}\right)+2p_{DIC}$, where p_DIC, as referred to the effective number of parameters, is a penalty for model complexity adjusting for overfitting.³² The log predictive density measures the model fit. In addition to DIC, WAIC introduced by Watanabe has the preferable property by averaging over the posterior distribution instead of conditioning on a point estimate.³² As with Akaike information criterion (AIC) or Bayesian information criterion (BIC) used in a frequentist framework, models with smaller DIC, DIC3, or WAIC values are considered to be preferable.

Apart from examining the in-sample model fit, we further measure predictive accuracy with new data by comparing the root mean (predictive) squared error (RMSE). We fit an ordinary linear regression model as a reference for accuracy comparison. We compute RMSE for each individual and average over all individuals. To determine the extent to which LCMs outperform a linear regression model, we compute the ratio of RMSE by dividing an RMSE of LCM by an RMSE of the linear regression model, of which value <1 indicates better predictive performance. We compare the average ratio as well as the likelihood of RMSE ratio < 1 across all subjects. Given a series of model fit and RMSE statistics, we choose the LCM models that seemingly predict the best and repeat MCMC samplings based on the entire data. Posterior summaries on model parameters are made incorporating the entire study sample.

4 |. PREDICTION OF GAB IN RI MEDICAID PREGNANT WOMEN

The study cohort consisted of 10,043 mother-newborn pairs between January 2008 and September 2015. Descriptive statistics of demographic characteristics and risk factors of the study sample were presented in Table A1 and corresponding ICD-9-CM codes were presented in Table A2. Although information on neonatal outcomes such as birth weight and Apgar score, which is designated to evaluate perinatal characteristics in newborn infants,³³ were documented in birth certificates, we chose not to include them in model fitting since such data were rarely collected in administrative claims databases. Data pre-process was conducted in SAS 9.4 (SAS Institute Inc, Cary, NC) and model fitting was conducted in R version 3.5.3.³⁴

We fitted a standard linear model as reference and a series of LCMs with the number of components ranging from 1 to 6. A mixture model with more than 6 clusters appeared to encounter overfitting problems in the sense that one or multiple clusters were allocated with too few cases to fit the pre-specified covariates. We ran 40,000 iterations for each LCM model, with 15,000 runs discarded as burn-in. After inspecting the empirical autocorrelation function, to mitigate autocorrelation in the trace plots, we thinned out samples by keeping every 25th draw. We examined trace plots for each parameter within each class and found that stationarity was well achieved and no obvious evidence of “label switching”. All models were fitted with training data accounting for 60% of the total study sample. As to alleviate uncertainty arising from the selection of training data, we performed cross-validation (CV) by repeating all analyses with 50 random datasets.

We presented DIC, DIC3, and WAIC statistics with 95% Monte Carlo uncertainty intervals from 50 independent training datasets in Table 1. The six-class mixture showed the lowest DIC, DIC3 and WAIC in 41 training sets out of 50, with the lowest average value, suggesting the best in-sample model fit. Furthermore, to assess the out-of-sample model fit, we computed and presented the RMSE for a linear regression model and latent class models utilizing the remainder of the entire sample (40%) (Table 2). Apart from one-class LCM, LCMs with two or more classes had more favorable RMSE values ranging from 1.835 to 2.145 than the standard linear regression model (RMSE=2.336). RMSE ratio and the likelihood of RMSE ratio < 1 were compared across all mixture models. Similarly, mixture models with at least two classes presented higher predicative accuracy compared with a linear regression in terms of RMSE and RMSE ratio (Table 2). A majority of mixture models yielded lower RMSE over 95% times compared with the linear model. Again, a six-class mixture was preferred with the lowest average RMSE (1.835) as well as the average RMSE ratio (0.524), which suggests maximization of predictive precision. It is worth noting that a three-class mixture appeared to have the highest likelihood of having RMSE ratio < 1 (0.974) which implies the best accuracy. In this regard, we further discussed mixture models with 3 and 6 classes using the entire data to conclude posterior summaries.

TABLE 1.

Model selection statistics for gestational age at birth.

Number of classes	DIC (95% CI)	DIC3 (95% CI)	WAIC (95% CI)
1	48739 (47581, 49601)	23299 (22733, 23722)	23302 (22736, 23724)
2	44784 (43785, 45789)	21151 (20664, 21654)	21165 (20680, 21664)
3	43516 (42715, 44335)	20483 (20057, 20861)	20504 (20072, 20884)
4	37016 (31479, 42353)	10329 (7743, 20235)	10387 (7769, 20059)
5	32231 (30420, 35863)	5887 (4332, 9000)	5911 (4345, 9020)
6	29317 (26774, 32979)	4410 (2658, 6730)	4441 (2684, 6769)

TABLE 2.

Predictive accuracy for gestational age at birth.

Model description	RMSE (95% CI)	Average RMSE ratio (95% CI)	% of RMSE ratio <1 (95% CI)
Linear Regression	2.336 (2.317, 2.358)	1 (reference)	1 (reference)
1-class LCM	2.334 (2.327, 2.366)	1.005 (1.005, 1.006)	0.435 (0.421, 0.446)
2-class LCM	2.145 (2.001, 2.547)	0.796 (0.718, 0.830)	0.965 (0.945, 0.976)
3-class LCM	1.976 (1.781, 2.504)	0.655 (0.614, 0.717)	0.974 (0.960, 0.985)
4-class LCM	1.900 (1.773, 2.232)	0.614 (0.553, 0.655)	0.960 (0.938, 0.976)
5-class LCM	1.863 (1.708, 2.375)	0.566 (0.517, 0.597)	0.960 (0.938, 0.981)
6-class LCM	1.835 (1.667, 2.359)	0.524 (0.498, 0.569)	0.960 (0.945, 0.976)

We plotted the mixture and class-specific posterior predictive density of the 3-class and 6-class model in Figure 1 and 2. The mixture posterior predictive density of the 3-class mixture resides on the distribution of true GAB almost perfectly with a smooth and continuous shape. The 6-class model, in contrast, appears to be spiky around integers indicating the number of complete gestational weeks. It indeed indicates better fit as compared to a smooth shape shown in the 3-class model, since the true GAB data are integers. The class-specific posterior distributions exhibit the features of each latent class. For a 3-class mixture (Figure 1), class 1 shows a flat curve with small probabilities, which covers nearly the entire span including extremely low GAB. Class 2 is characterized by large gestational age which likely represents full-term deliveries, followed by class 3 that is featured with near preterm and term births. As opposed to the 3-class model, classes of the 6-class model are not distinguished as clearly in terms of location and shape. To some extent, the 6-class model can be interpreted as a second layer of classification on the basis of the 3-class model. Class 3 and class 5 appear to cluster preterm births, while class 5 has a wider and flatter curve. Full-term births with around 39-40 complete weeks characterize class 2 and class 4. As with class 3 in the 3-class model, class 6 in the 6-class model features near preterm and term deliveries (Figure 2).

FIGURE 1.

Mixture and class-specific posterior predictive distributions of the 3-class model.

FIGURE 2.

Mixture and class-specific posterior predictive distributions of the 6-class model.

We presented the class-specific posterior summaries on a selection of model parameters in Table 3 and 4. Quadratic and cubic polynomials for continuous variables (e.g., maternal age) were not included in final models due to statistical insignificance and a decrement in model fit. As expected, risk factors manifested varying effects on GAB across classes in both 3-class and 6-class models. The covariate coefficients showed divergent class-specific posterior distributions (Figure A1 and A2). In particular, in the 3-class model, the coefficient of oligohydramnios appeared to be negative (−0.22; %95 CI:−0.39, −0.04) among preterm or “early” full term newborns (class 3), yet positive (0.30; %95 CI:0.15, 0.44) among “late” full term newborns (class 2). Maternal obesity was associated with an increase in GAB in class 1 (2.84; %95 CI:0.58, 5.12) and exhibited null effect in class 2 and 3. Likewise, antenatal hemorrhage attributed to a significant decrease in gestational age merely in class 1 (−3.41; %95 CI:−6.40, −2.89). In the 6-class mixture, a significant increase in GAB associated with oligohydramnios was observed in class 3 (0.76; %95 CI:0.57, 0.95), class 4 (0.48; %95 CI:0.43, 0.51), and class 6 (1; %95 CI:0.99, 1.01) with varying magnitude of effect. Infection of the amniotic cavity appeared to be associated with an increase among class 3 (0.93; %95 CI:0.65, 1.23) and class 6 (1; %95 CI:0.97, 1.03) but a decrease among class 1 (−0.65; %95 CI:−0.74, −0.55). A significant decrease in GAB linked with severe pre-eclampsia was present in class 1 (−2.99; %95 CI:−3.04, −2.94), class 2 (−3.99; %95 CI:−4.00, −3.97) and class 4 (−3.53; %95 CI:−3.58, −3.50), in contrast, class 6 (1; %95 CI:0.98, 1.02) showed a significant increase and class 3 and 5 showed null effect. On the other hand, some risk factors (e.g., stay in NICU and indicator of early/threatened labor or preterm-related disorders) exhibit a relatively consistent effect in GAB across classes and models.

TABLE 3.

Posterior mean and 95% credible interval of model parameters for the three-class model

Class (%)	Parameter (Covariate)	Posterior Mean	95% Credible Interval
1 (1.81%)	β11 (Intercept)	35.222	(32.053, 38.290)
	β1,2(Severe pre-eclampsia)	0.079	(−4.057, 4.340)
	β1,3(Cervical incompetence)	−4.046	(−6.687, −1.646)
	β1,4(IAC)	−0.546	(−3.391, 2.326)
	β1,5 (Placental abruption)	−0.582	(−4.378, 2.911)
	β1,6(Oligohydramnios)	−1.278	(−4.301, 1.559)
	β1,7(Alcohol abuse)	1.974	(−1.356, 5.455)
	β1,8(Diabetes)	1.135	(−1.991, 4.472)
	β1,9(Obesity)	2.844	(0.578, 5.119)
	β1,10(Antenatal hemorrhage)	−3.413	(−6.403, −2.889)
	β1,11(Stay in NICU)	−0.001	(−0.019, 0.019)
	β1,12 (Maternal age (linear))	0.011	(−0.111, 0.128)
	${\sigma }_1^2$(error)	21.055	(15.982, 27.185)
2 (46.52%)	β2,1(Intercept)	39.850	(39.682, 40.004)
	β2,2(Severe pre-eclampsia)	−4.058	(−4.531, −3.588)
	β2,3(Cervical incompetence)	−0.245	(−0.533, 0.026)
	β2,4(IAC)	0.268	(−0.002, 0.528)
	β2,5 (Placental abruption)	0.109	(−0.198, 0.427)
	β2,6(Oligohydramnios)	0.296	(0.149, 0.442)
	β2,7(Alcohol abuse)	−0.213	(−0.474, 0.030)
	β2,8(Diabetes)	−0.424	(−0.560, −0.288)
	β2,9(Obesity)	0.024	(−0.074, 0.123)
	β2,10(Antenatal hemorrhage)	−0.025	(−0.157, 0.119)
	β2,11 (Stay in NICU)	−0.069	(−0.073, −0.066)
	β2,12(Maternal age (linear))	−0.011	(−0.017, −0.006)
	${\sigma }_2^2$(error)	0.601	(0.549, 0.652)
3 (51.67%)	β3,1(Intercept)	39.081	(38.870, 39.308)
	β3,2(Severe pre-eclampsia)	0.327	(0.018, 0.662)
	β3,3(Cervical incompetence)	−0.395	(−0.739, −0.095)
	β3,4(IAC)	0.064	(−0.279, 0.408)
	β3,5(Placental abruption)	−0.709	(−1.086, −0.349)
	β3,6(Oligohydramnios)	−0.221	(−0.392, −0.043)
	β3,7(Alcohol abuse)	0.316	(0.016, 0.609)
	β3,8(Diabetes)	−0.303	(−0.489, −0.125)
	β3,9(Obesity)	−0.033	(−0.165, 0.101)
	β3,10(Antenatal hemorrhage)	−0.152	(−0.353, 0.028)
	β3,11 (Stay in NICU)	−0.162	(−0.174, −0.153)
	β3,12(Maternal age (linear))	−0.016	(−0.023, −0.009)
	${\sigma }_3^2$(error)	1.537	(1.441, 1.627)

Abbreviations. IAC: infection of the amniotic cavity; NICU: neonatal intensive care unit.

TABLE 4.

Posterior mean and 95% credible interval of model parameters for the six-class model

Class (%)	Parameter (Covariate)	Posterior Mean	95% Credible Interval
1 (7.37%)	β1,1 (Intercept)	41.006	(40.977, 41.033)
	β1,2(Severe pre-eclampsia)	−2.992	(−3.044, −2.942)
	β1,3(Cervical incompetence)	−13.988	(−14.099, −13.878)
	β1,4(IAC)	−0.652	(−0.736, −0.554)
	β1,5(Placental abruption)	−0.978	(−1.027, −0.931.971)
	β1,6(Oligohydramnios)	−0.001	(−0.022, 0.019)
	β1,7(Alcohol abuse)	−1.000	(−1.028, −0.976)
	β1,8(Diabetes)	−0.006	(−0.025, 0.037)
	β1,9(Stay in NICU)	−0.087	(−0.088, −0.087)
	${\sigma }_1^2$(error)	0.005	(0.004, 0.005)
2 (21.03%)	β2,1(Intercept)	39.999	(39.991, 40.007)
	β2,2(Severe pre-eclampsia)	−3.9851	(−4.004, −3.965)
	β2,3(Cervical incompetence)	−3.995	(−4.015, −3.973)
	β2,4(IAC)	0	(−0.014, 0.014)
	β2,5 (Placental abruption)	−3.001	(−3.011, −2.982)
	β2,6(Oligohydramnios)	0.002	(−0.006, 0.009)
	β2,7(Alcohol abuse)	−1.000	(−1.011, −0.990)
	β2,8(Diabetes)	0.001	(−0.007, 0.010)
	β2,9 (Stay in NICU)	−0.091	(−0.092, −0.091)
	${\sigma }_2^2$(error)	0.002	(0.001, 0.002)
3 (12.24%)	β3,1(Intercept)	37.095	(36.875, 37.330)
	β32, (Severe pre-eclampsia)	−0.392	(−0.819, 0.012)
	β3,3(Cervical incompetence)	3.306	(2.887, 3.684)
	β3,4(IAC)	0.928	(0.652, 1.232)
	β3,5(Placental abruption)	0.835	(0.244, 1.351)
	β3,6(Oligohydramnios)	0.756	(0.565, 0.946)
	β3,7(Alcohol abuse)	3.548	(3.258, 3.818)
	β3,8(Diabetes)	0.074	(−0.104, 0.256)
	β3,9 (Stay in NICU)	−0.161	(−0.170, −0.153)
	${\sigma }_3^2$(error)	0.357	(0.287, 0.438)
4 (32.36%)	β4,1(Intercept)	38.999	(38.994, 39.005)
	β4,2(Severe pre-eclampsia)	−3.533	(−3.575, −3.501)
	β4,3(Cervical incompetence)	0	(−0.011, 0.011)
	β4,4(IAC)	0.001	(−0.011, 0.014)
	β4,5(Placental abruption)	−0.002	(−0.014, −0.044)
	β4,6(Oligohydramnios)	0.476	(0.428, 0.512)
	β4,7(Alcohol abuse)	−0.999	(−1.008, −0.990)
	β4,8(Diabetes)	0	(−0.004, 0.004)
	β4,9 (Stay in NICU)	−0.146	(−0.147, −0.143)
	${\sigma }_4^2$(error)	0.0009	(0.0009, 0.001)
5 (11.03%)	β5,1(Intercept)	38.872	(37.917, 39.862)
	β5,2(Severe pre-eclampsia)	−0.925	(−2.074, 0.179)
	β5,3(Cervical incompetence)	−2.777	(−3.736, −1.860)
	β5,4(IAC)	0.021	(−0.865, 0.943)
	β5,5(Placental abruption)	−1.305	(−2.496, −0.142)
	β5,6(Oligohydramnios)	−0.144	(−0.800, −0.554)
	β5,7(Alcohol abuse)	−0.081	(−1.048, 0.904)
	β5,8(Diabetes)	−0.486	(−1.324, 0.304)
	β5,9 (Stay in NICU)	−0.059	(−0.068, −0.051)
	${\sigma }_5^2$(error)	11.151	(10.130, 12.229)
6 (15.97%)	β6,1(Intercept)	38	(37.990, 38.011)
	β6,2(Severe pre-eclampsia)	1.000	(0.980, 1.020)
	β6,3(Cervical incompetence)	0	(−0.015, 0.017)
	β6,4(IAC)	1.000	(0.972, 1.028)
	β6,5(Placental abruption)	−2.000	(−2.025, −1.975)
	β6,6(Oligohydramnios)	1.000	(0.991, 1.009)
	β6,7(Alcohol abuse)	−1.000	(−1.014, −0.986)
	β6,8(Diabetes)	0	(−0.010, 0.009)
	β6,9 (Stay in NICU)	−0.162	(−0.174, −0.153)
	${\sigma }_6^2$(error)	0.0019	(0.0018, 0.0021)

As a supplementary assessment of covariate effects, we evaluated the posterior predictive density on hypothetical patients having a fixed set of characteristics. A conceptual young health subject exhibited a nearly symmetric distribution centered at 39-40 weeks (Figure 3 (a)). With the presence of long-stay in NICU (i.e. 12 days), the posterior distributional shape shifted toward left and appeared to be right skewed (Figure 3 (b.1)). Similarly, risk factors such as early/threatened labor or preterm-related disorders, substance or tobacco use, older age with chronic medical conditions, and placental abruption shifted the density to lower GAB with right skewness, compared to the reference (Figure 3 (b.3–6)). For severe cases complicated with multiple obstetrical complications or combined with admission to NICU, the posterior density became highly skewed with a second peak centered at 36-37 weeks (Figure 3 (b.7–8)). Hypertension complicated with preeclampsia dramatically changed the shape in which another mode concentrated at very low GAB emerged (Figure 3 (b.2)). For a healthy subject with multiple gestations, the posterior density concentrated at a lower GAB as compared to a healthy singleton pregnancy (Figure 4 (a)). For subjects with mild conditions (i.e., substance use, old age, and placental abruption), posterior density shifted slightly toward left yet the shape remained unchanged (Figure 4 (b.3–6)). When fitting a 6-class model, a similar pattern of covariate effects was observed though the posterior predictive density appeared spiky. Long stay in NICU and hypertension complicated with preeclampsia appeared to be the driving factors that led to a dramatic change in posterior density by lowering GAB (Figure 5 and 6 (b.1–2)). For multiple gestation, the distributional shapes resembled those for singleton gestation, yet concentrated at a lower GAB (Figure 5 and 6).

FIGURE 3.

Posterior predictive density of a 3-class model for (a) a pseudo healthy pregnant woman (reference) and (b) pseudo young pregnant women with varying complications.

FIGURE 4.

Posterior predictive density of a 3-class model for (a) a pseudo healthy multifetal pregnant woman (reference) and (b) pseudo multifetal pregnant women with varying complications.

FIGURE 5.

Posterior predictive density of a 6-class model for (a) a pseudo healthy pregnant woman (reference) and (b) pseudo pregnant women with varying complications.

FIGURE 6.

Posterior predictive density of a 6-class model for (a) a pseudo healthy multifetal pregnant woman (reference) and (b) pseudo multifetal pregnant women with varying complications.

5 |. DISCUSSION

In this study, we proposed latent class models for predicting the gestational age at birth in administrative healthcare utilization data. The results demonstrated that mixture models (≥ 2 classes) provided significant improvement in predicting a skewed continuous variable as compared to a standard linear model. Within a Bayesian framework, the mixture model with only one component, i.e., Bayesian linear regression model, was not preferred since it showed inferior prediction performance yet more computationally intensive than an ordinary linear regression. By allowing the number of classes to increase from 2 to 6, model fit and prediction accuracy appeared to increase, which is likely due to the fact that the more clusters were assigned, the more subtle subgroup heterogeneity was captured. However, overfitting issues might persist despite imposing penalties on the number of parameters to model fit statistics. In the exploratory analysis, we examined mixture models with ≥7 classes, in which we observed that too few “ones” have been allocated to one or multiple classes for covariates with relatively low prevalence. Despite a slight improvement in model fit, we did not present results for mixture models with ≥7 classes given the consideration of clinically relevant interpretation and MCMC chain convergence.

Besides superior predictive accuracy, the mixture models have revealed several advantages over traditional regression models. First, it has the flexibility of classifying subjects into latent clusters based on their demographic or clinical features. As such, it allows for the identification and investigation of unknown heterogeneous subgroups in the overall population. In 3-class mixture, live births were partitioned into three different classes representing extreme preterm or preterm birth (class 1), mild preterm or “early” full birth (class 3), and “late” term birth (class 2), which roughly aligned with the clinically defined categories: extreme preterm birth (< 34 weeks), preterm birth (34-37 weeks) and term birth (≥ 37 weeks). Women who developed placental abruption, cervical incompetence, intrauterine infections, oligohydramnios, polyhydramnios, and vaginal bleeding were more likely to be in class 1 than women without obstetrical complications. Women who underwent induction of labor were with a higher likelihood of belonging to class 3 than those who didn’t. Furthermore, the 6-class mixtures can be considered the subcategorization of the 3-class mixture. A majority of cases allocated to class 1 of the 3-class mixture fell into class 5 in the 6-class mixture. The class 2 and 4 from the 6-class mixture came from class 2 of the 3-class mixture while the class 3 and 6 from the 6-class mixture belonged to class 3 of the 3-class mixture. We also noticed that compared to maternal chronic co-morbid conditions, obstetrical complications occurring throughout pregnancy posed a more significant impact on class-membership allocation. More specifically, women who experienced obstetrical risks were more likely to fall into the preterm term category than those who did not. Besides, another advantage lies in the capabilities of investigating class-specific covariate coefficients. A majority of risk factors included have been observed to be associated with premature birth or adverse pregnancy outcomes. However, few has been examined in terms of their influence in heterogeneous subgroups. Under a mixture framework, we are able to draw posterior summaries on coefficients within each class. Lastly, as opposed to ordinary linear regression models, this modeling approach considered individual variabilities as an unknown parameter and incorporated it into the posterior sampling scheme, thus measuring uncertainties associated with individual variabilities.

Challenges remain in determining the optimal number of latent classes. In our analysis, the selection of the number of mixture components relied on information criteria approaches (i.e., DIC, DIC3, and WAIC), which have been proven to be a validated model selection criterion and widely used in a finite mixture model context.^15,35 When taking the prediction precision into account, both the 3-class model and the 6-class model showed preferred features. The 6-class model appears to be most precise by minimizing average RMSE, whereas the 3-class model maximized accuracy by having the highest likelihood of outperforming a normal linear model. The 95% CIs of RMSE statistics for the two models appeared to overlap. The takeaway, in this particular case, is that it is hard to choose one model that utterly outperforms the others. Instead of fitting mixture models repetitively with different numbers of components, an infinite component mixture model, also known as Dirichlet process mixture, might be a proper alternative that not only identifies latent classes but also considers uncertainties arising from the unknown number of clusters.^36,37 As opposed to assigning a Dirichlet prior, it sets a prior on Dirichlet process distributions.^36,37 However, the challenge is that the resulting number of components from the Dirichlet process mixture might be too large to translate into clinical relevance.

The study has several limitations worth mentioning. Administrative claims data were inherently prone to misclassification arising from mis-specified or inaccurate entry of medical claims coding. To mitigate such bias, we adopted the diagnosis codes that have been reported or validated if available. We also crosschecked certain risk factors identified on the basis of diagnosis codes with information available on birth certificates. For instance, we compared the prevalence of tobacco use at baseline or in pregnancy based on ICD-9-CM codes with the rate of self-reported smoking in pregnancy, which were both comparably high. However, these findings highlight the issue of generalizability since Medicaid enrollees generally represent a class of low-income individuals with relatively higher incidence of risk factors and co-morbidities. The specific limitation of external generalizability is however unrelated to the internal validity of the current study.

Second, maternal Medicaid data and infant vital statistics data were linked based on infants’ identification rather than mothers’ identification. As a result, we alternatively assumed the independence of repeated pregnancy outcomes for each individual. When the mother-level identification data are available, the correlation between multi-pregnancy or multi-gestation can be taken into account, for example by fitting a random effect at mother-level in the model. Third, we extracted the information on stay in neonatal intensive care from birth certificate. Alternatively, information on this variable can be reliably identified in administrative claims data via procedure codes (e.g., 99295, 99296).⁸ However, in the absence of infants’ medical claims, this variable might be underestimated by including partial maternal medical claims. Further model fitting and calibration are encouraged when offsprings’ medical claims from Medicaid database are available. Fourth, we assumed that the ultrasound-based estimation of gestational age was correctly measured and documented. Therefore, there is a chance for measurement error, albeit minimal. Lastly, data on gestational age were truncated into the number of completed weeks, and we considered the full weeks of gestation as a continuous variable. A more flexible approach is to add an unknown quantity μ_i ∈ [0, 1) to the observed number of completed weeks to represent true gestational age and assign a uniform or beta prior for μ_i.¹²

Taken together, our study provides a novel solution to a current research hurdle. Administrative healthcare claims databases typically provide adequate sample size and follow-up period to detect statistical or clinical significance, even for rare outcomes or exposures. However, inaccurate estimation of gestational age can potentially introduce bias and thus limiting the use of routinely collected health utilization data for obstetric research. Importantly, our proposed models showed superior predictive performance, which might help to addresss bias associated with GAB estimation. Also, the proposed mixture models can be fitted easily with Gibbs samplers specified in Section 4.2. The natural semi-conjugate priors simplify MCMC chains and reduce computational time notably. The application of LCMs can also be extended to investigate associations of risk factors (e.g., medication use or hazardous exposures) and gestational age or other pregnancy-related outcomes.

APPENDIX

TABLE A1.

Descriptive statistics of baseline characteristics

Characteristics	Mean (SD) or n (%) (N=10,043)	Characteristics	Mean (SD) or n (%) (N=10,043)
Maternal age at birth †	38.4 (2.13)	Deliver-related procedures
Obstetrical complications		Cesarean delivery	2,849 (28.37)
Cervical incompetence	161 (1.60)	Induction of labor	1,108 (10.14)
Rhesus isoimmunization	591 (5.89)	Comorbid conditions
Isoimmunization from blood group incompatibility	64 (0.64)	Diabetes	634 (6.31)
Polyhydramnios	208 (2.07)	Hypertension	840 (8.36)
Oligohydramnios	569 (5.67)	Chronic renal disease	164 (1.63)
Infection of the amniotic cavity	188 (1.87)	Epilepsy	128 (1.28)
Severe pre-eclampsia	231 (2.30)	Asthma	1,639 (16.32)
Pre-eclampsia/eclampsia	620 (6.17)	Anemia	2344 (23.34)
Placental abruption	176 (1.75)	Septic infection	765 (7.62)
Premature rupture	856 (8.52)	Psychiatric conditions
Previa placenta	542 (5.40)	Depression	3160 (31.47)
Threatened miscarriage	146 (1.45)	Anxiety or PTSD	2,549 (25.38)
Vaginal bleeding	259 (2.58)	Bipolar disorder	634 (6.31)
IUGR	1,907 (18.99)	Psychosis	163 (1.62)
Antepartum hemorrhage	746 (7.43)	Obstetric characteristics
Postpartum hemorrhage	338 (3.37)	Multiple gestation	93 (0.93)
Cardiac events	55 (0.55)	Substance abuse	644 (6.41)
Fibromyalgia	206 (2.05)	Tobacco abuse	1887 (18.79)
Known or suspected Fetal abnormalities	4896 (48.75)	Alcohol abuse	297 (2.96)
Stay in NICU†	1.7 (10.2)	Obesity	1329 (13.23)
Disorders*/early or threatened labor	3182 (31.68)

Abbreviations. IUGR: intrauterine growth restriction; NICU: neonatal intensive care unit; PTSD: post-traumatized stress disorder.

^*:disorders relating to short gestation or low birth weight.

^†:being identified without using ICD-9-CM code; stay in NICU was identified from information relating to “days in NICU” on birth certificates .

TABLE A2.

ICD-9-CM diagnosis codes

Characteristics	ICD-9-CM code	Characteristics	ICD-9-CM code
Obstetrical complications		Deliver-related procedures
Cervical incompetence	654.5x, 761.0x	Cesarean delivery	669.7x
		Cesarean delivery†	740.xx, 741.xx, 742.xx, 749.9x
Rhesus isoimmunization	656.1x	Induction of labor†	731.xx, 734.xx
Isoimmunization from blood group incompatibility	656.2x	Comorbid conditions
Polyhydramnios	657.xx, 761.3x	Diabetes	250.xx, 648.0x, 648.8x
Oligohydramnios	658.0x, 761.2x	Hypertension	401-405.xx, 642.0x 642.1x, 642.2x, 642.7x
Infection of the amniotic cavity	658.0x	Chronic renal disease	646.2, 581-588.xx
Severe pre-eclampsia	642.5x	Epilepsy	345.xx
Pre-eclampsia/eclampsia	642.6x, 642.7x	Asthma	493.xx
Placental abruption	641.2x	Anemia	280-289.xx
Premature rupture	658.10, 658.11, 658.13	Septic infection	038.xx
Previa placenta	641.0x, 641.1x	Psychiatric conditions
Threatened miscarriage	632.xx, 634.xx	Depression	296.xx, 300.4x, 301.1, 309.0x 309.1x, 309.28, 311.xx
Vaginal bleeding	623.8x, 636.2x	Anxiety or PTSD	309.2x, 309.4x, 309.81, 313.0x 300-300.3x, 300.5-300.9x
IUGR	656.5x	Bipolar disorder	296.0x, 296.4x, 296.4-296.8x
Antepartum hemorrhage	641.1-641.3x	Psychosis	291.0x, 292.81, 295.0-295.7x
	641.8x, 641.9x		295.9x, 296.24, 296.34, 296.44
			294.54, 297.1x, 298.8x, 298.9x
Postpartum hemorrhage	666.0-666.3x	Obstetric characteristics
Cardiac events	410.xx, 427.5x	Multiple gestation	651.xx, V272-V277.xx
Cardiac events†	379.1x, 996.0x, 996.3x	Substance abuse	292.xx, 305.2-305.9x
			340.xx, V654.2x
Fibromyalgia	729.1x	Tobacco abuse	305.1x
Known/suspected Fetal abnormalities	655.xx	Alcohol abuse	291.xx, 303.xx, 305.0x
Disorders*/early or threatened labor	765.xx, 644.0x, 644.2x	Obesity	278.xx, 649.1x

Abbreviations. IUGR: intrauterine growth restriction; PTSD: post-traumatized stress disorder.

^*:disorders relating to short gestation or low birth weight.

^†:procedure code.

FIGURE A1.

Posterior summaries of class-specific covariate coefficients in the 3-class model.

FIGURE A2.

Posterior summaries of class-specific covariate coefficients in the 6-class model.