A comparison of care management delivery models on the trajectories of medical costs among patients with chronic diseases: 4-year follow-up results

Abstract

Care management is becoming increasingly offered in the U.S. as a means of helping patients better manage their chronic diseases and possibly avoid worsening or exacerbations of their conditions, which can result in potentially avoidable and costly healthcare services. Since care management can be provided using different intervention methods and by different entities, we sought to compare different models of care management delivery on their long-term medical savings. Specifically, we compare health plan provided care management to provider delivered care management. Evaluation of the effectiveness of care management programs can be challenging because it can take time for patients to make recommended changes and then demonstrate healthcare savings associated with those changes. In this study, we modeled the unknown form of the time-varying program effects using a spline-based technique in the Bayesian framework, where the number and locations of knots were treated unknown and learned via reversible jump Markov chain Monte Carlo. We also addressed additional modeling challenges from features seen in our healthcare cost data such as highly right-skewed outcomes with non-constant variances and extra zeros. To provide a more robust analysis, we incorporated a follow-up period of up to 4 years that is longer than the most of the published studies on care management. The results of this work demonstrate that cost savings do accrue with specific models of care management. In particular, the embedded model of care management was significantly more effective than the health plan provided care management in controlling medical costs after 2 years of engagement, and the savings increased over time. This information should help policy makers and employer groups achieve a better understanding of the potential value of care management.

Appendices

Appendix 1: Posterior computation

Let $\varvec{\varTheta }_k$ represent the current model with k knots located at $\varsigma _1,\ldots , \varsigma _k$ (i.e, $\varvec{\varTheta }_k=(\varsigma _1,\ldots , \varsigma _k)$). We describe the steps of sampling from the posterior conditionals given $\varvec{\varTheta }_k$ as follows:

• 1. Generate $\varvec{\beta }^{(k)}\sim MVN(\varvec{\varSigma }_1^{-1} \mathbf{L}_1,\; \varvec{\varSigma }_1^{-1})$, where

  $$\begin{aligned} \varvec{\varSigma }_1&= {} \frac{1}{\sigma _e^2 (1-\delta ^2)}\sum _{g=1}^{n_G}\sum _{i\in g}\sum _{j=1}^{n_i} \varvec{\varPsi }^{(k)}(t_{igj}) \varvec{\varPsi }^{(k)}(t_{igj})' +\frac{1}{\sigma ^2_\beta } \mathbf{I}_{{{M_\beta ^{(k)}}}}\;\; \text{ and } \\ \mathbf{L}_1&= {} \frac{1}{\sigma _e^2 (1-\delta ^2)}\sum _{g=1}^{n_G}\sum _{i\in g}\sum _{j=1}^{n_i} (Y_{igj}^*- u_g - v_{ig} - \sigma _e \delta Z_{igj}^*) \varvec{\varPsi }^{(k)}(t_{igj}). \end{aligned}$$

• 2. Generate the variables

  $$\begin{aligned} H_{igj}^*\sim & {} N(\xi _{igj},\; 1), \\ Y_{igj}^*\sim & {} N(\vartheta _{igj},\; \sigma _e^2 (1-\delta ^2)),\;\; \text{ and } \\ Z^*_{igj}\sim & {} N\left( {\left( Y^*_{igj}-\vartheta _{igj}\right) \delta }/{\sigma _e},\; 1-\delta ^2\right) \end{aligned}$$

  for $i=1,\ldots , n, j=1,\ldots , n_i$, and $g=1,\ldots , n_G$. The generated variable $H_{igj}^*$ should be truncated at 0 to $[0, \infty ]$ if $H_{igj}>0$ and to $[-\infty , 0]$ if $H_{igj}=0$.

• 3. Generate $\varvec{\alpha }\sim MVN(\varvec{\varSigma }_2^{-1}{} \mathbf{L}_2,\; \varvec{\varSigma }_2^{-1})$, where

  $$\begin{aligned} \varvec{\varSigma }_2&= {} \sum _{g=1}^{n_G}\sum _{i\in g}\sum _{j=1}^{n_i} \varvec{\varLambda }(t_{gj}) \varvec{\varLambda }(t_{igj})' + \frac{1}{\sigma ^2_\alpha } \mathbf{I}_{M_\alpha }\;\; \text{ and } \\ \mathbf{L}_2&= {} \sum _{g=1}^{n_G}\sum _{i\in g}\sum _{j=1}^{n_i} (H^*_{igj} - w_g - s_{ig})\varvec{\varLambda }(t_{igj}). \end{aligned}$$

• 4. Generate variances

  $$\begin{aligned} \sigma ^2_\alpha\sim & {} IG(a_1 + {M_\alpha }/{2},\; b_1 + {\varvec{\alpha }' \varvec{\alpha }}/{2})\;\; \text{ and } \\ \sigma ^2_{\beta }\sim & {} IG(a_2 + {M_{\beta ^{(k)}}}/{2},\; b_2+ {\varvec{\beta }^{(k)\prime }\varvec{\beta }^{(k)}}/{2}). \end{aligned}$$

• 5. Using the Metropolis–Hastings, generate the skewness variable $\delta$ from the conditional distribution;

  $$\begin{aligned} \pi (\delta \mid \cdot ) \propto (1-\delta ^2)^{-\frac{N}{2}}\exp \left[ {-\frac{\sum _{g=1}^{n_G}\sum _{i\in g} \sum _{j=1}^{n_i} \left( Y^*_{igj} - \vartheta _{igj}-\sigma _e \delta Z^*_{igj}\right) ^2}{2\sigma _e^2 (1-\delta ^2)}}\right] I_{\left( -1<\delta < 1\right) }, \end{aligned}$$

  where N is the total number of observations.

• 6. Generate the practice-level random effects $(w_g, u_g)$ for $g=1,\ldots , n_G$ from the conditional distributions

  $$\begin{aligned} w_g \mid u_g\sim & {} N\left( \frac{\left[ \sum _{i\in g}\sum _{j=1}^{n_i} H_{igj}^*- \varvec{\varLambda }(t_{igj})'\varvec{\alpha }- s_{ig}\right] - \overline{a}_{12} u_g}{M_g^*+ \overline{a}_{11}},\; \frac{1}{M_g^*+ \overline{a}_{11}}\right) \;\; \text{ and } \\ u_g \mid w_g\sim & {} N\left( \left[ \frac{\left( \sum _{i\in g}\sum _{j=1}^{n_i}Y_{igj}^*- \varvec{\varPsi }^{(k)}(t_{igj})'\varvec{\beta }^{(k)}-v_{ig}\right) }{\sigma _e^2(1-\delta ^2)} - \overline{a}_{12}w_g\right] S^{-1},\; S^{-1}\right) , \end{aligned}$$

  where $M_g^*$ is the total observations contributed by patients who belong to practice g and $S = {M_g^*}/[{\sigma _e^2(1-\delta ^2)}] +\overline{a}_{22}$. Note that $\overline{a}_{11}$, $\overline{a}_{12}$, and $\overline{a}_{22}$ are the (1,1), (1,2), and (2,2) entries form matrix $\mathbf{A}^{-1}$, respectively.

• 7. Generate the patient-level random effects $(s_{ig}, v_{ig})$ for all i from the conditional distributions

  $$\begin{aligned} s_{ig} \mid v_{ig}\sim & {} N\left( \frac{\sum _{j=1}^{n_i}(H_{igj}^*- \varvec{\varLambda }{(t_{igj})}'\varvec{\alpha }- w_g)-\overline{b}_{12}v_{ig}}{M^*_{ig} + \overline{b}_{11}},\; \frac{1}{M^*_{ig} +\overline{b}_{11}}\right) \;\; \text{ and }\\ v_{ig} \mid s_{ig}\sim & {} N\left( \left[ \frac{\sum _{j=1}^{n_i}(Y_{igj}^*- \varvec{\varPsi }^{(k)}(t_{igj})'\varvec{\beta }^{(k)}-u_{g}- \sigma _e\delta Z_{igj}^*)}{\sigma _e^2(1-\delta ^2)} - \overline{b}_{12}s_{ig}\right] S^{-1},\; S^{-1}\right) , \end{aligned}$$

  where $M^*_{ig}$ is the total observations contributed by the ith patient and $S = {M^*_{ig}}/{\sigma _e^2(1-\delta ^2)} + \overline{b}_{22}$. Note that $\overline{b}_{11}$, $\overline{b}_{12}$, and $\overline{b}_{22}$ are the (1,1), (1,2), and (2,2) entries form matrix $\mathbf{B}^{-1}$, respectively.

• 8. The covariance matrices for the practice-level and the patient-level random effects are generated from

  $$\begin{aligned} \mathbf{A}\sim & {} IW\left( v+n_G,\; I + \sum _{g=1}^{n_G} \varvec{\rho }_g \varvec{\rho }_g'\right) \;\; \text{ and } \\ \mathbf{B}\sim & {} IW\left( v+n,\; I+\sum _{g=1}^{n_G} \sum _{i\in g} \varvec{\eta }_{ig}\varvec{\eta }_{ig}'\right) , \end{aligned}$$

  respectively.

Dimension changing of the parameter space

For the dimension changing of the parameter space, we implement RJMCMC algorithm to update $\varvec{\varTheta }_k$. Denote the probabilities of three types of transitions from $\varvec{\varTheta }_{k}$ to the next model $\varvec{\varTheta }_{k^*}$: $b_{k}$ for the birth step, $d_{k}$ for the death step, and $\omega _k$ for the relocation step. By following the Dension’s paper (Dension et al. 1998), the probabilities are given as

$$\begin{aligned} b_k&= {} 0.4 \times \text{ min }\left\{ 1,\; {{Poi}(k+1)}/{{Poi}(k)}\right\} , \\ d_k&= {} 0.4 \times \text{ min }\left\{ 1,\; {{Poi}(k-1)}/{{Poi}(k)}\right\} ,\;\; \text{ and }\\ \omega _k&= {} 1-b_k-d_k. \end{aligned}$$

when $K = 1$ in the current model, we choose either for the birth step or relocation step with equal probabilities.

To transit from $\varvec{\varTheta }_k$ to $\varvec{\varTheta }_{k^*}$, we employ the jumping proposal in the spirits of Botts’s paper (Botts and Daniels 2008). For the birth step, we firstly choose one knot $\varsigma _{\breve{j}}$ randomly from the set of existing knots and generate a new knot $\varsigma ^{new}$ from a truncated normal distribution centered at $\varsigma _{\breve{j}}$ with standard deviation 1 such that $T^{{min}}<\varsigma ^{new}< T^{{max}}$. In this case, $\varvec{\varTheta }_{k^*} = (\varsigma _1, \ldots , \varsigma _{k}, \varsigma ^{new})$, where $k^*= k+1$ and the proposed probability is

$$\begin{aligned} J(\varvec{\varTheta }_k \rightarrow \varvec{\varTheta }_{k^*}) = b_k \times \frac{1}{k} \sum _{\varsigma _{\breve{j}} \in \varvec{\varTheta }_k} TN(\varsigma ^{new}\mid \varsigma _{\breve{j}}). \end{aligned}$$

For the death step, we randomly choose a knot from the set of existing knots and delete it. Let $\varsigma _{\breve{j}}$ be such a knot. In this case, $\varvec{\varTheta }_{k^*} = (\varsigma _1, \ldots , \varsigma _{\breve{j}-1}, \varsigma _{\breve{j}+1}, \ldots , \varsigma _{k})$, where $k^*= k-1$ and the proposed probability is

$$\begin{aligned} J(\varvec{\varTheta }_k \rightarrow \varvec{\varTheta }_{k^*}) = d_{k} \times \frac{1}{k}. \end{aligned}$$

For the relocation step, we firstly choose one knot $\varsigma _{\breve{j}}$ randomly from the set of the existing knots and generate a new knot $\varsigma ^{new}$ as described in the birth step to replace $\varsigma _{\breve{j}}$. In this case, $\varvec{\varTheta }_{k^*} = (\varsigma _1, \ldots , \varsigma _{\breve{j}-1}, \varsigma ^{new}, \varsigma _{\breve{j}+1}, \ldots , \varsigma _{k})$, where $k^*= k$ and the proposed probability is

$$\begin{aligned} J(\varvec{\varTheta }_k \rightarrow \varvec{\varTheta }_{k^*}) = \omega _{k} \times \frac{1}{k} \times TN(\varsigma ^{new}\mid \varsigma _{\breve{j}}). \end{aligned}$$

To determine whether to move from $\varvec{\varTheta }_k$ to $\varvec{\varTheta }_{k^*}$, we calculate the acceptance probability as

$$\begin{aligned} \varDelta (\varvec{\varTheta }_k \rightarrow \varvec{\varTheta }_{k^*}) = \min \left\{ 1, \frac{\varPi _{\varvec{\varTheta }_{k^*}}}{\varPi _{\varvec{\varTheta }_k}} \times \frac{J(\varvec{\varTheta }_{k^*} \rightarrow \varvec{\varTheta }_{k})}{ J(\varvec{\varTheta }_{k} \rightarrow \varvec{\varTheta }_{k^*})}\right\} , \end{aligned}$$

where $\varPi _{\varvec{\varTheta }_k} \propto \mid \varvec{\varSigma }_1 \mid ^{-1/2} \exp ({\mathbf{L}_1 \varvec{\varSigma }_1^{-1} \mathbf{L}_1}/{2})$ and $\varvec{\varSigma }_1$ and $\mathbf{L}_1$ are defined in the step 1 in the posterior computation.

Appendix 2: Simulation

The simulation study was carried out to compare the performance among seven competing models that differ by the degree of complexity in terms of how robust each of the methods is to sensible changes under different data-generating scenarios: (1) positive outcomes are highly right skewed and processes generating non-zero outcomes and magnitude of them are highly correlated, (2) positive outcomes are highly right skewed and processes generating non-zero outcomes and magnitude of them are uncorrelated, (3) positive outcomes are slightly skewed and processes generating non-zero outcomes and magnitude of them are highly correlated, and (4) positive outcomes are slightly skewed and processes generating non-zero outcomes and magnitude of them are uncorrelated.

Under each scenario, we generated 250 samples. Each sample has $n=500$ subjects and each subject has $n_i=13$ observations at time $-1.5, -1, \ldots , 4, 4.5$. We randomly assigned subjects to $n_G=100$ groups. The outcomes $Y_{igj}$ for $i=1,\ldots , n, j=1,\ldots , n_i$, and $g=1,\ldots , n_G$ were sampled via

$$\begin{aligned} \text{ Probit }\left[ \Pr (H_{igj}=1)\right]&= {} \varvec{\varLambda }(t_{igj})'\varvec{\alpha }+ w_g + s_{ig} \;\; \text{ and } \\ \log Y_{igj}\mid H_{igj}=1\sim & {} SN\left( \varvec{\varPsi }^{(K)}(t_{igj})'\varvec{\beta }^{(K)} + u_g + v_{ig},\; \sigma ^2_e = 1,\; \varpi =\frac{\delta }{\sqrt{1-\delta ^2}}\right) \end{aligned}$$

for $t_{igj}=-1.5, -1.0, \ldots , 4, 4.5$. Without loss of generality, the covariate vectors $\varvec{\varLambda }(t_{igj})$ and $\varvec{\varPsi }^{(K)}(t_{igj})$ only included time effects, where $\varvec{\varLambda }(t_{igj})' = (1,\; t_{igj},\; t^2_{igj})$ and $\varvec{\varPsi }^{(K)}(t_{igj})'= \left( 1,\; t_{igj},\; t^2_{igj},\; (t_{igj}-\varsigma _1)^2_{+},\; (t_{igj}-\varsigma _2)^2_{+},\; (t_{igj}-\varsigma _3)^2_{+}\right)$ included $K=3$ piecewise quadratic truncated polynomials located at $\varsigma _1 = -1, \varsigma _2 = 0.5$, and $\varsigma _3 = 3.5$. We chose $\varvec{\alpha }= (1,\; 0.05,\; -0.01)'$ to produce a moderate number of zero outcomes. The vectors $\varvec{\beta }^{(K)}\sim MVN\left( \mathbf{0},\; \mathbf{I}_{\text{ card }\left( \varvec{\varPsi }^{(K)}\right) }\right)$ were sampled to depict true time trends.

Depending on the scenario, we generated the skewness parameter $\delta \sim U[0.8, 1)$ for scenarios 1 and 2 and $\delta \sim U[0, 0.2]$ for scenarios 3 and 4; random effects for group $(w_g,\; u_g)'$ and subject $(s_{ig},\; v_{ig})'$ were both simulated from a bivariate normal distribution with zero mean and covariance $\begin{pmatrix} 1 &{} \rho \\ \rho &{} 1 \end{pmatrix}$ where $\rho \sim U[0.75, 1)$ for scenarios 1 and 3 and $\rho = 0$ for scenarios 2 and 4.

We fit samples generated under each scenario with seven models: (1) log normal distribution, (2) gamma distribution, (3) independent probit log normal distribution, (4) correlated probit log normal distribution, (5) independent probit three-parameter generalized gamma distribution, (6) correlated probit three-parameter generalized gamma distribution, and (7) our proposed model. The parametrization used for three-parameter generalized gamma distribution is consistent with the formulation in Manning et al. (2005). We calculated the E to O ratio defined to be the ratio of expected to observed values of Y. With all the E to O samples, we calculated longitudinal root mean square error (LRMSE) as the square root of the mean square error between E to O ratios and 1 across all the time measurements.

We found models 1–6 produced some very large E to O ratios at points where they fail to capture the true curvatures, especially when the time trend drastically drops or increases. In view of this, we interpret the results in two ways. Firstly, we calculated the percentage of times extreme E to O ratio happens. The ratio is considered as extreme if it is larger or equal than 10. Secondly, we calculated LRMSE and its 95 % confidence interval based on non-extreme ratios. In this way, we conclude the performance of each model in how well they fit unusual and usual time trends. The simulation results are provided in Table 4.

Table 4 Simulation results to compare the performance among seven competing models under four data-generating scenarios

Although using simple framework to illustrate complex processes has its intuitive appeal, when it is built on assumptions with unreasonable connections to the reality, the results are often biased. As Table 4 shows, simplified models (models 1 and 2) are proven to be unrealistic choices as they fail to sufficiently account for excessive zeros and potential skewness. Compared to all the other models considered, they are at higher risk of reaching false conclusions. On the other hand, complex models fold in more information and generally perform better. Given scenarios 1 and 2 where skewness is present, models 5 and 6 outperform models 3 and 4 as the third parameter in the generalized gamma distribution helps explain the inherent skewness. Besides, under scenarios 1 and 3, models incorporating correlation perform better. However, the results from models 1–6 reveal varying degrees of sensitivity to different scenarios as they appear to perform relatively better under certain scenario than the others. In contrast, model 7 shows substantially improved estimates and more consistent LRMSE values.

We also fit our application data with models 1–6 and compared the results with the estimated difference in changes of costs $\hat{\zeta }(t)$ shown in Table 3 to understand how the net benefit from the care management program changes if a different model is applied for the analysis. As Fig. 2 shows, models 1 and 2 capture trends that are way off, giving overestimated or underestimated results in either PDCM embedded or centralized models. Model 3 would lead to a different conclusion on the PDCM centralized effect as it trends upward over time. Model 4 would overestimate the effect from the PDCM centralized group in the early years of engagement given the confidence band sits below that of $\hat{\zeta }(t)$. Models 5 and 6 seem to better approximate $\hat{\zeta }(t)$ and would have similar conclusions about the program effects.

It appears as though models 5 and 6, compared to models 1–4, better fit the application data, assuming the real time effects can be modeled parametrically still limits its applicability in cases when this assumption is violated as seen in the simulation study.

Fig. 2

The plot captures the estimated difference in changes of costs and its 95 % confidence band from models 1 to 6 (in black) and the proposed model $\hat{\zeta }(t)$ (in gray) for (1) $G = 1$ (PDCM embedded model) versus $G = 3$ (HPDCM model) in the upper panel and (2) $G = 2$ (PDCM centralized model) versus $G = 3$ (HPDCM model) in the lower panel. The x-axis measures the time from the engagement in years and the y-axis represents the estimated difference in changes of costs