Claims data-driven modeling of hospital time-to-readmission risk with latent heterogeneity

Abstract

Hospital readmission risk modeling is of great interest to both hospital administrators and health care policy makers, for reducing preventable readmission and advancing care service quality. To accommodate the needs of both stakeholders, a readmission risk model is preferable if it (i) exhibits superior prediction performance; (ii) identifies risk factors to help target the most at-risk individuals; and (iii) constructs composite metrics to evaluate multiple hospitals, hospital networks, and geographic regions. Existing work mainly addressed the first two features and it is challenging to address the third one because available medical data are fragmented across hospitals. To simultaneously address all three features, this paper proposes readmission risk models with incorporation of latent heterogeneity, and takes advantage of administrative claims data, which is less fragmented and involves larger patient cohorts. Different levels of latent heterogeneity are considered to quantify the effects of unobserved factors, provide composite measures for performance evaluation at various aggregate levels, and compensate less informative claims data. To demonstrate the prediction performances of the proposed models, a real case study is considered on a state-wide heart failure patient cohort. A systematic comparison study is then carried out to evaluate the performances of 49 risk models and their variants.

Appendix: E-M algorithms derivations

Maximizing (3) consists of two steps repeating iteratively, namely the expectation step (i.e., E-step) and the maximization step (i.e., M-step). In the E-step of iteration r, a conditional expectation is computed for Eq. 3, i.e., $\text {Q}(\boldsymbol {\beta },\lambda _{0}(t),\boldsymbol {{\gamma }})=\text {E}_{\boldsymbol {{\gamma }}|\mathbf {D},\boldsymbol {\beta }^{(r)},\lambda _{0}(t)^{(r)}}(\text {L}(\boldsymbol {\beta },\lambda _{0}(t)|\mathbf {D},\boldsymbol {{\gamma }}))$, which can be explicitly expressed as

$$ \text{Q}(\boldsymbol{\beta},\lambda_{0}(t),\boldsymbol{{\gamma}}) =\text{Q}_{1}(\boldsymbol{\beta},\lambda_{0}(t))+\text{Q}_{2}(\boldsymbol{{\gamma}}), $$

(6)

where β^(r) and λ₀(t)^(r) are updated values of β and λ₀(t) at iteration r. The two separate parts can be expressed by

$$\begin{array}{@{}rcl@{}} \text{Q}_{1}(\boldsymbol{\beta},\lambda_{0}(t))&=& \sum_{j = 1}^{N_{\text{J}}}\sum_{i = 1}^{n_{j}}[\delta_{ij}\{\log\lambda_{0}(t_{ij})+\boldsymbol{\beta}^{\text{T}}\mathbf{X}_{ij}\\&&+\text{E}({\gamma}_{j}|\mathbf{D},\boldsymbol{\beta}^{(r)},\lambda_{0}(t)^{(r)})\} \\ &&-{\Lambda}_{0}(t_{ij})\exp(\boldsymbol{\beta}^{\text{T}}\mathbf{X}_{i}\\ &&+\log\text{E}(\exp({\gamma}_{j})|\mathbf{D},\boldsymbol{\beta}^{(r)},\lambda_{0}(t)^{(r)}))] \end{array} $$

(7)

and

$$ \text{Q}_{2}(\boldsymbol{{\gamma}})=\sum_{j = 1}^{N_{\text{J}}}\text{E}(\log(p({\gamma}_{j}))|\mathbf{D},\boldsymbol{\beta}^{(r)},\lambda_{0}(t)^{(r)}). $$

(8)

To evaluate Q(β,λ₀(t),γ), the conditional expectation term $\text {E}(g({\gamma }_{j})|\mathbf {D},\boldsymbol {\beta }^{(r)},\lambda _{0}(t)^{(r)}) = \int g({\gamma }_{j})p({\gamma }_{j}|\mathbf {D},\boldsymbol {\beta }^{(r)},\lambda _{0}(t)^{(r)})d{\gamma }_{j}$ need to be computed, where g(γ_j) ∈{γ_j, exp(γ_j), log(p(γ_j))}. Monte Carlo simulation based on the conditional distribution p(γ_j|D,β^(r),λ₀(t)^(r)) can be performed to compute these expectations numerically [39].

In the M-step of iteration r, the coefficient vector β and baseline readmission rate function λ₀(t) are updated by maximizing Q₁(β,λ₀(t)). When λ₀(t) is modeled parametrically (e.g., Weibull, Exponential, etc.) with the parameters 𝜃_λ, Q₁(β,𝜃_λ) can be directly maximized based on numerical optimization methods, such as the Newton-Raphson method, which is readily available in many computing packages and routines. When λ₀(t) is modeled non-parametrically with Cox baseline specification, directly maximizing Q₁(β,λ₀(t)) is not straightforward. A partial likelihood approach is considered to update β by treating λ₀(t) as nuisance parameters and maximizing the partial likelihood function Q1′(β), which is explicitly given by

$$\begin{array}{@{}rcl@{}} \text{Q}_{1}^{\prime}(\boldsymbol{\beta}) &=& \sum_{j = 1}^{N_{\text{J}}}\sum_{i = 1}^{n_{j}}\delta_{ij}[\boldsymbol{\beta}^{\text{T}}\mathbf{X}_{i} \\ &&-\log\sum_{t_{lj}\geq t_{ij}}\exp\{\boldsymbol{\beta}^{\text{T}}\mathbf{X}_{l}\\&&+\log\text{E}(\exp({\gamma}_{j})|\mathbf{D},\boldsymbol{\beta}^{(r)},\lambda_{0}(t)^{(r)})\}]. \end{array} $$

(9)

Equation 9 resembles the partial log-likelihood for the Cox proportional hazard model and thus can be maximized based on the readily available packages for estimating the Cox proportional hazard model [40]. After maximizing $\text {Q}_{1}^{\prime }(\boldsymbol {\beta })$, β at iteration r can therefore be updated as β^(r+ 1). In the meantime, λ₀(t)^(r+ 1) and its counterpart of cumulative baseline function Λ₀(t)^(r+ 1) can be calculated non-parametrically with the Nelson-Aalen estimator at iteration r [41].

Due to the augmentation techniques in the E-M method, latent variables γ_j’s can be directly estimated by maximizing Q₂(γ). Different distributions, such as Gamma and Lognormal distributions, can be assumed for the latent variables γ_j’s. For instance, when Gamma distribution is assumed, each latent variable ${\gamma }_{j}^{(r + 1)}$ at iteration r can be updated as

$$ {\gamma}_{j}^{(r + 1)} =\frac{1/({\sigma}^{2})^{(r)}+\sum_{i = 1}^{n_{j}}\delta_{ij}}{1/({\sigma}^{2})^{(r)}+\sum_{i = 1}^{n_{j}}{\Lambda}_{0}(t_{ij})^{(r + 1)}\exp((\hat{\boldsymbol{\beta}}^{\text{T}})^{(r)}\mathbf{X}_{ij})}, $$

(10)

where Λ₀(t)^(r+ 1) and (σ²)^(r) are updated cumulative baseline function and variance on the latent variable, respectively.