# A Predictive Model for Readmission of Patients with Congestive Heart Failure: A Multi-hospital Perspective

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10 depends on the medical treatments that she receives, her health status and prior hospitalization history. Consequently, researchers in the marketing literature have proposed that the Erlang-2 distribution be used to model inter-purchase times, as it more closely resembles customer purchase behavior (Chatfield and Goodhardt 1973, Gupta 1991, Jeuland et al. 1980, Morrison and Schmittlein 1981, Fader et al. 2005). Hence, the Erlang-2 distribution is relevant in our context of patient readmission behavior as it assumes that timing of the next admission is conditionally dependent on the duration of the previous admission. The Erlang-2 distribution takes the form of f ( x;2, λ) = λ xe for x, λ 0, where x is a continuous i random variable and λ is the (readmission) rate parameter. We follow Gupta's (1991) general approach which specifies patient i's probability, or hazard function, to pay a hospital visit in time period t, given a set of time-varying covariates, X t, as 2 λx where is the baseline hazard at time t (Cox 1972). λ t X 1 (, ) t γ = λ λ φ() ht X e t t t t (2) We follow Seetharaman and Chintagunta's (2003) formulation of the continuous time hazard model to incorporate time-varying covariates in the Erlang-2 distribution. The patient-level survivor function of the inter-admission time distribution between the (j-1) th and the j th admission is specified as: tj tj j j 1 t = j j u du j t j 1 tj 1 St ( t, X ) ( 1+ λ φ( ) ) exp[ λ φ( u) du] = (1 + λψ( t, t )) exp[ λψ( t, t )] j j j 1 j j j 1 (3) where (4) The individual probability density function during the time interval ( t, t ], given a covariate j 1 j vector X t, then follows j 2 j 1 λj γ1 t λ j j φ j ψ j j 1 λψ j j j 1 f( t t,, X ) = ( t ) ( t, t ) exp[ ( t, t )]. j (5) The overall likelihood at the patient level is simply the product of equation (5) over j as: J L(T i, γ 1 λ, X t ) = i j=1 f(t j t j 1 λ j, γ 1, X tj ) + τ i (6) 10

11 where τ i ~ N(μ, σ 2 ) represents patient-level random effects to capture the unobserved heterogeneity of individual patients (where µ and σ represent the mean and standard deviation of the random effect normal distribution function, respectively). We note that incorporating time-varying covariates into the Erlang-2 Gamma model in this manner is a new methodological contribution to the literature. Although Fader et al. (2004) also incorporate time-varying covariates into the Erlang-2 distribution, they only model the grouped duration data where time is grouped into weeks and the covariates are assumed to be constant during a week's interval. Wooldridge (2010) argues that this treatment is unsuitable for multi-spell data, where the event can occur multiple times during the chosen time interval. Since patient admissions can occur at any time and multiple times during a certain time period (e.g. a month), and considering that the covariates (comorbidities) may change at any time, we need to model readmissions along a continuous time frame. Further, to account for the fact that the rate of patient visits may differ across patients, we adopt the common mixture distribution for λ (Winkelmann 2010) which is assumed to be gamma distributed with α r r 1 λα 1 shape parameter r and scale parameter : g( λ r, α) = αλ e Γ () r. The flexibility associated with the Gamma distribution allows it to fit various shapes of distributions due to its additive and conjugate properties. Gupta (1991) observes that the most appropriate specification for inter-purchase time is a model that features an Erlang-2 inter-purchase process with gamma-distributed purchase rates (to account for customer heterogeneity). Based on the assumptions that inter-admission times are distributed according to (5) and that unobserved heterogeneity in λ follows a gamma distribution, we specify the likelihood function as, (7) We further consider the possibility that, after every admission, a patient can become inactive with a dropout probability of p from a geometric process. However, each patient is likely to have a different p, the cause of which may not always be observed, such as death or relocation outside the region, which leads to a data truncation problem. For a geometric process, this unobserved heterogeneity is commonly modeled 11

12 through a beta mixing function for the binary-outcome geometric processes (Winkelmann 2010). Taken together, this yields the beta-geometric distribution (Fader et al. 2005) which is specified as: P(dropout after jth admission) = p(1 p) j 1 (8) where the heterogeneity in dropout probabilities follows a beta distribution, with parameters a and b a b indicating the relative propensity of dropping out or not: f( p ab, ) = p (1 p) Bab (, ) Thus a patient may be inactive either after T (i.e. no observation is made between the last admission and the end of the period) or right after the last admission. We model these cases as follows: i. A patient is inactive after T: 3. (9) ii. A patient becomes inactive right after the last admission J: J J 2J L( λ t1,, tj, T, inactive at time τ ( tj, T]) = λ φ( tj) ψ( tj, tj 1) exp λ ψ( tj, tj 1) (10) j= 1 j= 1 This yields the likelihood function: (11) Expectation over the distribution of λ yields the likelihood function, (12) where ( r+ 2 J) r J J Γ ( r+ 2 J) α A1 = φ( tj) ψ( tj, tj 1) α + ψ( tj, tj 1 + ψ( T, tj ) (13) j= 1 Γ() r j= 1 ( r+ 2 J) r J J Γ( r+ 2 J) α A2 = φ( tj) ψ( tj, tj 1) α + ψ( tj, t j 1) (14) j= 1 Γ() r j= 1 Taking expectation over λ and p yields the individual likelihood function, 3 For a full derivation of equations 9-15, see Appendix A in the Online Supplement. 12

13 (15) Hence, the log-likelihood of N patients, who have at least one readmission, is specified as: N LL = log( L ( r, α, a, b, γ Xi( t), Ji, tj, T )). i i 1 1 i= 1 Combining it with the logit hurdle component, for a patient with no readmission after the initial admission (J=0), the overall likelihood function simplifies to (16) r r 1 αγ r αγ α ψ e r ( T,0) L = 0 exp( λψ ( T,0) ) (1 + λψ ( T,0)) dλ = Γ( r) α + ψ( T,0) α + ψ( T,0) (17) Altogether, this yields a BG/EG Hurdle model, the likelihood function of which is given by N 1 di d (1 θ0 ) i 1 di θ0 L 1 d 1 i i= 1 (1 L0 ) L = (18) where, θ 0 = P( J = 0) and d = 1 min{ J,1} i i i i The log-likelihood of the BG/EG Hurdle model is therefore N LL = ( d log θ + (1 d ) log(1 θ ) (1 d ) log(1 L ) + (1 d ) LL, i= 1 i 0i i 0i i 0i i 1i where the first two terms of the right hand side refer to the hurdle, while the last two terms are the likelihood of positive count of readmissions Contrast with the Literature We now recap the key methodological contributions that differentiate our study from prior studies in the literature. Table 1 provides a summary of the key differentiators of our study in contrast with the prior literature, where we draw upon studies from multiple disciplines. With respect to research design, there are several distinguishing features of our paper. First, none of the prior studies have focused on the association between health IT and patient readmission risk. Second, although the marketing literature has numerous studies that integrate the estimation of the propensity and timing of consumer purchases, the prior healthcare (19) 13

15 observe that this is a major improvement in our study compared to previous studies which have been restricted to studying patient readmission data from hospitals that belong to a single hospital system (Silverstein et al. 2008). Our data records 65,188 admissions which originate from 40,983 distinct patients with CHF as their primary diagnosis. Among these patients, 70% had a single admission, while 30% (12,211) experienced multiple admissions as shown in Figure B.1 in Appendix B. Table B.1 in the Appendix provides a description of our sample data. Our data captures several patient demographic characteristics including gender, racial profile and discharge age. Among all patients, 52% (21,281) are female, 72% (29,320) are Caucasian, and 21% (8,686) are African Americans. The average discharge age is sixty-nine years, with 66% (27,134) of the patients being sixty-five years or older. The hospital health IT usage data are drawn from the HIMSS Analytics database for the corresponding four-year period, i.e to After consulting with health IT practitioners, and based on the intensity of health IT usage among our sample of 67 hospitals, we identify 45 applications that are commonly used in treatment of CHF patients. We omit various IT applications that are only relevant to general hospital management, and focus instead on clinical and administrative functions associated with CHF. Exploratory factor analysis (EFA) on this group of applications results in selection of 18 health IT applications, where the factor scores are greater than the threshold of 0.6 for significance. Such clustering of HIT has been previously employed in healthcare settings to group health IT applications according to their primary functionality (Bhattacherjee et al. 2007; Himmelstein et al. 2010; Bardhan and Thouin, 2013). We code each HIT application as a binary variable where one indicates that it has been implemented and operational, and zero indicates otherwise. Using EFA with Varimax rotation, we identify three major classes of health IT applications: Administrative IT, Clinical IT, and Cardiology IT. Administrative IT consists of health information management applications, chart tracking, revenue cycle management, and patient billing applications. Clinical IT comprises of hospital-wide clinical systems such as EMR, operating room IT, and CPOE applications. Cardiology IT, which are primarily used to treat CHF patients, include cardiology information systems, cath lab systems, echocardiology and computerized tomography (CT) 15

17 types. For each admission, hospitals record the admission type and the risk of patient mortality. The standard admission type is coded into six classes (class 1 denotes emergency). Risk mortality is coded on a scale from 1 to 4, which indicates the patient death risk as minor, moderate, major, or extreme, respectively. Congestive heart failure is likely to be accompanied by other comorbidities. Frequent comorbidities associated with CHF include diabetes mellitus, hypertension, peripheral vascular disease, chronic pulmonary disease, renal failure, anemia, alcohol abuse, drug abuse, and ischemic heart disease (Ross et al. 2008). We control for these comorbidity variables which are identified by the Elixhauser index (Elixhauser et al. 1998) based on the ICD-9-CM (International Classification of Diseases) diagnosis codes. Other hospital characteristics may also affect patient readmission rates. We include hospital-level control variables such as the number of beds, case mix index, and teaching/non-teaching hospital attributes. Num_beds represents the number of beds available for use in each hospital, and represents hospital capacity. We control for the hospital case mix index (CMI) which accounts for the average severity of patients disease case mix. The teaching/non-teaching indicator represents the academic status of the hospitals. Table 2 provides definitions as well as descriptive statistics on our model variables. 5. Empirical Analysis We first discuss identification of the potentially endogenous HIT variables and then present the results of our empirical estimation, starting with the baseline results and followed by the BG/EG hurdle estimation Identification of the Health IT Effects It is likely that our health IT variables which we construct might be subject to endogeneity. For example, having higher level of readmissions may prompt a hospital to implement HIT (i.e. simultaneity). Identification of the causal effect of health IT can be challenging, because it is hard to isolate it from a hospital s efforts to improve patient outcomes such as readmission reduction. To address endogeneity, we identify two instrumental variables (IV): average level of health IT in (external) peer hospital systems, and the difference in a hospital s level of health IT between the current and the previous year. The first IV is defined as the average level of HIT among peer hospitals, after excluding those other hospitals within the 17

18 same health system as the focal hospital. The second IV is derived by taking the difference in the values of the HIT variables across two consecutive years. Using the difference of an endogenous variable as an IV was proposed by Arellano and Bond (1991) and has become a well-accepted method to account for endogeneity. Both IVs are correlated with the current year s level of HIT of the focal hospital since (a) the hospital is likely to monitor and follow the HIT applications that its peer hospitals have implemented (due to peer pressure), resulting in correlated hospital-level IT decisions and (b) the previous year s HIT (and by construction the derived difference) ought to be correlated with the level of HIT in the current year. At the same time, these two IVs are unlikely to be systematically determined by an individual patient s readmission at the focal hospital. In other words, competition in the local healthcare market may drive health IT implementations such that the IT infrastructure of competing hospital systems may influence the focal hospital s decision to implement HIT. However, it is unlikely that HIT implementation decisions of peer hospitals will be associated with an individual patient s readmission rate at the focal hospital. Likewise, a hospital s current level of HIT may depend on its level in the prior year, but prior year levels (and the derived difference) predates patient outcomes in the current year and thus is unlikely to be systematically co-determined. Hence, both variables fulfill the criteria for IV estimation. We also test the strength and exogeneity of our IV. The F-value of the Administrative IT, Clinical IT, and Cardiology IT variables, in the first stage, are , , and 578.l3 respectively, confirming the strength of these IVs. Furthermore, the Hansen s J test provides a test statistic of J= (p = ), supporting the exogeneity of these IVs Baseline Model Results For patient-level analysis, our objective is to develop a better understanding of the determinants of readmission propensity within 30 days of discharge from the previous admission. The dependent variable in the logit model is measured as a binary variable which takes a value of one in the presence of a 30-day readmission, and zero otherwise. The dependent variable in the proportional hazard model is measured as the time interval between two consecutive admissions that occur within a 30-day window. If a readmission occurs outside the 30-day window, we treat the subsequent visit as a new admission, a common practice in research and practice (Joynt and Jha 2012). We present our estimation results for the logit and Cox 18

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