Measuring hospital spatial wingspan by using a discrete choice model with utility-threshold

Policy makers increasingly rely on hospital competition to incentivize patients to choose high-value care. Travel distance is one of the most important drivers of patients’ decision. The paper presents a method to numerically measure, for a given hospital, the distance beyond which no patient is expected to choose the hospital for treatment by using a new approach in discrete choice models. To illustrate, we compared 3 hospitals attractiveness related to this distance for asthma patients admissions in 2009 in Hérault (France), showing, as expected, CHU Montpellier is the one with the most important spatial wingspan. For estimation, Monte Carlo Markov Chain (MCMC) methods are used.

It is often advocated that competition between hospitals may improve efficiency and 2 quality of health care [1]. In fact, in theory, due to increasing competition, hospitals 3 decrease production costs and improve the quality of health care delivered so as to 4 attract new patients. Factors of hospital choice include price, quality, distance or travel 5 time (easy access), waiting time, provider network and others. Their relative 6 importance differs according to the market characteristics and the regulatory context [2]. 7 Early studies identified distance or travel time as the major factor negatively affecting 8 hospital choice [3], [4] even if sensitivity to distance varies with patient characteristics 9 (age, ethnicity, income and religion), admission types (i.e., stronger effect of distance for 10 common procedures) and hospital type (i.e., weaker effect for admissions to larger 11 specialized hospitals) [5]. 12 As in many European countries, French hospitals are financed through Diagnosis 13 Related Groups based on a prospective payment system. Regarding acute care, this 14 scheme has been fully implemented in 2005 for private for-profit hospital budgets and in 15 2008 for public hospital budgets. The reform was intended to improve efficiency and 16 fairness in financing and also to increase competition between and within the public and 17 private sectors [6]. Information about performance and quality of hospitals is not yet perceived by the patients and the general practitioners is partially reflecting the quality 24 of care and easy access. In these circumstances, travel distance certainly plays a key 25 role in patients' decision [8]. Further, in areas where there are many hospitals within 26 short distance, the attractiveness of a hospital (easy access, care quality, . . . ) can be 27 measured by the distance travelled by patients to get there. That is to say, longer the 28 distance travelled, the more attractive the hospital. 29 In this context, a spatial modeling on hospitals choice may be very useful to propose 30 a classification of hospitals (most to least attractive). From a policy perspective, these 31 findings have implications for regulation. For instance, the effect of making the quality 32 of care information publicly available may not have the expected impact in areas where 33 facilities are scarce and distant from one to another. More generally, the predominant 34 effect of distance on hospitals choice should be acknowledged and taken into 35 consideration before implementing any reforms aimed at incentivizing hospitals 36 competitiveness.

37
In this work, we present a method to numerically measure the distance beyond which 38 no patient is expected to choose the hospital for treatment by using a new approach in 39 discrete choice models. This distance is used to compare hospitals attractiveness.

42
The paper is organized as follows. Section Data describes the data we used for 43 illustration while Section Proposed Approach explains the approach we introduced 44 to cope with the issue we encountered with the RUMs. An application of the approach 45 is presented in Section Illustration and the paper ends with a brief discussion.

47
The data we used in this work, were obtained in a fully anonymized and de-identified 48 manner from the Programme de Médicalisation des Systèmes d'Information (PMSI).

49
The PMSI is an administrative dataset recording all inpatient admissions in French 50 private and public hospitals covering all social health insurance schemes. It is intended 51 to describe the medical activities of hospitals. Information about the patient and the 52 hospital of stay such as the patient desease, his place of residence, location of the 53 hospital, type of hospital, are held in the dataset.

54
For our case study, we used data about asthma hospital patients stays in 55 Languedoc-Roussillon, a region known to have a high competitive health care market in 56 France. Located in the South of France, ( Fig 1), the region consists of five departments: 57 Aude, Gard, Hérault, Lozère and Pyrénées-Orientales. It extends from 42 • to 44.5 • 58 Latitude North and between 1.5 • to 5 • Longitude East. The Languedoc-Roussillon 59 region has a population of about 3 million people, which includes both rural and 60 metropolitan areas. It is the second region with the highest population growth (1.1%) 61 in France (about 0.4%) due essentially to immigrants and more than 16% of the 62 population are retirees. In 2009, 1,289 hospital patient stays have been recorded for asthma in this region as 64 reported in [9]. The study focused on admissions in 3 hospitals of Hérault: one is private for-profit and 2 are not-for-profit. given therapy); therefore discrete choice models can be used to model the data. Since it 77 seems reasonable to consider that a patient chooses an hospital when obtaining a 78 certain utility of doing so, Random Utility models (RUMs) [10] can be used.

79
The concept of these models is that for any given alternative j from J alternatives in 80 competition, an agent n is expected to obtain a certain utility U nj of choosing the 81 alternative. This utility U nj is often decomposed in the sum of 2 parts: i.e, U nj = V nj + nj . By considering nj , j = 1, . . . , J, random, thus assigning them a 87 distribution f and under the assumption that the agent chooses the alternative with the 88 greatest utility, the probability of choosing an alternative j by the agent n can be 89 derived as According to the specification of f (.), different specific models are obtained. For 91 example, the Logit model [11] is obtained under the assumption that nj are 92 independent copies of a Gumbel distribution, i.e, This model is by far the most widely used among RUMs due to some of its properties.

94
But through the model use, limitations arise: independence of unknown parts may not 95 always hold; Logit can not handle some cases of taste variation or some panel data. 96 Various models have been developed to cope with these limitations. We can cite the Nested-Logit model [12]. It is derived under the assumption that the cumulative joint where B k is a subset of the J alternatives partitioned in K non-overlapping subsets  Probit ( [13] for binary case) is developped under the assumption that with n = ( n1 , . . . , nJ ) T and Σ the covariance matrix. The problem with this model is 108 that the probability of choice is not closed-form and hence, requires simulations.

109
Mixed Logit [14] and [15], more flexible [16], is so as if β is the unknown parameter 110 of V nj , then the choice probability is . where g can be any density function on β.

112
In this study, we have patients admissions for asthma in 3 hospitals of Hérault that 113 have been registered in 2009. To use RUMs, we define the utility that may obtain a 114 patient n from choosing hospital j as that is to say α j and β j are unknown parameters specific to the hospital j and x n − x j , the euclidean distance between patient n and hospital j locations. β j controls how V nj varies according to the distance. By assessing α j positive and utility decreasing (thus β j negative) by far one moves away the hospital j, the ratio where d j is the distance beyond which V nj ≤ 0. This distance d j called wingspan, can 117 be interpreted as the distance beyond which no patient is expected to choose the 118 hospital considering the distance since the representative utility is negative.

119
However, first, r j can not be estimated with RUMs. In fact, the constants α j can 120 not be estimated in the expression Only the difference α j − α i can be estimated; and there is an infinity of constants α j 122 and α i giving the same difference. One more problem in estimating of r j , is that for 2 123 hospitals j and i in a same locality, β j and β i can not be estimated but their difference 124 Clinique le Millénaire in our study, which are both located in Montpellier. These two 126 issues make that r j can not be estimated with the RUMs, though knowing this ratio is 127 needed in this study in order to compare the attractiveness of the hospitals.

128
Second, the assumption of utility-maximization in choosing a hospital for treatment 129 can be not true for some patients (bounded rationality). 130 So, to estimate r j and cope with the situation where patients do not maximize the 131 utility, we propose a new approach. In our approach, the utility is really considered 132 deterministic and the assumption of utility maximization, which is criticized by many 133 psychologists is released for a more loose assumption by introducing a utility-threshold: 134 any alternative whose utility for the agent n exceeds a certain threshold S n , can be 135 chosen by this agent.

136
The approach

137
In order to cope with the estimation issue of ratio mentioned above, we present another 138 approach where the assumption of utility maximization used to derive the choice 139 probability is released. In fact, when an agent is faced with a choice among several 140 alternatives, he is often not sure which alternative he should select, and does not always 141 make the same choice under seemingly identical conditions, as noted by Tversky [17].
142 Therefore, it is strongly possible that an agent does not always maximize the utility 143 when choosing an alternative. As an agent does not always choose the alternative with 144 the greatest utility for him and as it is unlikely he chooses an alternative that he thinks 145 useless or harmful for him, we can suppose that he chooses an alternative from the 146 moment the utility he may obtain considering some aspects covers his certain need 147 (Bounded rationality). It is this certain need that we call a threshold of utility. It is a 148 minimum of utility that may demand the agent from an alternative. It can vary due to 149 socio-demographic variables, the type of alternatives, and other unknown factors. Since 150 we don't know how exactly it varies, though that these socio-demographic variables can 151 make an agent to be more or less demanding, we treat it random and assign it a 152 distribution. Thus, any alternative probability choice of the agent considering some 153 aspects can be derived.

154
To formalize the idea, consider that we have an agent n facing choice among J 155 alternatives. As for random utility models, these alternatives are in finite number. They 156 do not have to be mutually exclusive, but just exhaustive in the sense that any choice of 157 the agent may be met by the combination of these alternatives. Let U nj be the utility 158 that may obtain the agent n considering some aspects from choosing the alternative j. 159 The agent can choose the alternative if only the utility he may obtain from doing so, 160 exceeds a minimum of utility (called utility-threshold) S n , i.e, U nj > S n . The S n is 161 treated as random and assigned a probability density function f . The probability of an 162 alternative j to be chosen by the agent n is defined as just The greater U nj is, compared to S n , the more likely the alternative is going to be 164 chosen. Therefore, the alternative with the greatest utility considering the aspects has 165 the most chance to be chosen.

166
Considering that it is more realistic for an agent to choose an alternative if having a 167 benefit, the threshold is expected to be positive. This utility-threshold can be supposed 168 as a conjonction of many independant factors (incomes, religion, education. . . ).

169
Therefore, we can assess that the logarithm of S n has a normal distribution with 170 parameters (µ n , σ n ), i.e, Thus, 172 P nj = Pr(U nj ≥ S n ) = F µn,σn (U nj ), where F µn,σn is the cumulative log-normal distribution with parameters (µ n , σ n ). But 173 in general, it is known that absolute utility can not be measured, so µ may not be 174 known. We know that from the constraint, exp(µ n ) is from zero to infinity. So we 175 divide U n by exp(µ n ) and re-state the probability choice so as where U r nj = U nj / exp(µ n ), a relative utility which is to be estimated. For Eq (10), in 177 fact, it is easy to show that for any x for a log-normal distribution.

179
The probability of an agent choice can be written as where y nj =1 if j is chosen by n and 0 otherwise.

181
For a sample of N agents choosing among J alternatives and considering that the S n 182 are independent, the likelihood for A, the set of observed choices, is For unkown parameters estimation, likelihood maximization can be used; but in this 184 paper, a Bayesian framework is adopted.  Model under the proposed approach 191 Consider that we have N patients who face choice among J hospitals. We suppose that 192 the only observed variable that may determine their choice is travel distance. Therefore, 193 we define the utility that may obtain the patient n considering the distance from 194 choosing hospital j as j is positive, x n , x j , respectively geographic coordinates of the patient n and 196 hospital j locations and x n − x j , the eucludean distance between these locations. The 197 parameter β * j exhibits the effect of distance. Since the utility of a hospital is supposed 198 to reduce as one moves away, the parameter β * j is expected to be negative. From the assumption that the threshold S n is distributed log-normal, then for U nj ≤ 0, the 200 probability that the patient n chooses the hospital j is zero. It means that beyond the 201 distance d j so as α * j + β * j .d j = 0, no patient is supposed to choose the hospital j for 202 treatment. This distance will be used to compare hospitals attractiveness. The bigger 203 this distance is, the more attractive the hospital is. By estimating the ratio we can subsequently estimate d j .

205
The main target is now to estimate the ratio r j for the 3 hospitals with 229 patients 206 stays (Table 1) by using the approach above in a Bayesian framework.

207
Denote A, the set of choices actually observed. To reduce over parameterization, we 208 assume that the parameters of the log-normal distribution are the same for all the 209 patients since people in a same community share almost the same characteristics, i.e, 210 (µ n , σ n ) = (µ, σ), ∀n. Then we define the relative utility, without impact on r j because Supposing that S n , n = 1, . . . , 229 are independent, then the probability to have A, 213 the set of the observed choices, when, σ, α j , β j , j = 1, 2, 3 are known can be expressed 214 as 215 π(A | α 1 , α 2 , α 3 , β 1 , β 2 , β 3 , σ) = 229 n=1 j {P nj } ynj (13) where P nj = F 0,σ (α j + β j . x n − x j ) and F 0,σ , cumulative log-normal distribution 216 with parameters (0, σ). y nj = 1 when the agent n chooses the alternative j and zero 217 otherwise.

218
It can be good to use joint distribution for some of the parameters to take into 219 account the presence of other alternatives but we suppose them all independent. Hence, 220 for each of α j , β j and σ, we set a prior distribution. For α j , a uniform distribution can 221 be used. But since it is unlikely for α j to be infinite and we don't know its exact upper 222 bound, thus, we prefer to use an Inverse-Gamma distribution with scale and shape 223 parameters equal to 1. Also an Inverse-Gamma is used for σ but the scale and shape 224 parameters equal 2 and for β j , a Uniform U (−2, 0), as prior distributions.

225
Estimation methods

228
To compute r j , a sample of the parameters has to be drawn from the posterior 229 distribution π(α, β, σ | A).

230
MCMC algorithms such as Gibbs sampling [18] are often used to have approximate 231 draws from this kind of distributions. For Gibbs sampling, parameters conditional 232 distributions given in Eq 14 whose detailed expressions are given in Appendix are to be 233 used. But, these distributions are not closed-form; which means that drawing from 234 these distributions is not straightforward.
So, we used Metropolis-Hasting steps [19] within a Gibbs sampler to update each 236 parameter. For proposal ditribution, normal distribution was used for β j , and a 237 log-normal was used for α j and σ, for j=1,2,3.

239
To collect sample draws, we ran 100,000 iterations with the MCMC codes. We removed 240 the first 20,000 iterations considered as burn-in period of the chain, then in the  Table 2; which means that this ratio can be 249 used for comparison. The table 3 gives the computed average distance at which the 250 utility provided by the hospital is zero; which means that beyond this distance, no 251 asthma sick is expected to come in the hospital according to the data. For recall this 252 distance d j is obtained by equaling α * j + β * j .d j to zero, that's to say 1 + r j .d j = 0. The results clearly show that the CHU Montpellier is the most attractive and with 254 an important wingspan that goes beyond the Languedoc-Roussillon, the administrive 255 zone of the center, even if this wingspan has to be relativized due to the distance used 256 and the assumptions made on the sample. Its attractiveness is not surprising. CHU

257
Montpellier is a well-known center providing a high level healthcare service in 258 Languedoc Roussillon and has big accomodation capacity. It has also easy access due to 259 transportation facilities: Highway A9, railways (TGV and TER) and many other roads. 260 The rank of CH Béziers is plausible because it is in reality the second health center in 261 Hérault with lower activity than CHU Montpellier. with what we expected for the attractiveness of the hospitals in competition.

267
Though the results obtained seem to be plausible, in Bayesian analysis, it is 268 important to know how sensitive they are to the choice of the prior distributions. In our 269 illustration, it is good to know how the r j is sensitive to the choice of the different 270 priors; mainly the choice of the priors of α j and β j . So, to check the sensitivity, we keep 271 one of 2 priors constant and make vary the hyperparameters of the other. For β j , we 272 used unif(-5,0), unif(-10,0) and unif(-20,0) and no sensible variation are observed. But 273 when we make vary the prior of α j , the mean of α j becomes smaller when its prior is 274 more informative and bigger in the opposite. And by the way, it changes the mean of β j . 275 Even if the change is not very big, the mean of r j seems to become bigger for CH 276 Montpellier and smaller for Clinque le Millénaire (which has a small number of 277 admissions), when the prior of α j is more informative as reported in Table 4. But when 278 using less informative prior for α j , we noticed that the ratio for CHU Montpellier not 279 stable. So, caution has to be used when choosing the prior of α j . 280 Table 4.

281
In this paper, we presented a method to measure hospital spatial attractiveness by using 282 a new approach in discrete choice models. This approach releases the assumption of 283 utility-maximization which seems sometimes not realistic for a more loose assumption. 284 The assumption supposes a threshold of utility above which an agent can choose an In this section, we give the derivation of parameters conditional distributions used in the 294 Gibbs sampler (see section Illustration). Then the explicit form of the distributions is 295 obtained by replacing every term of these distributions by its mathematical expression. 296 For recall, Bayes formula is π(θ | x) = π(x | θ)π(θ) π(x | θ)π(θ) dθ ∝ π(x | θ)π(θ), where π(θ | x) is called posterior distribution, π(θ) prior distribution and π(x | θ), the 297 likelihood.