Healthcare Quality by Specialists Under a Mixed Compensation System: An Empirical Analysis

ABSTRACT

We analyze the effects of a mixed compensation (MC) scheme for specialists on the quality of their healthcare services. We exploit a reform implemented in Quebec (Canada) in 1999. The government introduced a payment mechanism combining a per diem with a reduced fee per clinical service. Using a large patient/physician panel dataset, we estimate a multi‐state multi‐spell hazard model analogous to a difference‐in‐differences approach. We compute quality indicators from our model. Our results suggest that the reform reduced the quality of MC specialist services measured by the risk of rehospitalization and mortality after discharge.

1. Introduction

Over the past few decades, healthcare expenditures have increased substantially and now represent an important and rising share of gross domestic product (GDP) in many developed countries.

In this context, one may wonder whether the resources from the healthcare sector could be allocated more effectively, especially those provided by physicians. In Canada, which provides the institutional setting for this paper, physician spending represented 13.6% of total healthcare spending in 2022. This corresponds to the second largest category after hospitals (see Canadian Institute of Health Information CIHI ^2022a).

Therefore, it is clear that the design of physicians' payment mechanisms that provide effective quantity and quality of clinical services is a crucial priority for healthcare policy.

Three pure physician payment mechanisms are in place in Canada (CIHI ^2022b). These include fee‐for‐service (payment per service), capitation (payment per beneficiary), and salary (payment per unit of time). There are also mixed compensation (MC) systems that combine at least two payment mechanisms. The literature on the theoretical analysis of payment systems on physicians' behavior and health outcomes is numerous (e.g., Ellis and McGuire ¹⁹⁸⁶; Ma and McGuire ¹⁹⁹⁷; McGuire ²⁰⁰⁰; Choné and Ma ²⁰¹¹; Brekke et al. ²⁰²⁰). At the normative level, it concludes, among others, that a well‐designed MC system may improve social welfare compared with a pure payment system. One main reason is that an MC system can theoretically use more optimizing policy instruments than any pure payment system, at least when one ignores its potentially higher administrative costs. However, at the positive level, the empirical evidence regarding MC systems is rare.

The basic reason is the scarcity of natural experiments, such as reforms introducing an MC system, which can be used to analyze its effects on the quantity and quality of healthcare services. ¹

In this paper, we analyze a major MC physician payment reform introduced in Quebec (Canada) in 1999. More specifically, we examine its impact on the quality of healthcare services provided by hospital departments that switched from a fee‐for‐service (FFS) to an MC system. We develop and estimate a multi‐state multi‐spell (MSMS) hazard model with unobserved heterogeneity to do so. Several output‐based indicators for healthcare quality, such as readmission to hospitals and mortality rate in hospital and after discharge, are used.

The MC reform introduced for specialists by the Quebec government combines FFS and salary systems. ² Before 1999, most specialist physicians in Quebec (92%) received payments through an FFS system. Under the public MC system, specialists receive the same wage (per diem) ³ for time spent working in hospital, combined with a reduced fee per service (on average, 41% of standard fees). During a per diem, a specialist can perform not only clinical but also non‐clinical services such as administrative duties and teaching, which are unpaid under the FFS system. The MC system is optional, and unanimity by vote is required at the department level to adopt the system. Therefore, it may happen that, in a given hospital, some departments may choose to opt for the MC system, while in other departments, doctors are paid under the FFS system.

Using administrative panel data and based on a difference‐in‐differences approach, Dumont et al. (2008) have studied the effect of this reform on the quantity of clinical services delivered by MC doctors. According to their results, the latter reduced their clinical services by 6.15%. A crucial issue is whether the MC doctors substituted quality for the quantity of these services. Intuitively, such a reform may exert two opposite effects on the quality of clinical services provided by MC doctors. On the one hand, the payment by per diem does not penalize them (in terms of income) for additional time and effort per clinical service delivered. Moreover, partial fees reduce the opportunity cost of such behavior. Therefore, the reform may induce them to substitute quality for quantity of clinical treatments. In particular, MC doctors with a low marginal disutility of effort and who are highly altruistic, that is, whose welfare strongly depends upon their patients' utility, may be encouraged to spend more time and effort treating their patients. ⁴ On the other hand, the per diems may support minimum‐effort work standards as the payment is independent of the quality and quantity of clinical services provided. This effect is likely to be more significant for MC doctors who have a high marginal disutility of effort or a low degree of altruism toward their patients, especially in the absence of strong competition between doctors. ⁵ Note here that the presence of very long waiting lists under a user‐free universal system like the one in place in Quebec is likely to limit the strength of competition between physicians.

Regarding the previous studies on the effect of the MC reform on the quality of clinical services, results by Dumont et al. (2008) suggest that time per clinical service increased by 3.8% for specialists who adopted the MC system. Yet, this input‐based measure of quality does not provide any information concerning the actual effect of the MC system upon the health of patients. The output‐based approach addresses this problem by focusing on events that are correlated with the patient's health following a particular treatment. For example, Cutler (1995) measured mortality rates in hospitals (or within 1 year of discharge) and hospital re‐admission rates as measures of the quality of care that was received. Geweke, Gowrisankaran, and Town (2003) used mortality rates to measure hospital quality, controlling for non‐random hospital admissions (see also Fischer et al. ²⁰¹⁴, for a survey). Clemens and Gottlieb (2014) found that financial incentives affected the services that were provided to cardiology patients in the U.S., yet they found statistically insignificant impacts on patient health outcomes, as measured by mortality rates within 4 years of diagnosis.

To our knowledge, the study by Echevin and Fortin (2014), henceforth denoted EF, is the only one that provides an empirical analysis of the effect of the MC reform using an output‐based indicator of the quality of clinical services. These authors estimate a transition model between spells (or stays) in and out of the hospital, using panel data from a major teaching hospital in Quebec (Sherbrooke University Hospital Center, CHUS). Their results suggest that the hospital length‐of‐stay (LOS) of patients treated in departments that opted for the MC system increased by 4.2% (0.28 days). Yet, the reform did not generally affect the re‐admission rate of these patients to the same MC department with the same diagnosis.

Following EF, our study aims to analyze the effect of the MC reform on the quality of health services using an output‐based approach. However, we extend the EF paper in four crucial directions. First, using administrative panel data, we estimate an MSMS proportional hazard model (see Bijwaard ²⁰¹⁴, for a survey) analogous to a difference‐in‐differences approach. The observational unit that is treated, that is, that has adopted the MC system over the sample period, is the department (or specialty) of the treating physician in a hospital.

Second, a key challenge in evaluating the causal effect of the MC system on healthcare quality is the self‐selection of physicians into the compensation systems. For instance, doctors in the departments that have adopted the MC system may have stronger preferences for providing services to patients with substantial health problems, given that time (per diem) devoted to clinical services is not penalized at the margin when patients suffer from a serious condition. To account for this problem, we control for several diagnoses fixed effects and a co‐morbidity variable. Also, instead of using data from only a single hospital (CHUS), we use data from the total number of hospitals in Quebec (143) and introduce many hospital and specialty fixed effects.

Third, we introduce unobserved heterogeneity that is correlated across states at the individual level (see Bijwaard ²⁰¹⁴). As is well known, failure to account for unobserved heterogeneity may be the source of bias in the duration dependence on the conditional hazard rates.

Finally, we develop an econometric framework inspired by Athey and Imbens (2006) nonlinear approach to simulate the unconditional average effect of treatment on the treated of the reform on patients' duration (in days) in each state leading to a given transition (see subsection 3.2). This provides information for evaluating the reform's effect based on variables that may be most useful for policymakers.

Overall, and as is the case in EF, our empirical results suggest that the average risk of discharge to home is reduced (by 5.7%) in departments that have adopted the MC system. Also, according to our results and using our output‐based indicators, the quality of healthcare services decreased in departments that adopted the MC system, based upon two quality indicators. First, in contrast with EF, our results indicate that the average risk of rehospitalization (here, within 30 days, with the same diagnosis and in a similar department) increased by as much as 17.8% for patients who were treated in MC departments.

Second, the risk of death increased by 6.2% for patients at home within 1 year following discharge from an MC department.

The rest of the paper is organized as follows. Section 2 introduces the data and presents some descriptive statistics. Section 3 presents our MSMS hazard model. Section 4 presents our main results. Section 5 provides discussions and concludes.

2. Data

The data set that is used in the paper is based on the administrative files of Quebec Health Insurance Board (RAMQ), which is the agency responsible for paying physicians in Quebec. These files contain a record of every medical service performed (and billed to the government) by all physicians licensed to work in Quebec (and working in the public sector, a group representing about 98% of all physicians in 2018). Our data set is built from a sample based on patients from 1996 to 2016, including periods before and after the reform date (October 1, 1999). Indeed, as we shall see, the transition to an MC scheme was particularly marked in the early years of the reform and eased off in the second half of the 2000s. This is illustrated in particular by the rapid increase in the share of mixed remuneration in specialists' total medical remuneration (excluding per diem), followed by its capping at almost 29% of total medical remuneration (see Figure A1 in Appendix E). In 2011, approximately 50% of specialist physicians billed under MC at least once during the year, ⁶ with a slight decline observed in subsequent years. Our sample comprises all patients who were born in either April or October of an odd‐numbered year. This criterion guarantees the sample's representativeness within the limits of the RAMQ's data set. The sample is also limited to people who have had at least one hospital admission of at least one day's duration during the sampling period. This condition is essential in our model to obtain the necessary information on the payment system (FFS vs. MC) of the treating physician's department from the data. Note that we consider the last department where he was hospitalized if he is no longer in hospital.

There are 320,441 individuals in our sample. For each stay in hospital, we observe the date of the patient's admission, ⁷ their treating physician's specialty, diagnosis, and LOS before discharge to home or death. Moreover, data contains information on the patient's personal characteristics (age, sex, and date of death, when it applies). We match the patient data to information concerning the payment system of the treating physician's department. ⁸

It is important to note that while most departments were paid under FFS before the reform, not all of those who opted for MC adopted it as soon as the reform was implemented for the following reason. As mentioned earlier, unanimity by vote was required for the department to adopt MC. The fact that the compensation system is managed by departments rather than by hospital means that each department can make independent decisions at different times. This results in variations in the adoption dates of mixed compensation, as illustrated in Figure 1. Besides, no department returned to FFS after adopting MC during the sample period. Figure 1 shows a great disparity between specialties. Pediatrics and Psychiatry are the specialties that rapidly adopted the MC scheme.

FIGURE 1.

Proportion of Patients' Stays under Mixed Compensation. The adoption rate of mixed compensation is calculated on January 1 of each year. The reform was launched on October 1, 1999. Source: RAMQ and authors' computations.

Other medical specialties such as General Surgery, Internal Medicine, and Obstetrics & Gynecology are average or slightly above average, while medical specialties such as Urology or Gastroenterology have much lower adoption rates. The endogeneity in the adoption of the MC scheme is discussed below.

The information regarding the patient's transitions over the sample period starts with their first admission to hospital after the beginning of the sampling period. As mentioned above, three potential states may occur after the first spell or stay in hospital: home, re‐admission, and death. The data are structured to reflect these three states and the four transitions among the latter. Re‐admission is strictly defined as hospitalization in the same department with the same Diagnostic‐Related‐Group (DRG) and not for a simple follow‐up visit, ⁹ provided that the latter occurred within the last 30 days.

Any stay at home (after a stay in hospital) involves two transitions: Home to Readmission and Home to Death. The time spent in the first transition is thus $\min \left(30,T_h\right)$, where $T_h$ is the duration spent at home. If there has been no readmission (if $T_h$ > 30 or if the admission is in another DRG), the patient is right‐censored in the Home to readmission transition at the duration $\min \left(30,T_h\right)$. Note that when a patient is readmitted to hospital after 30 days, we consider this hospitalization a new admission. ¹⁰ The time spent in the Home to Death transition is $\min \left(365,T_h\right)$. If there is no death after hospitalization (i.e., if the patient is readmitted to hospital before 365 days at home, if he spends more than 365 days at home or even if the death occurs after 365 days), the patient is right‐censored in the Home to Death transition at the duration $\min \left(365,T_h\right)$.

Table 1 indicates that, on average, during our sample period, patients who were admitted to hospital have a 97.5% probability of being discharged to home and 2.5% of dying in hospital. On average, patients at home have a 2% probability of being readmitted to hospital (within 30 days after the previous stay in hospital in the same department and DRG) and 3.1% probability of dying at home (within 365 days) and a 94.9% probability of being censored after 365 days.

TABLE 1.

Transition average probabilities.

	Home	Re‐admission	Death	Censored
Hospital	0.975		0.025	0.000
Home		0.020	0.031	0.949

Source: RAMQ and authors' computations.

Table 2 provides descriptive statistics over our sample period broken down according to each of the four transitions and censored at home. We observe 691,360 stays in hospital when the patient is discharged to home, and 17,401 stays in hospital when they die in hospital. Also, there are 14,130 stays at home with re‐admission and 21,046 stays with death at home. The average LOS in hospital when the patients are discharged to home is 8.02 days, while it is 10.80 days when they die in hospital.

TABLE 2.

Descriptive statistics.

Statistics		Ad (RAd)	Ad (RAd)	Home	Home	Censored
		to home	to death	to RAd	to death	at home
Length of stay	Mean	8.02	10.80	13.77	107.23	299.92
	SD	19.17	25.69	8.50	105.06	121.62
	Min	1	1	1	1	1
	Max	4028	1147	30	365	365
Mixed Comp.	Mean	0.39	0.35	0.45	0.34	0.39
	Min	0	0	0	0	0
	Max	1	1	1	1	1
Treatment	Mean	0.62	0.54	0.67	0.52	0.62
	Min	0	0	0	0	0
	Max	1	1	1	1	1
Post Reform	Mean	0.81	0.82	0.78	0.83	0.81
	Min	0	0	0	0	0
	Max	1	1	1	1	1
Female	Mean	0.57	0.47	0.52	0.47	0.58
	Min	0	0	0	0	0
	Max	1	1	1	1	1
Age	Mean	48.76	72.70	46.44	72.54	48.86
	SD	24.03	14.90	24.75	14.50	23.83
	Min	0	0	0	0	0
	Max	109	108	107	109	108
Co‐morbidity Index (CCI)	Mean	1.05	4.10	1.76	4.21	0.93
	SD	1.93	3.10	2.51	3.27	1.76
	Min	0	0	0	0	0
	Max	18	16	18	16	17
Number of stays		691,360	17,401	14,130	21,046	656,184
Number of patients		317,279	17,401	9694	21,046	311,083
Stay$/$Patient		2.18	1.00	1.46	1.00	2.11

Source: RAMQ and authors' computations.

The average length at home when the patients are readmitted to hospital, is 13.77 days, while it is 107.23 days when they die at home. While 39% of patients who were admitted to hospital and left for home had an MC treating specialist, 45% of patients at home who were readmitted to hospital had an MC treating specialist during their former stay at hospital. In our sample, 57% of patients are female, and the average age of patients is 49.3 years. We also provide information about the average Charlson Co‐morbidity Index (Charlson et al. 1987) in each and overall transition. ¹¹ The average overall value of the CCI is 1.12 but reaches a value greater than 4 in transitions that end with death.

3. Estimation Strategy

Our econometric approach attempts to take into account three basic issues. First, our model must be intrinsically dynamic to allow the patient to move to various states over time. Our model focuses on three states: hospital, home, and death. For this purpose, we develop a recurrent MSMS proportional conditional hazard model with correlated unobserved heterogeneity (see Cameron and Trivedi ²⁰⁰⁵, ch. 17–19) and (Bijwaard 2014).

Second, we incorporate a difference‐in‐differences (DiD) approach in our model since one observes two basic groups in our data: the treated group of departments that are under MC at least once during the sample period and the control group of departments that are always under FFS. Third, since the decision of a department to adopt the MC system is optional, one must consider potential endogeneity problems in our analysis. Let us now analyze in greater detail how our econometric approach accounts for these three potential sources of bias.

3.1. A Multiple‐State Multiple‐Spell Hazard Model

Owing to our administrative data, we assume four possible states for an individual $i$ and four transitions (see Figure 2). ¹² The original state is the first Admission to hospital (Ad) after January 1, 1996. The three other states are: Re‐admission to hospital (RAd), ¹³ Home (H), and Death (D), ¹⁴ the latter being the absorbing state from which the individual cannot exit. The model is in quasi‐continuous time (i.e., the observation period is the day). Our model has four transitions: the transition from Admission to Home, the transition from Admission to Death, the transition from Home to Re‐admission, and the transition from Home to Death.

FIGURE 2.

States and transitions.

For the sake of simplicity, we do not distinguish $Ad\to D$ from $RAd\to D$ and $Ad\to H$ from $RAd\to H$. ¹⁵ In other words, we assume that the likelihood of being discharged or dying, conditional on the control variables, is independent regardless of whether hospitalization is the first admission or a re‐admission. One reason for making this assumption is that we control for the patient's CCI, which is assumed to reflect variations due to multiple hospitalizations for the same diagnosis if the patient's health condition becomes more critical. We define:

• $\mathcal{I}\mathcal{S}=\{\text{Ad},\text{RAd},\mathrm{H}\}$, the set of possible initial states for a transition.

• $\mathcal{T}=\{(1),(2),(3),(4)\}$ the set of transitions, and

• $\mathcal{T}(j)$ the set of transitions starting by the state $j\in \mathcal{I}\mathcal{S}$; that is, $\mathcal{T}(\text{Ad})=\mathcal{T}(\text{RAd})=\{(1),(2)\}$ and $\mathcal{T}(\mathrm{H})=\{(3),(4)\}$.

In our recurrent MSMS model, an individual may experience several states and the same state more than once. For example, individuals may be readmitted to hospital, go home, or readmitted again to hospital before dying (the absorbing state). The instantaneous hazard of a transition $r\in \mathcal{T}(j)$ is the instantaneous probability (or density) of leaving the state $j$ through the transition $r$, conditional on survival to time $t$.

3.1.1. The Multi‐State Multi‐Spell Proportional Conditional Hazard Function

In our analysis, the patient's MSMS proportional hazard of a transition $r\in \mathcal{T}(j)$ conditional on ${\mathbf{x}}_{(r)}(T)$ and ${\nu }_{(r)}$, can be written as:

$${\lambda }_{(r)}\left(t|{\mathbf{x}}_{(r)}(T),{\nu }_{(r)}\right)={\lambda }_{(r)0}(t)\exp \left({\mathbf{x}}_{(r)}^{\prime }(T){\mathit{\beta }}_{(r)}\right){\nu }_{(r)},$$ (1)

where ${\mathbf{x}}_{(r)}(T)$ is the vector of explanatory variables affecting the instantaneous hazard of transition $r$, $T$ is the calendar time (in days) on which the patient is leaving state $j$, ¹⁶ ${\nu }_{(r)}$ is the unobserved heterogeneity of transition $r$, and ${\mathit{\beta }}_{(r)}$ is the vector (to be estimated) of the parameters of the explanatory variables for the conditional hazard of the transition $r$.

Thus, Equation (1) includes three multiplicative regression factors that explain the conditional hazard that individual $i$ in state $j$ will go through the transition $r\in \mathcal{T}(j)$. We examine each of these three factors.

3.1.2. The Baseline Hazard ${\lambda }_{(r)0}(t)$

The hazard of the transition r (e.g., transiting from home to hospital) may depend upon the time that has elapsed in that initial state $j$ (e.g., the number of days at home). In state $j$, this dependence is captured by the baseline hazard, that is, ${\lambda }_{(r)0}(t)$. In the proportional hazard model, this is common to all individuals and only depends on $t$. This element is often modeled parametrically in the literature, with the distribution being assumed to belong to a given family of parametric distributions such as the Weibull distribution. To introduce greater flexibility, and following Han and Hausman (1990) and Meyer (1990), we propose an estimator which assumes that the baseline hazard is piecewise constant by time interval, so that it can be written as the step function ${\lambda }_{(r)0}\left(t,{\mathit{\alpha }}_{(r)}\right)$, where ${\mathit{\alpha }}_{(r)}$ is a vector of parameters to be estimated (see Table 3 for more details). ¹⁷

TABLE 3.

Piecewise constant intervals (measured per day) for each transition.

	Ad (RAd) $\to$ home	Ad (RAd) $\to$ death	Home $\to$ RAd	Home $\to$ death
${\alpha }_1$	[1, 2]	[1, 2]	[1, 3]	[1, 4]
${\alpha }_2$	[2, 3]	[2, 5]	[3, 6]	[4, 16]
${\alpha }_3$	[3, 4]	[5, 10]	[6, 8]	[16, 31]
${\alpha }_4$	[4, 5]	[10, 16]	[8, 12]	[31, 51]
${\alpha }_5$	[5, 6]	[16, 29]	[12, 16]	[51, 81]
${\alpha }_6$	[6, 8]	[29, Inf]	[16, 19]	[81, 121]
${\alpha }_7$	[8, 11]		[19, 22]	[121, 181]
${\alpha }_8$	[11, 18]		[22, 26]	[181, 261]
${\alpha }_9$	[18, Inf]		[26, 30]	[261, 365]

3.1.3. The Regression Function $\exp \left({\mathbf{x}}_{(r)}^{\prime }(T){\mathit{\beta }}_{(r)}\right)$

The hazard that is associated with going through transition $r$ may also depend upon some covariates other than time in state $j$. Therefore, the second element of the MSMS hazard model is a regression function that captures the effect of these variables. Since the probability of moving from one state to another is positive, we use an exponential function. In Subsection 3.2, we develop an expression for ${\mathbf{x}}_{(r)}^{\prime }(T){\mathit{\beta }}_{(r)}$, which allows us to estimate the average effect of treatment on the treated (ATT) of the reform on the log of the hazard for each patient's transition.

3.1.4. Unobserved Heterogeneity ${\nu }_{(r)}$

The third element of our MSMS conditional hazard model is unobserved and time‐invariant heterogeneity. Following van den Berg (1997); Bonnal, Fougère, and Sérandon (1997); Mealli and Pudney (2003) and Lacroix and Brouillette (2011), we model this as a mixed weighting of the values taken by iid random variables drawn from a standard normal distribution. The unobserved individual and transition heterogeneity are:

$${\nu }_{(r)}=\exp \left({\omega }_{(r)}\right),$$

$${\omega }_{(r)}={\psi }_{(r)}{\epsilon }_1+{\varphi }_{(r)}{\epsilon }_2,$$

where ${\epsilon }_1$ and ${\epsilon }_2$ are drawn independently from the standard normal distribution. The variables ${\psi }_{(r)}$ and ${\varphi }_{(r)}$ are loading factors. The correlation coefficient between transitions $r$ and $s$ can then be written as:

$$Corr\left({\omega }_{(r)},{\omega }_{(s)}\right)=\frac{{\psi }_{(r)}{\psi }_{(s)}+{\varphi }_{(r)}{\varphi }_{(s)}}{\sqrt{\left({\psi }_{(r)}^2+{\varphi }_{(r)}^2\right)\left({\psi }_{(s)}^2+{\varphi }_{(s)}^2\right)}}$$ (2)

From this equation, we see that our modeling of heterogeneity accounts for interdependence between the states. For identification, ${\psi }_{(r)}=1$, $\forall r\in \{1,2\}$ and ${\varphi }_{(r)}=1$, $\forall r\in \{3,4\}$. Therefore,

$${\omega }_{(r)}={\epsilon }_1+{\varphi }_{(r)}{\epsilon }_2,\forall r\in \{1,2\}$$

and

$${\omega }_{(r)}={\psi }_{(r)}{\epsilon }_1+{\epsilon }_2,\forall r\in \{3,4\}.$$

Appendix A details the simulated likelihood function used to estimate the model.

3.2. DiD Estimation in the Multi‐State Multi‐Spell Hazard Model

Our framework to estimate the MC effect can be analyzed using a linear difference‐in‐differences (DiD) model based on a general approach inspired by Athey and Imbens (2006). The econometric object that we model is the conditional hazard ${\lambda }_{(r)}\left(t|{\mathbf{x}}_{(r)}(T),{\nu }_{(r)}\right)$, expressed in $\log$. The set of treated units varies across time. Over a very long period, all hospital departments could adopt MC, and the set of non‐treated units would then be empty. Thus, we refer to non‐treated units in our sample as the departments that do not adopt MC, that is, before 1 January 2017. In the absence of reform (referred to as $N$), we assume that the regression function ${\mathbf{x}}_{(r)}^{\prime }(T){\mathit{\beta }}_{(r)}$ in the log of the conditional hazard function (see Section 3.1.1) is defined by

$$z_{(r)}{(T)}^N=h_{(r)}(T)+\underset{U_{(r)}(T)}{\underbrace{\sum _{k=1}^{K^D}{\beta }_{(r)k}^D{D}_k(T)+\sum _{k=1}^{K^I}{\beta }_{(r)k}^I{I}_k(T)+{{\tilde{\mathbf{x}}}^{\prime }}_{(r)}(T){\tilde{\mathit{\beta }}}_{(r)}}},$$ (3)

where $h_{(r)}(T)$ controls for time effect on the conditional hazard, $D_1(T)$, …, $D_{K^D}(T)$ are $K^D$ specialty dummy variables, $I_1(T)$, …, $I_{K^I}(T)$ are $K^I$ hospital dummy variables, ${{\tilde{\mathbf{x}}}^{\prime }}_{(r)}(T)$ includes other control variables, and ${\beta }_{(r)k}^D$, ${\beta }_{(r)k}^I$, ${\tilde{\mathit{\beta }}}_{(r)}$ are unknown parameters.

Here, an important remark is in order. For the sake of parsimony and to make the model tractable, we do not cross dummy variables for the specialties with those for the hospitals. This would imply a maximum of $33\times 143=4719$ parameters to estimate per transition. Rather, we assume the separability of the specialty dummies for which we are interested in the MC effect and the dummies for the largest hospitals (see below).

We include in $h_{(r)}(T)$ a linear combination of annual dummy variables as well as linear and quadratic quarterly trends. We also cross the quarterly trends with the specialties to take into account technological progress heterogeneity across specialties. We control for patients' observable characteristics in ${{\tilde{\mathbf{x}}}^{\prime }}_{(r)}(T)$ such as age, sex, and some patients' health characteristics when they are admitted to hospital (see below). We also include in ${{\tilde{\mathbf{x}}}^{\prime }}_{(r)}(T)$ departments' characteristics, such as sociosanitary regions (19 dummies) and the number of specialist physicians per department.

Moreover, we specify the term ${\mathbf{x}}_{(r)}^{\prime }(T){\mathit{\beta }}_{(r)}$ in the log of the conditional hazard function for the departments that have adopted MC (referred to as $I$) as

$$z_{(r)}{(T)}^I={\theta }_{(r)}MC(T)+\underset{z_{(r)}{(T)}^N}{\underbrace{h_{(r)}(T)+U_{(r)}(T)}},$$ (4)

where $MC(T)=1$ if the department of the treating physician has adopted MC before the date $T$ and $MC(T)=0$ otherwise. The parameter ${\theta }_{(r)}$ captures the effect of the MC on the expected log of the conditional hazard and the corresponding ATT for the transition $r$. Most importantly, this measure neither depends upon unobserved patient heterogeneity ${\nu }_{(r)}$ nor upon the control variables that are included in $z_{(r)}{(T)}^N$ (see Appendix D). For the sake of parsimony, we assume that the MC effect is constant in Equation (4), irrespective of the specialty and adoption date.

One important identification strategy, when using the DiD approach as applied to our hazard framework, is to assume that in the absence of the MC reform and given $h_{(r)}+U(r)(T)$, the hazard is the same for the treated and non‐treated units in a given time period (parallel trend assumption). While it is impossible to test this hypothesis, one can consider a modified version of Equation (3) before the reform, that is, on the period 1996‐01‐01 to 1999‐08‐31, which is given by:

$${\tilde{z}}_{(r)}(T)=\stackrel{\sim }{MC}{\tilde{\theta }}_{1(r)}Q(T)+\stackrel{\sim }{MC}{\tilde{\theta }}_{2(r)}Q{(T)}^2+h_{(r)}(T)+U_{(r)}(T),$$ (5)

where $\stackrel{\sim }{MC}=1$ for the specialty of the treating physician that will adopt MC later and $\stackrel{\sim }{MC}=0$ for the non‐treated departments. Finally, $Q(T)$ and $Q{(T)}^2$ respectively represent the linear and quadratic quarterly trends. Testing the hypothesis that the hazard is the same in a given time period before the reform, regardless of whether the department will adopt MC or not, is equivalent to jointly testing ${\tilde{\theta }}_{1(r)}=0$ and ${\tilde{\theta }}_{2(r)}=0$ when we also control for all other covariates.

3.3. Endogeneity in the Multiple‐Spell Multiple‐State Hazard Model

As mentioned above, our model may be subject to serious endogeneity problems. The basic reason is that the decision to adopt the MC system after the reform is optional at the department level. For instance, this may be the source of a selection bias, given that some variables affecting the department's decision to choose MC may also directly influence the quality of services delivered. To deal with this problem, and in the absence of instrumental variables, we introduce a large number of control variables in the model. These covariates attempt to account for unobservable attributes of hospitals and specialties that affect both the quality of clinical services and departments' preferences for payment systems. Also, we perform several parallel trend tests comparing MC and FFS doctors before the reform. Our results suggest that our estimates are generally robust to the endogeneity issue. Appendix B provides a more detailed discussion regarding our approach to treat this problem.

4. Results

Table 3 provides the piecewise constant intervals used in the baseline hazard of each transition. They have been selected after several trials as a function of the relative importance of observations in each interval.

Model 1 in Table 4 presents the most general specification, including unobservable heterogeneity (represented by four loading factors: two $\varphi$ and two $\psi$) and all explanatory variables. Also, this model includes as many as 80 dummies for the largest hospitals. Introducing a larger number of dummies for hospitals would render the model intractable, given that it is estimated using a simulated maximum likelihood approach.

TABLE 4.

Effect (in log) of the reform on patients' risk of transition (overall) ^a .

Variables	Model 1		Model 2		Model 3		Model 4		Model 5		Parallel trend
	Estimate	SD	Estimate	SD	Estimate	SD	Estimate	SD	Estimate	SD	p‐value
1. Ad(RAd) $\to$ home
Mixed compensation	−0.057	0.005	−0.047	0.003	−0.082	0.004	−0.082	0.004	−0.081	0.004	0.071
Female	−0.006	0.005	−0.018	0.003	−0.002	0.005	0.034	0.005	−0.002	0.005
Age	−0.003	0.000	−0.005	0.000	−0.001	0.000	0.000	0.000	−0.001	0.000
(Age/150)2	−0.021	0.001	−0.009	0.000	−0.022	0.001	−0.032	0.001	−0.022	0.001
$\varphi$	0.001	0.003	—	—	0.002	0.003	−0.005	0.003	0.002	0.003
2. Ad(RAd) $\to$ death
Mixed compensation	−0.003	0.023	0.004	0.022	0.017	0.021	0.008	0.021	0.016	0.021	0.954
Female	−0.079	0.017	−0.112	0.016	−0.077	0.017	−0.151	0.017	−0.077	0.017
Age	0.023	0.004	0.017	0.003	0.024	0.003	0.053	0.003	0.024	0.003
(Age/150)2	0.006	0.004	0.020	0.004	0.005	0.004	−0.025	0.004	0.005	0.004
$\varphi$	−0.342	0.019	—	—	−0.346	0.019	−0.344	0.017	−0.347	0.019
3. Home $\to$ RAd
Mixed compensation	0.178	0.027	0.186	0.026	0.222	0.026	0.231	0.026	0.219	0.026	0.189
Female	0.002	0.020	0.034	0.018	0.000	0.020	−0.039	0.020	0.000	0.020
Age	−0.009	0.002	−0.007	0.002	−0.015	0.002	−0.012	0.002	−0.015	0.002
(Age/150)2	0.005	0.003	−0.004	0.003	0.011	0.003	0.016	0.003	0.011	0.003
$\psi$	−0.199	0.015	—	—	−0.189	0.015	−0.283	0.015	−0.189	0.015
4. Home $\to$ death
Mixed compensation	0.062	0.022	0.048	0.020	0.069	0.021	0.060	0.021	0.065	0.021	0.887
Female	−0.219	0.017	−0.171	0.014	−0.220	0.017	−0.327	0.017	−0.219	0.017
Age	0.004	0.003	0.012	0.003	0.008	0.003	0.031	0.003	0.008	0.003
(Age/150)2	0.053	0.004	0.035	0.003	0.049	0.004	0.036	0.004	0.049	0.004
$\psi$	−0.608	0.016	—	—	−0.598	0.016	−0.789	0.016	−0.599	0.016
log(likelihood)	−2346071		−2302170		−2350172		−2368791		−2350320
Number of patients	320.441		320,441		320,441		320,441		320,441
Number of observations	1,400,121		1,400,121		1,400,121		1,400,121		1,400,121
Unobserved heterogeneity	Yes		No		Yes		Yes		Yes
Hospital FE	80 largest		80 largest		10 largest		10 largest		10 largest
Charlson co‐morbidity index	Yes		Yes		Yes		No		Yes
Department size (number of specialists)	Yes		Yes		Yes		Yes		Yes
12 specialty FE b	Yes		Yes		Yes		Yes		Yes
18 diagnoses FE	Yes		Yes		Yes		Yes		Yes
19 region FE	Yes		Yes		Yes		Yes		Yes
Linear & quadratic trends	Yes		Yes		Yes		Yes		No
Year FE	Yes		Yes		Yes		Yes		Yes

^a All specifications use a multi‐spell multi‐state proportional conditional hazard model. The baseline hazards are piecewise constant. In all models (except Model 2), unobserved heterogeneity is modeled using a mixed weighting of the values taken by iid random variables drawn from a standard normal distribution. The parallel trend p‐values correspond to Model 3. All the models (except Model 2) are estimated using simulated Maximum Likelihood. Model 2 does not account for unobserved heterogeneity and is estimated using Maximum Likelihood.

^b The 12 specialty fixed effects are: Cardiology, General surgery, Thoracic surgery, Gastroenterology, Obstetrics & Gynecology, Pulmonology, Internal medicine, ENT & Cervical surgery, Pediatrics, Psychiatry, Urology, Others (22 specialties).

Results for the first transition in Model 1 suggest that in MC departments, the patient's risk of being discharged to home decreases on average by 5.7%, compared with FFS departments (= ATT for the first transition). This means that patients whose treating doctor is under MC tend to stay longer in hospital than when their treating doctor is under FFS, ceteris paribus. Besides, results from the second transition in Model 1 indicate that the risk that a hospitalized patient dies in hospital is not significantly influenced by the system under which his treating doctor is paid.

Importantly, results from the third transition suggest that the risk of patient re‐admission with the same diagnosis and in a similar department within 30 days following discharge increases by 17.8% when their treating doctor is under MC. This suggests that the MC system has a strong and negative effect on the quality of healthcare services insofar as it increases the risk of re‐admission in hospital.

Results from the fourth transition indicate that the risk of death within 1 year after a patient's discharge to home increases by 6.2% when the treating doctor is under MC. Therefore, results from this transition would suggest that the MC reform reduced healthcare service quality.

Model 3 provides one way to check for the presence of endogeneity in the estimates of the reform's effect on the four transitions. This model reduces the number of hospital dummies to the ten largest hospitals. This corresponds to 36% of patients' transitions. Our results suggest that the estimated effect of being treated in an MC department increases (in absolute value) for all transitions in Model 3. The estimated negative effect increases from 0.057 to 0.082 in transition 1 and from 0.003 to 0.017 in transition 2. In the latter case, the estimated coefficients are not significant in any model. The estimated positive effect of being treated in an MC department slightly increases from 0.178 to 0.222 in transition 3 and from 0.062 to 0.069 in transition 4. Therefore, one can conclude that endogeneity affects the MC coefficient in transition 1, but it does not seem very important in the other transitions, at least based upon the number of hospital dummies.

Model 4 is similar to Model 3 except that an additional potential bias is introduced in the former model, given that the Charlson Co‐morbidity Index has been removed from the set of explanatory variables. When one compares Model 4 to Model 3, results indicate that the estimated effects of being treated by an MC doctor on transitions seem quite robust to the absence of the co‐morbidity index. The estimated effect of the reform on transition 1 does not change, while the effects of both models are not significant in transition 2. Also, the estimated effects of the reform on transitions 3 and 4 are very close.

Model 5 is quite similar to Model 3 except that linear and quadratic quarterly trends have been removed from Model 1. Results also suggest that the reform's effect on transitions in MC departments is very robust to such a change. Another way to (informally) check for the presence of endogeneity is to run parallel trend tests using the pre‐reform period for each of the four transitions, as explained above. For this purpose, we use Model 3 with the ten largest hospitals, as estimating a model with 80 hospitals, such as Model 1, is intractable when the sample is limited to the pre‐reform period. Results that are associated with the p‐value provided in the last column of Table 4 indicate that we cannot reject the parallel test hypothesis for all transitions at the 5% level.

Therefore, we can conclude that based on our complementary solutions to take endogeneity into account, it is apparently not a severe problem in our analysis of the effect of reform on the quality of healthcare services by MC specialists; an exception could be on the risk of transition from hospital to home (transition 1). This latter caveat originates from the sizable disparity between the estimated MC coefficients of Model 1 and Model 3 (quite large), where both coefficients are negative and significant. This difference could be interpreted by considering that in some smaller hospitals, the constraints on beds are greater, which could reduce the effect of the MC scheme on the duration of hospitalization while, at the same time, decreasing the incentive to opt for this contract.

Regarding the effects of control variables such as sex and age on the four transitions, the results that were obtained are consistent with our expectations. Being female increases time spent in hospital, reduces the risk of dying in hospital when hospitalized, and reduces the risk of dying at home within 1 year after being discharged from hospital. Likewise, being older increases the LOS in hospital at an increasing rate, yet increases the risk of dying in hospital when hospitalized, while reducing the risk of being readmitted within 30 days, ¹⁸ with the rate increasing the risk of dying within 1 year after being discharged.

In Model 1, the loading factor parameters $\varphi$ and $\psi$, which account for the unobserved heterogeneity, are significant in transitions 2 to 4, but not in transition 1 that is associated with the risk (per day) of a patient being discharged. To check whether the parameters of interest are robust to this problem, we present the results of Model 2, which assumes that the $\varphi$’s and $\psi$’s are zero (no unobserved heterogeneity). We see that the estimated MC coefficients do not change much when comparing Models 1 and 2. For instance, the reform's effect on the risk of being readmitted to hospital (transition 3) when the patient's treating physician is under MC is 0.178 in Model 1 and 0.186 in Model 2. The effect of MC on the risk of transition from home to death (transition 4) is 0.062 in Model 1 and 0.048 in Model 2. Note, however, that the hazards strongly vary between these two models in some transitions. As mentioned above, the basic reason is that failure to take unobserved heterogeneity into account may introduce a bias in the relationship between the hazard and duration. Thus, in Figure 3, one sees that in transition 1, the hazard without heterogeneity (Model 2) is larger than the hazard with heterogeneity (Model 1) for any given level of duration in the state (positive bias).

FIGURE 3.

Hazards with and without heterogeneity (Model 1 and Model 2). For simplicity's sake, all explanatory variables are fixed at zero.

We also show in Table 5 that the correlation between the various transitions is either positive or negative and statistically significant, given the results that are presented in the previous table. In particular, we observe that a longer hospital stay, ceteris paribus, is associated with a lower risk of death in hospital but associated with a higher risk of re‐admission and a higher risk of death outside of the hospital. This may be due to higher risks of complications following longer hospital stays, given patients' unobservable attributes. In addition, the risk of death in hospital is negatively correlated with the risks of re‐admission and death outside of hospital. Therefore, if a patient lasts in hospital before dying, even though they would not have died there, they are at greater risk of being readmitted or dying at home. Finally, the risk of re‐admission is positively correlated with the risk of death at home, which is expected, given the unobservable patient characteristics that can lead both to rehospitalization and to death.

TABLE 5.

Correlation between the transitions.

	Ad (RAd) $\to$ home	Ad (RAd) $\to$ death	Home $\to$ RAd	Home $\to$ death
Ad(RAd) $\to$ home	1.00	0.95	−0.19	−0.52
Ad(RAd) $\to$ death	.	1.00	−0.50	−0.77
Home $\to$ RAd	.	.	1.00	0.94
Home $\to$ death	.	.	.	1.00

Note: This table computes $Corr\left({\omega }_{(r)},{\omega }_{(s)}\right)$, the correlation between the unobserved heterogeneity of the transition (r) and that of the transition (s) (see Equation (2)). The transition (r) is indicated in the first row, and the transition (s) in the first column. A positive correlation means that a higher duration in the transition (r) is associated with a lower probability for the transition (s).

The Appendix (Dynamics) extends Model 1 in Table 4 to analyze some dynamic elements of the reform's effect. It shows that the effect of the reform on the quality of care provided by physicians under MC is negative and worsens over time, as these physicians accumulate years under this payment system. It also provides the ATTs in terms of expected duration in a given stay (rather than hazards of transition) and for each specialty included in the analysis (see Table A2). ¹⁹ Further robustness checks are presented in Appendix E. Firstly, Table A4 shows the estimation results using a Weibull hazard. This specification considers unobserved patient heterogeneity and the fixed effects of the 80 largest hospitals. We observe only minor differences from our more general model, particularly for out‐of‐hospital mortality and readmission rates, for which the results are very similar.

Our analysis also takes a closer look at the effects of the MC system, considering the size of the facility. As shown in Table A5, we categorize hospitals into small, medium, and large based on the number of stays, with each category accounting for a third of stays. The estimation results reveal variability in effects across different sizes, potentially influenced by the diverse representation of specialties in these hospitals. For example, the effects of the reform appear more negative on quality measured by 30‐day readmission for small and large facilities and less significant for medium‐sized facilities. As for 1‐year out‐of‐hospital mortality, the effect is only significant for large hospitals. This finding underscores the practical relevance of our research, as it suggests that the impact of the MC system is not uniform across all healthcare settings.

Another analysis, presented in Table A6 in Appendix E, focuses on early and late adopters of the MC system. It is important to note that specialties adopting the MC system earlier differed from those that adopted it later, highlighting potential factors that influenced the timing of adoption. However, contrary to expectations, the effects were not more significant for early adopters, who presumably had stronger motivations or more favorable adoption conditions than for late adopters. These findings indicate that residual selection bias is likely minimal in our analysis.

5. Discussions and Conclusion

Our paper is one of the first to evaluate the effect of a physician's mixed compensation system on the quality of their healthcare services based upon a natural experiment. This system was introduced in Quebec in 1999 and blended salary (per diem) and partial fee per clinical service. We develop an MSMS hazard model with correlated heterogeneity analogous to a difference‐in‐differences approach. Our results indicate that the reform reduced the quality of healthcare services provided by MC specialists. Our results suggest that the risk of rehospitalization of their patients within 30 days after discharge increased by 17.8%, and the risk of dying within 1 year after discharge increased by 6.2%. As a result, the average spell at home before rehospitalization decreased by 3 days, and the average spell at home before death after discharge decreased by 7.45 days. These results could be attributed to the fact that being partly paid by salary (per diem), independent of the quantity and quality of clinical services, may encourage some MC specialists to evade medical practice for alternative allocation of time in hospital, which is unpaid under the FFS system (such as administrative and teaching duties) and to reduce their effort when performing clinical services.

Our results show that the reform increased the LOS in hospital by 0.75 days (i.e., a 5.7% decrease in the probability of being discharged) for a patient treated by an MC specialist. Yet, the effect of MC on LOS is less directly interpretable in terms of quality. Referring to a time‐based approach, one could consider that the increase in the duration of hospitalization may have had beneficial effects on patients' health insofar as the MC doctor can take longer to cure the patient (e.g., Carey ²⁰¹⁵; Heggestad ²⁰⁰²). However, we can also consider the opposite effect: hospitalizations that are too long can be considered inappropriate because they can lead to complications for hospitalized patients and may be ineffective given the costs that they incur (e.g., Regenbogen et al. ²⁰¹⁷). The length of hospitalization directly correlates with complications following treatment, whether a simple procedure such as cataract surgery or a more complicated procedure such as heart bypass surgery. If the treatment is badly performed or is not of good quality, the duration of hospitalization will increase, given the complications that it causes.

The MC reform has had some positive effects on the healthcare system. In particular, one expects that it has reduced the number of unnecessary clinical services to some degree. Previous studies (e.g., see Dumont et al. ²⁰⁰⁸; Fortin, Jacquemet, and Shearer ²⁰²¹) indicate that the volume of clinical services significantly decreased following the reform. Yet, no evidence provides information regarding the proportion of this reduction that can be judged unnecessary.

The reform has enabled MC doctors, particularly those involved in more complex medical activities (such as pediatricians, psychiatrists, and surgeons), to allocate more time to their patients. This increased patient interaction has fostered better teamwork among healthcare professionals, which should have, in turn, positively influenced patient health outcomes. Additionally, the reform may have motivated some MC doctors to dedicate more time to training their students during per diem sessions, thereby enhancing the students' competency in providing healthcare. However, our findings indicate that the reform's effect does not fully align with certain expected goals, especially concerning the quality of clinical services as measured by output‐based indicators. To fully understand the reasons for the MC system's negative impact on health outcomes, it is essential to gather more clinical evidence, which should be carefully considered when deciding whether to retain, revise or abandon this remuneration scheme.

Our analysis is context specific and does not apply universally to all MC systems. One key advantage of the MC payment mechanism is its ability to use more policy instruments compared to pure FFS, salary, or capitation systems (Ma and McGuire ¹⁹⁹⁷). Public economics theory suggests that a policymaker generally needs as many instruments as there are objectives to achieve—in this case, the quality and quantity of clinical services. Therefore, the MC payment system is a logical approach. However, further research is needed to determine the optimal level of each instrument and to develop an MC system that effectively balances the needs of patients, physicians, and the healthcare system as a whole.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A. The Simulated Likelihood Function

The likelihood function of our model relies upon some basic concepts. First, the survival function, that is, the probability that the spell's duration (stay) in the original state $j$ is at least equal to $t$, can be shown to be:

$$S_j\left(t|\mathbf{x}(T),{\epsilon }_1,{\epsilon }_2\right)=\prod _{s\in \mathcal{T}(j)}\exp \left(-\exp \left({\mathbf{x}}_{(s)}^{\prime }(T){\mathit{\beta }}_{(s)}+{\psi }_{(s)}{\epsilon }_1+{\varphi }_{(s)}{\epsilon }_2\right){\int }_0^t{\lambda }_{(s)0}(\tau )d\tau \right).$$

Second, the conditional density of duration at time $t$, in the state $j\in \mathcal{I}\mathcal{S}$, followed by the transition $r\in \mathcal{T}(j)$ if there is no censure in the state (i.e., the likelihood of the state), is:

$$f_{(j)}\left(t|\mathbf{x}(T),{\epsilon }_1,{\epsilon }_2\right)={\left({\lambda }_{(r)}\left(t|{\mathbf{x}}_{(r)}(T),{\epsilon }_1,{\epsilon }_2\right)\right)}^c{S}_j\left(t|\mathbf{x}(T),{\epsilon }_1,{\epsilon }_2\right),$$ (A1)

where $c=1$ if the transition is effective and $c=0$ if the patient is censored in the state $j$. For example, after being discharged from hospital, if a patient does not die within 365 days or is not readmitted within 30 days, then we can only observe their stay in the state Home, but the patient is censored in the state. This censure is also effective though there is a new admission, but the latter is considered as readmission (maybe because the admission occurs after 30 days or because the admission is not associated with the same DRG). In this case, $c=0$ and the likelihood of the state in Equation (A1) is only the survival function.

As a patient passes through multiple states and transitions, their likelihood, which is denoted as $L_i\left(\mathit{\theta },{\epsilon }_{1,i},{\epsilon }_{2,i}\right)$, is the product of the likelihood in each state and for each transition, where $\mathit{\theta }$ is the vector of all parameters to be estimated. The subscript $i$ stands for the $i$‐th patient. As ${\epsilon }_{1,i}$ and ${\epsilon }_{2,i}$ are not observed, we have to maximize the following log‐likelihood function with respect to $\mathit{\theta }$:

$$\log L=\sum _i\log \left({\int }_{{\epsilon }_{1,i}}{\int }_{{\epsilon }_{2,i}}{L}_i\left(\mathit{\theta },{\epsilon }_{1,i},{\epsilon }_{2,i}\right)f\left({\epsilon }_{1,i}\right)f\left({\epsilon }_{2,i}\right)d{\epsilon }_{1,i}d{\epsilon }_{2,i}\right),$$ (A2)

where $f$ is the probability density function of the standard normal distribution.

We approximate this optimization program by maximizing the following simulated log‐likelihood function with respect to $\mathit{\theta }$:

$$\widehat{\log L}=\sum _i\log \left(\frac{1}{100}\sum _{m=1}^{100}{L}_i\left(\mathit{\theta },{\epsilon }_{1,i}^{(m)},{\epsilon }_{2,i}^{(m)}\right)\right),$$ (A3)

where ${\epsilon }_{1,i}^{(m)}$ and ${\epsilon }_{2,i}^{(m)}$, $m=1,\text{…},100$ are drawn from $\mathcal{N}(0,1)$.

Appendix B. Treatment of the Endogeneity Problem

In this appendix, we discuss the various ways we use to deal with the endogeneity issues in our model. First, owing to our panel data set, we introduce fixed effects for hospitals where patients have been (currently or lastly) admitted and for their treating doctor's specialty. This approach has a number of advantages. It allows us to account for the unobservable characteristics of the hospital that may influence both the decision of its departments to operate under MC and the quality of healthcare services delivered. For instance, peer effects and homophily between doctors within a hospital may be such that their preferences regarding professional behavior and choices are similar. Also, the fixed effect for the treating doctor's specialty allows us to account for the unobservable attributes of his specialty that may influence his preferences for the payment systems (e.g., pediatrists and psychiatrists prefer the MC system while radiologists prefer the FFS system), and his professional behavior.

Here, an important remark is in order. We did not include dummies for all 143 hospitals that are used in our model. To render the model more parsimonious and tractable, we limit the number of dummies in some specifications to the largest 80 hospitals since the latter includes as many as 99% of patients who have moved from one state to another. Also, we introduce 12 dummies for specialties (see note in Table 4). ²⁰ These 92 dummies (which involve 368 parameters to estimate since there are four transitions) allow us to account for unobservable time‐invariant characteristics of the 80 hospitals and 12 specialties, which in turn may influence their decision to adopt MC. Ideally, we should include ($80\times 12=960$ dummies) for each transition to account for the unobservable characteristics of each department in each hospital. However, such a model is not estimable since it would involve at least $960\times 4=3,840$ parameters to estimate.

Second, it is arguable that departments in which physicians prefer to treat patients with complex health problems have more incentive to adopt the MC system. The basic reason is that being paid with per diems and low fees per service under MC penalize a physician less than under FFS when their patients' treatments are time‐intensive. This may be the source of a serious selection bias in identifying the reform's effect on patients' transitions. For instance, the estimated effect of MC on the risk of dying when the patient is at home is likely to be overestimated as long as MC physicians treat patients with more complex diseases than under FFS. To take this heterogeneity problem into account, we have introduced a large number of control variables. In particular, we have included 18 diagnosis dummy variables per transition as an indicator of the patient's health when admitted to hospital. ²¹ Also, as discussed earlier, we have introduced the Charlson Co‐morbidity Index (CCI score) as an additional control variable.

Third, we have introduced a control variable for the number of specialists in the patient's department. This variable may affect the decision to adopt the MC system as it may be more difficult to reach unanimity when the department size is large. Yet, this variable may also directly influence the quality of healthcare services, given that the number of physicians who are substitutes (or complements) for the treating physician increases when the size of the patient's department increases. Note that the so‐called incidental parameter problem is not an issue here since the asymptotics are on patients and not on hospitals or departments, which are assumed to be fixed in the analysis.

Fourth, using Equation (5), we have performed parallel trend tests based on the estimated parameters when one focuses on the pre‐reform period. We compare the evolution of the (linear and quadratic) quarterly trends in departments that would later switch to MC vs those that would stay under FFS. This test is discussed at the end of Subsection 3.2 (see Equation (5)).

Section 4 provides a robustness analysis of our results, depending on the introduction (or not) of various control variables to account for endogeneity (see Table 4). It also presents the results of our parallel trend tests.

Appendix C. Dynamic Analysis and Duration

Dynamics

This subsection of the appendix (Dynamics) extends Model 1 in Table 4 to analyze some dynamic elements of the reform's effect. To reach this objective, we allow the MC coefficient (= ATT) to vary across the intervals of years that physicians who adopt MC would spend under this compensation system (denoted Exp). In particular, we study the reform's effect on the risk of the four transitions depending on how long the department of the treating physician has been under MC.

Regarding the first transition, results from Table A1 suggest that the patient's risk of being discharged to home decreases more and more as his treating physician, through his department, is under MC for a long time. When the physician has worked under MC for less than 2 years, the effect is −3.3% (and is significant), while the effect is −11.0% (and is significant) when the physician has been under MC for over 10 years. However, the effect of MC is never significant (given large SDs relative to the means) on the hospitalized patient's risk of dying (second transition), regardless of whether their treating physician's department is an early or later MC adopter.

The reform's effect on the risk of hospital re‐admission (third transition) is positive and significant when the treating physician is under MC for 2 years and more. The effect is increasing, that is, 21% when the physician spends between two and 4 years under MC and attaining a value of 34.7% when the physician has accumulated ten years or more under MC.

Finally, regarding the reform's effect on the patient's risk of dying within 1 year after discharge, it is positive and increases when the treating physician is under MC for more than 5 years. The effect is 6.8% when the physician is under MC for more than 5 years but less than 9 years; it reaches 20.2% when the physician is under MC for ten years or more.

In short, based on transition 3 and 4 results, the reform's effect on the quality of MC physicians' healthcare services is negative. Moreover, this effect increases (in absolute value), given that these physicians have been under MC for a long time. This may partly be explained by behavioral adjustments to the reform that will likely require more time.

TABLE A1.

Short‐ vs long‐term effects (in log).

Variables	Estimate	SD
1. Ad(RAd) $\to$ home
Exp: $<$ 2 years	−0.033	0.007
Exp: 2–4 years	−0.037	0.007
Exp: 5–9 years	−0.067	0.006
Exp: $\ge$ 10 years	−0.110	0.007
$\varphi$	0.000	0.003
2. Ad(RAd) $\to$ death
Exp: $<$ 2 years	−0.074	0.039
Exp: 2–4 years	−0.019	0.034
Exp: 5–9 years	0.014	0.031
Exp: $\ge$ 10 years	0.060	0.036
$\varphi$	−0.340	0.020
3. Home $\to$ RAd
Exp: $<$ 2 years	0.020	0.043
Exp: 2–4 years	0.210	0.038
Exp: 5–9 years	0.179	0.037
Exp: $\ge$ 10 years	0.347	0.042
$\psi$	−0.198	0.015
4. Home $\to$ death
Exp: $<$ 2 years	−0.020	0.038
Exp: 2–4 years	0.018	0.033
Exp: 5–9 years	0.068	0.030
Exp: $\ge$ 10 years	0.202	0.034
$\psi$	−0.606	0.016
log(likelihood)	−2345987
Number of patients	320,441
Number of observations	1,400,121
Unobserved heterogeneity	Yes
Hospital FE	80 largest
Charlson co‐morbidity index	Yes
Department size (number of specialists)	Yes
12 specialty FE	Yes
18 diagnoses FE	Yes
19 region FE	Yes
Linear & quadratic trends	Yes
Year FE	Yes

Note: This specification, corresponding to Model 1, uses an MSMS proportional conditional hazard model. The baseline hazards are piecewise constant. Unobserved heterogeneity is modeled using a mixed weighting of the values taken by iid random variables drawn from a standard normal distribution. The model is estimated using simulated Maximum Likelihood.

The Effect of the MC Reform on Duration

Thus far, our findings have primarily focused upon the ATT of the reform on the patient's risk of each transition (in log). However, this doesn't offer a clear insight into how the reform impacts the patient's expected unconditional duration of stay in a given state (with transition $r$). Such information is crucial for policymakers and hospital managers. This is particularly evident for the duration of hospitalization, as it directly correlates with resource allocation and costs. Similarly, the duration before rehospitalization after discharge pertains to costs and effective resource management. For instance, a higher frequency of patients readmitted within a week of discharge suggests a greater emphasis on post‐discharge care or follow‐up interventions during the initial days after leaving the hospital. Furthermore, the duration before death post‐discharge provides distinct viewpoints on patient outcomes and potential costs. A shorter duration between discharge and death might indicate premature discharges or insufficient post‐discharge support, leading to potentially avoidable costs linked to end‐of‐life care outside the hospital.

Appendix D provides the formal relationship between an unconditional duration expectation in a stay and the conditional hazard function. More heuristically, to simulate the ATT of the reform on a duration expectation, one would first calculate the density function of the duration, given that there is a formula that links the hazard to the density. The hazard is nothing more than the density divided by the expression one minus the distribution function. Note that for the case with heterogeneity, we need to integrate the density with respect to heterogeneity to obtain an unconditional density. The unconditional duration expectation is then obtained from integration in the same way that we would calculate an expectation by knowing the density function. Finally, we differentiate between the duration expectations when the treating physician is working in a department under MC minus the duration expectations when the department is under FFS to compute the ATT of the reform on the patient's duration expectations.

Table A2 provides results for unconditional ATT on the expected duration of stay (in days) for each state, overall specialties, and selected specialties. In addition, Table A3 in Appendix E presents results of the effect (in log) of the MC reform across specialties with parallel trend tests. In Table A2, we use Model 3 (10 hospitals) rather than Model 1 (80 hospitals), given that the latter is non‐tractable because of the large number of integrations that are required to perform in the model with heterogeneity and that many standard errors cannot be computed in the model without heterogeneity. For the sake of comparison, we also present Model 3 without heterogeneity.

When the patient is in hospital (with the transition to Home), being in a department under MC increases their duration at the hospital by 0.75 days on average (= ATT), and the effect is significant. This is consistent with the negative effect of the reform on the hazard of leaving the hospital to go home (see Table 4). The ATT is also significant and positive for almost all specialties (except Psychiatry and Urology, for which it is negative and significant). The five specialties for which the ATT is the largest are Others (1.80 days), Gastroenterology (1.53 days), Thoracic Surgery (0.85 days), Pulmonology (0.80 days) and Internal Medicine (0.35 days). The model's results without heterogeneity follow the same pattern (same sign) though the size may differ. For instance, the ATT for Gastroenterology is much higher (2.38 days).

Regarding the duration in hospital when the transition is to death, our results are also consistent with our expectations, indicating that the effect (unconditional ATT) of the reform is non‐significant on average. The duration before death decreases, and this effect is significant for ORL & Cervicofacial Surgery (−12.27), for Obstetrics & Gynecology (−10.31), and General Surgery (−1.87). In contrast, the duration before death significantly increases for Urology (6.81) and Others (1.48). As was the case for the ATT with the transition to Home, the results are quite similar in the model without heterogeneity.

The reform also affects the duration of stay at home after discharge and before re‐admission to the hospital with the same diagnosis. An important indicator of the effect of MC on the quality of healthcare services that are provided by treating specialists can be measured by its effect on this duration. According to our results, the reform decreased duration by 2.98 days on average before patient re‐admission when an MC specialist treated a patient. When significant, the ATT is negative for all selected specialties. The five specialties for which the ATT is the largest in terms of absolute value are Pediatrics (−9.71), Obstetrics & Gynecology (−8.99), General Surgery (−6.02), Psychiatry (−5.71), and Internal Medicine (−5.41). This again suggests that based on this quality indicator, the reform's effect negatively affected the quality of healthcare services by specialists.

Finally, the ATT of the reform on the duration of the period after discharge to death provides an additional measure of its effect on the quality of healthcare services by MC specialists. Our results suggest that the reform reduced by 7.45 days the duration of the period after discharge to death. When significant, the ATT is negative for each specialty. The specialties for which the ATT is largest (in absolute value) are General Surgery (−43.91), Urology (−28.70) and Internal Medicine (−16.35). Based on this indicator and in conformity with our hazard results, these results indicate that the MC reform negatively affects the quality of healthcare services by MC specialists.

Appendix D. ATT on the Log of the Hazard and Duration

As pointed out in Subsection 3.2, the average treatment effects on the treated (ATT) on the logarithm of the conditional hazard for the transition $r\in \mathcal{T}$ is the parameter ${\theta }_{(r)}$. Indeed, from Equations (1) and (4), the log of the conditional hazard for the departments that have adopted MC is given by

$$\log \left({\lambda }_{(r)}^I(t,MC)\right)=\log \left({\lambda }_{(r)0}(t)\right)+{\theta }_{(r)}MC(T)+z_{(r)}{(T)}^N+\log \left({\nu }_{(r)}\right).$$ (D1)

The ATT on the log of the hazard is then $\log \left({\lambda }_{(r)}^I(t,MC=1)\right)-\log \left({\lambda }_{(r)}^I(t,MC=0)\right)={\theta }_{(r)}$. The advantage of this measure lies in its simplicity. It depends neither upon the unobserved patient heterogeneity ${\nu }_{(r)}$, nor upon the control variables included in $z_{(r)}{(T)}^N$ (see Equation (3)). Importantly, it does not depend on whether we condition or not on ${\nu }_{(r)}$.

Computing the ATT on the duration is more challenging as the model becomes nonlinear (see Athey and Imbens ²⁰⁰⁶) and requires numerical integrations. Moreover, the unobserved heterogeneity term ${\nu }_{(r)}$ does not cancel out, as is the case of the log of the hazard. Therefore, we compute the ATT of the expected duration, where the expectation is taken with respect to the unobserved patient heterogeneity ${\nu }_{(r)}$ and to the control variable (i.e., $z_{(r)}{(T)}^N$).

Our approach to computing the unconditional ATT on the duration can be described in several steps.

Step 1. From Equation (A4), we first compute the survival conditionally on ${\nu }_{(r)}$ and $z_{(r)}{(T)}^N$. The survival function is given by

$$S_{(r)}^I(t,MC)=\exp \left\{-\exp \left\{{\theta }_{(r)}MC(T)+z_{(r)}{(T)}^N\right\}{\nu }_{(r)}{\int }_0^t{\lambda }_{(r)0}(\tau )d\tau \right\}.$$

Note that ${\int }_0^t{\lambda }_{(r)0}(\tau )d\tau$ has a closed form because ${\lambda }_{(r)0}(\tau )$ is a step function. In particular, if the baseline hazard equates ${\alpha }_k$ on the interval $\left[{\alpha }_k^{-},{\alpha }_k^{+}\right)$, for $k=1,2,\text{…}$, then ${\int }_0^t{\lambda }_{(r)0}(\tau )d\tau ={\sum }_{s=1}^k{\alpha }_s\left(\min \left\{t,{\alpha }_s^{+}\right\}-{\alpha }_s^{-}\right)$ for $t\in \left[{\alpha }_k^{-},{\alpha }_k^{+}\right)$.

Step 2. From the survival function $S_{(r)}^I(t,MC)$, we compute the density function of the duration conditionally on ${\nu }_{(r)}$ and $z_{(r)}{(T)}^N$. This density function is defined by

$$\begin{array}{ll}\hfill g_{(r)}^I(t,MC)& =-\frac{\partial S_{(r)}^I(t,MC)}{\partial t}.\hfill \\ \hfill g_{(r)}^I(t,MC)& =\exp \left\{{\theta }_{(r)}MC(T)+z_{(r)}{(T)}^N\right\}{\nu }_{(r)}{\lambda }_{(r)0}(t)S_{(r)}^I(t,MC).\hfill \end{array}$$

Step 3. We now take the expectation of $g_{(r)}^I(t,MC)$ to obtain the unconditional density function denoted $g_{(r)}^{I\ast }(t,MC)$, where the expectation is with respect to ${\nu }_{(r)}$ and $z_{(r)}{(T)}^N$. Unfortunately, $g_{(r)}^{I\ast }(t,MC)$ does not have a closed form.

Given that $g_{(r)}^{I\ast }(t,MC)$ is an expectation with respect to two variables, we use the law of iterated expectations and approximate it using a two‐stage approach. ²² In the first stage, we compute the expectation of $g_{(r)}^I(t,MC)$ with respect to the first variable ${\nu }_{(r)}$, and conditionally on the second variable $z_{(r)}{(T)}^N$. Let ${\widehat{g}}_{(r)}^I(t,MC)$ be this expectation. Since $z_{(r)}{(T)}^N$ and ${\nu }_{(r)}$ are independent, ${\widehat{g}}_{(r)}^I(t,MC)$ is approximated using simulations from the distribution of ${\nu }_{(r)}$, as is done for the log‐likelihood Equation (A3).

In the second stage, we compute $g_{(r)}^{I\ast }(t,MC)$, which is the expectation of ${\widehat{g}}_{(r)}^I(t,MC)$ with respect to $z_{(r)}{(T)}^N$. We approximate this second expectation using a sample mean. Indeed, for each observation under the MC, we can approximate $z_{(r)}{(T)}^N$ by replacing the parameters in Equation (3) with their estimates. Then, $g_{(r)}^{I\ast }(t,MC)$ can be approximated using the mean of ${\widehat{g}}_{(r)}^I(t,MC)$ in the subsample under the MC.

Step 4. From the unconditional density function $g_{(r)}^{I\ast }(t,MC)$, we obtain the expected duration given by

$$D^I(MC)={\int }_0^{\infty }\tau g_{(r)}^{I\ast }(\tau ,MC)d\tau .$$

Step 5. Finally, the ATT on the expected duration is $D^I(MC=1)-D^I(MC=0)$. We also compute the standard deviation of this ATT using the simulation approach proposed by Krinsky and Robb (1990).

TABLE A2.

Unconditional effect of the reform on durations (ATT in days).

	Model 3
	(With heterogeneity)		(Without heterogeneity)
	Estimate	Std. Err	Estimate	Std. Err
1. Ad(RAd) $\to$ home
Overall	0.75	0.04	0.84	0.04
Cardiology	0.11	0.06	0.12	0.07
Gen. Surgery	0.20	0.06	0.16	0.06
Thor. Surgery	0.85	0.31	0.34	0.29
Gastro.	1.53	0.25	2.38	0.27
Obs. & Gyn.	0.17	0.03	0.28	0.02
Pulmonology	0.80	0.24	1.11	0.24
Inter. Med.	0.48	0.16	0.79	0.18
ORL & CFS	0.14	0.09	0.33	0.08
Pediatrics	0.35	0.07	0.42	0.06
Psychiatry	−2.06	0.67	−4.15	0.61
Urology	−0.32	0.09	−0.28	0.09
Others	1.80	0.08	1.66	0.09
2. Ad(RAd) $\to$ death
Overall	−0.20	0.28	−0.25	0.26
Cardiology	0.32	0.44	0.32	0.42
Gen. Surgery	−1.87	0.89	−2.00	0.79
Thor. Surgery	−0.14	0.80	−0.49	0.72
Gastro.	−2.37	1.47	−2.35	1.47
Obs. & Gyn.	−10.31	3.78	−10.91	3.79
Pulmonology	−0.39	0.44	−0.59	0.44
Inter. Med.	−0.76	0.42	−0.29	0.45
ORL & CFS	−12.27	5.57	−10.18	5.90
Pediatrics	−10.66	5.94	−11.00	6.51
Urology	6.81	3.38	6.70	3.21
Others	1.48	0.43	1.08	0.48
3. Home $\to$ RAd
Overall	−2.98	0.34	−3.18	0.33
Cardiology	0.14	1.33	−0.29	1.19
Gen. Surgery	−6.02	1.00	−6.21	1.02
Thor. Surgery	−3.38	2.81	−3.52	2.85
Gastro.	−1.85	1.71	−0.78	1.95
Obs. & Gyn.	−8.99	1.19	−9.21	1.12
Pulmonology	−0.33	1.76	−1.38	1.63
Inter. Med.	−5.41	0.99	−5.33	1.00
ORL & CFS	−5.18	3.24	−5.56	3.00
Pediatrics	−9.71	1.45	−10.73	1.25
Psychiatry	−5.71	0.98	−5.14	0.93
Urology	4.16	3.04	3.73	3.23
Others	0.68	0.71	−0.16	0.70
4. Home $\to$ death
Overall	−7.45	2.55	−5.55	2.07
Cardiology	6.05	7.55	5.04	6.27
Gen. Surgery	−43.91	6.50	−38.60	5.65
Thor. Surgery	−15.64	16.19	−11.44	15.28
Gastro.	−2.80	10.54	1.94	10.11
Obs. & Gyn.	−25.42	13.12	−24.01	14.12
Pulmonology	10.53	7.57	8.94	7.74
Inter. Med.	−16.35	4.90	−14.63	4.20
ORL & CFS	−3.75	17.38	−13.16	15.86
Psychiatry	−35.95	20.84	−30.67	21.31
Urology	−28.70	14.58	−24.25	14.82
Others	6.64	3.50	7.56	3.02

Appendix E. Complementary Tables and Figures

FIGURE A1.

Share of remuneration and physicians under mixed compensation scheme. Source: RAMQ and authors' computations.

FIGURE A2.

Hospital stay duration and the rates of hospital death, 30‐day readmission, and one‐year death at home. Source: RAMQ and authors' computations.

TABLE A3.

Effect (in log) of the reform on patients' risks of transition (by specialty).

Variables	Model 1		Model 2		Model 3		Model 4		Model 5		Parallel trend
	Estimate	SD	Estimate	SD	Estimate	SD	Estimate	SD	Estimate	SD	p‐value
1. Ad(RAd) $\to$ home
MC: Cardiology	0.038	0.014	0.031	0.010	−0.020	0.013	−0.008	0.013	0.013	0.013	0.957
MC: Gen. Surgery	−0.014	0.010	−0.002	0.008	−0.034	0.010	−0.050	0.010	−0.056	0.009	0.890
MC: Thor. Surgery	−0.068	0.032	0.004	0.025	−0.095	0.032	−0.085	0.032	−0.041	0.027	0.874
MC: Gastro.	−0.144	0.029	−0.188	0.022	−0.171	0.029	−0.225	0.029	−0.143	0.027	0.587
MC: Obs. & Gyn.	−0.036	0.011	−0.056	0.007	−0.059	0.010	−0.069	0.010	−0.009	0.009	0.758
MC: Pulmonology	−0.102	0.031	−0.099	0.023	−0.101	0.031	−0.102	0.031	−0.124	0.030	0.099
MC: Inter. Med.	−0.024	0.016	−0.027	0.012	−0.043	0.015	−0.049	0.015	−0.065	0.013	0.742
MC: ORL & CFS	−0.031	0.026	−0.061	0.019	−0.042	0.026	−0.067	0.026	−0.038	0.024	0.199
MC: Pediatrics	−0.106	0.022	−0.109	0.015	−0.108	0.022	−0.120	0.022	−0.120	0.015	0.320
MC: Psychiatry	0.090	0.024	0.123	0.018	0.078	0.023	0.077	0.023	−0.113	0.015	0.581
MC: Urology	0.056	0.027	0.044	0.020	0.092	0.027	0.081	0.027	0.155	0.026	0.731
MC: Others	−0.119	0.009	−0.074	0.007	−0.167	0.009	−0.156	0.009	−0.196	0.008	0.092
Female	−0.007	0.005	−0.017	0.003	−0.002	0.005	0.032	0.005	−0.002	0.005
Age	−0.003	0.000	−0.005	0.000	−0.001	0.000	0.000	0.000	−0.001	0.000
(Age/150)2	−0.021	0.001	−0.009	0.000	−0.022	0.001	−0.030	0.001	−0.022	0.001
$\varphi$	0.001	0.003	—	—	0.002	0.003	−0.005	0.003	0.002	0.003
2. Ad(RAd) $\to$ death
MC: Cardiology	0.024	0.058	0.031	0.057	−0.042	0.054	−0.096	0.054	−0.052	0.052	0.659
MC: Gen. Surgery	0.093	0.060	0.110	0.060	0.138	0.058	0.221	0.058	0.096	0.050	0.797
MC: Thor. Surgery	0.109	0.155	0.190	0.152	0.030	0.153	0.018	0.153	−0.122	0.129	0.126
MC: Gastro.	0.177	0.120	0.163	0.117	0.186	0.119	0.309	0.118	0.122	0.108	0.698
MC: Obs. & Gyn.	0.570	0.223	0.605	0.222	0.619	0.224	0.817	0.220	0.553	0.174	NA
MC: Pulmonology	0.001	0.088	0.035	0.083	0.076	0.087	0.134	0.086	0.072	0.085	0.991
MC: Inter. Med.	−0.077	0.046	−0.094	0.043	0.070	0.042	0.116	0.041	0.112	0.037	0.376
MC: ENT & CFS	0.463	0.224	0.401	0.222	0.474	0.224	0.508	0.221	0.269	0.198	0.627
MC: Pediatrics	0.486	0.335	0.500	0.335	0.535	0.330	0.537	0.332	−0.082	0.211	NA
MC: Urology	−0.685	0.302	−0.675	0.300	−0.649	0.300	−0.637	0.300	−0.585	0.298	NA
MC: Others	−0.067	0.035	−0.056	0.034	−0.104	0.034	−0.180	0.034	−0.095	0.032	0.787
Female	−0.079	0.017	−0.112	0.016	−0.078	0.017	−0.153	0.017	−0.078	0.017
Age	0.023	0.004	0.016	0.003	0.024	0.003	0.052	0.003	0.024	0.003
(Age/150)2	0.006	0.004	0.020	0.004	0.005	0.004	−0.024	0.004	0.005	0.004
$\varphi$	−0.341	0.020	—	—	−0.345	0.020	−0.342	0.017	−0.349	0.019
3. Home $\to$ RAd
MC: Cardiology	−0.109	0.093	−0.099	0.091	−0.011	0.091	−0.022	0.090	−0.097	0.084	0.521
MC: Gen. Surgery	0.407	0.084	0.415	0.082	0.481	0.082	0.528	0.081	0.384	0.066	0.949
MC: Thor. Surgery	0.311	0.284	0.316	0.282	0.339	0.283	0.349	0.283	0.640	0.236	NA
MC: Gastro.	0.009	0.131	−0.046	0.125	0.118	0.130	0.214	0.129	0.283	0.118	0.728
MC: Obs. & Gyn.	0.637	0.096	0.644	0.095	0.729	0.093	0.755	0.093	0.396	0.071	0.658
MC: Pulmonology	−0.025	0.113	0.020	0.102	0.021	0.112	0.017	0.111	0.159	0.108	0.310
MC: Inter. Med.	0.397	0.083	0.401	0.080	0.416	0.080	0.437	0.080	0.390	0.065	0.069
MC: ORL & CFS	0.262	0.217	0.256	0.213	0.363	0.216	0.427	0.216	0.450	0.193	0.476
MC: Pediatrics	0.835	0.107	0.899	0.103	0.782	0.107	0.805	0.106	0.399	0.067	0.984
MC: Psychiatry	0.513	0.080	0.482	0.075	0.442	0.077	0.436	0.077	0.245	0.048	0.639
MC: Urology	−0.314	0.201	−0.314	0.197	−0.255	0.200	−0.217	0.199	−0.187	0.197	0.729
MC: Others	−0.112	0.049	−0.077	0.045	−0.044	0.048	−0.027	0.047	0.039	0.043	0.737
Female	0.000	0.020	0.032	0.018	−0.001	0.020	−0.040	0.020	0.000	0.020
Age	−0.010	0.002	−0.008	0.002	−0.016	0.002	−0.013	0.002	−0.016	0.002
(Age/150)2	0.005	0.003	−0.004	0.003	0.012	0.003	0.016	0.003	0.012	0.003
$\psi$	−0.195	0.015	—	—	−0.187	0.015	−0.280	0.015	−0.189	0.015
4. Home $\to$ death
MC: Cardiology	−0.061	0.064	−0.047	0.060	−0.054	0.060	−0.086	0.059	−0.061	0.058	0.780
MC: Gen. Surgery	0.331	0.052	0.294	0.047	0.350	0.050	0.388	0.049	0.195	0.043	0.883
MC: Thor. Surgery	0.245	0.149	0.209	0.138	0.139	0.147	0.143	0.145	0.133	0.128	0.509
MC: Gastro.	−0.037	0.114	−0.054	0.102	0.028	0.112	0.219	0.110	0.070	0.104	0.737
MC: Obs. & Gyn.	0.204	0.140	0.209	0.131	0.237	0.140	0.517	0.135	0.297	0.113	0.391
MC: Pulmonology	−0.153	0.093	−0.130	0.079	−0.110	0.092	−0.043	0.090	−0.076	0.090	0.738
MC: Inter. Med.	0.188	0.053	0.165	0.048	0.174	0.050	0.225	0.050	0.171	0.044	0.447
MC: ORL & CFS	0.046	0.143	0.123	0.129	0.031	0.142	0.022	0.139	0.062	0.131	0.482
MC: Psychiatry	0.339	0.181	0.306	0.176	0.293	0.178	0.303	0.177	0.152	0.114	NA
MC: Urology	0.233	0.119	0.218	0.108	0.204	0.116	0.212	0.115	0.210	0.114	0.375
MC: Others	−0.064	0.034	−0.065	0.030	−0.063	0.032	−0.141	0.032	−0.027	0.030	0.421
Female	−0.220	0.017	−0.172	0.014	−0.220	0.017	−0.328	0.017	−0.219	0.017
Age	0.004	0.003	0.012	0.003	0.008	0.003	0.032	0.003	0.008	0.003
(Age/150)2	0.053	0.004	0.035	0.004	0.050	0.004	0.036	0.004	0.050	0.004
$\psi$	−0.613	0.016	—	—	−0.603	0.016	−0.797	0.016	−0.603	0.016
log (likelihood)	−2345239		−2301265		−2349294		−2367719		−2350000
Number of patients	320,441		320,441		320,441		320,441		320,441
Number of observations	1,400,121		1,400,121		1,400,121		1,400,121		1,400,121
Unobserved heterogeneity	Yes		No		Yes		Yes		Yes
Hospital FE	80 largest		80 largest		10 largest		10 largest		10 largest
Charlson co‐morbidity index	Yes		Yes		Yes		No		Yes
Department size (number of specialists)	Yes		Yes		Yes		Yes		Yes
12 specialty FE	Yes		Yes		Yes		Yes		Yes
18 diagnoses FE	Yes		Yes		Yes		Yes		Yes
19 region FE	Yes		Yes		Yes		Yes		Yes
Linear & quadratic trends	Yes		Yes		Yes		Yes		No
Year FE	Yes		Yes		Yes		Yes		Yes

Note: All specifications use a multi‐spell multi‐state proportional conditional hazard model. The baseline hazards are piecewise constant. In all models (except Model 2), unobserved heterogeneity is modeled using a mixed weighting of the values taken by iid random variables drawn from a standard normal distribution. The parallel trend p‐values correspond to Model 3. The term NA indicates that the test could not be run for some specialties because the data set was too small. All the models (except Model 2) are estimated using simulated Maximum Likelihood. Model 2 does not account for unobserved heterogeneity and is estimated using Maximum Likelihood. We only estimate the MC effect for specialties of interest that have a sufficient number of observations. Pediatrics does not appear in the fourth transition because we only observe 14 deaths at home under FFS. Pediatrics is included in the category of “Others” in this case for this transition. Similarly, Psychiatry is also included in the “Others” category for the third transition, given that we have only eight observations under FFS.

TABLE A4.

Effect (in log) using the Weibull hazard.

Variables	Estimate	SD
1. Ad(RAd) $\to$ home
Mixed compensation	−0.067	0.005
Female	−0.007	0.005
Age	−0.003	0.000
(Age/150)2	−0.025	0.001
$\log (\alpha )$	0.374	0.001
$\varphi$	0.024	0.005
2. Ad(RAd) $\to$ death
Mixed compensation	−0.014	0.023
Female	−0.076	0.017
Age	0.011	0.004
(Age/150)2	0.017	0.004
$\log (\alpha )$	−0.194	0.006
$\varphi$	−0.542	0.022
3. Home $\to$ RAd
Mixed compensation	0.181	0.027
Female	0.000	0.020
Age	−0.009	0.002
(Age/150)2	0.003	0.003
$\log (\alpha )$	0.013	0.008
$\psi$	−0.221	0.012
4. Home $\to$ death
Mixed compensation	0.062	0.022
Female	−0.213	0.017
Age	0.006	0.003
(Age/150)2	0.050	0.004
$\log (\alpha )$	−0.481	0.006
$\psi$	−0.452	0.012
log(likelihood)	−2302315
Number of patients	320,441
Number of observations	1,400,121
Unobserved heterogeneity	Yes
Hospital FE	80 largest
Charlson co‐morbidity index	Yes
Department size (number of specialists)	Yes
12 specialty FE	Yes
18 diagnoses FE	Yes
19 region FE	Yes
Linear & quadratic trends	Yes
Year FE	Yes

Note: This specification, corresponding to Model 1, uses an MSMS proportional conditional Weibull hazard model. Unobserved heterogeneity is modeled using a mixed weighting of the values taken by iid random variables drawn from a standard normal distribution. The model is estimated using simulated Maximum Likelihood.

TABLE A5.

Effect (in log) by hospital's size.

Variables	Estimate	SD
1. Ad(RAd) $\to$ home
Small	0.043	0.008
Medium	−0.065	0.007
Large	−0.125	0.007
$\varphi$	0.001	0.003
2. Ad(RAd) $\to$ death
Small	0.047	0.053
Medium	−0.154	0.034
Large	0.076	0.030
$\varphi$	−0.342	0.019
3. Home $\to$ RAd
Small	0.225	0.043
Medium	0.127	0.041
Large	0.184	0.036
$\psi$	−0.199	0.015
4. Home $\to$ death
Small	0.043	0.042
Medium	−0.011	0.033
Large	0.133	0.031
$\psi$	−0.607	0.016
log (likelihood)	−2345882
Number of patients	320,441
Number of observations	1,400,121
Unobserved heterogeneity	Yes
Hospital FE	80 largest
Charlson co‐morbidity index	Yes
Department size (number of specialists)	Yes
12 specialty FE	Yes
18 diagnoses FE	Yes
19 region FE	Yes
Linear & quadratic trends	Yes
Year FE	Yes

Note: This specification, corresponding to Model 1, uses an MSMS proportional conditional hazard model. The baseline hazards are piecewise constant. Unobserved heterogeneity is modeled using a mixed weighting of the values taken by iid random variables drawn from a standard normal distribution. The model is estimated using simulated Maximum Likelihood.

TABLE A6.

Earlier and later adopter effects (in log).

Variables	Estimate	SD
1. Ad(RAd) $\to$ home
Adopt: 1999–2001	−0.075	0.006
Adopt: 2002–2004	−0.111	0.008
Adopt: 2005–2007	−0.020	0.008
Adopt: 2008–2016	0.082	0.011
$\varphi$	0.001	0.003
2. Ad(RAd) $\to$ death
Adopt: 1999–2001	−0.059	0.028
Adopt: 2002–2004	0.134	0.032
Adopt: 2005–2007	−0.051	0.049
Adopt: 2008–2016	−0.152	0.066
$\varphi$	−0.343	0.019
3. Home $\to$ RAd
Adopt: 1999–2001	0.089	0.031
Adopt: 2002–2004	0.411	0.041
Adopt: 2005–2007	0.267	0.052
Adopt: 2008–2016	−0.057	0.088
$\psi$	−0.197	0.015
4. Home $\to$ death
Adopt: 1999–2001	0.033	0.027
Adopt: 2002–2004	0.154	0.033
Adopt: 2005–2007	0.050	0.044
Adopt: 2008–2016	0.000	0.057
$\psi$	−0.609	0.016
log (likelihood)	−2345868
Number of patients	320,441
Number of observations	1,400,121
Unobserved heterogeneity	Yes
Hospital FE	10 largest
Charlson co‐morbidity index	Yes
Department size (number of specialists)	Yes
12 specialty FE	Yes
18 diagnoses FE	Yes
19 region FE	Yes
Linear & quadratic trends	Yes
Year FE	Yes

Note: This specification, corresponding to Model 1, uses an MSMS proportional conditional hazard model. The baseline hazards are piecewise constant. Unobserved heterogeneity is modeled using a mixed weighting of the values taken by iid random variables drawn from a standard normal distribution. The model is estimated using simulated Maximum Likelihood.

Data Availability Statement

This paper uses confidential data from the Régie de l'Assurance Maladie du Québec (RAMQ, the Health Insurance Organization of Quebec). The data can be obtained by filing a request directly with the Institut de la statistique du Québec (https://statistique.quebec.ca/research/#/accueil).