Els perfils dels sistemes de salut europeus

 Aquest article científic que trobareu a sota, ha estat generat per IA, en concret Claude de Anthropic. He tardat una mitja hora en fer-lo. He estat comparant els mateixos prompts entre Gemini, ChatGPT i Claude. Definitivament Claude és el millor model. ChatGPT s'hi acosta. Gemini necessita millorar. Almenys aquesta és la meva conclusió per aquesta tasca.

Llegiu-lo amb atenció. Es tracta d'una classificació dels sistemes de salut europeus basada en l'enquesta de caraterístiques estructurals. Habitualment es fa una classificació discreta, aquí hi ha una classificació continua basada en graus de pertinença. Es tracta d'una metodologia que vaig fer servir fa més de 3 dècades a la meva tesi doctoral per a classificar hospitals.

Avui mostro els resultats tal com han sortit de l'algoritme. La solució de dos perfils extrems sembla encertada, amb tres no millorava, a la discussió s'explica bé. Més endavant corregiré si cal, alguna coseta. És la primera vegada que faig un article sencer amb IA i m'he sorprès favorablement. Fa 30 anys, el programa d'ordinador GoM tardava més de 24 hores en fer els perfils, i després calia calibrar-ho i tornar-hi si no sortia bé. I després calia escriure l'article. Claude m'ha ofert una productivitat mai imaginada, increïble. Al final de l'article trobareu un resum en català fet amb NotebookLM, el que faig servir habitualment a aquest blog.

Quan sentiu parlar de la IA, no penseu en abstracte, penseu en coses concretes com aquesta i el canvi que representen i representaran encara més en el futur.

Mixed-membership profiles of health systems in 24 European countries: evidence from the OECD Health System Characteristics Survey 2023

Pere Ibern

CRES, Universitat Pompeu Fabra

 

Abstract

Typologies of health systems usually assign each country to a single category, although many systems combine features of several ideal types. We propose a probabilistic classification in which prototype profiles are estimated from the data and each country is described by its degree of membership in each profile. We analysed the 2023 OECD Health System Characteristics Survey for 24 European countries (356 categorical items; 58% of responses observed) and fitted a regularised Grade of Membership model. The number of profiles was chosen by repeated hold-out validation, and memberships were assessed in the full sample, out of sample (leave-one-country-out) and by bootstrap. Two profiles were supported (gain of 0.018 log-likelihood units per held-out response, SE 0.003; improvement in 95% of replicates); three or more did not improve prediction. The first profile, multi-payer social health insurance (Austria, Belgium, Switzerland, Czechia, Germany, Luxembourg, the Netherlands, Slovakia), combines multiple insurance funds, price negotiation between purchasers and providers, and self-employed physicians in private solo practices. The second, covering the other 16 countries, is characterised by public provision and a single main purchaser. The partition was stable across random starts and regularisation settings. The Netherlands, Belgium, France and Austria had the most mixed out-of-sample memberships. A three-profile solution separating tax-funded national health services from single-fund insurance was unstable and is reported as exploratory. Graded membership offers policy-makers a more informative basis than hard typologies for choosing peer countries and identifying hybrid systems.

Keywords: health system typology; mixed-membership models; Grade of Membership; Europe; Health System Characteristics Survey; cross-national comparison

1. Introduction

Comparing health systems requires reducing a large number of institutional features, such as who is covered by whom, how purchasers contract with providers, how professionals are paid and employed, and who owns hospitals, to a manageable set of types. Classical typologies distinguish tax-funded national health services, social health insurance systems built on earnings-related contributions, and private insurance-based systems [1,2]. Empirical classifications have since used regulation, financing and service-provision indicators to derive a small number of types for OECD countries, for example five types in the deductive classification of Böhm and colleagues [3] and the "worlds of healthcare" of Reibling, Ariaans and Wendt [4]. A recent latent-profile analysis has also classified OECD systems by financing structure [5].

Most of these approaches assign every country to exactly one category. This is a strong assumption. Systems reformed over several decades frequently combine features of different ideal types: a social insurance system can use public provision, a national health service can contract with private providers, and a single national fund can coexist with competing complementary insurers. A hard classification hides this heterogeneity and cannot express how close a country is to a type or to the boundary between two types.

Mixed-membership models offer a natural alternative. In the Grade of Membership (GoM) model [6,7], a small number of pure-type profiles are estimated from the data, and each unit is described by a vector of non-negative memberships summing to one. The same family of models underlies latent Dirichlet allocation for text [8], mixed-membership stochastic blockmodels for networks [9] and individual-level mixture models for multivariate categorical data in epidemiology [10]. To our knowledge, such models have not been applied to the institutional characteristics of health systems.

The objectives of this study are (i) to estimate prototype profiles of health-system characteristics from the OECD Health System Characteristics Survey (HSCS) 2023; (ii) to quantify the degree of membership of each country in each profile among European systems; (iii) to determine, using predictive validation, how many profiles the available data support; and (iv) to assess how much the resulting memberships depend on modelling choices and on the countries included.

2. Data and methods

2.1 Data source

We used the 2023 round of the OECD Health System Characteristics Survey (data flow DSD_HSCS2023@DF_HSCS2023) [11], a survey completed by national authorities that describes institutional features of health systems (the survey follows the earlier OECD survey of health-system institutional characteristics [12]). The extract analysed, downloaded from the OECD Data Explorer on 5 October 2026, contains 581 survey measures for 24 countries: Austria, Belgium, Czechia, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Latvia, Lithuania, Luxembourg, the Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland and the United Kingdom. The analysis deliberately focuses on European countries. Restricting the set to Europe compares systems that operate within a shared regulatory and policy environment (including the European single market, cross-border care rules and common fiscal and health-policy surveillance) and that share the historical lineages (Bismarckian insurance, Beveridgean national health services and the former centrally planned systems of Central and Eastern Europe) that structure most typologies. The profiles are therefore defined relative to this set of countries. Measures cover financing and coverage arrangements, purchasing and provider payment, hospital and primary-care organisation, workforce, digital health and governance. The survey uses skip logic, so many items are legitimately unanswered for a given country.

2.2 Construction of the analytical variables

Each survey measure was treated as a categorical item. We excluded free-text comment fields, numeric fields and responses longer than 110 characters. Missing codes (".." and blanks) were treated as unobserved. Response labels were harmonised (case, trailing punctuation and synonymous wordings such as "fee for services" and "fee-for-service"; "payment per case (DRG-like)" variants; "relative value scale" variants). Cost-sharing items were recoded as free at the point of care versus any cost-sharing, and co-payment exemption items were restricted to total, partial or no exemption. An item was retained if it was answered by at least four countries and had between two and six distinct harmonised categories. This yielded J = 356 items (201 yes/no; 250 binary in total; 106 with three to six categories; 880 category levels in all) belonging to 66 question blocks (the question number, e.g. Q14), with 4,919 observed responses out of 8,544 possible (57.6%), between 72 and 262 per country (Table 1).

Because some blocks contain dozens of near-duplicate items (the co-payment exemption matrix, Q14, alone contributes 33 items), each item received a weight w_j = 1/√n_b, where n_b is the number of retained items in its block, so that large blocks do not dominate the likelihood.

2.3 Mixed-membership (Grade of Membership) model

Let y_ij be the response of country i to item j, observed only if the country answered it. The model assumes K latent pure-type profiles. Profile k is described by a probability distribution λ_kj over the categories of each item, and country i by a membership vector g_i = (g_i1, …, g_iK) in the simplex. The probability of observing category l is Pr(y_ij = l) = Σ_k g_ik λ_kjl. Responses are independent given g and λ, and only observed responses enter the weighted log-likelihood Σ_i Σ_j w_j log Σ_k g_ik λ_kj(y_ij). Missing responses are therefore handled by the likelihood and require no imputation, under the assumption that which items a country answers is not informative about its profile conditional on the memberships.

Maximum-likelihood estimation with 24 countries and several hundred free profile parameters per profile drives memberships to the vertices of the simplex (pure assignment), which is an overfitting artefact. We therefore used penalised (maximum a posteriori) estimation with symmetric Dirichlet priors, Dir(α) on the memberships and Dir(β) on the category probabilities of each profile and item. The default values were α = 1.3, a mild shrinkage toward mixed memberships, and β = 1. The sensitivity of the results to both values is reported in Section 3.5.

2.4 Estimation

Parameters were estimated by the expectation-maximisation algorithm [13], in which the E-step computes the posterior probability that each observed response was generated by each profile, and the M-step updates memberships and profile probabilities from the weighted expected counts plus the prior pseudo-counts. Iterations stopped when the change in the objective was below 10^-8 or after 1,000 iterations. Because the likelihood is multimodal, each model was fitted from many random starts (at least 40, and 100 to 300 for the stability analyses) and the best objective was retained. Profiles were labelled by size and, for comparisons across fits, aligned by solving a linear assignment problem on weighted cross-entropy or on membership differences.

2.5 Choice of the number of profiles

The number of profiles K was chosen by predictive validation. In each of 20 replicates, 10% of the observed responses (cells of the country-by-item matrix) were randomly hidden, models with K = 1, …, 6 were fitted to the remaining data (best of three starts), and the weighted mean log predictive probability of the hidden responses was computed. The one-profile model (smoothed item-wise category frequencies across countries) was the baseline. For each K we report the paired gain over K = 1 across replicates, its standard error and the proportion of replicates in which the gain was positive. The procedure was repeated with 30% of responses hidden (10 replicates) as a more demanding check. Because the same hidden cells are used for all K within a replicate, differences are paired.

2.6 Degrees of membership and their uncertainty

We report three quantities for each country. First, the full-sample membership vector from the model fitted to all countries. Second, an out-of-sample membership obtained by leave-one-country-out: all of the country's responses are removed, the profiles are re-estimated from the other 23 countries and aligned to the full-sample profiles, and the country's membership is then estimated by maximum likelihood (flat prior) given those profiles. This quantity does not use the country to define the profiles and is not shrunk by the prior, so it expresses how closely the country resembles each prototype and is the preferred measure of mixedness. Third, a bootstrap over items: 50 resamples of the items (with replacement), refitting the model on each resample (best of five random starts, 400 iterations) and aligning labels to the full-sample solution. We report the 90% percentile interval of the membership in the dominant profile and the bootstrap support, defined as the proportion of resamples in which the dominant profile of the country is the same as in the full sample. The bootstrap reflects variability due to the choice of survey items (and, for K = 3, also imperfect optimisation in a multimodal likelihood); it does not capture variability due to the choice of countries, which the leave-one-country-out analysis addresses.

2.7 Robustness checks

•      Multi-start stability: the proportion of random starts reaching the best objective, and the agreement (adjusted Rand index, ARI [14]) of near-optimal solutions with the reported partition.

•      Sensitivity to regularisation: the ARI between the reported partition (dominant profile) and the partition obtained for α ∈ {1.0, 1.3, 1.6, 2.0} and β ∈ {0.5, 1, 2}.

•      Model-free check: the mean weighted mismatch dissimilarity between countries, computed over the items answered by both (at least 39 shared items for every pair), within and between profiles, with a permutation test (20,000 permutations) of the within-minus-between difference, and the agreement of the partitions with average-linkage hierarchical clustering of the same dissimilarities.

All analyses used custom code in Python 3.13 (NumPy, pandas, SciPy, scikit-learn) and Matplotlib for figures.

3. Results

3.1 Data description

Table 1 shows the number of retained items answered by each country. Coverage was reasonably even (between 133 and 262 responses for 23 countries), with the exception of Hungary, which answered only 72 of the 356 items; its memberships are correspondingly less precise.

Table 1. Number of retained items answered by each country (of 356).

Country	Answered	%	Country	Answered	%
Austria	213	60%	Lithuania	258	72%
Belgium	221	62%	Luxembourg	200	56%
Czechia	237	67%	Netherlands	243	68%
Estonia	262	74%	Norway	229	64%
Finland	162	46%	Poland	183	51%
France	202	57%	Portugal	177	50%
Germany	232	65%	Slovakia	230	65%
Greece	209	59%	Slovenia	214	60%
Hungary	72	20%	Spain	209	59%
Iceland	246	69%	Sweden	168	47%
Ireland	133	37%	Switzerland	208	58%
Latvia	222	62%	United Kingdom	189	53%

Percentages relative to the 356 retained items. The missing responses are mostly structural (skip logic).

3.2 Number of profiles

Predictive validation supported two profiles (Table 2, Figure 1). Relative to the one-profile baseline, K = 2 improved the held-out log-likelihood by 0.018 per response (SE 0.003) when 10% of the responses were hidden, in 95% of the 20 replicates, and by 0.006 (SE 0.002) when 30% were hidden. K = 3 did not improve on the baseline (−0.001, SE 0.004, with 10% hidden; −0.009, SE 0.002, with 30% hidden), and K ≥ 4 was consistently worse. With 24 countries, additional profiles are estimated from very few countries and fit idiosyncratic response patterns that do not generalise to hidden responses.

Table 2. Gain in held-out log-likelihood per response relative to the one-profile model.

K	10% hidden: gain (SE)	Replicates improved	30% hidden: gain (SE)	Replicates improved
2	+0.018 (0.003)	95%	+0.006 (0.002)	80%
3	−0.001 (0.004)	70%	−0.008 (0.002)	20%
4	−0.012 (0.005)	30%	−0.024 (0.004)	0%
5	−0.018 (0.005)	20%	−0.027 (0.002)	0%
6	−0.015 (0.005)	25%	−0.022 (0.005)	0%

Mean paired difference over 20 replicates (10% hidden) and 10 replicates (30% hidden); α = 1.3, β = 1. Positive values indicate better prediction than K = 1.

Figure 1. Predictive validation. Mean paired gain in held-out log-likelihood per response relative to K = 1, with ±1 standard error, for 10% and 30% of responses hidden.

On this evidence we regard K = 2 as the model supported by the data. We nevertheless also report K = 3, because it gives a more differentiated reading of the second profile and illustrates what a finer typology would look like, but we treat it as exploratory and discuss its limitations explicitly.

3.3 Two-profile solution

Sixteen countries had their dominant membership in profile P1 and eight in profile P2 (Table 3, Figure 2). Full-sample memberships of the dominant profile ranged from 0.93 to 0.98 (mean 0.96), partly reflecting the prior-based shrinkage. The out-of-sample memberships show where countries are closer to the boundary: the Netherlands (0.59 in P2), Belgium (0.70 in P2), France (0.71 in P1) and Austria (0.76 in P2) had out-of-sample memberships below 0.8, while the other 20 countries were at 0.82 or higher, and 16 of the 24 were at 0.96 or higher. The dominant profile obtained leave-one-country-out was the same as in the full sample for all 24 countries. In the item bootstrap the dominant profile was retained in 76% to 100% of resamples (mean 93%; at least 90% for 15 countries and at least 80% for 22), with the lowest support for the Netherlands (76%), Hungary (78%), Slovenia (80%) and France (82%).

Table 3. Degrees of membership, two-profile solution (K = 2).

Country	Dominant	Full sample		Out of sample		90% bootstrap interval	Bootstrap
	profile	P1	P2	P1	P2	(dominant, full)	support
Spain	P1	0.98	0.02	1.00	0.00	0.97–0.99	100%
Lithuania	P1	0.98	0.02	1.00	0.00	0.51–0.98	94%
Finland	P1	0.98	0.02	1.00	0.00	0.96–0.98	100%
Iceland	P1	0.97	0.03	1.00	0.00	0.91–0.98	100%
Sweden	P1	0.97	0.03	1.00	0.00	0.96–0.98	100%
Portugal	P1	0.97	0.03	1.00	0.00	0.94–0.98	96%
Estonia	P1	0.97	0.03	0.89	0.11	0.04–0.98	88%
United Kingdom	P1	0.97	0.03	1.00	0.00	0.82–0.98	98%
Latvia	P1	0.97	0.03	1.00	0.00	0.68–0.98	96%
Greece	P1	0.97	0.03	0.96	0.04	0.77–0.98	98%
Norway	P1	0.95	0.05	0.82	0.18	0.10–0.98	86%
Poland	P1	0.95	0.05	0.99	0.01	0.11–0.97	88%
Ireland	P1	0.94	0.06	0.83	0.17	0.14–0.97	90%
Slovenia	P1	0.94	0.06	0.85	0.15	0.06–0.97	80%
France	P1	0.93	0.07	0.71	0.29	0.07–0.97	82%
Hungary	P1	0.93	0.07	0.96	0.04	0.07–0.96	78%
Switzerland	P2	0.02	0.98	0.00	1.00	0.93–0.98	98%
Luxembourg	P2	0.02	0.98	0.00	1.00	0.96–0.98	100%
Germany	P2	0.03	0.97	0.01	0.99	0.94–0.98	100%
Slovakia	P2	0.03	0.97	0.00	1.00	0.96–0.98	100%
Czechia	P2	0.03	0.97	0.04	0.96	0.94–0.98	100%
Belgium	P2	0.05	0.95	0.30	0.70	0.09–0.98	88%
Austria	P2	0.05	0.95	0.24	0.76	0.10–0.97	88%
Netherlands	P2	0.06	0.94	0.41	0.59	0.03–0.97	76%

P1: public provision and single-payer financing. P2: multi-payer social health insurance. Memberships sum to 1 across profiles within each estimate. Out of sample: leave-one-country-out. Bootstrap: 50 item resamples; support is the share of resamples in which the dominant profile is unchanged. The bootstrap does not capture country-sampling variability.

Figure 2. Degrees of membership of the 24 countries in the two profiles: full-sample estimate (left) and out-of-sample estimate by leave-one-country-out (right).

Table 4 describes the profiles by the items that best discriminate them. Profile P2 (multi-payer social health insurance; Austria, Belgium, Switzerland, Czechia, Germany, Luxembourg, the Netherlands and Slovakia) is characterised, in all countries that answered, by multiple insurance funds or companies as the main source of basic coverage (100% of 8, against 0% of 16 in P1), by negotiation of prices between purchasers and providers (100% of 7 versus 33% of 12), by self-employed primary-care physicians (100% versus 33%) and specialists (100% versus 29%), and by outpatient specialist care delivered mainly in private solo practices (88% versus 6%). Profile P1 (the other 16 countries) is characterised by publicly employed primary-care physicians (60% versus 0%) and specialists (64% versus 0%), by the absence of purchaser-provider price negotiation (67% of 12 answering "no", versus 0% in P2), and by greater autonomy of nurses in advanced roles (90% of those answering allow independent prescribing, versus 0% in P2). P1 is internally mixed regarding the main source of coverage: 50% report a national health system and 44% a single health insurance fund.

Table 4. Selected discriminating features of the profiles: share of countries in each profile (by dominant membership) giving each response, with the number of countries answering in parentheses.

Item	Response	K = 2		K = 3
		P1	P2	P1	P2	P3
Main source of basic coverage	Multiple insurance funds	0% (16)	100% (8)	11% (9)	0% (8)	100% (7)
	National health system	50% (16)	0% (8)	22% (9)	75% (8)	0% (7)
	Single health insurance fund	44% (16)	0% (8)	67% (9)	12% (8)	0% (7)
Price negotiation between purchasers and providers	Yes	33% (12)	100% (7)	60% (5)	14% (7)	100% (7)
Primary care physicians predominantly	Self-employed	33% (15)	100% (8)	62% (8)	12% (8)	100% (7)
	Publicly employed	60% (15)	0% (8)	38% (8)	75% (8)	0% (7)
Outpatient specialists (community) predominantly	Self-employed	29% (14)	100% (8)	38% (8)	29% (7)	100% (7)
	Publicly employed	64% (14)	0% (8)	50% (8)	71% (7)	0% (7)
Outpatient specialist care delivered mainly in	Private solo practices	6% (16)	88% (8)	11% (9)	12% (8)	86% (7)
Fees based on a relative value scale	No	55% (11)	38% (8)	14% (7)	100% (5)	43% (7)
Advanced-role nurses may prescribe medicines	Yes, independently	90% (10)	0% (6)	100% (5)	80% (5)	0% (6)
Advanced-role nurses authorised to bill	Yes, for all services	33% (9)	0% (6)	60% (5)	0% (4)	0% (6)
	No	56% (9)	83% (6)	20% (5)	100% (4)	83% (6)

Items are survey questions Q2, Q24, Q29A, Q30A, Q19A, Q26A, Q50 (nurse prescribing) and Q51. Profiles are defined by the dominant membership in the full-sample fit. A blank item label continues the item above.

3.4 Three-profile solution (exploratory)

The three-profile solution (Table 5, Figure 3) splits P1 of the two-profile solution into two profiles and leaves the multi-payer profile essentially unchanged. The new P3 is the multi-payer social insurance profile (Austria, Switzerland, Czechia, Germany, Luxembourg, the Netherlands and Slovakia; Belgium moves out). P2 (Spain, Finland, the United Kingdom, Greece, Ireland, Iceland, Portugal and Sweden) is a tax-funded national-health-service profile: 75% of the 8 countries report a national health system as the main source of coverage, 75% publicly employed primary-care physicians, no country reports multiple insurance funds, and all five countries answering report that fees are not based on a relative value scale. P1 (Belgium, Estonia, France, Hungary, Lithuania, Latvia, Norway, Poland and Slovenia) is a single-fund insurance profile: 67% of the 9 report a single health insurance fund as the main source of coverage, price negotiation is common (60% of the 5 answering) and primary-care physicians are largely self-employed (62%), while nurses with advanced roles can independently prescribe (100% of the 5 answering) and bill (60%).

Table 5. Degrees of membership, three-profile solution (K = 3, exploratory).

Country	Dominant	Full sample			Out of sample			90% bootstrap	Bootstrap
	profile	P1	P2	P3	P1	P2	P3	interval (dominant)	support
Estonia	P1	0.95	0.03	0.02	0.22	0.78	0.00	0.02–0.96	64%
Lithuania	P1	0.95	0.03	0.02	0.40	0.60	0.00	0.02–0.97	56%
France	P1	0.94	0.03	0.03	0.77	0.11	0.11	0.04–0.95	76%
Latvia	P1	0.91	0.06	0.03	0.34	0.66	0.00	0.02–0.95	42%
Norway	P1	0.91	0.06	0.03	0.10	0.74	0.16	0.02–0.95	32%
Hungary	P1	0.91	0.04	0.05	0.17	0.82	0.01	0.04–0.94	60%
Slovenia	P1	0.90	0.06	0.04	0.25	0.59	0.16	0.02–0.95	50%
Poland	P1	0.90	0.06	0.04	0.22	0.78	0.00	0.03–0.94	50%
Belgium	P1	0.84	0.02	0.13	0.57	0.00	0.43	0.03–0.95	48%
Finland	P2	0.03	0.95	0.02	0.00	1.00	0.00	0.02–0.96	84%
Sweden	P2	0.03	0.95	0.02	0.00	1.00	0.00	0.03–0.96	76%
Spain	P2	0.03	0.95	0.02	0.00	1.00	0.00	0.02–0.97	72%
Iceland	P2	0.03	0.94	0.03	0.00	1.00	0.00	0.02–0.96	82%
Portugal	P2	0.04	0.93	0.03	0.04	0.96	0.00	0.03–0.96	62%
Ireland	P2	0.03	0.93	0.04	0.00	1.00	0.00	0.04–0.95	82%
Greece	P2	0.07	0.90	0.03	0.52	0.48	0.00	0.03–0.96	64%
United Kingdom	P2	0.08	0.88	0.04	0.43	0.57	0.00	0.02–0.96	66%
Switzerland	P3	0.03	0.02	0.95	0.00	0.00	1.00	0.03–0.96	78%
Germany	P3	0.03	0.02	0.95	0.07	0.00	0.93	0.03–0.96	88%
Luxembourg	P3	0.03	0.02	0.95	0.00	0.00	1.00	0.03–0.96	84%
Slovakia	P3	0.03	0.02	0.95	0.06	0.00	0.94	0.04–0.96	92%
Czechia	P3	0.03	0.02	0.95	0.00	0.04	0.96	0.03–0.96	84%
Netherlands	P3	0.03	0.04	0.93	0.13	0.28	0.59	0.02–0.95	50%
Austria	P3	0.04	0.04	0.92	0.00	0.33	0.67	0.05–0.94	74%

P1: single-fund insurance. P2: tax-funded national health service. P3: multi-payer social health insurance. Out of sample: leave-one-country-out. Bootstrap support: share of 50 item resamples in which the dominant profile is unchanged.

Figure 3. Degrees of membership of the 24 countries in the three profiles: full-sample estimate (left) and out-of-sample estimate by leave-one-country-out (right).

The out-of-sample memberships show that the three-profile solution is considerably less stable than the two-profile one. Twelve countries had an out-of-sample dominant membership below 0.8 and six below 0.6 (Greece 0.52, Belgium 0.57, the United Kingdom 0.57, Slovenia 0.59, the Netherlands 0.59 and Lithuania 0.60). Bootstrap support was also much lower than for K = 2: it averaged 67%, exceeded 90% for only one country (Slovakia, 92%) and was 50% or lower for six (Norway 32%, Latvia 42%, Belgium 48%, the Netherlands, Poland and Slovenia 50%). The dominant profile obtained leave-one-country-out coincided with the full-sample profile for only 16 of 24 countries (67%). The eight discrepancies are Greece (assigned to P1 rather than P2) and seven of the nine members of P1, namely Estonia, Hungary, Lithuania, Latvia, Norway, Poland and Slovenia, which when left out resembled the tax-funded national-health-service profile more than the single-fund profile (out-of-sample memberships in P2 of 0.59 to 0.82). Only Belgium and France remain assigned to P1 out of sample (0.57 and 0.77). Thus P1 of the three-profile solution is defined by a few countries and does not behave as a stable prototype; it is better read as a boundary region between national health services and multi-payer insurance than as a third distinct type.

3.5 Robustness

For K = 2 the solution was highly stable. The best objective was reached in 99 of 100 random starts, all near-optimal solutions gave an identical partition (ARI = 1), and the partition was identical (ARI = 1) in all 12 combinations of α and β examined (Table S1). Increasing α from 1.0 to 2.0 progressively softens the memberships (mean dominant membership from 1.00 to 0.89 at β = 1) without changing the partition. Item resampling changed the dominant profile of a country in at most a quarter of the resamples (Table 3, support column); the 90% intervals of the membership in the dominant profile are wide for the countries with lowest support (for example France and Ireland, whose lower limits are close to zero because the resamples that reassign them produce memberships near zero), and they reflect item sampling only.

For K = 3 the objective is multimodal: the best solution was reached by 4 of 300 starts (1 of 100 in the first stability run), and seven of the 100 starts were within one log-likelihood unit of the best, with a minimum ARI of 0.74 relative to the reported partition. The partition was reproduced exactly (ARI = 1) in 5 of 12 regularisation settings, was close in others (ARI 0.86), but was substantially different with β = 2 (ARI = 0.44), as well as with α = 1.0 and β = 1 or 2 (ARI 0.54 and 0.44; Table S1). The two- and three-profile partitions are nearly nested (ARI = 0.44 because the two-profile P1 is split in two): the seven countries of the three-profile multi-payer profile all belong to the two-profile multi-payer profile, as does Belgium, which the three-profile solution assigns instead to the single-fund profile.

The model-free check supports the existence of structure in both solutions: countries assigned to the same profile were more similar than countries assigned to different profiles (mean weighted mismatch 0.414 within versus 0.489 between for K = 2; 0.398 versus 0.471 for K = 3; permutation p < 0.0001 for both). However, average-linkage hierarchical clustering of the same dissimilarities agreed only modestly with the model-based partitions (ARI = 0.28 for K = 2 and 0.07 for K = 3), indicating that the GoM profiles are driven by a subset of discriminating items rather than by overall similarity across all items.

4. Discussion

Using a mixed-membership model on 356 institutional characteristics of 24 OECD countries, we found one robust contrast: multi-payer social health insurance systems, with multiple insurance funds, negotiated prices and a private, self-employed ambulatory sector, versus systems with public provision of primary care and a single main purchaser, whether a national health service or a single fund. This contrast corresponds to the classical Bismarck-Beveridge distinction that underlies most typologies [2,3], and it was supported by predictive validation, by the stability of the solution across random starts and regularisation settings, and by the leave-one-country-out analysis. Finer distinctions were not supported by the same criteria with the number of countries available.

The main contribution of the approach is graded membership. In the two-profile solution most countries are close to the prototypes, but the out-of-sample memberships single out the Netherlands, Belgium, France and Austria as cases that resemble the opposite profile to a considerable degree. This is consistent with the intermediate character often attributed to systems that combine regulated competition or mandatory insurance with strong public steering, but we caution that these memberships are statements about survey responses, not about the full institutional reality, and that we have not checked them against national documentation. The fact that France is assigned to the public/single-payer profile illustrates that the profiles describe the features asked about in the survey (purchasing, provider employment, price setting, nurse roles), which may differ from the classification based on the financing source alone.

The three-profile solution is instructive for what it shows and for what it cannot show. Its profiles correspond to recognisable types: a tax-funded national health service group (Spain, Portugal, Ireland, the United Kingdom, Greece and the Nordic countries Finland, Iceland and Sweden), a multi-payer social insurance group, and a group of single-fund insurance systems that includes most Central, Eastern and Baltic European countries together with Belgium, France and Norway. However, the single-fund profile did not hold up out of sample: when a country was left out, most of its members were closer to the national health service profile. Several of these countries moved from centrally planned, state-run health systems towards fund-based financing in the 1990s, and the data are compatible with a continuum between national health services and fund-based systems rather than with a separate type, but this interpretation requires evidence beyond the present data. We therefore recommend that the finer typology be considered a hypothesis for testing with a larger set of countries and not a result.

Compared with deductive classifications and with hard clustering, the mixed-membership approach has three practical advantages: it quantifies how much each country resembles each prototype; it yields profile-level response probabilities for every item, which makes the prototypes directly interpretable; and it handles the structural missingness that characterises institutional surveys without imputation. Its main cost is the need for regularisation when the number of units is small, because unpenalised estimates collapse to pure memberships. Because the degree of mixing in full-sample estimates depends on the prior, we recommend interpreting the out-of-sample memberships and not the full-sample ones when the purpose is to assess hybridity.

4.1 Policy implications

Three implications follow for health policy analysis in Europe. First, the choice of peer countries for benchmarking and performance comparison can be informed by profile membership: comparing a country with the members of its dominant profile, and with the hybrid cases for which memberships are split, controls for the institutional features that most differentiate systems. Second, the countries with mixed out-of-sample memberships (the Netherlands, Belgium, France and Austria) combine features of both profiles, which makes them natural sources of experience on how elements of one model, such as negotiated purchasing or public provision of primary care, operate within the other; they are candidates for policy-learning studies. Third, because the memberships are continuous, repeated rounds of the survey could be used to track whether reforms move a system toward or away from a prototype, which a hard classification would register only after a country switches category. The typology should be used to describe and to select comparators; it does not by itself support causal statements about the performance of a profile.

4.2 Limitations

•      Sample size. With 24 countries the number of profiles is weakly identified and the profiles depend on a small number of countries. The profiles are defined relative to this European set and should be re-estimated, not assumed, if the set of countries changes.

•      Measurement. The survey is completed by national authorities, and the interpretation of questions, in particular when a system has regional or multiple schemes, may differ across respondents. Free-text comments and numeric fields were not analysed.

•      Missing data. We assume that unanswered items are non-informative given the memberships. Skip logic makes some missingness structural, and some items are answered mostly in one type of system, which may reinforce the profiles.

•      Model. Items are assumed conditionally independent given memberships, although items within a block are related; block weights mitigate but do not remove this. The prior parameters were fixed rather than estimated; the partition for K = 2 was insensitive to them, but K = 3 was not. The bootstrap describes item sampling only, and for K = 3 it is also affected by the multimodality of the likelihood.

•      Selection of K. Predictive validation hides random responses of countries that remain in the fit; it does not evaluate prediction for entirely new countries, which is captured only by the leave-one-country-out memberships.

5. Conclusions

A mixed-membership model applied to the OECD HSCS 2023 identifies a robust two-profile structure in 24 European health systems, separating multi-payer social health insurance systems from systems with public provision and a single main purchaser, and shows that some countries, notably the Netherlands, Belgium, France and Austria, are best described by graded rather than categorical membership. A finer three-profile description is plausible but not statistically supported with the data currently available. Repeating the analysis with future survey rounds would allow the stability of the profiles and the movement of countries between them over time to be tested, and a larger set of countries would allow finer profiles to be examined.

Data availability

The survey data are publicly available from the OECD Data Explorer (OECD Health System Characteristics Survey 2023) [11]. The harmonised item matrix and the analysis code will be deposited in a public repository [repository and DOI to be added before submission].

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the authors used Claude (Anthropic) to write and run the statistical analysis code and to draft the manuscript text. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

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Supplementary material

Table S1. Agreement (adjusted Rand index, ARI) between the reported partition and the partition obtained under alternative prior settings.

α	β	ARI, K = 2	ARI, K = 3
1.0	0.5	1.00	0.86
1.0	1	1.00	0.54
1.0	2	1.00	0.44
1.3	0.5	1.00	1.00
1.3	1	1.00	1.00
1.3	2	1.00	0.44
1.6	0.5	1.00	0.86
1.6	1	1.00	1.00
1.6	2	1.00	0.44
2.0	0.5	1.00	1.00
2.0	1	1.00	1.00
2.0	2	1.00	0.44

Reported solution: α = 1.3, β = 1. Each cell is the best of 20 random starts.

PS. Resum de l'article en català fet amb NotebookLM

L'estudi elabora una tipologia contínua i probabilística dels sistemes de salut europeus mitjançant el model de grau de pertinença (Grade of Membership, GoM), superant les classificacions tradicionals que assignen rígidament cada país a una sola categoria discreta. A partir de les dades de l'enquesta de l'OCDE de 2023 (Health System Characteristics Survey) per a 24 països europeus, s'analitzen 356 variables categòriques sobre finançament, cobertura, contractació, organització de proveïdors i rols professionals. Mitjançant validació predictiva (repeated hold-out), es va determinar el nombre de perfils prototípics que millor descriuen la diversitat institucional de la regió.

La validació predictiva va confirmar que les dades donen suport a una solució de dos perfils principals (K = 2), mentre que afegir un tercer perfil o més no millora la capacitat predictiva del model:

• Perfil d'assegurança social multipagador (Bismarck): Agrupa 8 països (Alemanya, Àustria, Bèlgica, Eslovàquia, Luxemburg, Països Baixos, República Txeca i Suïssa). Es caracteritza per la presència de múltiples fons d'assegurança (100% dels països), la negociació de preus entre compradors i proveïdors (100%), metges d'atenció primària i especialistes majoritàriament autònoms (100%) i una atenció especialitzada ambulatòria prestada principalment en consultoris privats individuals (88%).
• Perfil de provisió pública i pagador principal únic: Inclou els altres 16 països de la mostra (com Espanya, Regne Unit, Suècia, Finlàndia, Portugal, França, Lituània, Estònia o Noruega). Es defineix per metges de primària (60%) i especialistes (64%) d'ocupació pública, absència de negociació de preus (67%) i una major autonomia de la infermeria avançada per prescriure medicaments de manera independent (90%). La font de cobertura d'aquest perfil es divideix entre serveis nacionals de salut (50%) i un fons d'assegurança únic (44%).

L'anàlisi fora de mostra (leave-one-country-out) permet identificar el grau de caràcter mixt o híbrid de determinats sistemes. Entre els països situats en posicions menys extremes i més properes a la frontera entre ambdós perfils destaquen els Països Baixos (0,59 en el perfil multipagador), Bèlgica (0,70), França (0,71 en el perfil públic) i Àustria (0,76). Aquests casos combinen la regulació de la competència o l'assegurança obligatòria amb un fort control o direcció pública.

Una solució exploratòria de tres perfils (K = 3) desglossa el perfil públic en un grup de servei nacional de salut finançat per impostos (com Espanya, Regne Unit, Portugal, Grècia o els països nòrdics) i un grup d'assegurança de fons únic (que inclou majoritàriament països de l'Europa central, de l'est i bàltics, juntament amb Bèlgica, França i Noruega). Malgrat oferir una lectura intuïtiva, aquesta classificació més fina va resultar instable en les proves fora de mostra i de bootstrap, comportant-se el grup de fons únic més com un continu o regió fronterera que no pas com un tipus independent.

L'ús de graus de pertinença continus ofereix als gestors i analistes de polítiques sanitàries un marc més precís per a la selecció de països comparables (peer benchmarking). A més, permet utilitzar els països híbrids com a font d'aprenentatge sobre la combinació d'elements institucionals i facilita el seguiment de les reformes del sistema de salut al llarg del temps.