Revealing institutional vulnerabilities in Nordic health systems through multidimensional scaling of COVID-19 mortality data

Abstract

This study introduces a novel application of Multi-Dimensional Scaling (MDS) to examine COVID-19 mortality trends across Nordic welfare states, addressing a significant gap in pandemic analysis. While existing literature has extensively compared Nordic COVID-19 responses through traditional epidemiological metrics, no previous study has employed MDS to visualize and analyze time-series mortality patterns in relation to institutional welfare structures. Drawing on excess mortality data from 2020 through 2024, this research reveals distinct national trajectories: Sweden experienced sharp increases in deaths and major outbreaks, Finland showed a steady decline, and Norway exhibited irregular patterns. The study’s methodological innovation lies in linking these MDS-derived patterns to varying degrees of market-oriented reforms across the Nordic welfare states, providing new insights into how institutional design shapes pandemic resilience. Excess mortality is measured using the P-score, which represents the percentage difference between observed and expected deaths based on pre-pandemic averages. Peak P-scores–3387.09% in Sweden (2020), 913.23% in Finland (2022), and 629.8% in Norway (2022) highlight the differing capacities of each system to respond to crises. This work fills a critical methodological gap by demonstrating how dimensionality reduction techniques can reveal institutional vulnerabilities in welfare systems during prolonged public health emergencies, offering new analytical tools for comparative welfare state research and pandemic preparedness planning.

Introduction

The COVID-19 pandemic has had substantial economic and health repercussions^1–5. It has particularly affected vulnerable groups, including low-income households and countries that depend on food imports, disrupting food systems through trade barriers, supply chain challenges, and diminished market access.

Against this backdrop, Nordic countries (with their strong welfare institutions and varying degrees of market-oriented reforms) offer a valuable case study. However, existing comparative analyses of Nordic COVID-19 responses have predominantly relied on conventional epidemiological comparisons and policy analysis frameworks^6,7. While these studies provide valuable insights into mortality outcomes and policy effectiveness, they lack sophisticated analytical tools to visualize complex temporal patterns and their relationship to underlying institutional structures. This represents a significant methodological gap in the literature, particularly given the multidimensional nature of pandemic impacts across different welfare regimes. Recent work has applied both mechanistic COVID-19 models and descriptive multidimensional scaling approaches^8,9, with the present study adopting the latter for comparative visualization.

This paper addresses this gap by introducing Multi-Dimensional Scaling (MDS) as a novel analytical framework for examining COVID-19 mortality patterns in relation to welfare state characteristics. Unlike previous studies that examine Nordic pandemic responses through descriptive statistics or regression analysis, this research employs MDS to create low-dimensional visualizations of high-dimensional mortality data, enabling the identification of distinct national trajectories that may not be apparent through conventional analytical approaches. The methodological innovation lies in linking these visualized patterns to institutional differences in market-oriented reforms, providing new insights into how welfare state structures influence pandemic resilience.

According to Pizzato et al.¹⁰, Europe saw an estimated 1,642,586 excess deaths, representing an 8.0% rise compared to expected mortality rates, mainly between 2020 and 2022. Italy, Poland, and Germany reported the highest total numbers of excess deaths, whereas Bulgaria, Lithuania, and Slovakia experienced the most significant relative increases. Age-standardized death rates across Europe varied considerably during the pandemic, ranging from 1.8 per 10,000 population in Sweden to 24.7 in Bulgaria. These disparities reflect deeper structural inequalities: countries with higher poverty and income inequality generally experienced elevated excess death rates, while those with greater GDP per capita, higher health expenditure, and stronger vaccination coverage saw lower mortality. This context situates Sweden’s relatively low death rate not only as a policy outcome, but also as one shaped by broader welfare and economic factors.

Among the European Nordic countries, Sweden presents a particularly notable case–not only for its relatively low average mortality compared to the European mean, but also for its divergence from Nordic peers. Specifically, Conyon and Thomsen⁶ examine the progression of COVID-19 in Scandinavia during 2020 and evaluate whether Sweden’s unique public health approach led to different infection and mortality rates compared to its neighbours. They focus on two main questions: how COVID-19 spread in Scandinavia and the impact of Sweden’s less stringent public health measures. In the absence of a vaccine and community herd immunity, non-pharmacological interventions (NPIs) were crucial. Denmark and Norway implemented strict lockdowns, including widespread business and school closures and significant travel restrictions. In contrast, Sweden adopted a more relaxed strategy, relying on community trust rather than enforced measures. This situation has provided a unique opportunity to analyze the effects of different lockdown strategies on health outcomes. New COVID-19 cases in Sweden surged sharply in March and November 2020, with a decline starting in early 2021. Sweden experienced higher daily COVID-19 deaths compared to Denmark and Norway, with a notable mortality spike during the first wave. By February 2021, Sweden’s COVID-19 mortality per capita remained high compared to its neighbours, and its reproduction rate of the virus stayed above one for most of the year. The Oxford University stringency index shows that the policy measures in Scandinavia had similar time-series trends, At the same time, mobility data indicated that Swedes maintained relatively higher interaction rates during the first wave. Testing rates were highest in Denmark, whereas Sweden had the highest positivity rate, surpassing the 5% threshold recommended by the WHO. In terms of vaccination, Denmark led in implementation as of early 2021, with Sweden and Norway following similar trends¹¹. Based on the empirical evidence presented, Conyon, and Thomsen⁶ maintain that lockdowns result in fewer COVID-19 cases and deaths. However, their study does not evaluate other trade-offs associated with lockdowns, such as economic impacts or effects on political liberty. Additionally, it does not fully consider potential unobserved differences between countries, as highlighted in the studies by Pettersen and Lars¹² and Yarmol-Matusiak et al.⁷. The latter study analyzed the early impact of COVID-19 from February to July 2020 across the Nordic countries. It found that Sweden experienced higher COVID-19 incidence rates across all age groups and a higher death rate related to COVID-19 compared to its Nordic peers. This higher death rate was only partially explained by demographic factors and was particularly pronounced in seniors’ care homes, contributing to elevated all-cause mortality. Yarmol-Matusiak et al.⁷ concluded that Sweden’s health system faced significant strain, revealing systemic issues in seniors’ care and challenges with contact tracing and testing. The initially looser government restrictions likely contributed to Sweden’s higher COVID-19 impact. In response, Sweden increased testing rates, imposed more restrictive measures, and expanded its intensive care unit capacity to better manage the epidemic.

Recent comparative research has examined why Nordic countries, despite shared welfare state traditions, responded differently to the COVID-19 pandemic. Laage-Thomsen and Frandsen¹³ used a structured case comparison and expert survey data from Denmark, Norway, and Sweden to explore whether national variation in pandemic responses reflected divergent expert preferences. Their findings suggest that policy differences were shaped more by politico-administrative organization than by expert dissensus. While valuable for understanding institutional dynamics, their approach is based on cross-sectional survey data and conventional regression techniques. Irfan et al.¹⁴ provide a quantitative analysis of the trade-off between public health policies and economic outcomes in the Nordic region. Using joinpoint regression on epidemiological and macroeconomic data from January 2020 to January 2021, they show that Sweden’s relaxed approach led to higher infection and mortality rates without clear economic gains. Though methodologically robust, their analysis treats mortality as a linear outcome and does not account for institutional variation. Saunes et al.¹⁵ analyze governance structures and pandemic responses across five Nordic countries using data from the COVID-19 Health System Response Monitor. Their study emphasizes variation in decentralization, emergency powers, and expert involvement, revealing differences in health system coordination. However, the focus remains on institutional arrangements rather than empirical outcome analysis. Hallberg et al.¹⁶ conduct a comparative study of epidemiological outcomes and policy responses in the five Nordic countries from 2020 to 2022. Using data on infections, hospitalizations, ICU admissions, and mortality, they link outcome trends to 13 policy measures via a stringency index. Their results highlight variation across waves, with Sweden showing the highest early burden and Denmark overtaking it later. While methodologically robust, the analysis relies on unidimensional time-series and policy indices.

In contrast to these prior comparative studies, this research introduces a novel methodological framework by applying Multi-Dimensional Scaling (MDS) to COVID-19 mortality data and health profiles across Finland, Norway, and Sweden. Rather than relying on static or unidimensional metrics, MDS enables spatial and temporal visualization of pandemic trajectories through dimensionality reduction of key indicators including total cases, new cases, total deaths, and deaths per million. This approach facilitates outcome-based comparisons that can uncover hidden patterns in mortality dynamics and relate them to structural features of welfare states, offering a complementary perspective on institutional resilience that has not been previously explored in the Nordic COVID-19 literature.

The analysis reveals distinct patterns: Sweden shows a sharp upward trend with a notable outlier indicating a surge in cases; Finland demonstrates a consistent downward trend in deaths per million; and Norway exhibits a more irregular pattern with two significant outliers in new cases per million. These results highlight the varying experiences and management of the COVID-19 pandemic in each country, offering insights into the effectiveness of public health interventions and the pandemic’s diverse trajectories across neighbouring nations. The analysis highlights how underlying health conditions and demographic factors shape population vulnerability to COVID-19, underscoring the need for context-specific health planning. It also compares policy effectiveness across Northern European countries, noting that proactive measures and sustained health system investments have been critical. Additionally, the study considers the implications of market-oriented reforms for the future resilience of Nordic welfare states. Excess mortality data from 2020 to 2024 reveals persistent public health challenges, with continued elevated death levels reflecting the pandemic’s lasting impact.

While numerous studies have assessed COVID-19 responses in the Nordic region^6,7,10, few have systematically examined how structural welfare reforms intersect with pandemic mortality patterns using advanced visualization techniques. Most existing comparative analyses rely on descriptive statistics, correlation analysis, or traditional regression approaches that may obscure complex temporal relationships and multidimensional patterns in the data. This study addresses that gap by applying MDS to link temporal mortality data with institutional variation, providing both methodological innovation and substantive insights into welfare state resilience during health crises.

The study makes three distinct contributions to the literature. First, it demonstrates the analytical power of MDS for comparative pandemic research, showing how dimensionality reduction can reveal patterns invisible to conventional statistical approaches. Second, it provides new empirical evidence linking market-oriented welfare reforms to pandemic vulnerability through spatial analysis of mortality trajectories. Third, it offers a replicable methodological framework that can be applied to other health crises and comparative institutional analyses, expanding the analytical toolkit available to researchers studying welfare state resilience and public health outcomes.

The structure of this article is as follows. Section “Material and methods” describes the dataset and introduces the MDS methodology. Section “Results” presents national mortality patterns and pandemic responses, followed by MDS visualizations of cross-country trajectories. Section “Discussion” examines welfare state reforms and their potential influence on pandemic outcomes. Section “Conclusions” concludes with reflections on the implications for the resilience of Nordic welfare systems.

Material and methods

This section presents the data analyzed, provides introductory notes on MDS, and details the software and hardware utilized in the study.

Data

We selected Finland, Norway, and Sweden due to their shared institutional characteristics as Nordic welfare states, high-quality and comparable mortality data, and similar demographic profiles. Although geographic isolation was not modeled directly, these countries also share low population densities and limited cross-border connectivity relative to central Europe. These structural similarities support a meaningful comparative analysis of COVID-19 mortality trajectories and public health responses.

The primary data source for COVID-19 indicators is Our World in Data (OWID), which compiles official statistics from national and international sources into a harmonized cross-country format. OWID was selected for its standardized structure, broad indicator coverage, and temporal consistency–making it particularly suited for longitudinal, comparative analysis. While we recognize the availability of high-quality register-based data in the Nordic countries, OWID facilitates reproducible analysis across countries. A major advantage of using the OWID dataset is its high level of completeness for the Nordic countries during the period under study (March 2020–December 2023). Key indicators such as mortality rates, case counts, and policy indices are consistently available for Finland, Norway, and Sweden over most of the study period. No imputation or interpolation was performed. Years or periods with unavailable data were retained as missing and excluded from summaries where applicable (e.g., Sweden 2024 in Table 2).

Table 2.

Yearly average of excess mortality (P-scores, %).

	FIN	NOR	SWE
2020	3.089	− 0.341	7.444
2021	7.588	3.518	1.320
2022	17.562	12.112	3.931
2023	14.522	7.890	4.212
2024	4.155	4.388	NA

The OWID project is managed by the Global Change Data Lab, a nonprofit organization affiliated with the University of Oxford. It was founded by social historian and development economist Max Roser and operates under the leadership of Hetan Shah¹¹. The analysis centers on key indicators such as excess mortality, new COVID-19 cases, and the government stringency index. These variables were selected based on relevance, reliability, and cross-national availability. Other indicators were excluded to streamline the analysis and improve the interpretability of the MDS results. A full list of considered variables is provided in Appendix List of considered variables.

Study design and analytical workflow

This study integrates descriptive statistics, excess mortality analysis, and Multi-Dimensional Scaling (MDS) to explore national differences in COVID-19 mortality and their relationship to welfare state structures in Finland, Norway, and Sweden. The analysis consists of the following steps:

1. Data Acquisition: COVID-19 mortality and health system indicators were obtained from¹⁷ and¹¹.

2. Data Preprocessing: Weekly incidence-based metrics were smoothed using 7-day moving averages. No interpolation, extrapolation, or imputation of mortality data was performed. Normalization was performed as follows. All per-capita indicators were converted to per-million units for descriptive and comparative analyses. Excess mortality P-scores were calculated as

   

   For subsequent multivariate analyses, including multidimensional scaling, all variables were standardized using z-score normalization (mean , standard deviation ).

3. Descriptive Comparative Analysis: National time series of total deaths, new deaths per million, and excess mortality were visualized and compared. Statistical summaries, including means, standard deviations, and peak values, were computed for each country.

4. Dimensionality Reduction and Visualization: Multidimensional scaling (MDS) was applied as a descriptive visualization tool to key time-varying COVID-19 metrics. Pairwise dissimilarities were computed using Euclidean distance on z-score standardized variables, and a two-dimensional configuration was retained for visualization. MDS stress values were reported to assess embedding quality.

5. Policy and Institutional Interpretation: Observed mortality patterns were discussed in relation to differences in public health governance and institutional context.

Figure 1 shows the overall analytical workflow of the study. The specific steps used in the MDS component are detailed separately in Fig. 2.

Figure 1.

Overview of the analytical workflow: from data collection to institutional interpretation.

Figure 2.

Core workflow of the multi-dimensional scaling (MDS) analysis of COVID-19 mortality trends. Robustness and uncertainty diagnostics arereported separately.

Multi-dimensional scaling

Multi-Dimensional Scaling (MDS) is a technique used for visualization that translates the distances or dissimilarities between objects into a lower-dimensional space to identify clusters. MDS is valuable across various domains, including marketing, ecology, molecular biology, and social networks, as it simplifies complex datasets and supports effective decision-making^18,19.

MDS highlights distances or dissimilarities between objects, whether they are tangible or conceptual²⁰. In MDS visualizations, objects that are more alike are placed closer together, while those that are less alike are spaced farther apart. Additionally, MDS serves as a dimensionality reduction technique, simplifying complex, high-dimensional data while retaining the relationships between data points²¹. Initially developed in psychometrics, MDS assigns numerical values to abstract concepts and maintains relative distances when reducing data from higher to lower dimensions, enabling visualization in two, three, or more dimensions²².

MDS is well suited for capturing dissimilarities in the temporal trajectories of mortality data, enabling a visual and non-parametric comparison of pandemic patterns across countries. Its effectiveness lies in revealing structural differences without relying on strong distributional assumptions.

Figure 2 outlines the core methodological workflow and analytical steps of the MDS analysis, while Appendix Mathematical setting describes the specific techniques used within the MDS framework²³. Robustness and uncertainty assessments are reported separately in Appendix Uncertainty and robustness analysis.

In this study, MDS is applied to time-varying COVID-19 mortality trajectories to enable a non-parametric comparison of pandemic dynamics across countries. The analysis comprises 46 aligned time points per country (Finland, Norway, and Sweden), yielding a total of 138 country–time observations. Each observation consists of 12 time-varying COVID-19 indicators, as listed in Table 3.

Table 3.

Considered variables.

Category	Variables	Variables description
Cases		Total cases
		new cases
		new cases (smoothed)
Deaths		total deaths
		new deaths
		new deaths (smoothed)
Per million metrics		total cases per million
		new cases per million
		new cases (smoothed) per million
		total deaths per million
		new deaths per million
		new deaths (smoothed) per million

Prior to dissimilarity computation, all variables were standardized using z-score normalization (mean , standard deviation ) to ensure comparability across metrics with different scales. Pairwise dissimilarities were computed using Euclidean distance in the standardized feature space, resulting in a dissimilarity matrix. The MDS procedure does not impose temporal smoothing, dynamic modeling, or probabilistic structure; temporal ordering is used solely for visualization of country trajectories.

Results

Health and demographic profile

Lupu et al.²⁴ identified that during the first wave of the pandemic, comorbidities, population age, and population density were crucial factors. In the relaxation period, government effectiveness and power distance emerged as significant, while in the second wave, education and population density were key. Based on these findings, we have selected the features listed in Table 1 for analysis. This table provides a clear comparison of key health indicators in Finland, Norway, and Sweden, providing information on various aspects of public health. Finland has the highest cardiovascular death rate (153.51 per 100,000) among the three, while Norway has the lowest (114.32), suggesting variations in heart disease prevention and treatment. In terms of diabetes prevalence, Finland also ranks highest (5.76%), with Sweden showing the lowest rate (4.79%), indicating possible differences in lifestyle and health interventions.

Table 1.

Health features and their measures for different countries.

Feature	Finland	Norway	Sweden
Cardiovascular Death Rate (per 100,000)	153.51	114.32	133.98
Diabetes Prevalence (%)	5.76	5.31	4.79
Female Smokers (%)	18.3	19.6	18.8
Male Smokers (%)	22.6	20.7	18.9
Median Age (years)	42.8	39.7	41
Aged 65 and Older (%)	21.8	17.3	20.3
Aged 70 and Older (%)	13.3	11.1	12.8
Population Density (per km)	18.14	14.46	24.72
Hospital Beds (per 100,000)	3.28	3.60	2.22

Smoking rates between sexes show some variations. Norway has the highest percentage of female smokers (19.6%), while Finland leads in male smoking rates (22.6%). Sweden’s smoking rates are lower in both categories, which may reflect the success of its anti-smoking campaigns. When it comes to population density, Sweden stands out with 24.72 people per km, while Norway has the lowest density at 14.46 people per km. This could influence the respective approaches of the countries to health care and public health policies, especially during a pandemic. The median age is relatively similar, with Finland having the highest (42.8 years), suggesting an older population compared to Norway (39.7 years) and Sweden (41 years). In terms of elderly populations, Finland also has the highest percentage of people aged 65 and older (21.23%), as well as those aged 70 and older (13.26%). Finally, the number of hospital beds per thousand people reveals that Norway has the most (3.6 beds), while Sweden has the fewest (2.22 beds), which may have implications for healthcare capacity, particularly during health crises like the COVID-19 pandemic.

COVID-19 response and mortality trends

Figure 3 presents the total deaths per million inhabitants for Finland, Norway and Sweden from early 2020 to December 2023. Initially, the data for Norway and Sweden in February 2020 contain gaps, and Norway also has missing values in the early dates. During early 2020, Sweden had significantly higher death rates compared to Finland and Norway, indicating different pandemic response strategies. Throughout 2020 and 2021, all three countries saw increasing death rates, with Sweden consistently reporting higher figures than Finland and Norway. The emergence of the Omicron variant in late 2021 triggered a surge in infections, leading to a further rise in deaths. By December 2022, Sweden’s death rate was 2019.73 deaths per million, while Finland’s was 1482.74 and Norway’s was 820.65. In 2023, death rates continued to increase in all three countries. Sweden’s rate rose substantially more than Finland’s and Norway’s, reaching 2452.86 deaths per million by December 2023, compared to 2023.71 for Finland and 1034 for Norway. This data highlights significant differences in mortality rates among the countries, particularly during the peak pandemic periods, suggesting varying public health responses and the ongoing challenges in Sweden’s management of the pandemic.

Figure 3.

Total deaths per million for Finland, Norway and Sweden (from 2020-04-05 to 2023-12-03). Source¹⁷ and authors’elaboration.

Excess mortality is measured using the P-score, defined as the percentage difference between observed deaths and expected deaths based on average mortality during the period 2015–2019. A P-score of zero indicates mortality equal to the historical baseline, positive values indicate excess mortality, and negative values indicate fewer deaths than expected. A P-score of 100 corresponds to mortality that is twice the expected baseline level.

Table 2 reports yearly average P-scores for Finland, Norway, and Sweden from 2020 to 2024, allowing for a comparative assessment of excess mortality patterns across the three Nordic countries.

Finland exhibits a moderate but clearly defined increase in excess mortality over the study period, with P-scores rising from 3.1% in 2020 to a peak of 17.6% in 2022, followed by a decline to 14.5% in 2023 and further to 4.2% in 2024. This pattern suggests a delayed but clearly observable mortality impact relative to the early pandemic phase, with elevated mortality persisting beyond 2021.

Norway shows comparatively lower excess mortality throughout the period. After a slightly negative P-score in 2020 (), excess mortality increases to 3.5% in 2021 and peaks at 12.1% in 2022, followed by a decline to 7.9% in 2023 and 4.4% in 2024. Overall, Norway’s excess mortality profile is characterized by smaller fluctuations and lower peak values than Finland.

Sweden experienced its highest excess mortality in the initial pandemic year, with a P-score of 7.4% in 2020. In subsequent years, excess mortality declined markedly and remained relatively stable, fluctuating between approximately 1.3% and 4.2% from 2021 to 2023. This pattern indicates that the largest mortality impact in Sweden occurred early in the pandemic, followed by comparatively limited excess mortality in later years. Data for Sweden in 2024 are not available, preventing a complete assessment for the final year.

The results indicate moderate excess mortality across the three countries, with distinct temporal profiles rather than extreme or persistent deviations from historical norms. Finland shows the largest mid-period increase, Sweden displays an early peak followed by stabilization, and Norway generally exhibits lower excess mortality throughout the study period. These findings are consistent with broader European evidence indicating elevated mortality among adults aged 45 years and older during the 2023–2024 winter season, coinciding with the co-circulation of COVID-19, influenza, and respiratory syncytial virus (RSV)²⁵.

COVID-19 impact across countries: a 2D MDS analysis of key metrics

This section analyzes the progression and severity of COVID-19 across Finland, Norway, and Sweden using three variants of Multi-Dimensional Scaling (MDS): classical (cMDS), nonmetric (nMDS), and robust (rMDS). MDS allows for visual comparison of pandemic trajectories based on key metrics such as smoothed new deaths and per-million indicators. The choice among these techniques depends on the dataset’s characteristics and analytical goals: cMDS is preferable for clean datasets that emphasize variance explanation; rMDS is more robust to outliers and data irregularities; and nMDS is best suited for preserving ordinal relationships when exact distances are less reliable. Together, these methods provide a comprehensive toolkit for exploring and interpreting multidimensional data.

In this study, we apply the said techniques to analyze COVID-19 mortality trends across Finland (FIN), Norway (NOR), and Sweden (SWE). The main analysis focuses on smoothed new death rates and per-million metrics. A complete list of the considered variables is provided in Appendix List of considered variables, Table 3. By reducing the dataset to two dimensions (Y1 and Y2), MDS enables a visual comparison of the countries’ relative positions with respect to key pandemic indicators. The resulting spatial configurations illustrate differences in the severity, timing, and progression of the pandemic across the three countries.

Figure 4 displays the results of Classic MDS, while Fig. 5 presents the results of Nonmetric MDS, and Fig. 6 illustrates the results of Robust MDS. The figures capture the primary variance across two dimensions. The x-axis represents the cumulative impact of the pandemic, encompassing factors such as total case numbers and deaths per capita. The y-axis captures secondary variance, potentially reflecting differences in the timing of outbreaks, healthcare capacity, or government responses.

Figure 4.

Classic metric multi-dimensional scaling (cMDS) analysis (with Euclidean distances) of COVID-19 metrics (e.g., total cases, new cases, total deaths, etc.) for Finland (FIN), Norway (NOR), and Sweden (SWE). The spread and positioning of the points suggest varying pandemic trajectories and policy responses, with Finland and Norway clustering closely due to similar containment strategies and lower case counts, while Sweden appears more isolated, reflecting its less restrictive approach and higher cumulative impact. March 2020–December 2023. Source¹⁷ and authors’ elaboration.

Figure 5.

Nonmetric multi-dimensional scaling (nMDS) analysis (with Euclidean distances) of COVID-19 metrics (e.g., total cases, new cases, total deaths, etc.) for Finland (FIN), Norway (NOR), and Sweden (SWE). The spread and positioning of the points suggest varying degrees of severity or different stages of the pandemic in each country. March 2020–December 2023. Source¹⁷ and authors’ elaboration.

Figure 6.

Robust (Nonmetric) MDS (rMDS) preserves rank order of inter-point distances, offering greater robustness to outliers, non-linearities, and irregular data compared to classical MDS. The spatial configuration reflects temporal and epidemiological similarity across countries, with variance spread more evenly across dimensions, avoiding dominance by a small subset. This makes rMDS particularly suited for analyzing COVID-19 metrics with uneven reporting or data anomalies. March 2020 - December 2023. Source¹⁷ and authors’ elaboration.

The relative positioning of the countries in the MDS space highlights differences in multivariate COVID-19 mortality patterns over time. Finland and Norway appear closer to each other in the embedded space, indicating greater similarity in the selected indicators, whereas Sweden occupies a more distant position, reflecting a distinct multivariate trajectory. These differences are descriptive and summarize patterns present in the underlying data without implying causal relationships. These spatial differences align with each country’s early pandemic response: Sweden avoided lockdowns and mask mandates, while Finland and Norway implemented stricter containment measures. Such policy contrasts likely contributed to the distinct trajectories observed in the MDS plots.

The two-dimensional MDS configuration achieved a Kruskal stress value of 0.0756, indicating a good-to-fair representation of the original high-dimensional dissimilarities. Shepard plot analysis showed strong monotonic agreement between original dissimilarities and embedded distances (Pearson ). Uncertainty analysis confirmed stability ( in leave-one-out validation), with no influential outliers detected (Appendix Uncertainty and robustness analysis, Table 7).

Table 7.

Summary of uncertainty quantification results.

Validation Method	n	Mean Stress	SD	95% CI
Original MDS	1	0.0756	–	–
Random starts	20	0.0791	0.0057	–
Subsampling (80%)	30	0.0741	0.0048	[0.066, 0.086]
Leave-one-out	30	0.0756	0.0002	[0.075, 0.076]
Distance metrics	3	0.0849	0.0161	–

All stress values < 0.10 indicate good to excellent fit.

Overall, the MDS plots demonstrate that countries with similar pandemic responses and healthcare dynamics cluster together, while those adopting different approaches, like Sweden, stand apart.

The comparison between cMDS, rMDS, and nMDS highlights important differences in how each method captures the structure of the dataset, particularly in the context of COVID-19 data. While cMDS and rMDS often produce similar group separations, they differ in how they handle variance and outliers, each offering distinct and complementary insights into the data.

The cMDS analysis, based on Euclidean distances, shows a wide dynamic range in its eigenvalues, ranging from approximately to . This broad spectrum indicates that cMDS captures extreme variations in the data, potentially amplifying the influence of outliers. The eigenvalue distribution in cMDS exhibits a distinct “elbow” pattern after the first few dimensions, suggesting that two to four dimensions account for most of the meaningful variance. This sharp decline highlights the dominance of a few dimensions in explaining the variance structure, making cMDS highly effective for visualizing clear group separations in datasets with minimal noise. However, its sensitivity to outliers can distort underlying relationships, as evidenced by sudden spikes in positive eigenvalues corresponding to extreme data points.

In contrast, rMDS, which also uses Euclidean distances but emphasizes the rank order of distances rather than their absolute magnitudes, presents a different pattern. The rMDS eigenvalues are skewed toward extreme negative values, extending to nearly , indicating reduced sensitivity to outliers by penalizing dimensions with lower rank consistency. The absence of a pronounced “elbo” in the rMDS eigenvalue plot suggests a smoother transition between dimensions, with variance distributed more evenly across them. This reflects a more robust scaling approach, as no single dimension dominates the variance explanation. Additionally, rMDS shows more consistent eigenvalue magnitudes with fewer abrupt changes, emphasizing the robustness of inter-point relationships. The large negative tail in the rMDS results underscores its focus on preserving the ordinal structure of the data, making it better suited for datasets with non-linear relationships, missing values, or anomalies, such as those often found in COVID-19 reporting.

nMDS, on the other hand, focuses on preserving the rank order of dissimilarities rather than their exact values. This approach makes nMDS particularly robust to noise and outliers, as it prioritizes the relative distances between points rather than their absolute magnitudes. While nMDS may not capture the extreme variations as prominently as cMDS, it provides a more stable representation of the data’s underlying structure, especially in the presence of anomalies. The nMDS results often align closely with rMDS in terms of group separations, but with a stronger emphasis on ordinal relationships, making it ideal for datasets where the exact distances are less reliable.

cMDS is well-suited for clean datasets with strong separation; rMDS offers greater resilience to outliers; and nMDS is effective for preserving relative distances in noisy or nonlinear data. Applying all three methods enhances the reliability and interpretability of complex pandemic data.

A full technical comparison–including eigenvalue distributions and diagnostic benchmarks across distance norms–is available in Appendix Supplementary MDS method comparison, supporting the robustness of the patterns discussed here.

MDS visualizations clearly differentiate Sweden’s pandemic trajectory from those of Finland and Norway, reflecting divergent public health strategies. These empirical patterns form the basis for the institutional analysis presented in the following section.

In summary, cMDS is well-suited for clean datasets and strong separation, rMDS is preferable when robustness to outliers is needed, and nMDS is ideal for preserving relative positions in noisy or nonlinear data. These differences confirm the value of applying multiple MDS methods when analyzing complex pandemic data.

Analysis of multidimensional scaling and COVID-19 data for Finland, Norway, and Sweden

Clustering

The scatter plot of the multidimensional scaling (MDS) coordinates provides insights into the clustering of COVID-19 data for Finland, Norway, and Sweden. A key observation is whether Sweden’s COVID-19 data forms a distinct cluster or overlaps with the points of Finland and Norway. If Sweden’s data points are close to those of Finland or Norway, it could suggest that the trends in COVID-19 cases, deaths, and other metrics in Sweden were similar to those in the other two countries. Conversely, if Sweden’s points form a separate cluster, this would indicate that the pandemic trajectory in Sweden differed significantly from that of Finland and Norway.

Relative positions

In analyzing the relative positions of the points, we observe whether Sweden’s data points are consistently higher or lower in either dimension compared to those of Finland and Norway. Higher or lower values in the MDS plot can reflect differences in the magnitude of metrics, such as total cases, deaths, or testing rates. If Sweden’s points are higher in one or both dimensions, this may indicate more severe outcomes or higher numbers of cases compared to Finland and Norway. Conversely, lower points could indicate that Sweden experienced relatively better outcomes or lower case numbers.

Temporal trends

Given that the coordinates are based on time-series data, it is possible to track how the COVID-19 situation in each country evolved over time. By analyzing the movement of the points in the MDS plot, we can observe how the pandemic progressed in each country. For instance, if a country’s points move consistently in one direction, this could indicate a steady trend, such as an increase or decrease in cases over time. Comparing the movement of the points for each country could reveal differences in how Finland, Norway, and Sweden responded to and managed the pandemic.

Country-specific observations

Sweden The MDS plot reveals an upward trend for Sweden, indicating a consistent increase in the MDS values over time. This trend suggests that Sweden’s COVID-19 situation worsened as the pandemic progressed, with metrics such as total cases or deaths increasing over time. Additionally, there is a significant outlier in the data at coordinates (1302179.431, 215380.1196). This outlier indicates a point in time where Sweden experienced an unusual event, such as a major surge in cases, changes in testing or reporting policies, or other anomalies that led to a sharp deviation from the general trend.

Finland Finland’s MDS points show a downward trend, reflecting a steady decrease in the MDS values over time. This suggests that Finland experienced an overall improvement in its COVID-19 situation, with a consistent decline in cases, deaths, or other metrics. Notably, Finland’s points show a regular pattern with no significant outliers or deviations, indicating a relatively stable progression in managing the pandemic. This regularity may be attributed to effective public health measures or a more controlled outbreak compared to Sweden and Norway.

Norway Like Finland, Norway also shows a downward trend in its MDS points, though the trend is less consistent. While Norway experienced a general improvement in its COVID-19 situation, there are two major outliers at coordinates (-128353.6516, -116374.6416) and (291850.8737, -98402.72516). These outliers suggest periods where Norway’s COVID-19 situation deviated significantly from the general trend, possibly due to spikes in cases or anomalies in data reporting. These deviations may reflect moments of heightened transmission or other factors, such as policy changes or variant outbreaks, that caused sharp changes in the data.

Overall Analysis The upward trend and large outlier observed for Sweden could indicate rapid increases in cases, deaths, or other metrics. This may reflect a combination of factors such as relaxed public health measures, high transmission rates, or delays in implementing interventions. In contrast, Finland’s regular downward trend suggests a more controlled progression, possibly due to more effective management of the pandemic or consistently lower transmission rates. Norway’s downward trend, with two sharp deviations, points to periods of significant change, such as spikes in cases or changes in public health responses.

Comparison of COVID-19 data across countries

Sweden: new cases per million

The time-series data on new cases per million in Sweden shows significant fluctuations. Early in the pandemic, the values were relatively low, starting from 1.26 cases per million. However, later in the timeline, there are sharp increases, with peaks of 1705.2, 3640.12, 4022.62, and 5304.42 cases per million. The most significant outlier is a massive spike of 18235.79 cases per million, which aligns with the large outlier in the MDS coordinates for Sweden. This suggests an unprecedented surge in cases, possibly due to the emergence of new variants, relaxed restrictions, or other contributing factors.

After each spike, the new cases per million drop significantly, indicating periods of recovery, likely driven by lockdowns, public health measures, or vaccination efforts. The data shows that while Sweden experienced waves of increased cases, there were also periods where the situation improved.

Norway: new cases per million

Norway’s new cases per million follow a similar pattern of fluctuations, though the country experienced periods of low case numbers early in the pandemic. Initial values remained below 250 cases per million, followed by a surge to 561.89 cases per million. Multiple spikes occurred, with peaks at 1866.2, 4706.91, and 4613.48 cases per million. However, the most dramatic outlier is a spike to 23755.28 cases per million, far exceeding previous values. This extreme outlier corresponds to one of the deviations in Norway’s MDS plot.

Following the major spike, Norway saw a rapid decline in cases, with values dropping to 12536.62 and later stabilizing at lower levels. This pattern reflects periods of both rapid transmission and subsequent recovery phases, influenced by public health interventions and the natural ebb and flow of pandemic waves.

Finland: new deaths per million

Finland’s data on new deaths per million reveals an initial phase with low mortality, followed by gradual increases. Early in the pandemic, deaths per million remained close to zero, but a significant surge occurred later, with peaks of 23.4 and 37.41 deaths per million. Finland’s largest spike occurred towards the end of the timeline, reaching 36.04 deaths per million. Unlike the extreme outliers seen in the new cases data for Sweden and Norway, Finland’s mortality rate remained moderate overall.

After the largest spike, Finland experienced fluctuations in new deaths per million, with values ranging from 13.2 to 29.25 before declining to 2.68 in the final data points. This decline suggests improved control of the pandemic, potentially due to vaccination campaigns or other interventions.

Discussion

The COVID-19 pandemic has highlighted the critical role of primary health care (PHC) systems in ensuring universal health coverage and delivering essential health interventions. In many low- and middle-income countries, PHC faces significant challenges, including insufficient funding, inequitable access, and high out-of-pocket costs, which perpetuate poor-quality care and a reliance on specialist services instead of accessible, community-based care. This situation further entrenches health disparities and limits the effectiveness of health systems in these regions²⁶.

In Europe, an independent commission found that many health systems struggled during the pandemic due to chronic underfunding and a lack of investment in the health workforce, problems that were exacerbated by the 2008 financial crisis²⁷. The World Health Organization (WHO) has drawn attention to the wide disparities in health outcomes across different social groups within European countries, attributing these inequalities to a combination of economic, social, environmental, and institutional factors^28,29. For example, reports such as the Marmot Review and the European Commission’s Report on Health Inequalities demonstrate that while overall health metrics, like life expectancy and infant mortality, have improved, persistent disparities remain between social groups and regions^30,31.

Health systems in Finland, Norway, and Sweden

The health systems in Finland, Norway, and Sweden have faced unique challenges and exhibited distinct strengths and weaknesses in responding to the COVID-19 pandemic. These differences are shaped by each country’s health infrastructure, policies, and historical evolution, influencing their effectiveness in managing the pandemic.

Espinosa-Gonzalez et al.³² identify several clusters of European health systems based on their approach to PHC. Cluster 2 includes countries like Greece, Ireland, Israel, Malta, Sweden, and Ukraine, characterized by centralized or partially centralized public financing and regulation of PHC, often with significant private sector involvement. In these countries, high private expenditure (over 30% of total health expenditure) or co-payment systems prevail, and gatekeeping mechanisms to access secondary care are often lacking. For instance, in some Swedish counties, patients face co-payments as high as €33³². Cluster 3, comprising Finland, Norway, Spain, and the United Kingdom, is marked by centralized public financing but devolved regulation of PHC. These countries also have lower levels of private sector involvement and implement gatekeeping systems that regulate access to secondary care, ensuring that patients first consult primary care providers before moving to specialized services³². In response to the liberalizing reforms of the 1980s, radical historians viewed these shifts as a revolutionary transformation that dismantled the social democratic welfare state³³.

In 2016 the Finnish government proposed a health system reform allowing patients to choose between public, private, and third-sector providers, all licensed by regional authorities. Despite the Confederation of Finnish Industries supporting the change, concerns arise about its efficiency. Critics argue that instead of overhauling an already efficient system, efforts should focus on improving its weaknesses, as Finland provides excellent healthcare at a relatively low cost compared to other Nordic and European nations.

Sweden’s universal, tax-financed healthcare system, designed to provide comprehensive services with minimal out-of-pocket costs, was initially established post-war by the social democratic party to ensure egalitarian access based on medical needs rather than financial means. However, market-oriented reforms in the 1990s and 2000s, which increased the role of private, for-profit providers, have seemingly weakened these egalitarian principles, contributing to the rise of voluntary private health insurance³⁴.

MDS patterns and institutional differences

The spatial divergence observed in the MDS plots–particularly Sweden’s distinct trajectory compared to Finland and Norway–can be understood in light of long-term institutional transformations that reflect deeper differences in public health governance and welfare state structure.

Recent scholarship shows that the Nordic welfare states have undergone varying degrees of marketization through quasi- and pseudo-market reforms, especially in eldercare, where legal and structural changes have fostered the expansion of for-profit provision^35–38. Sweden’s more extensive adoption of market-oriented reforms, alongside its decentralized governance model, likely contributed to fragmented coordination and inconsistent service delivery during the pandemic. This is reflected in Sweden’s early and steep mortality peak in the MDS results. In contrast, Finland and Norway retained stronger central oversight and integrated public health infrastructures, which may have enabled more coherent responses.

These institutional differences align with Blomqvist³⁴ and Diderichsen et al.³⁹, who document the erosion of egalitarian principles in Sweden’s welfare state. Finland and Norway’s centralized regulatory structures may have preserved service coherence. As Kuhlmann et al.⁴⁰ and Vabo et al.⁴¹ argue, hierarchical governance models facilitated faster mobilization and resource deployment during the crisis. The MDS configurations thus capture not only mortality divergence but also the effects of these institutional models.

Marketization and pandemic vulnerabilities

Recent research highlights that long-term care systems across the Nordic countries have undergone varying degrees of informalization and privatization, with Sweden showing particularly rapid growth in for-profit provision–challenging universalist principles and deepening inequalities^37,42. These structural divergences, as visualized in the MDS plots, reflect underlying institutional reforms that have shifted the balance of care responsibilities and shaped national capacities to respond to the pandemic.

These interpretations are supported by qualitative assessments of Sweden’s long-term care (LTC) sector. Szebehely⁴³ highlights chronic underfunding, staff shortages, reliance on hourly workers, and poor compliance with hygiene protocols in care home were structurally underprepared for the pandemic due to chronic underfunding, high proportions of untrained or hourly staff, poor hygiene compliance, and delays in PPE distribution and intervention. Although authorities emphasized protecting older adults, care home residents were initially overlooked. These systemic weaknesses reflect challenges tied to New Public Management-style reforms and align with the mortality patterns observed in the MDS plots.

Meagher and Szebehely⁴⁴ trace these issues to Sweden’s broader transition from a universalistic, publicly managed LTC model to one shaped by municipal autonomy and for-profit provision. While disability services expanded, eldercare coverage declined, increasing reliance on informal family care. Municipalization–enabled by local taxation–produced uneven regional outcomes. Legislative changes, such as the Municipal Act (1991) and Disability Act (1993), introduced “layered” institutional shifts that fragmented service provision. These structural reforms help explain the vulnerabilities exposed by the pandemic and Sweden’s distinct MDS trajectory.

Nordic marketization and welfare sustainability

Broader socioeconomic shifts have also challenged the egalitarian ethos of Nordic welfare states. Kvist et al.⁴⁵ report a marked rise in income inequality since the 1980s, driven by affluence among top income groups, labor market changes, and reduced redistribution. This erosion of equality exacerbates vulnerabilities during crises such as COVID-19, where unequal access to care, income security, and information amplify the effects of institutional fragmentation. These structural changes underscore the significance of our MDS findings: institutional design and social stratification intersect in shaping pandemic outcomes.

The move toward marketization in both quasi- and pseudo-market forms reflects a broader transformation in Nordic welfare regimes. Reforms such as Sweden’s “Friskolereformen” (1990s) and Norway’s “Helseforetaksreformen” (2001) exemplify this shift. Sweden’s reform path emphasized individual choice; Norway’s emphasized economic efficiency. Despite different motivations, both introduced a commodified logic into public service delivery, raising concerns about the long-term sustainability of the Nordic model⁴⁶.

As Lethbridge⁴⁷ notes, privatization has not delivered clear efficiency gains. Public provision shows better worker-client ratios, lower sickness and turnover rates. Austerity measures compound service exclusions, while the profit motive often leads to degraded labor conditions–undermining service quality. These findings echo broader concerns that extensive marketization can erode equity, cohesion, and resilience in public welfare systems.

Management, governance and communication

For Juranek and Zoutman⁴⁸, a key factor in Sweden’s higher death rates is the effectiveness of social distancing measures. The study compares Denmark and Norway, which implemented strict measures, with Sweden, which took a more lenient approach. By using Sweden as a counterfactual, it estimates that if Denmark and Norway had adopted Sweden’s lenient measures, hospitalizations would have peaked 15-20 days later, with peak hospitalizations increasing by 133% in Denmark and 231% in Norway, and ICU patients rising by 107% in Denmark and 140% in Norway. This analysis highlights the significant impact of social distancing policies on pandemic outcomes.

Aguilar-Guerra et al.⁴⁹ highlight that Denmark and Norway’s centralized ministerial governance system ensures agencies remain dependent on ministries for policy implementation, while Sweden’s decentralized system grants agencies independence from ministries. This difference led to three key findings: less government influence did not always yield better economic outcomes, reduced government interference was associated with higher morbidity and mortality rates, and certain public policies can have unexpected negative effects on the economy.

Bjorkdahl et al.⁵⁰ maintain that Sweden distinguished itself from Denmark and Norway by delegating most pandemic communication to its Public Health Agency rather than political leaders, resulting in a more technical, expert-driven rhetoric. This approach reflects Sweden’s tradition of separating administrative and political roles, differing from practices in Denmark and Norway. Despite this, all Scandinavian countries effectively aligned their rhetoric with their specific situations and maintained public trust. However, the varying impacts of COVID-19 across these countries underscore that rhetorical success does not necessarily equate to overall pandemic response success.

Policy implications and actionable insights

The MDS analysis and mortality patterns observed across Finland, Norway, and Sweden provide several actionable insights for health policy makers seeking to strengthen pandemic preparedness and welfare state resilience. First, the spatial divergence of Sweden in the MDS plots, coupled with its early mortality peak, underscores the importance of maintaining strong central coordination mechanisms in public health governance. Policy makers should consider establishing robust inter-agency coordination protocols that can be rapidly activated during health emergencies, avoiding the fragmentation that may arise from excessive decentralization. This includes developing standardized data sharing systems, unified communication strategies, and clear chains of command that transcend administrative boundaries between national, regional, and municipal levels.

Second, the differential mortality outcomes highlight the vulnerability of market-oriented reforms in health and social care during systemic crises. The institutional analysis suggests that countries with more extensive privatization and market delegation–particularly in eldercare–may face greater challenges in coordinating emergency responses and ensuring equitable access to protective measures. Policy makers should therefore conduct comprehensive reviews of their care systems to identify potential coordination gaps, strengthen oversight mechanisms for private providers, and establish contingency protocols that ensure consistent standards across public and private sectors. This may involve developing regulatory frameworks that require private providers to maintain emergency preparedness plans, participate in coordinated response efforts, and adhere to unified quality standards during health crises.

Third, the persistent excess mortality documented through 2024 across all three countries indicates that pandemic impacts extend well beyond the acute phase, requiring sustained policy attention and resource allocation. Health systems should invest in long-term monitoring capabilities to track excess mortality patterns, develop targeted interventions for vulnerable populations, and build resilient primary care infrastructures that can maintain essential services during disruptions. This includes strengthening community-based care networks, enhancing digital health capabilities, and ensuring adequate staffing and resource reserves to handle both routine care and emergency surges. Additionally, policy makers should consider how welfare state structures can be adapted to better support population health resilience while preserving the egalitarian principles that have historically defined Nordic social models.

Conclusions

This study combined time-series analysis and multidimensional scaling (MDS) to examine COVID-19 mortality trajectories in Finland, Norway, and Sweden. Across multiple indicators, the three countries exhibited clearly distinct patterns in the timing, magnitude, and progression of mortality during the pandemic. Sweden experienced an earlier and sharper peak, Finland showed a more gradual trajectory, and Norway displayed intermittent surges, highlighting meaningful heterogeneity within an otherwise institutionally similar region.

The MDS analysis provided a compact, multivariate summary of these differences and demonstrated that a two-dimensional representation preserves the main structure of the data with acceptable fidelity. While MDS is inherently a descriptive technique, the observed configurations offer a coherent visual account of cross-country differences that is robust to alternative specifications and perturbations, as documented in the uncertainty analyses.

Within this three-country comparison, the relative positioning of Sweden, Finland, and Norway is consistent with well-documented differences in public health governance and pandemic response strategies during the early phases of COVID-19. In particular, Sweden’s distinct trajectory contrasts with the more similar patterns observed for Finland and Norway. These associations do not constitute causal evidence but provide empirically grounded context for interpreting how governance arrangements and coordination practices may be linked to observed outcomes.

More broadly, the findings contribute to ongoing debates on the resilience of Nordic welfare states and the implications of decentralization and market-oriented reforms for crisis management. Rather than demonstrating direct effects, the analysis illustrates how differences in institutional organization can manifest in divergent pandemic trajectories, even among countries with comparable socioeconomic profiles.

Several limitations warrant consideration. The analysis is restricted to three countries and relies on aggregated mortality data, which may mask regional and demographic heterogeneity. Differences in reporting practices and baseline health conditions may further affect comparability. In addition, the use of MDS supports exploratory pattern identification but does not enable formal statistical inference.

Future research should extend this framework to a larger set of countries and incorporate more explicit measures of institutional design and governance. Combining registry-level data with formal causal methods would allow a more systematic evaluation of how public health systems and welfare-state arrangements shape outcomes during systemic crises.

List of considered variables

Table 3 lists all variables considered during the initial stage of the analysis, including both raw and derived indicators related to cases and deaths. However, only a subset–specifically, excess mortality, new COVID-19 cases, and the government stringency index–was retained for the final analysis to ensure conceptual focus and reduce dimensional noise.

Mathematical setting

We first provide some definitions and notation that are helpful in describing the various categories of MDS. Consider a dataset with -dimensional observations. In general, the MDS procedure measures the similarity between the objects in each category of data. Let us denote by the dissimilarity between two objects and , where

All dissimilarities are collected into an matrix , called the dissimilarity matrix (sometimes given in triangular form). Let be the lower -dimensional space of the data, with , in which the distance between and is denoted by . Usually, one chooses or to explore the analysis for interesting structures. The dissimilarities , calculated from the -dimensional data, and the distances in the lower-dimensional space are related by a suitable monotonic function , i.e.,

On the other hand, if the -th object in the reduced space has coordinates , then the distances are expressed as , for a general norm . The most commonly used norms are the following:

• Euclidean norm:

  
• Mahalanobis norm:

  

  where denotes the covariance matrix of .
• Minkowski norm:

  

  For the special case of , the Minkowski norm gives the city block distance. When , the Minkowski distance gives the Euclidean distance. In the special case of , the Minkowski distance reduces to the Chebyshev norm (i.e., ).

The form of specifies the MDS model. The ratio, or MDS interval⁵¹, is often used in applications, assuming , with two free parameters. Other common choices for include higher-order polynomial, logarithmic, and exponential functions.

Metric MDS (mMDS)

In the metric MDS (mMDS), the transformation must be a continuous monotonic function, which (theoretically) minimizes the following optimization problem:

1

The above objective function is called stress. From a numerical point of view, one can solve the problem (1) through some operations research techniques, e.g., gradient or steepest descent.

Classical MDS (cMDS)

If the dissimilarity measures and the distances are taken to be Euclidean, then a closed-form solution exists to find the configuration of points in a -dimensional space. This is the case of the so-called classical MDS (cMDS) method. For this reason, the mMDS methods are always referred to as non-classical metric MDS. The function is equal to the identity function, that is, . Gower⁵² gave the name of principal coordinates analysis (PCoA) to the classic MDS technique, because it uses ideas similar to those in principal component analysis (PCA).

Nonmetric MDS (nMDS)

Nonmetric MDS (nMDS) relaxes the properties of and assumes only the rank order of the dissimilarities be preserved, so that the transformation must obey the monotonicity constraint:

for any pair of objects . Because of this, nonmetric MDS is also known as ordinal MDS. Let us introduce the disparity quantities as

where are taken equal to the Minkowski distances. In this setting, one has to solve the following problem (often named as monotonic least squares problem):

2

nMDS needs to optimize two objectives simultaneously. This is usually done iteratively, as follows.

Algorithm 1.

nMDS pseudocode.

Robust nonmetric MDS (rMDS)

As a special case of the nMDS method, we recall the robust nonmetric version, in which the empirical cumulative distribution function (eCDF) of distances is preserved. The algorithm is similar to those of Algorithm 1, but in this case, the stress function to optimize is the p-value of the Kolmogorov-Smirnov test (or analogous tests) run between the eCDFs of and ⁵³. In⁵⁴, it is shown that the nMDS method performs well when outliers are present, even in relatively large numbers, and also performs comparably to other approaches when no outliers are present.

Supplementary MDS method comparison

This appendix provides additional detail on the comparative performance of classical, nonmetric, and robust MDS methods used in Section Results.

Eigenvalue distribution in MDS analysis

The eigenvalue distribution derived from the cMDS analysis provides critical insights into the variance structure of the dataset. The plot of the logarithm of the absolute values of the eigenvalues, sorted in descending order, reveals that a small number of dimensions capture the majority of the variance. Specifically, the first few eigenvalues stand out with significantly larger magnitudes, notably with values around 32.41, 25.01, 18.68, and 13.95. This substantial gap between the leading eigenvalues and the remaining ones indicates that the first two to four dimensions contain most of the meaningful information within the data. The sharp decline after these dominant eigenvalues, typically visualized as an “elbow” in the scree-type plot (Fig. 7), suggests an optimal point at which additional dimensions contribute diminishing returns in explaining the variance. In this case, the elbow appears around , signifying that dimensions beyond this point primarily capture noise or minor variations that do not substantially enhance the data’s explanatory power.

Figure 7.

Logarithm of the eigenvalues from cMDS analysis (with Euclidean distances) of COVID-19 metrics (e.g., total cases, new cases, total deaths, etc.) for Finland (FIN), Norway (NOR), and Sweden (SWE). March 2020–December 2023. Source¹⁷ and authors’ elaboration.

The application of a logarithmic transformation compresses the scale of the eigenvalues, allowing for a clearer visualization of this pattern by mitigating the overshadowing effect of the largest values. Additionally, if eigenvalues are strictly positive (within round-off error), the reduced data provides an exact reconstruction of D (the distances matrix), with inter-point distances matching those in D within round-off error. The smallest negative eigenvalue in this analysis is sufficiently low (equal to ), which supports the relatively good quality of the reconstruction.

Overall, this eigenvalue distribution supports the effectiveness of dimensionality reduction through MDS, confirming that a two- to four-dimensional solution adequately represents the underlying structure of the dataset. The low stress value associated with this solution further validates its goodness of fit, ensuring that the reduced dimensional representation accurately reflects the relationships within the original high-dimensional space.

Table 4 summarizes two key diagnostics derived from the cMDS analysis across different distance norms: the elbow point and the minimum eigenvalue. The elbow point marks the optimal embedding dimension where the cumulative variance explained reaches a practical plateau. This is analogous to the “elbow” observed in scree plots for PCA, where the inclusion of additional dimensions provides diminishing returns in capturing variance. The position of this elbow varies depending on the choice of distance metric, reflecting how different norms emphasize different aspects of the data geometry.

Table 4.

Elbow point and minimum eigenvalue from the cMDS method for different distance norms.

Norm type	Euclidean	Mahalanobis	City block	Minkowski	Chebyshev
Elbow	9	12	3	9	2
Minimum eigenvalue	-0.0119		− 5.3163	− 0.0191	− 1.5744

The minimum eigenvalue reflects the extent to which the pairwise distances in the reduced-dimensional space faithfully reconstruct the original distance matrix. In classical MDS (cMDS), eigenvalues correspond to the principal axes of variation in the data. Negative eigenvalues indicate minor distortions introduced during embedding, as the method forces the configuration into Euclidean space even if the original distance matrix is only approximately embeddable. Ideally, the minimum eigenvalue should be close to zero, indicating high reconstruction fidelity.

Across norms, the Euclidean, Minkowski, and Mahalanobis norms all yield relatively small (near-zero) minimum eigenvalues, consistent with accurate embeddings. In contrast, the City Block and Chebyshev norms produce larger negative eigenvalues, signaling greater incompatibility with the Euclidean embedding space imposed by cMDS. These results highlight how the choice of distance norm influences both the optimal dimensionality (elbow point) and the quality of the resulting embedding (minimum eigenvalue), providing valuable guidance for selecting an appropriate configuration for downstream analysis.

Shepard plot and nMDS analysis

Figure 8 displays the Shepard plot, a key visualization in nMDS. The x-coordinates of the blue circles represent the original dissimilarity values, while the y-coordinates correspond to the Euclidean distances in the two-dimensional space. The connected red points indicate the disparity values, which are monotonic transformations of the dissimilarities. Most points lie close to the first bisector line, demonstrating that the two-dimensional scaling provides a reasonably good representation of the higher-dimensional dissimilarities. The embedded box within the figure offers a helpful zoom into a specific interval of dissimilarities, further illustrating the alignment between disparities and dissimilarities.

Figure 8.

Shepard plot. The x coordinates of the blue circles correspond to their original dissimilarity values, and the y coordinates correspond to their Euclidean distances in two-dimensional space. The connected red points indicate the disparity values, which are a monotonic transformation of the dissimilarities. The box inside the figure provides an helpful zoom for a given intervals of dissimilarities. nMDS analysis for Finland (FIN), Norway (NOR), and Sweden (SWE). March 2020–December 2023. Source¹⁷ and authors’ elaboration.

The Shepard plot compares disparities (monotonically transformed dissimilarities) and dissimilarities (original distances between data points). Disparities are derived to preserve the rank order of distances in the lower-dimensional space, while dissimilarities represent the original high-dimensional distances. The close alignment between disparities and dissimilarities indicates a strong fit, with minimal deviations for most data points. For example, at lower values (e.g., 3,451.78 vs. 3,592.23 or 2,866.69 vs. 2,973.72), disparities slightly underestimate the dissimilarities, which is expected as the monotonic regression prioritizes rank order over exact magnitudes. At higher values (e.g., 2,713,617.93 vs. 2,713,387.89 or 1,348,891.14 vs. 1,345,454.62), the differences remain small relative to the magnitude of the values, confirming the robustness of the fit across a wide range of distances.

A few instances show more noticeable deviations, where disparities exceed dissimilarities or vice versa. These minor inconsistencies may arise from local non-linearities that the monotonic regression cannot perfectly accommodate without increasing overall model stress. However, given the complexity of the dataset–particularly COVID-19 metrics with potential non-linear relationships–these discrepancies are minimal and do not significantly impact the overall representation. Moreover, Fig. 9 shows the confidence ellipse (also called error ellipse) at the 95% confidence level 55. This provides a graphical representation of the uncertainty associated with the error between the original disparities and their corresponding Euclidean distances. As expected, approximately 95% of the data points fall within the ellipse. Since the dissimilarities are known, the elongated (high-eccentricity) shape of the ellipse suggests low uncertainty in the error estimation.

Figure 9.

Shepard plot. The x coordinates of the blue circles correspond to their original dissimilarity values, and the y coordinates correspond to their Euclidean distances in two-dimensional space. The connected red points indicate the disparity values, which are a monotonic transformation of the dissimilarities. The box inside the figure provides an helpful zoom for a given intervals of dissimilarities. nMDS analysis for Finland (FIN), Norway (NOR), and Sweden (SWE). March 2020–December 2023. Source¹⁷ and authors’ elaboration.

To further validate our analysis, we conducted a permutation test to compare the original dissimilarities and the disparities (see Table 5). We performed 3,000 permutations and found no significant difference in their means, with a p-value of 95.33%. Based on this result, only 7.78% of the sample showed statistically significant differences. The effect size, measured by Hedges’ , was extremely small (on the order of ), further indicating negligible divergence between the two distributions.

Table 5.

Permutation test carried out 3000 permutations between the original disparities and their simulated distances.

	Significant differences	Proportion of significant differences	Effect size (Hedges’g)	p-value
Permutation test results	735	7.78%	9.0355	0.9533

A p-value less than a fixed precision (e.g. 0.01) indicates that the null hypothesis is not rejected.

In conclusion, the Shepard plot and the comparison between disparities and dissimilarities confirm that the NMDS solution effectively preserves the rank order of distances in the reduced-dimensional space. The minor deviations observed support the robustness of the transformation, indicating that the chosen dimensionality captures the high-dimensional structure with minimal stress and distortion. This analysis underscores the appropriateness of NMDS for datasets with complex relationships, such as COVID-19 metrics across Finland (FIN), Norway (NOR), and Sweden (SWE).

Comparison of fit quality across MDS methods

Table 6 presents a comparative evaluation of the fit quality obtained using three Multi-Dimensional Scaling (MDS) techniques – classical MDS (cMDS), nonmetric MDS (nMDS), and robust MDS (rMDS) – applied under various distance norms. For cMDS and nMDS, the values reported in Table 6 represent the conventional stress, which quantifies how well the lower-dimensional configuration preserves the original pairwise dissimilarities between observations. Lower stress values indicate a better fit, meaning that the resulting low-dimensional configuration more accurately reflects the high-dimensional structure of the data.

Table 6.

Fit quality comparison across cMDS, nMDS, and rMDS under different distance norms.

Norm type	nMDS (stress)	cMDS (Stress)	rMDS (KS statistic)
Euclidean			0.2439
Mahalanobis	0.0083	0.1000	0.4767
City block	0.0065	0.1806	0.2535
Minkowski			0.2462
Chebyshev	0.0011	0.0242	0.2268

For cMDS and nMDS, values represent conventional stress. For rMDS, values represent the average Kolmogorov-Smirnov (KS) statistic, which quantifies the divergence between distance distributions in the original and embedded spaces.

For rMDS, the values reported in the table do not represent traditional stress. Instead, they correspond to the average Kolmogorov-Smirnov (KS) statistic, which quantifies the discrepancy between the empirical distribution of pairwise distances in the high-dimensional input space and the corresponding distribution in the low-dimensional embedding. In this context, lower KS values indicate that the overall distribution of pairwise distances has been better preserved, even if individual pairwise distances are not strictly maintained. This emphasis on distributional preservation renders rMDS particularly robust to the presence of outliers, noise, and non-standard data distributions.

The results indicate that nMDS consistently achieves the lowest stress values across all considered norms. Under the Euclidean and Minkowski norms, nMDS achieves extremely low stress values ( and , respectively), indicating an almost perfect embedding. These results confirm that nonmetric MDS is highly effective in preserving rank-order relationships between observations, particularly under these norms. In contrast, cMDS exhibits moderately higher stress values across all norms, with the lowest stress observed under the Chebyshev norm (0.0242). This suggests that cMDS can effectively preserve the maximum coordinate differences between observations, though its overall performance is more sensitive to the choice of distance metric. For rMDS, the KS statistic values reflect the preservation of the overall distance distribution rather than individual pairwise distances. Among all norms, rMDS performs best under the Chebyshev norm, with the lowest KS statistic (0.2268), indicating superior preservation of the global structure. However, the Mahalanobis norm results in the highest KS statistic (0.4767), suggesting that this norm may be less suitable for rMDS when the data contains complex correlations or outliers.

These results reflect the distinct objectives and strengths of each method. Nonmetric MDS excels at preserving the rank-order structure of dissimilarities, resulting in exceptionally low stress values under most norms. Classical MDS focuses on preserving absolute pairwise distances, resulting in moderately higher stress values that are sensitive to the chosen norm. Robust MDS, by prioritizing the preservation of the overall distance distribution through the KS statistic, is better suited for datasets with outliers, irregular structures, or non-linear relationships.

Uncertainty and robustness analysis

Prior to multidimensional scaling (MDS), all variables were standardized using z-score normalization to ensure comparability across indicators with different scales:

3

where denotes the value of variable j for observation i, and and are the mean and standard deviation of variable j computed across all 138 observations. Post-standardization diagnostics confirmed numerical accuracy, with a mean close to zero () and a standard deviation close to one (0.997).

Pairwise dissimilarities were computed using Euclidean distance on the standardized variables:

4

yielding a dissimilarity matrix (mean = 3.97; SD = 2.83).

To assess the numerical stability of the MDS configuration, we conducted a series of robustness checks examining sensitivity to initialization, data composition, distance metric choice, and dimensionality. These analyses were designed to evaluate embedding stability rather than to provide statistical inference or probabilistic uncertainty estimates. All computations were performed in MATLAB R2023a using a fixed random seed (42) to ensure reproducibility.

Robustness to initialization was evaluated by repeating the MDS procedure 20 times using different random starting configurations while holding the dissimilarity matrix fixed; all runs converged to solutions with similar stress values. Sensitivity to data composition was assessed using two complementary approaches: (i) subsampling validation based on 30 repetitions of MDS fitted to random 80% subsets of the data stratified by country, and (ii) leave-one-out analyses in which individual observations were sequentially removed and the MDS solution recomputed.

To examine dependence on the choice of distance metric, additional MDS solutions were computed using alternative dissimilarity definitions, including Euclidean, city-block, Chebyshev, and Minkowski distances. Furthermore, MDS solutions were computed for dimensionalities ranging from one to five dimensions to assess the trade-off between embedding quality and interpretability.

Table 7 summarizes stress values across all robustness checks. The original two-dimensional MDS solution achieved a Kruskal stress of 0.0756, which falls within the range typically interpreted as a good-to-fair representation. Stress values obtained under random starts, subsampling, and leave-one-out procedures remained close to the original value, indicating limited sensitivity to initialization choices or moderate perturbations of the data.

Leave-one-out analyses showed minimal variation in stress (SD = 0.0002), indicating that no single observation exerted a dominant influence on the configuration. Subsampling validation yielded comparable stress values (mean = 0.074; 95% CI: [0.066, 0.086]), consistent with stable embedding quality under changes in sample composition. Across alternative distance metrics, convergent solutions remained below a stress threshold of 0.10, suggesting that the configuration is not specific to a single metric choice.

Dimensionality assessment (Fig. 10B) revealed a monotonic decrease in stress with increasing dimensionality (2D: 0.076, 3D: 0.028, 4D: 0.006, 5D: 0.002). A two-dimensional solution was retained for visualization, as higher-dimensional embeddings offer diminishing gains in stress reduction while substantially reducing interpretability of temporal trajectories.

Figure 10.

Uncertainty and robustness diagnostics for the MDS analysis. (A) Stress distributions across validation methods. (B) Scree plot showing stress as a function of dimensionality,supporting the choice of a two-dimensional representation. (C) Comparison of stress values across distance metrics. (D) Original two-dimensional MDS configuration with countriescolor-coded. (E) Overlay of MDS solutions obtained from multiple randomstartingconfigurations. (F) Stress distribution obtained from subsampling validation.

Overall, the robustness analyses indicate that the MDS configuration is numerically stable under a range of perturbations, including initialization, data composition, distance metric choice, and dimensionality. These diagnostics support the use of the two-dimensional MDS solution as a descriptive visualization of multivariate COVID-19 mortality patterns. They do not imply statistical inference or causal interpretation beyond the exploratory scope of the method.

Figure 10 presents six complementary views of the uncertainty and robustness analysis, providing visual support for the stability summarized in Table 7.

Author contributions

G.O.: Conceptualization, Data curation, Investigation, Writing- Original draft preparation, Supervision, Funding acquisition, Project administration. G.O. and M.B. Formal analysis, Methodology, Software. G.O., M.B. and V.N. Validation, Investigation, Visualization, Reviewing and Editing.

Data availability

The data used and analyzed during the current study are available from the corresponding author upon request.

Declarations

Competing interests

The authors declare no competing interests.