2.1 Prevalence and magnitude of synergistic information

Before decomposing shared information into synergistic, unique and redundant components, we first examined how often pairs of variables jointly carried any information about a third variable. With V = 17 analysed variables treated symmetrically as potential targets, this yields distinct variable-pair–target triplets per dataset (i.e., all pairs among the remaining variables for each target). Across all age, sex and multimorbidity strata, 86–97% of these two-predictor–one-target triplets showed statistically significant joint mutual information under permutation testing with false-discovery-rate (FDR) control. In the full sample, 2,005 of 2,040 tested triplets (98.3%) had significant joint mutual information (FDR thresholds 0.042–0.048; Table A in S1 Appendix). These results indicate that joint dependence between variables is therefore widespread. We then examined how much of this shared information is truly synergistic, as opposed to being explained by redundant or unique contributions.

Across the 135 synergistic source pairs (pairs that have statistically significant synergy for at least one target), synergy constituted a small but widely varying fraction of TMI (Fig 1). The distribution of synergy fractions was right-skewed, with a median of 0.090; the interquartile range was 0.047–0.174. This indicates that, for most pairs, fewer than 10% of shared information bits arose from joint effects, whereas a minority of pairs exhibited substantially larger fractions. Synergistic effects are therefore selective rather than ubiquitous, motivating methods that explicitly identify these rare but informative interactions.

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Fig 1. Synergistic information constitutes a small but heterogeneous fraction of total mutual information (TMI).

Distribution of synergy fraction (synergy strength / TMI) across 135 source pairs. The right-skewed distribution indicates that a minority of relationships exhibit disproportionately strong synergistic contributions. The vertical dotted lines show the median (long dashes) and mean (short dashes).

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To assess the stability of these estimates, we computed bootstrap confidence intervals (1,000 replicates). Across the top-10 BUST pairs, 95% CIs for mean synergy were narrow (median width 0.005 bits) and all excluded zero (Fig L in S1 Appendix). At the individual-target level, 73–93% of targets per pair had synergy CIs that excluded zero, indicating that the synergistic signal is not driven by a small subset of targets.

2.2 Simulation validation

Three simulation experiments confirmed that the QE bias-corrected pipeline recovers known PID atoms and BUST scores under conditions matched to the empirical data (Methods Section 4.6; Section S5 of S1 Appendix). In the cardinality sweep (128 conditions, 50 replicates each), estimated PID atoms matched ground truths with mean absolute bias below 0.01 bits and RMSE below 0.05 bits for K ≤ 4, with 95% confidence interval coverage between 90% and 98%. At K = 5 with N = 500 a small positive synergy bias (≈0.035 bits) appeared but diminished at larger sample sizes. Under LASA-matched asymmetric cardinality profiles, all atoms were recovered with RMSE < 0.02 bits at N = 2,636, and regimes with zero true synergy produced negligible estimated synergy (bias < 0.001 bits at ). BUST scores were also faithfully recovered: synergy strength and total information had RMSE < 0.13 and < 0.008 bits respectively, and the relative ranking of pairs by all three BUST components was preserved. Breadth exhibited a conservative downward bias for broad-synergy pairs, meaning pairs classified as “broad” in the empirical BUST map genuinely possess broad profiles.

2.3 Breadth and uniformity of synergy across targets and strata

After estimating synergy for each source pair across all targets and strata, we summarised each pair on the BUST map (Fig 2). Breadth (B) quantifies how widely a pair’s synergistic information is distributed across targets, whereas Uniformity (U) quantifies how consistently that synergy appears across age, sex, and multimorbidity strata. Both scores are median-centred, so B = 0 and U = 0 represent a “typical” pair in this dataset. Pairs with B > 0 exhibit broader-than-typical synergy profiles (spread across more targets), whereas B < 0 indicates synergy concentrated on a smaller subset of targets. Similarly, U > 0 indicates more uniform-than-typical synergy across strata, whereas U < 0 indicates subpopulation-specificity (stronger variation across strata).

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Fig 2. BUST map summarises synergistic source pairs by breadth and uniformity.

Each point represents one source pair (n = 135). The x-axis is Breadth (B), measuring how widely a pair’s synergy is distributed across targets; the y-axis is Uniformity (U), measuring how consistently that synergy appears across age, sex, and multimorbidity strata (see Methods). Both axes are median-centred, so B = 0 and U = 0 correspond to a typical pair; dashed lines at zero define four archetypes: broad–uniform (BU), narrow–uniform (NU), broad–specific (BS), and narrow–specific (NS). Point size encodes synergy strength (bits) and colour encodes TMI (bits). The two most extreme pairs are annotated in each quadrant.

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The highest composite BUST score (min–max normalised to [0,1] across pairs) was observed for perceived health relative to peers (i.e., one’s own health rating compared to the estimated health of peers) with physical activity (composite = 1.000), which combined strong synergy (0.20 bits) with positive Uniformity (U = 0.27) and slightly below-median Breadth (B = -0.17). This indicates a synergy pattern that is expressed consistently across strata but is concentrated on a subset of targets. Other top-ranked pairs included alcohol use with grip strength (BU; composite = 0.945), pain symptoms with physical activity (NU; composite = 0.915; the highest strength at 0.20 bits), sleep quality with perceived health relative to peers, depression with perceived health relative to peers, sleep with physical activity, cognition with (perceived) general health, general health with grip strength, BMI with alcohol use, and sleep with alcohol use. These interactions form an interpretable pattern in which subjective comparative health, sleep, pain, mood, activity, body composition, cognition, and physical capacity jointly encode non-additive information about multiple targets, including functional limitations.

Composite BUST scores provide a compact entry point by ranking source pairs that jointly combine (i) non-trivial synergy strength, (ii) above-typical Breadth across targets, and (iii) above-typical Uniformity across strata (Methods Section 4.4). Importantly, BUST does not reduce each pair to a single number: it retains each pair’s Breadth and stratum-specific Uniformity components, which enables targeted follow-up views once the full set of BUST scores has been computed. We therefore complement the composite ranking with an explicit analysis of low-uniformity pairs, which isolates interactions whose synergy profiles vary most across age group, sex, or multimorbidity status.

Context-dependent synergies and drivers of non-uniformity.

Uniformity in the BUST representation is computed separately for each stratification variable (sex, multimorbidity status, and age group) and then aggregated for the 2D map. To make the “specific” (low-uniformity) region of the map interpretable in practice, we ranked source pairs by their stratum-specific uniformity components and visualised the lowest-uniformity examples (Fig 3). Each row shows a source pair, and each column reports the entropy-based uniformity score for that stratification, where lower values indicate greater between-stratum variation in the pair’s synergy profile.

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Fig 3. Low-uniformity pairs reveal which stratification drives context-dependent synergy.

(a) Pairs with the lowest multimorbidity-specific uniformity component. (b) Pairs with the lowest median uniformity across sex, multimorbidity, and age. Heatmaps show the entropy-based uniformity components for each source pair across sex, multimorbidity status (present vs. absent), and age group. Lower values indicate greater between-stratum variability in the pair’s synergy profile for that stratification, whereas higher values indicate more stable behaviour across strata. Row labels report each source pair and its synergy strength S (bits).

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Fig 3a focuses on multimorbidity status (present vs. absent) by showing the ten pairs with the lowest multimorbidity uniformity component. For these pairs, the multimorbidity column is systematically lower than the corresponding sex- and age-based uniformity components, indicating that their context dependence is driven primarily by differences between the multimorbidity strata rather than by sex or age.

Fig 3b provides a complementary, global view by selecting the ten pairs with the lowest median uniformity across sex, multimorbidity, and age. In this view, low overall uniformity arises for different reasons: for some pairs the lowest component is age-based, for others it is sex-based, and for others it is multimorbidity-based. This decomposition shows that “non-uniform” synergy is not a single phenomenon: the same overall label can reflect instability along different subgroup dimensions, which guides which stratified analyses are most relevant for follow-up.

2.4 System-level organisation of synergy

We then examined how these pairwise synergies are organised at the system level. Using composite BUST scores as edge weights, we constructed a weighted synergy network in which each variable is connected to every other variable. Despite this near-complete connectivity, hierarchical clustering of the associated distance matrix (1 minus the composite BUST weight) revealed a structured organisation consisting of four groups (Fig 4a). Waist–hip ratio (WHR) and blood pressure (mean arterial pressure, MAP) formed small metabolic and cardiovascular branches, respectively. A larger subjective-health and functioning cluster grouped anxiety symptoms, perceived health relative to peers, pain symptoms, physical activity, sleep quality and functional limitations. The remaining variables formed a cognition/comorbidity cluster linking alcohol use, BMI, depression, general health, hearing difficulties, cognition, memory complaints, memory problems and grip strength. These groupings align with established geriatric syndrome domains (i.e., malnutrion; cardiovascular ageing; psychosocial functioning; dementia, sarcopenia), suggesting that synergistic information recovers clinically meaningful higher-order structure.

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Fig 4. Hierarchical, network, and ranking perspectives on synergistic structure under the BUST representation.

(a) Dendrogram from Ward hierarchical clustering on a distance matrix derived from composite BUST edge weights. Coloured branches indicate clusters identified below a threshold of 0.7× the maximum linkage distance; grey branches represent inter-cluster connections. (b) Weighted BUST network retaining only edges above the 75th percentile of the composite weight distribution; isolated nodes are omitted. Node colours follow dendrogram cluster assignments. Edge thickness is scaled nonlinearly with the composite weight. Absence of an edge or node does not imply independence, but indicates that the pair’s composite weight fell below the plotting threshold. (c) Composite rankings using synergy strength as the magnitude term (BUS, x-axis) versus total mutual information (BUT, y-axis), with Breadth (B) and Uniformity (U) held fixed. Dashed lines mark the top-10 cutoff under each composite; shaded regions identify pairs ranking in the top 10 under each definition. Point size encodes synergy strength (S, in bits); colour encodes synergy fraction (S/TMI).

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To make the multivariate organisation visually explicit, we plotted a thresholded BUST network in which nodes represent variables and edges represent composite BUST scores (Fig 4b). For clarity, we retained only edges above the 75th percentile of the composite distribution (34 of 135 possible edges) and removed nodes with no such edges. This representation reveals a compact backbone centred on sleep quality, pain symptoms, perceived health relative to peers, physical activity, depression, grip strength and alcohol use.

2.5 Relationship between synergy and conventional dependence measures

We evaluated how PID-based synergy compares with traditional dependence measures such as TMI. If strong synergy simply reflected high overall information sharing, then ranking pairs by TMI would recover the same source pairs as ranking by synergy. We therefore constructed two composite scores that summarised each pair’s breadth and cross-stratum consistency but differed in their magnitude component: one used synergy strength and the other used TMI (Methods Section 4.4).

Synergy strength and TMI were only weakly correlated across the 135 source pairs (Pearson r = 0.31, p < 0.001), and the top-10 pairs under the synergy-based composite showed zero overlap with the top-10 pairs under the TMI-based composite (Fig 4c). Several high-TMI pairs, such as combinations of memory complaints and problems, had modest synergy fractions, whereas strongly synergistic pairs—such as alcohol use–grip strength, sleep–physical activity and perceived health relative to peers–physical activity—occupied mid-range TMI values. This confirms that synergy-based rankings are not recoverable from TMI alone and underscores the added value of PID relative to correlation or TMI screening. Several TMI-composite top pairs were tied, consistent with near-redundant variables yielding similar total-information profiles.

3.1 PID reveals information architecture beyond mutual information

The complete non-overlap between the top-10 synergy-based and top-10 TMI-based composite rankings demonstrates that bivariate PID provides non-redundant information relative to conventional mutual information analysis. High TMI pairs were not simply high-synergy pairs with additional redundant content; instead, they belonged to a distinct set of relationships whose shared information was dominated by redundant and unique components. Conversely, strongly synergistic pairs occupied mid-range TMI values and were not identifiable by total information content alone.

3.2 BUST archetypes recover expected constellations and highlight new candidates

The BUST map places individual source pairs into archetypes and helps link synergistic patterns to clinical narratives (Fig 2). Several of the strongest and most clinically interpretable pairs (i.e., pairs with plausible joint-state narratives in routinely assessed domains) fall in the broad–uniform (BU) and narrow–uniform (NU) quadrants, reflecting synergy that is expressed consistently across age, sex and multimorbidity strata. Examples include sleep with physical activity, sleep with perceived health relative to peers, alcohol use with grip strength, and depression with perceived health relative to peers. These pairs combine positive Uniformity with either broad or narrower Breadth and jointly encode information about many functional outcomes. Their positions on the BUST map are consistent with evidence that poor sleep, low physical activity, depression, problematic alcohol use and perceived poor health tend to cluster within individuals and are collectively associated with declines in physical and mental health. For instance, depression and excess weight have been shown to reinforce each other bidirectionally in longitudinal cohorts [18,19], and such mood–weight constellations often coexist with disturbed sleep and low activity in older adults. Perceived health relative to peers is particularly notable: self-rated health integrates information from multiple domains and predicts retirement due to disability and mortality beyond objective diagnoses [20,21]. Its repeated appearance in high-scoring BU and NU pairs suggests a role as an integrative “health signal” that synergises with behavioural and mood-related variables.

Pairs in the broad–specific (BS) quadrant exhibit synergy that spans multiple targets but varies across strata. These include sleep with alcohol use, pain with alcohol use and sleep with hearing difficulties. Their placement is consistent with evidence that alcohol-related effects on metabolic and behavioural functioning differ substantially across intake levels and population subgroups. Heavy drinking is more strongly linked with adiposity and cardiometabolic risk, whereas light or moderate consumption shows weaker or mixed associations [22,23]. Similar variation is reflected in population studies reporting associations between alcohol intake, abdominal obesity and health-related behaviours [24]. The BS quadrant, therefore, captures pairs whose joint behaviour is broad but modulated by demographic or multimorbidity context.

Pairs in the narrow–specific (NS) quadrant exemplify concentrated but population-dependent synergies. These include cognition with perceived health relative to peers and several combinations involving BMI, waist–hip ratio (WHR), or blood pressure. In these cases, synergy is consistent with subgroup-conditional coupling between subjective appraisal and objective markers: combinations of self-rated health with cognitive or physical capacity may carry joint information that is not captured by additive contributions alone, and the strength of this coupling can vary across strata. Similarly, configurations combining functional limitations with poorer perceived health reflect closely related constructs, but the presence of synergy suggests that their joint states can still encode non-additive information about other targets, in a way that differs across subgroups.

Overall, these quadrant-level patterns serve two purposes. First, they provide face validity by showing that BUST recovers clinically familiar constellations—such as links among sleep, perceived health, mood, physical activity, and alcohol use—in positions that match established knowledge. Second, the map highlights pairs whose interactions are less well characterised in routine multimorbidity analyses, including pain–alcohol use, anxiety–physical activity and sleep–alcohol use. These combinations are not typically examined together despite plausible mechanistic connections, and their location on the BUST map suggests synergistic behaviour that is strong yet not readily apparent from correlation or co-occurrence alone. Such pairs, therefore, represent promising candidates for deeper mechanistic or interventional study. This heterogeneity suggests that synergy-informed applications may require both population-wide constellations (e.g., alcohol use–grip strength, sleep–physical activity) and more targeted, domain-focused interactions (e.g., pain–physical activity, perceived health relative to peers–physical activity).

Low-uniformity pairs add a clinically relevant layer to this picture. A pair can show clear synergistic behaviour in the pooled sample while still behaving differently between subgroups, meaning that a single population-wide summary may not be appropriate for screening, risk stratification, or hypothesis generation. Decomposing uniformity into its sex-, age-, and multimorbidity-specific components helps distinguish interactions that are broadly stable from those whose joint implications differ between participants with versus without multimorbidity or across age groups. One interpretation is that multimorbidity status can change the context in which symptoms, behaviours, and physiological markers co-occur, so that the same two-variable constellation may encode different joint information depending on baseline health complexity. In practice, this argues for reporting not only a single BUST position but also which stratification dimension drives any observed context dependence when translating synergistic relationships to clinical assessment or to cohort-to-cohort comparison.

The heterogeneity of BUST archetypes among the strongest pairs also invites interpretation. Several high-ranking narrow–uniform pairs, such as perceived health relative to peers–physical activity, pain–physical activity and depression–perceived health relative to peers, combine relatively focused Breadth with consistent cross-stratum behaviour. These cases can often be understood through clinical mechanisms. For example, the synergy between pain and physical activity may reflect different coping patterns: individuals who remain active despite pain tend to exhibit more adaptive coping and better functional trajectories, whereas those who reduce activity may follow a more passive pattern associated with poorer outcomes. BUST does not distinguish these causal pathways, but it flags that the joint state of pain and activity is more informative about downstream targets than either measure alone. Likewise, narrow–specific constellations such as cognition–perceived general health or cognition–perceived health relative to peers may arise because cognitive performance aligns more directly with an individual’s own perceived health than with more cognitively demanding comparative assessments. These examples illustrate how BUST-derived synergy patterns can map onto clinically meaningful behavioural and cognitive processes, helping to generate mechanistic hypotheses about why certain constellations exhibit strong or population-dependent synergy.

3.3 Synergistic clusters mirror established geriatric domains

Our synergy-based clustering also aligned closely with established geriatric domains. The dendrogram separated a metabolic marker (WHR) and a cardiovascular marker (blood pressure) from a cluster of symptoms and behaviours (sleep quality, physical activity, pain symptoms, anxiety symptoms, perceived health relative to peers, and functional limitations), and from a broader cognition/comorbidity cluster (depression, perceived general health, hearing difficulties, cognition, memory complaints and problems, BMI, alcohol use and grip strength) (Fig 4a). This structure is consistent with epidemiological work showing that central obesity and elevated blood pressure are core components of cardiometabolic multimorbidity that predict cardiovascular events and later cognitive decline, effectively bridging metabolic, cardiovascular and neurological pathways [25–28].

Likewise, the close grouping of sleep disturbance, pain, mood and reduced activity echoes symptom-cluster studies in older adults, where co-occurring insomnia, chronic pain and depression are strongly associated with frailty and reduced physical function, and where combinations of such symptoms produce worse outcomes than any single symptom alone [29]. This clinical notion of “worse than the sum of the parts” is closely aligned with the synergistic patterns revealed by the BUST representation, even though the underlying statistical formalisms differ.

These clusters arise purely from the organisation of synergistic information and therefore complement, rather than replicate, additive or correlation-based structures. The cognition/comorbidity cluster, which links depression, hearing loss, subjective memory concerns, poorer perceived general health, and weaker grip strength, also mirrors current understanding of age-related cognitive vulnerability. Hearing impairment and depression have both been associated with increased dementia risk [30], and subjective cognitive complaints are common among patients attending audiology clinics, where multiple dementia risk factors can co-occur [31]. Grip strength and BMI further anchor this cluster in frailty: lower muscle strength and adverse body-composition profiles predict poorer physical and cognitive outcomes in later life [32–34]. Interpreted clinically, the high-weight backbone of synergistic relationships links subjective health, mood, behaviour and physical capacity - domains often jointly implicated in frailty and functional vulnerability - whereas cardiometabolic markers appear to contribute more selectively within this dataset. The emergence of these clinically recognisable constellations from purely synergistic information suggests that the BUST representation captures meaningful higher-order structure in the multimorbidity landscape.

3.4 Clinical practice and future directions

From a clinical perspective, our findings suggest that synergy-aware analyses can complement existing multimorbidity tools rather than replace them. Broad, relatively uniform pairs such as sleep with physical activity, perceived health relative to peers with physical activity, and depression with perceived health relative to peers behaved as integrative signals: their joint states carried information about a wide range of functional outcomes and did so consistently across demographic and multimorbidity strata. In practical terms, this supports the use of brief questions on sleep, perceived health, and mood, combined with simple anthropometric and performance measures, as efficient “multi-domain probes” in routine assessment. At the same time, more specific but robust synergies, such as alcohol use with grip strength or BMI, point to particular constellations that may warrant closer scrutiny in individuals who meet specific combinations of behavioural and physical risk factors.

Synergy can be read as a value-of-information signal: it quantifies how much additional predictive content a second measurement contributes given what the first has already revealed. This framing suggests three concrete translation pathways. First, assessment prioritisation: in time-constrained settings where comprehensive geriatric assessment is not feasible, BUST-ranked pairs could guide which cross-domain questions to ask together. High-composite pairs such as perceived health relative to peers with physical activity achieved the largest breadth and uniformity scores, suggesting that combining a brief comparative-health question with an activity probe yields more information across multiple targets than either question alone. Second, context-sensitive follow-up: when a clinician observes a joint state that belongs to a high-synergy pair - for example, a patient reporting both poor sleep and low physical activity - this combination could prompt targeted screening for outcomes that neither symptom predicts well in isolation (in this case, pain, grip strength, and functional limitations). Third, subgroup-aware interpretation: low-uniformity pairs signal that synergistic behaviour differs across strata. For instance, the alcohol use–grip strength pair showed context-dependent behaviour across multimorbidity strata; a clinician might therefore weight this combination differently in a patient with established multimorbidity than in an otherwise healthy individual.

These pathways remain illustrative: translating BUST output into actionable decision rules would require prospective validation, integration with existing risk models, and evaluation of whether synergy-ranked features predict patient outcomes better than simpler alternatives. Because our analyses are observational and based on a single cohort, we view these synergies primarily as hypothesis-generating rather than ready-made decision rules. Future work could use similar pipelines to screen other longitudinal studies, compare BUST patterns across healthcare settings, and embed synergistic pairs into prognostic models or clinical decision support tools. Ultimately, synergy-guided maps of where joint effects concentrate could inform the design of combination interventions - targeting sleep and activity, mood and weight, or alcohol use and physical capacity - that explicitly leverage interactions rather than treating each domain in isolation.

3.5 Study limitations

We pooled observations across LASA waves without explicitly modelling within-subject dependence. Waves are spaced approximately three years apart, and many of the variables we analyse (symptoms, signs, wellbeing indicators and functional limitations) change on much shorter time-scales, so repeated measurements are not near-duplicate records and can reflect genuinely different health constellations. Our focus is on population-level dependency structure rather than individual trajectories, and we therefore treat each record as an independent sample from a relatively stable underlying mechanism. Nonetheless, repeated participation introduces some within-person correlation and may lead to a mild underestimation of uncertainty.

Additional limitations relate to study design and measurement structure. The three-year interval between waves constrains temporal resolution, and the available variables may not capture all clinically relevant domains that contribute to synergistic behaviour. Moreover, our analyses rely on discretised versions of predominantly continuous variables, which can inflate or deflate higher-order structure depending on the choice of cutpoints. Continuous PID formulations exist that could, in principle, avoid discretisation altogether. Gaussian PID [35,36] provides closed-form decompositions when all variables are jointly Gaussian, and non-parametric mutual-information estimators based on k-nearest-neighbour statistics [37] could serve as building blocks for continuous PID in more general settings. However, our dataset contains an inherent mix of variable types: some predictors are binary (memory complaints, anxiety) or ordinal with few levels (perceived health, depression), while others are originally continuous (grip strength, BMI, blood pressure). Gaussian PID requires all variables to be continuous and jointly Gaussian - assumptions that are violated by binary and coarsely ordinal measures. Methods for PID with mixed discrete–continuous sources remain an open problem in information theory. Our discretisation strategy, while it inevitably discards some fine-grained variation in continuous variables, offers three practical advantages: it handles mixed variable types within a single estimation framework; it produces clinically interpretable categories anchored to established thresholds, and it is fully compatible with the well-characterised redundancy lattice used throughout our analyses. Moreover, the post-hoc sparse-state merging procedure (Section 4.3.1) ensures adequate cell counts for reliable estimation. A binning sensitivity analysis presented in S1 Appendix confirms that the main qualitative findings - including the identity of high-ranking BUST edges and the separation between synergy- and TMI-based rankings - are robust to alternative discretisation choices.

Generalisability is also limited. LASA is a community-based cohort of older Dutch adults, and the synergy patterns observed here may differ in younger populations, in clinical or hospital settings, or in cohorts with different disease burdens or measurement practices. The BUST representation and PID pipeline are directly transferable to such contexts, but the specific locations and magnitudes of synergies should be interpreted as cohort-specific until replicated elsewhere. At the same time, LASA is a population-based cohort sampled from municipal registers across three Dutch regions (Amsterdam, Zwolle and Oss), which supports generalisability of the main patterns to community-dwelling older adults in the Netherlands, even though replication in other cohorts and clinical settings remains necessary.

Despite these limitations, bootstrap resampling (1,000 replicates) confirmed that the top-ranked pairs’ synergy estimates are stable, with narrow 95% confidence intervals that exclude zero for all top-10 BUST pairs (Section S7 of S1 Appendix). Because our conclusions emphasise qualitative patterns - such as the divergence between synergy- and TMI-based rankings and the structure of high-weight BUST edges and cliques - rather than formal inference, the limitations described above are unlikely to alter the main results. Even so, future work could incorporate cluster-aware resampling, subject-level aggregation, or designs with finer temporal granularity to address these considerations more explicitly.

The PID framework generalises naturally beyond two sources. The number of partial information atoms is governed by the Dedekind numbers and grows super-exponentially with the number of sources: from 4 atoms for two sources, to 18 for three, 166 for four, and 7,579 for five [38,39]. While enumerating and estimating these atoms remains computationally feasible for up to approximately five sources, the binding constraint is the joint state space required for reliable estimation. With discrete variables whose levels range from 2 to 7, the number of cells in the joint contingency table grows exponentially: for example, five three-level sources and one three-level target produce 3⁶ = 729 joint states, yielding an average of fewer than four observations per cell in our largest stratum () - far below the minimum required for stable information-theoretic estimation. Even with four sources the situation is marginal (3⁵ = 243 cells, ∼11 observations per cell). Our focus on bivariate PID (two sources, one target) was therefore a deliberate design choice that balances the ability to detect irreducibly joint effects against the need for adequate statistical power given the sample sizes and variable cardinalities available in the LASA cohort. Future applications to substantially larger cohorts or to variables with fewer discrete levels could extend the analysis to three- or four-source PID and thereby capture higher-order synergies that bivariate decomposition cannot resolve.

Conclusions

This study introduces a bias-aware, interpretation-focused workflow for detecting and summarising synergistic relationships in cohort data. By combining permutation-screened bivariate PID with the Breadth–Uniformity–Synergy–Total (BUST) representation, network visualisation and clique analysis, we show how higher-order dependence can be mapped in a way that remains interpretable at the level of individual variable pairs. The approach is implemented in open, reusable code and is compatible with standard epidemiological designs, making it directly transferable to other multimorbidity and ageing studies.

Applied to LASA, the BUST representation reveals that synergistic information is present but concentrated: most pairs show small synergy fractions, whereas a minority exhibit clearly “more than the sum of the parts” behaviour. Synergistic pairs occupy distinct regions of the BUST map, with some acting as broad, relatively uniform constellations and others showing narrower or population-specific patterns across age, sex and multimorbidity strata. Relationships ranked highest by synergy are entirely distinct from those ranked highest by TMI, indicating that PID-based synergy captures information architectures that are not recoverable from correlation or TMI alone.

Variables such as sleep, BMI, depression, grip strength and perceived health relative to peers participate disproportionately in high-scoring generalist synergies and act as integrative hubs, while combinations involving alcohol use and physical capacity form more focused but robust specialist synergies. These patterns suggest that a small set of routinely collected measures, including brief questions on comparative health and mood alongside simple functional tests, can yield joint information about daily functioning and related targets that is not available from any single variable alone.

Although our analyses are observational and confined to one cohort, the workflow provides a practical template for future work. Synergy-guided variable maps could be used to refine screening batteries, to prioritise combinations of risk factors for mechanistic study, and to identify candidate constellations for multi-target interventions—for example, jointly addressing sleep and activity, mood and weight, or alcohol use and physical performance. More broadly, the results illustrate that multimorbidity and functional decline are shaped not only by the accumulation of individual deficits but also by how specific symptoms, behaviours and conditions interact, underscoring the value of bringing formal notions of synergy into clinical epidemiology.

Overview of the workflow

We developed an interpretation-first pipeline to detect, debias, and summarise synergistic information at scale. For each target, we (i) screen source–source–target triplets using permutation tests with false-discovery-rate control; (ii) estimate bivariate PID atoms with small-sample bias correction and redundancy-definition sensitivity; and (iii) compress triplet-level results to pair-level profiles via a Breadth–Uniformity–Synergy–Total (BUST) map, which quantifies how widely a pair’s synergy extends across outcomes, how consistent it is across strata, and how strong and information-rich it is overall. Data curation, discretisation, and stratification follow clinical guidance and cohort documentation.

4.1 Information-theoretic framework

Mutual information (MI) quantifies the reduction in uncertainty about a target after observing a source. For discrete variables,

where denotes Shannon entropy (log base 2), measured in bits. For two predictors , MI does not separate overlap from complementarity. Partial information decomposition (PID) partitions the joint information about Y into four non-negative components [38]:

interpreted respectively as information shared by both sources (redundancy), information available only from X₁ or only from X₂ (unique), and information that emerges only when both sources are observed together (synergy).

Defining the PID atoms.

The decomposition requires a definition of redundancy; all other atoms follow by subtraction. We adopt the measure of Williams and Beer [38], which defines redundancy as the outcome-wise minimum of the specific information each source provides:

where the specific information of outcome y given source X is

Intuitively, I_red captures, for each target outcome separately, the lesser of the two sources’ predictive contributions, then averages over outcomes. The remaining atoms are obtained by subtraction:

(1)(2)

4.2 Synergistic information

Synergistic information is the piece of predictive power for a third variable (target) that appears only when two source variables are observed together. If either source is considered in isolation, this component of the signal vanishes. It represents information that is carried jointly by both sources and cannot be recovered from either alone. Consequently, any model that assumes the total effect of two predictors can be written as a sum (additive) or a product (multiplicative/independent-action) will miss that irreducible joint, or synergistic, part.

Mutual information (MI) quantifies how much uncertainty about an outcome is reduced by observing a feature, without assuming a specific model or scale. Partial Information Decomposition (PID) separates the joint information from two features into redundant, unique, and synergistic parts, allowing us to test whether combinations matter beyond their individual contributions.

For example, if PID indicates that 18% of the information predicting an outcome is synergistic, then an additive or multiplicative model cannot recover that 18% from main effects alone; at minimum, an explicit interaction term or a more flexible model is required to capture the observed pattern.

4.3 Screening candidate source–target triplets

Before running PID, we screened every candidate triplet for evidence of any statistical dependence between the source pair and the target. For each triplet the null hypothesis is

i.e., the target Y_j is statistically independent of the joint source (X_a, X_b). The test statistic is the plug-in (maximum-likelihood) estimate of the joint mutual information computed from the observed contingency table. To build a null distribution we performed 500 random permutations of the target labels Y_j while holding the joint distribution of (X_a, X_b) fixed. Shuffling only the target preserves any dependence structure between the two sources, so the null distribution reflects the absence of genuine source–target information without destroying inter-source associations. For each permutation the plug-in MI was recomputed from the permuted data, and the p-value was defined as the proportion of permuted MI values greater than or equal to the observed MI. Across all triplets, we controlled the false-discovery rate at q = 0.05 using the Benjamini–Hochberg procedure [43]. Triplets whose adjusted p-value fell below 0.05 were retained for subsequent PID estimation; the remainder were excluded, thereby focusing computational resources on triplets with statistically supported source–target dependence.

4.3.1 PID estimation with bias correction.

Plug-in estimates of information-theoretic quantities are positively biased in finite samples. For multivariate measures such as O-information, this bias can distort the balance between redundancy and synergy [44]; for PID atoms specifically, Koçillari et al. [45] showed that the bias of synergy grows quadratically with the number of discrete variable levels, whereas biases in unique and redundant information grow more slowly. Correcting for finite-sample bias is therefore especially important for synergy estimation in variables with multiple categories.

The expectation of a plug-in information estimator admits a series expansion in inverse sample size [45]:

where is the population quantity of interest and a₁, a₂ are unknown, distribution-dependent constants. Quadratic extrapolation (QE) exploits this expansion by evaluating the estimator at multiple sample sizes and fitting a degree-2 polynomial in 1/N; the intercept recovers , removing both the first- and second-order bias terms. A linear (two-point) extrapolation would leave the component uncorrected - a material concern given the quadratic scaling of synergy bias with variable cardinality.

Concretely, for each PID atom we compute the plug-in estimate on three nested samples: the full dataset (N observations), a random subsample of N/2, and a random subsample of N/4. These three evaluation points at 1/N, 2/N and 4/N define a degree-2 polynomial whose intercept (at ) is the QE estimate.

To remove any residual estimator-specific bias, we combine QE with a target-shuffle null: we repeat the same three-point QE procedure on data in which the target variable has been randomly permuted (while preserving the joint of the two sources), and subtract the shuffled QE estimate from the original:

By shuffling only the target while keeping the sources fixed, the joint dependency structure among sources is preserved. This ensures that the null distribution reflects the absence of genuine source–target information without destroying the joint information shared between sources. We average over K = 30 independent target shuffles per subsample fraction to stabilise the null estimate.

We truncate negative atoms to zero for reporting. This preserves the non-negativity of reported synergies but breaks exact additivity of PID atoms, and it can inflate the prevalence of very small positive synergies near zero.

4.4 Breadth–Uniformity–Synergy–Total (BUST) classification

For each unordered source pair (X_a, X_b) we ask two related questions: (i) is its synergy spread across many targets (broad) or concentrated on a few (narrow)? (ii) are those synergistic effects similar across population strata (uniform) or do they vary (specific)? We answer (i) by quantifying how the pair’s synergy is distributed across targets and (ii) by quantifying how even that distribution is across strata. Both quantities are defined in terms of normalised synergy-based probability distributions and summarised using bounded deviation-from-uniformity measures.

Notation.

Let denote the set of all targets. For each unordered pair (X_a, X_b) and target , let s_j ≥ 0 be the bias-corrected PID synergy (Section 4.3.1), with when no positive synergy is detected. The pair’s total synergy strength is

and the associated synergy shares across targets are

defined whenever S > 0. For stratified analyses, let and denote the synergy and TMI (sum of all PID atoms) for target j in stratum s of stratification type t (sex, multimorbidity, or age group).

Breadth (B): synergy spread across targets.

We quantify how broadly synergy is distributed over targets using the normalised Shannon entropy of the synergy shares . For each pair with S > 0 we define

with when all synergy is concentrated on a single target and when synergy is perfectly uniform across all targets. To obtain a median-centred score that reflects the empirical behaviour of the data, we compute the median breadth across all pairs, H_B,med, and map H_B to a breadth score

clipped to [−1, 1] for stability. Thus denotes broadly distributed synergy (generalist behaviour), denotes highly concentrated synergy (specialist behaviour), and corresponds to the median breadth observed in the data. Pairs with S=0 do not receive a B score and are excluded from subsequent BUST plots.

As robustness checks, we also compute alternative breadth measures by replacing H_B with (i) 1 minus a normalised Gini coefficient of the synergy shares, and (ii) 1 minus a normalised Kolmogorov–Smirnov (KS) distance between the empirical cumulative distribution of w_j and the uniform distribution. These alternative B scores are reported in S1 Appendix and are highly correlated with the entropy-based B.

Uniformity (U): cross-stratum stability of synergy.

To characterise whether a pair’s synergy is uniform across strata or context-specific, we first form synergy-based distributions across strata for each target and stratification type. For target j and stratum s of type t we define the synergy fraction

so that whenever defined. For a given j and t, we collect the fractions across strata, add a small stabilising constant to each value to avoid zero cells, and normalise to obtain a probability vector

whenever at least two strata contribute non-zero information. For each target–type combination we then compute the normalised entropy

so that when synergy fractions are similar across strata of type t (uniform), and when synergy is concentrated in a subset of strata (specific to, e.g., a particular age group or sex).

Because some targets contribute little synergy overall and may be noisy, we aggregate these target-level uniformities using weights proportional to their full-population synergy s_j. For each stratification type t we define

summing over targets with well-defined and s_j > 0. The overall uniformity index for a pair is then obtained by averaging across available stratification types:

where is the subset of types (sex, multimorbidity, age group) with at least two informative strata. As for breadth, we median-centre this raw uniformity index using its empirical median H_U,med across pairs and define

clipped to [−1, 1]. Large positive U values indicate synergy fractions that are stable across strata (uniform, or “universal”), whereas large negative U values indicate synergy that is concentrated in specific demographic or strata (context-specific).

To directly address potential numerical instabilities of coefficient-of-variation based metrics when means are small, all U scores are built from bounded probability distributions with explicit -stabilisation, avoiding divisions by near-zero denominators. In sensitivity analyses we recompute uniformity using (i) 1 minus a normalised Gini coefficient and (ii) 1 minus a normalised KS distance for the , rather than entropy. These alternative U measures show high correlations with the entropy-based U and do not materially change the BUST map.

Synergy strength (S) and total information (T).

For each pair, the synergy strength is defined as the total amount of positive synergy across all targets,

measured in bits. The total information (T) is defined by summing TMI_j over all targets with non-zero synergy in the full population,

thereby capturing the overall information content carried by the sources about the outcome space, irrespective of whether it is synergistic, redundant, or unique. The BUST map uses (B,U) as a 2D embedding and encodes S (and, where available, T) through bubble size and colour to visualise which pairs are both broad, uniform and information-rich.

4.5 Analysing BUST networks

Composite score construction.

To prioritise source pairs that are simultaneously broad, cross-stratum robust, strongly synergistic, and information-rich, we constructed a composite BUST score for network analysis. First, the breadth and uniformity scores (B,U) obtained from the entropy-based metrics were already scaled to [−1, 1] as described above. Second, synergy strength S and total information T were each transformed to [−1, 1] by median-centring and applying a hyperbolic tangent transform to soften the influence of extreme values:

with omitted if reliable TMI was unavailable. All four dimensions were then linearly mapped from [−1, 1] to [0,1] and combined multiplicatively,

where each subscript 01 denotes the corresponding rescaled value in [0,1]. The multiplicative form acts as a soft AND-gate: a pair must score reasonably well on all dimensions (breadth, uniformity, synergy strength and total information) to attain a high composite value, while a low score in any dimension down-weights the overall importance. This avoids selecting pairs that are extreme on a single axis but weak on the others, and reflects our interest in identifying edges that are jointly broad, robust and information-rich. For network visualisations we defined “strong” BUST edges as those with composite scores above the 75th percentile of BUST_comp, and we confirmed that the resulting network structure is robust to alternative percentile cut-offs and to using Gini- or KS-based B and U in place of the entropy-based metrics (Section S2 of S1 Appendix).

Synergistic clique detection and stability analysis.

Stable synergistic cliques were identified using a threshold-based stability approach to avoid arbitrary cutoffs. We constructed a weighted graph where nodes represented variables and edges were weighted by the composite BUST scores integrating breadth, uniformity, synergy strength and total information (Section 4.5). Cliques (fully connected subgraphs of size ≥3) were enumerated independently at four thresholds (67th, 75th, 80th, 85th percentiles of composite scores). Cliques appearing in ≥3/4 thresholds were classified as stable. Each stable clique was characterised by its mean composite weight, average B and U scores, and total synergy strength, then ranked by a weighted importance score

This choice reflects three design considerations. First, we prioritised stability across thresholds (weight 0.4), because cliques that recur under different cut-offs are less likely to be artefacts of a particular threshold. Second, we favoured cliques that are larger (weight 0.3), as these capture broader constellations of jointly interacting variables. Third, we gave comparable weight to the mean composite edge weight (weight 0.3), so that cliques with consistently strong BUST edges are preferred over those formed by weaker links. These weights are heuristic but reflect a conservative preference for robust, sizeable and strongly connected cliques rather than small or unstable motifs driven by a single strong edge.

4.6 Simulation validation of the estimation pipeline

To verify that the QE bias-corrected PID estimation pipeline (Section 4.3.1) and the BUST summary scores (Section 4.4) recover known ground truths under realistic conditions, we conducted three sets of simulation experiments with analytically defined joint distributions whose PID atoms were computed numerically using the dit library [42].

4.7 Sensitivity to discretisation choices

To assess whether the BUST classification is sensitive to the choice of discretisation strategy, we re-ran the full PID pipeline on the LASA full-population dataset under five alternative binning schemes: (i) the clinical cut-points used in the primary analysis; (ii) equal-frequency tertiles; (iii) equal-frequency quartiles; (iv) equal-width terciles; and (v) a coarsened binary split at each variable’s median. For each scheme, we recomputed all PID atoms with QE bias correction, constructed BUST scores, and compared the resulting pair rankings using Spearman rank correlations of composite BUST scores and Jaccard overlap of the top-10 ranked pairs. Among the four non-binary schemes, Spearman correlations ranged from 0.83 to 0.96, and the equal-frequency tertile and quartile schemes were nearly interchangeable (, Jaccard top-10 = 0.82). The coarsened binary scheme was poorly correlated with all others (), consistent with the severe information loss from collapsing multi-level variables to two categories. Full results, including BUST map comparisons and rank correlation matrices, are provided in Section S6 of S1 Appendix.

4.8 Data and preprocessing

We analysed data from the Longitudinal Ageing Study Amsterdam (LASA), an ongoing population-based cohort designed to study physical, emotional, cognitive and social functioning in older adults in the Netherlands. The original cohort of 3,107 community-dwelling adults aged 55–85 years was recruited in 1992/93 and has been followed triennially ever since, with refreshment samples drawn in 2002/03 (n = 1,002) and 2012/13 (n = 1,023) to maintain population representativeness [17]. Data are collected through face-to-face structured interviews and self-completed questionnaires.

For the present analyses, we used data from the 1995/96–2021/22 waves (nine waves, each approximately three years apart). After excluding interviews with more than 50% missing values and applying random-forest imputation (see below), the analytic sample comprised 14,716 person-wave observations from 4,025 unique participants. We focused on variables that are central to multimorbidity and ageing research and that appear throughout the results: symptoms (sleep problems, pain), clinical signs (grip strength, body mass index, waist–hip ratio, mean arterial pressure), behaviours (physical activity, alcohol use), self-rated health indicators (perceived general health, perceived health relative to peers), cognition (MMSE score, memory complaints, memory problems), sensory function (hearing difficulties) and activities of daily living (functional limitations), anxiety and depression.

Variables included in the synergy analysis.

All variables except age, sex and multimorbidity (used solely for stratification) were included as candidates for both predictors and outcomes in the PID triplets. Treating variables symmetrically allows estimation of how any pair jointly informs each remaining variable, supporting the Breadth, Uniformity and Total components of the BUST representation. Table 1 summarises all variables included in the synergy analysis together with their domain classification, number of discrete levels after post-hoc merging (Section 4.3.1), and discretisation basis.

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Table 1. Clinical variables included in the PID synergy analysis.

https://doi.org/10.1371/journal.pcbi.1014386.t001

Memory complaints, memory problems, and MMSE are grouped under “Cognition” for analytic parsimony. Alternative classifications are defensible (memory complaints as a symptom, MMSE as a clinical sign, doctor-confirmed memory problems as a disease); the choice does not affect downstream PID or BUST calculations.

Data and code availability

The LASA data used in this study are not publicly available because they contain sensitive participant information and are distributed under governed access. Researchers may request access to LASA data for specific research questions by submitting an analysis proposal to the LASA Steering Group. If approved, access is provided under a data-sharing agreement. Details and the required proposal materials are available via the LASA website: https://lasa-vu.nl/en/request-data/.

The analysis pipeline is publicly available as the open-source Python package bust-pid at https://github.com/cillianhourican/bust-pid. The package implements permutation screening, QE-corrected PID estimation, BUST scoring, network construction, and visualisation, and includes reproduction scripts for the main paper figures (requiring LASA data access) as well as a synthetic-data demo.