Introduction

The Saudi MOH Statistical Yearbook 2024 publishes a table headed “Referrals including reverse referrals to general or specialized government hospitals according to the referring and receiving cluster in 2024 G” [1]. It has twenty health-cluster labels on each axis, a count at every intersection and printed marginal totals. The apparent geometry invites several very different readings. A diagonal cell may mean that the two printed cluster labels match, but it does not say whether the referring and receiving facilities are the same, whether a person traveled, or whether the referral was accepted. An off-diagonal cell identifies two unlike labels; without a firm axis key it cannot safely be called a transfer from one named cluster to the other. A useful first analysis is therefore one that preserves the table's exact accounting while declining to convert a cell count into a patient or service outcome.

Saudi e-referral research makes the distinction between data frames concrete. Alharbi and colleagues analyzed 755,145 Saudi Medical Appointments and Referrals Center (SMARC) e-referrals from 2023–2024, tabulating 13 administrative regions, business units, internal versus external referral and acceptance-related attributes [2]. Earlier nationwide SMARC analyses described more than 600,000 requests from 2020–2021 [3], modeled referral acceptance in those years [4], and compared referrals within versus between administrative regions [5]. A Saudi primary-care case study investigated an e-referral process and geographic accessibility at its own service level [6]. These studies differ in time frame, event definition, facility scope and geography from the MOH's 2024 twenty-cluster government-hospital table. Their results should not be inserted as missing cells, denominators or validation for it. The 2025 SMARC paper does not present this exact twenty-by-twenty cluster-pair matrix in its accessible full text; that observation defines a narrow analytic opening, not a claim to have invented referral research or discovered that most counts are diagonal.

A 2024 systematic review of e-consultation found possible improvements in access, but low-certainty evidence [7]. This does not establish a mechanism for the MOH table: interpreting concentration as effective gatekeeping would require referral origin, eligibility, disposition and patient follow-up, none of which are available in the MOH array. The more discriminating question is how the off-diagonal published counts are distributed among unordered pairs. This keeps the analysis informative when the row and column direction is uncertain. It also exposes how much of the off-diagonal total is between distinct cluster labels that share a Riyadh or Jeddah city name. Shared wording is not proof of a common catchment, nearby facilities or short travel.

Other network literatures show why data-generating processes matter. A scoping review found that patient-sharing networks assembled from administrative encounters vary in their definitions of a tie and their analytic unit [8]. Analyses of the US patient-referral network [9] and referral paths [10] rely on individual encounter sequences or shared-patient links—evidence not supplied by an aggregate yearbook grid. Hospital postacute referral concentration [11], patient-sharing network costs [12], the persistence of physician relationships [13], and physician- versus patient-induced network construction [14] likewise address distinct endpoints and data. They motivate precise naming of a matrix edge, not importation of their network statistics or outcome associations into Saudi data.

An actual referral pathway has stages absent from a static count grid. Electronic hospital-referral work has measured information completeness [15], direct-admission coordination [16] and specialist communication [17]; a pediatric referral study distinguished scheduling from completion [18]. None of those process measures can be reconstructed from a Table 4-63 count. A data-quality framework separates conformance and plausibility from fitness for a particular analytic use [19]. RECORD [20] and STROBE [21] supply useful reporting principles for source, selection, variables and limitations, although this is a documentary census rather than a patient-level cohort. We therefore asked: among the counts printed in the MOH 2024 twenty-cluster matrix, what fractions are on the diagonal, which unordered label pairs account for the largest off-diagonal counts, and how much of the off-diagonal total is between distinct clusters with the same Riyadh or Jeddah city name? We did not study patient access, completed care, referral efficiency, facility performance or outcomes.

Methods

Design, source and unit

We conducted a deterministic cross-sectional analysis of one official MOH 2024 yearbook workbook [1]. A frozen copy is bound to this study as official/Statistical-Yearbook-2024.xlsx, SHA-256 a482e1f88c13b24246312fdb4696e3f9c49f2dd9459d56ef585f47bb47032c83, identical to the linked MOH file at extraction. On worksheet 4-63, Table 4-63, the 400 numeric intersections occupy B6:U25, row totals V6:V25, column totals B26:U26, and the grand total V26. We retained twenty label positions, trimmed a leading space in the Hail column heading, and checked English column labels (B5:U5) against English row aliases (W6:W25) by positional order. Wording differs, for example “Riyadh Cluster I” versus “Riyadh First Health Cluster”; the row and column sequences nevertheless designate the same twenty labels. Labels were standardized for display only. Neither spelling normalization nor the official English names supplies a facility or population crosswalk.

The observation unit for all calculations was a published matrix cell, not a person, known single referral episode or hospital encounter. We refer to the source's entries as “published referral counts,” because its caption includes reverse referrals but does not define whether repeated requests or linked stages are counted once. The table does not identify accepted, denied, scheduled, completed or cancelled referrals. The twenty labels are health clusters, not Saudi Arabia's thirteen administrative regions; a city can contain more than one cluster. We made no substitution from SMARC records or another yearbook table. Source cells were read in their stored integer form; no percentages in this article are transcribed from a precomputed graphic.

Validation and orientation-invariant measures

Let M[i,j] denote the nonnegative integer at position i,j for i,j from 1 to 20 in the common label order. We asserted that each of the twenty sums over j equals its printed V total, each of the twenty sums over i equals its printed row-26 total, and their common grand sum equals V26. We separately confirmed that every cell is an integer at least zero and that each label occurs once on each axis after whitespace normalization. Passing these checks verifies transcription and internal workbook arithmetic, not the events in MOH source systems or the intended axis orientation.

The descriptive total was N = sum(M[i,j]). The diagonal count was D = sum(M[i,i]), and off-diagonal count O = N − D. We report D/N and O/N; a diagonal entry is only a same printed cluster-label position. For every two distinct labels i < j, we formed the unordered-pair count P[i,j] = M[i,j] + M[j,i]. There are 190 possible pairs, and the pair counts sum exactly to O. We sorted pairs by descending two-cell count, breaking count ties by the two printed English column labels alphabetically; Table 5 reports all 190 ranks, coordinates and source values. Both D and P are unchanged by transposing M. We do not report in-degree, out-degree, net exchange, directional arrows, path length or patient-flow centrality. The Arabic stacked headings at A4/A5 read receiving then sending clusters, while the English headings at W4/W5 read “Referred from” then “Referred to” in their stacked order. Those captions do not unambiguously assign one English directional label to numeric rows versus columns. Only an attributable MOH key can settle orientation.

The five-pair summary was chosen after seeing the matrix as an exploratory reporting cutoff, not as a source-defined or clinically meaningful threshold. We display the top-one, top-five, top-six and top-ten counts and shares to show sensitivity to that choice; in particular, we report the fifth–sixth gap. Separately, we defined four unordered pairs whose printed labels share a Riyadh or Jeddah city name: all three pairings among Riyadh I, II and III, and Jeddah I–II. Their combined count was subtracted from O to show how much of the off-diagonal total remains when these specific unlike-label pairs are set aside. This name-based partition was transparently calculated for description; no dated prospective protocol is asserted. It does not establish shared catchments, geographic boundaries, distance traveled or patients' residence. We made no inference from the remaining pairs about interregional travel. Table 1 defines the denominators; Table 2 supplies the largest source-pair positions; Table 3 presents the city-name sensitivity; Table 4 shows cutoff sensitivity. We did not calculate confidence intervals or hypothesis tests for this complete finite matrix of published cells.

Prior work and interpretation plan

Before drafting, we reviewed the full accessible 17-page 2025 SMARC article [2], checked the earlier Saudi reports [3–6] by indexed metadata and abstracts, and ran a finite PubMed/web search for the MOH matrix, cluster-pair and referral-flow terms. We did not treat abstracts as full-text verification of every contextual claim; a small claim-to-source ledger is retained for human authors. The 2024 e-consultation review [7] was used only as indirect context on access outcomes, not as validation of the matrix or its referral direction. The literature screen is not an exhaustive systematic review and cannot certify priority. We compared numerator definitions and geography rather than percentages across sources. The reporting approach adapts source-provenance and limitation items from RECORD [20] and STROBE [21], without claiming that participant selection, confounding or follow-up items apply to a table census. Exact cell locators, extraction code, the 190-pair CSV, source hash and local frozen workbook are retained in this study folder; the full 190-pair ledger is also reproduced in Table 5 of this DOCX. An external code location remains to be chosen by accountable authors.

Results

Matrix accounting and diagonal fraction

The 400 values and all forty printed marginal totals reconciled to V26 = 341,365. The twenty diagonal positions summed to 301,968, or 88.46% of N. Off-diagonal positions summed to 39,397, or 11.54%. Exactly 246 of the 380 off-diagonal positions were nonzero; at the unordered-pair level, 151 of 190 two-cell sums were positive. These counts describe the density of the published grid, not the number of referral routes, hospitals or people. Table 1 shows the arithmetic and the relevant denominators. A transpose of the original matrix preserves all reported results.

Table 1. Accounting identities and orientation-invariant summary of MOH Table 4-63

Source: official MOH 2024 yearbook worksheet 4-63, Table 4-63 [1], workbook SHA-256 in Methods. All 20 printed row totals and 20 printed column totals were recalculated exactly. Percentages use 341,365 as denominator and are rounded to two decimal places. “Diagonal” and “off-diagonal” describe label positions, not completed care or actual travel. The table includes reverse referrals; its event unit and directional axes were not independently verified.

The count's meaning is bounded by the source caption. It is defensible to say that 301,968 numbers are on like-labelled intersections. It would not be defensible to say 301,968 patients remained within a cluster, or that 39,397 people crossed a cluster boundary. The single N provides no rate denominator: it is not the number of patients who needed referral, the number of referral requests started, or the number of opportunities for acceptance. The diagonal result is therefore a property of the publication, not a quality-of-care indicator.

Largest unordered pairs

Summing the two opposing positions for each unlike-label pair makes the source's directional ambiguity irrelevant to the ranking. The five largest pairs contributed 17,496 of 39,397 off-diagonal counts (44.41%). The Jeddah I–II pair alone contributed 8,265 (20.98%). Riyadh I–III contributed 3,361; Riyadh II–III 2,110; Riyadh I–II 2,036; and Madinah–Tabouk 1,724. The sixth was Eastern–Al-Ahsa at 1,704, 20 fewer than the fifth; there is no tie at this display cutoff. Table 2 supplies the exact two source cells for the displayed top five, and Table 5 supplies source coordinates and both values for all 190 unordered pairs, including zero pairs. Their order within each pair follows the printed label index only and must not be read as a sending-to-receiving arrow.

Table 2. Five largest unordered pairs of different printed health-cluster labels

Source: MOH Table 4-63 [1], worksheet cells shown; denominator is all 39,397 off-diagonal published counts, not 341,365. Each pair's two positions are summed; no source-cell direction is assigned. The four Riyadh/Jeddah pairs join distinct health-cluster labels whose printed names share a city name. A sum may include reverse referrals, repeat events and unknown referral states; it is not a count of unique moving patients. The complete ranked 190-pair source-cell ledger is Table 5 of this article.

The concentration is not uniform across all unlike-label pairs: the top five occupy only five of 190 possible unordered pairs but contain 44.41% of the off-diagonal count. Because five was an exploratory display choice, Table 4 also shows how the share changes at one, six and ten pairs; there is no source-defined threshold or detected changepoint. These sums do not establish a hub-and-spoke service organization, because referral origin/destination, contributing facilities, catchment population and referral need remain unknown. In particular, the two unequal source cells in a pair must not be described as counterflows until the bilingual table heading is clarified by MOH.

Shared city-name sensitivity

Four of the top five pairs join distinct clusters whose printed labels share the Riyadh or Jeddah city name. Together they hold 15,772 counts, 40.03% of O and 4.62% of N. Removing these four name-based pairs leaves 23,625 other off-diagonal pair counts, 59.97% of O and 6.92% of N. Table 3 shows the calculation and makes the sensitivity's assumption visible. The remaining 23,625 must not be renamed “interregional transfers”: an unlike cluster label does not identify travel distance, and neither Saudi administrative-region geography nor patient residence is defined by the table.

Table 3. Distinct-cluster pairs sharing a Riyadh or Jeddah city name

Source and pair formula as Table 2 [1]. “Shared city name” is a partition based on the printed label strings, not a verified catchment, municipal or distance classification. Counts for the four pairs are subtracted once each from O = 39,397. The residual equals 6.92% of N = 341,365. Percentages were calculated from unrounded counts; displayed component percentages may not sum exactly. Neither subset is an accepted/completed referral or unique-patient denominator.

Descriptive rank-cutoff sensitivity

The five-pair cutoff captures nearly half the off-diagonal published counts, but its exact percentage changes materially if one more pair is included. One pair accounts for 20.98%, five for 44.41%, six for 48.73% and ten for 62.03%. The fifth and sixth sums differ by 20 counts, which is only 0.05 percentage points of O. The rank ordering is exact for the released workbook under the two-cell rule; neither the fifth/sixth boundary nor a ten-pair boundary has intrinsic clinical or policy meaning. Table 4 reports the values without treating the proportions as confidence intervals or temporal trends.

Table 4. Sensitivity of off-diagonal share to exploratory top-k display cutoff

Source: all 190 two-cell sums from MOH Table 4-63 [1], ranked by descending sum with printed-label alphabetical tie-break; Table 5 provides every pair and both source-cell coordinates. Shares use the full off-diagonal O = 39,397; values are rounded to two decimals. Fifth-ranked Madinah–Tabouk = 1,724 and sixth-ranked Eastern–Al-Ahsa = 1,704, a difference of 20. The displayed cutoffs are exploratory descriptions, not prespecified source thresholds or separate statistical groups. Source: MOH 2024 yearbook Table 4-63 [1], workbook SHA-256 a482e1f88c13b24246312fdb4696e3f9c49f2dd9459d56ef585f47bb47032c83. Each row sums source cells M[i,j]+M[j,i] for i < j; totals rank descending, with printed English labels breaking ties. The 190 pair sums equal O = 39,397, with 39 zero pairs. Cell order has no verified direction or unique-patient meaning.

Table 5. Complete 190-pair source-cell ledger

All 190 unordered pairs, including 39 zero pairs, are reproduced here from official worksheet 4-63 [1]; this in-manuscript ledger is not a public machine-readable deposit.

Table A1. All 190 unordered cluster-label pairs ranked by their two printed source positions

Discussion

Principal finding and what it does not show

The 2024 MOH referral table has a pronounced diagonal and an off-diagonal distribution in which a few unordered cluster pairs carry many published counts. The public diagonal observation alone adds little to the published table; external e-consultation literature [7]does not establish the meaning of these cells; the audit's incremental contribution is the complete, source-addressed 190-pair decomposition plus a transparent city-name and rank-cutoff sensitivity. A reader can recompute 17,496 as the five largest two-cell sums, check their rank against every other pair in Table 5, and reproduce 15,772 as the four Riyadh/Jeddah name-sharing pairs. These are descriptions of a 400-cell aggregate publication. They are not estimates of a hidden patient population and do not justify uncertainty intervals as though the printed matrix were a random sample.

The large Jeddah I–II and Riyadh I–III two-cell totals should prompt questions, not causal explanations. One plausible hypothesis is proximity or shared tertiary services; another is the way source systems assign clusters to facilities; still others include reverse-referral accounting, repeated requests, phased referrals or different center types. The table supplies no facility-level map, no timestamped referral sequence and no disposition. A diagonal entry could represent a transfer between two facilities under one cluster label. Conversely, an off-diagonal entry between two Riyadh-named clusters does not prove that the referring and receiving facilities are near each other. The results do not distinguish clinical need, care coordination, local gatekeeping, administrative assignment or data-entry practice. The source owner is the appropriate party to define each event and direction before such interpretations are attempted.

The closest Saudi study, Alharbi and colleagues' 2025 SMARC analysis, tabulated 372,926 e-referrals in 2023 and 382,219 in 2024, 755,145 altogether, using thirteen administrative regions and multiple workflow attributes [2]. Its overall 87.52% “internal” designation concerns the same administrative region across both years. Our 88.46% diagonal is the same health-cluster label in one MOH 2024 government-hospital matrix that explicitly includes reverse referrals. Although the two percentages look alike, neither numerator, denominator nor geographic criterion is shown to match. Even 382,219 versus 341,365 in 2024 cannot be interpreted as missing or extra referrals without a documented mapping between platforms, sectors, reverse-referral inclusion and extraction dates. The 2020–2021 SMARC internal/external and acceptance analyses [3–5] similarly ask different questions. Their richer records support questions this public table cannot answer; this paper supplies a transparent fine-grained pair profile that those articles do not display.

Published referral-network studies clarify a further trap. A shared-patient link can be inferred from longitudinal claims or encounters, but that link is not necessarily a formal referral [8,9,13,14]. A referral path can order encounters for the same person [10]; this matrix cannot. Postacute referral concentration can be related to spending only because linked administrative records include payment and beneficiaries [11,12]; here there is no cost or outcome field. Thus even familiar network terms such as “hub,” “centrality,” or “flow” would imply more than the present data. We used “unordered pair” and “published count” to preserve what is actually identifiable. If the axis orientation is later documented, a directed analysis could be conducted as a new study, ideally with event-level validation and a clear policy question.

Data quality and clinical claim ceiling

The workbook passes internal arithmetic checks: every cell is nonnegative, both marginal directions reconcile, and the national total agrees. This is useful for reproducibility but weak evidence of clinical validity. The framework of Kahn and colleagues separates data conformance from fitness for a particular use [19]. The present matrix is fit for the narrow purpose of describing printed count placement, not for scoring whether referrals worked. Electronic referral implementations show that completeness of clinical information, coordination and communication can be measured [15–17]; scheduling and completion are demonstrably different endpoints [18]. None can be derived from B6:U25. Even the word “referral” here should remain tethered to the yearbook caption, rather than expanded into completed specialty care.

The bilingual header deserves a dedicated source-owner query. Arabic labels on the left and English labels on the right of the stacked heading place “referred from” and “referred to” in an order that does not unambiguously tell an outside analyst which numeric axis is origin. The bilingual row aliases and English column headings align by ordered cluster identity, despite different wording; that supports the positional diagonal but cannot settle which off-diagonal position is i-to-j. Unordered two-cell sums survive an axis reversal exactly; marginal totals can be verified without calling one a source or destination. A nonexpert may find this caution overly conservative, but a wrong orientation would reverse every directional interpretation while leaving all arithmetic checks intact. No increase in model complexity can repair a missing axis definition.

The city-name sensitivity is similarly descriptive. We identified the four pairs by a stated label rule after inspecting the grid; no prospective selection is claimed. It is reasonable to note that the two distinct labels in each pair share “Riyadh” or “Jeddah,” and unreasonable to infer that the other 23,625 counts crossed administrative-region boundaries. Cluster polygons, referring-facility addresses and patient residence would be required for distance or catchment analysis. Moreover, the health-cluster labels are organizational units; a request counted at one site could concern care delivered at another. The association of label concentration with purchasing design in the online commentary [7] remains an untested proposition for a linked, defined dataset, not an inference this grid can test.

Strengths, limitations and next evidence

This audit reads the official workbook rather than a secondary graph; fixes the exact 400-cell universe and SHA-256; validates 40 printed marginals; discloses the pair formula and two-cell locators for all 190 pairs; and uses a statistic invariant to transposition. The complete source table is publicly locatable and the calculations require only sums and divisions. We did not fit a graph model, calculate uncertainty on a census, or claim an “optimal” concentration threshold. The five-pair cutoff was exploratory and the top-one, six and ten comparisons expose its sensitivity. The result remains reproducible if all rows and columns are transposed, and its count-based denominators are explicit in Tables 1–4 and Table 5.

The chief limitation is conceptual: the publication supplies a number for a cluster pair without a record-level data dictionary. We do not know its event unit, repeat handling, inclusion of reverse referrals, source and destination facility types, whether entries are attempted, accepted or completed, or when they were entered. We have no linked patient identities, procedure types, referral reasons, triage status, dates, capacity, distance or outcomes. A single 2024 snapshot cannot establish a temporal pattern or effect of Saudi health-sector reform. Source row/column labels do not certify catchment boundaries. The English spelling of a cluster is not an administrative-region identifier. We have not obtained MOH confirmation of the directional axes; an erroneous axis inference would be particularly consequential for policy. The finite literature screen and the 2025 SMARC full-text comparison do not certify first publication of a cluster analysis, especially in Arabic or grey literature. Table 5 makes the exact pair ordering inspectable in this DOCX, but code and CSV remain internal rather than a persistent public supplement.

An attributable MOH or referral-data-owner response should specify: the event counted in one cell; whether reverse referrals share the same inclusion rule; whether repeat requests, cancellations and multi-step referrals are counted more than once; whether source and destination are assigned by organization, facility or patient location; what acceptance or completion stage is included; the exact row/column orientation; and whether each cluster label has a dated organizational map. If those definitions support a linked event-level analysis, a later study can investigate direction, network paths and patient outcomes. If they remain unavailable, the present manuscript can still accurately report a published-table topology provided a human editor accepts that limited contribution.

Conclusion

The 2024 MOH twenty-cluster referral matrix prints 341,365 counts, of which 301,968 lie on its diagonal and 39,397 off it. Five unordered cluster pairs account for 17,496 off-diagonal counts under an exploratory cutoff; four pairs between distinct clusters whose printed labels share Riyadh or Jeddah account for 15,772. These source-cell facts are invariant to transposing the matrix. They neither establish a route taken by a patient nor measure acceptance, completed care, access, quality or health outcomes. A directional or performance interpretation requires the source owner's definition of the counted event and axes.