Sixty-Nine of a Hundred Thirteen

A cell printed as K (P%) can be checked with nothing but the row it sits in. In the appendix to PHTI's assessment of virtual musculoskeletal solutions, sixteen such cells do not survive that check — and in the clearest of them the missing denominator turns out to be printed fifty pages later.

Recomputed 25 August 2026 · source: PHTI, Virtual Solutions for Musculoskeletal Conditions — Appendices (PDF) · every row quoted below is pdfplumber's layout extraction of the page named beside it, not a text dump

1. The check needs no second source

Most claims about a published number die on access: you need the underlying study, the protocol, the raw file. This one does not. Wherever a table prints a count and a percentage in one cell, K (P%), and prints an n for that arm in the same row, the row makes an arithmetic claim about itself. Either some whole denominator divides K into P%, or none does; and either that denominator is the n beside it, or it is something the table never names.

Run over this appendix, the check finds cells of all three kinds. The controls matter as much as the hits: cells like 12 (12.5%) against n = 96 reconcile exactly, so what follows is not an instrument that reddens everything it touches.

2. The closed loop: a denominator recovered, then found in print

Mecklenburg 2018, chronic low back pain. Two outcomes, one arm:

C | VAS - MCID | 12 weeks | Chronic low back pain | Hinge Health | 113 | NR | 56 (81%) | NR p34, Appendix D-3: Other Pain Scales in Prospective Clinical Trials

C | ODI - MCID | 12 weeks | Chronic low back pain | Hinge Health | 113 | NR | 38 (55%) | NR p40, Appendix E-1: Function and Disability in Interventional Trials

56 of 113 is 49.6%, not 81%. 38 of 113 is 33.6%, not 55%. But the percentages are not arbitrary. Ask which whole denominators print 81% from 56: only 69. Which print 55% from 38: 68, 69 or 70. The two cells intersect at a single number, 69, and it is never printed in either row. Do the same on the control arm, 10 (28%) and 9 (25%) against n = 64, and the denominator lands on 36.

Fifty pages on, in a different appendix, both numbers appear:

ITT | Retention Rate | 12 weeks | Chronic low back pain | Hinge Health | 113 | NR | 69 (61.0)¹ Control | 64 | NR | 36 (56.3)¹ p93, Appendix L-1: Adherence in Interventional Trials

69 of 113 stayed; 36 of 64 stayed. The response rates in sections D-3 and E-1 are over the people who stayed, while the n printed beside them counts the people who started. A reader who divides as the table invites gets 49.6% where the report means 81%.

3. Where the denominator is never printed at all

Wang 2022a is the largest comparison in the set, and there the same two outcomes pin the hidden denominator down just as tightly — but nothing in the report ever names it.

PGIC | 12 weeks | Acute MSK pain | Hinge Health | 262 | NR | 95 (78.5%) | NR Control | 675 | NR | 53 (43.1%) | NR p36, Appendix D-4: Other Pain Scales in Observational Studies

ITT | Various - MCID | 12 weeks | Acute MSK pain | Hinge Health | 262 | NR | 94 (77.7%) | NR Control | 675 | NR | 62 (50.8%) | NR p44, Appendix E-2: Function and Disability in Observational Studies

95 → 78.5% and 94 → 77.7% each admit exactly one whole denominator, and it is the same one: 121. So the intervention column rests on 121 of the 262 people it names — 46%. The control column is worse and is not even self-consistent: 53 → 43.1% requires 123, 62 → 50.8% requires 122, two different denominators for one arm at one visit, and both are about 18% of the 675 printed beside them.

That asymmetry is the finding. 77.7% and 50.8% are set side by side as if they were the same kind of quantity over the arms they name. One is a rate among 46% of its arm, the other among 18% of its arm. Neither percentage of respondents appears anywhere in the appendix, so a reader cannot see the difference, let alone correct for it.

4. One cell has no whole denominator at all

ITT | Various – MCID | 12 weeks | Acute MSK pain | Hinge Health | 75 | NR | 29 (38.9%) | NR Control | 96 | NR | 12 (12.5%) | NR p44, Appendix E-2 (the same cell also appears on p36 as 29 (38.9))

29 out of 75 is 38.7%. For 38.9% you need a denominator between 74.45 and 74.65 — and no whole number lies there, under rounding up, rounding down, truncation or round-half-to-even. There is no population of 29 out of anything that prints 38.9%. The control cell in the very same row-pair, 12 (12.5%) against 96, is exact.

5. Sword Health: same shape, four more cells

C | ODI - MCID | 8 weeks | Chronic low back pain | Sword Health | 70 | NR | 23 (40.4%) | NR Control | 70 | NR | 19 (42.2%) | NR p41, Appendix E-1

Function - MCID | 8 weeks | Chronic shoulder pain | Sword Health | 41 | NR | 24 (61.5%) | NR Control | 41 | NR | 23 (65.7%) | NR p42, Appendix E-1

Both trials print equal arms — 70 and 70, 41 and 41 — and in both the recovered denominators are unequal: 57 against 45, and 39 against 35. The arms were balanced at entry and unbalanced at the point where these percentages were computed, and only the entry numbers are shown.

6. Kaia Health: the denominator is in the report, in another appendix

Drop-out Rate | 12 weeks | Acute low back pain | Kaia Health | 680 | NR | 253 (27.2) Control | 261 | NR | 51 (16.4) p93, Appendix L-1: Adherence in Interventional Trials

Read as printed, Kaia's drop-out rate is 253 of 680 — 37.2%, ten points worse than the 27.2% in the cell. The denominators that do print 27.2% from 253 run 927 to 933, and 51 → 16.4% gives 310 to 312. Appendix M-1 prints the top of each range:

Acute low back pain | Kaia Health¹ | 933 | 42.0 (12.4) | 65% | NR Control | 312 | 37.0 (12.6) | 64% | NR p121, Appendix M-1: Baseline Characteristics in Interventional Trials

933 − 680 = 253 and 312 − 261 = 51 exactly. The drop-out counts are the differences between the randomised and the analysed populations, and the percentages are over the randomised total — while the n column of the drop-out table shows the analysed one. (Both cells round up: 253/933 is 27.12% printed 27.2, and 51/312 is 16.35% printed 16.4 — consistent with the rest of the appendix, which does not hold to one convention. See section 8.) The two numbers a reader needs are in the document. They are ninety pages apart and the row that uses them names neither.

7. Every cell the check flagged

And the controls — cells of the same shape, in the same tables, that reconcile against their own n:

8. A smaller thing: the rounding is not one convention

On p93 alone, 36/64 = 56.25% is printed 56.3, rounded up, while 69/113 = 61.06% is printed 61.0, rounded down. Because of that, the denominators recovered above are taken as the union over four conventions — half up, half to even, truncation and ceiling — which is the weakest and safest claim available. Under any single convention several of the sets get smaller, never larger.

9. The instrument

Enter a cell and the n printed beside it. The tool lists every whole denominator that would print that percentage from that count.

10. What this does not say

It does not say the underlying trials are wrong, or that any vendor's product does not work. Percentages over completers are a normal thing to report; several of the rows above are even labelled C for completer analysis. The defect is in the table: the number that divides is not shown, the number that is shown does not divide, and the two are far enough apart to change the reading. Where a completer count exists in the report at all (section 2, section 6) it sits in a different appendix, under a different heading, with nothing pointing from one to the other.

It also does not rest on my reading of the layout. Each quoted row is the table extraction of the named page; the same script recomputes every figure on this page and fails if any cell, any n, or any recovered denominator moves.

Part of a series on published figures whose denominator is absent, wrong, or somewhere else: PHTI's diabetes assessment · a worked example, free · the reachability tool · index.