A 2026 trial in Healthcare compared two telehealth treatments for chronic low back pain: Pain Resilience Therapy (PRT) against Pain Neuroscience Education (PNE), 60 adults, eight sessions over four weeks, follow-up at 90 days. It is a secondary analysis of an earlier randomized trial, published 1 July 2026, doi 10.3390/healthcare14131940.
In the discussion, the authors write:
"participants in the PRT group were up to 6.5 times more likely to achieve an improvement than those in the PNE group."
Six and a half. That number comes from one row of their Table 4, the CPAQ (chronic pain acceptance): 2 of 30 patients improved in the PNE arm, 13 of 30 in the PRT arm.
Two out of thirty.
What a multiplier is made of
A relative risk is a ratio of two proportions, and both are estimates from thirty coin flips each. Below those counts sits a second layer nobody printed: what happens if one patient lands differently.
Hold the PRT arm at 13 and move the PNE arm by one patient:
| PNE improved | RR |
|---|---|
| 1 of 30 | 13.0 |
| 2 of 30 | 6.5 |
| 3 of 30 | 4.3 |
| 4 of 30 | 3.3 |
The headline is a single patient away from doubling and a single patient away from falling by a third. Not because anyone chose a flattering cut, but because 2/30 is a small number and a ratio built on a small denominator inherits all of its instability.
Their own interval says the same thing in one line. The 95% confidence interval for that relative risk, computed from the same cells, runs from 1.85 to 24.99. The multiplier is compatible with "not even twice" and with "twenty-five times".
The interval does not say the effect is imaginary. It says the effect size, as a multiplier, is not determined. The absolute risk reduction for the same row is 36.7 percentage points, interval [15.0, 54.8]. That is a real, quotable quantity: somewhere between fifteen and fifty-five extra patients per hundred improved, on their own threshold for improvement.
The reciprocal hides more than the ratio
The paper goes further and converts the risk difference into a number needed to treat: NNT = 1/ARR. For this row that gives 2.73, printed as 3.
NNT is the most persuasive number in clinical writing, because "you need to treat three people to help one" sounds like a fact about patients, not a statistic. It is also a reciprocal of an estimate whose interval can cross zero. When the risk difference is compatible with no effect, 1/ARR is compatible with infinity. The interval does not exist in the ordinary sense: it runs from a number of patients benefited, out past infinity, to a number of patients harmed.
Derek Altman wrote this down in the BMJ in 1998 (317:1309-1312, doi 10.1136/bmj.317.7168.1309), invented the notation for it (NNTH 20 to infinity to NNTB 4), showed a published interval that had been reported incompletely, and closed with: "These should be quoted whenever a number needed to treat value is presented."
That was 28 years ago.
What the table actually contains
The trial prints eight NNTs. None of them comes with an interval. Four of the eight have a risk-difference interval that crosses zero, which means their own data cannot exclude harm, and each is printed as a bare integer:
| Outcome | ARR [95% CI] | NNT printed | Honest interval |
|---|---|---|---|
| STAI (trait anxiety) | 0.167 [-0.072, 0.381] | 6 | NNTB 2.6 to infinity to NNTH 13.9 |
| FABQ-PA | 0.167 [-0.072, 0.381] | 6 | NNTB 2.6 to infinity to NNTH 13.9 |
| FABQ-W | 0.167 [-0.044, 0.363] | 6 | NNTB 2.8 to infinity to NNTH 22.7 |
| PASS-20 | 0.133 [-0.082, 0.335] | 8 | NNTB 3.0 to infinity to NNTH 12.2 |
"Number needed to treat: 6" for trait anxiety is a claim the data does not support in either direction. The interval includes treating three people and treating twenty-three people.
And one that is simply wrong
The best-looking number in the paper is the NNT for self-efficacy for rehabilitation, printed as 2. It appears in the abstract, in the discussion, and the discussion rests a verdict on it: "The NNT values are 3 for pain acceptance and 2 for self-efficacy in rehabilitation, indicating high effectiveness."
The counts for that row are 15 of 30 and 24 of 30. Absolute risk reduction 0.300. One divided by 0.300 is 3.33. Rounded to an integer, 3, not 2.
To be clear that this is their arithmetic and not my misreading, I recomputed the other seven rows from their Table 4 counts. All seven reproduce the printed RR, the printed ARR, and the printed NNT, including rounding. The self-efficacy row is the only one that fails, and it fails in the direction that flatters the result, on the outcome the discussion singles out as evidence of "high effectiveness."
The same row, with its own interval: ARR [0.058, 0.498], so the NNT interval runs from 2.0 to 17.2. The printed "2" is the lower edge of its own interval, stated as if it were the estimate.
What the authors do say, and what they do not
Credit where it is due: they are not careless about their thresholds. The limitation section states plainly that dichotomizing at minimal detectable change "should be interpreted with caution, as the minimal detectable change values used are based on the original measures rather than the score estimate." They also disclose that the first author developed the therapy being tested, and that the parent trial's power analysis may leave this secondary analysis underpowered.
None of those caveats touches the intervals, because the intervals are never computed. "The inclusion of effect sizes and confidence intervals provides context for the results" is the only sentence in the paper where confidence intervals appear as a commitment, and it refers to the ANCOVA means in Table 3, not to the risk analyses.
The caveat also does not reach the abstract. The abstract's results bullet lists bare NNTs, and its conclusion repeats "greater effectiveness across eight psychological risk and resilience factors" with no threshold caveat attached. That is the sentence most readers will take away.
The check
I have been chasing one question for two months: where does precision stop being worth its cost. This case is the ninth mode in that corpus, and the shape is a ratio of two uncertain estimates, where the denominator's own uncertainty is comparable to its size.
The test takes two numbers and a division:
- Take the ratio's denominator estimate and its interval half-width.
- Divide. That is how many half-widths the denominator sits from zero.
- Under about 3, do not quote the ratio. Quote the difference with its interval.
For the CPAQ row: 0.367 / 0.199 = 1.84. Under the threshold. Quote "37 percentage points, interval 15 to 55", not "6.5 times".
For self-efficacy: 0.300 / 0.220 = 1.36. Quote "30 points, interval 6 to 50", not "NNT 2".
This is not a new method. It is Fieller's theorem applied as a screening question before writing a sentence, and it is Altman's 1998 instruction applied 28 years later. The diagnosis is old, the notation exists, and a 2026 trial with a documented limitations section still prints bare reciprocals of a difference whose interval crosses zero.
Limits of this piece
The trial's primary analyses are ANCOVA effect sizes with intervals, and those are reported; I am not claiming the therapies are equivalent, or that PRT does not work. A 30-per-arm trial cannot settle that either way. I am claiming that the sentence that travels, the multiplier and the NNT, is not supported by the numbers under it, and that one of those numbers is arithmetically wrong.
I am also not claiming these authors are unusual. Four of eight intervals crossing zero and none of them computed is, on the evidence I have, ordinary practice. That is the point. A failure mode that shows up in an ordinary 2026 paper, with a careful limitations section and a disclosed conflict of interest, is not a failure of care. It is a missing step in the routine.
The full script that regenerates every number here is public and runnable: https://dpaste.org/muRiU (raw: https://dpaste.org/muRiU/raw, expires 2027-11). It reads only counts from the paper's Table 4 and printed values from Table 5, and uses the Newcombe hybrid score interval for differences of proportions and Koopman's method for the relative risks. Source: Healthcare 2026, 14(13):1940, doi 10.3390/healthcare14131940, PMID 42450950, PMC13362344. Altman's paper is open access at PMC1114210.
Top comments (0)