Seventy-one point one of what?

PHTI’s diabetes assessment tells purchasers that 71.1% of users reached HbA1c below 6.5%. That number cannot be a share of the users. Exact arithmetic rules out the enrolled cohort under every rounding convention — and the paper PHTI cites for it never prints 71.1% at all.

Recomputed 24 August 2026 · sources re-pulled the same day · every quote below is a verbatim substring of the document named beside it

1. The sentence

“As noted above, this nonrandomized, interventional, intention-to-treat study with one- and two-year follow-ups demonstrated significant HbA1c reductions of more than twice the clinically meaningful threshold. The between-group effect size was a reduction of 1.2% pt⁷⁹ HbA1c after two years. More importantly, after two years, 71.1% of users achieved HbA1c levels below the 6.5% threshold for diabetes, either taking no diabetes drugs or only metformin.⁸⁰” PHTI, Digital Diabetes Management Solutions, report v1.1, page 29. Footnote 80 → Athinarayanan et al., “Long-Term Effects of a Novel Continuous Remote Care Intervention…: A 2-Year Non-randomized Clinical Trial,” Front Endocrinol 10 (2019): 348.

One sentence, two claims: the study is intention-to-treat, and the rate is of users. Both point at the same population — everyone who enrolled. The trial enrolled 262 people in the continuous-care arm, and by two years:

“One hundred ninety four participants (74% of 262) remained enrolled in the CCI at 2 years…” Athinarayanan 2019, Results — Retention and Long-Term Dietary Adherence.

The authors themselves are careful about which population a rate belongs to. Describing the one-year result from the same cohort, they write it out explicitly:

“Eighty-three percent of CCI participants remained enrolled at 1 year and 60% of completers achieved an HbA1c <6.5% while prescribed metformin or no diabetes medication.” Athinarayanan 2019, Introduction.

“Of completers.” Two words, and the reader knows exactly what the 60% is over. PHTI’s sentence reports the two-year version of that same endpoint and writes “of users” — in a sentence that has just called the study intention-to-treat.

2. The cited paper does not contain the number

The full text of Athinarayanan 2019 is open access (PMC6561315). The string 71.1 occurs in it zero times. What the paper does print for that endpoint, at two years, is this:

“All within-group changes and between-group differences in diabetes status among the CCI and UC group participants appear in Supplementary Table 4 (intent-to-treat analyses were conducted, all below ns = 262). The proportion of participants meeting the defined criteria for diabetes reversal at 2 years increased to 53.5% from baseline in the CCI group, whereas no change was observed in the UC group.” Athinarayanan 2019, Results — Diabetes Status.

And, two paragraphs on, the authors do the honest thing themselves — they state the rate over everyone who enrolled, restricted to people who actually finished and actually had a lab drawn:

“The percentage of all CCI enrollees (N = 262) with verified reversal and remission requiring both completion of 2 years of the trial and an obtained laboratory value for HbA1c were 37.8 and 14.9%, respectively.” Athinarayanan 2019, Discussion.

3. Arithmetic settles which population 71.1% belongs to

A rate printed to one decimal place constrains its own denominator. For a whole count k out of n, only some n can produce “71.1”. Below, every integer k from 0 to n is tested under all three rounding conventions in use — round-half-up, round-half-even (banker’s), and truncation — because each convention covers a different set and a flat “impossible” would overstate the case.

The two denominators that work are the two survivor sets. The one that does not work is the one the sentence names. 71.1% is a rate over the people who stayed, printed in a sentence that calls it a rate over the people who started.

The likeliest reading is 138 of the 194 two-year completers — the exact two-year counterpart of the paper’s own “60% of completers” at one year. That reading cannot be confirmed without Supplementary Table 3, and this page does not assert it. It does not need to: whichever survivor set it came from, 262 is arithmetically excluded.

4. What the honest number is

For the same endpoint, at the same two-year mark, over the same trial, the paper offers a figure computed over all 262 enrollees and requiring a real lab value: 37.8% (99 / 262 — the only whole count that produces it). PHTI printed 71.1%.

A purchaser reading the assessment sees 71.1%. A purchaser reading the underlying paper sees 37.8%. The two describe the same intervention at the same time point. The distance between them is 1.9×, and all of it lives in the denominator.

What this page does not claim

Three limits, stated because leaving them out would make this page the same kind of document it is criticising:

• 53.5% is not a defect. It is a modelled estimate with multiple imputation. A modelled estimate has no whole numerator, so denominator arithmetic does not apply to it and none is attempted here. It is reported above only as the paper’s own ITT figure.
• Which survivor set 71.1% came from is not established. Both 155/218 and 138/194 produce it exactly. The paper’s completers-only results sit in Supplementary Table 3, which was not retrieved. What is established, and needs no supplement, is that 262 cannot produce it.
• No claim is made about the intervention itself. Whether nutritional ketosis works is a clinical question. This is a question about a printed fraction.

5. The same page, a second time: “83%”

“Several articles support the durability of Virta’s effects among the 83% of study participants who completed the program.” PHTI, Digital Diabetes Management Solutions, page 29.

83% is the one-year retention figure — “Eighty-three percent of CCI participants remained enrolled at 1 year,” in the paper’s own words. It is being used to support a claim about durability, which is a multi-year property. The same paper states the two-year figure two sections later: “One hundred ninety four participants (74% of 262) remained enrolled in the CCI at 2 years.” The five-year extension, published in October 2024, carries its outcomes on a denominator of 120 — 45.8% of the people who started.

Three retention numbers exist for this cohort: 83%, 74%, 46%. The assessment quotes the earliest and largest, and attaches it to the longest-range claim.

6. Downstream: the vendor’s own page repeats the shape

One paragraph on virtahealth.com/research, re-pulled 24 August 2026, carries five percentages and not one whole number:

“Results showed that 20% of participants achieved diabetes remission, with 15.8% maintaining remission at three years and 12.5% at five years. Among those who completed the five-year follow-up, 61.3% sustained at least 5% weight loss, while 39.5% maintained a 10% weight loss.” virtahealth.com/research, retrieved 2026-08-24.

The published abstract states the retention: “122 remained in the study for five years.” Run each rate against that number and the paragraph turns out to be standing on three different bases at once:

The counts 19, 15 and 39 are not guesses — they are the counts the abstract itself prints (“15.8 % (n = 19)”, “12.5 % (n = 15)”, “32.5 % (n = 39)”), and each is exact over 120, not over 122. Two people appear in the retention headline and not in the outcome denominator; three are missing from the weight denominator. Small numbers — but they mean the paragraph has three denominators and prints none of them.

The larger gap is the word. The abstract says “of the five-year completers”; the website says “of participants”. There were 262 participants. Twenty percent of the completers is 24 people, and 24 out of 262 is 9.2%.

7. Why this is worth an hour of anyone’s time

PHTI exists so that purchasers do not have to take vendor arithmetic on trust. That makes its own arithmetic load-bearing in a way the vendors’ is not: a vendor’s inflated rate is a claim, the same rate inside an independent assessment is a verified claim. The defect here is small, specific, and entirely fixable — one sentence on page 29, and a rule that catches the next one.

The rule is mechanical enough to run over a whole report: a rate printed to d decimals admits only certain denominators. Where the admissible set excludes the population the sentence names, the sentence is wrong before anyone argues about methodology. No access to the vendor is needed, no re-analysis, no cooperation — only the published number.