Before addressing this question, the notion of “unaccounted energy” introduced by Hall et al. warrants examination. In the reanalysis, they compared reported energy intake and expenditure in our study, identifying discrepancies as violating the “physical law of energy conservation.” But this treatment disregards substantial cumulative error arising from measurement of the various components of energy balance, each with recognized imprecision and temporal variation. In the recent study by Hall et al. of ultra-processed food [17], conducted in the optimal environment of a metabolic ward, mean energy discrepancy on one diet was large (382 kcal/day) and “unaccounted energy” exceeded 250 kcal/day for most participants (Fig. 1c).

Recognizing this inherent variability and imprecision in the measurement of both energy intake and TEE, we can see why exclusions involving their difference or ratio (as in Fig. 1b of Hall et al.) would produce highly misleading results. Whereas individuals with low energy intake relative to TEE might have been nonadherent (i.e., unobserved food intake), they would also tend to be at the upper end of the natural distribution for TEE (related to true biological differences or randomly distributed measurement variation). Therefore, eliminating them would deplete the cohort of those with the greatest TEE denominator, deflating the diet effect.

We can demonstrate this phenomenon in three ways. First, we conducted the converse analysis, sequentially eliminating individuals with “unaccounted energy” arising from low TEE to energy intake (here, TEE resides in the numerator). As illustrated in Fig. 1d, the diet effect now increases with the progressive threshold because individuals at the lower end of the TEE distribution are eliminated, leaving a residual cohort enriched for hyper-responders. However, these models, involving postrandomization variables inextricably linked to the outcome, violate a basic principle of statistical inference and should be discarded as fatally flawed.

Second, we divided the per protocol (weight stable) group into tertiles, based on the ratio of energy intake to TEE (Fig. 1e). In an unadjusted model, those in the lowest tertile (i.e., those eliminated in the analysis of Hall et al.) demonstrated a substantially larger diet effect. However, they were also more likely than those in the other two tertiles to have a baseline TEE above the median (OR 2.7 [95% CI 1.2–6.1], p = 0.02). With adjustment for baseline TEE and other relevant covariates, the differences between the tertiles for diet effect diminished.

Hall et al. modeled CO 2 production (rCO 2 ) in our cohort to circumvent the need for respiratory quotient (RQ), deviating from well-established DLW methodology and introducing severe bias against the low-carbohydrate diet. Because food quotient (FQ) equals RQ during weight (and body composition) stability, as applies to our per protocol group, a third approach is to conduct a sensitivity analysis examining how varying degrees of nonadherence would affect FQ and thereby TEE. As shown in Table 1, the low- versus high-carbohydrate diet comparison remained statistically significant through 50% nonadherence. Of particular interest, the diet effect relative to carbohydrate proportion remained remarkably stable throughout the range of assumed nonadherence, and consistently above the hypothesized 50 kcal/day for every 10% decrease in the proportion of energy as carbohydrate [3]. Moreover, among participants in the lowest tertile of energy intake to TEE (for whom estimates of FQ may be least accurate), the unadjusted change in rCO 2 was itself significantly greater on the low- versus high-carbohydrate diet (10.3 vs −47.0 L/day, p = 0.01). That is, the diet effect on TEE in this subgroup was so large as to require no assumptions about FQ, providing further evidence against nonadherence as an explanation for study findings.

Table 1 Sensitivity analysis of TEE. This analysis examines how potential nonadherence could influence the diet effect on TEE in the per protocol (weight stable) group, considering how FQ (used in DLW methodology) changes with macronutrient ratio Full size table

As stated in the BMJ article, our preliminary estimates of energy intake, used by Hall et al., “would tend to selectively underestimate those with high energy expenditure” and were not intended to be definitive. With more precise and accurate data [10], we found that energy requirements for weight stability (i.e., by calorie titration) showed a similar magnitude of effect (≈200–300 kcal/day) and hierarchical order (low > moderate > high carbohydrate) among diets as TEE, as predicted by the Carbohydrate-Insulin Model. Due to imprecision involved in these (and all) methods for determining outpatient energy intake and expenditure, the magnitude of effect should be interpreted cautiously.