Application of Bayesian analysis to the doubly labelled water method for total energy expenditure in humans

Rationale The doubly labelled water (DLW) method is the reference method for the estimation of free‐living total energy expenditure (TEE). In this method, where both 2H and 18O are employed, different approaches have been adopted to deal with the non‐conformity observed regarding the distribution space for the labels being non‐coincident with total body water. However, the method adopted can have a significant effect on the estimated TEE. Methods We proposed a Bayesian reasoning approach to modify an assumed prior distribution for the space ratio using experimental data to derive the TEE. A Bayesian hierarchical approach was also investigated. The dataset was obtained from 59 adults (37 women) who underwent a DLW experiment during which the 2H and 18O enrichments were measured using isotope ratio mass spectrometry (IRMS). Results TEE was estimated at 9925 (9106‐11236) [median and interquartile range], 9646 (9167–10540), and 9,638 (9220–10340) kJ·day−1 for women and at 13961 (12851–15347), 13353 (12651–15088) and 13211 (12653–14238) kJ·day−1 for men, using normalized non‐Bayesian, independent Bayesian and hierarchical Bayesian approaches, respectively. A comparison of hierarchical Bayesian with normalized non‐Bayesian methods indicated a marked difference in behaviour between genders. The median difference was −287 kJ·day−1 for women, and −750 kJ·day−1 for men. In men there is an appreciable compression of the TEE distribution obtained from the hierarchical model compared with the normalized non‐Bayesian methods (range of TEE 11234–15431 kJ·day−1 vs 10786–18221 kJ·day−1). An analogous, yet smaller, compression is seen in women (7081–12287 kJ·day−1 vs 6989–13775 kJ·day−1). Conclusions The Bayesian analysis is an appealing method to estimate TEE during DLW experiments. The principal advantages over those obtained using the classical least‐squares method is the generation of potentially more useful estimates of TEE, and improved handling of outliers and missing data scenarios, particularly if a hierarchical model is used.

Although at least three important works describing the principles and practices of the DLW method, striving to promote universal consistency, have been produced, there is still some non-uniformity in the calculations adopted by workers at different laboratories. This is particularly the case for corrections for fractionation (Assumptions 1 to 3 for space ratios and Assumption 5 are discussed by Coward and Cole 6 ). The major difficulty in dealing with fractionation is the estimation of the proportion of water that undergoes phase change (from liquid to vapour) before being lost from the body. This is to some extent dependent on the environment of subjects and their physical activity and this needs to be considered within a given experimental paradigm.
The approach adopted for the space ratio, however, is less open to modulation by the experimental environment. When body water is estimated from an isotope dilution experiment, the value obtained is an over-estimate by a factor of approximately 4% if 2 H is used or about 1% when 18 O is employed. For 2 H, this is attributed to the exchange with labile hydrogen atoms, principally from proteins and lipids. 7 The 18 O pool size exceeds that of the body water pool, not only because of the exchange with dissolved CO 2 and bicarbonate, 8 which is fundamental to the principle of the DLW method, but also because of exchange with bone mineral and other deep pools. The practical consequence of this is that neither the accessible 2 H nor the 18 O volumes of distribution (pools) are coincident with the total body water, and furthermore there is a measurable difference between the apparent volumes into which the two isotopes are distributed.
These issues are discussed by Coward 9 alongside the recommendation of Schoeller et al., 10 that a fixed ratio of 1.03 (later revised to 1.034 11 and further to 1.036 11 ) between the 2 H space (N H ) and the 18 Speakman 12 discusses comprehensively the correctness of this approach, with the tentative conclusion that in humans the fixed ratio approach should be used, but with a modified coefficient derived from the mean experimental ratio found for the given sub-population under study.
The International Atomic Energy Agency (IAEA) 13

| Non-Bayesian equations for R CO 2 determination
Non-normalized R CO2 has been calculated using the equation of where k and N refer to the rate constant and pool size, respectively, with subscripts to indicate the isotope.
However, normalized R CO2 has been calculated using the equation of Schoeller et al: 10 where k refers to the rate constant, N refers to the normalized pool size which fixes the space ratio at 1.03, and the subscripts indicate the isotope.
The fractionation factors f 1 , f 2 and f 3 are given as 0.941, 0.991 and 1.037, respectively.

| Re-parameterization of the DLW equations
For ease of model specification in the Bayesian environment, the first step is to re-cast the DLW equations such that the observed mass spectrometric enrichments are expressed in terms of the parameters of physiological relevance. Since it is assumed that for all stable isotopes employed, including 2 H and 18 O, elimination from the body follows first-order kinetics, the expression for the MS-derived enrichment of each isotope at time t is of the form: In deriving Equation 5, it is assumed that the usual method of combining the δ-values (‰) of the sample of body water, δ(t) with the basal (pre-dose) value δ b , and that of a diluted sample of the dose, δ dd , made by adding d grams to a quantity T, of naturally abundant water of known enrichment δ T is used. The actual dose administered to the subject is D grams, the isotope space is denoted N (mol), and the fractional rate constant of elimination labelled as k (day −1 ).
The DLW technique combines the data from the two isotopes 2 H and 18 O to derive essentially four parameters: the CO 2 product (R):* The water turnover, R W and the fraction of body fat F where the subject's body weight is W, and α 1 , α 2 , β 1 , β 2 , γ 1 and γ 2 are constants that depend upon the fractionation model employed. The parameter F is not necessary to calculate the TEE in this model, but its inclusion allows a further useful outcome from the 2 H dataset.
In this instance, we assumed a common respiratory quotient, RQ = 0.85 for all subjects, and therefore, TEE bears a constant ratio to R CO2 with a constant of proportionality equal to 532.
Using simple algebra (see section S1, supporting information): The derivation of Equations 11-14 allows values to be sampled from prior distributions of the physiologically relevant parameters to make predictions of the observed kinetics. This therefore allows the generation of Bayesian estimates of the model parameters that derive TEE.

| Choice of priors
In Bayesian analysis, the choice of priors for the physiological parameters of interest is of paramount importance; for the DLW model described here, vague (non-informative) priors have been adopted for the parameters R CO2 ; R w and F, These priors allow the iterations to adopt values for these parameters that are almost entirely data driven. Note that a slightly different approach is used for the space ratio S. According to our prior knowledge, we suggested that S had a prior distribution that was normal, with a mean For the additional hierarchical analysis (see section S2, supporting information), hyper-parameters (population parameters) adopt these distributions with the individual parameters drawn from them and associated with normal distributions: The between-subject variance for the space ratio again reflects the richness of prior information for this variable.  When isotope data for a cohort of 37 women and 22 men was analyzed using classical least-squares methods, the space ratio was found to vary between 1.010 and 1.069, with the majority falling in the 'acceptable range' judged by the criterion of Prentice. 9 Using

| RESULTS
Coward's analysis, the mean estimates of TEE obtained using  The effects of normalization on the individual estimates of TEE are shown in Figure 1. From Figure 2 it is apparent that normalizing the space ratio decreases the estimated TEE if the natural space ratio is less than the target normalization, whilst the TEE is increased if the space ratio is more than the target normalization. Furthermore, this effect is highly linear.
When the independent Bayesian method was applied, as expected, the continuous distribution of individual median estimates of space ratios decreased ( Table 2). A further reduction in the width of the distribution is achieved by specifying a hierarchical model ( Figure 3). Since the distribution of space ratios was not found to be gender-specific (Figure 2), the results for the women and men have been combined in Figure 3. The posterior distributions are drawn in

| Independent Bayesian model
Independent Bayesian modelling of TEE gives results that correlate highly with those from normalized non-Bayesian methods (overall r 2 = 0.96). This is expected as they are conditional on the data.
However, an informative comparison is obtained from a Bland-Altman plot. 25 These results indicate that, overall, there is little difference between the two methods. However, for some individuals the discrepancy between normalized non-Bayesian and independent Bayesian methods is not insignificant. The limits of agreement between the normalized non-Bayesian and Bayesian methods are much wider for men (−1296 to +1367 kJ·day −1 ) than for women (−750 to +660 kJ·day −1 ). Although an analogous compression is seen for the women, it is not of such magnitude (

| DISCUSSION
In this study we used a Bayesian approach for the estimation of human total energy expenditure (TEE) using doubly labelled water data obtained from 59 participants after incorporating prior information of the space ratio parameter.
In the analysis of methods used to deriveTEE from the isotope data, we have shown that R CO2 is a linear function of the differences in the isotope effluxes, regardless of the model used. Furthermore, since TEE is taken as proportional to R CO2 (i.e. the ratio of macronutrients oxidized is taken to be the same for all subjects), a similar linear relationship must also hold for energy expenditure. Therefore, we write: when the natural spaces are used. Similarly, normalization of the spaces leads to a relationship: where the prime denotes the normalization process, which may be summarized by: Therefore, we expect normalization to change the estimated TEE according to: In this expression, the term in square brackets can be regarded as an approximation to the average of the two isotope fluxes, which will be roughly invariant in any population.
On this basis, it might be expected that the application of a Bayesian analysis would produce estimates of TEE midway between those obtained from the natural and normalized methods. However, it must be borne in mind that the usual method of analyzing the disappearance curves uses logarithmic transformation followed by linear least-squares methods, whereas the formulation that we have used for the Bayesian analysis fits the curves in their exponential form. The question of whether logarithmic transformation is appropriate has been discussed previously, and it has been noted that the correct choice of data pre-treatment depends upon the error structure of the data, 26 which is determined by the balance between biological variation and analytical performance. Since the Bayesian approach generates posterior distributions for the fitted data points it is indeed richer in information than the least-squares method. This is illustrated in A second noteworthy point illustrated in Figure 6  When a Bayesian hierarchical method was used, even when split across men and women, the spread of space ratios was further reduced. In particular, the lower bound is pushed upwards (Figure 3). In principle, a Bayesian method is a stochastic approach where the parameter of interest has an assumed probability distribution (prior) which is updated by the observed dataset to generate the parameter's posterior distribution. If in an extreme case where the measurement error is zero and the underlying mechanistic model is true, the prior information about the parameter will be considered of zero weight. As such, the model will fit perfectly into the dataset and thus the estimation of the parameter (for example TEE) will be an error-free value resulting in the same estimation as if a leastsquares method is used. In any other case, the prior information used in the Bayesian method will play a role in the estimation of the posterior distribution and if the prior information is valid this will increase the estimation accuracy. In effect, when the laboratory precision is limited, the use of Bayesian methods could improve the estimations of TEE to that of a laboratory in which a high level of instrument precision is observed. Where multi-subject datasets are available, a hierarchical model can be further applied that results in an even more precise estimation of TEE.

| CONCLUSIONS
Bayesian analysis is an appealing approach to estimate population and individual total energy expenditure with the doubly labelled water method. The method offers a valuable approach to deal with outliers and missing data and gives a smaller unbiased estimate on the population dispersion, particularly if a hierarchical model is used. ORCID Michelle Venables http://orcid.org/0000-0002-9380-0060