Network Harmonization of Physical Activity Variables Through Indirect Validation

Harmonization of data for pooled analysis relies on the principle of inferential equivalence between variables from different sources. Ideally, this is achieved using models of the direct relationship with gold standard criterion measures, but the necessary validation study data are often unavailable. This study examines an alternative method of network harmonization using indirect models. Starting methods were self-report or accelerometry, from which we derived indirect models of relationships with doubly labelled water (DLW)-based physical activity energy expenditure (PAEE) using sets of two bridge equations via one of three intermediate measures. Coef ﬁ cients and performance of indirect models were compared to corresponding direct models (linear regression of DLW-based PAEE on starting methods). Indirect model beta coef ﬁ cients were attenuated compared to direct model betas (10% – 63%), narrowing the range of PAEE values; attenuation was greater when bridge equations were weak. Directly and indirectly harmonized models had similar error variance but most indirectly derived values were biased at group-level. Correlations with DLW-based PAEE were identical after harmonization using continuous linear but not categorical models. Wrist acceleration harmonized to DLW-based PAEE via combined accelerometry and heart rate sensing had the lowest error variance (24.5%) and non-signi ﬁ cant mean bias 0.9 (95%CI: − 1.6; 3.4) kJ·day − 1 ·kg − 1 . Associations between PAEE and BMI were similar for directly and indirectly harmonized values, but most fell outside the con ﬁ dence interval of the criterion PAEE-to-BMI association. Indirect models can be used for harmonization. Performance depends on the measurement properties of original data, variance explained by available bridge equations, and similarity of population characteristics.

Harmonization of exposure and outcome variables is an essential step when integrating different sources of data for the same analysis, such as in meta-analysis of published results, pooled or federated meta-analysis of individual-level data, and global surveillance of risk factors for disease. Analyses of this nature which use information from multiple sources are often constrained by the quality and compatibility of the original data (Fortier et al., 2010). Harmonization aims to bring together various types and levels of data which represent the same underlying construct (e.g., physical activity, energy intake, body fat percentage, etc.) in order to achieve compatibility when methods vary between studies or study phases (Granda & Blasczyk, 2011). The process does not strictly require that precisely the same original collection and processing methods are employed in each study (Fortier et al., 2010), but the harmonized data should be "inferentially equivalent" (i.e., their format, function, and meaning are the same) (Atkin et al., 2017). This inferential equivalence will depend upon the scientific context and the type of analysis being undertaken.
A common harmonization approach is conversion to the level of the least detailed information, for example transformation of continuous data to a binary categorization of low vs. high physical activity level (Kilpelainen et al., 2011). However, this approach loses the resolution of the more detailed data, and may therefore limit the power and scope of subsequent analyses. It is also unclear how well variables harmonized in this way relate to the latent true value of the exposure. An alternative approach to harmonization is to restrict analyses to only those studies which have assessed and expressed the exposure and outcome in the desired way. This maintains the detail of the contributing data, but-as highlighted by Aune, Norat, Leitzmann, Tonstad, and Vatten (2015)-greatly reduces the proportion of the available data that can be included in evidence synthesis. At best, this leads to loss of power. At worst, this leads to bias if the studies that are included with optimal data have specific characteristics.
Another approach to harmonization is to use validation studies which report the statistical (e.g., regression) models of relationships between values from the less precise methods and the latent true level of exposure, as assessed by a construct-specific gold-standard criterion method. A direct model permits transformation of original data to the desired harmonized format. The problem is that this mapping approach is often not possible because the ideal validation study employing gold-standard criterion methods has either not been conducted, does not report the necessary model, or is not applicable to the population or setting in question. This limitation may be more common in particular populations or settings, such as those in which the feasibility or cost of gold (or even silver) standard methods is prohibitive. Consequently, some populations and settings may be studied with unsatisfactorily harmonized data or excluded from analyses altogether.
When the ideal direct model is unavailable, a potential solution may be to use a combination of models in the same network, whereby estimates from the less precise but more feasible field method are mapped to a criterion using "bridge equations" which form an indirect route (Figure 1). This concept is analogous to network meta-analysis in which multiple comparisons can be inferred despite not being directly tested (Lu & Ades, 2004). "Network harmonization" adopts similar principles in that it utilizes existing, ideally published, validity data to derive a new indirect model without the requirement to conduct de novo fieldwork. By utilizing a combination of published bridge equations and existing datasets, this study examines such an approach by comparing the inferential equivalence of data harmonized to a gold standard format via both direct and indirect models.

Study Outline
We present a hypothetical analysis task requiring harmonization of, at-first-glance, incompatible starting point values to the format of values arising from a target criterion. One could use a direct model between the two sets of data to complete this task, but as mentioned above often an alternative approach is needed. Here, we use intermediary values from a third method, for which separate links (bridge equations) to the starting values and target criterion are available. The regression models between starting point values and the intermediate (Bridge Equation 1) and between the intermediate and the target criterion (Bridge Equation 2) are used to derive the indirect model as outlined in Figure 1. In order to assess the validity of this network harmonization approach, the indirect model is compared to the direct model.

Description of Data
Here, we use the example of harmonizing four distinct sets of starting point values to the target of total daily physical activity energy expenditure (PAEE) expressed in kJ·day −1 ·kg −1 as measured using the gold standard criterion. Although the gold standard assessment method for PAEE has been used in the present study, the following methods are applicable to other target variables (e.g., moderate-intensity activity) with their own gold standard, or to target variables assessed by methods which are not of gold standard. For simplicity, we will use linear models to describe the links between data.
Target Criterion Values. The gold-standard target criterion for assessing PAEE (kJ·day −1 ·kg −1 ) was the difference between total and resting energy expenditure as measured by the DLW method and two lab-based assessments of resting metabolic rate, coupled with allowance for the diet-induced thermogenic effect.

Bridge Equations
To examine different aspects of the performance of network harmonization using indirect models, one of seven variations on Bridge Equation 1 was combined with one of three variations on Bridge Equation 2. In addition, we examine the performance of metaanalyzing multiple indirect routes.
If available, we used published bridge equations. If relevant equations were unavailable but correlation coefficients and basic (mean and SD) summary statistics were, we derived the equations through back-transformation of standardized coefficients using the corr2data STATA command. If this was not possible, we used individual-level data from existing datasets to derive equations (pretending these were published validation studies), and subsequently used them alongside existing bridge equations sourced from published work.
Bridge Equation 1. Five variations on Bridge Equation 1 were derived from the Fenland Study, a population-based cohort study of 12,435 adults born between 1950 and 1975 and registered with general practices in Cambridgeshire, United Kingdom. We randomly split this dataset into five subsamples to represent five independent validation studies. Participants attended our research facility and completed RPAQ (Besson et al., 2010) and underwent treadmill testing for individual calibration (Brage et al., 2007) whilst fitted with a chest-worn combined heart rate and movement sensor (Actiheart, CamNtech Ltd, Papworth, UK) (Brage et al., 2005). At the end of the clinical assessment, they were instructed to wear this device continuously for six days and nights and carry on with their normal behaviors. Data from RPAQ were used to derive duration (minutes per day) of MVPA by summing duration reported participating in activities with intensity > 3.0 METs (Ainsworth et al., 2011), estimates of PAEE calculated as frequency × duration × intensity (Besson et al., 2010), and the four-level Cambridge Index (Golubic et al., 2014;Wareham et al., 2003). The ACC TRUNK signal from the combined heart and movement sensor was used in the format of mean daily trunk acceleration in m·s −2 , while the HR signal together with treadmill test data was used to derive an individually calibrated estimate of PAEE as previously described (Brage et al., 2007(Brage et al., , 2015. The two signals were also combined (ACCHR) to predict PAEE using branched equation modelling (Brage et al., 2004). The Fenland Study was approved by the Health Research Authority National Research Ethics Service Committee East of England-Cambridge Central, and participants provided written informed consent. Five variations on Bridge Equation 1 using self-report as the starting point were derived from the linear regression of: 1) ACC TRUNK on RPAQ MVPA; (2) HR PAEE on RPAQ MVPA; (3) ACCHR PAEE on RPAQ MVPA; (4) ACCHR PAEE on RPAQ PAEE; (5a) ACCHR PAEE on RPAQ Cambridge Index. To examine whether indirect harmonization is robust to variations in measurement protocol and population (i.e., a potentially non-ideal bridge equation), we used an additional linear regression of (5b) ACCHR PAEE on RPAQ Cambridge Index. This was derived from the similar short European Prospective Investigation into Cancer and Nutrition Study Physical Activity Questionnaire (short EPIC-PAQ) administered in the EPIC cohort across 10 European countries (Peters et al., 2012;Wareham et al., 2003) denoted by ACCHR EUROPE .
To contrast the harmonization process of self-report data as starting points with that of objective data as the starting point, two additional AC bridge equations derived from the linear regression of: (6) ACC TRUNK on ACC WRIST ; and (7) ACCHR PAEE on ACC WRIST , were obtained from published work (White, Westgate, Wareham, & Brage, 2016).
Bridge Equation 2. Three bridge equations were obtained from published data (Brage et al., 2015) using the linear regression of DLW-based total daily PAEE expressed in kJ·day −1 ·kg −1 on: (1) ACC TRUNK ; (2) HR PAEE; and (3) ACCHR PAEE, in doing so linking to the prediction output from a Bridge Equation 1 described above.

Direct Model
The inferential equivalence of harmonized PAEE values was assessed using gold standard DLW-based PAEE values from the UK Biobank Validation Study (BBVS) reported in detail elsewhere (White et al., 2019). Briefly, rate of carbon dioxide production (rCO 2 ) was measured using the 10-day DLW method of Schoeller et al. (1986) and converted to total energy expenditure (TEE) using the energy equivalents of CO 2 of Elia and Livesey (1988) in 100 participants. Resting metabolic rate (RMR) was measured on two separate days during clinic visits with a fifteen-minute rest test by respired gas analysis (OxyconPro, Jaeger, Germany), and scaled by a factor of 0.94 to account for RMR measurements being conducted in the afternoon rather than the morning (Haugen, Melanson, Tran, Kearney, & Hill, 2003). The closest measurement value (visit 1, visit 2, or their mean) by proximity to the withinperson median of predictions of RMR using three equations (Henry, 2005;Nielsen et al., 2000;Watson et al., 2014) was used in analysis. Total daily REE was calculated, with an additional adjustment of sleeping metabolic rate being 5% lower than awake resting metabolic rate (Goldberg, Prentice, Davies, & Murgatroyd, 1988). Dietinduced thermogenesis was estimated using macronutrient intake assessed by food frequency questionnaire as previously described (Brage et al., 2015;Jequier, 2002). The REE and diet-induced thermogenesis were subtracted from TEE and divided by body mass yielding an estimate of total daily PAEE in kJ·day −1 ·kg −1 .
The four variations on the starting point values described above were replicated in BBVS so that four corresponding direct models predicting the target criterion DLW PAEE could be derived. Participants completed RPAQ and the raw data were used to derive duration of MVPA, PAEE, and the four-level Cambridge Index. In addition, participants were fitted with a tri-axial accelerometer (AX3, Axivity, Newcastle, UK) on the wrist for 9 days and nights whilst continuing with their usual activities. The ACC WRIST signal was used to approximate acceleration as a result of human movement and expressed in milli-g (van Hees et al., 2013) Values + Alpha 1 ) + Alpha 2 . Formula 3 simplifies to give the following formulae for deriving the new coefficients: These formulae provide parameter estimates for the indirect model coefficients but not their standard errors. To propagate the uncertainty in the parameter estimates from Bridge Equation 1 and Bridge Equation 2 to the new indirect model, 10,000 values of each parameter were sampled from a normal distribution with mean equal to the observed parameter estimate and standard deviation equal to the standard error of that parameter estimate; the formulae above were then applied to the sampled values. The means and standard deviations of the resulting distributions for Beta Indirect and Alpha Indirect were used as the coefficient point estimates and standard errors, respectively. For the indirect model using the categorical Cambridge Index derived from RPAQ, the categorical data were replaced by one constant and three dummy variables to represent four levels. The above steps were then applied in the same way, but repeated for each of the four values of Beta Indirect .
We meta-analyzed the newly derived beta and alpha coefficients from each of three indirect models predicting PAEE from duration of MVPA, thus generating a fifth indirect combined prediction equation using all available information; this represents the scenario where harmonization is performed using multiple validation studies of the same instrument.

Analysis
The inferential equivalence of each permutation of the indirect model was assessed alongside an equivalent direct model derived from the linear regression of target criterion PAEE data on corresponding starting point data available for 100 participants in the BBVS. PAEE was predicted from four types of starting point data using direct and indirect models and compared with values from the observed criterion PAEE (i.e., the "true" PAEE exposure using the DLW method) by calculating the mean bias and 95% limits of agreement, root mean square error (RMSE), and Spearman correlation to assess the similarity with which individuals were ranked. Note in this evaluation scenario, the mean bias of directly mapped relationships is always zero. We derived the theoretical combined explained variance as the product of the r 2 values from the two linear bridge equations for each indirect model.
Finally, to demonstrate utility, we examined the associations between all PAEE estimates and body mass index (BMI) using multivariable linear regression adjusted for age and sex in a subset of 1695 participants in the Fenland Study. All data processing and analyses were performed in STATA/SE 14.2 (StataCorp, TX, USA).

Results
The characteristics of participants from each of the sources of data are described in Table 1. The participants in the Brage et al. (2015) study were younger and more active with lower BMI than participants in BBVS and the Fenland Study, including the subset reported in White et al. (2016). Participants in EPIC were less active than those in the Fenland Study. The criterion value of PAEE from the DLW method in the BBVS had a mean (SD) of 49.7 (16.2) kJ·kg −1 ·day −1 and a range of 8.6 to 90.8 kJ·kg −1 ·day −1 .
The combinations of Bridge Equation AC and Bridge Equation CB used to generate the indirect model are shown in Table 2 alongside their r 2 values. The newly derived indirect models are plotted alongside the direct models in Figure 2. Table 3 reports the coefficients and performance of models predicting target criterion DLW-based PAEE (kJ·kg −1 ·day −1 ) from a continuous estimate of MVPA duration (minutes·day −1 ) derived from RPAQ. The beta coefficients for indirect models were attenuated compared with the direct model beta (also see Figure 2). The correlation between the predicted and DLW-based PAEE values was preserved irrespective of harmonization method; however, at group-level the values from ACCHR and HR indirect models were significantly biased. Each model was characterized by a narrowing of the range of PAEE values, and this was particularly pronounced when using ACC TRUNK values as the intermediate method, the indirect model with the smallest combined explained variance, most attenuated beta coefficient, and widest limits of agreement. The HR indirect model resulted in predictions with the largest RMSE while the ACC TRUNK and ACCHR indirect model predictions were similarly precise. Coefficients generated by metaanalyzing the three indirect models had smaller standard errors and performance was similar to that of harmonization via ACCHR. Table 4 reports the coefficients and performance of models predicting DLW-based PAEE (kJ·kg −1 ·day −1 ) from a continuous estimate of total daily PAEE (kJ·kg −1 ·day −1 ) derived from RPAQ. Compared with the raw values of RPAQ-derived PAEE, the indirect and direct models reduced group-level mean bias and approximately halved RMSE. As for MVPA estimates, the correlation between RPAQ PAEE values and criterion PAEE was maintained regardless of whether values were raw, directly or indirectly harmonized. The ranges of the indirectly and directly harmonized values were much reduced compared with those from the raw RPAQ and those of the criterion PAEE. Table 5 reports the coefficients and performance of models predicting DLW-based PAEE (kJ·kg −1 ·day −1 ) using the categorical Cambridge Index derived from RPAQ. Correlations with criterion DLW-based PAEE were weaker, and differed for direct and indirect harmonization. For the direct model, the value of PAEE assigned to being "inactive" was greater than that assigned to being "moderately inactive" and "moderately active"; the indirect model values of PAEE for each category were ordered more intuitively (also see Figure 2). RMSEs for the indirectly and directly harmonized values were similar, and also similar to RMSEs using continuous RPAQ data described above, however the group-level values were again biased when using the Fenland Study data for  Abbreviations: ACC TRUNK = mean daily trunk acceleration; ACC WRIST = mean daily high pass filter vector magnitude wrist acceleration signal; ACCHR = combined heart rate and movement sensing; ACCHR EUROPE = combined heart rate and movement sensing from European population; DLW = doubly labelled water method; HR = heart rate; MVPA = moderate-to-vigorous physical activity; PAEE = physical activity energy expenditure; RPAQ = Recent Physical Activity Questionnaire; SE = standard error. *Constant and three dummy variables used to represent four-level Cambridge Index.
Bridge Equation 1. The indirect model derived using the Bridge Equation 1 from a less active European population showed unbiased group-level estimates of PAEE. Table 6 reports the coefficients and performance of models predicting DLW-based PAEE (kJ·kg −1 ·day −1 ) from ACC WRIST (milli-g). Compared with the RPAQ-derived models described above, models based on accelerometer data as starting point and mapping via other objective methods resulted in stronger correlations of estimated PAEE with DLW-based PAEE. The RMSEs observed for both the indirect and direct models were smaller than for any of the RPAQ models, and with higher combined explained variance; the range of PAEE values was also better preserved as shown in Figure 2.
To demonstrate utility of the different harmonization approaches, we examined the associations of PAEE with BMI in the Fenland study (n = 1695 subsample); Figure 3 shows the beta (95% confidence intervals) of these linear associations by harmonization method and starting data format, alongside that from the silver-standard ACCHR PAEE. PAEE estimates harmonized from ACC WRIST showed statistically significant inverse associations with BMI. There was a striking difference between models for the Cambridge Index, with indirectly harmonized PAEE showing statistically significant inverse associations with BMI and the directly harmonized PAEE showing no relationship. Other associations of indirectly harmonized PAEE values with BMI were similar but more uncertain when compared to the direct equivalent. All associations using PAEE harmonized from continuous RPAQ data were weak with 95% confidence intervals crossing zero. MVPA data indirectly harmonized to PAEE via ACC TRUNK values resulted in the widest confidence intervals.

Discussion
This study is the first to examine and evaluate an indirect validation technique for harmonization, an essential step in any study which combines data from different sources in the same analysis. Our findings indicate that indirect models can be employed to harmonize data to a compatible format in the absence of the ideal validation study, but that the harmonized values may still be biased at group-level, have a narrower value range compared to the Figure 2 -Comparison of direct and indirect harmonization models by starting data type. Abbreviations: ACC TRUNK = mean daily trunk acceleration; ACCHR = combined heart rate and movement sensing; ACCHR EUROPE = combined heart rate and movement sensing from European population; ACC WRIST = mean daily high pass filter vector magnitude wrist acceleration signal; HR = heart rate; MVPA = moderate-to-vigorous physical activity; PAEE = physical activity energy expenditure; RPAQ = Recent Physical Activity Questionnaire. criterion, and that gains in precision are dependent upon the variance explained by the contributing bridge equations. These findings reinforce the necessity to carefully consider the inferential equivalence of harmonized data and "the truth" according to the scientific context and the purpose of the analyses being performed (Fortier et al., 2010).
For analyses such as exposure-disease associations which primarily rely on relative validity, harmonization using indirect models is beneficial in that it increases precision while retaining the correlation between the original data and estimates of the latent truth from the gold standard criterion (when all bridge equations are linear). Our results for RPAQ-derived PAEE demonstrate that even when data pre-exist in a seemingly compatible format, improvements in precision are possible following harmonization.
Furthermore, even if no harmonization need be conducted, the newly derived indirect model could be used to perform measurement error correction by regression calibration (Keogh & White, 2014).
Using indirect models revealed some unexpected advantages. When harmonizing Cambridge Index categorical data to a continuous estimate of PAEE, the direct model assigned a higher PAEE value to level 1 than both level 2 and 3, whereas the indirect model was more logically ordered by mean PAEE in kJ·kg −1 ·day −1 using the relationships in the Fenland sample or the published EPIC data. The illogical ordering in the small sample likely contributed to the null PAEE-BMI association for directly harmonized PAEE, whereas the two logically ordered indirectly harmonized PAEE estimates both showed inverse relationships. Given that validation Note: Combined explained variance is the product of the explained variance of the two bridge equations. It is not the explained variance (r 2 ) of the indirect model. Abbreviations: ACC TRUNK = mean daily trunk acceleration; ACCHR = combined heart rate and movement sensing; BBVS = Biobank Validation Study; DLW = doubly labelled water; HR = heart rate; MVPA = moderate-to-vigorous physical activity; PAEE = physical activity energy expenditure; RMSE = root mean square error; RPAQ = Recent Physical Activity Questionnaire; SE = standard error. Note: Combined explained variance is the product of the explained variance of the two bridge equations. It is not the explained variance (r 2 ) of the indirect model. Abbreviations: ACCHR = combined heart rate and movement sensing; BBVS = Biobank Validation Study; DLW = doubly labelled water; PAEE = physical activity energy expenditure; RMSE = root mean square error; RPAQ = Recent Physical Activity Questionnaire; SE = standard error. studies using comparisons with a gold-standard criterion are often relatively small, mapping of categorical data from questionnaires to continuous metrics may be unduly influenced by incorrectly classified individuals with extreme values. In a larger study such as a prospective cohort like Fenland, the influence of incorrectly classified individuals on category means is diminished resulting in more logically ordered group means. Thus in certain scenarios and under some feasibility constraints, it is possible that indirect models using bridge equations sourced from larger studies with silver-standard assessment are actually preferable to harmonization using a direct model from a smaller study using the gold standard.
Although indirect harmonization is a potential solution in the absence of direct models, these findings also suggest investigators should make use of all available information-both direct and indirect-in a form of network harmonization analogous to network meta-analysis. We observed that beta coefficients of indirect models were attenuated compared to those from direct models, and that the level of attenuation is related to the variance explained by the bridge equations being combined. We calculated the theoretical combined explained variance which summarises the r 2 values of the two contributing bridge equations derived in separate studies; indirect models with a higher value tended to have less attenuated beta coefficients, greater precision and less narrowed range of resulting  values. Shrinking of the range of harmonized values towards the mean when compared to the original data and the criterion will have implications for dose-response analyses which aim to assess the shape of any relationship across the full exposure range. It was noticeable that the indirect model for harmonizing MVPA to PAEE via ACC TRUNK had the most attenuated beta coefficient resulting in a very narrow range of PAEE values and the widest confidence intervals for the applied example of studying the PAEE association with BMI. A key finding was that associations between PAEE and BMI were similar for data harmonized using both direct and indirect models; the two techniques may therefore be inferentially equivalent but depending upon the validity of the original method, neither may result in data inferentially equivalent to the true level of exposure. The utility of all harmonization equations is influenced by the quality of the data being harmonized and the pre-existing error of the methods used in the bridge equations. For analyses which depend upon absolute agreement between harmonized data and the latent truth, even greater caution is required as most values from indirect harmonization were biased at grouplevel. The bias was typically <15% and did not appear to be related to the combined explained variance. The biases observed may therefore reflect differences in the populations from which the bridge equations and the data undergoing harmonization are sourced. Perhaps most strikingly, the criterion measures of PAEE in the BBVS and Brage et al. (2015) differed on average by 22%. The Brage et al. (2015) participants were younger and had lower BMI so these are likely to be true differences, even though some of the bias observed may be due to minor differences in the criterion method used to estimate PAEE in the two studies, e.g. different RMR protocols. We examined the effect of using alternative bridge equations from potentially incompatible measurement tools and populations by deriving two indirect models for harmonizing the RPAQ Cambridge Index to PAEE. The indirect model using a Bridge Equation AC derived from the Fenland Study resulted in biased group-level estimates, whereas the 'non-ideal' Bridge Equation AC derived from published data using the EPIC-PAQ in the less active European population did not. It is possible that the lower PAEE in this latter population counteracted any bias resulting from the higher PAEE in the second Bridge Equation from Brage et al. (2015). The contrast may also be partially attributable to differences between RPAQ and EPIC-PAQ. Any incompatibility of the populations and assessment methods from which the two bridge equations are derived must therefore be considered alongside the data being harmonized.
Further work is necessary to examine the suitability of combining bridge equations from different populations and the generalizability of the resulting indirect models, but our findings suggest that harmonization is sensitive to differences in the true level of exposure between participants in the bridge equation studies and those in the dataset being harmonized.
One potential solution to population differences could be the addition of covariates to indirect models, or, if the sample is large enough, stratification of results by covariates. However, this would require some existing published bridge equations to be re-derived, and limit future harmonization to datasets with compatible covariates. Moreover, adding covariates to the prediction likely requires that these are always adjusted for in subsequent association analyses, as associations may otherwise be driven by the covariates (i.e., confounded). In addition and on a more practical level, it is often the case that regression equations are not published, and this limits harmonization efforts despite some of the necessary fieldwork having been conducted. Populating the network of bridge equations between methods, and making this discoverable, is therefore an important task. We have added beta and alpha coefficients alongside their standard errors of the bridge equations -Association (beta coefficients and 95% confidence intervals) between PAEE and BMI, by exposure estimation method (Fenland Study, n = 1695 subsample with wrist acceleration). Note: Associations are adjusted for age and sex. Abbreviations: ACC TRUNK = mean daily trunk acceleration; ACCHR = combined ACC TRUNK and heart rate sensing; ACCHR EUROPE = combined heart rate and movement sensing from European population; BMI = body mass index; ACC WRIST = mean daily high-pass filtered vector magnitude wrist acceleration; HR = heart rate; PAEE = physical activity energy expenditure. *In the absence of doubly labelled water-assessed PAEE in a large cohort, a silver standard (ACCHR) has been used for comparison for the cross-sectional association. used in the present study to the online repository Diet, Anthropometry, and Physical Activity Measurement Toolkit (www. measurement-toolkit.org) which enables sharing of such results and which should increase the visibility and use of populationspecific harmonization models.
A potential limitation of the current work is that only simple linear bridge equations were combined and that the participants in the studies used were all from the same region of the UK; the generalizability of the technique to other populations therefore remains unclear. While it was a strength of this work that we conducted indirect harmonization with several different combinations of methods, there are many more methods in use and the viability of this process with other combinations, particularly when both bridge equations have weak r 2 is difficult to judge. In this study we were able to assess the validity of indirect harmonization using the equivalent direct relationship from a study with gold-standard measures. This will not be possible in most other scenarios as direct models would be unavailable-the exact problem that indirect harmonization tries to resolve. In these circumstances it is not possible to assess the precision or mean bias of indirectly harmonized estimates; the variance explained by the two bridge equations provides an indication, but additional research is required to examine this formally.

Conclusion
In summary, indirect models can harmonize data to compatible format when direct models are not available, and can therefore improve the inclusivity and resolution of data in analyses integrating information from different sources. This network harmonization approach has greater validity when the original data and bridge equations are stronger (i.e., more variance explained). Further work is required to examine the sources of bias, address difficulties when generalizing population-specific equations, and increase both the number and discoverability of bridge equations in the network.