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9.4 Summary of theoretical comparison of methods

A general conclusion from the foregoing analyses is that the two-point method for calculating CO2 production is the most robust in the presence of systematic changes in CO2 and water production, provided that NO and ND are correctly measured, and that the non-linear regression method is most sensitive to systematic changes. Both multi-point methods perform better when there is a high degree of covariance in the 2H and 18O data. Evidence from most data sets (see Chapter 5) and physiological commonsense indicates that a degree of covariance will normally exist. However, in all cases where large systematic changes occur, initial isotope distribution spaces will be incorrectly estimated from the zero-time intercepts of fitted curves. Thus bias can occur with this procedure. On the other hand, at all levels of analytical error, combining slopes and intercepts substantially improves precision.

Figure 9.4. Residuals arising from a simulation with large random errors in isotope enrichments

Upper panel - Linear regression
Lower panel - Non-linear regression

Table 9.3. Effect of moderate and large random errors in isotopic enrichment on the precision of the doubly-labelled water method for measuring energy expenditure during a 2 half-life metabolic period

Method/Period of increase



Model

Fixed pool

Variable pool

Relative standard deviation (%)

Two-point

Moderate error 1

5.1

4.4

Large error

20.2

22.3

Linear regression

Moderate error

2.3

1.4

Large error

10.3

6.7

Non-linear regression

Moderate error

2.1

1.9

Large error

4.9

4.4

1 sdO = 0.18 + (dO/260) x 0.79, and sdD = 1.2 + (dD/600) x 2.

Large random error = 3 x moderate random error.

Table 9.4. Combined effects of random analytical error in isotopic enrichment and baseline isotopic abundance on the precision of the doubly-labelled water method for measuring energy expenditure during a 2 half-life metabolic period

Method/Period of increase

Model

Fixed pool

Variable pool

Relative standard deviation (%)

Two-point

Moderate error 1

5.9

6.0

Large error

21.8

24.5

Linear regression

Moderate error

3.6

3.8

Large error

13.4

13.1

Non-linear regression

Moderate error

3.0

3.7

Large error

8.9

11.5

Superscripts as in Figure 9.3.

The non-linear regression method further improves precision in the face of analytical error but this improvement is small and does not compensate for the potential inaccuracies in the presence of the particular systematic changes in the isotope turnover rates specified in this simulation. However, as stated in Chapters 4 and 5 the fitting procedure should be chosen after inspecting the residual plot. Using this approach only some of the simulations would be identified as appropriate material for this type of fit: e.g. the variable rCO2 (middle) and variable rH2O (end) simulations in which the non-linear regression method performed marginally better than linear regression (see Tables 9.1 and 9.2).

The apparent advantages of the two-point method portrayed by the above analyses may however be somewhat misleading. As has been mentioned, it was assumed that one space or the other (NO or ND) was correctly measured from plateau determinations and the relationship between them was such that ND/NO = 1.03. There are therefore two questions that must be asked. Firstly, what level of precision and accuracy can be achieved by making measurements at plateaus of enrichment: near to the start of an experiment? Secondly, are we justified in fixing a relationship at ND/NO = 1.03 for all conceivable subjects. As already indicated (Section 4.3.1) our knowledge in this area is relatively scanty. For American adults a combination of analytical accuracy and noise about plateau values gives a precision of about 1.5% for volume determinations using saliva samples at 3-6 hr after a dose (recalculated from ref 10) but we do not know if this time interval is appropriate in all physiological circumstances. This error need not worry us too much however since if ND/NO really does equal 1.03 estimates of CO2 production will only be incorrect by the % inaccuracy in the plateau determination of body-water.

Table 9.5. Energy expenditure values calculated on the assumption that ND/NO = 1.03 compared to true values obtained when ND/NO 1.03

True ND/NO

Deviation from true energy expenditure if ND/NO assumed to be 1.03

(%)

1.01

-11.0

1.02

- 5.8

1.03

0

1.04

+ 6.6

1.05

+14.1

What is of greater concern is the appropriateness of ND/No = 1.03. The evidence is that this is a reasonable average figure (see Section 4.3 and Table 9.6) but we really do not know its physiological range for all conceivable subjects. Table 9.5 indicates the problem for a case where kD and kO are 0.1050 and 0.1300 respectively. It can be seen that even small physiological variations away from ND/NO = 1.03 could cause a substantial inaccuracy in estimates of CO2 production.

9.5 Practical comparisons of the two-point and slope-intercept methods

We have shown in Chapter 4 and earlier in this chapter that the two procedures are quite different from the point of view of what is intended to be measured. The two-point method will produce the correct value for total flux between two time points even in circumstances where systematic variations occur to such an extent that the calculation of an average flux rate per day could almost be said to be inappropriate. In contrast a multi-point method only produces a correct result when variations in rate constants are random. The multi-point method does, however, provide an estimate of the variation about the daily average production rate which combines effects of instrumental and physiological variation (see Chapter 5).

There is little data that allows comparisons to be made between two-point and multi-point methodologies. In the comparison provided by Coward 11 there was no significant bias between the two methods (see Table 9.6). The SDs of the differences between the two procedures are slightly higher than would be predicted from the theoretical uncertainties of about 3% for the slope/intercept approach and about 4.5% for the two-point method. In this comparison the small average difference between the two procedures almost certainly originated from the fact that the volume ratio ND/NO averaged 1.037 in the slope/intercept method but was normalised to 1.03 in the two-point method.

In contrast Schoeller & Taylor 12 compared two-point methods in which either a plateau method was applied to measure ND and NO and the values normalised to ND/NO = 1.03, or individual ND and NO were calculated from the intercepts of disappearance curves generated between two points. In these circumstances the differences between the results obtained using each method can only be attributed to different estimates of ND and NO because slopes used were the same. For a period 0-7 days the intercept method produced an average value that was 15% lower than that obtained using the plateau procedure (see Table 9.7) and if values of kD and kO are taken to be 0.105 and 0.130 respectively this difference is equivalent to an average ND/NO ratio of about 1.06. In the worst case (Subject D) where the difference was 26%, the same assumptions produce an ND/NO ratio of 1.08. These differences from the value of 1.03 on which spaces were normalised are clearly important and contrast markedly with the data from Table 9.6 where mean ND/NO ratios were 1.037 0.012 SD and with the more extensive data in Table 4.2. It is impossible to say for certain whether marked deviations from the value of 1.03 are genuine physiological differences, in which case the use of 1.03 as a normalising factor is unlikely to be inappropriate, or whether such large ratios have a non-physiological origin, in which case normalisation is an adequate procedure.

Table 9.6. Comparisons between rate-constants for 18O and 2H disappearance (kO and kD, d-1) isotope distribution volumes (NO and ND, g) and carbon dioxide production rate FCO2, estimated using the slope-intercept and two-point methods (n = 50)





Slope/intercept

Two-point

(A-B)

Statistical significance of difference

(A)

(B)

Mean

SD

(t)

kO

Mean SD

0.1138
0.0144

0.1132
0.0144

0.0006

0.0026

1.01

NO

Mean SD

34624
3889

34671
4102

-48

929

-1.00

kD

Mean SD

0.0868
0.0134

0.0864
0.0137

0.0004

0.0027

1.11

ND

Mean SD

35893
4009

35706
4222

188

887

1.01

Mean difference in FCO2 (as % of A) = -1.9 (SD 7.4), paired-t = 1.82.

Table 9.7. Energy expenditure (MJ/day) for 7 subjects calculated using the two-point method with ND and NO estimated from plateau or intercept

Subject

Two-point/plateau

Two-point/intercept

A

11.0

9.2

B

13.6

10.4

C

11.5

10.3

D

8.9

6.6

E

9.9

9.2

F

10.5

9.6

G

11.5

10.3

Mean

10.99

9.37

SD

1.48

1.33

Data from Schoeller & Taylor 12.

9.6 Conclusions

It will now be evident that the main difficulty with the methodologies we have been discussing lies not with the measurement of slopes of isotope disappearance curves but with the estimates of volume.

The use of multi-point data with fitting procedures appropriate to the error structure will provide a good estimate of the average difference between rate constants when there is a high degree of covariance between 2H and 18O data and even in cases where only CO2 production is increased and there is no such covariance the errors are fairly small even where 30% changes in CO2 production occur that persist for one third of a total measurement period. If such a change happened at regular intervals, such as every third day as might occur with a recreational runner, errors are less than 1% and can be ignored. There is also the additional factor to consider that, in the case of any level of analytical error, multi-point methods will reduce errors to about half of those obtained with a two-point method assuming that great care is taken to minimise analytical error in the measurement of baseline abundance. Furthermore if no systematic physiological deviations from linearity occur negative covariance will further improve precision. However, in the presence of large systematic variations in either CO2 production or water output, bias will occur in estimates of CO2 production if isotope distribution spaces are measured from the zero-time intercepts of isotope disappearance curves. This error will not always be observed in an estimate of a regression coefficient as these are almost invariably better than 0.99 but will be observable in plots of residuals near time zero (see Figure 9.2). This illustrates the importance of drawing and inspecting a residual plot for all multi-point studies. When systematic deviations are detected, then it will be theoretically preferable to obtain intercept data from the analysis of a subset of data near to zero-time.

In many ways the conclusions that should be drawn from the inadequacies of the two-point methodology are very similar to those outlined for the multi-point method. Provided there is the assurance of analytical accuracy in the determination of a slope the problem is likely to lie with the measurement of volume. The adequacy of this value will only be improved by making several measurements at different times on a plateau near to zero-time; in other words on a similar subset of data to that suggested for the multi-point method when systematic variation occurs.

There remains the problem of the relationship between NO and ND. The balance of current evidence suggests a value for ND/NO of about 1.035 and if this relationship cannot be established during an experiment it is not unreasonable to use this average value, on the understanding that if the true value for any subject is 1% different from this, the error produced in the measurement of CO2 production will be about 5%. On the other hand if it is possible to measure both ND and NO with an accuracy of 0.5 - 1.5% in experiments it is theoretically preferable to use these values. However, experience from the literature and from the data exchange exercise prior to this meeting suggests that values of 1.03 (plus or minus some small SD) are not always found. Until consistent findings for particular populations indicate otherwise it is commonsense to treat ND/NO ratios differing markedly from these values with some suspicion. The IDECG Workshop recommended that 1.015 - 1.060 should be adopted as the acceptable range.

At this meeting Speakman suggested that a general rule might be applied when chosing methods for calculating energy expenditure measurements and his view fairly encapsulates all the arguments propounded in this chapter. That is, that with low isotopic enrichments and relatively small temporal variation in water turnover or carbon dioxide production it will be preferable to use a multi-point slope-intercept method. This is because relatively low enrichments put a premium on the analyses and, in the absence of much temporal variation, fitting single exponentials to data is a satisfactory procedure. When temporal variation is large, as it may often be in wild animals, a two-point method will certainly be preferable from both the theoretical and practical point of view 13. The theoretical reasons are evident, the practical reason is that wild animals are difficult to catch more than a few times in any measurement period. There is the further advantage that relatively high enrichments are often used in experiments such as these and this is an advantage when analytical precision is considered, especially in the two-point method.


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