A general conclusion from the foregoing analyses is that the twopoint method for calculating CO_{2} production is the most robust in the presence of systematic changes in CO_{2} and water production, provided that N_{O} and N_{D} are correctly measured, and that the nonlinear regression method is most sensitive to systematic changes. Both multipoint methods perform better when there is a high degree of covariance in the ^{2}H and ^{18}O 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 zerotime 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.
Upper panel  Linear regression
Lower panel  Nonlinear regression
Table 9.3. Effect of moderate and large random errors in isotopic enrichment on the precision of the doublylabelled water method for measuring energy expenditure during a 2 halflife metabolic period
Method/Period
of increase

Model 

Fixed pool 
Variable pool 

Relative standard deviation (%) 

Twopoint 

Moderate error ^{1} 
5.1 
4.4 
Large error ² 
20.2 
22.3 
Linear regression 

Moderate error 
2.3 
1.4 
Large error 
10.3 
6.7 
Nonlinear regression 

Moderate error 
2.1 
1.9 
Large error 
4.9 
4.4 
^{1} sd_{O} = 0.18 + (d_{O}/260) x 0.79, and sd_{D} = 1.2 + (d_{D}/600) x 2.
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 doublylabelled water method for measuring energy expenditure during a 2 halflife metabolic period
Method/Period
of increase 
Model 

Fixed pool 
Variable pool 

Relative standard deviation (%) 

Twopoint 

Moderate error ^{1} 
5.9 
6.0 
Large error ² 
21.8 
24.5 
Linear regression 

Moderate error 
3.6 
3.8 
Large error 
13.4 
13.1 
Nonlinear regression 

Moderate error 
3.0 
3.7 
Large error 
8.9 
11.5 
Superscripts as in Figure 9.3.
The nonlinear 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 r_{CO2} (middle) and variable r_{H2O} (end) simulations in which the nonlinear regression method performed marginally better than linear regression (see Tables 9.1 and 9.2).
The apparent advantages of the twopoint method portrayed by the above analyses may however be somewhat misleading. As has been mentioned, it was assumed that one space or the other (N_{O} or N_{D}) was correctly measured from plateau determinations and the relationship between them was such that N_{D}/N_{O} = 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 N_{D}/N_{O} = 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 36 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 N_{D}/N_{O} really does equal 1.03 estimates of CO_{2} production will only be incorrect by the % inaccuracy in the plateau determination of bodywater.
Table 9.5. Energy expenditure values calculated on the assumption that N_{D}/N_{O} = 1.03 compared to true values obtained when N_{D}/N_{O} ¹ 1.03
True
N_{D}/N_{O} 
Deviation from true energy expenditure if N_{D}/N_{O} 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 N_{D}/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 k_{D}
and k_{O} are 0.1050 and 0.1300 respectively. It can be seen that even small
physiological variations away from N_{D}/N_{O} = 1.03 could cause a
substantial inaccuracy in estimates of CO_{2} production.
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 twopoint 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 multipoint method only produces a correct result when variations in rate constants are random. The multipoint 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 twopoint and multipoint 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 twopoint method. In this comparison the small average difference between the two procedures almost certainly originated from the fact that the volume ratio N_{D}/N_{O} averaged 1.037 in the slope/intercept method but was normalised to 1.03 in the twopoint method.
In contrast Schoeller & Taylor ^{12} compared twopoint methods in which either a plateau method was applied to measure N_{D} and N_{O} and the values normalised to N_{D}/N_{O} = 1.03, or individual N_{D} and N_{O} 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 N_{D} and N_{O} because slopes used were the same. For a period 07 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 k_{D} and k_{O} are taken to be 0.105 and 0.130 respectively this difference is equivalent to an average N_{D}/N_{O} ratio of about 1.06. In the worst case (Subject D) where the difference was 26%, the same assumptions produce an N_{D}/N_{O} 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 N_{D}/N_{O} 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 nonphysiological origin, in which case normalisation is an adequate procedure.
Table 9.6. Comparisons between rateconstants for ^{18}O and ^{2}H disappearance (k_{O} and k_{D}, d^{1}) isotope distribution volumes (N_{O} and N_{D}, g) and carbon dioxide production rate F_{CO2}, estimated using the slopeintercept and twopoint methods (n = 50)


Slope/intercept 
Twopoint 
(AB) 
Statistical significance of difference 

(A) 
(B) 
Mean 
SD 
(t) 

k_{O} 
Mean SD 
0.1138 
0.1132 
0.0006 
0.0026 
1.01 
N_{O} 
Mean SD 
34624 
34671 
48 
929 
1.00 
k_{D} 
Mean SD 
0.0868 
0.0864 
0.0004 
0.0027 
1.11 
N_{D} 
Mean SD 
35893 
35706 
188 
887 
1.01 
Mean difference in F_{CO2} (as % of A) = 1.9 (SD 7.4), pairedt = 1.82.
Table 9.7. Energy expenditure (MJ/day) for 7 subjects calculated using the twopoint method with N_{D} and N_{O} estimated from plateau or intercept
Subject 
Twopoint/plateau 
Twopoint/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}.
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 multipoint 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 ^{2}H and ^{18}O data and even in cases where only CO_{2} production is increased and there is no such covariance the errors are fairly small even where 30% changes in CO_{2} 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, multipoint methods will reduce errors to about half of those obtained with a twopoint 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 CO_{2} production or water output, bias will occur in estimates of CO_{2} production if isotope distribution spaces are measured from the zerotime 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 multipoint 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 zerotime.
In many ways the conclusions that should be drawn from the inadequacies of the twopoint methodology are very similar to those outlined for the multipoint 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 zerotime; in other words on a similar subset of data to that suggested for the multipoint method when systematic variation occurs.
There remains the problem of the relationship between N_{O} and N_{D}. The balance of current evidence suggests a value for N_{D}/N_{O} 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 CO_{2} production will be about 5%. On the other hand if it is possible to measure both N_{D} and N_{O} 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 N_{D}/N_{O} 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
multipoint slopeintercept 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 twopoint 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 twopoint method.