sampling variability from a true, diurnal variation by an appropriate analysis (Received 14 April 1944)

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1 1 J. Physiol. (I945) 104,1 I-5 J / 6I2. III. II DIURNAL AND SAMPLING VARIATIONS IN THE DETERMINATION OF HAEMOGLOBIN By R. H. MOLE, From the Pathological Laboratory, Radcliffe Infirmary, Oxford (Received 14 April 1944) When repeated determinations of haemoglobin are made on an individual animal or man, variations of as much as 20-30% may be found during the course of the day. Since such variations undoubtedly lie outside the range to be expected from the error of replicate determinations, they are assumed to be diurnal in character (Dreyer, Bazett & Pierce, 1920; Rabinowitch, 1924). A diurnal variation, however, would be expected to be regular and recurrent, yet the recorded results show haphazard and irregular alterations. These might be the' result of variability in repeated sampling of an individual's blood, independent of any diurnal variation and due to technical factors in taking the blood and to physiological alterations in the composition of the blood which are not necessarily diurnal. It is possible to distinguish this sampling variability from a true, diurnal variation by an appropriate analysis of the results of repeated,haemoglobin determinations. The method is the analysis of variance as developed by Fisher (1941, 1942), and this paper illustrates the value of the method when applied to the results obtained by McCarthy & Van Slyke (1939) and by others. If haemoglobin determinations are made at a nu'mber of different times of day and on a number of different individuals, the variability of the results will be due in part to differences between individuals, in part to differences in time of day when the blood samples are taken, and in part to the remaining causes of variation which make up the natural variability or uncontrolled error of the observations. The relative magnitudes of the mathematical expressions for the three parts into which the variability may be analysed make it possible to decide whether there are real diurnal differences. The details of the arithmetical work will not be described here as they are fully treated by" Fisher. In this paper the conventional probability level P= 1/20 or 0*05 has been taken as the border line between significant and non-significant differences (Fisher). PH. CIV..L 1

2 2 R. H. MOLE The most carefully carried out observations on diurnal variations in haemoglobin seem to be those by McCarthy & Van Slyke (1939), who used the carbon-monoxide capacity method of Van Slyke & Hiller (1928). This is probably the most accurate method of haemoglobin estimation there is. Earlier work was exhaustively reviewed by Smith (1931), but the results were obtained by less precise methods and are less well adapted for statistical analysis. TABLE 1. McCarthy & Van Slyke's results on twelve normal subjects in ml. of carbon monoxide capacity per 100 ml. of blood 9a.m. 11 a.m. 2p.m. 5p.m. 8p.m. 11 p.m. 19*97 19* Difference between 9 a.m. and 11 p.m. means =0-84±0-18 (see Table 2). Difference between 9 a.m. and 5 p.m. means=0-42. THE RESULTS OF MCCARTHY AND VAN SLYKE Full details of the observations are given in the original paper. The highest and lowest measurements on any one individual occurred at almost any time of day and differed on the average by 1-3 vol. CO, or 7% haemoglobin. Throughout this paper 100% haemoglobin has been assumed to equal 18'5 vol. of gas-combining capacity, as on Haldane's scale. The complete table of results has had to be split into two for statistical analysis, since five individuals were examined on 2 days each, and thirteen on 1 day only. Of these last, one 11 p.m. result is missing, so th4t the complete results on the remaining twelve will be first considered. The average figures for each of the sampling times are given in Table 1 and show a steady, if small, decline through the day. The analysis of variance is set out in Table 2 and the diurnal variation in blood carbon-monoxide capacity between 9 a.m. and 11 p.m. is found to be highly significant when compared with the uncontrolled error of the observations. TABLE 2. Analysis of variance of results on twelve subjects by McCarthy & Van Slyke Variance 9 a.m.-ll p.m. freedom Variance ratio Probability (P) Individual differences Diurnal differences Less than 0001 Uncontrolled error Standard deviation of repeated observations =V/ =0-46 vol. CO. Standard error of difference between two means each derived from twelve observations 2x vol. CO. It is also worth while to examine the smaller changes between 9 a.m. and 5 p.m., since these are the limits of the ordinary working day in laboratory and hospital. An exactly similar variance analysis (not given here) yields a value of P greater than 0 05 so that the diurnal variation over this shorter

3 DETERMINATION OF HAEMOGLOBIN 3 period is not statistically significant: nor does it become significant with the increase in the number of observations which results from including in the analysis the measurements of the five subjects on the first of the 2 days on which they were examined. It is unfortunately not possible to include the second-day results of these five subjects, since there is a marked difference (P less than 0.05) between the uncontrolled error variances of the five first-day and five second-day sets of results due to the much smaller uncontrolled error of the latter (Table 3). This means that the first- and second-day sets TABLE 3. Analysis of variance of results on five subjects by McCarthy & Van Slyke Second day First day* A Dgeso 9 a.m.-5 p.m. freedom Variance freedom Variance Individual differences Diurnal differences Uncontrolled error * The analysis is complicated by the loss of two measurements from the first-day series. The missing values have been estimated by minimizing the sum of squares for error as Fisher (1942) has suggested. This allows the utilization of the other observations on the individuals concerned, but the analysis is, of course, not as precise as if the missing value had actually been observed. The actual estimates here are and The difference in uncontrolled error variance between the first- and second-day series is not due to the use of estimated values. are not comparable and should not therefore be grouped as if they were. Nevertheless, this reduction in error is important from the experimental point of view, for the smaller the uncontrolled error the smaller the diurnal difference which is significant. Without the method of variance analysis this reduction in variability in the second 5-day set of results went undetected, so that the conditions under which the sampling variability of haemoglobin measurement was brought so low, although of great interest, were not recorded. DISCUSSION The method of variance analysis has provided an exact measure of the random variability of the results and of the probability of the averaged diurnal variations of all the subjects having occurred as a consequence of this random variability. In the last resort the significance of any particular value of P is always a matter of opinion, but its actual value is a property of the data and is fixed by them. However strict the criterion of significance, the analysis shows that there is a real diurnal variation, even if its size is much smaller than has been supposed. Diurnal variation, defined as a regular change in haemoglobin through the day, is found from McCarthy & Van Slyke's results to be a steady decrease amounting to about 4 % between 9 a.m. and 11 p.m. This is the average result obtained by eliminating differences between individuals and variability due to technical and sampling factors and thus represents a physiological normal value. 1-2

4 4 R. H. MOLE In advanced heart failure Rabinowitch & Strean (1924) made the interesting observation that there was a much smaller variability than in normal people. It should be pointed out that it is possible that the time of day for maximum and minimum haemoglobin levels may be different for different individuals. The method of analysis used in this paper will obscure any such individual differences in diurnal variation which will, on the other hand, go to swell the uncontrolled error. It would be possible to detect such individual differences by an analysis of repeated serial estimations on several days for each individual. The duplicate series of McCarthy & Van Slyke have been shown to differ so considerably in the magnitude of the uncontrolled error that they should not be considered as a homogeneous group of data, but if they are so grouped and analysed for differences in diurnal variation between individuals no such individual differences are found. The data do not bear out the suggestion that each individual has a characteristic diurnal variation of his own, though a large enough series of repeated observations might show that this hypothesis was true. The uncontrolled error of McCarthy & Van Slyke's observations consists of technical errors in the performance of the estimations, sampling variability and any other unspecified factors which affect the results. The variability of repeated estimations of carbon monoxide capacity on the same sample of blood is very small (Van Slyke & ililler). 'It is therefore safe to assume that the uncontrolled error of 'McCarthy & Van Slyke's data is almost entirely due to sampling variability, with the reservation discussed in the last paragraph. The variation with repeated sampling will be due to local environmental and metabolic influences which affect the composition of the blood in the vein from which the blood is taken, and also to more general factors which alter the proportion of red cells to plasma in the circulating blood. None of these variations are necessarily diurnal although they may normally occur in a regular manner during the day in relation to such regularly recurring events as meals. Any such regularly occurring changes will have been included in the diurnal variation in the analysis and by definition. Lastly, it is perhaps worth pointing out that if the diurnal variation is disregarded and included in the random variability, the standard deviation of repeated estimations is 054 vol. of carbon monoxide. On the normally accepted basis of twice the standard error, this means that there must be a difference of nearly 8 % between two haemoglobin estimations on one individual before this difference can be attributed with safety to anything but experimental error. In most experimental and clinical work where haemoglobin measurements are used to assess an experimental or therapeutic procedure, there will be at least as big an experimental ertor as this, against which the changes in haemoglobin' must be interpreted. The magnitude of

5 DETERMINATION OF HAEMOGLOBIN 5 sampling and diurnal variations sets a limit, too, beyond which it becomes no longer useful to increase the accuracy of haemoglobinometer readings. The limit is reached when the standard deviation of repeated measurements is about 1% on Haldane's scale. SUMMARY 1. Evidence for diurnal variations in haemoglobin has been examined by the method of variance analysis. 2. In the best series of observations there was a significant fall in haemoglobin between 9 a.m. and 11 p.m. of 0-8 vol. of carbon monoxide capacity, or about 4 %. 3. There was no significant diurnal change during the working day from 9 a.m. to 5 p.m. 4. The variation in haemoglobin content with repeated-sampling of venous blood has a standard deviation of 0 54 vol. of carbon monoxide, or 0-46 vol. after diurnal variation has been allowed for. REFERENCES Dreyer, G., Bazett, H. C. & Pierce, H. F. (1920). Lancet, 2, 588. Fisher, R. A. (1941). Stati8tical MethoS for Re8earch Workers. Edinburgh. Fisher, R. A. (1942). The Design of Experiment&. Edinburgh. McCarthy, E. F. & Van Slyke, D. D. (1939). J. biol. Chem. 128, 567. Rabinowitch, L M. (1924). J. Lab. din. Med. 9, 120. Rabinowitch, I. M. & Strean, G. (1924). Arch. intern. Med. 34, 124. Smith, C. (1931). Arch. inten. Med. 47, 206. Van Slyke, D. D. & Hiller, A. (1928). J. bil. Chem. 78, 807.