Determination of the heterophil (H) to lymphocyte (L) ratio (H/L) has become widely used as an abridged assessment of avian leukocytes to provide information about the hematologic homeostasis of birds.^{1, 2} The H/L ratio is calculated from the relevant leukocyte proportions obtained from a differential leukocyte count, most commonly assessing 100 cells. Typically, differential leukocyte counts are performed on a Romanowsky stained blood film using optical microscopy (“visual-optical count”).³ This method is well established; however, there has been little consideration of measurement uncertainty and its effect on H/L ratio values. Possolo⁴ defined measurement uncertainty as “the doubt about the true value of a measurand that remains after making a measurement.” Considerations of the measurement uncertainty associated with various laboratory measurands have been published.^5-7 To the authors’ knowledge, however, there has been little consideration of the measurement uncertainty of the H/L ratio. This letter outlines how the principles of measurement uncertainty have been applied to the H/L ratio and illustrates the effects that measurement uncertainty can have on such a calculation and its associated interpretation.

The definitive document which describes the evaluation of measurement uncertainty (MU, also referred to as uncertainty in measurement) is the Evaluation of measurement data—Guide to the expression of uncertainty in measurement usually referred to as the GUM.⁸ The general procedures for determining MU as described in the GUM and the approach for determining the uncertainty of the H/L ratio as outlined by Rümke⁹ have here been combined to provide a suggested procedure for the measurement uncertainty of a ratio obtained from differential leukocyte counts. The effect of the propagation of uncertainty arising from the individual heterophil and lymphocyte counts to produce a ratio (R), where R = H/L and r is the Pearson correlation coefficient between H and L, is given by the standard uncertainty (u) of the ratio (u(R)). This uncertainty can be determined from the GUM propagation of uncertainty equation:

u^{2} (R) = {(\frac{\partial R}{\partial H})}^{2} u^{2} (H) + {(\frac{\partial R}{\partial L})}^{2} u^{2} (L) + 2 r_{H, L} (\frac{\partial R}{\partial H}) (\frac{\partial R}{\partial L}) u (H) u (L)

(1)

For the ratio R = H/L,

\frac{\partial R}{\partial H} = \frac{1}{L}

and

\frac{\partial R}{\partial L} = \frac{- H}{L^{2}}

. Substituting these terms back into Equation 1 gives:

u^{2} (R) = \frac{u^{2} (H)}{L^{2}} + \frac{H^{2} \times u^{2} (L)}{L^{4}} - 2 r_{H, L} \frac{H \times u (H) \times u (L)}{L^{3}}

(2)

Equation 2 is the basis for determining u(R), but the actual values for u(H), u(L), and u(R) also depend on the probability density distributions (PDFs) appropriate to the data to which they refer. Thus, in addition to any correlation between H and L, the form of the PDFs for the various terms requires careful consideration.

It is commonly assumed that all terms have Gaussian PDFs; however, specific situations require the consideration of alternative input and output density distributions.¹⁰ In particular, the assessment of uncertainty for a ratio is complicated by the fact that the PDF of a ratio is usually non-Gaussian, even when the two input variables are actually Gaussian. That is, even if two input variables are themselves considered Gaussian, it is incorrect to assume that their ratio also follows a Gaussian density distribution.

Notably, the law of propagation of uncertainty (section 5 of the GUM),⁸ described by Equation 1 above, does not in itself depend on a particular input distribution. Nor does it provide any information with regard to the distribution of the output quantity. Knowledge of the distributions for all input quantities is required to provide their appropriate standard uncertainties, while knowledge of the distribution for the output quantity is required to correctly apply the calculated standard uncertainty and provide a coverage interval with the correct coverage probability. Numerous publications discuss the distribution of the input quantity and its relevance,^10-12 but much less is available with regard to the distribution of the output quantity. Holmes and Buhr¹³ considered the error propagation in ratios derived from uncorrelated input variables with Gaussian distributions and illustrated the flaw in assuming that a Gaussian distribution also applies to the ratio. Rümke⁹ described the imprecision of ratios calculated as the quotient of two percentages observed in differential leukocyte counts and recommended that coverage intervals be published with all such values. Thus, in determining the MU of the H/L ratio, the number of leukocytes counted, the correlation between heterophils and lymphocytes, the distribution of the two input measurands, and the distribution of the ratio all need to be considered. The total number of leukocytes counted in a differential count has been shown to affect the variability in the number of individual cell types counted¹⁴ and, consequently, will affect the MU. While a larger number of leukocytes can be counted to reduce variation, the current study focussed on 100-cell differential leukocyte counts as the most common number of leukocytes counted in published studies. Stephens et al.¹⁵ considered the effect of counting a much larger number of leukocytes using an automated hematology analyzer rather than the limited number of leukocytes feasible in manual differential counting. Unfortunately, such automated approaches are not practical for avian differential leukocyte counts due to the impact of nucleated erythrocytes and thrombocytes.

The correlation between measurands has also been shown to be an important consideration in the propagation of uncertainty.^{5, 7, 16} In the context of a differential leukocyte count with a finite number of cells counted, the inclusion of one cell in the count is at the expense of another cell, and the over-representation of one leukocyte type will result in the under-representation of another leukocyte type.⁴ As heterophils and lymphocytes typically comprise the two most numerous leukocyte types in birds, it transpires that in a 100-cell differential leukocyte count, an increase in the proportion of one of these cell types is typically at the expense of the other. Consequently, a negative correlation is likely, and this was observed in the current study (r = − 0.8391). Accounting for such correlation in the propagation of uncertainty has not been well explored in the hematological context; however, applicable examples from laboratory medicine have been published.⁷

The method proposed for determining the coverage interval of the H/L ratio follows the procedures outlined by Rümke⁹ and supporting studies.^{17, 18} The procedure for a 100-cell differential cell count depicts the proportion of a particular cell type as being drawn from a binomial distribution. The use of a binomial distribution for such a counting procedure is particularly suitable when individual measurements are integer counts distributed into two (or more) categories. In contrast to procedures that rely on a Gaussian distribution for result data, a binomial coverage interval may be determined for a single measurement (eg, a single 100 cell count). In this situation, the variance for a binomial distribution is determined by the integer count of a particular cell type counted: for example, if N counts are taken, the variance of the counts is m(1-m/N), where m is the integer count of a particular cell type, and N is the number of cells counted. In addition, another important distinction between a Gaussian distribution and a binomial distribution is that the ratio of two binomial distributions is an approximately log-Gaussian distribution, thus allowing the logarithm of the ratio to be treated as a Gaussian (or approximately Gaussian) distribution. With these comments as background, the standard uncertainty of the H/L ratio can be determined from Equation 1, which in turn provides the following relationship:

u^{2} (\log (R)) = \frac{1}{(H^{2} / L^{2})} \times u^{2} (R)

(3)

where H is a given heterophil count and L the lymphocyte count, with R = H/L their ratio. The probability density distribution of the ratio is significantly skewed as expected, but the logarithm of the ratio is approximately Gaussian as described previously.^{9, 17, 18} In addition, as shown by Equation2, u²(R) can be obtained as follows:

u^{2} (R) = \frac{1}{L^{2}} \times u^{2} (H) + \frac{H^{2}}{L^{4}} \times u^{2} (L) - 2 r_{H, L} \frac{H \times u (H) \times u (L)}{L^{3}} .

(4)

Combining Equations 3 and 4 by substituting u²(R) from Equation 4 into Equation 3 gives:

u^{2} (\log (R)) = \frac{1}{H} + \frac{1}{L} - \frac{2}{N} - 2 r_{H, L} \sqrt{(\frac{1}{H} - \frac{1}{N}) (\frac{1}{L} - \frac{1}{N})} .

(5)

Equation 5 states that log(R), or log (H/L), has an associated standard uncertainty (u(log(R)), or u(log(H/L)); with a factor of 1.96, a 95% coverage interval for log(H/L) is given by:

\log (H / L) \pm 1.96 \times u (\log (H / L)) .

(6)

Taking the exponent of Equation 6 gives the coverage interval for the H/L ratio:

95 % coverage interval for H / L = (H / L) \exp (\pm 1.96 u (\log (H / L)))

(7)

The lower bound for the interval is given by the negative sign in the exponent, with the upper bound being given by the positive sign.

To practically illustrate this approach to calculating MU, 30 distinct 100-cell differential leukocyte counts were undertaken on an archival Romanowsky-stained avian blood film (Galah, Eolophus roseicapilla). Leukocytes were classified according to Clark et al.,³ namely heterophils, lymphocytes, monocytes, eosinophils, and basophils. The H/L ratio was calculated by dividing the number of heterophils by the number of lymphocytes.¹ The uncertainty of the H/L ratio was evaluated using the equations outlined above. Data analysis was performed using Microsoft Excel¹⁹ and IBM SPSS.²⁰ All calculations were performed to at least five decimal places to eliminate rounding errors. A rounding of actual values was performed at the end of the calculations.

The 95% coverage intervals for each of the 30 individual replicates are shown in Table 1. As outlined above, the ability to determine a coverage interval for a single measurement is a feature of the binomial distribution. In this particular example, it can also be seen that the upper interval (upper bound minus ratio value) is always larger than the lower interval (ratio value minus lower bound), which confirms the skewed nature of the distribution. In addition, the mean of the 30 individual ratios is 5.75 with a “significant” correlation between the H and L values (correlation coefficient − 0.8391), a mean lower bound of 3.22, and a mean upper bound of 10.37. The shortest coverage interval is 2.0 to 5.1 (with a range of 3.1), which occurs for replicate number 23 (H/L ratio = 3.2); while the longest coverage interval is 5.3–25.9 (with a range of 20.6), which occurs for replicate number 12 (H/L = 11.7).

TABLE 1. Differential Leukocyte Counts (100 cells), heterophil to lymphocyte ratios and 95% coverage intervals

Replicate	Heterophil %	Lymphocyte %	Monocyte %	Eosinophil %	Basophil %	H/L	u(log(H/L))	Lower H/L	Upper H/L	Range H/L
1	75	22	3	0	0	3.4	0.2388	2.1	5.4	3.3
2	84	14	1	1	0	6.0	0.2855	3.4	10.5	7.1
3	86	10	4	0	0	8.6	0.3346	4.5	16.6	12.1
4	92	8	0	0	0	11.5	0.3642	5.6	23.5	17.8
5	80	15	5	0	0	5.3	0.2813	3.1	9.3	6.2
6	77	22	1	0	0	3.5	0.2360	2.2	5.6	3.4
7	80	18	2	0	0	4.4	0.2568	2.7	7.4	4.7
8	89	10	1	0	0	8.9	0.3301	4.7	17.0	12.3
9	76	17	7	0	0	4.5	0.2698	2.6	7.6	5.0
10	82	14	4	0	0	5.9	0.2883	3.3	10.3	7.0
11	83	12	5	0	0	6.9	0.3098	3.8	12.7	8.9
12	82	7	9	2	0	11.7	0.4046	5.3	25.9	20.6
13	87	11	0	2	0	7.9	0.3176	4.2	14.7	10.5
14	85	13	2	0	0	6.5	0.2948	3.7	11.7	8.0
15	81	14	3	0	2	5.8	0.2897	3.3	10.2	6.9
16	81	18	1	0	0	4.5	0.2554	2.7	7.4	4.7
17	81	12	6	1	0	6.8	0.3126	3.7	12.5	8.8
18	81	12	5	2	0	6.8	0.3126	3.7	12.5	8.8
19	78	17	3	2	0	4.6	0.2671	2.7	7.7	5.0
20	82	15	1	2	0	5.5	0.2785	3.2	9.4	6.3
21	74	21	3	2	0	3.5	0.2458	2.2	5.7	3.5
22	76	20	4	0	0	3.8	0.2490	2.3	6.2	3.9
23	74	23	2	1	0	3.2	0.2349	2.0	5.1	3.1
24	78	14	7	1	0	5.6	0.2938	3.1	9.9	6.8
25	85	12	0	3	0	7.1	0.3069	3.9	12.9	9.0
26	76	21	2	1	0	3.6	0.2430	2.2	5.8	3.6
27	82	18	0	0	0	4.6	0.2540	2.8	7.5	4.7
28	77	22	1	0	0	3.5	0.2360	2.2	5.6	3.4
29	81	16	2	1	0	5.1	0.2711	3.0	8.6	5.6
30	77	21	1	1	0	3.7	0.2417	2.3	5.9	3.6

The current study clearly illustrates the variation (uncertainty) in a heterophil to lymphocyte ratio obtained from a 100-cell differential leukocyte count; it also highlights the need to report the MU and to consider the effect that MU might have when comparing results. The current study focused on the MU associated with the H/L ratio for an individual bird. However, in the broader context of studies of the H/L in a population of birds, the MU will contribute to the total uncertainty along with other sources of uncertainty, such as biological variation. In such studies, the MU needs to be considered when attributing the components of the total variation.

APPENDIX A: Derivation of binomial uncertainty equations for the H/L ratio

The general equation which provides for the propagation of uncertainty is fully described in the GUM. For a function such as y = f (x) or y equals function x, in which f(x) contains two terms (x₁ and x₂), both of which are mutually correlated with the Pearson correlation coefficient

r_{x_{1}, x_{2}}

, the GUM propagation of uncertainty equation is:

u^{2} (y) = {(\frac{\partial y}{\partial x_{1}})}^{2} u^{2} (x_{1}) + {(\frac{\partial y}{\partial x_{2}})}^{2} u^{2} (x_{2}) + 2 r_{x_{1,} x_{2}} (\frac{\partial y}{\partial x_{1}}) (\frac{\partial y}{\partial x_{2}}) u (x_{1}) u (x_{2}) .

(8)

But in the current situation, x₁ represents the heterophil count (or H), x₂ represents the lymphocyte count (or L) and y represents the heterophil to lymphocyte ratio R = H/L. Thus the uncertainty of the H to L ratio (R) is given by:

u^{2} (R) = {(\frac{\partial R}{\partial H})}^{2} u^{2} (H) + {(\frac{\partial R}{\partial L})}^{2} u^{2} (L) + 2 r_{H, L} (\frac{\partial R}{\partial H}) (\frac{\partial R}{\partial L}) u (H) u (L) .

(9)

For R = H/L:

\frac{\partial R}{\partial H} = \frac{1}{L}

and

\frac{\partial R}{\partial L} = \frac{- H}{L^{2}}

. Substituting these terms back into Equation 9 (Equation 2) gives:

u^{2} (R) = \frac{u^{2} (H)}{L^{2}} + \frac{H^{2} \times u^{2} (L)}{L^{4}} - 2 r \frac{H \times u (H) \times u (L)}{L^{3}}

(10)

u^{2} (R) = \frac{1}{L^{2}} \times u^{2} (H) + \frac{H^{2}}{L^{4}} \times u^{2} (L) - 2 r \frac{H \times u (H) \times u (L)}{L^{3}} .

(11)

In the particular case of the H/L ratio, R is considered as log-Gaussian and is obtained from the ratio of two binomial distributions.

Again, using the GUM propagation of uncertainty equation for y = f(x), where f(x) has only one term without correlation, the variance in y (u²[y]) is given by:

u^{2} (y) = {(\frac{\partial y}{\partial x})}^{2} u^{2} (x) .

(12)

But if function f(x) = f(R) = log R, or y = log R, then:

u^{2} (\log (R)) = \frac{1}{R^{2}} \times u^{2} (R)

(13)

u^{2} (\log (H / L)) = \frac{1}{(H^{2} / L^{2})} \times u^{2} (H / L) .

(14)

From Equation 11 above, where R = H/L:

u^{2} (H / L) = \frac{1}{L^{2}} \times u^{2} (H) + \frac{H^{2}}{L^{4}} \times u^{2} (L) - 2 r \frac{H \times u (H) \times u (L)}{L^{3}} .

(15)

Substituting u²(H/L) as shown in Equation 15 into Equation 14, gives:

u^{2} (\log (H / L)) = \frac{1}{(H^{2} / L^{2})} \times (\frac{1}{L^{2}} \times u^{2} (H) + \frac{H^{2}}{L^{4}} \times u^{2} (L) - 2 r \frac{H \times u (H) \times u (L)}{L^{3}})

(16)

= \frac{L^{2}}{H^{2}} \times (\frac{1}{L^{2}} \times u^{2} (H) + \frac{H^{2}}{L^{4}} \times u^{2} (L) - 2 r \frac{H \times u (H) \times u (L)}{L^{3}}) .

(17)

Multiply out and simplify Equation 17:

u^{2} (\log (H / L)) = \frac{L^{2}}{L^{2} H^{2}} \times u^{2} (H) + \frac{L^{2} H^{2}}{L^{4} H^{2}} \times u^{2} (L) - 2 r \frac{L^{2} H \times u (H) \times u (L)}{L^{3} H^{2}}

(18)

= \frac{1}{H^{2}} \times u^{2} (H) + \frac{1}{L^{2}} \times u^{2} (L) - 2 r \frac{u (H) \times u (L)}{L H}

(19)

For a binomial distribution, however, the variance (u²) equals m(1-m/N).

That is: u²(H) = H(1-H/N) and u²(L) = L(1-L/N). Substituting into Equation 19 gives:

u^{2} (\log (H / L)) = \frac{1}{H^{2}} \times H (1 - \frac{H}{N}) + \frac{1}{L^{2}} \times L (1 - \frac{L}{N}) - 2 r \frac{\sqrt{H (1 - \frac{H}{N})} \times \sqrt{L (1 - \frac{L}{N})}}{L H}

(20)

But

\sqrt{x} \times \sqrt{y} = \sqrt{xy}

, thus:

u^{2} (\log (H / L)) = \frac{1}{H^{2}} \times H (1 - \frac{H}{N}) + \frac{1}{L^{2}} \times L (1 - \frac{L}{N}) - 2 r \frac{\sqrt{H (1 - \frac{H}{N}) \times L (1 - \frac{L}{N})}}{L H}

(21)

= \frac{1}{H^{2}} \times H (1 - \frac{H}{N}) + \frac{1}{L^{2}} \times L (1 - \frac{L}{N}) - 2 r \sqrt{\frac{H (1 - \frac{H}{N}) \times L (1 - \frac{L}{N})}{L^{2} H^{2}}}

(22)

= \frac{1}{H^{2}} \times H (1 - \frac{H}{N}) + \frac{1}{L^{2}} \times L (1 - \frac{L}{N}) - 2 r \sqrt{\frac{(1 - \frac{H}{N}) \times (1 - \frac{L}{N})}{L H}}

(23)

= \frac{1}{H^{2}} \times H (1 - \frac{H}{N}) + \frac{1}{L^{2}} \times L (1 - \frac{L}{N}) - 2 r \sqrt{\frac{1}{H} (1 - \frac{H}{N}) \times \frac{1}{L} (1 - \frac{L}{N})}

(24)

Cancel and simplify terms to give Equation 5; that is:

u^{2} (\log (H / L)) = \frac{1}{H} + \frac{1}{L} - \frac{2}{N} - 2 r \sqrt{(\frac{1}{H} - \frac{1}{N}) (\frac{1}{L} - \frac{1}{N})}

(25)

Enter actual H and L values into Equation 25 (Equation 5); take the square root and use Equation 26 (Equation 6) to give the 95% coverage interval for log(H/L):

\log (H / L) \pm 1.96 \times u (\log (H / L))

(26)

Taking the exponent of Equation 26 (Equation 7) gives the coverage interval for the H/L ratio:

95 % confidence range for H / L ratio = (H / L) \exp (\pm 1.96 u (\log (H / L)))

(27)

The lower bound for the interval is given by the negative sign in the exponent, with the upper bound being given by the positive sign.

REFERENCES

1Gross WB, Siegel HS. Evaluation of the heterophil/lymphocyte ratio as a measure of stress in chickens. Avian Dis. 1983; 27: 972-979.
10.2307/1590198
CAS PubMed Web of Science® Google Scholar
2Davis AK, Maney DL, Maerz JC. The use of leukocyte profiles to measure stress in vertebrates: a review for ecologists. Funct Ecol. 2008; 22: 760-772.
10.1111/j.1365-2435.2008.01467.x
Web of Science® Google Scholar
3Clark P, Boardman WSJ, Raidal SR. Atlas of clinical Avian hematology. Wiley-Blackwell; 2009: 40-49.
Google Scholar
4Possolo A. Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Technical Note (NIST TN). : National Institute of Standards and Technology. 2015: 14 [online], 10.6028/NIST.TN.1900 (Accessed March 29, 2021)
10.6028/NIST.TN.1900
Google Scholar
5Farrance I, Frenkel R. Uncertainty of measurement: a review of the rules for calculating uncertainty components through functional relationships. Clin Biochem Rev. 2012; 33: 49-75.
PubMed Google Scholar
6Frenkel RB, Farrance I. Uncertainty in measurement: procedures for determining uncertainty with application to clinical laboratory calculations. Adv Clin Chem. 2018; 2018(85): 149-211.
10.1016/bs.acc.2018.02.003
Google Scholar
7Rigo-Bonnin R, Canalias F. Measurement uncertainty estimation for derived biological quantities. Clin Chem Lab Med. 2021; 59: e1-e7.
10.1515/cclm-2020-1003
CAS PubMed Google Scholar
8 Bureau International des Pois et Measures. Evaluation of measurement data – guide to the expression of uncertainty in measurement (GUM). JCGM 100:2008. Available at: http://www.bipm.org/en/publications/guides/gum.html (accessed March 29, 2021).
Google Scholar
9Rümke CL. The imprecision of the ratio of two percentages observed in differential white blood cell counts: a warning. Blood Cells. 1985; 11: 137-140.
CAS PubMed Web of Science® Google Scholar
10Coskun A, Oosterhuis WP. Statistical distributions commonly used in measurement uncertainty in laboratory medicine. Biochem Med. 2020; 30: 5-17. doi:10.11613/BM.2020.010101
10.11613/BM.2020.010101
Web of Science® Google Scholar
11 CLSI (2012). Clinical and laboratory standards institute (CLSI). Expression of measurement uncertainty in laboratory medicine: approved guideline. CLSI; 2012, document EP29-A.
Google Scholar
12Farrance I, Frenkel R. Uncertainty in measurement: a review of Monte Carlo simulation using Microsoft excel for calculation of uncertainties through functional relationships, including uncertainties in empirically derived constants. Clin Biochem Rev. 2014; 2014(35): 37-61.
Google Scholar
13Holmes DT, Buhr KA. Error propagation in calculated ratios. Clin Biochem. 2007; 40: 728-734.
10.1016/j.clinbiochem.2006.12.014
PubMed Web of Science® Google Scholar
14Rümke CL, Bezemer PD. Kuik DJ Normal values and least significant differences for differential leukocyte counts. J Chronic Dis. 1975; 28(11–12): 661-668.
10.1016/0021-9681(75)90077-6
PubMed Google Scholar
15Stephens L, Hintz-Prunty W, Bengtsson HI, Proudfoot JA, Patel SP, Broome HE. Impact of integrating Rumke statistics to assist with choosing between automated hematology analyzer differentials vs manual differentials. J Appl Lab Med. 2017; 1: 357-364.
10.1373/jalm.2016.021030
PubMed Google Scholar
16Farrance I, Frenkel R. Measurement uncertainty and the importance of correlation. Clin Chem Lab Med. 2021; 59: 7-9.
10.1515/cclm-2020-1205
CAS Google Scholar
17Katz D, Baptista J, Azen SP, Pike MC. Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics. 1978; 34: 469-474.
10.2307/2530610
Web of Science® Google Scholar
18Koopman PAR. Confidence intervals for the ratio of two binomial proportions. Biometrics. 1984; 40: 513-515.
10.2307/2531405
Web of Science® Google Scholar
19 Microsoft Corporation. Microsoft Excel [Internet]. 2018. https://office.microsoft.com/excel
Google Scholar
20 IBM Corporation. IBM SPSS Statistics for Windows, Version 27.0. : 2020
Google Scholar

Volume51, Issue2

June 2022

Pages 196-200

Measurement uncertainty in the heterophil to lymphocyte ratio of birds

APPENDIX A: Derivation of binomial uncertainty equations for the H/L ratio

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Measurement uncertainty in the heterophil to lymphocyte ratio of birds

APPENDIX A: Derivation of binomial uncertainty equations for the H/L ratio

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