Correspondence Anal. Chem. 1997, 69, 789-790
A Heuristic Derivation of the Horwitz Curve Richard Albert†
Food and Drug Administration, 25 Lennon Court No. 26, South Boston, Massachusetts 02127 William Horwitz*
Food and Drug Administration, HFS-500, Washington, D.C. 20204
The Horwitz curve is a simple exponential relationship between the relative standard deviation among laboratories to concentration, C, expressed in mass/mass units. Examination of almost 10 000 interlaboratory data sets shows that the curve is more or less independent of analyte, matrix, method, and time of publication, over the range from pure materials, C ) 1 (100%), to trace polychlorinated aromatic contaminants (PCCs), C ≈ 10-12. The functional relationship can be derived simply by assuming that the infinitesimal fractional change in standard deviation is proportional to the infinitesimal fractional change in C, by integrating, and by determining the constant of integration from empirical results. Mycotoxin and PCC data show that the limit of measurement in the interlaboratory environment is C ≈ 10-9, where results become uninterpretable because of the appearance of excessive numbers of false positive and false negative values. Lower values are possible only because of the extraordinary specifications for quality control for these analyses. The Horwitz curve is a very simple exponential relationship of the variability of chemical measurements in the interlaboratory environment to the concentration of the analyte, more or less independent of analyte, matrix, method, and time of publication.1 The curve was characterized by Hall and Selinger2 as “one of the most intriguing relationships in modern analytical chemistry.” The relationship is now supported by the examination of almost 10 000 individual data sets from method precision studies using typically eight laboratories analyzing five individual test samples of commodities ranging from agricultural products to geological specimens, with analytes ranging from aflatoxins through zeralanone, at concentrations, C, in mass/mass units, ranging from pure compounds (g/g; C ) 1) to ultratrace contaminants (ng/kg; C ) 10-12). The Horwitz curve is useful as the initial estimate of expected among-laboratory variability prior to the performance of an interlaboratory study.3 The curve is also helpful in interpreting the results of method- and laboratory-performance studies and in * Address correspondence to this author. Tel: +1202-205-4346/4046. Fax: +1202-401-7740. E-mail:
[email protected]. † Scientist Emeritus. (1) Horwitz, W. Anal. Chem. 1982, 54, 67A. (2) Hall, P.; Selinger, B. Anal. Chem. 1989, 61, 1465. (3) Analytical Methods Committee. Analyst 1995, 120, 2303.
setting initial limits for quality control purposes. “Acceptable” performance usually provides variability values within one-half to twice the value predicted by the equation from the concentration. Within-laboratory variability is expected to be one-half to two-thirds the among-laboratory variability. We recently reviewed the statistical parameters of the methods for the determination of polychlorinated environmental contaminants (biphenyls, dioxins, furans) in environmental media where the analytical performance is somewhat better than predicted from the historically based curve.4 We attribute this improvement to the unrealistic operating conditions utilized in these studies, as well as to the extensive requirements for rigorous cleanup and the extraordinary specifications for quality control for these analyses. Such restrictions no longer qualify the work as capable of being performed in the ordinary analytical laboratory. These analyses require very specialized instrumentation operated in a clean-room environment. We have also shown that the interlaboratory limit of quantitation for analytical chemistry, in general, is in the single-digit parts-per-billion (C ) 10-9) region, below which the results become uninterpretable because of the appearance of excessive numbers of false positive and false negative results. Even under such uncommon nonroutine conditions, the Horwitz curve provides at least a benchmark that shows how much better than expected the analytical results are. We present here a simple, plausible derivation of this intriguing relationship. In making replicate estimates of the concentration, C, of an analyte in a matrix, the chemist usually finds that multiple estimates are not identical but tend to spread out around some central value designated as the “mean” or “average.” This spread can be characterized in several ways, but the most frequent measure is the statistical function known as the standard deviation, σ. The standard deviation is effectively the square root of the average squared deviation of each estimate from the central value. The tighter the clustering of the estimates about the central value, the smaller is the value of σ. Knowledge of σ is important because the standard deviation affects the degree of reliance that can be placed on analytical measurements.5 The interesting question is how σ might depend on C. Analytical chemistry is directed toward a meaningful estimation of C, but a complete interpretation requires a statement of the (4) Horwitz, W.; Albert, R. J. AOAC Int. 1996, 79, 589. (5) Thompson, M. Analyst 1995, 120, 117N. (6) Stewart, I. From Here to Infinity; Oxford University Press: New York, 1996; p 242.
S0003-2700(96)00837-2 This article not subject to U.S. Copyright. Publ. 1997 Am. Chem. Soc.
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uncertainty (the complement of reliability) of that estimate, a fact often neglected in reporting chemical measurements. The scatter of most chemical measurements appears to be a reasonable fraction of the measurement, i.e., for concentrations of the order of magnitude of g/100 mL or g, the range of scatter of “wellbehaved” measurements, e.g., measurements in statistical control, will be a fraction of that base unit. Similarly, if the measurement is in mg/kg, the range of scatter of the measurements is a fraction of mg/kg and certainly not in the range of g/kg. In other words, σ “tracks” C: as C decreases, so does σ. It is observed experimentally that σ decreases less rapidly than C, so the relative variability, e.g., the coefficient of variation or relative standard deviation, RSD, increases for lower values of C. How much does σ change with C? A reasonable assumption is that, for a small percentage increase in C, there will be a proportional small percentage increase in σ. If C is doubled, one does not expect σ to double, nor does one expect σ to increase by a factor proportional to 2. However, for a small percentage change in C, e.g., 1%, it is reasonable to expect a correspondingly small percentage change in σ. Furthermore, these two percentage changes can be assumed to be proportional. Algebraically,
dσ/σ ) R dC/C
(1)
where R is the (positive) proportionality constant. (dσ/σ) is an infinitesimal fractional change in σ, and dC/C is an infinitesimal fractional change in the (true) C. Integrating both sides of eq 1 and multiplying both sides by the appropriate constant to convert to the base 10 yields
log10 σ ) R log10 C + β
(2)
where β is the constant of integration. The derivation of eq 2 depends only on the simple assumption that the relationship given in eq 1 is reasonable. To obtain values of the constants R and β, we need to know specific values for σ at only two different concentration levels. At the highest concentration level, for pure materials, where C ) 1.00, we find empirically that about two-thirds of the interlaboratory estimates are quite close to 1.00, falling, e.g., into the range 0.98-1.02. Such a spread corresponds to σR ) 0.02, where the subscript R indicates among-laboratories (r would indicate within-laboratory). Substituting this empirical value into eq 2 yields a value for β:
log10 0.02 ) R log10 1.00 + β ) R(0) + β ) -1.699 (3) At a more moderate concentration level, e.g., 1%, where C ) 0.01, two-thirds of the estimates are found to be between C ) 0.0096 and C ) 0.0104. This situation corresponds to σR ) 0.0004. Thus,
log1 0 0.0004 ) R log10 0.01 - 1.699 ) -2R - 1.699 ) -3.398, so R ) 0.8495 Therefore,
log10 σR ) 0.8495 log10 C - 1.699, or σR ) 0.02C0.8495 The coefficient of variation or relative standard deviation, RSDR(%), in percent, is
RSDR(%) ) (σR/C) × 100, so RSDR(%) ) 2C0.8495C-1 ) 2C-0.1505
(4)
That eq 4 is, indeed, the Horwitz curve can be seen by taking the logarithm (base 2) of both sides, yielding 790
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log2(RSDR(%)) ) log2 2 + log2 C-0.1505 ) 1 + 3.3223[-0.1505] log10 C where 3.3223, the logarithm (base 2) of 10, is the factor needed to convert a logarithm from the base 10 to a logarithm to the base 2. Carrying through the multiplication gives
log2(RSDR(%)) ) 1 - 0.5 log10 C Raising 2 to the power of each side of the equation gives
2log2(RSDR) ) RSDR(%) ) 2(1-0.5log10C) which is the original form of the Horwitz equation as derived empirically by drawing in a line by eye that represented the empirical data reasonably well. Equation 4 is the same equation in a more compact form. In words, this equation states that, starting with “pure materials” (C ) 1.0, g/g), the amonglaboratories relative standard deviation in percent, RSDR(%), of 2.0% at that point increases by a factor of 2 for every two decades of decrease in concentration. A convenient reference and check point for programmable calculators is, if C ) 10-6 (µg/g; ppm), RSDR ) 16%. The RSDR curve approaches infinity as the concentration approaches zero. Near zero concentration, the influence of the analyte on the measured value decreases, and the effect of the blank or matrix increases. Because this effect arises from the individual laboratory environment, it occurs at different concentrations in different laboratories. This effect leads to the high values for RSDR (>50%) and to the typical discrepancies in the reports of positive and negative values from the different laboratories on the identical test sample.4 Application of Fractal Theory. An alternative explanation of the Horwitz curve may lie in a finding from fractal theory described by Stewart6 for the measurement of the perimeter of the coastline length of Great Britain. As smaller yardsticks are used, ever smaller, subtler indentations are noted, which contribute to the estimate of the perimeter. It was found empirically that the estimate of the perimeter varies as the yardstick size raised to a power. As resolution is improved, i.e., as smaller and smaller differences in length are distinguished (equivalent to concentration in our case), larger estimates are obtained. We find this result, especially the exponential relationship, suggestive and worthy of further exploration. Note that the derivation given in this paper deals with the fractional change dC/C, which, in effect, is a measure of resolution or discrimination. Thus, it is not changes in concentration per se that cause changes in the standard deviation; rather it might be the improved resolution that the chemist achieves in response to lower concentrations that drives the changes in the standard deviation of the concentration estimates.
Received for review August 16, 1996. Accepted November 25, 1996.X AC9608376
X
Abstract published in Advance ACS Abstracts, January 15, 1997.
