Dr. Pardue responds:

Department of Chemistry, 1393 BRWN BLDG., Purdue University, West Lafayette, IN 47907, Fax 1-765-496-1200, E-mail pardue{at}chem.purdue.edu

To the Editor:

My original paper (1) focused largely on quantitative features of the IUPAC (slope) interpretation of sensitivity (2) in response to a challenge (3) related to the utility and advantages of that interpretation. The rejoinder (4) does not refute the quantitative utility of the slope interpretation (1)(2) but asks "not only when but why the 'slope' interpretation emerged". This paper addresses these questions with more emphasis on the more general response/stimulus interpretation (1) of which the slope interpretation (2) is a special case.

To understand my intent, readers should consider the terms "dictionary definition(s)" and "formal definition(s)" interchangeable in any and all references to my use of these terms (1).

formal definitions (5)(6)
Quantitative parts of the formal definitions of sensitivity (Table 1 , items 1b and 2a are themselves embodiments of the response/stimulus interpretation of sensitivity.

Regarding a suggestion (3) that the quantitative ("ancillary") part of the Oxford English Dictionary (OED) definition is "of more recent origin", Mr. Alan Hughes, Chief Science Editor of the Oxford English Dictionary, informed me (A.M. Hughes, personal communication) that both parts of the OED definition (Table 1 , items 1a and 1b) were written at the same time. It is also his opinion, as the senior scientist on the Supplement in which they first appeared, that the two parts of the definition are consistent with one another and that "the 'ratio' is a way of expressing the 'degree to which' in a quantitative way", see also Table 1 , item 2b. He mentioned a 1944 citation ("Sensitivity is merely the ratio of effect/cause... ") as an example of a quantitative expression of sensitivity (see also Table 1 , item 2b).

These dictionary definitions are themselves unequivocal statements of the response/stimulus interpretation of sensitivity; it is difficult to understand how, in the absence of preconceived ideas, they can be interpreted in any other way.

historical precedence
Although citations (4) from Yeats and Maxwell are too implicit to support either interpretation, other comments by Maxwell (7)(8) are sufficiently explicit to clarify his interpretation of the word "sensitive".

Whereas the authors (4) associate the word "measure" with Maxwell's description of a "sensitive galvanometer", Maxwell (4)(7) actually uses the words "indicate the existence of a current" in this context and, in the following paragraph, uses the word measure only in the context of a "standard galvanometer" used to measure currents accurately. Moreover, whereas the authors imply that "measurement uncertainty" (one of their "two factors" (4)) is an inherent part of sensitive and "sensitivity", Maxwell does not mention uncertainty in the description of a sensitive galvanometer (Table 1 , item 3) but includes it in the description of a standard galvanometer used to measure currents accurately. In fact, by devoting every part of the design of a sensitive galvanometer to maximum deflection, no part can be devoted to controlling measurement uncertainty. It should also be noted that Maxwell's use of "a small" EMF, is consistent with "small amounts" in the OED definition (Table 1 , item 1a) and inconsistent with "the smallest amount" inherent in the imprecision interpretation (3).

Another statement, "The galvanometer is only required to be sensitive enough to detect the existence ... of a current without in any way determining its value... " (8) is even more explicit; Maxwell did not associate sensitive with the ability to measure or quantify a stimulus (Table 1 , item 3a) but rather with the ability "to detect its existence", a subtle difference with profound consequences.

Finally, the statement (8), "a galvanometer is most sensitive when its deflection is small... " makes it clear that Maxwell expected the sensitivity of a galvanometer to change with the amount of deflection, a feature inherent in the response/stimulus interpretation (1)(9)(10) and excluded in the "imprecision" interpretation. The response/stimulus interpretation of sensitivity (1)(5)(6)(9)(10) is consistent with all these comments by Maxwell; the imprecision interpretation is not.

analogy
Formal definitions of sensitivity (5)(6) and a calculus-based derivative (11) are very similar; each involves changes in a dependent variable for "small changes" in an independent variable. Application of arguments inherent in the imprecision interpretation of sensitivity (3)(4) would lead to the conclusions that a derivative is the smallest detectable value of the independent variable, that the derivative of any function can have only one value, and that a derivative is not useful in the absence of knowledge of the uncertainty in the independent variable. Application of reasoning involved in the response/stimulus interpretation results in all the familiar properties of derivatives.

In determining a derivative or sensitivity, the use of small changes in independent variables ensures that tangents rather than chords are obtained.

applicability to alternative situations
Fig. 1 , which illustrates the effects of temperature on two options for a kinetic-based determination (12), is used to compare the applicability of the two interpretations to a different situation. The slope interpretation leads to the conclusion that option a is more sensitive to temperature than option b. The imprecision interpretation would involve calculation of the standard deviation of the temperature at zero temperature, _[T]0, a quantity of little or no relevance to this or other types of stimulus/response studies, of which there are many.

View larger version (15K): [in this window] [in a new window]   Figure 1. Effects of temperature on two options for a kinetic-based determination. Ordinate, results for each option normalized to the value at 37 °C to permit direct comparison of slopes.

As illustrated by this example, perceived difficulties associated with comparing sensitivities for a single stimulus characterized by different responses (4) can be resolved by using relative values of responses.

There are other problems with the _[D]0 criterion. For example, in initial-rate determinations of enzyme activity, the imprecision of the fixed signal with zero enzyme present would likely be a poor indicator of the imprecision of rate measurements with small amounts of enzyme present because rate measurements are influenced more than fixed signals by variables such as instrumental noise, temperature, pH, ionic strength, inhibitors, etc.

In this and other analogous situations (e.g., small analyte peaks superimposed on trailing edges of large peaks (13)), _[D]0 would be an indirect indicator of analytical performance at best and could easily lead to misleading conclusions.

Even for situations in which the _[D]0 criterion is a valid indicator of system performance near zero concentration, it may not be particularly relevant to the problem of interest. For example, the potassium concentrations of clinical interest (3.5–5.3 meq/L) are so far from the spectroscopic limits of detection (LOD 0.00025 meq/L) that the quantitative resolution in the clinical range is much more relevant than the limit of detection or the imprecision at zero concentration.

responses to selected comments
Although few if any data, including _[D]0, are useful in isolation (4), all who select chromatographic or spectroscopic peak maxima for quantitative applications because slopes of calibration plots are largest at the maxima use the slope as "a useful indicator of analytical performance" (4).

Quantitative resolution, QR (1), is not the same as imprecision (4) because the error term, _r, (Eq. 2 in (1)) includes effects of both random and systematic errors.

Because the slope of a calibration plot is the change in response per unit of change in the stimulus, an increase in the slope of a calibration curve must necessarily represent a concomitant increase in the ability of an instrument to indicate a slight change in condition (4); nothing in my paper (1) indicated otherwise (4).

To determine the sensitivity at any value of a stimulus for any device (including an analytical balance (3)(4)) with or without numerical readout, one simply records readings for two or more values of the stimulus differing by small amounts and computes the quotient of the change(s) in the readout divided by the corresponding change(s) in the stimulus, as described in dictionary definitions (5)(6). Nothing in the response/stimulus interpretation requires one to dismantle an instrument to determine its sensitivity.

The slope interpretation of sensitivity is simply a way to quantify the subjective meaning of the term and in no way interferes with the conventional use of the words, sensitivity or sensitive. In contrast, association of these words for "living things and measuring systems" with "two factors" (4), namely, measurement error divided by the response per unit of stimulus, associates these terms exclusively with measured amounts of stimuli (e.g., _[D]0 (3)), a result with profound consequences. For example, using this interpretation, the terms could no longer be used in a subjective sense such as purely qualitative chemical analyses, alarms, light activated door openers, remote controls, responses of our skin to sun, etc., for which the sole function is to detect, sense, or respond to events or stimuli without in any way measuring their magnitudes (8).

The comment related to "minimal information" (1), far from de-emphasizing the importance of "precision" (4), emphasizes the association of sensitivity with complete precision and error profiles rather than just one value (_[D]0) of just one factor (imprecision) that influences the error profile.

Any confusion associated with the use of sensitivity results from and resides with those who convolute the term with measurement error. By treating sensitivity and measurement error separately, the IUPAC interpretation leads to a perfectly logical conclusion that a low sensitivity coupled with small measurement error can result in a small detection limit. However, by convoluting these two factors (4), the imprecision interpretation leads to a semantic contradiction. For example, substituting the equality, sensitivity = "the imprecision of the zero dose measurement (_[D]0)" (3), into the statement "... maximal sensitivity is achieved when the imprecision of the zero dose measurement (i.e., _[D]0) is least... " (3), yields, "maximal sensitivity is achieved when sensitivity is least". It is difficult to understand how this conclusion or replacement of "standard deviation" with another term as a quantitative descriptor of imprecision will reduce semantic confusion.

Finally, the second sentence under "Linear dynamic range" (1), should read, "If one knows the sensitivity of a method and the values of the signal or other measurement objective corresponding to the lower and upper ranges of linearity, then the concentrations corresponding to the lower and upper limits of the linear range can be computed as the quotient of each signal divided by the sensitivity". Having informed the authors of this correction, it surprising that the meaning remains unclear.

summary
The response/stimulus and IUPAC interpretations result from literal applications of explicit statements in dictionary definitions (5)(6) and the scientific literature dating to the latter part of the 19th century (1)(7)(8)(9)(10). They do not convolute factors that are best considered separately for both subjective and quantitative uses of the terms, and they yield straightforward, unambiguous information for any situation for which response/stimulus data can be obtained.

Readers, editors, and official committees are not expected to be "intimidated" (4) by the teachings of anyone, regardless of the area of expertise. However, given that this discussion focuses on analytical systems (3), it seems reasonable that all should value the opinions of analytical chemists whose teachings do not focus on a single application but are formulated to be applicable to the widest possible range of situations.

References

1. Pardue HL. The inseparable triad: analytical sensitivity, measurement uncertainty, and quantitative resolution. Clin Chem 1997;43:1831-1837. [Abstract/Free Full Text]
2. . IUPAC. Analytical Chemistry Division. Nomenclature, symbols, units and usage in spectrochemical analysis-II. Data interpretation. Anal Chem 1976;48:2294-2296.
3. Ekins R, Edwards P. On the meaning of "sensitivity". Clin Chem 1997;43:1824-1831. [Abstract/Free Full Text]
4. Ekins R, Edwards P. On the meaning of "sensitivity": a rejoinder. Clin Chem 1998;44:xxxx-xx.
5. The Oxford English dictionary, 2nd ed. Vol. XIV. Oxford: Clarendon Press, 1989:986..
6. Webster's new international dictionary, unabridged. Springfield MA: G & C Merriam Co., 1961:2068..
7. Maxwell JC. A treatise on electricity and magnetism, 3rd ed. Oxford: Clarendon Press, 1882:351,360..
8. Maxwell JC. An elementary treatise on electricity, 3rd ed. Oxford: Clarendon Press, 1881:186,194..
9. Treadwell FP, Hall WT. Analytical chemistry, Vol. 2. New York: Wiley, 1919:7–8..
10. Kolthoff IM, Sandell EB. Textbook of quantitative inorganic analysis 1949:202-203 Macmillan New York. .
11. Websters seventh new collegiate dictionary, Springfield, MA: G & C Merriam Co., 1963:223..
12. Lim KB, Pardue HL. Error-compensating method for enzymatic determination of DNAs. Clin Chem 1993;39:1850-1856. [Abstract]
13. Foley JP. Systematic errors in the measurement of peak area and peak height for overlapping peaks. J Chrom 1987;384:301-313.