to eq (5) as the absolute value of the ratio of the | slope K=(M2-M1)/(Q2—Q1) to the standard deviation of M, M. The larger the sensitivity, the more useful will be the test method M for the characterization of Q. It should be noted, however, that in the general case, K is no longer constant but varies with the value of Q. Thus, even in cases in which the experimental error (measured by σ) remains constant, the sensitivity may vary with the value of Q. Only when the error is proportional to K is the sensitivity constant.

If the properties under consideration cannot be expressed by means of a single criterion Q, it is not possible to determine the absolute sensitivity of a method of test. It is possible, however, to determine the relative sensitivities of two or more methods used to characterize these properties. This important application of the sensitivity concept can best be

does not involve the quantity Q, and the sensitivity ratio can be used to compare the measurement of tensile stress [9] and the measurement of strain [4]. The relationship between these two methods of measurement for a GR-S synthetic rubber compound, according to Roth and Stiehler [4], is given by the equation: SE" C (8)

where S represents tensile stress, E represents strain, and n and C are constants for any particular type of vulcanizates.

If the logarithmic derivative is taken, it follows that

criterion Q exists, and two alternative measuring methods M and N, both related to Q, are to be compared. For example, density and refractiveindex methods for determining the bound styrene in GR-S may be compared without knowing the actual percentage of bound styrene. Let and be the sensitivities corresponding to the two methods. From eq (5) it follows that the ratio of the sensitivities is given by

measurements of tensile stress would detect variations in the vulcanizates better than measurements of strain. However, Roth and Stiehler [4] show that the error of measurement of strain is much smaller than that of the usual measurement of tensile stress; hence, the sensitivity of strain measurements is

greater.

From eq (9) it follows that the slope of the strain versus tensile-stress curve is

(7)

Thus K' is the slope of a curve of M plotted as a function of N. From eq (5) it follows that the dimension of sensitivity is that of 1/Q, since σ has the dimension of M, and K is of dimension M/Q. On the other hand, the ratio of the sensitivities of alternative test methods given in eq (6) is dimensionless. This fact, as well as eq (7), shows that the comparison of two methods, by means of the ratio of their sensitivities, does not necessitate a knowledge of their relation to the theoretical Q. All that is required is a knowledge of their mutual relationship.

In the case of bound styrene, the relation between density and refractive index can be established from a series of samples of different bound styrene contents without a knowledge of bound styrene in any sample. Of course, the bound styrene content could be determined by some absolute method, and the absolute sensitivities of the refractive index and density methods for measuring this property could be established.

In the case of stress-strain measurements, on the other hand, the characteristic-degree of vulcanization-cannot be represented by a single quantity Q and consequently no absolute sensitivities for either method can be calculated. Nevertheless, relation (6), with K' given by (7), can be applied, since it

=

(10)

This expression is found to exceed unity, as shown in table 1, which lists data pertinent for the calculation of the sensitivity ratio, for tensile-stress and strain values obtained in three different plants and for two cures [10]. It should be noted that the ratio of the two sensitivities varies with the degree or time of cure, since the factor E/nS decreases as vulcanization progresses. The advantages of the strain test are therefore greatest for tests on vulcanizates that are undercured. The data also show that the greater sensitivity of the strain test is due to its better reproducibility.

It should be noted that the application of the sensitivity criterion in comparing two test methods implies that a definite functional relationship exists between the properties measured by the two methods. This restriction is not introduced by the sensitivity concept, but rather a limitation inherent in any valid comparison. If a characteristic Q can be adequately measured by two different methods M and N, both methods must be functions of Q and therefore functionally related to each other. In many cases, M and N, in addition to depending on Q, will also depend on other factors not common to both. A comparison of M and N for the determination of is then only valid under conditions in which the results yielded by M and N are solely governed by variations in Q, i. e., all noncommon factors must be held constant for all samples involved in the comparison. Failure to satisfy this condition will result in data of M and N that may well show significant correlation, but not necessarily a definite functional relationship either with each other or with the characteristic Q.

It is also important to note that the functional relationship assumed to exist between the methods M and N need not be known for the application of the sensitivity criterion.

In the example shown in table 1, the numbers of degrees of freedom used in the estimation of the standard deviations ranged from 38 to 48. Examining the data of plant A and the 100-minute cure, for which there were 48 degrees of freedom for each standard deviation, Fo, at the 5 percent level of lower confidence limit of the sensitivity ratio equals

Ratio

where K' is the slope of the curve of M versus N in the region of the curve at which the comparison is made. If this ratio exceeds unity, M is superior to N. Since, in general, both K' and the quantities

σ and σχ will be determined experimentally, the

ratio can only be approximated, and its estimate will be subject to random fluctuations.

In practice it is fortunately quite often the case that the two tests are carried out on the same sample or in such a manner that their relationship is known with much higher precision than either of the two measurements. Thus, a comparison of the relative merits of measuring the rate of tread wear of tires by weight loss or by depth loss can be made by measuring both losses on the same tire. While either of these experimental quantities depends on highly variable climatic and road conditions, the relation between the two is practically free from these effects because both are obtained under the

same identical conditions.

From this value it can be concluded that strain, even in the least favorable of the cases examined, is at least as sensitive as stress, and most likely more sensitive.

If the experimental error in the estimate of the slope K' is not negligible, the above test of significance is not valid. In such cases, the correct statistical procedure for testing the significance of the sensitivity ratio depends on the type of relationship between the two test methods (linear, quadratic, logarithmic, etc.) as well as on the design of the experiment used to establish the relationship. No attempt is made in this paper to deal with the statistical theory for these more complex situations.

5. Effect of Scale of Measurement

There exist many cases in which measurements of physical or chemical properties can be expressed in more than one scale. For example, in measuring the light-absorption characteristics of materials, the results can be expressed either in optical density or in percentage transmittance. Another example is the measurement of refractive indices: In many instruments, a scale is provided that allows the direct reading of the refractive index rather than the angles of refraction and of incidence. In these cases the different scales of measurement correspond to functionally related quantities, but the functions

relating them are not linear. An important advantage of the sensitivity concept is its nondependence on the scale of measurement. The standard deviation, being expressed in the same units as the measurement, has a value that depends on the unit and scale in which the measurement is expressed. The coefficient of variation, which is defined as the ratio of the standard deviation to the mean value, is nondimensional, because both these quantities are expressed in the same units. However, except for scales that are proportional to each other, the coefficient of variation is dependent on the scale in which the measurement is expressed.

Consider, for example, the logarithmic transformation of a measurement y:

It is evident, from this formula, that the coefficient of variation of z, σ,/2, is in general different from that of y, oly. It can be shown that the only transformation that leaves the coefficient of variation rigorously unaltered is a proportional transformation: z=ky, i. e., a simple change of units. (To the extent that the approximate expression σ2 = dz/dyo, is applicable [for details see 12, secs. 27.7 and 28.4]-the coefficient of variation is also unaltered under the transformation z=k/y.)

On the other hand, the sensitivity of the transformed variable z, for any transformation

[1] R. G. Newton, Proc. Second Rubber Technol. Conf., p. 233, Institution of the Rubber Industry, London (W. Heffer & Sons, Ltd., Cambridge, England, 1948). [2] M. C. Throdahl, J. Colloid Sci. 2, 187 (1947); Rubber Chem. and Technol. 21, 164 (1948).

[3] J. H. Dillon, Physics 7, 73 (1936); Rubber Chem. and Technol. 9, 496 (1936).

[4] F. L. Roth and R. D. Stiehler, J. Research NBS 41,, 87 (1948) RP1906; India Rubber World 118, 367 (1948); Rubber Chem. and Technol. 22, 201 (1949).

[5] J. M. Buist and O. L. Davies, Trans. Inst. Rubber Ind. 22, 68 (1946); Rubber Chem. and Technol. 20, 288 (1947).

[6] R. G. Newton, J. R. Scott, and R. W. Whorlow, Proc. Intern. Rheol. Congr. II, 204; III, 61 (1948).

[7] E. Reichel, Z. anal. Chem. 109, 385 (1937). [8] W. E. Deming, Statistical adjustment of data, ch. III (John Wiley & Sons, Inc., New York, N. Y., 1943). [9] ASTM Book of Standards, pt. 6, D-412-51T, p. 134 (American Society for Testing Materials, Philadelphia, Pa., 1952).

[10] Private communication from the Office of Synthetic Rubber, Reconstruction__Finance Corp.

[11] R. L. Anderson and T. A. Bancroft, Statistical theory in research (McGraw-Hill Book Co., New York, N. Y., 1952).

[12] H. Cramer, Mathematical methods of statistics (Princeton University Press, Princeton, N. J., 1946). WASHINGTON, February 8, 1954.