Anal. Chem. l g 8 3 , 55, 1424-1426
1424
1
m/z 344 Fey-
-340 345 350 m/z
Flgure 3. Portion of a negative ion EH mass spectrum showlng the
isotope cluster of Fey- ( m / r 344). NaI. While the intensity of these ions is as much as 21% of the analyte ion intensity, their absolute abundance is small. Registry No. H4Y, 60-00-4; H,YR, 85479-87-4; H2YR2, 85479-88-5;HYR3,85479-89-6;YR4,85479-90-9;Fey-, 15275-07-7; Fe2(SO4),,10028-22-5;glycerol, 56-81-5.
LITERATURE CITED (1) Segur, J. 8. In “Glycerol”; Miner, C. S.,Dalton, N. N., Eds.; Reinhold: New York, 1953; Chapter 7. (2) Weast, R. C. “Handbook of Chemistry and Physics”, 55th ed; CRC Press: Cleveland, OH, 1974. (3) Barber, M.; Bordoli, R . S.;Elliott, G. J.; Sedgwlck, R. D.; Tyler, A. N. Anal. Chem. 1982, 54, 645A-657A. (4) Martin, S. A.; Costello, C. E.; Biemann, K. Anal. Chem. 1982, 5 4 , 2362-2366. ( 5 ) Burllngame, A. L.; Aberth, W.; Straub, K. M. Anal. Chem. 1982, 5 4 , 2029-2034. (6) Stlmpson, B. P.; Evans, C. A., Jr. J . Necfrost. 1978, 5 , 411-430. (7) Stlmpson, 8. P.; Evans, C. A., Jr. Biomed. Mass Spectrom. 1978, 5 , 52-63.
(8) Simons, D. S.;Colby, B. N.; Evans, C. A,, Jr. Jnt. J . Mass Spectrom. Ion PhyS. 1974, 15, 291-302. (9) Field, F. H. J . Phys. Chem. 1982, 86, 5115-5123. (10) Chan, K. W. S.; Cook, K. D. J . Am. Chem. SOC. 1982, 104, 5031-5034. (11) Stimpson, E. P.; Simons, D. S.; Evans, C. A,, Jr. J . Phys. Chem. 1978, 82, 660-670. (12) Schwarzenbach, G. “Complexometrlc Titrations”; Interscience: New York, 1957; p 8. (13) Lal, S. T. F.; Evans, C. A., Jr. Org. Mass Spectrom. 1978, 13, 793-734 . - - . - .. (14) Lai, S. T. F.; Evans, C. A., Jr. Biomed. Mass Spectrom. 1979, 6 , 10-14. (15) Lai, S. T. F.; Chan, K. W.; Cook, K. D. Macromolecules 1980, 13, 953-956. Dyer, J. R. “Applications of Absorptlon Spectroscopy of Organic Compounds”; Prentice-Hall: Englewood Cliffs, NJ, 1965; Chapter 3. Basolo, F. In “The Chemlstry of the Coordination Compounds”; Ballar. J. C., Jr., Ed.; Reinhold: New York, 1956; Chapter 12. Chan, K. W. S.; Cook, K. D. Anal. Chem. 1983, 55, 1306-1309.
Kelvin W. S. Chan Kelsey D. Cook* University of Illinois School of Chemical Sciences and Materials Research Laboratory 44 Roger Adams Laboratory, Box 49 1209 West California Street Urbana, Illinois 61801
RECEIVED for review January 24,1983. Accepted March 24, 1983. This research was supported by the National Science Foundation (Grant DMR-80-24632jointly funded by the U.S. Army Research Office). The Materials Research Laboratory is supported in part by the National Science Foundation (Grant DMR-80-20250).
Lowest Limit of Reliable Assay Measurement with Nonlinear Calibration Sir: A recent paper by Oppenheimer, Capizzi, Weppelman, and Mehta ( I ) shows the degree of sensitivity of the “lowest limit of reliable assay measurement” (LLORAM) to the choice of criteria and to the methods of calculation employed. In particular these authors demonstrate the necessity of taking into account any nonuniformity of measurement variance along the calibration (standard) curve if this occurs. Their work focuses on three lowest limit criteria all of which had been introduced earlier by Currie (2): the critical level LC,, the lowest response above which an observed response may be reliably recognized as detected; the determination limit LDw,the a priori actual response which may be expected to lead to detection as specified by the level LC,; and the determination limit LQ,, the lowest response for which the coefficient of variation (CV), i.e., relative standard deviation, is sufficiently small for quantitative determination. Also defined are the analyte quantities XC,, XD,, and X Q , corresponding to Lcw, LDw,and LQ,, respectively, through the calibration curve. Currie’s (2)original formulations apply to cases of uniform variance of the response variable and Oppenheimer et al. ( I ) remove the restriction of uniformity by introducing least-squares weighting factors. However, both Oppenheimer ( I ) and Currie ( 2 ) treat only techniques which generate linear calibration lines. While this condition is a very common special case, there are a number of analytical methods which yield decidedly nonlinear response vs. analyte curves and one purpose of this paper is to extend the generality of the LLORAM procedures one step further to these cases. We will consider lowest limit estimates derived from nonlinear
calibration curves having arbitrary form and having been constructed from standard sample measurements exhibiting nonuniform variance. Yanagishita and Rodbard (3) have studied this same problem for the particular technique of radioimmunoassay. A second purpose here is to suggest an alternative specification for the determination limit which may be more useful than the above definition in some circumstances. Currie’s (2) original developments were based on net responses ynet,i.e., the signal obtained after subtraction of the “blank” from the “gross” response. This subtraction has the effect of shifting the calibration line, ynetvs. x , to pass through the origin or to within statistical proximity of the origin. Also the standard deviation snetof the net response is generally not zero so that the coefficient of variation snet/ynetincreases perhaps without limit as ynetdecreases toward zero. Hence there is usually no problem finding some net signal LB such that the coefficient of variation is less than some arbitrary value CQ (say, 0.10) for ynet> LQ, and greater than CQ for ynet< LQ,. However, it is statistically advantageous not to subtract the blank from the other standard sample responses but simply to include the blank as one of the points used to construct the calibration curve, ycd vs. x . When this is done, the calibration curve generally does not pass near the origin and the coefficient of variation based on yd does not necessarily become very large as ycd decreases. Hence it may be impossible to find a determination limit LQ, corresponding to an arbitrary relative precision CQ. Such a case is shown later in the illustrative example. An alternative procedure suggested by reviewers
0003-2700/83/0355-1424$01.50/00 1983 American Chemical Soclety
ANALYTICAL CHEMISTRY, VOL. 55, NO. 8, JULY 1983
of this paper is to base the determination limit not on the precision of the response but on the precision of the estimated assay. If the calibration curve does not cross the x axis except perhaps within statistical proximity of the origin, this option avoids the above difficulty. It also has the further appeal that assay precision, not response precision, is the primary concern to analyelts and so is a more desirable criterion for a determination limit. One would like to define a limit analogous to L, such that the relative standard deviation of a predicted assay be sufficiently small, but this definition leads to calculational difficulties. The statistical uncertainty of the assay depends both on the variance of the unknown sample response and on the variance inherent in the calibration curve. A standard error estimate for an assay determined through a nonlinear calibration curve can be approximated by a Monte Carlo method ( 4 ) . But to prescribe a fixed value CQfor the coefficient of variation and solve for the corresponding assay is awkward. Alternatively, taking into account that most analysts do not require a determination limit calculated to high precision, we offer the following suggestion: By use of digital computer techniques already reported (5),it is possible to calculate upper and lower confidence limits X u and XL, respectively, for an assay X determined from a measurement Y through a nonlinear calibration curve. X is not generally centered in the X u , X z interval because its distribution function is not symmetrical even if the Y distribution function is symmetrical ( 4 , 6). Nevertheless, suppose under extraordinary circumstances am assay X , characterized by a normal (Gaussian) distribution is found. The corresponding confidence limits woluld be X,u = X, ts, and XnL - ts, where t is the appropriate Student’s t statistic and s, is the standard error estimate of X,. The coefficient of variation CV(X,) in this case would be s,/X, = ( X n u- X n L ) / 2 t X n . We propose to define an “effective” coefficient of variation CV’(X) of a nonnormally distributed assay X by
0 . 2 1 . 2
’
(1)
where tg6is the two-sided Student’s t statistic for the 95% confidence level. A specific confidence level must be chosen here because t does not cancel from this expression if X is not normal. We note that CV’, like C V , is an expression of relative precision and CY’ approaches CV in the limits both as the calibration curve approaches linearity and its variance becomes small. To find a determination limit based on eq 1we will calculate CV’(X) values for assays corresponding to arbitrary response levels which span the entire range of the calibration curve and then construct a table or plot of CV’(X) vs. X or Y. A glance at this table or plot would perhaps be sufficient to establish an approxiimate assay or response level that has satisfactory precision. A more precise determination limit corresponding to a prescribed C, could be estimated by tabular interpolation or by a curve-fitting procedure on the plot.
COMPUTATIONAL METHODS Brief suimmaries of the computations required to find the four LLOFlAM quantities are described here. These procedures result from merging equations and techniques detailed in Oppenheimer’ie (1)Appendix with those detailed in ref 51. The critical level and detection limit are illustrated in Figure 1 of ref 1 and also in Figure 1 of the present paper. The critical level Lcwis the intersection of the y axis with the upper confidence band of the calibration curve and Xcr7 is the assay projected through the calibration curve corresponding to a single response measurement having value Lew. Lcwis calculated as the quantity y,&) + t,var[yCd(x)J with x = 0 and where var[y,&)] i s given by one of the forms off eq 8 of ref 5. Then X,, is calculated as the solution of the equation Lew = y,&) using a Newton-Raphson iteration.
‘
.
*
8
6
’
‘ ia
DOSE
0.30
W v)
z
0
%
0.28
W
u
I
+
CV’(X) = (X, - XL)/2Xt,,
’
*
4
1425
0
0.2
1
’CW
0.4
D ’W
0.6
DOSE
Flgure 1. Response v8. dosage data used in the illustrative example and shown in Table I are the dlscrete points in Figure 1A. The solid lines are the calibration curve, eq 3. The dashed curves are upper and lower confidence bands. Figure 1B is the low dosage region of Figure 1A.
The assay detection limit X h is the lower confidence limit of the assay corresplonding to the response Le,. Thus X ~ D , is the solution of the equation Lew= yd(x) - tpvar[yd(x)l and this solution also requires an iterative procedure. The response detection limit LD, is ycd(XDw). The determination limiting assay XQ, is found as the numerical solution of the equation CQyc&) = (~ar[y,&)])’/~ for some prescribed coefficient of variation C ., Then the response LQ, is yd(XQw). To estimate a determination limit based on the effective coefficient of variation of the assays, we first choose a series of responses Yithat span the range of interest for the determination limit. For each such Yiare solve the equation Y, = yd(x) for the corresponding assay X, and also solve either eq 10a or 10b of ref 5 with t , = tggand with Ni = 1. Each such solution yields a pair of confidence limits which identify with Xu and X b Substituting thetie results into eq 1 above we calculate a series of CV’(Xi) values which are tabulated or plotted. Whereas for the linear case examined by Oppenheimer ct al. ( I ) , noniterative results can be calculated exactly for Xcw,Lcwand approximately for the other two limits, the nonlinear calibration curve requires iterative solutions for all LLORAM results. But since the construction of the calibration curve itself requires a digital computer effort, these additional iterations do not demand any increased computational power.
ILLUSTRATIVE EXAMPLE To show the determination of LLORAM values in a case characterized by both nonlinear calibration and nonuniform
1426
ANALYTICAL CHEMISTRY, VOL. 55, NO. 8, JULY 1983
0.5
Table I. Data for Illustrative Example ( 7 ) dose response response X ia Tib variance x l o 3 0.268 0.295 0.313 0.354 0.431 0.494 0.554 0.601
0 0.5 1 2 4 6 8 10
0.4 CVCY) OR CV'CX)
0.080 0.044 0.036 0.045 0.11 0.16 0.38 0.40
a Micrograms of standard protein per tube. 12 replicated absorbance measurements.
0.3 0.2
.
0.1
CV'CX) '
.
.
.
X
,
,
Mean of
I
0.2
0.4
measurement variance, we exhibit in Table I a summary of spectrophotometric protein assay data using aqueous bovine serum albumin standards as measured by Long et al. (7). Each standard dosage xi was replicated 12 times and the means j$ of the 12 responses are shown along with the response variances. The data points plotted in Figure 1A are the j$ vs. xi values and these obviously cannot be regarded as linearly related. Also the variance of the response is nonuniform, passing through a minimum near dose xi = 1pg and increasing rapidly at higher dosages. We decided to model the variance by a quadratic function of the response which turns out to be
+
var(y) = (0.592 - 3 . 3 6 ~ 5 . 1 5 ~ ~ X)
(2)
This function adequately represents the essential features of the variance behavior. A polynomial calibration curve with a priori unspecified degree is selected by x2 testihg (8) at the 95% confidence level as described in ref 5 and this is found to be 0.270
.
X
0.6
DOSE,
ycal=
I
i
.
u
'
a
CVCY) . * ( , * * . , . . . * + . 1 * l + +
+ 0 . 0 4 4 5 ~- 0 . 0 0 1 1 4 8 ~ ~
(3)
which is plotted as the solid curve in both parts of Figure 1. LLORAM values are found as follows: To find a critical level Xcw,Lcwwe have selected a 95% confidence level for the upper confidence band of the calibration curve. The appropriate Student's t statistic depends on the number of degrees of freedom in the variance function. In this case we will follow Rodbard et al. (9) and pool all the degrees of freedom associated with the data from which eq 2 was derived. Having utilized 96 data points to find three parameters in the variance function, we have 93 degrees of freedom. The value of t(l.a)in Oppenheimer's (1)eq 2 is 1.661 which is the one-sided upper 5 percentage points of the t distribution. At a dose x = 0 this upper confidence band, shown as the upper dashed curve in both parts of Figure 1, corresponds to a response Lcw of 0.283. Then using the calibration curve itself, this response projects to a dosage XcW of 0.297 pg as drawn in Figure 1B. T o calculate a detection limit XDw,LDwwe again use t = 1.661 to construct a lower 95% confidence band to the calibration curve as drawn as the lower dashed curves in Figure 1. By projecting Lcw = 0.283 through this lower band, we find XDw= 0.567 pg and the predicted response for this dosage is 0.295 = LD,. The determination limit XQ,,LQ,is that point along the calibration curve a t which the coefficient of variation CV(y) is a sufficiently small fraction of the response. We have calculated this coefficient of variation from eq 2 for a range of responses spanning the range of the calibration curve and
Flgure 2.
variation
0.8
.0
x
Effective coefficient of variation CV'(X) and coefflcient of calculated for the Illustrative example.
CV&)
show these results as the lower curve in Figure 2. We observe that in this example CV(y) is virtually constant. The higher relative precision that one would expect a t higher response levels if the response variance were uniform is lost because of the rapidly increasing actual variance. Consequently, there is no meaningful Lgwdetermination limit in this case. On the other hand, the effective coefficient of variation CV'(X) as defined by eq 1 is better behaved. Using t,, = 1.986 corresponding to a two-sided 95% confidence level and 93 degrees of freedom, we calculate CV'(X) values as shown as the upper curve in Figure 2. This quantity is also essentially constant except for X less than about 1pg where the relative precision deteriorates rapidly. If one could accept a precision of, say, 20% of the assay for quantitative determination, we see from Figure 2 that this limit occurs a t a dose near 0.8 kg and the measurement response of a single sample having this dose would be about 0.3. Digital computer listings of FORTRAN IV program codes which carry out the nonlinear calibration, nonuniform variance assay calulations were offered to readers of ref 5 . Those programs have been revised to perform the LLORAM calculations described above. Listings may be obtained free of charge on request to the author.
ACKNOWLEDGMENT Raw data from ref 7 were kindly made available by Leonard Oppenheimer of Merck Sharp & Dohme Research Laboratories, Rahway, NJ. LITERATURE CITED (1) Oppenhelmer, L.; Caplzzl, T. P.; Weppelman, R. M.; Mehta, H. Anal. Chem. 1983, 55, 638-643. (2) Currle, L. A. Anal. Chem. 1968, 4 0 , 586-593. (3) Yanagishita, M.; Rodbard, D. Anal. Biochem. 1978, 8 8 , 1-19. (4) Schwartz, L. M. Anal. Chem. 1976, 48, 2287-2289. (5) Schwartz, L. M. Anal. Chem. 1979, 51, 723-727. (6) Schwartz, L. M. Anal. Chem. 1977, 4 9 , 2062-2068. (7) Long, R. A,; Weppelman, R. M.; Taylor, J. E.; Tolman, J. E.; Olson, G. Biochemistry 1981, 2 0 , 7423-7431. (8) Browniee, K. A. "Statistical Theory and Methodology In Science and Englneering"; Wiley: New York, 1960. (9) Rodbard, D.; Lenox, R. H.; Wray, H. L.; Ramseth D. Clin. Chem. (Winston-Salem, N.C.)1978, 22, 350-358.
Lowell M. Schwartz Department of Chemistry University of Massachusetts Boston, Massachusetts 02125 RECEIVED for review February 1, 1983. Accepted April 13, 1983.
