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chem. 1991, 63,1270-1270
between single and parallel columns. (The higher efficiency observed when the Carbowax column is operating in parallel may be the result of two factors: (a) owing to the lower flow rate, only one-third of the total injected sample is channeled through this column, and (b) the lower carrier velocity of 20 cm/s is near the optimum level for 0.5pm liquid film thickness.) CONCLUSIONS There are a number of ways in which the difference in carrier velocities and capacity factors between two parallel capillary columns of dissimilar liquid phase may be set 80 that the resulting chromatograms do not overlap. Simply using two columna of unequal length or unequal inside diameter or both is not as satisfactory as using columns of unequal liquid-phase thickness and adding a retention gap to the column with the thicker liquid phase. Samples containing several components of low polarity can be satisfactorily resolved on the two-column system if the low thickness phase is the polar column (Carbowax 20M). The reverse holds for high polarity samples. Samples of intermediate polarity and narrow polarity range, or containing fewer components, can be satisfactorily resolved on either system. LITERATURE CITED (1) Schombwg, 0.;Husmann, H. ClwometoqepMe 1975,8 , 36. (2) Jennlngs, W. (3. Ana&?!caI Gas chrometobxephy; Academic Press: New York, 1980:Chapter 1.
(3) WtSCh. W. TwOdknsnsknel T e c h n i q ~ .R M t I n Cepillarv Qas Clwometopaphy; Huethb Betlag: Heidelberg. Germany, 1981;pp 3-56. (4) Mdven. U. F.; Cooper, W. J.; W a n . M.; Maz. R. J . &$I Res. chromaw.1984, 7,639-40. (5) Jennlngs, W. G.; Shlbamoto, T. OuelldpbrveAna!~sl6of Flew and Frawnce VdenVes by &I? Capby c)vcwnatOgaphy;Academlc Press: New York, 1980,chepter I. (6) TSI, M. Y.; CatherlM. C. Ckh. chem. 1989, 35, 1989-1991. (7) Tsal, M. Y.; Oliphant, C.; Josephson, M. W. J . Cfiromatogr. 1985, 347,l-10. (8) Watts, V. W.; Simonlck, R. F. J . Ana/. Toxlc4. 1088, 10, 198-204. (9) Perrigo, B. J.; Peel, H. W.; Ballantyne, D. J. J. chrometog. 1985, 341.81-88. (10) Baker, J. K. Anal. Chem. 1977. 49, 906-910. (11) Dulnker, J. C.; Schulr, D. E.; Petrlck, 0. Anal. Chem. 1988. 60, 478-482. (12) Schulte, E.; Malisch, R. Fresenius' Z . Anal. Chem. 1989, 311, 545-555. (13) Sefe, S.;Bandlera. S.; Sawyer, T.; Robertwon. L.; Sage, L.; Parkinson, A.; Thomas, P. E.; Ryan, D. E.; Reik, L. M.; Levkr, W.; Denomme, M. A.; Flgb, T. EHP. EnvLon. Heellh Perspect. 1985,60,47-56. ( 14) Leece, B.; Denomme, M. A.; Towner, R.; Angela, Ll S. M.; Sage, S. J . T o x h l . &&on. Health 1965, 16, 379-388. (15) M y . M. J. E., In Gas " I t O g * e p h y ; Coates. V. J., Noebels, H. J., Gerguson, I. S., E a . ; Academic Press: New York, 1958; pp 1-13. (16) Hlnshaw, J. LC-GC 1989, 7,237-240. (17) Enre, Leslle Open Tubhr cduvnns I n Ges Chromatography; Plenum Press: New York, 1965;Chapter 2. (18) Jennlng, W. G. Ana&tkal Gas chrometogrephy; Academlc Press: New York, 1980;Chapter 6.
RECEIVED for review January 7,1991. Accepted March 28, 1991.
Confidence Limits for the Abscissa of Intersection of Two Least-Squares Lines Such as Linear Segmented Titration Curves Kenneth N. Carter, Jr.,* Dan M. Scott,' Jon K. Salmon, and Gregory S. Zarcone Division of Science, Northeast Missouri State University, Kirksville, Missouri 63501
The confidence limits for the abscissa of the Intersection of two Ieadsquarw lines, such as linear segmented tltratlon curves, are cakuiated by udng a rknple method, based on the Interuction of confidence bands, that always overedlmates the wldth of the confidence Interval, and by'udng a more accurate computatlonally Intenrlve method based on Integratlon of the transformed bivariate normal distribution (Le.,Creasy's method). Numerical results obtalned by a p plying the forogolng methods to experhntai and simulated data are compared wlth thoro obtained by Fkller's theorem and H o r d o r propagation of variance as previously applied in the primary chemical Mrature. A dkcrepancy k resolved by correcting the poollng of varlance In the prior work. Rebtlonrhlpr between the various methods are shown. Fldkr's method, wlth proper pooling, gives reliable results unless the data are very noisy, whereas Creasy's method glves finlte inciuslve confldence intervals even when Fkiier's method falls.
INTRODUCTION For many analytical methods, including conductometric, amprometric, and spectrophotometrictitrations and the Gran
* To whom corres ondence should be addressed.
De artment o f Industrial Engineering, University of Nebrasia-Lincoln, Lincoln, NE 68588.0518. 0003-2700/91/0383-1270$02.50/0
plot method for potentiometric titrations, the titration curve consists (essentially)of two joined lines, the changeover point being the endpoint. This end point is estimated by the abscissa of intersection of the least-squares straight lines that are fit to the experimental points for each segment. We will aasume that the assignment of each point to one line segment or the other is known, and that the abscissa of intersection will therefore lie between the largest x value of one set and the smallest x value of the other. (We will later examine this very big assumption.) At the point of intersection, the two lines y1 = al + blx and yz = a2 + b2x have the same ordinate, from which it follows that the abscissa of intersection, henceforth denoted by uppercase X,is given by Random errors in the points produce uncertainty in the slopes and intercepts of the lines and therefore in the point of intersection. The uncertainty of the end point can be expressed as a confidence interval for the abscissa of intersection. The probability that a confidence interval contains the true value is equal to the confidence level (e.g., 95%) chosen. Our initial cursory search of the literature via the DIALOG Chemical Abstracts database in the mid 1980s did not reveal any published methods. Our first approach, based on intersecting confidence bands about each line, gave intervals that were clearly too wide, but 0 1991 American Chemical Soclety
ANALYTICAL CHEMISTRY, VOL. 83, NO. 13, JULY 1, 1991
Table I. Statistics for the Illustrative Example of Ref 1
Line 1
Line 2
nl = 6
n2 = 7 Ew1= 5.065 x lW" E w ~ 4.423 X lo-' SII(l)= 5.8987 X lo-' suc2,= 7.0636 x 10-3 X2 25.746 i1 8.786 92 = 0.4690 91 0.5881 X = 16.3668 mL b2 = 0.010368574 bl = -4.028 534 503 a2 = 0.202059423 a1 = 0.838776416 V , = 1.44379 X Vz 4.05454 X lo-"
V-
= 8.6694 X lo-""
Calculated by Using Nonpooled Values for var 9, = 2.851 X var 92 9.168 X var b, = 2.448 X 10" var b2 = 5.740 X var a, = 2.174 X 10" var a2 = 3.896 X
V 10" lo4 lo4
Calculated by Using Vpoolda var Y1 = 1.7116 X var Y2 = 1.960 X var bl = 1.4697 X 10" VBT bz 1.2273 X 10-8
var a, = 1.3057 X 10" var Aa = 9.637 X 10" cov (Aa,Ab)= -4.451 X sx = 0.03891
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execution introduce further confusion. Having several methods for comparison, fast computers for reliable execution, and good access to the literature helps us clear up the confusion and suggest reliable guidelines for use. Our simple conservative method proved useful mainly for catching mistakes. On the other hand, our integration method gives excellent agreement with Fieller's method for small to moderate variances and a lower bound when the variances are huge. Fieller's method is usually quite adequate, but the integration method has the advantage of allowing comparison of precision even in the cases where the variance is so large, or the confidence level chosen so high, that other methods fail to give a finite interval.
THEORY Intersecting Confidence Bands. Our first method is crude and conservative (i.e., pessimistic). Around each line, confidence bands may be calculated for the mean response by using the standard expression (15)
VU a2 = 8.3312 X lo4 var Ab = 2.697 X 10"
Fieller Quadratic?
+
1.513311 19 X 10-3p- 4.95359444 X 10-2X 4.053592 20 X lo-' = 0 95% confidence limita from the quadratic: 16.455 and 16.279 mL
"Emended from ref 1. suggested the more exact approach of integrating the density function. We had just worked out the math for this integration when a second, more thorough search of DIALOG (in 1986) revealed work by Schwartz and Gelb (1) on statistical uncertainties of end points at intersecting straight lines for titrations. They present two methods: firsborder propagation of variance for the abscissa of intersection and application of Fieller's theorem (2-4), previously unknown to us. In their example, the ordinate variance can be considered on a priori grounds to vary systematicallyas a function of position along the curve, so that weighted least-squares analysis is appropriate. We recommend reference to details of their discussion that are not repeated here. A mistake in their expression for pooling the variance estimates is easily corrected. (The emended values are given in our Table I.) In a monograph on chemometrics, Sharaf et al. (5) cite a book by Lark et al. (6) that applies Fieller's theorem correctly to the intersection of lines with constant variance (homoscedastic data). Lark et d. cite Kastenbaum (7)and Fisher (8, 9), who in turn cite Fieller's theorem. Searching for citations of Fieller's theorem (using SCISEARCH) revealed treatments of the problem in the statistical, biometric, and chemical literature. Fieller has presented a more detailed proof (3) of his theorem. Creasy gives an alternative approach based on direct integration (IO),which is contested by Fieller (4) and others (11). Wallace (12) presents a selective review of the Fieller-Creasy problem. Jandera (13) does not cite Fieller but rather considers as limits the extremes of the area in common to the intersecting confidence bands or diverging linear approximations thereof. In addition to reviewing earlier methods, Liteanu (14)calculates limits as weighted averages of the abscissas of the confidence hyperbolas at the ordinate of intersection. Application of different published methods to the same sets of data yielded results differing by more than a factor of 2. Approaches in the chemical literature vary in the rigor of their statistical foundations. Some use simplifications no longer necessary where microcomputers are available. Mistakes in
(where M~ is the mean response, 9 is the ordinate of the regression line, talPis student's t for n - 2 degrees of freedom a t the (1- a)level of confidence (Le., leaving an area of a / 2 to the right); s is generally referred to as the standard deviation of the response about the line, but in our examples it is calculated by using weights in the numerator and is therefore simply the square root of the value V that appears in eq 12 (see below); &wi is the s u m of weights, which simply reduces to n, the number of points, if nonweighted least-squares analysis is used; x is the value of the abscissa a t which the bands are being constructed; x = Cwixi/Cwi is the weighted mean of the abscissas of the data points; and S,, = Ewi(xi Actually, these bands are for the ordinate of the true line at only a single point. If confidence bands for the entire line are desired, the critical constant (2R&,-2)1/2should be substituted for tu/z,resulting in wider bands (16). (Note that tUlzis equal to (F$,-z)1/2.) The bands for some finite segment of the line will be intermediate in width. Methods exist for the construction of such bands, but they are more difficult and not worth the trouble here, since the main purpose of finding a conservative (too wide) confidence interval was achieved without them. (Intervals obtained by using (2&,-2)1/2 are rigorously conservative. Use of tUlPas the critical constant, though not rigorous, resulted in conservative 95% confidence intervals for the data we examined.) If both lines are found within their respective bands, then their point of intersection must lie within the area in common between the bands. If the line segments are assumed independent, the joint probability will simply be the product of the confidence levels for the individual bands. Hence, bands of probability P = 0.9747 intersect to give an area of probability P = 0.950. If the extremes of such an area are projected onto the x axis, a pessimistic estimate of limits on the abscissa is obtained. The source of the pessimism is easy to see by inspection of Figure 1: What we have obtained is actually a simultaneous estimate for X and Y,whereas all we care about is the estimate for X alone, which will be much narrower. To obtain the true probability for X alone, the probability that the intersection occurs at values of Y above or below the bounded area should be also be included. This is accomplished by our method of integrating the statistical density function of X,discussed in the following sections. Before proceeding, we note the superficial similarity of the foregoing method to the methods of Jandera et al. (13) and Liteanu et al. (14). As do we, Jandera takes as limits the
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ANALYTICAL CHEMISTRY, VOL. 63,NO. 13, JULY 1, I991 1
0.90 _I
0.20
v
0.0
6.0
10.0
16.0
20.0
26.0
Titrant Volume, mL
30.0
36.0
Figure 1. Simulated conductometrlc tltration curve with 95% COnfldence intervals for the end point calculated by using (A) intersecting 97.47% bands, wlthout pooling of variance, and (6)Fieller's theorem, wlth proper pooling. Ordinate variance Is 1000 times that of the data of ref 1, so the Inherent scatter in the data Is about 31 times as great. Note in Table I I I that agreement between values from Fieller's and Creasy's methods Is good even for these rather noisy data.
extremes of the area of intersection. In contrast, he assumes that the confidence level for the abscissa will be the same as that for the individual bands, but a rigorous statistical justification for this is not apparent. Liteanu calculates the limits as weighted averages of the abscissas of the confidence hyperbolas at the ordinate of intersection. The approach appears to be the same one that is used for the "discrimination problem", in which, given one or more observations of y, confidence intervals for the corresponding unknown value of the independent variable x are constructed by application (16, 17) of Fieller's theorem. Chemists have used the method to calculate confidence limits for readings from calibration curves (6,17). Liteanu's method could be understood as a similar application of Fieller's theorem to each of the lines individually. However, the form of the equations used by Liteanu requires that the true ordinate of intersection be known exactly, and such is generally not the case. Regardless of this, the rationale for taking an average of the limits so obtained is unclear. Simple geometrical arguments (18) show that averaging can lead to unrealistic widening of the intervals in some cases. Density Function of X . The statistical density function q ( X ) of the abscissa of intersection gives the probability density for the occurrence of the abscissa of intersection at any point. We show the derivation of this function below. The definite integral of the density function gives the probability that the true value of the abscissa of intersection lies between the limits of integration. We want the density of X = -Aa/Ab. For purposes of simplifying the notation in this section we define z = -Aa and u = Ab so that X = z / u . The notation p, indicates the "population mean" of z. If the random errors of the ordinates of the data are normal, then by the addition theorem (19) for the normal distribution, z and u will each be normally distributed. Because each depends on both line 1 and line 2, z and u will be correlated. Their joint density function is the bivariate normal density
We must integrate this bivariate normal appropriately to obtain q ( X ) . This is not yet the integration of q ( X )itself but
rather an integration of f(z,u) to obtain q ( X ) . The density function of X is
where the integration corresponds to integration over all values of u (i.e., Ab) and z (Le., -Aa) such that the point of intersection is X. The factor Iul is the Jacobian for the change of variables, the absolute value being necessary to assure that the transformed integral is everywhere nonnegative, as must be the case for a meaningful density function (20). Some details of the following manipulations were checked with the symbolic algebra program MACSYMA, a large symbolic manipulation program developed at the MIT Laboratory for Computer Science and supported since 1982 by Symbolics, Inc. of Burlington, MA. To perform the integration, we first collect the coefficienta of like powers of u to obtain an integral of the form l:me-(auP+bu+c)
I4 du
(5)
where
The integral is recast into a sum of recognizable integrals as follows:
The two integrals of expression 7b differ only in the sign preceding the coefficient b. The technique of integration is the same for both, and we show it here for the first of these.
The first of the integrals in eq 8b is of the form Jet dg, and the second is standard form 7.4.2 of Abramowitz and Stegun (21),which contains the complementary error function. Together, the definite integrals of eq 8b evaluate to
ANALYTICAL CHEMISTRY, VOL. 63, NO. 13, JULY 1, 1991
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is not exact. They obtain (1) var X = The solution of the second of the two integrals in expression 7b is the same, except that b is prefixed by a minus sign wherever it occurs to odd powers. By applying the relationships erfc(arg) = 1- erf(arg) (104 erfc(-arg) = 1 - erf(-arg) = 1
+ erf(arg)
(lob)
and multiplying by the factor that preceded the original integral of eq 4b, we obtain the following expression for the density (Le., statistical distribution) of the abscissa of the intersection:
var Aa
+ X2 var Ab + 2X cov (Aa,Ab)
(13)
-
The standard error estimate of X is then sx = d v a r X,and the confidence limits at the ( 1 - 4 level are approximately
X
f
taj2SX
(14)
Application of Fieller’s Theorem. Fieller’s theorem has been applied to calculating confidence limits of X for lines with uniform (6, 7, 9, 14) and nonuniform (1) variance. Fieller’s theorem gives confidence limits for a ratio X = -Aa/ Ab, yielding the following equation (1)
al + blX = a2 + b2X f tal,*
(15)
where var (11) This result is essentially the same as Creasy’s (IO),though her results look different because she converted to polar coordinates prior to obtaining her final expressions. In both cases the density function is cumbersome and intractable to integration in closed form: Hers contains an infinite series, o w the error function. However, numerid integration proves straightforward. Estimation of Parameters. Values of the population parameters such as a* in the equation for q ( X ) must be estimated by the corresponding statistics. Because replicate experiments will not in general be available to allow direct calculation of these statistics, they must be obtained by propagation of variance (1,and ref 20, section 10.6) from the statistics of the regression. Reference 1may be consulted for the relevant rudiments of weighted least squares (19),nonweighted least squares being simply a special case thereof. We follow Schwartz’s (I) notation where practical. The variance of the ordinates of points fit to a leashquares line may be expressed as u2cy,x) =
VdJ(Y,x)
(12) where V is calculated as the sum of squares of the weighted residuals divided by the degrees of freedom, and 4Cy,x) is a function, expressible in y or x , that takes heteroscedasticity (i.e., nonconstant variance) into account. As in ref 1, our ordinates are given by y = ( l / R ) ( x o+ x i ) , where R is the resistance, xo is the initial volume of the analyte solution, and xi is the volume of titrant added. The conductance measurements themselves are treated as having uniform variance (11, so the function 4(yi,xi)= ( x o + xJ2 describes the heteroscedasticity of y, and the weighting factors are given by wi = 1/4(xJ. Usually, some such prior knowledge must be used, as the form of the variance function cannot generally be determined from a small set of data (22). In some instances, tests of null hypotheses involving the scedasticity might be appropriate (23). The parameters = var Aa, a,2 = var Ab, and p = cov -az2(-Aa,Ab)/dvar Aadvar Ab are calculated by first-order propagation of variance, which is exact in these cases, because the Aa and Ab are linear functions of random variables, so that all higher order terms in the Taylor expansions are identically zero. First-Order Propagation of Variance for var X. Schwartz and Gelb apply first-order propagation of variance to obtain an expression for var X,noting that because X involves the ratio of random variables first-order propagation
t
= var Aa
+ X2 var Ab + 2X cov (Aa,Ab)
(16)
Equation 15 can be rewritten as a quadratic in X,the roots of which are considered to be the confidence limits for the true value (E or px) of the endpoint X. The value of t used is ta12.fora confidence level of (1 - a). Reference 1 uses the notation t, throughout for this critical value of t a t the (1a)level of confidence, but despite the different conventions of notation, they and we are using the same numerical values of t. The left-hand side of eq 15 plotted versus X is simply line one, while the right side gives bands curving away from each side of line two. The bands curve because var c is a function of X. The confidence limits are the intersections of the bands with the line. If the variance is large enough or the segments close enough to being collinear, then instead of each band intersecting with line one, thereby giving a finite interval that includes the end point, a single band may intersect twice, giving a semiinfinite interval (16,17). Or the bands may diverge so rapidly that no intersection occurs at all, and the entire x axis is included in the confidence interval at the level of confidence chosen. Only if a finite inclusive interval is obtained are the confidence limits useful. An alternative arrangement of the quadratic (6, 7)
[ ( A u )-~ t,/22 var Aa]
+ 2X[AaAb - t,/22cov (Aa,Ab)] + X2[(Ab)2- tal? var Ab] = 0 (17)
reveals that the first and last groups of symbols enclosed in braces have the form of hypothesis tests,i.e., two-tailed t tests, for significant difference of intercepts and significant difference of slopes, respectively. When the hypothesis test for different slopes fails, the coefficient of x2 becomes negative and no real roots are obtained, so Fieller’s confidence interval embraces the entire x axis at the chosen level of confidence. (References 5-7 used bl - b2 where we have defined Ab = b2 - bi.1 NUMERICAL IMPLEMENTATION General Considerations. We programmed the methods in TURBO Pascal version 3 (Borland International, Inc.), aiming for accuracy and readability of code, rather than speed of execution or ease of use. Values for t were calculated by using equation 26.7.5 of Abramowitz and Stegun (21). Methods involving confidence bands were also programmed into a spreadsheet for graphical display. Most calculations were performed on a 16MHz IBM PS/2 Model 80 Computer with an 80387 numeric coprocessor, but all methods can be run on an original 4.77-MHz IBM PC clone with 640K of RAM and no coprocessor. Only the integrationswere time consuming: For 1ppt resolution, 27 s were required on the Model 80 Computer, 22.2 min on the PC, and 2 min on an 8MHz PC with 8087. A user-hostile fault-intolerant MS-DOS version of the major methods is available in both source
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 13, JULY 1, 1991
intervals resulting from application to small sets of data could seem qualitatively reasonable in the absence of comparison. Use of the correct expression for pooling
Table 11. Confidence Intervals for Data of Ref 1 (Comparison of Methods to Establish Correct Pooling) (Line 1: n = 6, v = 4. Line 2: n = 7, v = 5) eq for method
method
eq for
pooling
width of 95% interval
Fieller a a 0.367" first order a a 0.367' Fie11er 17 18 0.363 first order 13 18 0.363 2 none intersecting bandsb 0.358 intersecting bandab 2 20 0.294 11 20 Creasy x ( t / z ) 0.176 Fieller 17 20 0.176 first order 13 20 0.176 11 20 Creasy 0.153 11 none Creasy 0.152 "Values reported in ref 1. *Using tal* as the critical constant. Using (2&,,-2)1/2 88 the critical constant gave intervals of 0.470 and 0.371 mL with no pooling and pooling using eq 20, respectively. and compiled form from the corresponding author. Blank magnetic media (5.25 in.1360 kbyte, 3.5 in.1720 kbyte, or 3.5 inJ1.44 Mbyte) and a stamped self-addressed return mailer should be provided by the requester, together with specificationsof the system on which the program will be run. The programs and parameters for generating simulated data are likewise available on request, but these do require the coprocessor. Method of Integration. The error functions were evaluated by using rational approximation 7.1.26 of Abramowitz and Stegun (21),having an error less than or equal to 1.5 X lo-', adequate for our purpose. For numerical integration of q ( X ) ,Simpson's threepoint rule was applied incrementally over a grid fine enough to provide resolution of 1 or 2 ppt (or better) relative precision in the calculated interval. Bode's five-point method gave no significant improvement, strongly supporting adequacy of the simpler method. Provided the density function is unimodal and monotonically decreasing in the neighborhood of the limits, simple pictorial reasoning shows that an interval of minimum width will be obtained if integration is carried out to equal values of q ( X ) ,that is, to equal heights of the density function, on each tail. Other criteria (such as integrating to obtain equal probabilities on each side of the experimentally obtained X ) will give total intervals as wide or wider. APPLICATION AND COMPARISON O F METHODS Correction of Pooling. Various methods were applied to the conductometric titration data of Schwartz and Gelb (1). Estimates of the 95% confidence interval, given in Table 11, vary by more than a factor of 2. How is it that the previously published ( 1 ) intervals from Fieller's theorem with pooled variance are wider than those from the method of intersecting bands without pooling? The equation for pooling in ref 1 may be recast in simpler notation as
where SSE is the weighted sum of squares of the residuals (Le., the sum of squares of errors), given by
SSE
Cwi[jj + b(xi - a) - yi]' i
(19)
But this V is simply an estimate, weighted by degrees of freedom, of the total sum of squares of error per line, which will increase without limit as more and more data points are used. This in turn will yield estimates for var b and var 9 that do not diminish in size as a result of increasing degrees of freedom but rather approach constant values, behavior which is clearly unrealistic. The more complicated original form of the expression may have obscured the mistake, and confidence
SSEl+ SSE2 V = nl + n2 - 4
(20)
gives intervals from Fieller's theorem that are smaller than thaw from the method of intersectingbands. Because Creasy's method does not directly take into account the effect of limited degrees of freedom, it gives intervals that are too narrow. However, provided the distribution of X is adequately close to Gaussian, the application of the correction factor t / z produces excellent agreement with intervals from Fieller's theorem. (Here z represents the critical value of the standard normal distribution for the chosen confidence level.) Significant Figures. We found that 'guard figures" must be included in the coefficients of X in eq 17 in order to avoid errors in the roots and subsequently calculated confidence intervals. Because calculation of confidence intervals involves a subtraction of the lower root from the upper root, significant figures may be lost in the process. To obtain confidence intervals to three accurately calculated figures, we had to take the coefficients to considerably more than three figures. Our reporting of intervals to three or four figures is simply for clarity in presenting behavior of the computational methods. The statistical uncertainty in the error estimates themselves is considerable, making the reporting of such exact estimates statistically unjustified for the small sets of data we are considering. Determining Optimal Sets of Data. If the data have much scatter or if systematic curvature is present, a priori assignment of points to one line or the other may be unreliable and unjustified. Seber (24) recommends that the estimated changeover point be used to check the assumed assignment. Suppose there are m experimental points x(1) < 4 2 ) < ... < x(m),with points through x ( n ) used to estimate line 1, and points r(n + 1) and beyond used to estimate line 2. Then an assumption is being made that that changeover point lies between x(n) and x(n + 1). If the estimate does not lie between these values, then there is reason to question the assignment of points. (Note that this is quite different from a situation involving two possibly overlapping samples falling on different lines.) McCullough and Meites (25) describe a computer code for assignment and rejection of points, with the criteria and numerical values used with them chosen specifically for titrations. Vieth (26) describes alternative methods for assignment based on minimizing the residual s u m of squares but does not treat elimination of points exhibiting systematic deviation. Even in the absence of systematic deviations, we cannot be guaranteed reliable assignment of points occurring within the confidence interval. Such uncertainty in assignment might add further error not taken into account by the reported confidence level. For titrations, there is the further problem of eliminating any dubious points near the intersection that exhibit systematic deviation from linearity. Schwartz and Gelb (1) suggest that the optimum set of points is that which yields the minimum value for width of the calculated confidence interval. While holding the number of points on one line constant, they add or delete doubtful points in sequence on the other line until a relative minimum in the confidence interval, calculated by Fieller's theorem, is obtained. Then that number of points is held constant, while points are added to or deleted from the line previously held constant. It may be necessary to repeat the process for each line in turn until a self-consistent answer is obtained. Unfortunately, the outcome of this search strategy depends on initial decisions, as illustrated by application to the data
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Table 111. Comparison of Methods for Simulated Data with Increasing Variance 95% confidence intervals:‘ upper half-width/mL, lower half-width/mL
conductancea variance cv/% V O 1O’V0 lOzV, 103V0 i04v0 106Vo 1@V0
0.207 0.654 2.07 6.54 20.7 65.4 207
methodb first order (eq 13)
Creasy (eq 11)
Creasy X ( t / z ) (eq 11)
Fieller (eq 17)
intersecting bands (eq 2)
0.04434, 0.04434 0.1402, 0.1402 0.4434, 0.4434 1.402, 1.402 4.434,4.434 14.02, 14.02 44.34, 44.34
0.0424, 0.0425 0.134, 0.134 0.424, 0.426 1.34, 1.36 4.57, 4.79 43.3, 45.4 116, 114
0.0443, 0.0444 0.140, 0.140 0.443, 0.445 1.40, 1.43 4.77, 5.01 45.3, 47.4 121,119
0.044 32, 0.04436 0.1400, 0.1405 0.4416, 0.4461 1.393, 1.439 4.667, 5.218
0.072 21, 0.072 28 0.2281, 0.2288 0.7195, 0.7261 2.264, 2.331 7.582, 8.383
a V, = 8.669 X lo-”, the same as the pooled V for the data of ref 1. cv = the average coefficient of variation or average relative standard deviation, expressed as a percentage. cv = (100%)(~/*)/0.0045.bEach branch had 16 points, so there are 28 degrees of freedom. Pooling using eq 20 was employed throughout. The correction factor for Creaay’s method used the following values: t = 2.049, z = 1.960, t / z = 1.045. ‘The upper and lower half-widths give the distances of the upper and lower confidence limits from the point estimate X = 16.365 mL. Thus, the total width of a confidence interval is the sum of upper and lower half-widths, which are not, in general, equal.
of ref 1. Beginning with eight points on line 1(rl = 4-16 mL) and ten points on line 2 ( x 2 = 17-32 mL), first deleting points from line 1 while holding line 2 constant, we obtained a self-consistent optimal set of six points on line 1 and eight points on line 2, giving an end point of 16.342 mL and a confidence interval of width 0.161 mL, which is the global minimum. However, starting with five points per line and first altering line 1,we obtained an optimal set of four points on line 1and six on line 2, giving an end point of 16.292 mL and a confidence interval of width 0.199 mL. A prior subjective judgment that the point at 14 mL is not dubious might have ruled out this set. In general, though, it appears that the only assured course may be an exhaustive search. Since the points are only added or deleted in order (no skipping is allowed), the number of trials would be simply the number of dubious points on line 1times the number of dubious points on line 2. Effect of Variance and Degrees of Freedom. To facilitate investigation of the effects of variance and degrees of freedom, simulated data seta with normal errors were generated. For the set plotted in Figure 1, the variance multiplier V is lo00 times the correctly pooled V for the data of ref 1. The confidence bands, calculated by using eq 2 without pooling, are a t the 97.47% level and yield an area of intersection a t the 95% level. The confidence interval from Fieller’s theorem, with proper pooling, is shown for comparison. Table I11 shows that a t higher variances, the limits from Fieller’s theorem become less and less symmetrically disposed about the experimental point of intersection, as do those from Creasy’s method. The intervals from Creasy’s method exhibit less asymmetry, but the existing asymmetry is indicative of nonnormality so that the correction factor t / z is less and less effective. Furthermore, the widths of intervals from Fieller’s theorem are greater than or equal to those from Creasy’s method. Firsborder propagation of variance yielded intervals that agreed in overall width with Fieller’s to 1% ,even for the data of Figure 1that have a scatter roughly 31 times that of Schwartz’s data. Of course, any finite variance would lead to larger and larger intervals as the lines approached colinearity. For data with small to moderate variance the term P / a in eq 11was too small to evaluate, but it became important with increasing variance. For example, the values of c were about 1.10358 X l@and 1.10387 X 10-l for the entries in Table I11 with V = V, and V = 106V,,, respectively. The variance estimates of bj and y j for line j bear an approximately inverse relationship to the number of points nj. Increasing the number of points also reduces the value talP. For the simple case nl = n2 = nj, the overall effect is such that the width of the confidence interval varies roughly as
2.00 1.50
I
” 1.00
i
-Aa 0.60 0.00 -50
I/
-1.00 -.040
0.000
0.080
0.040
0.120
Ab
Figure 2. A 25% joint confidence elnpSe for -he and pb for lines wRh data having an ordinate variance 1 000 000 times that of the data of
ref 1. Because of the different scales on the axes,symmetrical angular displacements do not appear symmetrical, and a line with an apparent slope of 45’ has an actual slope of about 18.7 51-‘. 0.040
‘i
L
‘
0.030
J
0.000 0.0
6.0
10.0
16.0
20.0
26.0
30.0
I .O
X
Flgurr 3.
Creasy and Fieller density functions and 25% confidence
limits for the abscissa of intersection of lines wkh data having ordinate variance 1 000000 times that of the data of ref 1. X = 16.365 mL is marked by the arrow. A and B mark limits using equal q ( X ) as the criterion for integration. C marks limits using equal probability on each side of X as the criterion for integration.
For example, the abscissa of intersection of the 16-point lines with relative standard deviation 6.54% has a 95% confidence interval of width 2.8322 mL, with upper and lower subintervale of widths 1.3934 and 1.4388 mL, respectively. Taking 61 points/line reduced the width of the confidence interval to 1.4452 mL, with upper and lower subintervals of widths 0.7164 and 0.7288 mL, which are more nearly equal. Relationship Between Fieller’s Theorem and Creasy’s Method. A relationship between Fieller’s theorem and
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ANALYTICAL CHEMISTRY, VOL. 63,NO. 13, JULY 1, 1991
Creasy’s method asserted in the statistical literature (11,12) is illustrated with the help of Figure 2, which shows a joint confidence ellipse (5) for -Aa and Ab and limits for X. (In these calculations, and those associated with Figure 3,we have treated the variances as if they were known exactly, that is, as if Y = m.) Each possible value of X corresponds to the slope of a unique line passing through the origin of Figure 2, and limits for X correspond to the slopes of a pair of such lines. Such a pair divides the plane into sectors. Sector A is defined as the one containing the values of -Pa and Ab actually obtained in the experiment. In sector A the sign of Ab is the same as that observed,whereas in sedor C the sign is opposite. Creasy’s method corresponds to integrating the probability in sectore A and C. Barnard and Fisher (in ref 11) each assert that Fieller’s theorem gives results equivalent to those that would be obtained by subtracting the probability in sector C from that in sector A. Therefore, for the same level of confidence, Fieller’s intervals must be wider than Creasy’s whenever there is significant probability in sector C. Subtraction cancels some of the messier terms and produces the following expression for Fieller’s density function:
(Fisher also mentions the alternative possibility of simply integrating sector A alone but does not give general guidelines for when this would be appropriate.) The Creasy-Fieller Paradox. The discrepancy between the Fieller and Creasy methods at extremely high variances or confidence levels is known as the Creasy-Fieller or Fiel1erCreasy paradox (12,27). It has been suggested that proper practice depends on the specifics of the problem under consideration and cannot be settled purely by reference to abstract mathematics (12). Those who disagreed (11) with Creasy’s method emphasized that at the origin the ratio is undefined (11,12) and that the significant possibility of an undefined ratio would render meaningless the results for some problems, such as the biological assays for which Fieller’s theorem was originally presented (2). However, in our context of two least-squares lines, a denominator of zero has a perfectly plausible interpretation, namely the possibility that the true lines for the data in Figure 1 are parallel or even collinear, as could occur, for instance, in a conductometric titration with no analyte present! The question is, should a finite interval for the abscissa of intersection be reported for a given level of confidence if equivalence or parallelism of the lines cannot be ruled out at that level of confidence? Since Creasy’s method counts the probability that the true lines may intersect with Ab different in sign from that observed (and since the origin and coordinate axes of the (Ab, -Aa) continuous probability space are of lower dimension than the space, so that they have essentially zero probability), the method will never fail to give a finite interval, as long as the numerical implementation is sufficiently accurate to avoid excessive roundoff. The coefficients in the Fieller quadratic (eq 17)have the form of twetailed testa,which can fail for any finite confidence level if the variance is large enough. However, a one-tailed test for difference in slopes could never fail for confidence levels less than 50%. Furthermore, in the absence of a priori information, there will be at least a 50% probability that the sign of the Ab actually observed is the true one. Considering only the probability that the sign of Ab is the same as that observed amounts to integrating only sector A. We might expect that the sign of Ab would be known from the chemistry of the reaction employed in the titration, so that a one-tailed test might be appropriate. If the expected sign agrees with
that observed, we suggest that linear segmented titration curves might be an appropriate case for integrating sector A alone. The density function is
Intervals from eq 22 agree to three significant figures with those from Creasy’s method (i.e., eq 11) for the first five variances of Table 111. For V = lo6Vo,Fieller’s quadratic fails to give a finite interval at the 85% confidence level, whereas use of eq 22 gives limits up to the 92% level of confidence. At the 84% level of confidence, the widths of confidence intervals given by Fieller’s theorem, sector A integration, and Creasy’s method are 108,38.3,and 31.1 mL, respectively. (The t / z correction was applied to the intervals obtained by integration.) It should be emphasized that the joint normal probability density gives information about the distribution of the statistics -Pa and Ab (and, therefore, X ) , not about the distribution of the parameters P - ~ PAb, , and PX. Applied to data, the formalism allows inferences regarding the probable location of the parameters, but it says nothing about their distribution or frequency of occurrence. For instance, the frequency with which chemical samples are actually titrated for which p-b and P A b are identically equal to 0 is simply not addressed by the formalism. More general discussion is found in statistical literature dealing with relevant ongoing controveries in statistical inference, in which various fiducialist, frequentist, and Bayesian viewpoints are invoked (27). Table 111suggests that the data would have to be far worse than most titration data before the disputed considerations would make any practical difference. Comparison of the Creasy and Fieller Density Functions. Figure 3 shows Creasy (sector addition, eq 11) and Fieller (sector subtraction, eq 21) density functions as the top and bottom curves, respectively. Confidence limits on X at the 25% level are indicated by dotted lines. The noncentrality of both distributions is evident in this example for which the intersecting least-squares lines have bl not equal to -b2 and in which the data exhibit enormous standard deviations. Finer comparison than possible in the figure reveals that neither distribution is symmetrical about its maximum, though the Creasy distribution is closer to being symmetrical. However, for a test set of points exhibiting mirror symmetry with respect to a vertical line at X and fit with nonweighted least squares, both density functions appeared symmetrical and central. Comparison of Criteria for Limits of Integration. Because the integration methods do not take degrees of freedom into account directly, we here used z rather than t in Fieller’s quadratic to facilitate discovery of any other differences. Limits were determined from the Fieller density function using two different criteria for integration: integration to equal heights of the density function on each tail and to equal probabilities on each side of the value of X obtained from the least-squares lines. Table IV shows that it is the latter criterion that gives limits agreeing closely with Fieller’s quadratic, while the former criterion gives intervals of narrower total width, as expected. If the space of Figure 2 is reparameterized in terms of the polar coordinates 6 and r, then the ratio -AalAb corresponds to 6,and r is simply a nuisance parameter. Fieller’s limits can be expressed (12,B) as Bo f w , that is, as symmetrical angular limits about the observed angle Bo. What our numerical integration seems to show is that these symmetrical angular
ANALYTICAL CHEMISTRY, VOL. 63,NO. 13,JULY 1, 1991
Table IV. Comparison of Confidence Intervals Using Various Density Functions and Criteria for Integrationa
method Fieller's quadratic' with z sector A - C (to equal P)d sector A - C (to equal q)# Fieller's quadratic' with t sector A C (to equal P)d sector A + C (to equal q)a sector A only (to equal P)d sector A only (to equal 9)' propagation of variancef
+
half-widths of confidence intervals 25.0% 35.5% lowerb upper* lower upper
10.47 10.50 8.96 10.69 3.97 2.80 5.56 4.36 6.96
8.46 8.49 9.73 8.61 3.64 4.71 4.94 6.00 6.96
113.39 119.65 68.44 408.03 6.11 4.70 9.33 7.61 10.07
31.76 32.25 45.56 39.81 5.37 6.60 7.70 9.12 10.07
a Same data aa for Figure 3 and entry 7 of Table III: lines with V = 106V,. Statistics of lines are given in Table V. No t / z cor-
rection factors were applied to tabulated values that were computed by integration. b Lower and upper give the distances of the lower and upper confidence limita from the point estimate X = 16.365 mL. Thus, the total width of a confidence interval is the sum of lower and upper. 'Included for comparison. dDensity function integrated so as to include equal probabilities on each side of X. #Density function integrated to equal values of q ( X ) . 'Included for comparison: t,/,sY (as in eq 14). Table V t(,.s,H.O+)
t(r-~,a.6% = )0.46531
Z(H,O%)
~(36.6%)=
0.32135 = 0.31820 U&
UAb
0.46029
= ~ p =. 1.6832677532403+000 = 8 A b 8.3353724259793402
= -Aa = 6.3671682386603401 @u = Ab = 3.8907303018893402 p = r = 8.6768680619753-001 X = 1.6364969413503+001 line 1
n = 16 X U J=~ 1.3922120884033403 S,, = 2.949575556857E-002 S , = 1.2377447205523403 S , -8.4166949252193404 2 = 7.1042277283313+000 9 = 6.3605515535793401 SSE = 1.2137274502373-003 V = 8.6694817874113405 var b = 2.9392302791683403 var a = 2.1061437607003401 var 9 = 6.2271272169103402 b -2.8535274865743-002 a = 8.3877624629463401
line 2
n = 16
C U J=~1.0199852496313403 S ,, = 2.1627003310133=002 ,S = 1.2160455026383-003 SZy= 2.2431588720123404 i = 2.5161088615973+001 7 = 4.6303094191763-001 SSE = 1.2137188919413-003 V = 8.6694206567263405 var b = 4.0086092984793403 var a = 2.6227674486173+000 var 9 = 8.4996549296983402 b 1.0372028153153-002 a = 2.0205942242853-001
limits do not correspond to the confidence interval of minimum width. This might seem to indicate that even if Fieller's distribution were preferred to Creasy's, numerical integration with sector subtraction would be preferable to the use of eq 17. However, degrees of freedom must be taken into account, and the correction factor t / z may be inadequate for this purpose when asymmetry is extreme. A completely rigorous approach for small sets of data would make use of a joint distribution in terms of statistical estimates rather than parameters, but, as Creasy remarks, "this distribution appears somewhat complicated and has not yet been worked out" (IO). Importance of Weighting and Heteroscedasticity. For the data of ref 1, the use of weighted versus nonweighted least squares makes very little difference in either the confidence interval for X or the value of X itself. With nonweighted fitting, X = 16.372 mL and the width of the confidence interval is 0.170 mL. However, the use of weighting can be important when the heteroscedasticity is more pronounced.
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Franke and de Zeeuw (23)give illustrative examples for standard addition data where neglect of heteroscedasticitycan lead to serious errors, but they suggest using nonweighted least squares if in doubt, because, when applied to data that in fact exhibit a constant relative standard deviation, an overestimate of the interval width is obtained-a less serious problem than false optimism. The weighting function we used describes a scedasticity intermediate between constant standard deviation and constant relative standard deviation. The fact that the width of our interval from nonweighted fitting, 0.170 mL, was actually smaller than that calculated with weighted fitting might be due to the fact that for the data of ref 1,the variance of the points on the second segment is actually smaller than that of the points on the first segment, though the weighting function we used would imply that the contrary is more commonly the case. Differences in End Points. In a titration of a mixed solution of a strong and a weak acid, for which three linear segments are obtained, the quantity of weak acid may be determined by the difference of the first and second end points. Schwartz and Gelb (I) report that an attempt to apply Fieller's theorem fails, because the equation for e involves products of random variables, but that first-order propagation of variance can be applied to the end point difference AX to obtain an estimate of sW Our Table I11 indirectly suggests that this "small-variance confidence interval" (I) of width 2t./gax will be quite reliable for good data. CONCLUSIONS AND RECOMMENDATIONS Our method of intersecting bands is unduly pessimistic, especially the form using (2F&z)1/z. The other band methods cited each involve two or more simplifications or nonrigorous assumptions. Errors from these may sometimes cancel to give fairly accurate results, but such cancellation cannot be relied upon in general, and discrepancies can be large. For titrimetric data with small to moderate variance, there was close agreement between confidence limits calculated by Creasy's method, Fieller's theorem, and fmt-order propagation of variance for X. Greater accuracy and firmer statistical justification make these methods preferable to methods based on intersecting confidence bands. Disputed aspects of statistical inference that distinguish between these methods become relevant only if scatter is huge or if the segments are sufficiently close to collinear. Likewise, the choice of criterion for limits of integration appeared to make only a minor difference for good data. Because the calculations are less complicated than those for integration, we recommend the use of either Fieller's quadratic or first-order propagation of variance, for which ref 1serves as a good guide once pooling is corrected. However, the frequency with which we encountered arithmetical errors, shaky conceptual foundations, and algorithmic blunders (not all mentioned here) in the literature and in our own early efforts suggests that new implementations of any method should be checked carefully. LITERATURE CITED Schwartz, Lowell M.; Geib, Robert I. Anal. Chem. 1984, 56, 1487-1492.
Fieller, E. C. J . R . Stet. Soc. Suppl. 1040, 7, 1-54. Fieiier, E. C. 0 . J . phenn. phemcol. 1944, 17, 117-123. Fleller, E.
C. J . R . Stat. Soc. 8 1954, 16, 175-185.
Sharaf, Muhammad A.; Illman. Deborah L.; Kowalskl, Bruce R. CheChemical Analysis 82; Wiley: New York, 1986; pp
mometrlcs;
129-131.
Lark, P. D.; Craven, B. R.; Bosworth, R. C. L. The Hendllng of Chedcel Data; Pergamon Press: Oxford, 1968;Chapter 4. (A worked
numerical example Is provided.) Kastenbaum, Marvin A. BiOmetrbs 1959, 15, 323-324. Fisher, Ronald A. Statistical Methods for Research Workers, 13th ed.; Olhrer 8 Boyd Edinburgh, 1963;Section 26.2. Fisher, Ronald A. StetisthxlMethcds for Research Workers, 1 lth ed.; Hafner Publishing Co.: New York, 1950;Section 26.2. Creasy, Monica A. J . R . Stet. Soc. B 1954, 76, 186-194.
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Anal. Chem. 1001, 63,1278-1281 Irwln, J. 0.;et al. J . R . stet. Soc. 6 1954, 76, 204-222. Wallace, Davld L. I n R . A . FMwr: An m d b n ; Fkmbwg, Stephen E., Hlnkby, Davld V., Ed.; Lecture Notes in Statktlce 1; Spring er-Verlag: New York, 1980 pp 119-147. Jandera, Pavel: Kdde, Stanlslav; KoW. Stanklav. Talente 1970, 77, 443-454. Llteanu, Candln; Rk& Ion; Llteanu, Victor. Talent8 1978, 25, 593-596. Walpde, Ronald E.; Myers, Raymond H. PrObebUtty and StetkHCs for Eng/ne8fs end sc&nt/sts, 2nd ed.; Macmillan: New York, 1978. Mlller, Rupert G., Jr. Shrnnltenews Statktkil Inference, 2nd ed.; Sprlnger-Verlag: New York, 1981. Schwartz. Lowell M. Anal. Chem. 1977, 49, 2082-2068. Carter, Kenneth N., Jr. Unpublished work, Northeast Missouri State University, 1990. Brown&, K. A. steLMcel Thecxy and Melhodhkgy h S c h c e and Englneerlng, 1st ed.; Wlby: New York, 1960. Kendall, Mawice G.; Stuart, Alan. The Advenced of StetkMw, 2nd ed.; Hafner Publishing Co.: New York. 1963: Volume 1. Sectbns 1.35, 11.3, 11.9, and 11.11.
(21) Abramowlb, Milton, Stegun, Irene A., Eds. Hencbdr of M ” h l Fmcthms; Dover: New York, 1984. (22) Schwartz. Lowell. M. Anal. Chem. 1979, 57, 723-727. (23) Franke, J. P.; de Zeeuw, R. A.; Hakkert. R. Anal. Chem. 1978, 50, 1374- 1380. (24) Seber, G. A. F. Linear Regression Analysis; Wlley: New York, 1977; pp 205-209. (25) McCullough, John 0.; Meites. Louis. Anal. Chem. 1975, 47, io8 I-1084. (26) Vbth, Ell~&6th. J . A@. physkl. 1989, 87. 390-396. (27) Wllkinson, G. N. J . R . Stet. Soc. E 1977. 39, 119-171. (28) Koschat. Martln A. Ann. Stet. 1987, 75, 482-488.
RECEIVED for review July 31,1990. Accepted March 14,1991. This work was supported in part by research startup funds and internal academic year research grants from Northeast Missouri State University.
Nitrogen Oxide Gas Sensor Based on a Nitrite-Selective Electrode Stacy A. O’Reilly, Sylvia Daunert, and Leonidas G. Bachas* Department of Chemistry and Center of Membrane Sciences, University of Kentucky, Lexington, Kentucky 40506-0055
Incorporatlon of dicyanocobait( 111) a,b,c,d,o,f,g-heptaProWlcobyrlnate In -4 pdy(vkryl-) “khas resulted in the development of electrodes that are sebdlvefor M e . An N0,gasbenror wasprepared by placlng this electrode behind a microporous gabpermeable membrane. NO, Is generated In the sample at pH 1.7 and, after crosdng the gacpermeabk membrane, Is trapped as nitrite by an Internal solutlon buffered at pH 2 5.5. Thls sensor Is different from the conventlonai Severlnghaus-type sensor, whlch employs a flat-bottom pH electrode. The latter senses changes in the pH of an internal unbuffered solutlon as NO, diffuses across the gas-permeabie membrane. The Severinghaus-type sensor exhlMls severe Interforewes from weak WpophHlc aclL that can cross the gas-penneabk membrane and affect the pH ol the Internal solution. The descrbd NO, sensor does not suffer from such interferences and exhibits btter detection IimHs.
INTRODUCTION In the commercially available Severinghaus-typegas sensors, a pH electrode is placed behind a gas-permeable membrane (GPM) (1-4).Gases in the sample diffuse through the GPM and change the pH of a thin film of an internal solution that is ”sandwiched” between the GPM and the pH electrode. In the case of the nitrogen oxide gas sensor, the internal solution is unbuffered and contains a relatively high concentration of sodium nitrite (5).This arrangement facilitates a direct relation between the partial pressure of NO2 (or NO) in the sample and the [H+] of the thin film (6). An inherent limitation of the commercial nitrogen oxide sensor is that other species (e.g., salicylic, acetic, and benzoic acids) can cross the membrane and alter the pH of the internal solution (6). Such an interference effect is common to all Severinghaus-type gas sensors (4). T o reduce such interferences, Meyerhoff and co-workers replaced the pH electrode behind the GPM with a polymer0003-2700/9110363-1276$02.50/0
membrane-based ion-selective electrode (ISE). Specifically, they developed ammonia, sulfur dioxide, and carbon dioxide sensors by using internal electrodes that were selective for ammonium (3,sulfite (81,and carbonate (9),respectively. In addition, they reported an air-segmented continuous-gassensing arrangement to measure NO, (IO). In the last system, NO, was generated in a flowing stream and was trapped behind a GPM by a flowing solution, which was pH 2.8 and contained hydrogen peroxide. The generated nitrate ions were detected by a flow-through nitrate-selective electrode. Finally, Coetzee and Gunaratna reported a chlorine gas sensor based on a solid-state chloride-selective electrode (11). This paper describes an NO, gas sensor that employs a nitrite-selective electrode, which is located behind the GPM and functions as the sensing element. This arrangement, along with the use of a buffered internal solution at a pH much higher than that of the sample solution, results in an improved detection limit (equal to 4 X lo-’ M) and selectivity when compared to the commercially available sensors, as well as the previously reported gas sensors.
EXPERIMENTAL SECTION Reagents and Apparatus. Vitamin BI2,2-(N-morpholino)ethanesulfonic acid (MES), tridodecylamine, dibutyl sebacate, sodium salicylate, sodium benzoate, and all the inorganic salts were obtained from Sigma (St. Louis, MO). Sodium acetate was procured from J. T. Baker (Phillipsburg,NJ). Chromatographic grade poly(viny1chloride) (PVC) was purchased from Polyscience (Warrington,PA). Bis(2-ethylhexyl) sebacate (DOS, purum) was purchased from Fluka (Ronkonkoma, NY). Tetrahydrofuran (THF), hydrochloric acid, and sulfuric acid were obtained from Fisher Scientific (Fair Lawn, NJ). 1-Propanolwas p u r c W from Aldrich (Milwaukee, WI). Sodium tetraphenylborate was obtained from Kodak (Rochester, NY). All standard solutions and the buffers were prepared with deionized (Milli-Q, Millipore, Bedford, MA) distilled water. Dicyanocobalt(II1) a,b,c,d,e,f,g-heptapropylcobyrinate, the ionophore used to prepare the nitrite-selective electrode, was synthesized by following the procedure of Murakami et al. (12). A Fisher Accumet 810 digital pH/mV meter was used to monitor the voltages. The potential was recorded on a Linear 0 1991 American Chemical Society
