ANALYTICAL CHEMISTRY, VOL. 51,
NO. 6 , MAY 1979
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Calibration Curves with Nonuniform Variance Lowell M. Schwartz Department of Chemistry, University of Massachusetts, Boston, Massachusetts 02 725
analyst and so must be estimated as the sample variance s2 from the experimental data a t hand. Equation 2 of Reference 1 is used to calculate s2 by pooling the replicate measurements made on all the samples. An important ramification of the homoscedastic condition is that s2 then enters into the variance on the right-hand side of Equation 1 as shown explicitly in Equations 5 , 9, and 1 2 of Reference 1. Yet if the y , and Y, have nonuniform variance, there exists no such common population variance 2,Equation 2 of Reference 1is irrelevant, and we must find another means of calculating the variance in Equation 1. A second problem arises if the calibration curve is to be constructed as a polynomial function ( I ) for which the degree is selected using statistical fitting criteria. If a value of s2 is available from Equation 2 or Reference 1,then the appropriate polynomial is selected as the one with smallest degree for which the variance of residuals of the y , data points about y , d ( x ) is near s2. Alternatively, if s2 is not available because of insufficient replication, polynomials of increasing degree are fitted arid F-tests on the successive residual variances can select the minimum degree for which lack-of-fit has been eliminated. The F statistic, however, is the ratio of two sample variances drawn from a common population and this procedure, as well as the former alternative, rests upon the existence of a uniform variance. Without this condition, some other niethod must be developed to select the appropriate polynomial degree.
Methods are shown for Calculating statistical uncertainties in the form of confidence limits for an assay obtained through a calibration curve of arbitrary form when the precision or Variance of the data is nonuniform along the curve. Illustrative examples are based on photometric and chromatographic analyses. The ramifications of ignoring variance nonuniformity are discussed.
In a recent communication ( I ) , it was shown how to find confidence limits for the quantitative analysis of an unknown sample determined by projecting its response measurement through a nonlinear calibration curve of arbitrary form. Several calculational methods were discussed but all were bslsed on an assumption that the precision of the data points was uniform along the calibration curve, the so-called “homoscedastic” or uniform variance condition. Although this assumption is a valid one for many analytical procedures, there are several for which it is not. These include all methods based on counting measurements ( Z ) , either photon or radioactive, and also photometric and chromatographic analyses under certain conditions. I t is the purpose of this paper to offer means of extending the previously discussed methods to cases such as these, i.e., those for which the data points along the calibration curve are inherently nonuniformly precise, the “heteroscedastic” or nonuniform variance condition. An extensive literature exists on statistical treatments of this same problem in radioligand assay ( 3 , 41, i.e., radioimmunoassay (RIA) and immunoradiometric assay (IRMA).
THE VARIANCE F U N C T I O N T h e heteroscedastic condition implies that the variances of points change in a regular way along the calibration curve and so the population variance a t a particular point ( r I 2 can in principle be expressed as a function of y , or x,. The analyst must have prior knowledge that the data have nonuniform variance, for experience has shown that this condition cannot be determined from a limited number of measurements ( 3 ) . The paper by Rodbard et al. ( 4 ) serves as a typical illustration of the extensive statistical experimentation required to establish the nature of the variance function. T h e prior knowledge may be theoretical as may be the case in a counting method for which the variance of a count, if determined by the Poissonian distribution, is exactly equal to the expected count. On the other hand, the variance function may be the result of extensive replication experiments. In either case, this functivn serves as the basis of the right-hand side of Equation 1 and so the associated number of degrees of freedom is infinite if the function is theoretical or very large otherwise. We will also allow the possibility that the variance function involves an unknown multiplying factor. Finally, we will regard the variance of measurements y , as being inherently a function of the measurement Q rather than of the concentration ( x ) . It is important to make this choice even though either could be regarded as the independent variable for this function since y and x are interdependent through the calibration curve. Had we regarded (r2 as a n inherent function of x , u I 2 ( x , )would be determinate (not random) because x , values are not random variables. T h e choice that u2 be a function of y makes the calculations more difficult but is more realistic. The variance of a measurement logically depends on the measurement itself, not on some other
S U M M A R Y O F U N I F O R M VARIANCE TREATMENT An understanding of the complications caused by nonuniformity of variance requires a brief reiteration of the principles involved in calculating confidence limits in the homoscedastic case. T h e notation used here is the same as that used previously (1). A calibration curve y,,l(x) is constructed by measuring the responses s., of n standard samples having concentrations x , . Each such standard may be measured repeatedly n, times. Then samples having unknown concentrations X I are each measured N,times generating response data Y,. An unknown X I is found in the usual way by projecting the mean Y,of the N , replicates through the calibration curve and onto the x-axis. If both y , and Y , are normally (gaussian) distributed random variables, then confidence limits for X, a t a prescribed level of significance a are calculated by finding two values of X which satisfy the equation
[E
-
Y , , ~ ( X )= ] ~ta2var[yL- Y ~ ~ X ) ] (1)
T h e statistic t , is selected from a Student’s t-table using the level a and the number of degrees of freedom inherent in the variance. T h e development of Equation 1 requires no assumption as to the uniformity of variance along the calibration curve. It is now assumed that d standard samples y , and unknown responses Y, have t h e same precision so that each such measurement is a random variable drawn from a population having variance u2. In many cases u2 is not known to the
e 1979 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 6 , MAY 1979
experimental factor that influences the level of the measurement. The difficulty arises because to evaluate a12(y)for a given standard sample using y = yi, the mean of a few replications, leads to spuriously erratic $(y) behavior. Instead a2(y) should be smoothed by using the expected value of y, Le., t h e mean of a great many replications. Because of practical limitations, however, this information is not available and so the next best recourse is to base a?(y) on the predicted value of y for a given x i , which is ycJxi). But a t the start of a calculation, the function ycd(x) involves a t least unknown parameter values and perhaps also a yet undetermined number of terms. Thus an iterative procedure is required since yc&) depends on evaluations of ui2(y) and cri2 depends on yd(xi). Nevertheless, we have a variance function a2(y) which is either completely specified or of the form U2(Y)
= V4b)
(2)
where 4(y) is a completely specified function and V is a constant multiplier whose value in any particular calibration experiment is unknown.
DETERMINING THE CALIBRATION CURVE The calibration curve is constructed from the n calibrating points 9; vs. x i by the method of least-squares. Weighting factors wi must reflect the number of replications niand also the variance a t the locality of the point so that T h e sum to be minimized is
(4) and minimization is understood to be with respect to adjustable parameters in the yd(x) function. Brownlee (5) shows how to calculate the parameters a and b and the parameter variances for the particular case of a linear calibration line ycd = a b ( x - a ) and with variances a? completely known (5, Section 11.12) or with variances involving an unknown multiplying factor (5, Section 11.13). Brownlee obtains the parameter variances using analysis-of-variance which we will follow. Previously ( I ) we offered three strategies for dealing with general nonlinear calibration curves: a linear segmented approximation, a three-parameter nonlinear function of specified form, and a polynomial function of adjustable degree but, in the following development, we abandon the linear segmented method because of its crudity.
+
TWO- OR THREE-PARAMETER FUNCTION OF SPECIFIED FORM As previously ( I ) we assume a three-parameter equation ycd = 9 + cl(x - a ) c z [ f ( x ) - where f ( x ) is a nonlinear function specified by the analyst as representing the curvature of his particular line and c1 and cz are adjustable parameters to be determined by multiple regression techniques. I t will also be convenient sometimes to omit the linear term and use the two-parameter model yd = 9 + cz[f(x) If this is done, several formulas given in Reference 1 are changed and these are t h a t in addition to c1 = cov(cl, cz) = 0, c2 = s,/s, (5) and
+
n
n.
var (c,) = a2/Sjf (6) When the variance is nonuniform, these equations and those for the three-parameter model require some modifications. T h e weighting factors wi (simply equal to ni for uniform variance) are given by Equation 3 with a? evaluated as u2[y,.&J] or by 4[Yd(xi)]if the multiplier Vis unknown. The formulas for c1 and c2 are unchanged, but the constants uz which appear in the formulas for var (cl), var (c,) and cov (cl,
c2) are replaced by unity if the variance function is completely specified (5, Section 11.12). If Vis not known a priori, it must be estimated from the sum of weighted residuals about the calibration curve and the values of V so determined then replace the constant a2 in the parameter variance formulas. Following Brownlee ( 5 , Section 11.13),if p is the number of terms in the model equation (2 or 3), the sum S of Equation 4 is distributed as the product of V and the chi square statistic with ( n - p ) degrees of freedom, VX2(n- p ) . The expected value of S is V ( n - p ) and, therefore, V = S / ( n - p ) is the estimate of uz required. T h e sum S is evaluated with the calibration curve as found by regression. But, as mentioned above, an iteration is necessary since the parameters c1 and c p cannot be found without w ,and w,cannot be calculated without a2[4’~(x,)]which involves c1 and c2. It is, nevertheless, a simple matter to program a computer to begin with provisional w,= n,, find c1 and cz on this basis, recalculate w,= n,/u,2, and reiterate until c l and cq cease to change.
POLYNOMIAL FUNCTION OF UNSPECIFIED DEGREE A calibration curve is sought in the form of a polynomial of degree m expressed either as a power series or as a linear combination of orthogonal polynomials. The degree is selected to fit the curvature of the calibrating data to statistical criteria which, for data of uniform variance, are based on that variance. If the variances are nonuniform and completely specified, the sum S of Equation 4 follows a chi square distribution with ( n - m - 1) degrees of freedom provided that the scatter of data points about the calibrating polynomial is random, Le., has no systematic error component due to lack-of-fit. Therefore, we may adopt a strategy, analogous to F-testing in the homoscedastic case, of calculating a series of calibrating polynomials with degrees from 1 to ( n - 2) and calculating S for each. T o test a given polynomial, we make the null hypothesis that the corresponding sum S is distributed as x2, adopt, say, a 95% confidence level for the test, and compare S with x2(n - m - 1) from a table of probability points a t cy = 0.05. If the tabulated x 2 exceeds S, we reject the null hypothesis and assume lack-of-fit as the cause. The polynomial of least degree for which x2 IS is selected. The power series or orthogonal polynomial coefficients and their variances are then available from the polynomial fitting routines. Again each polynomial in the test series requires an iteration involving w , as described but this is not difficult. A less tractable problem arises though if the multiplier V is not known and is, therefore, omitted from w,.The resulting sum S is distributed as Vx2(n - m - 1) which quantity is useless for x2testing. This author knows no method of circumventing this problem and hopes t h a t some reader with greater knowledge of statistical techniques will offer a solution. Meanwhile, we cannot proceed to select the degree of a calibrating polynomial from data a t hand when the response variances are nonuniform yet dependent on an unknown multiplying factor. CALCULATING CONFIDENCE LIMITS Two confidence limits for an assay X, are found as two appropriate values of X which satisfy Equation 1. A most important feature of this equation is that the right-hand side the variance of the mean of N , response includes var (YJ, measurements made on the unknown. Whereas for a homoscedastic case, this quantity depends only on N , and the constant 2, for a heteroscedastic case it depends on the location of YLalong the calibration curve. This dependence is expressed as (7)
ANALYTICAL CHEMISTRY, VOL. 51, NO. 6, MAY 1979
which shows its explicit dependence on X,a confidence limit yet to be found. The other variance term on the right-hand side of Equation 1 depends on the nature of the calibration curve and on whether or not a n unknown multiplier V is involved in the variance function. If cr2(y) is completely specified without an unknown V, a three-parameter calibration curve leads to
I
725
IO
lvONUNIFORM V A R I A N C E
I .5
1.0
a two-parameter model leads to
and a polynomial of degree m to m
var [ycJX)l =
C [P,(X)I2var
(8c) (a,) J=o In these expressions, weighting factors are W , = ni/a2[ycal(xi)I (94 If, however, a2(y) = V@(y)and V is found from the residuals of calibrating points about the line, then weighting factors are wi = ni/@[~cal(xJl (9b) and Equations 8a and 8b are modified by multiplying the right-hand sides by V. These options are summarized by writing Equation 1 in the alternative forms A calculation
[r, - Y c a , t w I z
=
is done using Equations loa, 9a, and any one of 8a, 8b, or 8c or it is done with Equation lob, 9b, and 8a or 8b. Either form of Equation 10 can be solved numerically or graphically as the intersection of the horizontal line y = yiwith two confidence bands as shown previously ( I ) .
ILLUSTRATIVE EXAMPLE 1 Although photometric analysis is normally done under conditions which yield a linear calibration line due to Beer’s law and uniform variances, we have deliberately chosen an experiment which is more complicated so that it may serve as a n example of the methods described here. The system is aqueous copper(I1) chloride, which in the near ultraviolet has an absorption band severely deviating from Beer’s law so t h a t a calibration curve of absorbance A vs. formal concentration a t a fixed wavelength in this band is distinctly nonlinear as is shown in Figure 1. The variance of an absorbance measurement is due to many sources of variability but we can classify these into those associated with sample preparation and those arising within the spectrophotometer itself. If standard samples are prepared by diluting a known stock solution into volumetric flasks using transfer pipets of differing volumes, the variance due to this operation is more-or-less uniform if the pipet volume range is not excessive. However, nonuniform variance is introduced if the photometer noise level is roughly independent of transmittance because the absorbance is a logarithmic transformation of transmittance and, hence, its variance cannot be uniform. In order to ensure that the overall variance
0 1
0 2 0 3 CONCENTRATION
Figure 1. Illustrative Example 1: Potentiometric calibration curves fit to the data given in Table I. Figure l a is a cubic polynomial obtalned by F-testing ( 1 ) but ignoring variance nonuniformity. Figure 1b is the quadratic polynomial Equation 12 fit to the same data by X*-testingand taking into account t h e variance function Equation 11
of absorbance vs. concentration be nonuniform, one simply reduces the spectrophotometer slit width to increase the dynode voltage which increases the noise level. When the variance due to this noise is large enough compared to the sample preparation variance, the overall variance is nonuniform and the calibration data are thus heteroscedastic. T h e proper statistical procedure for determining the variance function is to prepare many (perhaps 20 or more) replicate standard samples at each of several (perhaps 6 or more) concentration levels and record a single instantaneous absorbance reading A for each such sample. These measurements (perhaps 120 or more) are then studied to deduce the manner in which var ( A ) actually varies with the A level and this would constitute the desired variance function. Since the purpose here is not to find a truly accurate representation of the variance function but to demonstrate a calculational method, this procedure was shortcut in the following manner. A series of 20 replicate solutions was prepared a t the same concentration level and the absorbance of each of these was recorded using a Beckman ACTA C I11 spectrophotometer set a t 330 nm and with the slits sufficiently wide (0.40 mm) that the noise level was minimal and easily time-averaged. The variance sC2 of these 20 was 43. X lo4, and this was presumed to be due only to sample preparation variability. Then a second series of seven solutions ranging in absorbance from 0.05 to 2. was measured at 330 nm with the slit width reduced to 0.025 mm. In this series, however, 20 instantaneous absorbance readings were recorded from the fluctuating digital display for each solution. T h e variances sn2(A)so recorded were due to spectrophotometric noise and appeared to increase exponentially with absorbance level. A plot of log s,2(A)vs. A yielded a good straight line represented by sn2(A)= 17.7 X lo* exp(3.05 A ) . We then hypothesized that a calibration curve constructed by making a single instantaneous absorbance reading under narrow slit conditions for each of several standard solutions prepared as in the first series above would have a variance function given by
+
a2(A) = sC2 sn2(A)= 43.
X
+ 17.7 X exp(3.05 A ) (11)
The calibrating data recorded in this way are given in Table
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 6, MAY 1979
Table I. Illustrative Example 1: A Photometric Calibration Experimenta calibrating data concenabsorb. absorbconcentration, ance, ance, tration, Yi Xi 95% confidence limits Xi Yi
I 0.050 0.100 0.150 0.200 0.250 0.300 0.350
0.042 0.170 0.361 0.606 0.913 1.339 1.840
(interval)
“Unknowns” Treated Using Variance Function Equation 11 0.172 0.619 1.318
0.1017 0.2014 0.2995
0.0954 0.1965 0.2915
0.1079 0.2064 0.3092
(0.0125) (0.0099) (0.0177)
0.172 0.619 1.318
0.0993 0.2029 0.2985
0.1092 (0.0203) 0.100 0.2082 (0.0106) 0.200 0.3019 (0.0069) 0.300 at 330 nm using a Beckman ACTA C I11
Table 11. Illustrative Example 2 : Gas Chromatographic Quantitative Analysisa calibrating data injected peak replipeak charge, height, cations height, charge, Yi Xi 95% confidence limits Xi Yi ni
I 1.433 3.00 4.90 6.10 7.85 9.51 12.78 13.89 15.97
1
1 3 4 4
t
1
(interval)
true Xi
“Unknowns” Treated Using Variance Function Equation 1 3 1.433 9.24 16.47
0.551 3.358 5.956
0.486 3.052 5.407
0.625 3.734 6.630
(0.139) (0.682) (1.223)
0.564 8.44 5.73
“Unknowns” Treated by Methods of Reference 1 Ignoring Variance Nonuniformity
1.433 9.24 16.47 a Data are taken from Reference 7 , Figure 4.
0.532 3.362 5.980
I together with response measurements also taken singly on three “unknowns” with narrow slit width setting. T h e determinations of unknown concentrations and 95% confidence limits were calculated from these data using a polynomial calibrating curve and the variance function in Equation 11. This polynomial is ycal = -0.01072
0.100 0.200 0.300
“Unknowns” Treated by Methods of Reference 1 Ignoring Variance Nonuniformity
0.0889 0.1976 0.2950 a Single instantaneous absorbance measurements of aqueous CuCl, are taken spectrophotometer with slits set at 0.025 mm.
0.564 1.107 1.715 2.27 2.28 3.44 4.61 5.16 5.73
true Xi
+ 0 . 4 4 0 7 ~+ 1 3 . 3 4 ~ ’
(12)
and is plotted in Figure l b . A polynomial of degree 2 was chosen because with four degrees of freedom (seven data points less three parameters), the 5% tail of the chi square distribution is greater than x2 = 9.5 whereas this polynomial yielded a weighted sum of squares S = 3.1. The conclusion is t h a t lack-of-fit cannot be justified, whereas when a polynomial of degree 1was tried, S greatly exceeded x2. Equation 12 was used to find the unknown concentrations for the three unknown responses and Equation 10a was used to find the 95% confidence limits on these concentrations. The results are shown in Table I. In Equation loa, = 1.960 was selected corresponding to infinite degrees of freedom because of c 2 ( A )of Equation 11 represents the combined results of 160 measurements. Finney (6) discusses the problem of combining heteroscedastic variances and advises that using infinite degrees of freedom cannot be seriously in error when this many measurements are taken.
ILLUSTRATIVE EXAMPLE 2 Bocek and Novak (7) report that gas chromatographic quantitative analysis may under certain conditions involve nonuniform variance. The second illustrative example is based on such a case where the calibration curve is constructed by
0.213 3.065 5.675
0.843 3.658 6.292
(0.630) (0.593) (0.617)
0.564 3.44 5.73
measuring the peak heights corresponding to different injected molar charges of benzene. The charge is varied by injecting equal volumes u of different solutions of benzene in toluene having known concentrations MI. If the charges q, = M,u have variances s: due only to a uniform variance s,’ in the injected volume, then sq2 = MI2su2= (s,*/u2)qI2 showing the proportionality between sq2and q12. If the peak height measuring procedure itself introduces negligible variance, and if peak heights h, are proportional to q r , it follows that the variance in h, is proportional to hL2.This case is Variant I b discussed in Reference 7 and illustrated there as Figure 4 (mislabelled as Variant 2b). We have enlarged Figure 4 and have read off 33 visible calibrating values. These are grouped together into nine sets having virtually the same q level, each set is regarded as a replication series, and these will constitute nine calibrating points. The members of each series are averaged and these values are shown in Table I1 and plotted in Figure 2. For our purposes the charge q corresponds to x and height h to y , which has nonuniform variance given by the a priori variance function rJ>2
= Vy’
(13)
where the multiplier V will be regarded as unknown although its value can be estimated as about 1.3 X from information given in Reference 7. To fit a calibration curve to these data, we recognize the obviously linear relationship between x and y and so fit a two-parameter model equation with f ( x ) = x. The calibrating line so obtained was yca1= -0.0993 2.782~ (14)
+
and the variance of weighted residuals was V = 1.80 X
15
10
l5 I0
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SSY C O N F I D E N C E
N3NUNIFORM VARIANCE TREATIlENT
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assays X i given in Table I obtained by projecting the “unknown” absorbances through the respective curves. In all three cases, the difference between the Xivalues obtained by the two methods is a small fraction of the statistical uncertainties as represented by the 95% confidence intervals. Similarly, in the chromatographic example, F-testing yields the linear calibration curve drawn as the middle line in Figure 2a. The similarity between this line and the heteroscedastic calibration line below in Figure 2b is visually evident and &gain an examination of the corresponding X i values given in Table I shows how little these are affected. Variance nonuniformity may be more important in the confidence limit calculations and this is more obvious in the chromatographic example. The two outer lines drawn in both Figures 2a and 2b are plots of
Y = Y , ~ ( x )f t , b r [Yi- y,d(~)ll (14) for the unknown sample with Yi= 9.24. The intersections 2
4
CHARGE
Figure 2. Illustrative Example 2: Chromatographic calibration data from Figure 4 of Reference 7 as summarized in Table 11. Figure 2a shows treatment according to procedures in Reference 1 ignoring variance nonuniformity; Figure 2b uses heteroscedastic methodology. The middle line in both Figures 2a and 2b is the calibration line and the bounding lines are plots of Equation 14 which together with the y = 9.24 line comprise the graphical solutions for 9 5 % confidence limits
We selected three of the original 33 data points to serve as “unknowns” and calculated 95 7’0 confidence limits for these using Equations 8b and 10b with = 2.365 corresponding to the seven degrees of freedom inherent in V. These confidence limits are given in Table 11.
DISCUSSION If a calibrating procedure is known to be heteroscedastic but the analyst does not know the variance function, the effort required to establish this function is generally greater than t h a t required to find the calibration curve. Therefore, it is of interest to discuss the effects of nonuniform variance on calibration results in order to understand the conditions under which these effects are important. Firstly, there is one point of agreement among statistics texts and that is the minimal effect of weighting factors on fitted regression curves. Unless the variance nonuniformity is quite severe, the curve fitted to calibration data is likely to be nearly the same, whether or not the variance nonuniformity is included in the weighting factors. T o show this effect in the two illustrative examples, we have repeated all the calculations using methods of Reference 1. In the photometric experiment, the calibrating polynomial obtained by F-testing turns out to be a cubic and this is plotted along with the calibrating data in Figure la for comparison with the heteroscedastic curve shown in Figure l b . The similarity between these two curves is also evident by examining the
of these bands with the horizontal y = 9.24 represent graphical solutions to Equations 1 or lob. T h e difference between homoscedastic and heteroscedastic treatments is pronounced. In Figure 2a, the spacing between these bands increases slightly a t both extremities as the variance in Equation 14 is at a minimum near the center of the calibrating points. However, in Figure 2b the bands diverge as y increases and this is due to the y 2 dependence of the variance. The trend of the 95% confidence intervals in Table I1 shows the same effect. These intervals increase markedly with increasing Yi value, but if variance nonuniformity is ignored, they vary much less. A similar discussion of the photometric example is complicated by the variation in local slope of the calibration curve but the following conclusions can be drawn from both. An analyst wishing to avoid the labor of a heteroscedastic treatment will sacrifice statistical reliability by ignoring variance nonuniformity but will generally obtain roughly the same assays X iand roughly the same confidence intervals for samples falling in the region of the calibration curve where the variance is an average over the nonuniform range. This region typically falls somewhere in the mid-range of the curve and so confidence limits obtained near the extremities of the curve are likely to be severely in error. Interested readers will be sent FORTRAN program listings on request to the author.
LITERATURE CITED (1) L. M. Schwartz, Anal. Chem., 48, 2062 (1977). (2) L. M. Schwartz, Anal. Chem., 5 0 , 980 (1978). (3) D. J. Finney, BiOmetks. 32,721 (1976)and owr references cited therein. (4) D. Rcdbard, R. H. Lenox, H. L. Wray, and D. Ramseth, Clin. Chem. ( Winston-Salem,N.C.), 22, 350 (1976)and other references cited therein. ( 5 ) K . A. Brownlee, “StatisticalTheory and Methodology in Science and Engineering”, 1st ed.. John Wiley and Sons, New York, 1960. (6) D. J. Finney, “Statistical Method in Biological Assay” 2nd ed.,Hafner Publishing Co., New York, 1964, Chapter 14. (7) P. Bocek and J. Novak, J . Chromatogr., 51, 375 (1970).
RECEIVED for review November 6,1978. Accepted February 5 , 1979.
