Variable internal standard calibration

Spwhchimica Ah,

lDti1,Vol. 17, pp. 575 to 578. Perwmon Prew Ltd. Printed in Northern Irehnd

RESEARCH NOTE Variable internal standard calibration (Received

23 January 1961)

IN A recent note, CALDER[l] describes how statistical procedures can be applied to increase the reliability of analytical and correction curves in the variable internal standard method. Using sets of standard mixtures, each set differing only in internal standard concentration, the correction curve is drawn as the differences between the mean log IT/IS for the set used in constructing the analytical curve and the mean log IT/Is values from the other sets, Table 1. Log x, x,

=

log c,

= log c,

= 1.0 1.5 2.0 2.5 3.0

I

intensity

Set A 0.301 -0.068 $0.410 + 0.928 + I.405 + 1.909

I

ratios

B 0.544 -0.310 +0.186 + 0.652 + 1.166 + 1.685

(Y)

from

c 0.699 + + + +

0.464 0.005 0.538 1.010 1.537

standard

I

mixtures

D 0.845 -0.628 -0.152 + 0.392 + 0.868 + 1.395

~

E 1.000 -0.795 -0.269 + 0.206 $0.746 + 1.198

’

F 1.114 -0.879 -0.389 +0.091 + 0.634 + 1.092

plotted against internal standard concentration C,. I, is the intensity of the analytical line and I, the intensity of the internal standard line. The linear part analytical curve is established as the regression line of log IT/Is against log concentration of the analytical element, log C,. In a subsequent note CALDER[2] describes improvements to this in which regression lines are calculated for each set and their slopes tested to show no significant .differences. The analytical curve is then established as having the mean slope. In this same note it is also pointed out that by using log C, the correction curves also become linear with respect to log IT/IS over a certain range of C,. Linear correction curves have been used in this laboratory for some years. It is clear that regression techniques could also be applied to constructing the correction curves but as log IJI, is, over the range of C, and C, being considered, a linear function of log C, and log C,, the problem resolves to one of simple multiple regression. By using this technique equal weight is given in evaluation to each value of log IT/IS, log C, and log C,. This is not the case when taking mean values of a number of individual alopes. In addition fewer calculations are necessary for the multiple regression than for six single regressions. The data in Table 1 will be used to demonstrate how analytical and correction curves are established by multiple regression. The slope of the regression line of Y on X,, independent of X,, is given by

[l]

[2]

A. B. CALDER,&&~OC~~WL A. B. C~~~~~,Spect~+ochim. 7A-(4

PP.)

Acta 15, 280 (1959). Acta 16, 391 (1960).

Research Note

and the slope of the regression line of Y on X,, independent of X,, is given by b

(J%2:12)G%Y) - ~~~~~mw) y2J =

(lzx12@x22) - (Xx1x2)2

in which the terms represent the corrected sums of squares or products of the appropriate parameters, e.g. (zrIs) = Z(X12) -

y

N being the number of X, values. Using sets of standard samples of the type illustrated in Table 1, the value of (ZxIz2) becomes 0, and the two slopes can be obtained from the simpler functions GX2Y) -___ -~ (%Y) b1.2 - (Xx12)** *.and b2.1 - (z;x22) The equation for the multiple regression line is then Y = I + &:a (X, -

%) + k-2,(X,

-

5,)

in which ij, El and 5, are the means of the YI, X, and X2 values, respectively. Using the data in Table 1, the regression equation becomes Y = 0.4700 + 0.9986(X, - 2.0) - 0*9967(X, which simplifies to Y = 0*9986X, - 0.9967X2 - 0.7792

0.7505) (1)

The values of the slopes of the single regression lines for analytical and correction curves are calculated as: Correction curve slope

Analytical curve slope set A 0.9898 B 0.9940 c 1*0014 D 1.0132 E 1,0002 P 0.9930

log c,

Mean 0.9986

Mean

= 1.0 - 1.0157 1.5 -

0.9871

2.0 - 1.0140 2.5 - 0.9426 3.0 - 1.0130 -0.9945

It can be seen that the dependence of log IT/I, on log C, in multiple regression is the same as the mean of single regressions due to the latter showing no wide discrepancies. For the correction curve, however, the multiple regression gives a higher value as a result of less weight being placed on the value’ 0.9426 for log C, = 2.5 than in the case for the mean of the individual regressions. Accepting a value of log C, = 0.699 (e.g. 5 per cent Fe,O,) for the analytical curve, this has the equation Y = 0.9986 XI - 1.4759 and can be drawn through the points obtained for Y by inserting log C,(X,) 3.0 in this equation. 576

= 1.0 and

Research Note

Points for the correction curve are obtained by taking X, = O-301 and 1.114 and X, = 1.0 and subtracting from the resultant Y values, the Y value used for the analytical curve with X, = l+O and X, = O-699. Identical values would be obtained if X, = 3.0 were used. Also in his second note Calder derives an equation for the evaluation of log C, from a log IT/I8 value without recourse to graphical means. In so doing he assumes the slopes of analytical and correction curves to be unity. He gives the equation log c,

=

(

low 5 + logC,+log2 “% 1 setc

(2)

where C, is the value of C, when log IT/IS = 0. If this assumption regarding the slope is made then sets of standard mixtures are no longer necessary over this range of C, and C,. It is obvious that (log C#5’,&.,t c in equation (2) is a constant and the equation can be written (3)

X,=K+X,+Y log C, = K + log Cs + log I&

that is

(34

This equation is merely a rearrangement of the multiple regression equation (1) putting b11.2 and bypnlto unity. Hence all that is required to evaluate K in either equation (2) or (3) is a single mixture with known values of C, and C, for which the value of log IT/Is is obtained. Having then established K, routine analysis is effected by determining the log IT/Is value ( Y) and the log C, value (X,) by the independent method to give the result for log C, (X,). Standard mixtures are necessary, however, to establish the shape of any toe or shoulder to the curve, to define the limits of the linear portion and to establish whether in fact the slopes are close to unity. While it is probable that the slopes of analytical and correction curves will not differ significantly from unity when medium to high concentrations are being determined or when background correction is employed, it is considered that slopes should be tested for any departure from unity. This is particularly necessary at low concentrations and when background correction is not employed. In multiple regression, the variance of the two slopes is obtained from the expressions

(&%1.2=

GY2)-

b1.2(~%?/) iv-2

1

- bY2.1GW)

and

(Q12) - mpJ2 [ G%2)(~x22)

1

To test whether the slopes differ significantly is achieved by evaluating 1-h

and

t1= -G&l.,

1 - by.1 tz =

b%)*2.1

and referring these values to a table of the distribution oft with N - 2 degrees of freedom. In the general case when slopes are not unity, equation (1) can be re-written X =Kt-b$.J2t-Y 1 b P1.2 or

log C, = K, + K, log C, + K, log I& 577

(4)

Research Note

where K1, K, and K3 are constants. This equation can then be used for the evaluation of C, without involving graphical procedures. It is believed that the treatments described above can be of value if a logical sequence is followed. First it is necessary to establish the range of C, and Ca over which the log ITIIB: log C relationship is linear, before applying regression techniques. Secondly, when regression techniques have been used, it should be demonstrated that slopes do not differ significantly from unity when equation (3a) can be used. If the slopes differ from unity then the more general equation (4) is applicable. G. T. hAMBEEast African Agriculture and Forestry Research Organisation P.O. Box 21, Kikuyu, Kenya

578