Theoretical Aspects One Point Calibration

GUN. CHEM. 30/7,

1163-1167(1984)

Theoretical Aspects of One-Point Calibration:Causes and Effectsof Some Potential Errors, and Their Dependence on Concentration
G.J. Kemp
The potential errors of the one-point calibration technique are described mathematically. I examine the theoretical causes

Table 1. Symbols Used in This Paper
dx,dx’x SD(4 CV(x)
S, C
Sa, Pa
Sn,,
,

and effectsof analytical errors in absorbance and of errors in determining calibration concentration, and discuss techniques for minimizing their impact. The dependence of these

errorson the calibrationconcentrationand on the size of the
result sconsidered, i and some conclusions are drawn about

Absolute and relative bias (inaccuracy) in x. Standard deviation and coefficient of variation of x; squaresof these expressed as SD2(x), etc.

the choice of calibrationconcentration.
Additional Keyphrases: statistics
-

Sampleand standard oncentrations. c Concentrations sampleand ofthepreceding ofa
sample, affected by carryover. For unimodal distribution, the mean and the bot-

analytical error

tom and topreference limits.
Si, S2 For bimodal distribution, the lower and upper modalconcentrations.

analyzers the principle of one-point calibration is used. After the instrument is zeroed on reagent or diluent, it then aspirates in turn a calibration standard of assigned concentration C (see Table 1) and then a series of unknown samples, which may be patients’ specimens or qualitycontrol sera. If the absorbances given by the standard and a sample are A, and A5, respectively, then thesample concentration, S, is estimated by
S = (As/A,) C

In various

A3, AC,Ab A’, A(t)
‘4am,

Absorbances ofsample, standard, blank. and
Absorbances corrected for blank, and measured at time t, respectively. Absorbances of sample at concentrations Sn,, S1, etc. Unimodal “relative width’; bimodal “relative separation.” Constant and proportional errors.
Maximum

Asi

T, G B, E
Br,,,Em

(1)

values ofbaseline slope and drift.

This method, particularly useful for the measurement of analytes in complex matrices, has been used in a variety of analytical systems, including peak-drawing continuous-flow instruments, true equilibrium chemistries, and (with modification) some kineticmethods for the measurement of concentrations or enzyme activities. The validity this procedureof depends on the familiar assumptions of zero blank, linear calibrationurve, and c similarity of response to analyte in standards and in unknown samples. Failure to meet these conditions is a source of error; one example that has been discussed elsewhere (1) is non-zero reagent blank. All these assumptions are tested

P, 0, f U, V, W
K, Kr

Nonlinear error constants. General errorconstants. Equilibrium constants.

k, k’ K,, n h, r

“Analytical” constants.
Turbidity constants.

zi, z2

Carryover constantnd reagent concentration. a Bichromatic wavelengths.

during method development, but it may be difficult to know how large a deviation from ideal can be tolerated unless we know how the overall error is related to its components. In this paper I attempt to provide atheoretical analysis of these errors, their effects on overall accuracy and precision, and their dependence on the concentrations of sample and
standard.

Iii examining the dependence of these errors on the values of C and S, it is useful to consider two idealized distributions of values of S for patients. Unimodal distribution. Unimodal distribution is the “normal” distribution, with a mean valueSm, and it has a conventional reference interval, whose top and bottom limits (S and Sb) are, respectively, two population standard deviations above and below Sm. We can characterize it by a quantity T, which measures its “relative width.” By definition, T = 2 (population standard deviation)/Sm (3a) where the factor 2 is chosen to simplify subsequent sions. From this, we can write the following:...