The formula for nonlinearity requires two
equations, one for the curve that best fits the empiri-
cal calibration data and one for the ideal straight line
fit of the data. Kroll and Emancipator recommend,
and themselves use, polynomial equations to derive
the best fit curve. Polynomial equations are a good
choice because they can be readily fit to the data by
weighted multiple linear regression and they are also
easy to integrate. The ideal straight line fit is the
line that results in the minimum value for nonlinear-
ity. Formulas for calculating the slope and intercept
of this line can be found in Emancipator and Kroll
(1993). Note that these equations refer to the
calibration curve and, as such, relate the calibrator
concentration to the signal magnitude. The y data in
a linearity study, therefore, are signal magnitudes.
Luque de Castro
et al.
properly used the strength of
the fluorescence signal as the y variable in the
linearity study of their method,
A series of eight standard solutions with
concentrations between 0.1 and 20.0 µmol/L
were prepared from the phosphate solution
described in
Materials and Methods
. The
equation of the analytical signal obtained by
triplicate injection of these standards into the
FI manifold was as follows: fluorescence
intensity = 27.5 + 49.2 [P
i
] (µmol/L)
Some authors report using measurement results
rather than signal magnitudes as the y data. This is
improper and, even more to the point, paradoxical in
that the calibration curve is the matter under study;
without a calibration curve there cannot be a
measurement curve and, in turn, there cannot be
measurement results.
Assessment of analytical quality
The quality of an analytical method is assessed
through a characterization of the method trueness
and the method precision.
Trueness.
As discussed earlier, trueness is the
closeness of agreement between the true value of an
analyte and the average value obtained from a large
number of replicate measurements. To evaluate the
trueness of a method it is, therefore, necessary to
know the true analyte value in a sample. Indeed, it
is necessary to know the true value in a number of
samples with analyte concentrations that vary across
the proposed range of measurement of the method.
This requirement can be met in either of two ways.
If the method evaluators have access to a reference
method for the measurement of the analyte, the true
concentrations can be determined in clinical samples
from the evaluators’ laboratory. Otherwise, the
evaluators can use certified reference material for
which the true concentration of the analyte of inter-
est have been determined by the certifying agency
through the use of reference or definitive methods.
The steps in the characterization of the trueness
of a laboratory method are depicted in Figure 2.7.
Between-run replicate measurements are made of the
samples with known analyte concentration (top
graph). At least 5 replicate measurements should be
performed and 10 is preferred. The average value
of each set of replicates is computed and the differ-
ences between the averages and the true values are
calculated. The differences, which represent the
bias of the method at the sampled analyte concentra-
tions, are plotted (middle graph). In order to
characterize the bias at all concentrations within the
range of measurement, a linear bias model is fit to
the data using weighted linear regression. Bias is
classified as being constant if the constant term of
the model is nonzero. It is classified as proportional
if the slope of the model is nonzero. In the example,
the bias shows a mixed pattern.
The graph of the bias model is referred to as a
bias profile (Keller and Passing 1989). The bias
profile can be graphed in terms of absolute bias
(middle graph) or relative bias, which is the ratio of
the bias to the analyte concentration expressed as
percent (bottom graph). It is especially useful to
graph the bias profile in relative terms because
method bias criteria, which are rules for deciding if
a method shows adequate trueness for clinical
purposes, are usually expressed in relative rather
than absolute terms. A bias criterion can therefore
be plotted on the same graph as the bias profile and
the interval over which the method meets the crite-
rion can be easily appreciated. An example of this is
shown in the bottom graph. The bias criterion
depicted is a relative bias of less than 10%. The
relative bias of the method satisfies the criterion at
analyte concentrations greater than 19 units.
Recovery.
If method evaluators do not have
access to a reference or definitive method and if
there are no readily available certified reference
materials, method trueness cannot be evaluated by
the approach outlined in the preceding section.
Trueness must then be evaluated by means of a
recovery study. A known amount of analyte is
Laboratory Methods
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