resulting data set. The variance of the replicate
results for each run is calculated using the formula,
var
j
=
(
x
i
−
mean
j
)
2
n
−
1
where
x
i
is the
i
th replicate result in run
j, mean
j
is
the mean of the replicates in run
j,
and
n
is the
number of replicates assayed per run. If only two
replicates are assayed per run,
n-1
is set equal to 2
not to 1. This biases the analysis somewhat, so it is
better to assay three or more replicates per run.
Within-run variance is calculated as the average of
the individual run variances so,
SD
within-run
=
var
j
N
where
N
is the number of runs. The variance of the
individual run replicate means is computed using the
formula,
var
means
=
(
mean
j
−
overall mean
)
2
N
−
1
where
overall mean
is the mean value of the individ-
ual run replicate means. Then (Box
et al.
1978),
SD
between-run
=
var
means
−
SD
within
−
run
2
n
A point that needs to be mentioned in any
discussion of method precision is that within-run
precision can always be improved by measuring
samples in duplicate or triplicate and reporting the
average value of the results. With replicate
measurements, the imprecision decreases by a factor
equal to the square root of the number of replicates.
For duplicate measurements, the within-run impreci-
sion is 0.71 times as large as with single measure-
ments and with triplicate measurements, it is 0.58
times as large. This approach is costly in that fewer
samples can be run per batch but the resultant
improvement in method precision may significantly
increase the clinical utility of the method.
Resolving power and detection limit.
The
resolving power of a method is expressed in terms of
the minimum distinguishable difference in concen-
tration, D
min
. Using the formula,
D
min
= z
c
SD
within-laboratory
2
the resolution profile for a method can be calculated
from the imprecision model, giving,
D
min
= z
c
(b0 + b1 concentration)
b2
2
where the model parameters
b0
,
b1
, and
b2
apply to
within-laboratory imprecision.
The detection limit can be calculated as the
smallest concentration that solves the equation
concentration =
z
c
(b0 + b1 concentration)
b2
2
An approximate solution can be found graphically
from a plot of the resolution profile. It is the
concentration at which the resolution profile inter-
sects the line of identity. Using this starting value,
the exact solution can be found using an iterative
root solving algorithm such as Newton’s method
(Sadler
et al.
1992). Such algorithms are now
widely available in computer spreadsheet programs.
An alternative and more direct way of calculat-
ing the detection limit is to use the imprecision data
from the replicate sets with concentrations that are
near the detection limit. Typically, the zero concen-
tration replicate set and the lowest concentration
replicate set are used. The variances of the data sets
are calculated (var
zero
and var
low
) as is the pooled
variance,
var
pool
=
n
1
var
zero
+
n
2
var
low
n
1
+
n
2
−
2
where
n
1
and
n
2
are the number of replicates in the
zero concentration and low concentration data sets,
respectively. The detection limit is calculated using
the following formula (Rodbard 1978, Büttner
et al.
1980b),
detection limit = blank - t
c
var
pool
1
n
1
+
1
n
2
where
blank
is the mean value of the zero concentra-
tion replicate set and
t
c
is the confidence coefficient
as found with the t distribution. For a 95% confi-
dence level and equal numbers of replicates in each
data set,
t
c
equals 1.734 for 20 replicates, 1.701 for
30 replicates, 1.686 for 40 replicates, and 1.677 for
50 replicates.
Analytical range
The analytical range, or working range, is the
range of analyte concentrations for which the method
satisfies all of the following criteria: the bias of the
method is within acceptable limits, the precision of
the method is within acceptable limits, and, if appro-
priate, the calibration curve is acceptably linear.
The range is determined by reference to the bias
profile, the imprecision profile, and the findings of
the linearity study.
Laboratory Methods
2-22
