Download Limitations in Analytical Accuracy, Part II: Theories to Describe the

20 Spectroscopy 22(2)
w w w. s p e c t r o s c o p y o n l i n e . c o m
February 2007
Chemometrics In Spectroscopy
Limitations in Analytical
Accuracy, Part II: Theories to
Describe the Limits in Analytical
Accuracy and Comparing Test
Results for Analytical Uncertainty
This column continues from the previous discussion on the limits of analytical accuracy
and uncertainty. It introduces several tools used for evaluating and comparing analytical
results.
Jerome Workman, Jr. and Howard Mark
Limits in Analytical Accuracy
Y
ou might recall from our previous column (1) how
Horwitz throws down the gauntlet to analytical scientists, stating that a general equation can be formulated for the representation of analytical precision based
upon analyte concentration (2). He states this as follows:
CV(%) 2(10.5logC)
where C is the mass fraction as concentration expressed
in powers of 10 (for example, 0.1% analyte is equal to C 103).
A paper published by Hall and Selinger (3) points out an
empirical formula relating the concentration (c) to the coefficient of variation (CV), also known as the precision ().
They derive the origin of the “trumpet curve” using a binomial distribution explanation. Their final derived relationship becomes
CV c 0.15
50
They further simplify the Horwitz trumpet relationship in
two forms as follows:
CV(%) 0.006c0.5
and
0.006c0.5
They then derive their own binomial model relationships
using Horwitz’s data with variable apparent sample size.
CV(%) 0.02c0.15
and
0.02c0.85
22 Spectroscopy 22(2)
Both sets of relationships depict relative error as inversely proportional to
analyte concentration.
In yet a more detailed incursion
into this subject, Rocke and Lorenzato
(4) describe two disparate conditions
in analytical error: concentrations
near zero and macro level concentrations, say greater than 0.5% for argument’s sake. They propose that analytical error comprises two types,
additive and multiplicative. So their
derived model for this condition is
x e
where x is the measured concentration; is the true analyte concentration; and is a normally distributed
analytical error with mean 0 and standard deviation . It should be noted
that represents the multiplicative or
proportional error with concentration
and represents the additive error
demonstrated at small concentrations.
Using this approach, the critical
level at which the CV is a specific
value can be found by solving for x
using the following relationship:
(CVx)2 (
x)2 ()2
where x is the measured analyte concentration as the practical quantitation level (PQL used by the U.S. Environmental Protection Agency [EPA]).
This relationship is simplified to
x
w w w. s p e c t r o s c o p y o n l i n e . c o m
February 2007
1
2
(CV 2 ση )
where CV is the critical level at which
the coefficient of variation is a preselected value to be achieved using a specific analytical method, and is the
standard deviation of the multiplicative
or measurement error of the method.
For example, if the desired CV is 0.3
and is 0.1, then the PQL or x is computed as 3.54. This is the lowest analyte
concentration that can be determined
given the parameters used.
The authors describe the earlier
model as a linear exponential calibration curve as
y e
where y is the observed measurement
data. This model approximates a consistent or constant standard deviation
model at low concentrations and approximates a constant CV model for
high concentrations. In this model, the
multiplicative error varies as e
.
certainty range of 5.014 to 5.714 with
an uncertainty interval of 0.7. Therefore, if we have a relatively unbiased
analytical method, there is a 95%
probability that our true analyte value
lies between these upper and lower
concentration limits.
Detection Limit for
Concentrations Near Zero
Comparison Test for a Single Set
of Measurements Versus a True
Analytical Result
Finally, detection limit (D) is estimated using
3σ
D r
where is the standard deviation of
the measurement error measured at
low (near zero) concentration, and r is
the number of replicate measurements
made.
Uncertainty in an Analytical
Measurement
By making replicate analytical measurements, one can estimate the certainty of
the analyte concentration using a computation of the confidence limits. As an
example, given five replicate measurement results as: 5.30%, 5.44%, 5.78%,
5.00%, and 5.30%, the precision (or
standard deviation) is computed using
the following equation:
r
s
(x − x )
(r − 1)
i 1
2
i
where s represents the precision, means summation of all the (xi –x)2
values, xi is an individual replicate analytical result, –x is the mean of the
replicate results, and r is the total
number of replicates included in the
group (this is often represented as n).
For the previous set of replicates, s 0.282. The degrees of freedom are indicated by r 1 4. If we want to
calculate the 95% confidence level, we
note that the t-value is 2.776. So the
uncertainty (U) of our measurement
result is calculated as
U x t •
s
r
The example case results in an un-
Let’s start this discussion by assuming
we have a known analytical value by
artificially creating a standard sample
using impossibly precise weighing and
mixing methods so that the true analytical value is 5.2% analyte. We make
one measurement and obtain a value
of 5.7%. Then we refer to errors using
statistical terms as follows:
Measured value: 5.7%
“True” value: 5.2%
Absolute error: Measured value True value 0.5%
Relative percent error: 0.5/5.2 100
9.6%
Then we recalibrate our instrumentation and obtain the results: 5.10, 5.20,
5.30, 5.10, and 5.00. Thus, our mean
value (x–) is 5.14.
Our precision as the standard deviation (s) of these five replicate measurements is calculated as 0.114 with n 1
4 degrees of freedom. The t-value
from the t table, 0.95, degrees of
freedom as 4, is 2.776.
To determine if a specific test result
is significantly different from the true
or mean value, we use the test statistic
(Te):
Te x −
• n
s
For this example, Te 1.177. We note
that there is no significant difference
in the measured value versus the expected or true value if Te t-value.
And there is a significant difference
between the set of measured values
and the true value if Te t-value. We
must then conclude here that there is
no difference between the measured
set of values and the true value, as
1.177 2.776.
24 Spectroscopy 22(2)
Comparison Test for Two Sets of
Measurements
If we take two sets of five measurements using two calibrated instruments and the mean results are x–1 5.14 and x–2 5.16, we would like to
know if the two sets of results are statistically identical. So we calculate the
standard deviation for both sets and
find s1 0.114 and s2 0.193. The
pooled standard deviation s1,2 0.079. The degrees of freedom in this
case is n1 1 equals 5 1 4. The tvalue at 0.95, d.f. 4, is 2.776.
To determine if one set of measurements is significantly different from
the other set of measurements, we use
the test statistic (Te):
Te1, 2 w w w. s p e c t r o s c o p y o n l i n e . c o m
February 2007
x 1 x 2
1
s•
n1 n 2
For this example, Te1,2 0.398. So if
there is no significant difference in the
sets of measured values, we would expect Te t-value, because 0.398 2.776. And if there is a significant difference between the sets of measured
values, we expect Te t-value. We
must conclude here that there is no
difference between the sets of measured values.
Calculating the Number of
Measurements Required to
Establish a Mean Value (or
Analytical Result) with a
Prescribed Uncertainty (Accuracy)
If error is random and follows probabilistic (normally distributed) variance
phenomena, we must be able to make
additional measurements to reduce
the measurement noise or variability.
This is certainly true in the real world
to some extent. Most of us with some
basic statistical training will recall the
concept of calculating the number of
measurements required to establish a
mean value (or analytical result) with
a prescribed accuracy. For this calculation, one would designate the allowable error (e), and a probability (or
risk) that a measured value (m) would
be different by an amount (d).
We begin this estimate by computing the standard deviation of measurements. This is determined by first calculating the mean, then taking the
difference of each control result from
the mean, squaring that difference, dividing by n 1, then taking the
square root. All these operations are
included in the equation:
n
s
(x x )
(n 1)
i 1
2
i
where s represents the standard deviation; means summation of all the (xi
–x)2 values; xi is an individual control
result; –x is the mean of the control results; and n is the total number of control results included in the group.
If we were to follow a cookbook approach for computing the various parameters, we would proceed as follows:
● Compute an estimate of (s) for the
method (see previous);
● Choose the allowable margin of
error (d);
● Choose the probability level as
alpha (), as the risk that our measurement value (m) will be off by
more than d;
● Determine the appropriate t-value
for t1 /2 for n 1 degrees of freedom.
● Finally the formula for n (the number of discrete measurements
required) for a given uncertainty is
as follows:
⎛ t 2 •s 2 ⎞
n ⎜ 2 ⎟ 1
⎝ d ⎠
Problem Example: We want to learn
the average value for the quantity of
toluene in a test sample for a set of hydrocarbon mixtures.
s 1, 0.95, d 0.1. For this
problem, t1 /2 1.96 (from t table),
and thus n is computed as follows:
⎛ 1.96 2 •12 ⎞
n ⎜⎜
2
⎟⎟ 1 385
⎝ 0 .1
⎠
So if we take 385 measurements, we
conclude with a 95% confidence that
the true analyte value (mean value)
will be between the average of the 385
results and 0.1, or –x 0.1.
The Q-Test for Outliers (5–7)
We make five replicate measurements
using an analytical method to calculate basic statistics regarding the
method. Then we want to determine if
a seemingly aberrant single result is
indeed a statistical outlier. The five
replicate measurements are: 5.30%,
5.44%, 5.78%, 5.00%, and 5.30%. The
result we are concerned with is 6.0%.
Is this result an outlier? To find out, we
first calculate the absolute values of
the individual deviations, as in Table I.
Thus the minimum deviation
(DMin) is 0.22; the maximum deviation 1.00; and the deviation range (R)
is 1.00 0.22 0.78. We then calculate the Q-test value as Qn using:
Qn D Min
R
This results in the Qn of 0.22/0.78 0.28 for n 5.
Using the Q-value table (90% confidence level as Table II), we note that if
Qn Q-value, then the measurement
is not an outlier. Conversely, if Qn Q-value, then the measurement is an
outlier.
So because 0.28 0.642, this test
value is not considered an outlier.
Summation of Variance from
Several Data Sets
We sum the variance from several separate sets of data by computing the
variance of each set of measurements;
this is determined by first calculating
the mean for each set, then taking the
difference of each result from the
mean, squaring that difference, and dividing by r 1 where r is the number
of replicates in each individual data
Table I: Absolute values of individual
deviations
Compute Deviation
5.30–6.00
5.44–6.00
5.78–6.00
5.00–6.00
5.30–6.00
Absolute Deviation
0.70
0.56
0.22
1.00
0.70
26 Spectroscopy 22(2)
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February 2007
Table II: Q-value table (at different confidence levels)
n:
3
Q(90%): 0.941
Q(95%): 0.970
Q(99%): 0.994
4
5
6
7
8
9
10
0.765
0.829
0.926
0.642
0.710
0.821
0.560
0.625
0.740
0.507
0.568
0.680
0.468
0.526
0.634
0.437
0.493
0.598
0.412
0.466
0.568
set. All these operations are included
in the following equation:
r
s
2
(x x )
(r 1)
i 1
2
i
where s2 represents the variance for
each set; means summation of all
the (xi –x)2 values; xi is an individual
result; –x is the mean of each set of results; and r is the total number of results included in each set.
The pooled variance (s2p) is given as
s 2 s 22 ... s 2k
s 2p 1
k
where s2k represents the variance for
each data set, and k is the total number of data sets included in the pooled
group.
The pooled standard deviation p is
given as:
σ p s 2p
References
(1) J. Workman and H. Mark, Spectroscopy 21(9), 18–24 (2006).
(2) W. Horwitz, Anal. Chem. 54(1),
67A–76A (1982).
(3) P. Hall and B. Selinger, Anal. Chem.
61, 1465–1466 (1989).
(4) D. Rocke and S. Lorenzato, Technometrics 37(2), 176–184 (1995).
(5) J.C. Miller and J.N. Miller, Statistics for
Analytical Chemistry (second edition)
(Ellis Horwood, Upper Saddle River,
New Jersey, 1992), pp. 63–64.
(6) W.J. Dixon and F.J. Massey, Jr., Introduction to Statistical Analysis (fourth
edition), W.J. Dixon, Ed. (McGraw-Hill,
New York, 1983), pp. 377, 548.
(7) D.B. Rohrabacher, Anal. Chem. 63,
139 (1991).
Circle 44
Erratum
An astute reader, Charles R. Lytle,
PhD, pointed out a correction and
a clarification in our previous column (1). The Correction: Equation
1 is missing the denominator (r).
The Clarification: For Equations 1
and 2, r represents the number of
replicates (subsamples) for n as the
sample number; subscripts i and j
are indices for subsamples. The narrative is referring to multiple analyses of the same sample, better referred to as subsamples of the same
analytical sample.
(1) Spectroscopy 21(9), 18–24
(2006).
Jerome Workman,
Jr. serves on the Editorial
Advisory Board of
Spectroscopy and is director of research and technology for the Molecular
Spectroscopy &
Microanalysis division of Thermo Fisher
Scientific Inc. He can be reached by e-mail
at: [email protected].
Howard Mark
serves on the Editorial
Advisory Board of
Spectroscopy and runs a
consulting service, Mark
Electronics (Suffern, NY).
He can be reached via email at:
[email protected].