Anal. Chem. 1997, 69, 226-234
Comparison of Signal-to-Noise Ratios Edward Voigtman
Department of Chemistry, Lederle GRC, Box 34510, University of Massachusetts at Amherst, Amherst, Massachusetts 01003-4510
The probability distribution (density) function for the experimental signal-to-noise ratio (SNR) defined as xj /s, where xj is the sample mean and s is the customary sample standard deviation, has been derived and found to be in excellent agreement with accurate Monte Carlo simulation results. The SNR probability distribution function is a hypergeometric function which has no closed-form expression in elementary functions. The same applies to the probability distribution function for the relative standard deviation. In contrast, the probability distribution function for the approximate SNR defined by µ/s′, where µ is the population mean parameter and s′ ≡ s[(N - 1)/ N]1/2, has a closed-form expression but is inaccurate for small numbers of measurements. The experimental SNR is a biased estimator of the true SNR, but the bias is easily correctable. Monte Carlo simulation methods were used to derive critical value tables for comparison of experimental SNRs and relative standard deviations. The critical value tables presented herein are accurate to about 1% for confidence levels of 75%, 90%, and 95%, and to about 5% for 99% confidence level.
use in the generation of the desired critical value tables. However, it is easy (albeit tedious) to use accurate Monte Carlo calculations to generate the tables, which are presented here in both onetailed and two-tailed forms and which supplant the tables published by Williams.
The comparison and optimization of experimental signal-tonoise ratios (SNRs) are of considerable importance in all disciplines dealing with analytical measurements because of the fundamental connection between SNR and detection power, i.e., limits of detection. Since the experimental SNR is a random variate, defined in the simplest case as the quotient of two independent random variates (see below), the properties of the experimental SNR may be determined by examining its relevant probability distribution function. Similarly, comparison of two experimental SNRs requires consideration of the probability distribution function (PDF) for a quotient of two experimental SNRs. Understanding these PDFs and their implications is the key to understanding the performance of SNRs as figures of merit and optimization criteria. Williams realized that experimental SNRs could be compared via a statistical test analogous to the standard F test used to compare sample variances.1 Toward this end, he computed a set of four critical value tables for use in comparing experimental SNRs. In the course of rederiving his equations and regenerating his published tables with higher accuracy, it became apparent that Williams had made approximations at several crucial points in his derivation; therefore, it was an open question as to how close his tables might be to the correct, but then unknown, critical value tables. In this paper, the PDFs for the most commonly used definition of the SNR and the quotient of two such SNRs are presented. Unfortunately, the latter PDF is too complicated for convenient
where the customary sample standard deviation, denoted by s, is defined by
(1) Williams, R. R. Anal. Chem. 1991, 63, 1638-1643.
226 Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
THEORY Suppose that N replicate measurements (denoted by xi, for i ) 1, 2, 3, ..., N) are made from an infinite, Gaussian-distributed population having population mean parameter µ and population standard deviation parameter σ. The number of degrees of freedom, denoted ν, is N - 1, unless otherwise noted. To avoid unnecessary use of absolute value signs, it is assumed that µ is a positive real number, and to avoid having a SNR that is intrinsically close to zero, it is assumed that µ is at least several times larger than σ. It is also assumed that N is a relatively small value (230), since this is typical in practice. If the true SNR is defined as µ/σ, then the most commonly used experimental SNR test statistic is defined as
SNR ≡ jx/s
x
(1)
N
∑(x - jx)
s≡+
2
i
i)1
N-1
(2)
and jx is the sample mean, defined by jx ≡ (x1 + x2 + ... + xN)/N. The necessary PDF for jx is [ref 2, p 175]
pjx(xj) )
2 2 1 e-(xj-µ) /2σe σex2π
(3)
where σe ≡ σ/N1/2 is the population standard error. The PDF for s may be obtained from the PDF of the variance [ref 2, eq 8.3.15], where the variance is denoted as v (not to be confused with ν):
pv(v) )
2 νν/2 v(ν-2)/2e-νv/2σ σ 2 Γ(ν/2)
ν ν/2
(4)
by noting that, if y ≡ +x1/2, where x is a random variate having (2) Keeping, E. S. Introduction to Statistical Inference, 2nd ed.; Dover Publications, Inc.: New York, 1995 (republication of the book originally published by D. Van Nostrand Co., Princeton, NJ, 1962). S0003-2700(96)00675-0 CCC: $14.00
© 1997 American Chemical Society
Table 1. PDF Expressions for x j /s, x j /s′, x/se, and s/x j , where SNR ≡ x j /s, SNR′ ≡ x j /s′, SNRe ≡ x j /se, and RSD ≡ s/ x j : All Four PDFs Have the General Integral Form pr(r) ) K∫∞0 uνe-βµ2-γu du pR(R)
K
R
pSNR(SNR)
β
νν/2(ν + 1)1/2
SNR
2
σν+12(ν-1)/2Γ(ν/2)π1/2 pSNR′(SNR′)
(ν + 1)(ν+1)/2
SNR′ σ
pSNRe(SNRe)
2
Γ(ν/2)π
(ν + 1)(ν+1)/2νν/2
SNRe σ
pRSD(RSD)
ν+1 (ν-1)/2
ν+1 (ν-1)/2
2
Γ(ν/2)π
2
σ
ν+1 (ν-1)/2
2
Γ(ν/2)π
2
e-(ν+1)µ /2σ 1/2 2
2
e-(ν+1)µ /2σ 1/2
νν/2(ν + 1)1/2Rν-1
RSD
2
e-(ν+1)µ /2σ
2
1/2
2
e-(ν+1)µ /2σ
u
γ
ν + (ν + 1)R2
(ν + 1)µR -
(ν + 1)(1 + R2)
(ν + 1)µR -
(ν + 1)µR -
se
σ2
2σ2 ν + 1+ νR2 2σ2
s′
σ2
2σ2 (ν + 1)(ν + R2)
s
σ2
2σ2
(ν + 1)µ -
jx
σ2
Figure 1. Monte Carlo simulation program used to determine the PDF and CDF histograms, and critical values, for R, Re, and Rr.
px(x) as its PDF, then [ref 3, p 126] py(y) ) 2ypx(y2). Hence, since s is the positive square root of the variance,
ps(s) )
ν
ν/2
σν2(ν-2)/2Γ(ν/2)
2
sν-1e-νs /2σ
2
is [ref 3, eq 5-7]
pz(z) ) ypx(yz)
(6)
(5)
These equations are easily derived from first principles and have long been redundantly available in the literature. Note that jx and s are independent random variates [ref 3, p 250]. Thus, to obtain the PDF expression for the SNR, it is necessary to find the PDF for the quotient of two independent random variates. One way to proceed is as follows: assume x is a random variate having PDF given by px(x) and assume y is a constant greater than zero. Then, the PDF for z, where z ≡ x/y, (3) Papoulis, A. Probability, Random Variables, and Stochastic Processes; McGraw-Hill: New York, 1965.
A moment’s thought reveals that y may be replaced by a distributed sum of constants, i.e., a discrete PDF, resulting in the PDF for z being a weighted summation of terms of the form in eq 6:
pz(z) =
∑∆yp (y )y p (y z) y
i
i x
i
(7)
i
Then, in the continuous limit, the summation is replaced by integration, and the result is
pz(z) )
∫ p (y)yp (yz) dy ∞
0
y
x
Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
(8) 227
Figure 2. Typical R PDF and CDF histograms for ν1 ) ν2 ) 5.
Figure 3. Comparison of theoretical PDFs and million event histogram data for s and s′, with N ) 3.
With z ≡ SNR, x ≡ jx, and y ≡ s, eq 8 becomes
pSNR(SNR) )
∫ p (s)sp (sSNR) ds ∞
0
s
jx
(9)
Substitution of eqs 3 and 5 into eq 9 yields
∫
pR(R) ) K 228
∞ ν -βu2-γu ue 0
du
Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
(10)
where R, K, β, γ, and u are given in the first row of Table 1. Also given are the PDFs for three related statistics: jx/s′, jx/se, and s/xj, where s′ ≡ s[(N - 1)/N]1/2 and se is the standard error, defined as s/N1/2. Note that s/xj is the relative standard deviation (RSD). The integral in eq 10 may be expressed analytically in terms of hypergeometric functions [ref 4, eq A.1.52]. However, the (4) Middleton, D. Introduction to Statistical Communication Theory; Peninsula Publishing: Los Altos, CA, 1987 (republication of the book originally published by McGraw-Hill, New York, 1960).
Figure 4. Comparison of theoretical PDFs and million event histogram data for xj/s and µ/s′, with N ) 3.
Figure 5. Comparison of theoretical PDFs for xj/s and µ/s′, with N ) 21.
PDFs for the SNR and the three related statistics considered herein cannot be expressed, in closed form, in terms of elementary functions. Fortunately, it is easy to evaluate eq 10 by numerical integration. In the results presented below, the integration algorithm was simply a 1001 step “rectangular” approximation. Having obtained the PDF for jx/s, it is possible to define the quotient of a pair of experimental SNRs as a test statistic analogous to F, i.e.,
R ≡ SNR2/SNR1
(11)
where SNR1 ) jx1/s1 and SNR2 ) jx2/s2. The resulting PDF for R is then
pR(R) )
∫ SNR [∫ s p (s )p (s SNR ) ds ] × [∫ s p (s )p (s RSNR ) ds ] dSNR ∞
0
∞
1
0
1 s1
1
jx1
1
2 s2
2
jx2
2
1
1
∞
0
1
2
1
(12)
which must itself be integrated and inverted to yield the desired Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
229
Table 2. Correction Factors and Expectation Values for x j /s, When µ ≡ 9, σ ≡ 3, with ν Degrees of Freedoma
Table 3. Correction Factors and Expectation Values for s, When σ ≡ 3, with ν Degrees of Freedoma
ν
E[xj/s]
correction factor
ν
E[xj/s]
correction factor
ν
E[s]
correction factor
ν
E[s]
correction factor
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
15.423 5.2395 4.1417 3.7588 3.5682 3.4537 3.3776 3.3233 3.2827 3.2512 3.2259 3.2053 3.1882 3.1736 3.1612 3.1504 3.1410 3.1327 3.1253 3.1189 3.1127 3.1074 3.1025 3.0980 3.0939
0.195 0.573 0.724 0.798 0.841 0.869 0.888 0.903 0.914 0.923 0.930 0.936 0.941 0.945 0.949 0.952 0.955 0.958 0.960 0.962 0.964 0.965 0.967 0.968 0.970
26 27 28 29 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 200 300 400 500 700 1000
3.0902 3.0867 3.0835 3.0805 3.0777 3.0663 3.0578 3.0512 3.0460 3.0417 3.0382 3.0352 3.0326 3.0304 3.0285 3.0268 3.0253 3.0239 3.0227 3.0113 3.0075 3.0056 3.0045 3.0032 3.0023
0.971 0.972 0.973 0.974 0.975 0.978 0.981 0.983 0.985 0.986 0.987 0.988 0.989 0.990 0.991 0.991 0.992 0.992 0.992 0.996 0.998 0.998 0.999 0.999 0.999
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2.3937 2.6587 2.7640 2.8200 2.8546 2.8781 2.8951 2.9079 2.9180 2.9261 2.9327 2.9382 2.9429 2.9469 2.9505 2.9535 2.9562 2.9586 2.9608 2.9627 2.9645 2.9661 2.9676 2.9689 2.9702
1.2533 1.1284 1.0854 1.0638 1.0509 1.0424 1.0362 1.0317 1.0281 1.0253 1.0229 1.0210 1.0194 1.0180 1.0168 1.0157 1.0148 1.0140 1.0132 1.0126 1.0120 1.0114 1.0109 1.0105 1.0100
26 27 28 29 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 200 300 400 500 700 1000
2.9713 2.9724 2.9733 2.9743 2.9751 2.9786 2.9813 2.9834 2.9850 2.9864 2.9875 2.9885 2.9893 2.9900 2.9906 2.9912 2.9917 2.9921 2.9925 2.9963 2.9975 2.9981 2.9985 2.9989 2.9993
1.0097 1.0093 1.0090 1.0086 1.0084 1.0072 1.0063 1.0056 1.0050 1.0046 1.0042 1.0038 1.0036 1.0033 1.0031 1.0029 1.0028 1.0026 1.0025 1.0012 1.0008 1.0006 1.0005 1.0004 1.0002
a The correction factor, defined as (µ/σ)/E[x j/s], is independent of µ/σ because E[xj/s] is proportional to µ/σ for a given ν. Note that ν ) N - 1, where N is the number of replicate measurements.
critical values of R. Lack of a closed-form expression for the PDF for jx/s effectively guarantees that quotients of two such ratios are cumbersome multiple integrals, as in eq 12, which are very time consuming to process numerically. Therefore, it was decided to bypass eq 12 until more computing power becomes available and instead use Monte Carlo techniques to generate accurate histographic approximations of the desired R PDF. Then, simple running integration on the bin contents of the R histogram yields the approximate R cumulative distribution function (CDF) histogram (i.e., ogive), from which the critical R values are found by numerical inversion. MONTE CARLO SIMULATION Detailed Monte Carlo computer simulations were used to verify the derived SNR PDF expression for various choices of µ, σ, and ν and to generate the histographic approximations of the R PDFs for 324 choices of ν1 and ν2. All simulations were performed using the Extend simulation program (Imagine That, Inc., San Jose, CA) augmented with blocks from the LightStone 95 libraries (LightStone Labs, Northampton, MA) and several additional customprogrammed blocks. Most of the simulations were performed on a Macintosh IIfx computer (Apple Computer, Cupertino, CA) with 20 MB RAM; the remainder were done on a Macintosh PowerBook 180 with 8 MB RAM. All simulation numerical results, and numerical evaluations of functions and integrals, were saved as ASCII text files and graphed with Igor Pro (Wavemetrics, Lake Oswego, OR). Unless otherwise noted, we follow Williams1 and arbitrarily assume that µ ≡ 9, σ ≡ 3, and N ≡ 3. With the exception of the simulations used to prepare the critical value tables (see below), all of the simulations involve selection of N samples from Gaussian-distributed populations, 230
Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
a The correction factor, defined as σ/E[s], is independent of σ because E[s] is proportional to σ for a given ν. Note that ν ≡ N - 1, where N is the number of replicate measurements.
computation of jx, s, jx/s, s′, and so on, and histogram binning the resulting statistics. Each simulation is comprised of 1 million steps, ensuring that the resulting million event histograms were highly precise and, therefore, of diagnostic quality. Although the software actually used was very convenient for the task, any general purpose computer language suffices, provided it can properly generate the requisite Gaussian variates. Therefore, for reasons of economy of presentation, the simulation programs (with one exception described below) are not shown. Monte Carlo computer simulations were used to obtain histographic approximations of the PDFs and CDFs associated with quotients of experimental SNRs, for 324 examined ν1, ν2 pairs (i.e., 2 e ν1 e 19 and 2 e ν2 e 19). The simulation program used to generate the critical values is shown in Figure 1, which has, as its heart, the block labeled “Monte Carlo SNR ratios”. For each simulation step, it generated ν1 + 1 Gaussian variates from a population where µ was arbitrarily chosen as 30 and σ as 3, and computed jx1, s1, and se1. Similarly, it also generated another ν2 + 1 Gaussian variates, again from a population with µ ) 30 and σ ) 3, and computed jx2, s2, and se2. Then, three output ratios were computed:
R≡
Re ≡
jx2/s2 jx1/s1
x
jx2/se2 )R jx1/se2 Rr ≡
(13) ν2 + 1 ν1 + 1
s2/xj2 1 ) s1/xj1 R
(14)
(15)
Table 6. Critical Values of R for a One-Tailed Test at 90% Confidence Level ν2 ν1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
3.00 3.00 3.05 3.05 3.05 3.06 3.09 3.04 3.06 3.09 3.09 3.11 3.06 3.08 3.09 3.06 3.07 3.10
2.36 2.34 2.31 2.30 2.31 2.31 2.29 2.29 2.30 2.30 2.30 2.28 2.29 2.28 2.29 2.31 2.29 2.28
2.10 2.05 2.05 2.03 2.01 2.00 1.99 1.99 1.98 1.98 1.98 1.98 1.98 1.97 1.96 1.97 1.97 1.98
1.96 1.91 1.88 1.86 1.85 1.84 1.83 1.83 1.82 1.82 1.81 1.81 1.81 1.82 1.80 1.80 1.80 1.80
1.87 1.82 1.80 1.77 1.76 1.75 1.73 1.73 1.72 1.71 1.72 1.71 1.70 1.70 1.69 1.70 1.69 1.69
1.82 1.76 1.73 1.71 1.69 1.68 1.66 1.66 1.65 1.65 1.64 1.64 1.63 1.63 1.62 1.62 1.62 1.62
1.77 1.73 1.69 1.66 1.64 1.63 1.61 1.60 1.61 1.59 1.59 1.58 1.58 1.57 1.57 1.57 1.56 1.56
1.74 1.69 1.65 1.62 1.61 1.59 1.57 1.57 1.56 1.56 1.55 1.54 1.54 1.54 1.53 1.53 1.53 1.52
1.72 1.66 1.62 1.60 1.57 1.56 1.55 1.54 1.53 1.52 1.51 1.51 1.50 1.50 1.50 1.50 1.49 1.49
1.70 1.64 1.60 1.57 1.56 1.53 1.52 1.52 1.50 1.50 1.50 1.49 1.48 1.48 1.47 1.47 1.48 1.47
1.68 1.62 1.59 1.57 1.53 1.52 1.50 1.50 1.48 1.48 1.47 1.46 1.46 1.46 1.45 1.45 1.44 1.44
1.67 1.61 1.57 1.54 1.52 1.50 1.49 1.47 1.47 1.46 1.45 1.45 1.44 1.43 1.43 1.43 1.42 1.42
1.66 1.60 1.55 1.52 1.51 1.49 1.48 1.46 1.45 1.45 1.44 1.43 1.43 1.42 1.42 1.42 1.41 1.41
1.65 1.59 1.54 1.51 1.49 1.48 1.46 1.45 1.45 1.43 1.43 1.42 1.41 1.41 1.41 1.40 1.40 1.39
1.64 1.58 1.53 1.51 1.48 1.46 1.45 1.44 1.43 1.42 1.42 1.41 1.40 1.39 1.39 1.39 1.38 1.38
1.64 1.57 1.53 1.50 1.47 1.45 1.44 1.43 1.42 1.41 1.40 1.40 1.39 1.39 1.38 1.38 1.37 1.37
1.63 1.56 1.51 1.48 1.46 1.45 1.43 1.42 1.41 1.40 1.40 1.39 1.38 1.38 1.37 1.37 1.37 1.36
1.62 1.55 1.51 1.49 1.46 1.44 1.43 1.41 1.40 1.40 1.39 1.38 1.38 1.37 1.37 1.36 1.36 1.36
a
Numerator degrees of freedom ≡ ν2 ≡ N2 - 1.
Table 7. Critical Values of R for a Two-Tailed Test at 90% Confidence Level ν2 ν1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
4.37 4.34 4.39 4.41 4.39 4.38 4.48 4.41 4.41 4.46 4.44 4.46 4.39 4.43 4.45 4.38 4.44 4.47
3.13 3.07 3.02 3.02 3.02 3.01 2.98 2.97 2.95 2.96 2.97 2.93 2.96 2.94 2.98 2.98 2.97 2.95
2.67 2.59 2.57 2.52 2.50 2.49 2.47 2.45 2.44 2.44 2.44 2.44 2.43 2.43 2.42 2.43 2.43 2.42
2.43 2.34 2.30 2.25 2.24 2.21 2.20 2.19 2.18 2.18 2.16 2.17 2.16 2.17 2.15 2.15 2.14 2.15
2.29 2.19 2.15 2.10 2.08 2.07 2.04 2.05 2.03 2.01 2.02 2.00 1.99 2.00 1.97 2.00 1.98 1.98
2.20 2.09 2.05 2.01 1.98 1.96 1.93 1.93 1.92 1.91 1.90 1.90 1.88 1.88 1.87 1.87 1.88 1.87
2.13 2.04 1.99 1.93 1.91 1.89 1.86 1.85 1.85 1.83 1.82 1.81 1.81 1.80 1.79 1.79 1.78 1.79
2.09 1.98 1.92 1.87 1.85 1.82 1.80 1.80 1.78 1.77 1.76 1.75 1.75 1.74 1.73 1.73 1.73 1.73
2.05 1.94 1.87 1.84 1.80 1.78 1.76 1.75 1.73 1.72 1.72 1.71 1.69 1.69 1.69 1.68 1.67 1.67
2.01 1.91 1.84 1.80 1.77 1.74 1.72 1.71 1.70 1.69 1.68 1.67 1.66 1.66 1.64 1.65 1.65 1.64
1.98 1.88 1.82 1.80 1.74 1.71 1.69 1.69 1.67 1.66 1.64 1.64 1.63 1.62 1.62 1.61 1.60 1.61
1.97 1.86 1.79 1.75 1.72 1.69 1.67 1.65 1.65 1.63 1.62 1.61 1.60 1.60 1.59 1.59 1.58 1.58
1.95 1.85 1.77 1.72 1.70 1.67 1.65 1.64 1.62 1.61 1.60 1.59 1.59 1.57 1.57 1.57 1.56 1.55
1.94 1.83 1.76 1.71 1.68 1.65 1.64 1.61 1.61 1.59 1.59 1.57 1.56 1.56 1.55 1.55 1.54 1.53
1.92 1.81 1.74 1.70 1.66 1.64 1.61 1.60 1.59 1.57 1.57 1.55 1.55 1.54 1.53 1.54 1.52 1.52
1.91 1.81 1.74 1.69 1.65 1.63 1.61 1.59 1.58 1.56 1.55 1.54 1.53 1.53 1.52 1.51 1.51 1.50
1.90 1.79 1.71 1.67 1.64 1.62 1.59 1.57 1.56 1.55 1.53 1.53 1.52 1.51 1.50 1.50 1.50 1.49
1.89 1.78 1.71 1.67 1.63 1.61 1.58 1.56 1.55 1.54 1.53 1.52 1.51 1.50 1.49 1.49 1.49 1.48
a
Numerator degrees of freedom ≡ ν2 ≡ N2 - 1.
and each output was passed to an associated histogram block for binning in one of 10 000 bins between 0 and an appropriately chosen upper histogram limit. The number of bins per histogram was set rather high to minimize interpolation error in the computation of the critical values. The upper bin limits were chosen by running 10 000 step simulations for each of the 324 ν1, ν2 pairs and examining the resulting PDF and CDF histograms to ensure that at least 99.65% of the events were captured in the histogram binning limits. Then, each simulation was rerun with 100 000 simulation steps, a number chosen because the a priori preferred number of steps, 1 million, was found to be prohibitively time consuming on the Mac IIfx. Figure 2 shows PDF and CDF histograms for R that were obtained for ν1 ) ν2 ) 5, with 100 000 simulation steps and 125 histogram bins, a number chosen to avoid the “artificial noisiness” that accompanies use of 10 000 bins. At the end of each simulation, the histogram blocks displayed their PDF and CDF histograms, and the critical values for seven different confidence levels were automatically calculated, with
linear interpolation as necessary, and then rounded to three decimal places and displayed on the model worksheet, which was then printed. Thus, the output consisted of 324 printed sheets, of which a typical one is shown in Figure 1. The total simulation time for the 324 simulations of 100 000 steps each was about 63 h. The total number of Gaussian variates generated, neglecting the 10 000 step runs, was 745 200 000. Note that although critical value tables were prepared for Re and Rr, as defined in eqs 14 and 15, these are not included, per se, in the present paper. RESULTS In this section, we compare the theoretical results with the results of detailed simulations. Starting with the PDFs for the several definitions of sample standard deviation, Figure 3 shows the histogram results for s and s′, for N ≡ 3, together with corresponding theoretical PDFs computed from eq 5 and Williams’s eq A1,1 respectively. The agreement is excellent, the quality of the histograms being almost that of the numerically Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
231
Table 8. Critical Values of R for a One-Tailed Test at 95% Confidence Level ν2 ν1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
4.37 4.34 4.39 4.41 4.39 4.38 4.48 4.41 4.41 4.46 4.44 4.46 4.39 4.43 4.45 4.38 4.44 4.47
3.13 3.07 3.02 3.02 3.02 3.01 2.98 2.97 2.95 2.96 2.97 2.93 2.96 2.94 2.98 2.98 2.97 2.95
2.67 2.59 2.57 2.52 2.50 2.49 2.47 2.45 2.44 2.44 2.44 2.44 2.43 2.43 2.42 2.43 2.43 2.42
2.43 2.34 2.30 2.25 2.24 2.21 2.20 2.19 2.18 2.18 2.16 2.17 2.16 2.17 2.15 2.15 2.14 2.15
2.29 2.19 2.15 2.10 2.08 2.07 2.04 2.05 2.03 2.01 2.02 2.00 1.99 2.00 1.97 2.00 1.98 1.98
2.20 2.09 2.05 2.01 1.98 1.96 1.93 1.93 1.92 1.91 1.90 1.90 1.88 1.88 1.87 1.87 1.88 1.87
2.13 2.04 1.99 1.93 1.91 1.89 1.86 1.85 1.85 1.83 1.82 1.81 1.81 1.80 1.79 1.79 1.78 1.79
2.09 1.98 1.92 1.87 1.85 1.82 1.80 1.80 1.78 1.77 1.76 1.75 1.75 1.74 1.73 1.73 1.73 1.73
2.05 1.94 1.87 1.84 1.80 1.78 1.76 1.75 1.73 1.72 1.72 1.71 1.69 1.69 1.69 1.68 1.67 1.67
2.01 1.91 1.84 1.80 1.77 1.74 1.72 1.71 1.70 1.69 1.68 1.67 1.66 1.66 1.64 1.65 1.65 1.64
1.98 1.88 1.82 1.80 1.74 1.71 1.69 1.69 1.67 1.66 1.64 1.64 1.63 1.62 1.62 1.61 1.60 1.61
1.97 1.86 1.79 1.75 1.72 1.69 1.67 1.65 1.65 1.63 1.62 1.61 1.60 1.60 1.59 1.59 1.58 1.58
1.95 1.85 1.77 1.72 1.70 1.67 1.65 1.64 1.62 1.61 1.60 1.59 1.59 1.57 1.57 1.57 1.56 1.55
1.94 1.83 1.76 1.71 1.68 1.65 1.64 1.61 1.61 1.59 1.59 1.57 1.56 1.56 1.55 1.55 1.54 1.53
1.92 1.81 1.74 1.70 1.66 1.64 1.61 1.60 1.59 1.57 1.57 1.55 1.55 1.54 1.53 1.54 1.52 1.52
1.91 1.81 1.74 1.69 1.65 1.63 1.61 1.59 1.58 1.56 1.55 1.54 1.53 1.53 1.52 1.51 1.51 1.50
1.90 1.79 1.71 1.67 1.64 1.62 1.59 1.57 1.56 1.55 1.53 1.53 1.52 1.51 1.50 1.50 1.50 1.49
1.89 1.78 1.71 1.67 1.63 1.61 1.58 1.56 1.55 1.54 1.53 1.52 1.51 1.50 1.49 1.49 1.49 1.48
a
Numerator degrees of freedom ≡ ν2 ≡ N2 - 1.
Table 9. Critical Values of R for a Two-Tailed Test at 95% Confidence Level ν2 ν1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
6.26 6.21 6.25 6.28 6.26 6.19 6.34 6.32 6.32 6.45 6.28 6.28 6.20 6.32 6.32 6.24 6.34 6.38
4.04 3.97 3.86 3.89 3.84 3.84 3.82 3.81 3.76 3.82 3.81 3.75 3.75 3.79 3.81 3.82 3.78 3.75
3.31 3.18 3.16 3.09 3.06 3.04 3.00 2.99 2.96 2.97 2.97 2.98 2.95 2.93 2.93 2.96 2.94 2.96
2.93 2.79 2.73 2.69 2.66 2.63 2.61 2.60 2.57 2.57 1.55 2.57 2.55 2.57 2.53 2.53 2.53 2.54
2.74 2.58 2.53 2.46 2.43 2.41 2.38 2.38 2.36 2.33 2.34 2.32 2.32 2.30 2.29 2.30 2.30 2.28
2.58 2.45 2.37 2.32 2.29 2.24 2.22 2.22 2.19 2.18 2.17 2.16 2.15 2.14 2.14 2.12 2.14 2.12
2.49 2.36 2.28 2.21 2.19 2.15 2.11 2.09 2.10 2.07 2.05 2.05 2.05 2.03 2.03 2.01 2.02 2.01
2.44 2.28 2.19 2.13 2.09 2.06 2.03 2.02 2.01 1.99 1.97 1.96 1.96 1.95 1.94 1.93 1.93 1.94
2.37 2.20 2.13 2.07 2.04 2.00 1.97 1.95 1.94 1.91 1.91 1.91 1.89 1.88 1.87 1.86 1.87 1.86
2.32 2.17 2.09 2.01 1.98 1.94 1.92 1.90 1.89 1.88 1.86 1.85 1.84 1.84 1.82 1.82 1.83 1.81
2.28 2.13 2.05 2.01 1.94 1.90 1.88 1.87 1.85 1.84 1.82 1.81 1.80 1.80 1.78 1.77 1.77 1.77
2.25 2.10 2.02 1.96 1.91 1.88 1.84 1.83 1.82 1.80 1.79 1.77 1.76 1.75 1.74 1.74 1.74 1.73
2.22 2.08 1.98 1.92 1.89 1.85 1.83 1.80 1.78 1.77 1.75 1.75 1.74 1.73 1.71 1.72 1.71 1.70
2.21 2.06 1.97 1.90 1.86 1.82 1.80 1.78 1.77 1.74 1.73 1.72 1.70 1.70 1.69 1.69 1.68 1.67
2.18 2.03 1.94 1.89 1.84 1.80 1.77 1.76 1.74 1.70 1.71 1.70 1.69 1.67 1.67 1.67 1.66 1.65
2.18 2.02 1.94 1.87 1.83 1.78 1.77 1.74 1.73 1.72 1.69 1.67 1.67 1.66 1.65 1.64 1.64 1.63
2.17 2.00 1.91 1.84 1.81 1.77 1.73 1.72 1.70 1.69 1.67 1.66 1.65 1.64 1.63 1.63 1.62 1.61
2.15 2.00 1.90 1.85 1.79 1.76 1.73 1.70 1.69 1.68 1.66 1.65 1.64 1.62 1.62 1.61 1.61 1.59
a
Numerator degrees of freedom ≡ ν2 ≡ N2 - 1.
evaluated theoretical expressions. Hence, it is possible to use accurate histograms in lieu of the theoretical expressions, with very small error. Note also that, as N increases, s and s′ converge because each approaches σ as N goes to infinity. Figure 4 shows the N ≡ 3 histogram results for jx/s, together with the corresponding theoretical PDF computed by numerical integration of eq 10. Also shown are the N ≡ 3 histogram results for µ/s′, together with the corresponding theoretical PDF computed from Williams’s eq A6.1 As may be seen, the agreement between the PDFs and the associated simulation histograms is again excellent. Figure 5 is the same as Figure 4, except that all of the results pertain to N ≡ 21 and, for clarity, histogram results are omitted. It is evident that, as N increases, jx/s and µ/s′ converge because each approaches µ/σ as N goes to infinity. Of course, the same applies to ratios such as jx/s′, µ/s, etc. It is equally clear from Figures 3-5 that, for small N, s is not well-approximated by s′ and, more importantly, jx/s is not well-approximated by µ/s′. 232
Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
The PDF for jx/s is easily seen to be positively biased and asymmetric, particularly at small values of N. The expectation value of jx/s, computed as
E[xj/s] )
∫ (xj/s)p ∞
-∞
j/s)d(xj/s) jx/s(x
(16)
is greater than the true SNR given by µ/σ, confirming the positive bias. Computation of the expectation value of jx/s, as a function of ν, involves straightforward numerical integration as per eq 16. The results, computed for the arbitrary values µ ≡ 9 and σ ≡ 3, are shown in Table 2, which also shows the correction factor, defined as (µ/σ)/E[xj/s], needed to correct the bias in experimentally determined values of jx/s. The correction factor is independent of µ/σ because E[xj/s] is proportional to µ/σ for a given ν. The correction factors also apply to jx/se, because
Table 10. Critical Values of R for a One-Tailed Test at 99% Confidence Level ν2 ν1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
9.9 9.9 10.0 10.0 10.2 9.8 10.2 10.2 9.9 10.2 9.9 9.8 10.0 10.2 10.0 10.0 10.0 10.0
5.7 5.5 5.3 5.4 5.4 5.2 5.2 5.3 5.1 5.2 5.2 5.2 5.1 5.2 5.3 5.2 5.2 5.2
4.3 4.1 4.1 3.9 4.0 3.9 3.9 3.9 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.7 3.8 3.8
3.7 3.5 3.4 3.3 3.3 3.3 3.2 3.2 3.2 3.2 3.1 3.2 3.1 3.1 3.1 3.1 3.1 3.1
3.4 3.1 3.1 3.0 3.0 2.9 2.9 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.7 2.8 2.7 2.7
3.1 2.9 2.8 2.8 2.7 2.7 2.6 2.6 2.6 2.6 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
3.0 2.8 2.7 2.6 2.6 2.5 2.5 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.3 2.4 2.3
2.9 2.7 2.6 2.5 2.5 2.4 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.2 2.2 2.2 2.2
2.8 2.6 2.5 2.4 2.3 2.3 2.3 2.2 2.2 2.2 2.2 2.2 2.2 2.1 2.1 2.1 2.1 2.1
2.7 2.5 2.4 2.3 2.3 2.2 2.2 2.2 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.0
2.7 2.4 2.4 2.3 2.2 2.2 2.1 2.1 2.1 2.1 2.1 2.0 2.0 2.0 2.0 2.0 2.0 2.0
2.6 2.4 2.3 2.2 2.2 2.1 2.1 2.1 2.0 2.0 2.0 2.0 2.0 2.0 1.9 1.9 1.9 1.9
2.6 2.4 2.3 2.2 2.1 2.1 2.1 2.0 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.9 1.9 1.9
2.6 2.4 2.2 2.1 2.1 2.1 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.8
2.5 2.3 2.2 2.1 2.1 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.9 1.8 1.8 1.8 1.8 1.8
2.5 2.3 2.2 2.1 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.8 1.8 1.8 1.8 1.8 1.8 1.8
2.5 2.3 2.2 2.1 2.0 2.0 1.9 1.9 1.9 1.9 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
2.5 2.3 2.1 2.1 2.0 2.0 1.9 1.9 1.9 1.9 1.8 1.8 1.8 1.8 1.8 1.8 1.7 1.7
14 2.8 2.6 2.5 2.4 2.3 2.3 2.2 2.2 2.1 2.1 2.1 2.1 2.1 2.1 2.0 2.1 2.1 2.0
15 2.8 2.6 2.4 2.3 2.3 2.2 2.2 2.1 2.1 2.1 2.1 2.1 2.0 2.0 2.0 2.0 2.0 2.0
16 2.8 2.5 2.4 2.3 2.2 2.2 2.1 2.1 2.1 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.9
17 2.8 2.5 2.4 2.3 2.2 2.2 2.1 2.1 2.1 2.0 2.0 2.0 2.0 2.0 1.9 1.9 1.9 1.9
18 2.7 2.5 2.4 2.2 2.2 2.1 2.1 2.0 2.0 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.9 1.9
19 2.7 2.5 2.3 2.2 2.2 2.1 2.1 2.0 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.9 1.9 1.9
a
Numerator degrees of freedom ≡ ν2 ≡ N2 - 1.
Table 11. Critical Values of R for a Two-Tailed Test at 99% Confidence Level ν2 ν1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 a
2 14.0 13.7 14.3 14.4 14.3 13.7 15.2 14.5 14.3 14.5 14.5 14.4 14.2 14.3 14.0 14.3 14.0 14.2
3 7.3 7.1 6.6 6.8 6.7 6.5 6.6 6.6 6.5 6.5 6.7 6.7 6.6 6.6 6.7 6.6 6.6 6.5
4 5.2 4.9 4.9 4.7 4.7 4.8 4.7 4.7 4.5 4.5 4.5 4.5 4.5 4.4 4.5 4.5 4.5 4.5
5 4.3 4.1 4.0 3.8 3.8 3.8 3.8 3.7 3.7 3.7 3.7 3.7 3.6 3.6 3.6 3.6 3.6 3.6
6 3.9 3.6 3.5 3.4 3.3 3.3 3.3 3.2 3.2 3.2 3.2 3.2 3.1 3.1 3.2 3.2 3.1 3.1
7 3.6 3.3 3.2 3.1 3.1 3.0 3.0 3.0 2.9 2.9 2.9 2.9 2.9 2.8 2.9 2.8 2.8 2.8
8 3.3 3.1 3.0 2.9 2.9 2.8 2.7 2.7 2.7 2.7 2.7 2.6 2.7 2.6 2.6 2.6 2.6 2.6
9 3.2 2.9 2.8 2.7 2.7 2.6 2.6 2.6 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.4 2.4 2.4
10 3.1 2.8 2.8 2.6 2.6 2.5 2.5 2.5 2.4 2.4 2.4 2.4 2.4 2.3 2.3 2.3 2.3 2.3
11 3.0 2.8 2.7 2.5 2.5 2.4 2.4 2.4 2.3 2.3 2.3 2.3 2.3 2.3 2.2 2.2 2.2 2.2
12 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.3 2.3 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.1 2.2
13 2.9 2.7 2.5 2.4 2.4 2.3 2.3 2.2 2.2 2.2 2.2 2.2 2.2 2.1 2.1 2.1 2.1 2.1
Numerator degrees of freedom ≡ ν2 ≡ N2 - 1.
(µ/σe)/E[xj/se] ) (µxN/σ)/{xNE[xj/s]} ) (µ/σ)/E[xj/s] (17) If the bias is to be kept below 10%, N must be at least 9. For N ) 30, the bias is only 2.6%. For s, Kenney and Keeping state that “It is not unbiased, but it is the square root of an unbiased estimate of σ2 ” [ref 5, p 173]. The expectation value of s is given by
E[s] ≡
∫ sp (s) ds ) σx(2/ν) ∞
0
s
Γ[(ν+1)/2] Γ[ν/2]
(18)
where eq 5 was substituted into the integral in eq 18. The results in Table 3 show that the bias is negative but considerably smaller in magnitude than that of jx/s. As expected, s is less biased than s′.1 (5) Kenney, J. F.; Keeping, E. S. Mathematics of Statistics, 2nd ed.; D. Van Nostrand Co.: Princeton, NJ, 1951; Part II.
Critical Value Tables for R. Tables 4-11 are critical value tables for R, where R is defined in eq 13. [Tables 4 and 5, for 75% confidence, are present in the Supporting Information for this paper.] The entries in all of the tables have been rounded to two decimal places, except for the 99% confidence tables, where only one decimal place is given. This is thought to best reflect the accuracy of the critical values, which is believed to be about 1% for all of the tables except the 99% confidence tables, where the accuracy is believed to be about 5%. Several tests, with as many as 4 million simulation steps per ν1, ν2 pair, are consistent with this accuracy estimate. The errors can be reduced by increasing the number of simulation steps per ν1, ν2 pair, but this was infeasible on the available computer. As to usage, the tables are used in precisely the same way as standard F tables: two experimental SNRs are ratioed so that population 2 has the larger experimental SNR, i.e., the R test statistic is always greater than 1. The null hypothesis and alternate hypotheses are Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
233
H0: (µ1/σ1) ) (µ2/σ2)
(19)
Ha1: (µ2/σ2) > (µ1/σ1)
one-tailed
(20)
Ha2: (µ2/σ2) * (µ1/σ1)
two-tailed
(21)
If the R test statistic is less than or equal to the tabulated critical value for the relevant number of degrees of freedom in the numerator and denominator, the null hypothesis cannot be rejected; if it exceeds the critical value, the null hypothesis is rejected and the applicable alternate hypothesis cannot be rejected. Critical Value Tables for Re. Eight tables were computed but are not included here (see below) because they are unnecessary: the relationship between R and Re in eq 14 also applies to their critical values. This was verified by direct comparison of corresponding entries in the R and Re tables. In practice, these Re tables would be used in exactly the same way as the R tables described above and would be applicable when the experimental SNR is approximated as jx/se, as is customarily the case when the SNR “increases with N1/2 ”, and it is desired to compare two such ratios. Critical Value Tables for Rr. These eight tables were computed but are not included here (see below) because they are simply the transposes of Tables 4-11, respectively. As for the Re tables above, this was verified by direct comparison of corresponding entries in the R and Rr tables. Again, the Rr tables would be used in exactly the same way as the R tables and are applicable whenever two experimental RSDs are to be compared.
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Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
CONCLUSION All of the PDFs studied herein are biased and asymmetric, and in each case the bias and asymmetry decrease as the number of replicate measurements increases. As a rough guideline, 2030 replicate measurements suffice to give tolerably small bias in experimental SNR determinations, and it is quite easy to correct for SNR bias using the correction factors given in Table 2. Similarly, bias in s is easily correctable using the factors given in Table 3. Although Williams’s R tables are now supplanted by Tables 4-11, the conclusions he drew from his tables still appear largely applicable, mutatis mutandis. For this reason, more than any other, Tables 4-11 are designated as R tables. ACKNOWLEDGMENT This work was caused by a homework problem that was almost assigned. SUPPORTING INFORMATION AVAILABLE Tables 4 and 5, critical values of R for one- and two-tailed tests at 75% confidence level (2 pages). Ordering information is given on any current masthead page.
Received for review July 9, 1996. Accepted October 21, 1996.X AC960675D X
Abstract published in Advance ACS Abstracts, December 15, 1996.
