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Informational Analysis ofVariance Applied to Method-Comparison. A ComparativeStudyCostel S[acaron]rbu aa Department of Analytical Chemistry , “Babeş-Bolyai” University , RO-3400, Cluj-Napoca,RoumaniaPublished online: 17 Aug 2006.

To cite this article: Costel S[acaron]rbu (1997) Informational Analysis of VarianceApplied to Method-Comparison. A Comparative Study, Analytical Letters, 30:5,1051-1063, DOI: 10.1080/00032719708002317

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ANALYTICAL LETTERS, 30(5), 1051-1063 (1997)

INFORMATIONAL ANALYSIS OF VARIANCE APPLIED TO

METHOD-COMPARISON. A COMPARATIVE STUDY

Key words: analysis of variance, informational energy, statistical tests, method-

comparison

Costel S r b u

Department of Analytical Chemistry, "Babe$-Bolyai" University,

RO-3400 Cluj-Napoca, Romania

ABSTRACT

The results obtained in the determination of mercury in solid wastes by AAS using two preparation methods' ase compared through statistical parametric and

non-paramehc tests, linear regression and informational analysis of variance. The

informational analysis of variance (LANOVA) method is a distribution-free

procedure valid under minimal assumptions.It is not duenced by the range of the

data and has very satisfactory robustness properties. Applying this algorithm to

compare the effeciiveness of the traditional water-bath digestion method with the

microwave hes t ion method discussed in ref. l, it was possible to assess the

proportional errors introduced by microwave digestion method concerning the

analysis of mercwy in waste samples.

1051

Copyright 0 1997 by Marcel Dekker. Inc.

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1052 SARBU

INTRODUCTION

The informational energy (IE) concept and its detailed theoretical study as

well as its implications in the field of mathematics, called "informational statistics",

was introduced by Onicescu' and Onicescu and $tefinescu3.

It is well known by chemometricians from information theory that the

Shannon entropy may be calculated considering the probability, pi, associated to

each of the states of a system%

Orucescu observed that H(A) is the mean value of the logarithm to base 2 of

all probabilities and he addressed the question of whether the mean value of

probabilities

could not be a function "with similar characteristics of representation like Shannon's

entropy". Onicescu named it informational energy (IE).

Accordmg to Mihoc's considerations7 concerning the estimation of IE, if the

probabilities pi (i=1,2, ..., n) of a finite set of states are estimated by the relative

ii-equencies, $ of a real experiment, then the empirical IE may be calculated with

the following expression:

Equations 2 and 3 give information concerning the degree of organization of a

system or the mode of partition of its elements. Defined in this way, IE reveals

some remarkable properties. First, it reaches its minimum value when all the

probabilities are equal (ply2= ...=pn), i.e., the case of totally unorganized systems:

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ANALYSIS OF VARIANCE 1053

'@I.,, ....,p n) = 1''

E@l.p,.....pn) = 1 ( 5 )

(4) If pk = 1 and pirk = 0, i.e., the case with well organized systems, then 1E is

Hence the possible values for IE are between I/n and 1.

IE describes with the same success as Shannon's entropy the uniformity or

diversity of a system, process or phenomenon. It should. however. be remembered

that H(A) is a logarithrmc quantity, so that IE appears to be more sensitive in a

certain way than the entropy to modifications of the system. Moreover. this

informational function permits the calculation of some parameters of interest in

analyhcal chemistry such as the mformational correlation (IC) and the informational

correlation coefficient (ICC)~-"'.

INFORMATIONAL ANALYSIS OF VARIANCE

The objective in an analysis of variance (ANOVA) is to isolate and assess

sources of variation associated with independent experimental variables and to

determine how these variables interact and affect the response.

Usually, in analyhcal chemistry, the majority of ANOVA methods are used

to investigate the significance of the difference between the overall mean of q

subpopulations and an assumed value po for the population mean. Two different

null hypothesis are tested; the first being that q subpopulations have the same mean

p, = p2 = ... = b, where >O (0 I i I q) are means of q statistical populations.

The second is that the overall mean is equal to the assumed value p=po. The data

in tlus case will be arranged so that each column represents a different level of the

factor being tested.

This homogeneity concerning the means may be tested also using the

informational energy concept"-' l .

The one-way layout

Suppose some factor A, which we consider as having some effect on a

response variable of interest y, has q levels. An experiment is set up in which n

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1054 SARBU

measurements are made of the response y at all levels. The levels q are called

treatments or controlled factors, there being q controlled factors in the experimental

design. Each yij result can be written as a sum of a constant p (the general mean),

ai, a term which measures the effect of the factor A at the ith level, and an error

term eij, called the residual error or residual. The linear (or additive) model

y.. 'J = p + ai + eij (6)

can be written for the one-way layout. It is necessary now to test the null hypothesis

H,: pL1 = p2 = ... = p 9' Let 5 be a new random variable with q levels each having an associated

probability pi:

Now, it is possible to observe that the null hypothesis H, is equivalent to the

hypothesis H*: p, = pz = ... = pq = l/q. The H* hypothesis is true when E(6) = l/q,

i.e., when the informational energy of the random variable 6 is minimal.

If we define pi as

and substitute Eqn. 5 into Eqn. 4 and after that Eqn.4 in Eqn.2, the empirical

informational energy of random variable 4 is given by the following expression:

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ANALYSIS OF VARIANCE 1055

As the numerator of the expression for .&Y([) contains a sum of squares of the random

variables, it is possible using the theorems of classical repartition to construct a

criterion for testing the hypothesis H*: E([) = l/q12

If E") = .&Y((5) the null hypothesis is accepted, and on the other hand, if I&)

z .&Y"ta the null hypothesis is rejected, hence the effect of factor A is taken as

significant.

The two-way layout

Let us cansider the case in which an experiment must be set up to study the

effects of two factors A and B on a response varable y. Factor A has q levels

whereas factor B has m levels. For each combination of levels, we measure the

response yi by carrying out n observations. In cases with no replications and if we

assume that there is no interaction between the two factors, one may adopt a linear

model:

y . U = p + ai + p. J + % (10)

The hypothesis H, (ai=O), i.e., the factor A has no significant effect, is equivalent

to the hypothesis t

9 H : p l = p z = . . .=p

his is equivalent to H ~ * : E ~ ~ ) = l/q.

The estimated informational energy, .&Y(Ey concerning the probabilities pi is

given by Eqn. 6. The null hypothesis is then accepted when E([) = iT(E)l hence all ai

values are equal to zero; the effect of factor A is not significant.

The hypothesis pj = 0 (i = 1,2,--.,m), i.e., the factor has no S i d c a n t effect,

is equivalent to the hypothesis * I 1 H : p l = p 2 = . . .=pm

where

I y., Pj = 7

c Y., j = l

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1056 sAmu

and which is equivalent to the hypothesis

H,*: E ( ~ ) = l/m.

The estimated informational energy concerning the probabilities pj' is given by

m

If E",) z ift9) the null hypothesis is rejected, hence the effect of factor B is

sigruficant .

RESULTS AND DISCUSSION

To illustrate the p o t d of the informational analysis of variance presented

above we refer to the data discussed in ref. 1 concerning the effectiveness of

traditional water-bath digestion used in U.S.EPA method 7471 and microwave

digestion method 305 1.

The results obtained in the determination of mercury(ppm) in solid wastes

by AAS using the two preparation sample methods (see Table 1) are compared

through statistical parametric and non-parametric tests, informational analysis of

variance and Linear regression.

Paired t-Test

The t-)est6.13 is particularly suitable for the statistical treatment of samples

of &gMy varying composition. The t value is evaluated through the parameter Di

calculated as the difference between the results obtained with the two methods for

the same sample with regard to the sign, and the mean D of all the individual Di

differences:

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ANALYSIS OF VARIANCE 1057

Table 1. Determination of mercury @pm) in solid wastes by A A S using two

standard methods of sample preparation discussed in'.

Sample Method 305 1 Method 747 1

a b c Mean a b c Mean Di

1 7.12 7.66 7.17 7.32 5.50 5.54 5.40 5.48 +1.84

2 16.1 15.7 15.6 15.8 13.1 12.8 13.0 13.0 +2.80

3 4.89 4.62 4.28 4.60 3.39 3.12 3.36 3.29 +1.31

4 9.64 9.03 8.44 9.04 6.59 6.52 7.43 6.84 +2.20

5 6.76 7.22 7.50 7.16 6.20 6.03 5.77 6.00 +1.16

6 6.19 6.61 7.61 6.80 6.25 5.65 5.61 5.84 +0.96

7 9.44 9.56 10.7 9.90 15.0 13.9 14.0 14.3 -4.40

8 30.8 29.0 26.2 28.7 20.4 16.1 20.0 18.8 +9.90

where

For the pairs of means in Table 1, a t value of 1.43 1 was calculated. The tabulated

value of the t-distributionat the the 95% confidence level and 7 degrees of freedom

is t = 2356. It can be concluded that at this confidence level there is no significant

difference between the two methods of sample preparation. This was the conclusion

expressed in ref. 1.

Wilcoxon Matched-pair Signed-rank Test

All the Di values calculated (with regard to the sign) as the difference

between the means obtained with the two methods for the same sample are first

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1058 SARBU

ranked without regard to the sign, starting with the smallest value. Then the sign of

D, is considered. The null hypothesis of equivalence of the methods is taken

according to which the sum (T+) of all the ranks for the positive D, is close to the

sum for the negative D,(T-). The smaller the value of

T = min (T+,T-)

the larger the significance of the diference6, The values of T calculated fi-om the

data in Table 1 are TC = 29 and T = 7 . The tabulated value for T (nl = n, = 8) at

the 95% confidence level is 4. This value is lower than Tndn = 7. and it can be

concluded that also according to this test. there is no significant difference between

the two methods.

Mann-Whitney U-Test

The Mann-Whitney U-test"13 is a non parametric test for the comparison

of methods 1 and 2, through n I and n ,measurements performed with the two

methods. All the data are ranked by assigning rank 1 to the lowest, rank 2 to the

second and so on. For the two series of data the sums of the ranks R , and R, and

the parameters U , and U, are calculated:

The smaller of the two calculated U values is compared with the value tabulated for

the U distribution. The calculated values are U I = 24 and U, = 40. As the tabulated

value for n, = n2 = 8 at the 95% confidence level. U = 13, is lower than the

calculated U value, U I = 24, it is concluded that also according to this test there is

no statistically significant difference between the results of the two methods.

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ANALYSIS OF VARIANCE 1059

Informational analysis of variance

The null hypothesis in this case is equivalent to the hypothesis

H*: p1 = pz = ... = ps.

The probabilities p are calculated with Eqn. 6. This is equivalent to the hypothsis

H*: E(g) = I/q or E(g) = 1/8 = 0.125.

The empirical informational energy associated with the probabilities pi is

given by

8

2 = 0.228 (17) 603.6849 - ~r~

As E(g) + &F([,, the difference between the two methods of sample preparation is

sigruScant and it is concluded that there is an important method effect, which means

that one method shows a bias. This contradictory result in comparison with the

other parametric and non-parametric test is confirmed by ordinary linear regression

and principal components regression, respectively (the next section).

Linear Regression Analysis

It is well known that the application of the paired t-test in comparing two

methods over a wide range of concentration is inappropriate because the validity of

this parametric test rests on the assumption that any errors, either random or

systematic, are independent of the concentration. The preferred methad in such

cases is linear regression (y = a + bx). Applying the ordinary linear regression

method (LS) to the data in Table 1 we obtained the statistical results presented in

Table 2.

Hypothesis tests for the regression parameters were carried out using the

familiar T-tests pr~cedure~~ '~ . For example, if we are intersted in testing H,: a = 0,

so that this could be regarded as testing the significance of the intercept, then a T-

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1060 sARsu

Table 2. Regression analysis results concerning the comparison of the two

methods discussed in ref. 1.

Parameter Estimate Standard T Probability

Error Value Level

Intercept (a) 2.279 1.7529 1.300 0.24 126

Slope (b) 0.619 0.13 13 4.716 0.00327 ................................................................................ 95% Confidence limits for a:

95% Confidence limits for b:

Standard error of estimate :

Correlation Coefficient : r = 0.8874

R- squared :

a + ts, = 2.279 + 4.282

b + tsb = 0.619 * 0.3 18

s.,~ = 2.71715

R2 = 78.75%

statistic can be constructed using a, the sample estimate in the usual way, i.e. by T

= (a-O)/std.error. The significance levels (P-values) given in the last column of

Table 2, pertain to T-tests that the corresponding coefficients are zero. For a the P-

value is 0.24126 indicating that the intercept does not differ significantly from the

"ideal" value of 0. On the contrary the P-value for b of 0.00327 suggests a high significant value for slope. It is also possible to calculate the confidence intervals

for the parameters of regression which confirm that the intercept does not differ

significantly from 0 (the confidence interval of intercept includes 0) whereas the

value of slope (0.619) is significant different from the "ideal" value of 1 (the

confidence inerval of slope does not include 1).

An overall assessment of the quality of the linear regression is provided by

the coefficient of determination, R-squared. When multiplied by 100, it gives the

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ANALYSIS OF VARIANCE 1 0 6 1

percentage of the variability observed in the results obtained with the tested method

which is explained by the linear regression on the more precise method.

All the statistical results presented in Table 2 (including the value of R-

squared namely 78.75) dustrate the presence of the proportional errors introduced

by the microwave digestion method. Much more, the correlation coefficient

between the mean values obtained using the traditional water-bath digestion method

and the differences , Di, (r = 0.9323) illustrates once more the presence of the

proportional errors.

Owing to the lack of reliability of the standard regression model's estimates

of the constant and proportional errors (bias) in some analytxal situations

alternative models have been proposed in recent years'"''.

To take into account the variance on both methods we have computed the

principal component regression (PCR) because the standard deviation of

measurements was shown to be the same'. The results obtained concerning the

confidence intervals of intercept (a = 1.836 * 4.293) and slope (b = 0.668 * 0.322),

respectively, confirm also the conclusion regardmg the presence of a proportional

bias. The same result was obtained using the Abbe statistical text?" and a new

approach discussed recently by Nilsson2' (the next sections).

Abbe Text

In the case of this text, used also when the standard deviation of the methods

compared is the same, one calculates

where Di and D are the differences between the means of the two methods and the

mean of them, respectively. By applying tius text at the 5% confidence level and 8

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1062 sAmu

degrees of freedom one obtained the same results, i.e. the presence of a proportional

bias, because the calculated A value (0.2714) is lower than the tabulated value

(0.4912).

Mean square succesive difference approach

This algorithm discussed recently by Nilsson2' considers the differences zj

= lnyi - lnx,, i = 1,2, ..., n ordered a c c o r h g to the obtained mean concentrations of

the two methods, (9 + yJ2, or the concentration of the reference method, 3, if it has a better precision or is supposed to give conventionally true values. Pooling the

estimate from all the consecutive differences gives the variance estimate based on

the mean square succesive difference

which should be an estimate of the same variance as

l n s = -E (2 , - ; )2 (20)

n - 1 , = I

when the bias function is constant. If there is a gradual change in the bias function,

s2 will tend to be larger than sMSSD2 and we can use this method to test whether the

bias function is constant or not. One of the main conclusions of this approach is that

if sMSSD < 0.8s, the bias function is not constant. Applying this procedure in our

case we have obtained sMSSD < 0.8s because kssD2 = 0.0118 and s2 = 0.0216.

Even if we eliminate the highest difference the result obtained is the same, i.e. the

presence of the proportional bias introduced by the microwave digestion method.

CONCLUSIONS

A new approach for the analysis of method comparisons over a wide range

of concentrations was discussed and compared with parametric and non-parametric

tests and also with two linear regression methods namely LS and PCR.

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ANALYSIS OF VARIANCE 1063

Computing parametric and non-parametric tests. no differences were

observed between the traditional water-bath digestion and the alternative microwave

technique. On the contrary the regression methods LS, PCR and also informational

analysis of variance proved a significant difference. The same result was obtained

applying a new approach discussed recently in the analytical literature which

appears to be very similar with the Abbe test. Hence, it can be concluded that the

microwave technique introduces a proportional bias.

REFERENCES

1. 2. 3.

4. 5.

6.

7.

8. 9. 10. 1 1 . 12. 13.

14. 15. 16. 17. 18.

19. 20. 21.

R. Maw, L. Witry and T. Emond, Spectroscopy, 9,39( 1994). 0. Onicescu, ('. R.Acad.Sci.,Ser. A , 263, 84 1( 1966). 0. Onicescu and V. Stefanescu, Infbrmational Stalisircs, Editura Tehnica. Bucharest 1979. 0. Onicescu, Rev.Statist., 11, 4( 1966). V. $te finescu, Applications of Informational Energv and ('orrelalion. Editura Academiei, Bucharest 1979. D.L. Massart, B.G.M. Vandeginste, S.N. Deming, Y. Michotte and L. Kaufman, Chemomezrics: a Textbook, Elsevier, Amsterdam( 1988). V. Ste fanescu, Applications of Informational Enera and ( 'orrelairon, Editura Academiei, Bucharest, 1979, Anexa 2. C. Stirbu and H. Nqcu, Rev.Chim. (Bi~harest). 41, 276( 1990). C. Stirbu and H . Nqcu, Rev. Roum. Chim., 37,945( 1992). D. Dumitrescy C. Skbu and H.Pop, Anal. I x t t . . 26. 123( 1994). C. Stirbu, Anal.Chim.Acta, 271, 269( 1993). I. VBduva, The Analysis of Variance, Editura TehnicS,Bucharest( 1970). J.C. Miller and J.N. Miller, Statislics,fiw Analyiical ('hemisty, Horwood. New York (1988). M. Thomson, Analyst, 107, 1 l69( 1982). H. Passing and W. Bablok, J. Clin. Chem. Clin. Biochem., 21, 709( 1983). H. Passing and W. Bablok, J. Clin. ('hem. Clin. Biochem.. 22,43 I ( 1984). H. Passing and W. Bablok, J. Clin. Chem. C'lin. Biochem., 26, 783( 1988). C. Hartmann, J. Smeyers-Verbeke and D.L. Massart, n Analusis, 21. 125 ( 1 993). A.H. Kalantar, B.R. Gelb and J.S. Alper, Yalanta, 42, 597( 1995). C. Stirbu, V. Liteanu and D. Pop, STUDIA(CHEMIA), XXXVII, 13( 1992) G. Nilsson, J. Chemom., 5 , 523(1991).

Received: January 17, 1996 Accepted: December 15, 1996

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