Validation of Analytical Methods Using aRegression Procedure

Heinz W. Zwanziger† and Costel Sarbu*,‡

Fachbereich Chemiesund Umweltingenieurwesen, Fachhochschule Merseburg, Germany, and Faculty of Chemistry andChemical Engineering, Babes-Bolyai University, RO-3400 Cluj-Napoca, Romania

The evaluation and validation of analytical methods andinstruments require comparison studies using samplematerial for testing accuracy and precision. In analyticalchemistry, the commonly accepted means of analyzingdata from method comparison studies is least-squaresregression analysis, a model which has limitations. In thispaper, the results from ordinary least-squares and manyother regression approaches recommended in the litera-ture were compared with a new regression procedure thattakes into account the errors in both variables (methods).After a discussion of the properties of the regressionprocedure, recommendations are given for carrying outa method comparison study using informational analysisof variance. The efficiency of the regression procedureproposed is demonstrated by applying it to different datasets from published literature.

Analytical measurements are of vital importance in many fieldsof activity, including diagnosis and treatment of different diseases,environmental protection, producing and marketing of some usefulmaterials, and the performance of many scientific studies. There-fore, experimental data resulting from chemical or instrumentalmeasurements should be reliable; i.e., they first should beunbiased and precise.

From a chemometric point of view, the validation of a newanalytical method or an improved method, considering, forexample, time of analysis, price, and convenience, must ensurefirst the integrity and quality of that method concerning bias,precision, limits of detection and determination, range of linearity,selectivity, and the transferability of the method.1-4 The greatvariety of techniques used in analytical chemistry emphasizes theneed for establishing an efficient and reliable comparison meth-odology. The results of a large number of samples over the wholemeasurement range can be evaluated by a paired t-test or byregression. Regression procedures are preferred in many casesbecause they not only are less subject to statistical problems butalso deliver more information, since the paired t-test is strictlyvalid only for the detection of absolute systematic errors. More-

over, the t-test assumes that the errors are independent of theconcentration and have a normal distribution.5,6

To date, different regression procedures have been used forchemometric evaluation of data from method comparison studies.Each one has specific theoretical requirements for the data. It isobvious that the reliability of a procedure depends largely on towhat extent the data can meet these requirements.7,9

Under the premise of a linear relationship between the twomethods in the form of

the estimated values for a0 and a1 are tested against the nullhypothesis a0 ) 0 and a1 ) 1. If the estimated values differ onlyby chance from 0 and 1 at a predefined significance level, thenthe methods are considered to be equal. A slope significantlydifferent from unity indicates a proportional error (i.e., a matrixeffect), and an intercept different from zero implies a constanterror (i.e., a blank problem).

The linear regression based on ordinary least-squares pre-sumes that error(Y) . error(X). For the measurements ofmethod Y, analytical errors are allowed, so that repeated measure-ments of one sample will scatter perpendicularly to the x-axisaround their expected value on the regression line. The param-eters a0 and a1 of the regression equation are determined byminimizing the sum of squared distances (residuals) betweenmeasurement points and the regression line. The method of least-squares is sensitive to extreme data points, which may result inbiased values of a0 and a1. A change in the assignment of themethods to the variables of the regression procedure results innew parameters which cannot be converted into the old ones bythe regression equation.

A more appropriate approach for method comparison seemsto be the structural relationship model which allows error termsfor both variables. The orthogonal regressionsalso known as theDeming proceduresis based on the assumption that the standarddeviation of the measurement errors is the same for both methodsand the standardized principal component procedure, considering

† Fachbereich Chemiesund Umweltingenieurwesen.‡ Babes-Bolyai University.

(1) Youden, W. J.; Steiner, E. H. Statistical Manual of the AOAC; AOAC:Washington, DC, 1975.

(2) Cardone, J. M. J. Assoc. Off. Anal. Chem. 1983, 66, 1257.(3) Miller, J. N. Analyst 1991, 116, 3.(4) Alexandrov, Yu. I. Analyst 1996, 121, 1137.

(5) Massart, D. J.; Vandeginste, B. M. G.; Deming, S. N.; Michotte, Y.; Kaufman,L. Chemometrics: A Textbook; Elsevier: Amsterdam, 1988; p 88.

(6) Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 2nd ed.; EllisHorwood: Chichester, 1988; p 101.

(7) Passing, H.; Bablok, W. J. Clin. Chem. Clin. Biochem. 1983, 21, 709.(8) Hartmann, C.; Smeyers-verbeke, J.; Massart, D. L. Analusis 1993, 21, 125.(9) Kalantar, A. H.; Gelb. B. R.; Alper, J. S. Talanta 1995, 42, 597.

y ) a0 + a1x (1)

Anal. Chem. 1998, 70, 1277-1280

S0003-2700(97)00926-8 CCC: $15.00 © 1998 American Chemical Society Analytical Chemistry, Vol. 70, No. 7, April 1, 1998 1277Published on Web 02/25/1998

that the standard deviation of the measurement errors is different,but proportional.10-12 The parameters a0 and a1 of the regressionfunction are calculated in these cases by minimizing the sum ofsquared distances to the calibration line. Extreme data pointshave also a strong influence on the values of a0 and a1 and canlead to biased estimates. A change in the assignment of themethods to the variables does not alter the results of the methodcomparison. Linear regression and the principal componentprocedures require normally distributed error terms and samplepopulations.

The procedures of from Theil17 and Passing and Bablok7,11 usedfor method comparison studies are less sensitive to the underlyingdistribution of the data. In particular, they are resistant todeviating data points, which are likely to produce biased resultsfor procedures using least-squares estimators.

A NEW REGRESSION PROCEDUREWhen two analytical methods are compared, because both,

more and less, are affected by errors, practically it is not importantwhich is X and which is Y. In other words, we may write as welly ) f(x) or x ) f(y). In this situation, a more illuminating andintuitive alternative is to consider the implicit form of the linearfunction. It is well-known from analytical geometry that thegeneral linear function

represents a straight line when the coefficients A and B aredifferent from zero. Considering B * 0, then (2) can beformulated as follows:

In this case, the distance of any point (xi,yi) to the line (2) willhave the following expression:

The resulting fitting problem will be defined by the condition

The minimum of S follows from the approach to zero of thederivative of S with respect to A, B, and C. From the resultingsystem, considering M(x) ) 0, M(y) ) 0, and C ) 0, we obtain

Now, it is easy to observe that -A/B in (6) is the slope of thelinear equation

resulting from (3). After substitution of m in (6) and taking [M(y2)- M(x2)]/M(xy) ) w, we obtain a quadratic equation,

Taking into account only the positive value of m and consideringthat the centroid of x and y must satisfy the equation of the straightline (7), we can calculate n from the following equation:

By comparing (2) and (7) and taking C ) 1, we can obtainimmediately A and B.

The ratio of A and B could indicate the (dis)similarity betweenthe analytical methods compared. Thus, it is now possible tocompare analytical methods, considering either the ratio A/B orB/A (the equations are symmetrical) or, much more simply theirvalue. Considering the last possibility, we have to stress that, inthe ideal case, when the two methods produce practically equalresults, A and B will have the same absolute value. To thecontrary, in the other case, the higher the difference between Aand B, the more different will be the two methods. However, at

(10) Feldmann, U.; Schneider, B.; Klinkers, H. J. Clin. Chem. Clin. Biochem.1981, 19, 121.

(11) Bablok, W.; Passing, H. J. Clin. Lab. Aut. 1985, 7, 74.(12) Ripley, B. D.; Thomson, M. Analyst 1987, 112, 377.(13) Sarbu, C. Anal. Chim. Acta 1993, 271, 269.(14) Sarbu, C. Anal. Lett. 1997, 30, 1051.(15) Onicescu, O. C. R. Acad. Sci. Ser. A 1966, 263, 841.(16) Onicescu, O.; Stefanescu, V. Informational Statisitics; Editura Tehnica:

Bucuresti, 1979.(17) Theil, H. Proc. K. Ned. Akad. Wet., Ser. A53 1950, 386.(18) Maw, R.; Witry, L.; Emond, T. Spectroscopy 1994, 49, 39.

Ax + By + C ) 0 (2)

y ) - AB

x - CB

) mx + n (3)

di2 )

(Axi + Byi + C)2

A2 + B2 (4)

S )1

A2 + B2∑i)1

n

(Axi + Byi + C)2 ) min (5)

Table 1. Relevant Data Sets Concerning MethodsComparison Studies Discussed in This Paper

Example 1X Y X Y X Y X Y

8.71 7.35 0.90 1.25 1.81 2.31 0.40 1.297.01 7.92 9.39 6.58 1.27 1.88 0.00 0.373.28 3.40 4.39 3.31 0.82 0.44 1.98 2.165.60 5.44 3.69 2.72 1.88 1.37 10.21 12.531.55 2.07 0.34 2.32 5.66 7.04 4.64 3.901.75 2.29 1.94 1.50 0.00 0.00 5.66 4.660.73 0.66 2.07 3.50 0.00 0.49 19.25 15.863.66 3.43 1.38 1.17

Examples 2 and 3X Y X Y X Y X Y

2.34 2.48 1.35 1.04 0.38 0.35 1.96 1.801.20 1.22 2.04 1.89 7.27 7.75 0.33 0.231.88 2.14 1.97 1.90 0.28 0.14 5.54 5.210.08 0.0026 1.02 0.75 1.55 1.61 1.85 1.760.12 0.0023 1.45 1.16 0.06 0.013 3.40 3.621.12 1.05 28.20 29.40 1.50 1.63 4.19 4.071.60 1.42 22.60 23.70 1.06 1.05 0.04 0.048

22.40 26.30 22.37 23.30 5.19 5.50 3.02 3.062.16 1.99 27.00 29.50 0.33 0.054 1.33 1.351.34 1.06

Example 4X Y X Y X Y X Y

7.32 5.48 4.60 3.29 7.16 6.00 9.90 14.3015.80 13.00 9.04 6.84 6.80 5.84 28.70 18.80

M(y2) - M(x2)M(xy)

) B2 - A2

AB) B

A- A

B(6)

y ) mx + n (7)

m2 - wm -1 ) 0 (8)

y ) mx + yj - mxj (9)

1278 Analytical Chemistry, Vol. 70, No. 7, April 1, 1998

this moment, one problem remains: how to test objectively andrationally the significance of the difference between the absolutevalues of the two coefficients A and B. This can be done using,for example, informational analysis of variance, as demonstratedin the next section.

INFORMATIONAL ANALYSIS OF VARIANCEThe informational analysis of variance (IANOVA) method,13,14

based on informational energy (E ) ∑i)1n pi

2),15,16 is a distribution-free procedure that is valid under minimal assumptions. It is notinfluenced by the range of the data and has very satisfactoryrobustness properties. The null hypothesis in this case isequivalent to the hypothesis Ho: pA ) pB. The probabilities p arecalculated using the following relations:

The Ho hypothesis is true when E ) 1/2, i.e., when the informa-tional energy is minimal and the two coefficients A and B are equalin their absolute values. As was shown above, the null hypothesisis accepted if E ) ε, where E ) 1/2 represents the theoreticalinformational energy, and ε ) (A2 + B2)/(A + B)2, representsempirical informational energy. On the other hand, if E * ε, thenull hypothesis is rejected. Hence, the difference between thetwo compared methods is taken as significant.

RESULTS AND DISCUSSIONSome relevant studies concerning the methods comparison in

well-cited papers are considered to compare the advantages ofthe new regression method described above.

Illustrative Example 1. To illustrate the characteristics ofperformance of the algorithm, taking into account the errors inboth methods in the case of small deviations from homoscedas-ticity or in the presence of outliers, we refer to data discussed byRipley and Thomson12 concerning a set of 30 pairs of determina-tions of arsenate(V) ion in natural river water (Table 1). The xvalues are determined by selective reduction and atomic absorp-tion spectrometry, whereas the y values came from cold trappingand atomic emission spectrometry. The quoted values are

concentrations varying from 0 to 20 µg L-1. Considering theresults obtained (Table 2) using four different calibration methods,namely ordinary least-squares (OLS), weighted regression (WLS),iterated weighted least-squares (IWLS), and a maximum likelihoodfunctional relationship (MLFR) algorithm, Ripley and Thomsonconcluded that the best results were produced by MLFR.

Comparing the results obtained by computation of the implicitlinear function method (ILF), it is easy to notice that ILF is closerto the results obtained by the Deming method, also included inTable 2. Passing and Bablok’s linear regression algorithm gavevery biased parameters; in fact, it is the least sensitive to outliers.The empirical informational energy associated with the prob-abilities pA and pB is given by

As E ) ε1, it is concluded that, according to this test, there is nostatistically significant difference between the results of the twomethods.

Illustrative Examples 2 and 3. The second example in thepaper of Ripley and Thomson is determination of beryllium in rockand soil reference samples. The x values were obtained by aninductively coupled plasma atomic emission spectrometric (ICP-AES) method after fusion with lithium metaborate and dissolutionin dilute nitric acid; the y values were determined by an AASmethod after acid decomposition and solvent extraction (Table2).

The first batch of specimens, with concentrations in the range0.1-2.5 µg L-1, gave the results presented in Table 2. The authorsconcluded from this sample that there is no difference betweenthe methods at these concentrations, and here the IWLS resultsare a good approximation to MLFR. In this case, we have toemphasize again a very good agreement between the Demingmethod and ILF on one hand and between these methods andMLFR on the other hand.

The techniques were then applied to the 37 specimens in Table1, with concentrations in the range 0.0-30 µg L-1. Here, all theresults are in very good agreement. Weighted regression of y

Table 2. Regression Analysis Concerning the Comparison of the Methods in Table 1

fitted lines

regression method example 1 example 2 example 3 example 4

linear regression (OLS) y ) 0.544 + 0.845x y ) -0.162 + 1.063x y ) -0.194 + 1.084x y ) 2.279 + 0.619xx ) -0.299 + 1.089y x ) 0.203 + 0.899y x ) 0.193 + 0.920y x ) 0.413 + 1.402y

weighted regression (WLS) y ) 0.005 + 0.890x y ) -0.081 + 0.839x y ) -0.066 + 0.795xx ) -0.167 + 0.631y x ) 0.171 + 0.918y x ) 0.148 + 0.937y

iterated weighted y ) 0.109 + 0.904x y ) -0.135 + 1.048x y ) -0.153 + 1.062xregression (IWLS) x ) -0.083 + 0.871y x ) 0.161 + 0.925y x ) 0.152 + 0.936y

maximum likelihood functionalrelationship (MLFR)

y ) 0.106 + 0.973x y ) -0.167 + 1.076x y ) -0.161 + 1.068x

Deming method y ) 0.412 + 0.881x y ) -0.194 + 1.087x y ) -0.202 + 1.085x y ) 1.402 + 0.698x

implicit linear function(ILF) 2.040x -2.329y + 1 ) 0 -5.545x + 5.090y + 1 ) 0 -5.365x + 4.943y + 1 ) 0 3.856x -5.77y + 1 ) 0y ) 0.429 + 0.876x y ) -0.196 + 1.089x y ) -0.202 + 1.085x y ) 1.733 + 0.668xx ) -0.490 + 1.141y x ) 0.180 + 0.918x x ) 0.186 + 0.921y x ) -2.593 + 1.496y

Passing and Bablok y ) 0.859 + 0.588x y ) -0.204 + 1.007x y ) 1.087 + 1.024x y ) -2.442 + 1.108x

pA ) AA + B

and pB ) BA + B

(10) ε1 ) ∑i)1

2

pi2 )

(A2 + B2)

(A + B)2)

9.584

19.08) 0.502 (11)

Analytical Chemistry, Vol. 70, No. 7, April 1, 1998 1279

on x is worse, but the remaining lines are practically the same.Applying the informational analysis of variance, it can be

concluded that, also in these cases, there is no significantdifference between the two analytical methods because theequality E ) ε is obtained in each case (ε2 ) 0.500 and ε3 ) 0.500).

Illustrative Example 4. The last example reported in thispaper refers to the data discussed in ref 17 concerning theeffectiveness of traditional water bath digestion used in U.S. EPAmethod 7471 and microwave digestion method 3051. The resultsobtained in the determination of mercury (in the ppm range) insolid wastes by AAS using the two preparation sample methods(see Table 1) are compared through OLS, the Deming method,the Passing and Bablok method, and ILF. By examining theresults obtained (see Table 2), we can conclude that ILF and theDeming method are in very close agreement; again the Passingand Bablok procedure provides very different results. As in thecase E * ε (E ) 0.500 and ε4 ) 0.520), the difference betweenthe two methods of sample preparation is significant, and it isconcluded that there is an important method effect, which means

that one method shows a bias. This conclusion, i.e., the presenceof proportional errors introduced by microwave digestion method,concerning the analysis of mercury in waste samples was alsorecently demonstrated.14

CONCLUSIONSA new approach for the analysis of method comparison studies

over a wide range of concentrations was discussed and comparedwith more or less common regression procedures. All the resultsobtained show that the implicit linear function model (ILF) is aneffective method for parameters estimation and methods com-parison and can replace the least-squares method and otherproposed approaches. For any given set of data, an evaluationbased on an informational analysis of variance test allows a morereliable bias detection.

Received for review August 25, 1997. Accepted January7, 1998.

AC970926Y

1280 Analytical Chemistry, Vol. 70, No. 7, April 1, 1998