This article was downloaded by: [Temple University Libraries]On: 20 November 2014, At: 20:16Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

Analytical LettersPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/lanl20

A Comparative Study ofRegression ConcerningWeighted Least SquaresMethodsCostel S[acaron]rbu aa Department of Analytical Chemistry , Babes-BolyaiUniversity , RO-3400, Cluj-Napoca, RomaniaPublished online: 16 Aug 2006.

To cite this article: Costel S[acaron]rbu (1995) A Comparative Study of RegressionConcerning Weighted Least Squares Methods, Analytical Letters, 28:11, 2077-2094,DOI: 10.1080/00032719508000026

To link to this article: http://dx.doi.org/10.1080/00032719508000026

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

ANALYTICAL LETTERS, 28( 1 I ) , 2077-2094 (1995)

A COMPARATIVE STUDY OF REGRESSION CONCERNING

WEIGHTED LEAST SQUARES METHODS

Key words: Calibration, robust and weighted regression

Costel Sdrbu

Department of Analytical Chemistry, Babe$-Bolyai

University, RO-3400 Cluj-Napoca, Romania

ABSTRACT

The weighted least squares method using 1/x2 as a weighting

factor is described and compared with conventional ordinary and

weighted least squares and robust regression. Applications of

these different methods to the relevant data sets demonstrates

that the performance of the procedure discussed in this paper

exceeds that of ordinary least

often exceeds, that of weighted

INTRODUCTION

The quantitative analyt

experimental data obtained with

squares method and equals, and

or robust methods.

cal chemistry is based on

accurate measurements of various

physical measuring quantities. The treatment of data is mainly

done on the basis of simple stoichiometric relations and chemical

s q u i librium constants. In instrumental methods of analysis the

quantity of the component is calculated from measurement of a

2011

Copyright 0 1995 by Marcel Drkker, Inc .

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

measuring physical property which is related to the mass or the

concentration of the component.

Relating, correlating, or modeling a measured response based

on the concentration of the analyte is known as the field of

calibration. Calibration of the instrumental response is a

fundamental requirement for all i n s t r u m e n t a l a n a l y s i s t e c h n i q u e s .

In a statistical terms, a calibration refers to the establishment

of a predictive relation between the controlled or independent

variable (e.g. the concentration of a standard) and the

instrumental response. The common approach to this problem is to

use the unweighted linear least squares methods. The

conventional ordinary least squares analysis ( O L S ) is based on

the assumption of an independent and normal errors distribution

with uniform variance (homoscedastic) . Much more common in

practice, however, are heteroscedastic results, where the y-

direction error is concentration dependent.

In practice the actual shape of the error distribution

function and its variance are usually unknown, so we must

investigate the consequences if the conditions stated above are

not met. In general, the method of least squares does not lead

to the maximum likelihood estimate. In the spite of the fact that

least squares is not optimal, there is justification for using

it in the cases where the conditions are only approximatively

met. In particular, the Gauss-Markov theorem states that, if the

errors are random and uncorrelated, the method of least squares

gives the best linear unbiased estimate of the parameters,

meaning that of all functions for each parameter is a linear

function of the data points, lest squares is one for which the

variances of the parameters are smallest.

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

COMPARATIVE STUDY OF REGRESSION 2079

Nevertheless, if the tails of experimental error

distribution contain a substantially larger proportion of the

total area than the tails of a Gaussian distribution, the "best

lineartt estimate may not be very good, and there will usually be

a procedure in which the parameters are non linear functions of

the data that gives lower variances for the parameters estimates

than does least squares, that is the robust and resistant method.

A procedure is said to be robust if it gives parameter

estimates with variances close to the minimum variance for a wide

range of error distribution. Least squares is very sensitive to

the effects of large residuals, so the results are distorted if

large diferences between the observed data and the model

predictions are present with frequencies substantially greater

than those in a Gaussian distribution. Least squares is therefore

not robust. A procedure is resistant if it is insensitive to the

presence or absence of any small subset of the data, in practice

it applies particularly to small number of data points that are

wildly discrepant relative to the body of the data - so called

outliers. There are several reasons why data may be discrepant,

a gross error of measurement being only the most obvious. Another

is fact that certain data points may be particularly sensitive

to some unmodeled (or unadequately modeled) parameter, or from

another point of view, particularly sensitive to some systematic

error that has not been accounted for in the experiment.'

While suitable statistical tests concerning the nature of

errors and the goodness of fit are available in the analytical

literature they are often ignored and many data sets have

appeared which violate the assumptions requested for applying the

clasical least squares method.8i9

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

2080 sAmu

The purpose of the present study was to investigate the

performance of the weighted least squares method using 1/x2 as

a weighting factor which appears to be more efficient to overcome

the difficulties underlying above. The results were applied to

relevant calibration data discussed in analytical literature.

TEHORETICAL CONBIDERATIONB

Let us consider a set of N observations, yi, that have been

measured experimentally, each subject to some random error due

to the finite precision of the measurement process. We consider

that each observation is randomly selected from some population

that can be described by a statistical function with a mean and

variance. We may assume that the values of model parameters

obtained by chance that maximize the likelihood, will be a good

estimate of true values of these parameters if the model

corresponds to a good description of physical reality.

In explicit terms, we assume that yi = Mi(x) + ei, where

M(x) represents a model function and the ei are random errors

distributed according to some density function, fi(x). In the

case of most analytical measurement sthe value of the observation

is not influenced by other observations of the some quantity, or

of different quantities, so that the row data may be assumed to

be uncorrelated, and their joint distribution is therefore the

product of their individual marginal distributions.

The likelihood function, then is given by

N L = n fi [Yi - w, ( x ) 1

i=l

Because fi is a probability density

everywhere greater than or equal to

function, it must be

zero, and thus have a real

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

COMPARATIVE STUDY OF REGRESSION 208 1

algorithm. The logarithm is a monotonically incresing function

of its argument, so the maximum value of L corresponds also to

the maximum values of ln(L). Therefore we have

Gauss co1 sidered the case where the error distribution is

Gaussian, that is

fi(Ri) = ( 2 n ) - ’ I 2 0i-l exp[-(l/2) (Ri/ai)2], (3)

where Ri = [yi - Mi(x)], and uiz is the variance of the ith

observation. In this case

The second and third terms

its maximum value when

are independent of x, so ln(L) have

N

(5) S = (R , /a , ) ’ i=l

is a minimum.

Therefore, if the error distributions are Gaussian, and

observations were weighted by the reciprocals of their variances,

the method of least squares gives the maximum likelihood estimate

of the parmeters. We have to observe that the weighted least

squares method (WLS) is more general than the conventional

ordinary least squares method ( O L S ) . The OLS is a particular

result in the case of homoscedasticity when ui = u .

In recent years a great deal of work has been done on

determining what properties a robust and resistant procedure

should Obviously, if the the error distribution is

Gaussian, or very close to Gaussian, the procedure should give

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

2082 sAmu

results very close to those given by least squares. This suggests

that the procedure should be much like least squares for small

values of the residuals. Because the weakness of least squares

lies in its overemphasis of large values of the residuals, these

should be deemphasized (downweighted), or perhaps even ignored.

For intermediate values of the residuals the procedure should

connect the treatments of the small residuals and the extremely

residuals in a smooth or fuzzy fashion.

Concerning the weighted least squares method there are some

practical problems.

First, the variance in the data is not generally known and,

even when as many as five or six replicates are made, the

estimate of the variance is poor. Second, the variance, if

determined, is only known for the standards used in the

calibration; consequently, variances at intermediate values have

to be interpolated using a weighted function of the

concentration. Lastly, we have to remark that if the variance

estimate at any level of concentration is inaccurate, then

weighted least squares regression may produce regression

estimates that are more inaccurate than those produced by OLS.

Considering the pertinent observation about the pattern of

a possible variance function, that is the variance of y is

proportional to x2, some authors16-18 have addressed the question

of whether the inverse of the xi2 could not be an weighting

factor "with similar characteristics of weighting like the

inverse of the variance". Taking into account this statement we

have to replace wi with 1/xi2 in the expressions of slope (6) and

intercept ( 7 )

corresponding to the ordinary weighted least squares method:

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

COMPARATIVE STUDY OF REGRESSION 2083

N N N N

.. _. C W i C W i xf - c c w i Xi)Z 1.1 1-1 i-1

where

N N

are the coordinates of the weighted centroid. After elaboration,

we obtain the following expressions for slope (8) and intercept

(9) corresponding to this weighted least squares method,

namely X weighted least squares method (XWLS).

It is interesting to observe that the coordinates of centroid in

this case have the following expressions

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

2084

and will be much more close to the origin comparative with

ordinary least squares method. Moreover, the standard deviations

of slope and intercept calculated with equation (10) and (11)

respectively, will be also much smaller applying this more

general approach.

I Y

In the last two equations s represents the standard deviation of

residuals and it was calculated as usual using weighted

regression.

RESULTS AND DISCUSSION

Computing the relevant examples of heteroscedasticity

discussed by Garden and all9, using the inverse of variance as a

weighting factor (wi = l /s i2) (case l), and Miller and Miller’

(case 2) which used wi = si-’(Zsi-’/n) we obtained the results

presented in Table 1. The case 3 and 4 shown also in Table 1

refer to the data computing by Phillips and Eyringg to illustrate

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

COMPARATIVE STUDY OF REGRESSION 2085

the insensitivity of the technique of iteratively reweighted

least squares (IRLS) to erroneous observations (case 3 ) and the

case 4 , respectively compare the results given by XWLS with the

results reported by Aarons” using OLS, WLS and an extended least

squares method (ELS) to study the reproducibility for ibuprofen

assay. Considering the results (see Table 1) and taking into

account the major disavantage of the WLS and ELS method, namely

difficulty of obtaining a good estimate of variance and the

computation for ELS and IRLS the XWLS method appears to be the

most suitable. The performance of XWLS exceeds that of

coventional ordinary least squares method and equals or often

exceeds that of weighted and robust regression.

To illustrate the characteristics of performance ofthe XWLS

method in the situations of small deviations from

homoscedasticity or in presence of outliers, we refer to the data

discussed by Rajk6’ concerning the determination of Mo, Cr, Co,

Pb and Ni in sub-surface and drinking water by ICP-AES (Table 2).

Considering the results obtained (Table 3 ) using eight different

calibration methods namely, ordinary least squares (LS), least

sum of absolute residuals (LSA), least maximum absolute residuals

(LMA), iteratively reweighted least squares with tuning constants

6 and 9 (IRLS6 and IRLS9), most frequent values (MFV), single

median (SM) , repeated median (RM) , least median of squares (LMS) , Rajkb concluded that the best results were given by LMS. MFV

yielded appropriate results for measurements of Mo, Cr, Pb, Co and

Ni(221.6 nm). The calibration line calculated by RM was

acceptable for Cr, Pb and Ni measured at both wavelengths, and

by IRLS6 for Mo and Co. SM was good for Pb and Ni(231.6 nm) and

LSA for only Mo. IRLS9 and LS gave nearly the same results. LMA

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

2086 sAmu

Table 1. Comparison of conventional ordinary least squares (OLS),

weighted least squares method (WLS), iteratively reweighted least

squares (IRLS) and an extended weighted least squares (ELS) with

X weighted least squares method (XWLS) computing data sets

from2,9,19.20.

................................................................. Method Case 1 Case 2 Case 3 Case 4

OLS

WLS

IRLS

ELS

XWLS

a0 0.15 0.0091

a0 a1

a0 a1

a1 0.95 0.0738

0.19 0.92

a0 0.01 0.0090 -0.04 0.98 0.0737 1.09

0.0057 0.0199

0.0072 0.0197

0.0073 0.0196

Table 2. Calibration data measured by ICP-AES

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

COMPARATIVE STUDY OF REGRESSION 2087

Table 3. Estimated parameters obtained by the methods mentioned

in the text for the data in Table 2

................................................................. Method Mo Cr co Pb Ni* Ni**

LS a, -3.1 -11.9 17.2 9.8 0.5 21.9 al 814.5 858.5 846.1 413.0 886.3 798.7

al 802.5 845.6 859.0 407.1 884.1 807.1 LMA a, -6.3 -12.8 21.7 8.7 -3.4 19.8

.................................................................

LSA a, 8.2 -2.4 1.3 13.5 2.7 19.0

al 802.1 864.0 862.1 412.7 889.8 797.7 IRLS6 a, 11.1 -11.7 -1.5 10.0 0.9 22.2

IRLS9 a, -1.9 -11.8 15.7 9.9 0.6 22.0 al 802.3 858.0 862.1 412.9 885.8 799.1

al 813.6 858.3 847.2 412.9 886.1 798.8 MFV a, 9.4 1.3 -2.0 10.3 6.4 22.9

al 803.9 840.8 862.1 412.1 880.4 801.3 SM a, 5.3 -9.2 0.4 9.7 5.1 26.7

al 808.4 854.6 860.2 412.9 880.0 782.7 RM a, 2.1 -1.6 2.3 10.4 6.2 28.9

al 813.5 844.5 857.7 410.8 870.9 765.2 LMS a, 12.1 2.4 -1.3 11.8 9.0 29.4

al 802.5 839.6 862.0 407.1 848.6 761.1 XWLS a, 12.1 -21.4 -2.1 9.9 6.9 29.4

al 751.7 894.8 926.6 413.9 863.6 772.7

al 836.2 852.9 857.0 412.7 872.1 799.0 WLS a, -19.6 -7.8 5.0 10.3 6.7 22.0

____________________-------------------------------------------- *measured at wavelength 221.6 nm measured at wavelength 231.6 nm * *

gave very biased parameters, in fact it was the most sensitive

to the outliers.

Comparing the results obtained by computation of the XWLS

method, presented also in Table 3, it is easy to observe that the

XWLS is more closer to the LMS. Much more in the same table were

enclosed the results obtained by ordinary weighted least squares

method (WLS) using the reciprocal of variances as weighting

factors. The variances were calculated only for two replicates

available in the data of Rajk6 and are shown in Table 4.

Concerning the WLS we remark good results for Co, Pb,

Ni(221.6mn), especially in the cases of heteroscedasticity (see

si2 in Table 4).

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

2088 sARBu

In addition, for a more realistic comparison of the ten

methods of regression presented in Table 3 , we calculated the

estimates of sample concentration considering two values of the

signal y, 100 and 600, respectively. A careful examination of

results presented in Table 5 illustrates that the performance of

the XWLS equals that of LMS in some cases, and exceeds some of

the other methods.

A high discrepancy appears, however, for Mo and Co at 600

arbitrary units and also for Cr at 100 units. These results

confirm one more time that weighted methods provide, generally,

more accurate estimates of unknown samples at lower

concentration, and introduce the question if the LMS method is

realy the best in all these cases (see also Table 7 and 8).

As we have emphasized above the weighted centroid (%,:J is

much closer to the origin of the graph than the unweighted

centroid (x,y), and the weighting given the points nearer the origin - and particularly to the first point, which has the smallest error - ensures that the weighted regression line has an intercept very close to the first point (see Table 6).

Moreover, the weighted regression gives a smaller standard error,

which is more appropriate and allows us to detect a smaller bias

in the intercept that isreally present. These conclusions are

well illustrated in Table 6 .

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

COMPARATIVE STUDY OF REGRESSION 2089

Table 5. The estimated concentrations corresponding to 100 and

600 signal value, respectively obtained by each method

in Table 3.

................................................................ Method Mo Cr co Pb Ni* Ni,,

100 100 100 100 100 100 ................................................................ LS 0.123 0.130 0.098 0.218 0.112 0.098 LSA 0.114 0.121 0.115 0.213 0.110 0.100 LMA 0.133 0.131 0.091 0.221 0.116 0.101 IRLS6 0.111 0.130 0.118 0.218 0.112 0.097 IRLS9 0.125 0.130 0.100 0.218 0.112 0.098 MFV 0.113 0.117 0.118 0.218 0.106 0.096 SM 0.117 0.128 0.117 0.219 0.108 0.094 RM 0.121 0.120 0.114 0.218 0.108 0.093 LMS 0.110 0.116 0.118 0.217 0.107 0.093 XWLS 0.117 0.136 0.110 0.218 0.108 0.091 WLS 0.143 0.138 0.111 0.217 0.107 0.098 ................................................................

600 600 600 600 600 600 ........................................................ LS 0.740 0.713 0.689 1.429 0.676 0.724 LSA 0.737 0.712 0.697 1.441 0.676 0.719 LMA 0.756 0.709 0.671 1.433 0.678 0.727 IRLS6 0.734 0.713 0.698 1.429 0.676 0.723 IRLS9 0.740 0.713 0.690 1.429 0.676 0.724 MFV 0.735 0.712 0.698 1.430 0.674 0.720 SM 0.736 0.713 0.697 1.430 0.676 0.733 RM 0.735 0.712 0.697 1.435 0.682 0.746 LMS 0.733 0.712 0.698 1.445 0.697 0.750 XWLS 0.782 0.700 0.690 1.436 0.687 0.739 WLS 0.741 0.724 0.694 1.429 0.680 0.723 ___-____________________________________------------------------- *measured at wavelength 221.6 nm measured at wavelength 231.6 nm * *

Table 6. Estimated standard deviations of intercept, s,,, and

slope, sal, obtained by OLS, XWLS and WLS and the

coordinates of centroid (k,Fw) in the case of XWLS.

................................................................. Method Mo Cr co Pb Ni* Ni**

OLS S,, 9.66 4.83 12.22 3.03 4.68 5.08 *sal 16.87 8.43 21.32 5.28 8.18 8.86

sal 7.20 3.60 9.10 2.26 3.49 3.78 WLS Sao 7.62 3.31 8.62 1.96 3.71 3.28

sal 22.84 9.37 22.62 5.50 29.19 8.79 1.00 1.00 1.00 1.00 1.00 1.00

-2.14 9.93 6.93 29.45 yw 12.12 -21.40

-_________-_____________________________------------------------

XWLS s,, 14.41 7.20 18.21 4.51 6.98 7.57

* x w __---___________________________________------------------------- *x 10-6

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

2090 sARBu

Referring to the results shown in Table 6 it is interesting

to observe that the coordinates of the centroid are the

coordinates of the first point in the graph. The value 1 ~ 1 0 - ~ of

is due to the replacing of 0 with 1x10 -5 for x to working the -

program.

To evaluate the linearity of the methods studied, it is a

good opportunity to compare the different quality coefficients

(QC) used in the analytical literature to judge the goodness of

fit of a regression line. In this order we present in Table 7 the

values obtained for QC, (12), defined as’,

where yi and pi are the responses measured at each datum and

those predicted by the model in Table 3 , respectively and N is

the number of all data points, QC, used by when measured, yi,

replace estimates pi at the denominator”, and also QC, and QC,,

respectively referring to the mean signalz3 instead of the

signal itself and the mean of estimated signal 9 , respectively. The smaller the QC, the better the fit of the model.

The main conclusion drawn from the results in Table 7 by

comparing with the statements of Rajk6’ is that QC, and QC,,

respectively are most suitable to evaluate the goodness of fit

at least for the data computed in this paper. The quality

coefficient QC, and QC,, respectively appear to be a better

solution when comparing methods based on the same algorithm, i.e.

least squares. Rajk6 appreciated the QC, criterion as a pleasant

solution but he has not used it in his paper.

Taking into account the contradictory values of QC and the

diversity of the methods concerning their algorithm we have

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

COMPARATIVE STUDY OF REGRESSION 209 1

LS LSA LMA IRLS6 IRLS9 MFV SM RM LMS XWLS LS LSA LMA IRLS6 IRLS9 MFV SM RM LMS XWLS LS LSA LMA IRLS6 IRLS9 MFW SM RM LMS XWLS LS LSA LMA IRLS6 IRLS9 MFV SM RM LMS XWLS

QCl

QC2

QC3

QC4

273.4 37.2

160.1 21.5

413.5 28.8 81.0

288.5 19.7 19.2 69.2 21.1 85.3 20.9 63.3 19.7 30.4 44.6 23.2 22.1 4.6 5.1 5.6 5.4 4 . 6 5.2 4.9 4.8 5.5 9.2 4.6 5.1 5.5 5.5 4.6 5.3 5.0 4.8 5.7 9.5

43.4 425.5 36.9 45.3 44.5

930.2 72.5 688.0 531.9

6.0 23.7 47.4 21.5 24.3 24.0 56.8 30.5 49.6 59.5 6.0 2.3 2.9 2.4 2.2 2.2 3.3 2.3 3.0 3.5 4.9 2.3 2.9 2.4 2.2 2.2 3.4 2.3 3.0 3.5 4.8

60.6 142.4 59.0 30.2 61.9 15.3

349.5 105.1 36.1 11.8

497.1 87.2

612.4 19.8

457.7 12.6 64.1

115.3 22.1 11.6 5.3 6.3 6.2 6.6 5.4 6.7 6.4 6.3 6.4

10.7 5.3 6.2 6.4 6.4 6.4 6.5 6.3 6.1 6.4

10.3

19.4 20.0 23.1 19.1 19.2 18.6 19.8 18.5

19.2 22.9 42.0 19.3 23.5 23.1 24.8 22.1 25.3 32.3 23.2 2.7 3.0 2.8 2.7 2.7 2.7 2.7 2.7 3.0 2.7 2.7 3.0 2.8 2.7 2.7 2.7 2.7 2.7 3.0 2.7

18.3

735.5 86.1

163.7 384.1 536.9

6.8 20.2 8.1

4.6 49.8 32.9 79.6 46.9 48.6 6.0

14.4 6.8

17.2 4.6 2.0 2.1 2.2 2.0 2.0 2.3 2.2 2.6 4.9 3.2 2.0 2.1 2.2 2.0 2.0 2.3 2.2 2.6 4.9 3.3

12.8

18.6 29.6 26.2 17.7 18.4 15.6 6.2 3.2

3.2 2.9

13.8 19.0 17.6 13.3 13.7 12.1 5.5 3.1 3.1 2.6 2.3 2.5 2.4 2.3 2.3 2.4 3.0 4.9 5.3 3.8 2.3 2.9 2.4 2.3 2.3 2.4 2.9 4.8 5.3 3.8 ________________________________________-------------------------

*measured at wavelength 221.6 nm measured at wavelength 231.6 nm **

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

2092 sAmu

introduced two new quality coefficients namely QC, and QC,

respectively referring to the maximum of absolute residuals

(mad rd ) and the mean of the absolute residuals, ri,

respectively at the denominator in Equation 12. The results

obtained by computing QC, and QC,, respectively are shown in

Table 8. It is easy to observe a very good agreement of the

values in Table 8 with the Rajk6 statements especially for QC,.

Overall, it may be stated that these new quality coefficients

concerning the goodness of fit proposed in this paper confirm our

main conclusions and are in a good agreement with the statements

in the analytical literature.

We have to remark that in the case of QC, the higher the QC

values the better the fit of the model.

Moreover, the QC, criterion appears to be really a pleasant

solution because it is more sensitive and reliable and takes

values approximately within 1 and 2 (without multiplication by

100).

CONCLUSIONS

The weighted least squares method ( X W L S ) compared in this

paper uses the inverse of square of X i , namely 1 / X i 2 , as a

weighting factor in linear regression analysis. Given the present

results and many others computed by the author and unpublished,

it seems that the XWLS method is the best one for dealing

with problems of heteroscedasticity. The XWLS method produces

accurate and precise estimates of the parameters of calibration

line and is efficient over a broad range of error distributions.

The performance of XWLS has been shown to exceed that of

conventional ordinary least squares method and equals or often

exceeds that of weighted and robust regression. The method is

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

COMPARATIVE STUDY OF REGRESSION 2093

Table 8. The goodness of fit for the studied methods appreciated

by the quality coefficient QC, and QC,, respectively. ................................................................. Criterion Method Mo Cr co Pb Ni Ni** .................................................................

LS 56.85 71.01 63.38 59.23 61.29 64.29 LSA 49.20 49.95 52.91 61.38 57.59 63.05 LMA 86.29 81.25 102.59 71.98 77.37 83.47 IRLS6 48.94 69.81 52.82 58.51 60.71 62.08 IRLS9 55.09 70.40 61.18 59.10 61.01 63.58 MFV 48.91 49.61 58.37 59.31 56.58 57.10 SM 49.54 58.88 52.85 60.99 59.30 57.78 RM 50.93 49.73 53.00 64.25 49.77 52.00 LMS 49.06 49.73 52.81 57.69 47.63 52.00 XWLS 59.61 54.52 64.02 56.35 46.81 53.51 LS 1.33 1.16 1.29 1.25 1.26 1.22 LSA 1.64 1.68 1.96 1.40 1.34 1.41 LMA 1.10 1.16 1.07 1.29 1.24 1.20 IRLS6 1.65 1.16 2.02 1.25 1.27 1.22 IRLS9 1.36 1.16 1.33 1.25 1.26 1.22 MFV 1.64 1.83 2.03 1.25 1.44 1.30 SM 1.54 1.23 1.98 1.25 1.36 1.36 RM 1.45 1.72 1.93 1.27 1.47 1.88 LMS 1.66 1.84 2.01 1.32 1.71 1.96 XWLS 1.29 1.49 1.32 1.26 1.64 1.56

QC5

Qc6

................................................................. *measured at wavelength 221.6nm and 231.6 nm**, respectively

simple and easy to apply. It would seem, therefore, that their

application in routine analysis may be worthwhile.

It is also important to observe that the new quality

coefficients proposed in this paper allow us to do a more

realistic analysis of the linearity of calibration lines.

1.

2.

3 .

REFERENCES

D.L. Massart, B.M.G. Vandeginste, S.N. Deming, Y. Michotte

and L. Kaufman, Chemometrics: A Textbook, Elsevier,

Amsterdam(l988), pag. 75.

J.C. Miller and J.N. Miller, Statistics for Analytical

Chemistry, 2nd, edn., Ellis Horwood, Chichester(l988), pag.

101.

J. Agterdenbos, Anal. Chim. Acta, 108, 315(1979).

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014

2094 sAmu

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

J. Agterdenbos, Anal. Chim. Acta, 132, 127(1981).

J.N. Miller, Analyst, 116, 3(1991).

G. Klimov, Probability Theory and Mathematical Statistics,

Mir Publishers Moscow(1986), pag. 272.

R. Rajk6, Anal. Lett., 27, 215(1994).

M. Thompson, Analyst, 119, 127N(1994).

G.R. Phillips and E.M. Eyring, Anal. Chem., 55, 1134(1983).

L.M. Schwartz, Anal. Chem., 49, 2062(1977).

L.M. Schwartz, Anal. Chem., 51, 723(1979).

R.C. Rutan, and P.W. Carr, Anal. Chim. Acta, 215, 131(1988).

Y. Hu, J. Smeyers-Verbeke and D.L. Massart, J. Anal. At.

Spectrom., 4, 605(1989).

R. Wolters and G. Kateman, J. Chemom., 3, 329(1989).

P. Vankeerberghen, C. Vandenbosch, J. Smeyers-Verbeke and

D.L. Massart, Chemom. Intell. Lab. Syst., 12, 3(1991).

L. Galan, H.P.J. van Dalen and G.R. Kornblum, Analyst, 110,

323 (1985).

J.N. Miller, Spectroscopy Europe, 5(6), 22(1992).

P.L. Bonate, LC-GC, 10(6), 448(1991).

J.S. Garden, D . S . Mitchell and W.N. Mills, Anal. Chem., 52,

2310(1980).

L. Aarons, J. Pharm. Biomed. Anal., 2, 395(1984).

J. Knegt and G. Stork, Fresenius' 2. Anal. Chem., 270,

97 (1974) . P. KoScielniak, Anal. Chim. Acta, 278, 177(1993).

W. Xiaoning, J. Smeyers-Verbeke, D.L. Massart, Analusis, 20,

209 (1992) .

Receivec!: February I & , 1 0 0 5 Accepted : ?!arch ? P , 1'j01

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

20:

16 2

0 N

ovem

ber

2014