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Regression by least median squares, a methodological contribution to titration analysis
Chemometrics and intelligent laboratory systems ELSEVIER
Chemometrics and Intelligent Laboratory. Systems 27 (1995) 231-243
Regression by least median squares, a methodological contribution to titration analysis M.C. Ortiz-Ferngmdez *, A. Herrero-Guti6rrez Departamento de Quimica Anal[tica, Facultad de C. Y. ZA. y Ciencias Qufmicas, Pza. Misael Bahuelos, 09001 Burgos, Spain
Received 7 July 1994; accepted 19 October 1994
Abstract
Robust regression by least median squares, LMS, which is able to determine linear tendency provided that at least 50% of the experimental points comply with this tendency, has been applied to data from various titrations. The procedure based on the LMS regression provides an objective criterion for the determination of the end point; it can be reproduced and automated completely, thus contributing to a more versatile titration analysis. Under normal working conditions, concentrations with variation coefficients between 2% and 4% have been determined in amperometric titrations, between 0.5% and 5% in conductometric titrations of strong/weak acid and its mixtures, and around 1.5% in spectrophotometric and potentiometric titrations.
1. Introduction
Least median squares (LMS) regression has been used primarily in problems of calibration and of the calculation of detection limits as a m e t h o d for detecting and foreseeing the presence of outlier data [1-3]. T h e mathematical theory and basic results of the LMS m e t h o d may be consulted in Ref. [4]. The p r o g r a m P R O G R E S S [4] implements the LMS regression and eases its everyday use in the analytical laboratory. T h e b o o k is also a manual for P R O G R E S S . A Tutorial of the LMS regression can be found in Ref. [2]. T h e mathematical properties of L M S regression are of great interest in chemical problems,
* Corresponding author.
such that its use is p r o p o s e d in Ref. [5] to establish a linear range of calibration at the same time as it anticipates the presence of outlier data in routine analysis using carbon paste electrodes. In Ref. [6] it is used to objectively determine the transition time in c h r o n o p o t e n t i o m e t r y which leads to a significant improvement in the precision of the analytical measures performed using this technique. In Ref. [7] the efficiency of LMS regression is shown when Cu is determined, by stripping voltammetry, at trace levels (10-'~-10 m M) with the possible loss of linearity in the response. In this case the importance of assuming linearity results in a more effective evaluation of the detection limit for the technique p r o p o s e d by the analyst. However, the possibilities for the application of the LMS regression to chemical problems do not end with calibration. The aim of this work is
0169-7439/95/$09.50 ~) 1995 Elsevier Science B.V. All rights reserved SSDt 0169-7439(94)00071-9
232
M.C. Ortiz-Ferndndez, A. Herrero-Gutidrrez /Chemometrics and Intelligent Laboraton, Systems 27 (1995) 231-243
therefore to demonstrate that the ability of LMS to automaticaIly diagnose outliers is of considerable interest in the field of titrations.
g,
2. Theory It is well known that titrimetric analysis is based on determination of the end point, that is to say, the point at which the volumetric reaction is completed. The titration may be followed through the change in some physical property of the solution [8-11]. In the next section the following changes will be considered: (i) The electrical conductivity of the solution, conductometric titration. (ii) The current that passes through the titration cell between an indicator electrode and a depolarized reference electrode at a suitable applied e.m.f., amperometric titration. (iii) The absorbance of the solution, spectrophotometric titration. (iv) The duly linearized potential, Gran's graphics, between an indicator electrode and a reference electrode, potentiometric titration. In all titrations outlined above the theoretical titration curve is formed by two or more linear sections, and the equivalence point is the point where the adjacent segments intersect. However, the experimental line presents curvature as well as possible outliers, which makes the determination of linear segments difficult. Use of the LMS regression improves reliability and the ability to reproduce the determination of the end point, because it allows one to objectively define the segments of the straight line given by a titration.
3. Why the LMS regression? Each segment of a titration curve has as its model the equation Y=O% +alx
+e
(1)
where the predictor variable, x, is the volume added; the response variable, y, is the corrected signal in its case (current, conductivity, absorbance or pH) and lastly e is the random error that will affect the experimental response.
o ¢D
\
,
VV' Added volume Fig. 1. Influence of the curvature on the determination of the end point in titration analysis. V' is the end volume calculated when the first linear section has been fitted by LS (dashed line). V is the end volume calculated when the first linear section has been fitted by LS without outlier data detected by m e a n s of LMS (solid line).
Once n experimental values (Xi, yi ) have been obtained, the least squares regression, LS, estimates the parameters oq) and ~1 using the values a 0 and a~ such that they prove min ~ [ y i - ( a o + a l x i ) ] 2 i
(2)
1
If some datum lies outside the linear tendency, either because it is experimentally erroneous or because it belongs to a curved area, its contribution to the sum of squares is important, causing an error in the values a 0 and aj for those which reach the minimum [12,13]. Fig. 1 shows how just one anomalous point can shift the intersection point of the two segments, appreciably modifying the equivalence volume determined. When it can be accepted that the random error e has a normal distribution, independent of the value of the variable x and with constant variance, the LS regression provides the most probable parameters (slope and intercept) and the estimations obtained are unbiased and of lesser variance. The presence of anomalous points causes the distribution of the errors to move notably away from a normal distribution, as it introduces asymmetries and high cumulative
M.C. Ort&-Ferndndez, A. Herrero-Gutigrrez /Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
probability falling away from central values, which cancels the high inferential capacity of the LS regression. In general, robust regression methods have been constructed to highlight the problem of anomalous points. The LMS method estimates the parameters a 0 and a 1 of the Model (1) using the values a 0 and a I which give
minmedian{[yi-(ao+a,xi)] 2, i = 1 , 2 ..... n}
(3) As the median is less sensitive to the presence of a large residual, the presence of outlier data does not greatly modify the parameters in which the minimum, described in (3), is reached. Objectively, (i) the LMS regression reaches a maximum of 50% for the breakdown point, which means it can stand up to 50% of outlier data without the estimation of the parameters being affected; (ii) the influence function is bounded both for the X axis and the Y axis, which means that its insensitiveness to outliers both in the case where these present an anomalous abscissa (leverage point) and when it is its ordinate which has an anomalous value; (iii) the LMS regression has the ability to adjust exactly [14]. In other words, if at least 50% of the data concord with a linear model, it will be found by LMS. It should be r e m e m b e r e d that the curvature between straight lines of a titration produces the same effect as outliers a n d / o r leverages on each line that models the linear segments. This leads not only to unwanted changes in the estimated values of the slopes and intercept terms, but also to an overestimation of the residual variance in these adjustments. Hence, an incorrect evaluation of the significance of the coefficients and of the estimations made with the adjusted models will follow [12]. The mathematical properties of the LMS regression in relation to the titration curves allow one to (i) determine the true linear relation in each section, without the need to establish a priori those experimental points that will form it; (ii) decide, with regard to the linear relation established, which points do not belong to it, in
233
other words, determine objectively the points which form each section; at the same time, the detection of possible erroneous experimental data is guaranteed; (iii) perform the LS regression without outliers under the most favourable conditions to verify the hypothesis about e and an optimal estimation (that of least variance between the unbiased ones) of the parameters of the linear model. Thus, this work proposes the following procedure for the estimation of a titration curve: First step. Apply the LMS regression to the different segments of the titration curve, including in each one points beyond those that could be considered to belong to the segment under study. Second step. Eliminate from each section the outliers, based on criteria of LMS and on the habitual criteria [4] in this type of regression: SR (standardized residuals) greater than 2.5 a n d / o r R D (resistance to the diagnosis) greater than 2.5. Third step. Do the LS regression on each section using the remaining data. Fourth step. Calculate the intersection points using the equations obtained in the third step. Its abscissas are the end volumes looked for.
4. Experimental procedure Analytical-reagent grade chemicals were used without further purification. All the solutions were prepared with deionised water obtained in a Barnstead N A N O Pure II system. The additions were done with a Socorex micropipette.
4.1. Amperornetric titrations A P A R 263 potentiostat/galvanostat connected to a Model 303A static mercury drop electrode was used for the amperometric measurements. Sulphate solution was prepared in 0.5 M N a N O 3 containing 20% ethanol (to suppress the solubility of the lead sulphate) and trace of gelatine as maximum suppressor. The lead solution to be titrated was prepared in 0.05 M HC104 containing trace of gelatine as maximum suppressor.
234
M.C. Ortiz-lZ~'rndndez, A. Herrero-GutErrez /Chemornetrics and Intelligent Laboratoo' Systems 27 (1995) 231-243
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M.C. Ortiz-Ferndndez, A. Herrero-Guti~rrez/ Chemometrics and Intelligent Laboratory S4'stems 27 (1995) 231 243 10 m l o f t h e s a m p l e s o l u t i o n , c o n t a i n i n g sufficient s u p p o r t i n g e l e c t r o l y t e to s u p p r e s s the mig r a t i o n e f f e c t , w e r e p l a c e d in t h e cell w i t h t h e e l e c t r o d e s a n d t h o r o u g h l y d e o x y g e n a t e d f o r 12 m i n by p a s s i n g n i t r o g e n ( 9 9 . 9 9 % ) . A f t e r t h a t , a suitable potential was applied between the electrodes and the diffusion current was measured. A p p l i e d p o t e n t i a l s w e r e f o r t h e t i t r a t i o n o f P b 2+ w i t h C r O 2 - 0.1 M , E = - 0 . 9 4
V (vs. A g / A g C I ) ,
a n d f o r t h e t i t r a t i o n o f SO42- w i t h P b ~+ 0.1 M , - 1 . 2 V (vs. A g / A g C 1 ) . N i t r o g e n w a s b u b b l e d t h r o u g h t h e s o l u t i o n a f t e r e a c h a d d i t i o n f o r 10 s. Each addition was measured four times, and f o l l o w i n g t h i s t h e m e a n w a s c a l c u l a t e d t o give a replication. Five replications were performed for each of the amperometric titrations. A correction f o r t h e d i l u t i o n e f f e c t m a y , h o w e v e r , b e m a d e by m u l t i p l y i n g t h e v a l u e s o f t h e c u r r e n t by t h e f a c t o r
(V+ v)/V, the
in w h i c h V is t h e o r i g i n a l v o l u m e o f s o l u t i o n a n d v is t h e v o l u m e o f r e a g e n t
added.
Table 2 Experimental data and standardized residuals from the LMS regression. Amperometric titration of 10 ml of solution containing Pb 2+ with CrO 2 0.1 M (replication 1 in Table 1) Index
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Titrant
Corrected
Standardized residuals
gl
current IxA
First line
0 100 200 300 400 500 .600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800
311.75 290.37 255.00 223.51 188.96 157.92 126.29 98.73 78.48 57.11 49.25 71.37 113.59 158.90 207.48 261.91 310.59 355.97 417.72
- 0.50 1.07 0.43 0.41 0.10 0.05 - 0.10 0.50 2.24 3.80* 7.48 ~ 15.86 * -
-
235
Table 3 Experimental data and standardized residuals from the LMS regression. Amperometric titration of 10 ml of solution containing SO 2 with Pb e* 0.1 M (replication 4 in Table I) Index
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Titrant
Corrected
Standardized residuals
~1
current ixA
First line
0 100 21111 3/111 40(1 500 600 700 800 900 10110 1100 1200 1300 1401/ 1500 161/0 1700 1800 1900 2000
1.86 3.67 4.44 5.02 6.78 8.18 12.03 17.04 31.29 51.72 70.31 95.40 120.14 146.30 171.17 199.38 226.49 255.06 288.21 317.13 336.110
0.83 0.79 0.1 0.50 - 1/.57 - 0.83 0.19 1.83 8.32 * 18.07 * 26.86 * 39.05 * -
-
-
Second line 5.76 * 3.95 1.63 1./18 0.43 0.18 -0.43 -0.13 -0.13 0.26 1.91 2.411 0.16
* indicates the experimental data eliminated by LMS regression.
4.2. Conductometric titrations
Second line -
13.68 * 5.112 * 1/.97 0.01 - 0.48 - 0.47 0.45 0.48 0.01 2.05
* indicates the experimental data eliminated by LMS regression.
Conductometric measurements were perf o r m e d w i t h t h e aid o f a C r i s o n c o n d u c t i v i t y m e t e r 524. T h e c o n d u c t i v i t y w a s m e a s u r e d a f t e r each addition of a volume of the titrant, and the p o i n t s t h u s o b t a i n e d w e r e p l o t t e d to give a g r a p h that consists of two or m o r e straight lines inters e c t i n g at t h e e n d p o i n t s . T h e c o r r e c t i o n o f volu m e w a s a l s o d o n e by m u l t i p l y i n g t h e v a l u e s o f t h e c o n d u c t i v i t y by t h e f a c t o r ( V + v ) / V .
4.3. Spectrophotometric titrations Spectrophotometric measurements were perf o r m e d w i t h a L a m b d a 3B U V / V i s s p e c t r o m e t e r e q u i p p e d w i t h q u a r t z cclls o f 1 c m p a t h l e n g t h . E D T A s t a n d a r d s o l u t i o n w a s p r e p a r e d by a p p r o p r i a t e d i l u t i o n o f E D T A 0.1 v o l u m e t r i c s t a n dard solution (Merck) with deionised water.
236
M.C. Ortiz-Ferndndez, A. Herrero-Gutigrrez / Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
Sample solutions were prepared by mixing and Bi 3+ stock solutions and appropriate amounts of E D T A in volumetric flasks of 100 ml and then diluting the mixtures with water• Each sample contained fixed amounts of Cu 2+ and Bi 3+ but different amounts of EDTA. Absorbance was measured at 745 rim. At this wavelength Bi 3+, Cu 2+ and E D T A did not absorb, neither did the more stab[e bismuth complex
generated in the first branch of the titration. Once all Bi 3+ has been consumed copper complex began to be formed and absorbance started to rise, until the copper equivalence point was achieved• It provided two well-defined end points.
C u 2+
4.4. Potentiometric titrations Potentiometric measurements were performed by using a Crison micro pH 2002.
3 ~'5
450
)0
~
300
~ 2 .~5
+
.~0 150 O
~'5 0 300
600
900
1200
A d d e d v o l u m e o f CrO42
1500
0
1800
400
800
1200
1600
2000
A d d e d v o l u m e of P b 2÷ / ,uL
/ ,uL
(c)
,51
> +
0
3
6
9
12
A d d e d v o l u m e of N a O H
15
3
/ mL
6
9
12
A d d e d v o l u m e of N a O H
15
/ mL
0.06 -
-0.9
(el 0.04 -
o
/
.D < 0.02 -
-/
/
o
/
-0.6 7 "I-
¥
+
> -0.3
0.00 0
500
1000
1500
Added volume
2000
2500
of E D T A / p L
3000
0.0
2
4
6
8
10
A d d e d v o l u m e of HCL / m L
Fig. 2. Experimental data and lines adjusted by LS without outlier data. Amperometric titrations: (a) Pb 2+ with CrO 4 , and (b) SO 4 with Pb 2+ (data in Tables 2 and 3, respectively). Conductometric titrations: (c) HAc with NaOH and (d) a mixture of HAc and HCI with N a O H (data in Tables 4 and 5, respectively). Spectrophotometric titration: (e) a mixture of Cu 2+ and Bi 3+ with E D T A (data in Table 6). Potentiometric titration (Gran's Graphics): (f) C20, ~ with HCI (data in Table 7).
M.C. Ortiz-Ferndndez, A. Herrero-Guti&rez / Chemometrics and Intelligent Laboratory Systems 27 (19951 231-243 The potentiometric weak
titration of a solution of a
b a s e (20 ml C 2 0 4 N a
2) w i t h a s t r o n g
(HC1 0.216 M) was performed. using
Gran's
treated
The
d a t a (t:, p H ) h a v e b e e n
perimental by
transformations. means
of
the
p a i r s o f exlinearized by
The
[ H +]
following
point
values obtained
determination
of the end point.
Numeric
calculations
were
were
was
1988 v e r s i o n
486/33
computer.
implemented
by
of the program
gram for RObust
The using
LMS the
PROGRESS
reGRESSion)
re-
April (Pro-
[4].
5. Results and discussion
of
and V the initial volume of the
solution. The
MCS
gression
was
~ ) [ H +] a f t e r it; t, b e i n g t h e v o l u m e
the titrant added,
Tandon
functions:
v / [ H +] w a s a p p l i e d b e f o r e t h e e q u i v a l e n c e and (V+
acid
237
used for the
Table
1 shows the parameters
and coefficient of determination) various titrations
performed
on
a
of the proposed
(slope, intercept obtained
for the
after steps one, two and three procedure.
Table 4 Experimental data and standardized residuals from the LMS regression. Conductometric titration of 75 ml of solution containing HAc with NaOH 0.097 M (replication 1 in Table 11 Index
Titrant ml
Corrected conductivity mS
Standardized residuals
0 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 26 27 28 29 30 31 32
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 15.50 16.00
0.14 0.13 0.15 0.18 0.22 0.27 0.29 0.34 0.40 0.44 0.48 0.53 0.57 11.62 0.66 0.70 0.79 0.91 1.03 1.15 1.22 1.37 1.49 1.65 1.77 1.87 2.00 2.12 2.24 2.36 2.47 2.58 2.68
10.43 3.99 1.48 0.46 0.15 - 0.(19 2.18 1.88 - 0.02 - 0.46 - 0.96 - 0.02 0.23 0.28 0.00 0.04 4.98 14.12 23.03 -
* indicates the experimental data eliminated by LMS regression.
First line * *
Second line -
* * *
5.69 1.66 - 0.23 -0.28 - 0.43 -0.10 - 2.81 - 1.47 - 1.46 0.46 0.37 - 0.80 - 0.10 - 0.42 - 0.38 - 0.46 - 1.28 - 1.86 - 3.02
M.C. Ortiz-Ferndndez, A. Herrero-Gutidrrez / Chemometrics and Intelligent Laborato~' Systems 27 (1995) 231-243
238
included; similarly, for the second step the last 10 points have been taken with the certainty that the equivalence volume is beyond the first of them. In such a way it is possible to detect outliers and the linear zone. As it is not necessary to assign experimental pairs to some step in the neighbourhood of the end point, the procedure for obtaining the straight lines is guaranteed to be objective. In Fig. 2a, associated with this titration, much curving can be seen. This is a consequence of the evolution of the chemical reaction, the Pb 2+ is electroactive and when CrO 4 is added, the prc-
Table 2 shows the experimental data of an amperometric titration in which 10 ml of Pb 2+ is titrated with CrO 2 0.1 M [15]. The second and the third columns give the pairs of experimental data v o l u m e / c o r r e c t e d current for 18 successive additions of titrant. Columns 4 and 5 show the standardized residuals given by the LMS regression for the linear segments before and after the end point. The asterisk indicates that this datum was not used in the LS regression. According to the procedure proposed for the first step, the first 12 experimental data that reach a volume of titrant beyond the equivalence point have been
Table 5 Experimental data and standardized residuals from the LMS regression. Cnnductometric titration of a mixture of 50 ml of HAc and 5 0 ml of HCI with N a O H 0.097 M (replication 2 in Table 1 ) Index
*
Titrant ml
Corrected conductivity
0
0.00
1.85
1
0.50
1.72
2
1.00
1.59
3
1.50
4
2.00
5
2.50
6
3.00
7
Standardized residuals mS
First line
Second line
Third line
- 0.50
-
0.06
-
1.34
-
1.48
0.50
-
1.35
0.39
-
1.23
0.06
-
1.10
- 0.50
3.50
0.97
-
1.28
-
-
8
4.00
0.85
- 0.50
-
-
-
-
-
-
-
-
9
4.50
0.74
1.85
10
5.00
0.65
7.62
*
11
5.50
0.61
22.24
*
12
6.00
0.62
13
6.511
0.63
14
7.00
0.67
-
15
7.50
0.69
-
16
8.00
0.73
-
17
8.50
0.75
-
5.32
*
-
-
2.52
x
_
-
- 0.26 0.47
-
-0.50
-
0.32
-
-0.59
18
9.00
0.79
-
0.32
19
9.50
0.82
-
- 0.50
20
10.00
0.85
-
-
11.50
-
21
10.50
0.88
-
22
11.00
0.94
-
-/).26 4.46
*
18.73 * 3.04 "
23
11.50
1.02
-
12.90
~
0.40
24
12.00
1.10
-
21.45
*
25
12.5(/
l. 19
-
-
- 3.67
26
13.00
1.28
-
-
1.38
-
-
1.83
27
13.50
1.37
-
28
14.00
1.45
-
- I).23
29
14.50
1.54
-
-/).41")
30
15.00
1.63
-
31
15.50
1.72
-
indicates the experimental data eliminated by LMS regression.
0.40
- 0.23 -
0.4[)
*
M.C. Ortiz-Ferndndez, A. Herrero-Guti&rez /Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
239
which step each point belongs, which is very compromising in the case of additions 9 and 10, for which LMS detects the linear tendencies independently of the intensity of the curvature. The experimental results obtained in one of the amperometric titrations of SO 2 with Pb 2+ 0.1 M are shown in Table 3. As in the previous case, there is notable curving around the end point, see Fig. 2b, caused by redissolution of the precipitate despite the fact that alcohol was added to the medium to avoid this [16]. The LMS regression detects the curve by identifying additions 8, 9, 10 and 11 outside the linear tendency of the first step and additions 8 and 9 outside the linear
cipitate forms, which causes a decrease in the recorded current. After the equivalence point, it is the CrO 2 that is electroactive. However, around the equivalence point partial solubilisation of the PbCrO 4 occurs, which causes an increase in the current recorded with regard to what is theoretically expected. The LMS regression detects this anomalous behaviour in relation to the linear tendency and diagnoses as outliers in the first step the three points which precede the end point (additions 9, 10 and 11) while in the second step it is the initial points, 9 and 10, which are diagnosed as being outside the linear tendency. No prior decision is needed to point out to
Table 6 Experimental
data and standardized
residuals
c o n t a i n i n g C u e+, B i 3~ a n d a p p r o p r i a t e Index
T i t r a n t b~l
from the LMS
amounts of EDTA Absorbance
regression.
Spectrophotometric
0.1 M as titrant ( r e p l i c a t i o n
Standardized First line
titration
of 100 ml of a mixture
2 in T a b l e 1)
residuals S e c o n d line
T h i r d line
0
0
0.0116
0.05
-
-
1
100
0.0122
0.96
-
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2
200
0.0120
- 0.64
3
300
0.0126
0.26
-
-
4
400
0.0127
- 0.40
-
-
5
500
0.0130
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6
600
0.0134
0.16
-
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700
0.0139
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-
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800
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0.0154
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11
1100
0.0197
14.67 *
12
1200
0.0221
21.20 *
13
1300
0.0295
14
1400
0.0292
15
1500
16
3.86 *
-
1.59
-
-0.50
-
0.41
-
-0.41
-
3.32 *
-
-
0.05
-
0.0329
-
0.41
-
160/)
0.0363
-
0.50
-
17
1700
0.0387
-
18
1800
0.0441
-
1.59
-4.60
19
1900
0.0457
-
0.05
- 1.66
20
2000
0.0471
-
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21
2100
0.0466
-
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- 0.29
22
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0.0470
-
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0.29
23
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-
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-0.32
-
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0.88
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1,66 0.10 0.10
*
240
M.C. Ortiz-Ferndndez, A. Herrero-Guti~rrez / Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
tendency of the second, as can be seen in Table 3. Table 4 shows the experimental results of one of the conductometric titrations of H A c with N a O H 0.097 M. The first 19 additions were taken to adjust the first segment of the titration curve and the last 19 for the second. The diagnosis of outliers made with the regression indicates that the first two additions lie outside the linearity as corresponds to the initial curve which shows the titration of weak acids; the last three are not accepted as part of the segment preceding the equivalence point. Three outliers are identified in the second step: addition 14, which belongs to the first; 20, presumably an incorrect addition of titrant and addition 32. Notice that addition 15 was accepted as belonging to both steps, which does not mean that this is the equivalence point; it is much more precise to estimate it using all the points which respond to the linear model for each segment. This is shown in Fig. 2c with the two lines fitted by least squares to the points diagnosed as being correctly aligned.
The proposed procedure does not depend on the number of rectilinear sections which make up the titration curve. Table 5 summarizes the data of the conductometric titration of a mixture of strong acid and weak acid titrated with N a O H 0.097 M. Again, the curve between the first and second sections shows the advantage of using the LMS regression. In this case, points 10, 11 and 12 are diagnosed as outliers for both segments. The same occurs with point 22 between the second and third segments. Furthermore, the presence of an outlier not related to any curve is detected, addition 25, which must be interpreted as an experimental error. In Fig. 2d one can see the intersection of the three sections adjusted by means of an LS regression, which gives rise to the determination of the end volumes corresponding to the HC1 and to the HAc respectively. The titration of various metals with E D T A (or another complexing agent) is one of the most interesting applications of spectrophotometric titration. Table 6 is an example of the successive titration of bismuth and copper in a single titra-
Table 7 E x p e r i m e n t a l d a t a a n d s t a n d a r d i z e d r e s i d u a l s f r o m the L M S r e g r e s s i o n . P o t e n t i o m e t r i c t i t r a t i o n o f 20 ml of a s o l u t i o n c o n t a i n i n g C z O 2 with H C I 0.22 M ( r e p l i c a t i o n 1 in T a b l e 1) Index
T i t r a n t ml
L,/[H +]
(V+ u)[H + ]
0 1 2 3 4 5 6 7 8 9 10 ll 12 13 14 15 16 17 18 19
0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8,00 8.50 9.00 9,50 10.00
31547.87 25118.87 21188.06 16635.28 13738.52 10402.11 7312.53 4488.07 2530.53 1285.19 871.69 673.21 579.31 507.10 451.91 410.28 397.57 366.64 361.17 346.73
0.0003 0.0008 0.0015 0.0026 0.0040 0.0066 0.0112 0.0214 0.0436 0.0973 0.1609 0.2317 0.2973 0.3727 0.4564 0.5460 0.6093 0.7119 0.7759 0.8652
Standardized residuals First line
* i n d i c a t e s t h e e x p e r i m e n t a l d a t a e l i m i n a t e d by L M S r e g r e s s i o n .
1.59 0.30 0.10 - 0.38 - 0.13 -0.07 0.10 0.38 1.03 2.00 3.33 * 4.76 * -
S e c o n d line
13.11 9.58 6.30 3.57 2.29
1.45 0.95 0.21 - 0.08 0.01 0.37 - 0.48 0.48 -0.33 0.01
* * * *
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M.C. Ortiz-Ferndndez, A. Herrero-Gutdrrez / Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
tion with E D T A 0.1 M. The titration curve (Fig. 2e) presents three segments of which the first and third are practically horizontal since, at the chosen wavelength of 745 nm, neither the bismuth complex, nor the E D T A absorb significantly. This explains why the coefficient of determination, which is the percentage of variance explained by the LS regression once anomalous data have been eliminated, is low for the first and third sections, as can be seen in Table 1. The detection of outliers shown in Table 6 also indicates this effect; the central section shares various experimental points with each of the other two segments. Several times during this work it has been highlighted that the proposed procedure does not require the prior determination of which experimental data should form part of any segment, or of one or two consecutive segments. The advantage of this is clearly shown in the case of a potentiometric titration to which the Gran's method is applied [17]. This method involves performing a different transformation on the experimental data (~', pH) according to whether one is dealing with the segment prior to the equivalence point or after it. The procedure based on the detection of outliers by LMS does not require this: both transformations are applied to all experimental data and each of the two responses are adjusted separately by LMS to define their linear pattern. Table 7 shows the data corresponding to the titration of a weak base (C2042-) with HC1 0.22 M. The potentiometric curve shows a barely appreciable jump that makes locating the equivalence point more difficult. Even when the data have been transformed its location is difficult, as can be seen in Fig. 2f. Table 8 summarizes the concentrations calculated from the end volume obtained as the intersection of the straight lines described in Table 1; all the replications of the titrations performed are noted. Some statistical data are also included in the same table which allow one to evaluate the accuracy in the determination of the concentration. The coefficients of variation obtained from 0.45% to 5.13% agree with what one would expect from each analytical technique a n d / o r type of analyte to be titrated in the mixtures. It should
be pointed out that the replications take into account the day to day variability. They were not performed consecutively and the equipment was started up for each case, thus simulating the day to day use of these analytical techniques even for mixtures and substances whose end point of titration is difficult to determine. The precision of the procedure is theoretically assured because the parameters of the straight lines found are the most exact; the methodological advantage of the process is clear with regard to the habitual method, based on adjusting the linear segments directly with the LS regression, since any anomalous point or curve in the neighbourhood of the equivalence point considerably modifies the slope and independent term and thus the concentration calculated.
6. Conclusions The robust regression method LMS provides an objective criterion in the determination of linear segments in titration curves. As a result, the proposed procedure allows the determination of an end point which can be reproduced both in the titration of pure substances as well as in that of mixtures under normal conditions. As no prior decision need be made by the analyst the process is capable of complete automation. What is more, it includes the detection of outliers produced by experimental errors or in the acquisition of data, thus removing their influence on the titration.
References [1] D.L. Massart, L. Kaufman, P.J. Rousseeuw and A. Leroy, Least median of squares: a robust method for outlier and model error detection in regression and calibration, AnaIvtica Chimica Acta, 187 (1986) 171-179. [2] P.J. Rousseeuw, Tutorial to robust statistics, Journal of Chemometrics, 5 (1991) 1-20. [3] P.J. Rousseeuw, Least median of squares regression, Journal of the American Statistical Association, 79 (1984) 871-880. [4] P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection, Wiley, New York, 1987. [5] M.C. Ortiz, J. Arcos, J.V. Juarros, J. L6pez-Palaciosand L.A. Sarabia, Robust procedure for calibration and cal-
M.C. Ortiz-Ferndndez, A. Herrero-Guti&rez / Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
[6]
[7]
[8]
[9]
culation of the detection limit of trimipramine by adsorptive stripping voltammetry at a carbon paste electrode, Analytical Chemistry, 65 (1993) 678-682. M.C. Ortiz, J. L6pez-Palacios, M.J. Arcos, L.A. Sarabia, M.G. Piangerelli and D. Cingolani, Determinaci6n del Tiempo de Transici6n en Cronopotenciometr{a mediante Regresi6n Robusta, l I | Jornadas Cientfficas de Electroquimica del Grupo de la S.E.Q.A. y R.S.E.Q., Valladolid, Comunicaci6n 5-3, 1991. A. Herrero, M.C. Ortiz, J. Arcos and J. L6pez-Palacios, Optimization of the experimental parameters in the determination of Cu(II) by differential pulse anodic stripping voltammetry and evaluation of the characteristic curves of detection, Analyst, 119 (1994) 1585 1592. I.M. Kolthoff and P.J. Elvin, Treatise on Analytical Chemistry, Part L Theory and Practice, Vol. 2, Wiley, New York, 2nd edn., 1979. A,1. Vogel, Textbook of Quantitative b~organic Analysis, Longman, London, 4th edn., 1978.
243
[10] P.H. Rieger, Electrochemistry, Prentice-Hall, Englewood Cliffs, N J, t987. [11] E.A.M.F. Dahmen, Electroanalysis. Theory and Applications in Aqueous and Non-Aqueous Media and in Automated Chemical Control, Elsevier, Amsterdam, 1986. [12] N. Draper and H. Smith, Applied Regression Analysis, Wiley, New York, 2nd edn., 1981. [13] R.D. Cook and S. Weiberg, Residuals and Influence in Regression, Chapman and Hall, New York, NY, 1982. [14] F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, Robust Statistics. The Approach Based on Influence Functions, Wiley, New York, 1986. [15] J. Heyrovslo) and P. Zuman, Practical Polarography, Academic Press, London, 1968. [16] J.T. Stock, Amperometric Titrations, Krieger, New York, NY, 1975. [17] O. Budevsky, Foundation of Chemical Analysis, Ellis Horwood, Chichester, 1979.
Chemometrics and Intelligent Laboratory. Systems 27 (1995) 231-243
Regression by least median squares, a methodological contribution to titration analysis M.C. Ortiz-Ferngmdez *, A. Herrero-Guti6rrez Departamento de Quimica Anal[tica, Facultad de C. Y. ZA. y Ciencias Qufmicas, Pza. Misael Bahuelos, 09001 Burgos, Spain
Received 7 July 1994; accepted 19 October 1994
Abstract
Robust regression by least median squares, LMS, which is able to determine linear tendency provided that at least 50% of the experimental points comply with this tendency, has been applied to data from various titrations. The procedure based on the LMS regression provides an objective criterion for the determination of the end point; it can be reproduced and automated completely, thus contributing to a more versatile titration analysis. Under normal working conditions, concentrations with variation coefficients between 2% and 4% have been determined in amperometric titrations, between 0.5% and 5% in conductometric titrations of strong/weak acid and its mixtures, and around 1.5% in spectrophotometric and potentiometric titrations.
1. Introduction
Least median squares (LMS) regression has been used primarily in problems of calibration and of the calculation of detection limits as a m e t h o d for detecting and foreseeing the presence of outlier data [1-3]. T h e mathematical theory and basic results of the LMS m e t h o d may be consulted in Ref. [4]. The p r o g r a m P R O G R E S S [4] implements the LMS regression and eases its everyday use in the analytical laboratory. T h e b o o k is also a manual for P R O G R E S S . A Tutorial of the LMS regression can be found in Ref. [2]. T h e mathematical properties of L M S regression are of great interest in chemical problems,
* Corresponding author.
such that its use is p r o p o s e d in Ref. [5] to establish a linear range of calibration at the same time as it anticipates the presence of outlier data in routine analysis using carbon paste electrodes. In Ref. [6] it is used to objectively determine the transition time in c h r o n o p o t e n t i o m e t r y which leads to a significant improvement in the precision of the analytical measures performed using this technique. In Ref. [7] the efficiency of LMS regression is shown when Cu is determined, by stripping voltammetry, at trace levels (10-'~-10 m M) with the possible loss of linearity in the response. In this case the importance of assuming linearity results in a more effective evaluation of the detection limit for the technique p r o p o s e d by the analyst. However, the possibilities for the application of the LMS regression to chemical problems do not end with calibration. The aim of this work is
0169-7439/95/$09.50 ~) 1995 Elsevier Science B.V. All rights reserved SSDt 0169-7439(94)00071-9
232
M.C. Ortiz-Ferndndez, A. Herrero-Gutidrrez /Chemometrics and Intelligent Laboraton, Systems 27 (1995) 231-243
therefore to demonstrate that the ability of LMS to automaticaIly diagnose outliers is of considerable interest in the field of titrations.
g,
2. Theory It is well known that titrimetric analysis is based on determination of the end point, that is to say, the point at which the volumetric reaction is completed. The titration may be followed through the change in some physical property of the solution [8-11]. In the next section the following changes will be considered: (i) The electrical conductivity of the solution, conductometric titration. (ii) The current that passes through the titration cell between an indicator electrode and a depolarized reference electrode at a suitable applied e.m.f., amperometric titration. (iii) The absorbance of the solution, spectrophotometric titration. (iv) The duly linearized potential, Gran's graphics, between an indicator electrode and a reference electrode, potentiometric titration. In all titrations outlined above the theoretical titration curve is formed by two or more linear sections, and the equivalence point is the point where the adjacent segments intersect. However, the experimental line presents curvature as well as possible outliers, which makes the determination of linear segments difficult. Use of the LMS regression improves reliability and the ability to reproduce the determination of the end point, because it allows one to objectively define the segments of the straight line given by a titration.
3. Why the LMS regression? Each segment of a titration curve has as its model the equation Y=O% +alx
+e
(1)
where the predictor variable, x, is the volume added; the response variable, y, is the corrected signal in its case (current, conductivity, absorbance or pH) and lastly e is the random error that will affect the experimental response.
o ¢D
\
,
VV' Added volume Fig. 1. Influence of the curvature on the determination of the end point in titration analysis. V' is the end volume calculated when the first linear section has been fitted by LS (dashed line). V is the end volume calculated when the first linear section has been fitted by LS without outlier data detected by m e a n s of LMS (solid line).
Once n experimental values (Xi, yi ) have been obtained, the least squares regression, LS, estimates the parameters oq) and ~1 using the values a 0 and a~ such that they prove min ~ [ y i - ( a o + a l x i ) ] 2 i
(2)
1
If some datum lies outside the linear tendency, either because it is experimentally erroneous or because it belongs to a curved area, its contribution to the sum of squares is important, causing an error in the values a 0 and aj for those which reach the minimum [12,13]. Fig. 1 shows how just one anomalous point can shift the intersection point of the two segments, appreciably modifying the equivalence volume determined. When it can be accepted that the random error e has a normal distribution, independent of the value of the variable x and with constant variance, the LS regression provides the most probable parameters (slope and intercept) and the estimations obtained are unbiased and of lesser variance. The presence of anomalous points causes the distribution of the errors to move notably away from a normal distribution, as it introduces asymmetries and high cumulative
M.C. Ort&-Ferndndez, A. Herrero-Gutigrrez /Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
probability falling away from central values, which cancels the high inferential capacity of the LS regression. In general, robust regression methods have been constructed to highlight the problem of anomalous points. The LMS method estimates the parameters a 0 and a 1 of the Model (1) using the values a 0 and a I which give
minmedian{[yi-(ao+a,xi)] 2, i = 1 , 2 ..... n}
(3) As the median is less sensitive to the presence of a large residual, the presence of outlier data does not greatly modify the parameters in which the minimum, described in (3), is reached. Objectively, (i) the LMS regression reaches a maximum of 50% for the breakdown point, which means it can stand up to 50% of outlier data without the estimation of the parameters being affected; (ii) the influence function is bounded both for the X axis and the Y axis, which means that its insensitiveness to outliers both in the case where these present an anomalous abscissa (leverage point) and when it is its ordinate which has an anomalous value; (iii) the LMS regression has the ability to adjust exactly [14]. In other words, if at least 50% of the data concord with a linear model, it will be found by LMS. It should be r e m e m b e r e d that the curvature between straight lines of a titration produces the same effect as outliers a n d / o r leverages on each line that models the linear segments. This leads not only to unwanted changes in the estimated values of the slopes and intercept terms, but also to an overestimation of the residual variance in these adjustments. Hence, an incorrect evaluation of the significance of the coefficients and of the estimations made with the adjusted models will follow [12]. The mathematical properties of the LMS regression in relation to the titration curves allow one to (i) determine the true linear relation in each section, without the need to establish a priori those experimental points that will form it; (ii) decide, with regard to the linear relation established, which points do not belong to it, in
233
other words, determine objectively the points which form each section; at the same time, the detection of possible erroneous experimental data is guaranteed; (iii) perform the LS regression without outliers under the most favourable conditions to verify the hypothesis about e and an optimal estimation (that of least variance between the unbiased ones) of the parameters of the linear model. Thus, this work proposes the following procedure for the estimation of a titration curve: First step. Apply the LMS regression to the different segments of the titration curve, including in each one points beyond those that could be considered to belong to the segment under study. Second step. Eliminate from each section the outliers, based on criteria of LMS and on the habitual criteria [4] in this type of regression: SR (standardized residuals) greater than 2.5 a n d / o r R D (resistance to the diagnosis) greater than 2.5. Third step. Do the LS regression on each section using the remaining data. Fourth step. Calculate the intersection points using the equations obtained in the third step. Its abscissas are the end volumes looked for.
4. Experimental procedure Analytical-reagent grade chemicals were used without further purification. All the solutions were prepared with deionised water obtained in a Barnstead N A N O Pure II system. The additions were done with a Socorex micropipette.
4.1. Amperornetric titrations A P A R 263 potentiostat/galvanostat connected to a Model 303A static mercury drop electrode was used for the amperometric measurements. Sulphate solution was prepared in 0.5 M N a N O 3 containing 20% ethanol (to suppress the solubility of the lead sulphate) and trace of gelatine as maximum suppressor. The lead solution to be titrated was prepared in 0.05 M HC104 containing trace of gelatine as maximum suppressor.
234
M.C. Ortiz-lZ~'rndndez, A. Herrero-GutErrez /Chemornetrics and Intelligent Laboratoo' Systems 27 (1995) 231-243
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M.C. Ortiz-Ferndndez, A. Herrero-Guti~rrez/ Chemometrics and Intelligent Laboratory S4'stems 27 (1995) 231 243 10 m l o f t h e s a m p l e s o l u t i o n , c o n t a i n i n g sufficient s u p p o r t i n g e l e c t r o l y t e to s u p p r e s s the mig r a t i o n e f f e c t , w e r e p l a c e d in t h e cell w i t h t h e e l e c t r o d e s a n d t h o r o u g h l y d e o x y g e n a t e d f o r 12 m i n by p a s s i n g n i t r o g e n ( 9 9 . 9 9 % ) . A f t e r t h a t , a suitable potential was applied between the electrodes and the diffusion current was measured. A p p l i e d p o t e n t i a l s w e r e f o r t h e t i t r a t i o n o f P b 2+ w i t h C r O 2 - 0.1 M , E = - 0 . 9 4
V (vs. A g / A g C I ) ,
a n d f o r t h e t i t r a t i o n o f SO42- w i t h P b ~+ 0.1 M , - 1 . 2 V (vs. A g / A g C 1 ) . N i t r o g e n w a s b u b b l e d t h r o u g h t h e s o l u t i o n a f t e r e a c h a d d i t i o n f o r 10 s. Each addition was measured four times, and f o l l o w i n g t h i s t h e m e a n w a s c a l c u l a t e d t o give a replication. Five replications were performed for each of the amperometric titrations. A correction f o r t h e d i l u t i o n e f f e c t m a y , h o w e v e r , b e m a d e by m u l t i p l y i n g t h e v a l u e s o f t h e c u r r e n t by t h e f a c t o r
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added.
Table 2 Experimental data and standardized residuals from the LMS regression. Amperometric titration of 10 ml of solution containing Pb 2+ with CrO 2 0.1 M (replication 1 in Table 1) Index
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Titrant
Corrected
Standardized residuals
gl
current IxA
First line
0 100 200 300 400 500 .600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800
311.75 290.37 255.00 223.51 188.96 157.92 126.29 98.73 78.48 57.11 49.25 71.37 113.59 158.90 207.48 261.91 310.59 355.97 417.72
- 0.50 1.07 0.43 0.41 0.10 0.05 - 0.10 0.50 2.24 3.80* 7.48 ~ 15.86 * -
-
235
Table 3 Experimental data and standardized residuals from the LMS regression. Amperometric titration of 10 ml of solution containing SO 2 with Pb e* 0.1 M (replication 4 in Table I) Index
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Titrant
Corrected
Standardized residuals
~1
current ixA
First line
0 100 21111 3/111 40(1 500 600 700 800 900 10110 1100 1200 1300 1401/ 1500 161/0 1700 1800 1900 2000
1.86 3.67 4.44 5.02 6.78 8.18 12.03 17.04 31.29 51.72 70.31 95.40 120.14 146.30 171.17 199.38 226.49 255.06 288.21 317.13 336.110
0.83 0.79 0.1 0.50 - 1/.57 - 0.83 0.19 1.83 8.32 * 18.07 * 26.86 * 39.05 * -
-
-
Second line 5.76 * 3.95 1.63 1./18 0.43 0.18 -0.43 -0.13 -0.13 0.26 1.91 2.411 0.16
* indicates the experimental data eliminated by LMS regression.
4.2. Conductometric titrations
Second line -
13.68 * 5.112 * 1/.97 0.01 - 0.48 - 0.47 0.45 0.48 0.01 2.05
* indicates the experimental data eliminated by LMS regression.
Conductometric measurements were perf o r m e d w i t h t h e aid o f a C r i s o n c o n d u c t i v i t y m e t e r 524. T h e c o n d u c t i v i t y w a s m e a s u r e d a f t e r each addition of a volume of the titrant, and the p o i n t s t h u s o b t a i n e d w e r e p l o t t e d to give a g r a p h that consists of two or m o r e straight lines inters e c t i n g at t h e e n d p o i n t s . T h e c o r r e c t i o n o f volu m e w a s a l s o d o n e by m u l t i p l y i n g t h e v a l u e s o f t h e c o n d u c t i v i t y by t h e f a c t o r ( V + v ) / V .
4.3. Spectrophotometric titrations Spectrophotometric measurements were perf o r m e d w i t h a L a m b d a 3B U V / V i s s p e c t r o m e t e r e q u i p p e d w i t h q u a r t z cclls o f 1 c m p a t h l e n g t h . E D T A s t a n d a r d s o l u t i o n w a s p r e p a r e d by a p p r o p r i a t e d i l u t i o n o f E D T A 0.1 v o l u m e t r i c s t a n dard solution (Merck) with deionised water.
236
M.C. Ortiz-Ferndndez, A. Herrero-Gutigrrez / Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
Sample solutions were prepared by mixing and Bi 3+ stock solutions and appropriate amounts of E D T A in volumetric flasks of 100 ml and then diluting the mixtures with water• Each sample contained fixed amounts of Cu 2+ and Bi 3+ but different amounts of EDTA. Absorbance was measured at 745 rim. At this wavelength Bi 3+, Cu 2+ and E D T A did not absorb, neither did the more stab[e bismuth complex
generated in the first branch of the titration. Once all Bi 3+ has been consumed copper complex began to be formed and absorbance started to rise, until the copper equivalence point was achieved• It provided two well-defined end points.
C u 2+
4.4. Potentiometric titrations Potentiometric measurements were performed by using a Crison micro pH 2002.
3 ~'5
450
)0
~
300
~ 2 .~5
+
.~0 150 O
~'5 0 300
600
900
1200
A d d e d v o l u m e o f CrO42
1500
0
1800
400
800
1200
1600
2000
A d d e d v o l u m e of P b 2÷ / ,uL
/ ,uL
(c)
,51
> +
0
3
6
9
12
A d d e d v o l u m e of N a O H
15
3
/ mL
6
9
12
A d d e d v o l u m e of N a O H
15
/ mL
0.06 -
-0.9
(el 0.04 -
o
/
.D < 0.02 -
-/
/
o
/
-0.6 7 "I-
¥
+
> -0.3
0.00 0
500
1000
1500
Added volume
2000
2500
of E D T A / p L
3000
0.0
2
4
6
8
10
A d d e d v o l u m e of HCL / m L
Fig. 2. Experimental data and lines adjusted by LS without outlier data. Amperometric titrations: (a) Pb 2+ with CrO 4 , and (b) SO 4 with Pb 2+ (data in Tables 2 and 3, respectively). Conductometric titrations: (c) HAc with NaOH and (d) a mixture of HAc and HCI with N a O H (data in Tables 4 and 5, respectively). Spectrophotometric titration: (e) a mixture of Cu 2+ and Bi 3+ with E D T A (data in Table 6). Potentiometric titration (Gran's Graphics): (f) C20, ~ with HCI (data in Table 7).
M.C. Ortiz-Ferndndez, A. Herrero-Guti&rez / Chemometrics and Intelligent Laboratory Systems 27 (19951 231-243 The potentiometric weak
titration of a solution of a
b a s e (20 ml C 2 0 4 N a
2) w i t h a s t r o n g
(HC1 0.216 M) was performed. using
Gran's
treated
The
d a t a (t:, p H ) h a v e b e e n
perimental by
transformations. means
of
the
p a i r s o f exlinearized by
The
[ H +]
following
point
values obtained
determination
of the end point.
Numeric
calculations
were
were
was
1988 v e r s i o n
486/33
computer.
implemented
by
of the program
gram for RObust
The using
LMS the
PROGRESS
reGRESSion)
re-
April (Pro-
[4].
5. Results and discussion
of
and V the initial volume of the
solution. The
MCS
gression
was
~ ) [ H +] a f t e r it; t, b e i n g t h e v o l u m e
the titrant added,
Tandon
functions:
v / [ H +] w a s a p p l i e d b e f o r e t h e e q u i v a l e n c e and (V+
acid
237
used for the
Table
1 shows the parameters
and coefficient of determination) various titrations
performed
on
a
of the proposed
(slope, intercept obtained
for the
after steps one, two and three procedure.
Table 4 Experimental data and standardized residuals from the LMS regression. Conductometric titration of 75 ml of solution containing HAc with NaOH 0.097 M (replication 1 in Table 11 Index
Titrant ml
Corrected conductivity mS
Standardized residuals
0 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 26 27 28 29 30 31 32
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 15.50 16.00
0.14 0.13 0.15 0.18 0.22 0.27 0.29 0.34 0.40 0.44 0.48 0.53 0.57 11.62 0.66 0.70 0.79 0.91 1.03 1.15 1.22 1.37 1.49 1.65 1.77 1.87 2.00 2.12 2.24 2.36 2.47 2.58 2.68
10.43 3.99 1.48 0.46 0.15 - 0.(19 2.18 1.88 - 0.02 - 0.46 - 0.96 - 0.02 0.23 0.28 0.00 0.04 4.98 14.12 23.03 -
* indicates the experimental data eliminated by LMS regression.
First line * *
Second line -
* * *
5.69 1.66 - 0.23 -0.28 - 0.43 -0.10 - 2.81 - 1.47 - 1.46 0.46 0.37 - 0.80 - 0.10 - 0.42 - 0.38 - 0.46 - 1.28 - 1.86 - 3.02
M.C. Ortiz-Ferndndez, A. Herrero-Gutidrrez / Chemometrics and Intelligent Laborato~' Systems 27 (1995) 231-243
238
included; similarly, for the second step the last 10 points have been taken with the certainty that the equivalence volume is beyond the first of them. In such a way it is possible to detect outliers and the linear zone. As it is not necessary to assign experimental pairs to some step in the neighbourhood of the end point, the procedure for obtaining the straight lines is guaranteed to be objective. In Fig. 2a, associated with this titration, much curving can be seen. This is a consequence of the evolution of the chemical reaction, the Pb 2+ is electroactive and when CrO 4 is added, the prc-
Table 2 shows the experimental data of an amperometric titration in which 10 ml of Pb 2+ is titrated with CrO 2 0.1 M [15]. The second and the third columns give the pairs of experimental data v o l u m e / c o r r e c t e d current for 18 successive additions of titrant. Columns 4 and 5 show the standardized residuals given by the LMS regression for the linear segments before and after the end point. The asterisk indicates that this datum was not used in the LS regression. According to the procedure proposed for the first step, the first 12 experimental data that reach a volume of titrant beyond the equivalence point have been
Table 5 Experimental data and standardized residuals from the LMS regression. Cnnductometric titration of a mixture of 50 ml of HAc and 5 0 ml of HCI with N a O H 0.097 M (replication 2 in Table 1 ) Index
*
Titrant ml
Corrected conductivity
0
0.00
1.85
1
0.50
1.72
2
1.00
1.59
3
1.50
4
2.00
5
2.50
6
3.00
7
Standardized residuals mS
First line
Second line
Third line
- 0.50
-
0.06
-
1.34
-
1.48
0.50
-
1.35
0.39
-
1.23
0.06
-
1.10
- 0.50
3.50
0.97
-
1.28
-
-
8
4.00
0.85
- 0.50
-
-
-
-
-
-
-
-
9
4.50
0.74
1.85
10
5.00
0.65
7.62
*
11
5.50
0.61
22.24
*
12
6.00
0.62
13
6.511
0.63
14
7.00
0.67
-
15
7.50
0.69
-
16
8.00
0.73
-
17
8.50
0.75
-
5.32
*
-
-
2.52
x
_
-
- 0.26 0.47
-
-0.50
-
0.32
-
-0.59
18
9.00
0.79
-
0.32
19
9.50
0.82
-
- 0.50
20
10.00
0.85
-
-
11.50
-
21
10.50
0.88
-
22
11.00
0.94
-
-/).26 4.46
*
18.73 * 3.04 "
23
11.50
1.02
-
12.90
~
0.40
24
12.00
1.10
-
21.45
*
25
12.5(/
l. 19
-
-
- 3.67
26
13.00
1.28
-
-
1.38
-
-
1.83
27
13.50
1.37
-
28
14.00
1.45
-
- I).23
29
14.50
1.54
-
-/).41")
30
15.00
1.63
-
31
15.50
1.72
-
indicates the experimental data eliminated by LMS regression.
0.40
- 0.23 -
0.4[)
*
M.C. Ortiz-Ferndndez, A. Herrero-Guti&rez /Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
239
which step each point belongs, which is very compromising in the case of additions 9 and 10, for which LMS detects the linear tendencies independently of the intensity of the curvature. The experimental results obtained in one of the amperometric titrations of SO 2 with Pb 2+ 0.1 M are shown in Table 3. As in the previous case, there is notable curving around the end point, see Fig. 2b, caused by redissolution of the precipitate despite the fact that alcohol was added to the medium to avoid this [16]. The LMS regression detects the curve by identifying additions 8, 9, 10 and 11 outside the linear tendency of the first step and additions 8 and 9 outside the linear
cipitate forms, which causes a decrease in the recorded current. After the equivalence point, it is the CrO 2 that is electroactive. However, around the equivalence point partial solubilisation of the PbCrO 4 occurs, which causes an increase in the current recorded with regard to what is theoretically expected. The LMS regression detects this anomalous behaviour in relation to the linear tendency and diagnoses as outliers in the first step the three points which precede the end point (additions 9, 10 and 11) while in the second step it is the initial points, 9 and 10, which are diagnosed as being outside the linear tendency. No prior decision is needed to point out to
Table 6 Experimental
data and standardized
residuals
c o n t a i n i n g C u e+, B i 3~ a n d a p p r o p r i a t e Index
T i t r a n t b~l
from the LMS
amounts of EDTA Absorbance
regression.
Spectrophotometric
0.1 M as titrant ( r e p l i c a t i o n
Standardized First line
titration
of 100 ml of a mixture
2 in T a b l e 1)
residuals S e c o n d line
T h i r d line
0
0
0.0116
0.05
-
-
1
100
0.0122
0.96
-
-
2
200
0.0120
- 0.64
3
300
0.0126
0.26
-
-
4
400
0.0127
- 0.40
-
-
5
500
0.0130
-0.43 -
6
600
0.0134
0.16
-
7
700
0.0139
0.43
-
8
800
0.0136
- 1.48
9
900
0.0144
0.05
10
1000
0.0154
2.21
11
1100
0.0197
14.67 *
12
1200
0.0221
21.20 *
13
1300
0.0295
14
1400
0.0292
15
1500
16
3.86 *
-
1.59
-
-0.50
-
0.41
-
-0.41
-
3.32 *
-
-
0.05
-
0.0329
-
0.41
-
160/)
0.0363
-
0.50
-
17
1700
0.0387
-
18
1800
0.0441
-
1.59
-4.60
19
1900
0.0457
-
0.05
- 1.66
20
2000
0.0471
-
- 1.68
21
2100
0.0466
-
- 5.14 *
- 0.29
22
2200
0.0470
-
-
7.77 *
0.29
23
2300
0.0474
-
-
24
2400
0.0466
-
-
-0.88
25
251/0
0.0465
-
-
- 1.27
26
2600
0.0474
-
-
27
2700
0.0471
-
-
- 0.49
28
2800
0.0466
-
-
-
29
2900
0.0476
-
-
30
301/0
0.0477
-
-
* indicates the experimental
data eliminated
-
by LMS regression.
-0.32
-
0.88
0.88
0.29
1,66 0.10 0.10
*
240
M.C. Ortiz-Ferndndez, A. Herrero-Guti~rrez / Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
tendency of the second, as can be seen in Table 3. Table 4 shows the experimental results of one of the conductometric titrations of H A c with N a O H 0.097 M. The first 19 additions were taken to adjust the first segment of the titration curve and the last 19 for the second. The diagnosis of outliers made with the regression indicates that the first two additions lie outside the linearity as corresponds to the initial curve which shows the titration of weak acids; the last three are not accepted as part of the segment preceding the equivalence point. Three outliers are identified in the second step: addition 14, which belongs to the first; 20, presumably an incorrect addition of titrant and addition 32. Notice that addition 15 was accepted as belonging to both steps, which does not mean that this is the equivalence point; it is much more precise to estimate it using all the points which respond to the linear model for each segment. This is shown in Fig. 2c with the two lines fitted by least squares to the points diagnosed as being correctly aligned.
The proposed procedure does not depend on the number of rectilinear sections which make up the titration curve. Table 5 summarizes the data of the conductometric titration of a mixture of strong acid and weak acid titrated with N a O H 0.097 M. Again, the curve between the first and second sections shows the advantage of using the LMS regression. In this case, points 10, 11 and 12 are diagnosed as outliers for both segments. The same occurs with point 22 between the second and third segments. Furthermore, the presence of an outlier not related to any curve is detected, addition 25, which must be interpreted as an experimental error. In Fig. 2d one can see the intersection of the three sections adjusted by means of an LS regression, which gives rise to the determination of the end volumes corresponding to the HC1 and to the HAc respectively. The titration of various metals with E D T A (or another complexing agent) is one of the most interesting applications of spectrophotometric titration. Table 6 is an example of the successive titration of bismuth and copper in a single titra-
Table 7 E x p e r i m e n t a l d a t a a n d s t a n d a r d i z e d r e s i d u a l s f r o m the L M S r e g r e s s i o n . P o t e n t i o m e t r i c t i t r a t i o n o f 20 ml of a s o l u t i o n c o n t a i n i n g C z O 2 with H C I 0.22 M ( r e p l i c a t i o n 1 in T a b l e 1) Index
T i t r a n t ml
L,/[H +]
(V+ u)[H + ]
0 1 2 3 4 5 6 7 8 9 10 ll 12 13 14 15 16 17 18 19
0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8,00 8.50 9.00 9,50 10.00
31547.87 25118.87 21188.06 16635.28 13738.52 10402.11 7312.53 4488.07 2530.53 1285.19 871.69 673.21 579.31 507.10 451.91 410.28 397.57 366.64 361.17 346.73
0.0003 0.0008 0.0015 0.0026 0.0040 0.0066 0.0112 0.0214 0.0436 0.0973 0.1609 0.2317 0.2973 0.3727 0.4564 0.5460 0.6093 0.7119 0.7759 0.8652
Standardized residuals First line
* i n d i c a t e s t h e e x p e r i m e n t a l d a t a e l i m i n a t e d by L M S r e g r e s s i o n .
1.59 0.30 0.10 - 0.38 - 0.13 -0.07 0.10 0.38 1.03 2.00 3.33 * 4.76 * -
S e c o n d line
13.11 9.58 6.30 3.57 2.29
1.45 0.95 0.21 - 0.08 0.01 0.37 - 0.48 0.48 -0.33 0.01
* * * *
M.C. Ortiz-Fern~ndez, A. Herrero-Guti~rrez /Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
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M.C. Ortiz-Ferndndez, A. Herrero-Gutdrrez / Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
tion with E D T A 0.1 M. The titration curve (Fig. 2e) presents three segments of which the first and third are practically horizontal since, at the chosen wavelength of 745 nm, neither the bismuth complex, nor the E D T A absorb significantly. This explains why the coefficient of determination, which is the percentage of variance explained by the LS regression once anomalous data have been eliminated, is low for the first and third sections, as can be seen in Table 1. The detection of outliers shown in Table 6 also indicates this effect; the central section shares various experimental points with each of the other two segments. Several times during this work it has been highlighted that the proposed procedure does not require the prior determination of which experimental data should form part of any segment, or of one or two consecutive segments. The advantage of this is clearly shown in the case of a potentiometric titration to which the Gran's method is applied [17]. This method involves performing a different transformation on the experimental data (~', pH) according to whether one is dealing with the segment prior to the equivalence point or after it. The procedure based on the detection of outliers by LMS does not require this: both transformations are applied to all experimental data and each of the two responses are adjusted separately by LMS to define their linear pattern. Table 7 shows the data corresponding to the titration of a weak base (C2042-) with HC1 0.22 M. The potentiometric curve shows a barely appreciable jump that makes locating the equivalence point more difficult. Even when the data have been transformed its location is difficult, as can be seen in Fig. 2f. Table 8 summarizes the concentrations calculated from the end volume obtained as the intersection of the straight lines described in Table 1; all the replications of the titrations performed are noted. Some statistical data are also included in the same table which allow one to evaluate the accuracy in the determination of the concentration. The coefficients of variation obtained from 0.45% to 5.13% agree with what one would expect from each analytical technique a n d / o r type of analyte to be titrated in the mixtures. It should
be pointed out that the replications take into account the day to day variability. They were not performed consecutively and the equipment was started up for each case, thus simulating the day to day use of these analytical techniques even for mixtures and substances whose end point of titration is difficult to determine. The precision of the procedure is theoretically assured because the parameters of the straight lines found are the most exact; the methodological advantage of the process is clear with regard to the habitual method, based on adjusting the linear segments directly with the LS regression, since any anomalous point or curve in the neighbourhood of the equivalence point considerably modifies the slope and independent term and thus the concentration calculated.
6. Conclusions The robust regression method LMS provides an objective criterion in the determination of linear segments in titration curves. As a result, the proposed procedure allows the determination of an end point which can be reproduced both in the titration of pure substances as well as in that of mixtures under normal conditions. As no prior decision need be made by the analyst the process is capable of complete automation. What is more, it includes the detection of outliers produced by experimental errors or in the acquisition of data, thus removing their influence on the titration.
References [1] D.L. Massart, L. Kaufman, P.J. Rousseeuw and A. Leroy, Least median of squares: a robust method for outlier and model error detection in regression and calibration, AnaIvtica Chimica Acta, 187 (1986) 171-179. [2] P.J. Rousseeuw, Tutorial to robust statistics, Journal of Chemometrics, 5 (1991) 1-20. [3] P.J. Rousseeuw, Least median of squares regression, Journal of the American Statistical Association, 79 (1984) 871-880. [4] P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection, Wiley, New York, 1987. [5] M.C. Ortiz, J. Arcos, J.V. Juarros, J. L6pez-Palaciosand L.A. Sarabia, Robust procedure for calibration and cal-
M.C. Ortiz-Ferndndez, A. Herrero-Guti&rez / Chemometrics and Intelligent Laboratory Systems 27 (1995) 231-243
[6]
[7]
[8]
[9]
culation of the detection limit of trimipramine by adsorptive stripping voltammetry at a carbon paste electrode, Analytical Chemistry, 65 (1993) 678-682. M.C. Ortiz, J. L6pez-Palacios, M.J. Arcos, L.A. Sarabia, M.G. Piangerelli and D. Cingolani, Determinaci6n del Tiempo de Transici6n en Cronopotenciometr{a mediante Regresi6n Robusta, l I | Jornadas Cientfficas de Electroquimica del Grupo de la S.E.Q.A. y R.S.E.Q., Valladolid, Comunicaci6n 5-3, 1991. A. Herrero, M.C. Ortiz, J. Arcos and J. L6pez-Palacios, Optimization of the experimental parameters in the determination of Cu(II) by differential pulse anodic stripping voltammetry and evaluation of the characteristic curves of detection, Analyst, 119 (1994) 1585 1592. I.M. Kolthoff and P.J. Elvin, Treatise on Analytical Chemistry, Part L Theory and Practice, Vol. 2, Wiley, New York, 2nd edn., 1979. A,1. Vogel, Textbook of Quantitative b~organic Analysis, Longman, London, 4th edn., 1978.
243
[10] P.H. Rieger, Electrochemistry, Prentice-Hall, Englewood Cliffs, N J, t987. [11] E.A.M.F. Dahmen, Electroanalysis. Theory and Applications in Aqueous and Non-Aqueous Media and in Automated Chemical Control, Elsevier, Amsterdam, 1986. [12] N. Draper and H. Smith, Applied Regression Analysis, Wiley, New York, 2nd edn., 1981. [13] R.D. Cook and S. Weiberg, Residuals and Influence in Regression, Chapman and Hall, New York, NY, 1982. [14] F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, Robust Statistics. The Approach Based on Influence Functions, Wiley, New York, 1986. [15] J. Heyrovslo) and P. Zuman, Practical Polarography, Academic Press, London, 1968. [16] J.T. Stock, Amperometric Titrations, Krieger, New York, NY, 1975. [17] O. Budevsky, Foundation of Chemical Analysis, Ellis Horwood, Chichester, 1979.