Confidence limits for the abscissa of intersection of two least-squares lines such as linear segmented titration curves

Carter, Kenneth N.; Scott, Dan M.; Salmon, Jon K.; Zarcone, Gregory S.

doi:10.1021/ac00013a017

Cited by 11 publications

(8 citation statements)

References 26 publications

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“…25 To obtain the coverage interval, it is necessary to solve an equation that depends on the parameters of the two regression straight lines, i.e. the slopes and their associated variances, the independent terms and their associated variances and a covariance term, because there is a correlation between the slope and the independent term in each one of the straight lines:…”

Section: Analytical Sciences October 2003 Vol 19mentioning

confidence: 99%

“…The standard deviation 19 associated with the maximum signal value, ss,max, in the estimated polynomial curve is calculated by: (24) where sresid is the corresponding residual standard deviation, Xo is the matrix formed by the independent variables (x, x 2 , x 3 , ···) when substituting the value of the maximum, Xo T is the transpose matrix of Xo: (25) and the matrix (X T X) -1 is obtained from the variance-covariance matrix (V) of the regression coefficients: (26) where the main diagonal (sa 2 , sb 2 , sc 2 , ···, sz 2 ) represents the variances associated with the regression coefficients (a, b, c, ···, z, respectively) and the rest of the matrix is composed of the covariances among these coefficients.…”

Section: Analytical Sciences October 2003 Vol 19mentioning

confidence: 99%

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Data Analysis in the Determination of Stoichiometries and Stability Constants of Complexes

et al. 2003

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The determination of stoichiometries and stability constants of complexes by means of UV-visible spectrophotometry applying traditional methods does not evaluate the quality of the values obtained, since the classic application of these methods does not provide coverage intervals. However, the use of chemometric techniques in different steps of the application of these methods makes it possible to obtain not only a real value of the characteristics of the complexes but a validation of such a value. In this paper a methodology is proposed that combines some traditional methods, three different regression models (LMSR, LSR and LSPR) and a small number of mathematical algorithms. This methodology is able to estimate, in a simple and rigorous way, the stoichiometry and stability constant of a complex and its corresponding uncertainties.

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Section: Analytical Sciences October 2003 Vol 19mentioning

confidence: 99%

Section: Analytical Sciences October 2003 Vol 19mentioning

confidence: 99%

Data Analysis in the Determination of Stoichiometries and Stability Constants of Complexes

et al. 2003

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“…If the ordinates y i are assumed to have Gaussian (normal) distribution, the least squares parameters as well as Δa, and Δb are also normally distributed [68]. However,x I , which even is regarded as the ratio of two normally distributed variables, is not normally distributed and, indeed, becomes more and more skewed [69] as the variance levels increase. For sufficiently small variance though,…”

Section: Confidence Interval On the Abscissa Of The Point Of Intersecmentioning

confidence: 99%

“…Note that becausex I involves the ratio of random variables, first-order propagation of variance is not exact [69]. Evidently,x I is a random variable not normally distributed unless sðx I Þ is small enough.…”

Section: Confidence Interval On the Abscissa Of The Point Of Intersecmentioning

confidence: 99%

“…Conductometric titration of 100 mL of a mixture of acids with potassium hydroxide 0.100 M Table 5 shows the data [conductance (1/R), volume (x)] published by Carter et al [69]; Schwartz and Gelb [65]) and corresponding to the conductometric titration of a mixture of acids, perchloric acid and acetic acid with potassium hydroxide 0.100 M as titrant agent. The points recorded belong to the three branches of the titration curve; the first (branch A) corresponds to the neutralization of perchloric acid, the second (branch B) to the neutralization of acetic acid, and the third (branch C) to the excess of potassium hydroxide.…”

Section: Application To Experimental Systemmentioning

confidence: 99%

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Intersecting Straight Lines: Titrimetric Applications

Martín¹,

Martin²,

Asuero³

2017

Advances in Titration Techniques

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Plotting two straight line graphs from the experimental data and determining the point of their intersection solve a number of problems in analytical chemistry (i.e., potentiometric and conductometric titrations, the composition of metal-chelate complexes and binding interactions as ligand-protein). The relation between conductometric titration and the volume of titrant added leads to segmented linear titration curves, the endpoint being defined by the intersection of the two straight line segments. The estimation of the statistical uncertainty of the end point of intersecting straight lines is a topic scarcely treated in detail in a textbook or specialized analytical monographs. For this reason, a detailed treatment with that purpose in mind is addressed in this book chapter. The theoretical basis of a variety of methods such as first-order propagation of variance (random error propagation law), Fieller's theorem and two approaches based on intersecting confidence bands are explained in detail. Several experimental systems described in the literature are the subject of study, with the aim of gaining knowledge and experience in the application of the possible methods of uncertainty estimation. Finally, the developed theory has been applied to the conductivity measurements in triplicate in the titration of a mixture of hydrochloric acid and acetic acid with potassium hydroxide.

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