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AGUSTÍN GARCÍA ASUERO AN. R. ACAD. NAC. FARM.

interest in the field of chemical and physical measurements, which are widely
applied in the pharmaceuticals and related sciences.

Key words: Least squares.—Replicated observations.—Straight lines.

EXTENSIVE ABSTRACT

Calibration, method comparison and parameter evaluation in chemical
and pharmaceutical analysis

Several aspects of least squares, particularly in regard to the use of replication,
error analysis and weigthing and data transformations, appears to be poorly un-
derstood by a number of experimenters. Examples are given from older and more
recent literature where experimental data were processed in an incorrect way from
the point of view of statistics. As a matter of fact, however, the statistical methods
most commonly misapplied by analytical chemists are correlation and regression.
There is no doubt of the importance of these topics, which are closely related to
very basic operations, e.g., calibration and the comparison of two analytical me-
thods applied to a range of test material. In analytical chemistry as well as in other
quantitative sciences it is often necessary to fit a mathematical equation or model
to experimental data. Common situations that may be described by functional
relationships include calibration curves relating measured value of response to a
property of material, comparison of analytical procedures, relationships in which
time is the x-variate and parameter estimation methods. Parameters of the appro-
ximating function, however, are frequently derived using the least-squares metho-
dology.

To demonstrate that a least squares criterion is valid it is necessary to assume:
i) that the errors, ei, are random rather than systematic, with mean cero and
variances si2 = s2/wi (where s is a constant and wi is the weight of point) and follow
a gaussian distribution; this distribution is so common that is also referred to as
the normal one; ii) that the independent variable, i.e., x, the abscissa, is known
exactly or can be set by the experimenter either; iii) the observations, yi, are in an
effective sense uncorrelated and statistically independent, i.e. for cov(ei, ej) = 0 for
i = j, with means equal to their respective expectaions or true values, E{yi} = ?i; and
(iv) that the correct weights, wi, are known. The least squares criterion gives in-
deed poor results, however, if the observations are incorrectly weighted or if the
data contain «outliers», i.e., very poor observations at higher frequency than allo-
wed for by the normal distribution. When the conditions are met, the parameter
estimates found by minimization of a least squares criterion are best unbiased
linear estimates of the regression parameters.

Real data are often subject to problems that make the use of classical statistics
based on the normal distribution, difficult. The main practical problem probably
is the occurrence of outliers. Another difficulty can be that the distribution of the
data is not normal. The normality assumption is, in fact, quite reasonable to expect
the yi to be independent in many situations if they are the results of separate

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