Linearity, more than r2: a practical example

Abstract

The work intended to show through a simple practical example, how
the evaluation of the coefficient of determination r2, cant be considered the best
parameter to assess the linearity of an analytical procedure, and its single determination
is not enough for this purpose. A sandwich ELISA assay, employed to control
the process of production of IFNa2bHuLe was chosen as the model for the
study. Four different curves were tested in three assays, done in different days and
by different operators. The data were analyzed using the linear regression method.
The Logs of concentrations (LogC) were plotted versus the quotients of absorbances
and concentrations (OD/C), for each
curve and studied concentration. As a
complementary method to measure the
goodness-of-fit to linear model, the concentrations
of different points of the assayed
curves were estimated, interpolating
its mean absorbances in the regression
curves, and the calculated values
were compared with the theoretic,
expected values, through the percentages
of recovery. Although the calculated
r2 were > 0,95 for all of the assayed
curves, there was no lack of fit, with significance
levels of 5 and 1 %, only for
two of them. However, only for one of
these linear, well adjusted curves, the
95 % confidence interval of the slope was
included in the range of tolerance for its
variation in the linear range, defined by
the ASTM as ± 5 %. The plot of linearity
allowed visualizing the variations of the
slope, and the associated lack of fitting
to the linear model, detected for two of
the assayed curves. It was shown how
for the extremes values of concentration
of these two curves; the slope (OD/C) had
bigger variations than for concentrations
in the central zone of the curves.
This variation of the slope for the smallest
evaluated concentrations in these
curves, leads to bigger errors when the
linear regression curves are used to estimate
concentrations in this zone,
which is a result of the lack of fitting.
This lack of fitting was very well visually
expressed in the low percentages of
recovery obtained when the smallest
concentrations of the curves were estimated
through its lineal regression
curves. The plot of linearity is useful for
the evaluation of linearity, allowing visualizing
the trend of the slope in the
range of explored concentrations, and
offering a first idea about the lineal
range of the method. Although the calculation
of r2 although is the most frequently
used method in the check-up
of linearity, offering a measure of the
quality of the adjustment that isachieved with the regression, is not
enough for this purpose. The analysis
should be supplemented with some
other alternatives, as the plot of linearity
first, the calculation of the percentages
of recovery for the different concentrations
of the curve, and finally, confirmed
through a test for measuring the
goodness-of-fit to a linear model.