# Understanding the Linearity of a Calibration Plot

The essence of
a calibration plot is the linear relation between two variables. How do you
decide the degree of linearity to rely on interpretations based on the
calibration

plot. The points on a calibration line will seldom fall on a
perfect straight line. Normally you would use your ruler to draw a straight
line which passes through most of the points.
Our earlier
article Guidelines on generation and interpretation of calibration plots dealt
with considerations for establishing reliability of calibration. It also dealt
with common mistakes that should be avoided in generation and interpolation or
extrapolation of the calibration plots.

Two measures
of expressing linearity of relationship between two variables are :

Correlation
analysis – which applies to two independent factors X and Y i.e if X increases
does Y also increase, decreases or does not change at all, and

Regression
analysis – a change in X will have a corresponding change in Y but change in Y
will not have change in X.

In a
calibration plot where all points do not fall in a straight line the linear
regression is applied. Line of regression minimizes the distance of residuals
in the Y direction between the line and the individual points and passes
through the centroid (mean values of X and Y) of the data.

The linear regression
line uses method of least squares to establish the relationship between two
variables as the best fit straight-line. Most instruments yield a linear
response only over a specific concentration range beyond which the response is
nonlinear. Choice of correct region is important to minimize errors due to
nonlineraity

Coefficient
Correlation, r

Linear
coefficient expresses degree of linearity between two variables X and Y. It
lies in this range +1 to -1. The positive sign refers to a positive linear
relationship between the two variables and negative sign refers to a negative
relationship between the variables.

r value close
to 1 indicates a strong positive relationship that is an increase in value of X
is accompanied by a corresponding increase in Y.

r value close
to -1 indicates a strong negative correlation that is an increase in value of X
is accompanied by a corresponding decrease in value of Y.

r = 0
indicates that there is no relation between the two variables.

r = -1
indicates perfect negative relation between the variables.

Coefficient of
Variance, r^2

Often instead
of r the coefficient of variance r^2 is used. It indicates the percentage of
variation in Y associated with variation in X

r^2 lies
between 0 and 1

For example if
r= 0.98 then r^2 is 0.96 which means that 96% of the total variation in Y can
be explained by the linear relationship between X and Y. The remaining 4% of
variation in Y remains unexplained

r^2 is a
measure of how well the regression line represents the data

Source: lab-training

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