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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