Quant Regression Equations
This document summarises the equations used by Analyst to calculate the regression
curves used for Quantitation.
In the equations given below, “x” represents the analyte concentration for standards and
“y” represents the corresponding peak area or height. More precise definitions are given
in the following table where:
Ca = actual analyte concentration
Cis = internal standard concentration
DF = dilution factor
Aa = analyte peak area
Ais = internal standard peak area
Ha = analyte peak height
His = internal standard peak height
IS Used?
yes
no
Area Used?
yes
no
yes
no
x
Ca / Cis / DF
Ca / DF
y
Aa / Ais
Ha / His
Aa
Ha
Weighting Factors
The following table shows how the weighting factor (w in the equations below) is
calculated for each of the seven weighting types.
Weighting Type
None
1/x
1 / x2
1/y
1 / y2
ln x
ln y
Weight (w)
Always 1.0.
If |x| < 10-5 then w = 105, otherwise w = 1 / |x|.
If |x| < 10-5 then w = 1010, otherwise w = 1 / x2.
If |y| < 10-8 then w = 108, otherwise w = 1 / |y|.
If |y| < 10-8 then w = 1016, otherwise w = 1 / y2.
If x < 0 an error is generated, otherwise if x < 10-5 then w = ln 105,
otherwise w = |ln x|.
If y < 0 an error is generated, otherwise if y < 10-8 then w = ln 108,
otherwise w = |ln y|.
Regressions
This section gives the equations for each of the regression types. In the equations below
x, y and w are as defined above. All sums are calculated over all standards (with the
exception of those standards which are marked as “not used”).Linear
The linear calibration equation is:
y=mx+b
The slope and intercept are calculated as:
m = ( w wxy - wx wy ) / Dx
b = (  wx2 wy - wx wxy ) / Dx
and the correlation co-efficient is calculated as:
r = ( w wxy - wx wy ) / ( Dx Dy)
where:
Dx = w wx2 - ( wx )2
Dy = w wy2 - ( wy )2
Linear Through Zero
The liner through zero calibration equation is:
y=mx
The slope is calculated as:
m = wxy / wx2
and the correlation co-efficient as:
r = wxy / ( wx2 wy2 )
Mean Response Factor
The mean response factor calibration is:
y=mx
This is the same equation as for the linear though zero case. However the slope is
calculated differently as:
m = wy/x / wand the standard deviation as:
 = ( nD / ( n - 1 ) ) / w
where:
D = w * wy2/x2 - ( wy/x )2
Note that points whose x value is zero are excluded from the sums.
Quadratic
The quadratic calibration equation is:
y = a2 x2 + a1 x + a0
The polynomial co-efficients are calculated as:
a2 = ( b2 / b0 – b5 / b3 ) / ( b1 / b0 – b4 / b3)
a1 = b5 / b3 – a2 b4 / b3
a0 = ( wy – a1 wx – a2 wx2 ) / w
and the correlation co-efficient is then calculated as:
r = (1 – w * [w(y – a2x2 – a1x – a0)2] / Dy)
where:
Dy = w wy2 – ( wy )2
b0 = wx / w - wx2 / wx
b1 = wx2 / w - wx3 / wx
b2 = wy / w - wxy / wx
b3 = wx2 / wx - wx3 / wx2
b4 = wx3 / wx - wx4 / wx2
b5 = wxy / wx - wx2y / wx2
Power Function
The power function calibration equation is:
y = a xp
The equations for the linear calibration are used as described above to calculate the slope
(m) and intercept (b), except that “x” in those equations is replaced by “ln x” and y
replaced by “ln y”. When this is done a, p and the correlation co-efficient (r) are
calculated as:
a = exp ( b )
p=m
r=r