How to determine if two calibration curves are parallel, within 5%

Question

I have complex matrices and I need to make sure the matrix does not affect the measurement of my analyte. To do that, I need to prepare samples of my analyte in a solvent and of the matrix reinforced with my analyte. Both curves are going to have 5 concentration points, each done 3 times and, for sake of argument, let's assume both have comparable levels of precision. Then, I linearize these two calibration curves and check if they are parallel, which is where my question lies.

How do I prove that both curves are parallel (meaning no matrix effects are present), within a margin of 5%?

I have found the following method:

1. Get the y-data of the matrix data and apply them to the calibration function of the solvent and get an "apparent concentration" of the matrix.
2. Perform a linear regression of the concentration values of the solvent with the apparent concentrations of the matrix.

Then, I have an intercept a and a slope b. A value of b different from 1 indicates a concentration dependent bias. I have both standard deviations of the fit, stdev_a, stdev_b, so I have to determine the confidence interval (95%) of b and check if their values are withing these confidence intervals.

I have found two ways to do that in Excel and I am unsure which one is correct. For example, the slope:

• Do I use the conf = CONFIDENCE.T(0.05, stdev_b, N-4), get b+-conf, and check if 1 is within this interval b+-conf?
• Do I use t = T.INV(0.95,N-4), multiply that by the stdev_b, get the range b+-t*stdev_b and check if 1 is within this range?

I have used N-4 for the degrees of freedom, since it's two linear regressions, each consuming one degree of freedom, correct? In this case, N-4 == 11.

In summary:

• Which function do I use to calculate the confidence intervals for this example in Excel, and
• Is this method of checking parallelism correct?
• Is there an easier/simpler or more correct way of doing this parallelism check?

Answer 1

Define the angular coefficients of the two resulting lines and evaluate the parallelism between them. Perform the statistical evaluation of the data. For the parallelism test between the curves, considering the equations of the lines to be compared as: $y_1 = a_1 + b_1x + \varepsilon$ and $y_2 = a_2 + b_2x + \varepsilon$, the hypothesis for comparison of the lines slopes is given by: $$H_0 : b_1 = b_2$$ $$H_1 : b_1 \neq b_2$$ When the null hypothesis ($H_0$) is true, the regression lines become: $$y_1 = a_1 + b_3x + \varepsilon$$ $$y_2 = a_2 + b_3x + \varepsilon$$ Considering a significance level of $5\ \%$, the null hypothesis will be true when the test returns a P-Value above $0.05$.

More information in the link: http://www.real-statistics.com/regression/hypothesis-testing-significance-regression-line-slope/comparing-slopes-two-independent-samples/

Answer 2

I think that this is one of those in-between questions. Before migrating, let's define the chemistry better.

(1) Roughly what sort of precision are you expecting? What sort of instrumental precision, what sort of wet chemistry precision, and what sort of sampling precision?

Trying to figure if one error source dominates, or if all three contribute significant error.

(2) You said "Both curves are going to have 5 concentration points, each done 3 times..." I assume you made three different solutions of the sample, then spiked 4 different aliquots each with 4 different amounts of the analyte. Thus you end up with 5 solutions for some sort of instrumental analysis. Correct?

So the 15 solutions analyzed depend on three weighings of sample. Also I assume that you're using 4 different volumes. In other words you the spike an aliquot with a one ml pipette, and the second spike is with a two ml pipette, not two shots from the one ml pipette, and so on...

So I assume that you assume that the spiking is exact and you're not worried about the spiking errors.

A point here is that you seem to accept that the intercepts of the lines will be different. So is there something in the sample matrix that causes the blank value to vary? So although you have 5 points on the line, you seem to only have three different matrix solutions.

(3) When you say you "linearize these two calibration curves", I assume you mean that you used linear least squares to compute slope and intercept of each.

(4) Exactly what do you mean by 5% error.

I assume that you mean a 5% chance of a type I error in slope. But I don't think that is what you want to test. I think you spiked the sample solution with 4 different concentrations of the analyte. So what you really want to know is the difference between the sample values at the end of the line, not variation in the slope itself.

The difference between the non-spiked values will be a lot more sensitive than test on variabilities. You have very small statistics and the variabilities are going to be huge.

The gist here is also considering a type II error and/or using multiple determinations to drive the error to 5%.