# Does my data show quadratic regression in a calibration curve?

This data was taken for my science fair project, and I would like to know the relationship between spectrophotometer absorbance and cell density (taken manually) of algae after a toxicity test. The different levels of toxicity are IC 0 (no chemical), IC 25, IC 50, IC 100 (the most chemical). The IC value is basically the estimated about of the chemical needed to destroy a certain amount of the population. I took the manual density and absorbancy of four samples of different levels of toxicity.

IC value/Cell densities(cells/ml)/absorbancy:

IC 100/28000/0.408

IC 50/42600/0.302

IC 25/70000/0.01

IC 0/138000/0.79

Based off of these points, I would like to make a calibration curve. But when I do this, the highest r^2 is quadratic. Is this an acceptable calibration curve for finding other densities based off of known absorbances? Or would it have to be linear with a very low r^2?

• Using higher order interpolations (such as a quadratic) will generally provide a larger correlation coefficient; however this does not mean the quadratic is a better fit unless there is a theoretical reason for why the dependent and independent variables should be related through a quadratic equation. In your case, absorbance is typically linearly related to concentration. – bobthechemist Dec 28 '13 at 12:18

As you have correctly assumed, the relationship between absorbance and absorbant concentration is linear (law of Lambert-Beer): $$A = \varepsilon \, l \times c$$

This should give you a straight line through (0,0) and your data.

Now, for a graphical analysis of data, let's not look at the numbers themselves but on what kind of a graph it gives us. So after 2 minutes with gnuplot I get something like this: Now, there are two things that I note here:

1. The absorbance of the $\mathrm{IC_{25}}$ value is very very low (nearly 0) which indicates some wrong measurement. Maybe the cells weren't in the pathway of the light beam, only the medium?
2. You have four data points.

My advice is the following:

• Double check your measurements (the absorbance and the cell density). It would be even better if you did several measurements for the different inhibitor concentrations and then calculate the mean (and standard deviation if you so desire).
• Generate more data. It is very wonky to try and justify a linear regression through these 4 data points (although mathematically allowed). However, if all intermittent points are equally chaotically distributed along a straight line the linear relationship still holds (with a fairly high $R^2$). What this does to the interpretation of the data is a completely different story.
• Unfortunately, I don't have enough time to take more measurements. What do you mean about the intermittent points? Is there any relationship that I can use here? – Naomi Gruber Dec 28 '13 at 18:58
• With intermittent points I mean more measurements that eventually might give you the linear relationship. Since you don't have enough time to make more measurements, I suggest you point out the problems I wrote above and call it a day. It's not satisfying but the only way I see here. Maybe talk to your supervisor about it? – tschoppi Dec 29 '13 at 15:45

Your absorbency values seem anomalous compared with the cell density. One would assume that as cell density increased, so would absorbency. Your absorbency decreases first and then leaps up to a maximum. I can't say from your data whether there is any relationship between cell density and absorbency. On the other hand, there seems to be an inverse-square relationship between IC value and cell density. Hope that helps.

• What is an inverse-square relationship? – Naomi Gruber Dec 28 '13 at 18:56
• There is an inverse-square relationship between the force of gravity and the distance between two masses. In other words, $F=\frac{k}{r^2}$ This link has more information: csep10.phys.utk.edu/astr162/lect/light/intensity.html – Greg Dec 29 '13 at 4:04