# The difference between a referenced & an experimental data set (both 20 data). Application: protein quality (PDCAAS / AAS) in Nutritional Science

I'll start with the actual statistical question, which is rather short. Then, I will expand a bit by showing the context from which the question is coming (which is nutritionology).

# The more abstract statistical question

The number of data couples is 20 (the context is about amino acids, which can differ in number, depending on the scientific definition; but in the nutritional course sketched below, the amount of amino acids has been numbered 20). Each data couple exists of 2 numbers: 1 is a reference & 1 is an experimental number.

The point of the statistics here is to find a mathematical measure to say something about the difference from the experimental number set to the referenced number set, by comparing each of their experimental numbers to their respective referenced numbers.

So, the statistical goal; again, in other words; is to "measure" the difference between 20 data (from the experimental number set) to their respective 20 data (from the referenced number set); and to say something about the global difference.

So, considering I am not a statistician; one of the simplest measures come to my mind. Such as to calculate the difference between each of the number-couples of the data sets & then to sum all of these differences for each data-couple. Then, perhaps, to extrapolate big "outliers" / "peaks", one could multiple (e.g. by 1.5) for each number-couple that is further apart then, let's say a certain amount of mg's.

A small question on the side: could this simple measure above be taken by taking the integral of the referenced number set & taking the integral of the experimental number set; and then take the difference of them. Or is taking an integral not defined for a discrete (not continuous) number set? Rather, should I prefer the solution from the paragraph above (which you might perhaps, in a loose-weave, call an "integral"-counterpart for discrete mathematics)?

Rather, thus; the question is: is there a better way to get a mathematical measure to find a desired measure which I have tried to describe? Which part of elemental statistics am I missing to come up with a more elegant solution?

# The nutritional context (perhaps not necessary to answer the statistical question)

The context where this question comes from is nutritional science, via following a course on edX called Introduction to Nutrition – Food for Health. To my limited knowledge, this is one of the most excellent MOOCs on nutritional science at the moment, together with "Food for Thought", hosted at the same platform.

To sketch the context, I will show a question which I posted on the discussion-forum of this first MOOC (which is normally visible to enrolled students). The main professor of the course was "Sander Kersten", which is a Professor of Molecular Nutrition at Wageningen University & an Adjunct Professor at Division of Nutritional Sciences at Cornell University. He has answered my original question, as can be read below. The numbers (like 6.7 refer to sections in the MOOC).

The context is about measuring protein quality.

In short: the mathematical measure is calculated as follows. There is an "average" human body protein which is taken as a reference. Then, for another protein to be scored, its amino acid composition is compared to this reference; and there is looked for its (limiting) amino acid which diverges the most (to less mg's of amino acids in the test protein) from the referenced mg's of this amino acid. Then, there is corrected for digestibility; but this is less important for the question which I am then sketching. Namely: I wondered whether it really was the case that the PDCAAS (or the AAS) didn't incorporate multiple limiting amino acids & then: whether there couldn't be found a better statistical measure to speak about the general protein quality of a protein, in reference to the amino acid composition of a referenced protein.

Here is the full original context, which reveals all elements needed (to my knowledge) to understand the context:

Introduction to Nutrition – Food for Health (edX) >> Discussion >> Week 6: Proteins and Health >> Questions on Proteins and Health >>

_______________(asked ± 2/2015 by Vincent Verheyen)_______________

## Measures of protein quality: PDCAAS doesn't incorporate "multiple" limiting amino acids?

We saw in 6.7: Reading: limiting amino acid:

The limiting amino acid is the amino acid that is least abundant in a dietary protein source in comparison with the average human body protein.

Then we saw in 6.7: Reading: measures of protein quality, that a current way of measuring protein quality is the:

PDCAAS (Protein Digestibility Corrected Amino Acid Score).

And we were explained that:

The Amino Acid Score (AAS) for any particular protein is calculated by comparing the level of the limiting amino acid in the protein in question to the level of the same amino acid in a reference protein. The PDCAAS is the AAS corrected for digestibility. (Thus, the formula for calculating the PDCAAS is AAS x digestibility where AAS equals mg of limiting amino acid in 1 g of test protein / mg of same amino acid in 1 g of reference protein). The PDCAAS has a maximum value of 1.0 (or 100). Proteins having values higher than 1.0 are rounded off to 1.0.

So, then I began reading the suggested article: Protein quality evaluation twenty years after the introduction of the protein digestibility corrected amino acid score method, since I found it strange that this measure PDCAAS seems not to incorporate the difference in quality between e.g. the following types of protein:

• a protein that has 1 amino acid that is very poor in 1 amino acid.
• another protein that is very poor in 2 amino acids.

I haven't read the article all the way through, but I've read all the issues with their measure PDCAAS as well as its parent measure AAS that are noted in S186-S187, or Table 1; and this table doesn't seem to explicitly refer to this limitation.

Am I not understanding PDCAAS correctly, or is there indeed a lot of room for improvement of their measure?

As I understand it now, this is not assessed by the PDCAAS, but I would argue that incorporating such "multiple limiting amino acids", so to speak; might give a more reasonable overall assessment of the protein quality; while at the same time, I understand of course that metrics can use arbitrary definitions, each providing different information.

_______________(answered ± 2/2015 by Sander Kersten)_______________

You correctly point out that PDCAAS and AAS only take into account the one limiting amino acid. A certain protein may be low in more than one amino acid but only one amino acid can be truly limiting.