## Polynomials : do not accept them !

A great many Agricultural ‘response’ experiments, whether they are responses to fertilisers or to seeding rates, have perhaps 6 treatments at most; the trial area is gobbled up by the (generally 3) replicates of each treatment.

The few treatments and the implicit statistical errors (noise) very often generate a pretty murky picture.  To clear things up a bit smooth curves are often put through the data points.

Sadly and all too often the curves found in spreadsheet programs are used   No one can blame field workers ; everyone is in a hurry these days and everyone knows how to use a spreadsheet and not many people have either the time,training or finances to put their results into a curve fitting program.

The problem:  most crops respond to increases in quantity of supply in a diminishing way ; The first amount produces half the full response ,the next amount a quarter of the response , the next an eight an so on and so on. Often known as a ‘Law of Diminishing returns’ . Unfortunately the ‘trend line’ functions given to us in spreadsheets cannot easily replicate this.

The most commonly fitted spreadsheet trend line is the ‘polynomial order 2’  This curve is the trajectory of canon balls flying through a vacuum (a parabola)- it is symmetrical and always wishes to return to the ground IE zero yield.

In the graph below the red line is a (perfect!) Law of Diminishing returns crop response (red points and curve)  fitted by  a ‘polynomial order two’ IE a Parabola (the black curve):

The actual data suggests that the crop does not respond in any meaningful way to more than 200 units of inputs , yet the polynomial suggests that  ~325 units are needed. ie 50% of the expensive inputs are wasted 😦

The ‘Law Of Diminishing Returns’ (LODR) was elucidated over 100 years ago and today we realise that often an additional (linearly) increasing or decreasing amount has to be added to to it to match the crop responses that experimenters see.

The example below shows a crop that perhaps is fallen flat when the seeding rate gets too thick –  the data follows the LODR but has an amount removed from its yield that is proportional to the seeding rate:

In this case the polynomial order two overestimates the seeding requirement for maximum yield by a factor of about two ( data; max yield at ~125 seeds. Polynomial fit max yield at ~250 seeds). The symmetrical canon-ball trajectory

In other cases the yield can keep increasing with increasing seeding rate or nutrient supply , perhaps as a result of weeds or a lack of foresight on the part of the experimenter:

In this case the  ‘polynomial order 2’ underestimates the required seeding rate.

I hope that you can see from these examples that the Polynomial fit gives highly misleading results. Sadly it is all to easy to use 😦

What you may also notice is that the polynomial’s maximum hardly changes in any of these three very contrasting crops !!

NB:  higher order polynomials are sometimes used but with sparse and noisy agricultural data these often behave erratically and lead to obviously and visually objectionable  fits.