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MERN calculation and the criteria to determine the optimum model when fitting

I wrote a blog post about the models used to fit the data of yield nitrogen fertilizer response in my winter cereal project:

There are five models in it: quadratic model, exponential, square root, quadratic plateau and linear plateau model. However, the reason why we need to fit these models is that we need to calculate the MERN (Most Economic Rate of Nitrogen), and based on different models, the most MERNs are various.

  • The following is the general way to calculate the MERN:

where N refers to the nitrogen fertilizer rate, Y(N) refers to the corresponding yield under that nitrogen fertilizer rate, p is the assumed unit price of dry biomass of forage yield, and w is the assumed unit price of the nitrogen fertilizer. Let


F(N) is maximized when the first derivative of F(N) equals zero, so


so that MERN is calculated by solving this equation:

  • The necessity of selecting the optimum model:

an example of how MERNs are various based on different models, using the data from a site.

various MERN

The MERNs range from 94.23 to 68.52, and the yield at MERN ranges from 1.079 to 1.101 in this site, and the yield’s range can be much bigger in other sites. The diversity of MERNs suggests that in some cases, the farmers may apply too much nitrogen fertilizer using improper model, which would lead to the waste and pollution of N fertilizer, while in other cases, the farmers may apply too little nitrogen fertilizer using improper model, which would result in less yield of forage.

  • The criteria to determine the optimum model:

Traditionally, a lot of researchers tend to use the coefficient of determination to measure the goodness of these models, but many papers point out that in some cases, the R square of these models are very close and all high enough. For instance, if the R squares are 0.8543, 0.8678, 0.8462, 0.8514, 0.8401 respectively, there’s no significant difference between them, and they are all close to 1. It’s not a good idea to draw a conclusion of which is better directly from the coefficient of determination.

Similarly, some use criteria like root mean square error, residual mean square etc. , which have similar function and magnitude with R square.

In general, one important criterion to evaluate a regression model is its normality, and thus the normality of the residuals, which is the observed yield minus predicted yield based on the regression model, is regarded as a significant criterion in some paper.

residuals when fitting a square root model

In the plot above, the data of residuals are plotted with the corresponding nitrogen treatment. If the data point is above the x-axis, then the regression model tends to underestimate the yield, while if the point lies beneath the x-axis, then the model tends to overestimate the yield. Therefore, under the same treatment, when the data points lie symmetrically between two sides of the x-axis, the normality of the residuals are good enough, or the regression model has systematic bias.

deviation from mern

Also, after calculation of MERN, some use the nitrogen treatment deviation from MERN instead of nitrogen treatment as the x-coordinator. It has similar function, but focuses more on whether there’s any systematic bias around MERN.

In addition to using residual plots to visualize the goodness of the normality, some use Shapiro-wilk test to quantify the normality of the residuals. The null hypothesis of this test is that the population is normally distributed, so the more the p-value is close to 1, the better the normality is. Similarly, some others use Kolmogorov test to compare the normality, but this test is thought to be less powerful than Shapiro-wilk test.

Eventually, some would calculate the potential economic loss. The general steps of calculating potential economic loss are followed:

  • Assume that one model is the correct model, calculate the MERN and yield at MERN based on this “correct model”
  • Calculate the profit of the “correct model” by using the formula: (yield at MERN*price of forage)-(MERN*price of N fertilizer)
  • Calculate the profit of the other models by using the same formula. The MERN here is still based on the “correct model”, but the yield at MERN is calculated using the “incorrect models” to predict yield at MERN of “correct model”.
  • Calculate the difference between the profit between the “correct model” and the “incorrect models”.
  • Evaluate the risk of choosing the “incorrect models”.
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