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Nutrient Response Curves and Economic Optimum Dose

Source: R/nutrient_functions.R
nutrient_response.Rd

Fits nutrient response models and finds the dose for maximum yield and the economic optimum dose, where the marginal return equals the price ratio. Quadratic, linear-plateau, quadratic-plateau and Mitscherlich models are available; model = "all" fits all four and compares them.

Usage

nutrient_response(
  dose,
  yield,
  model = "quadratic",
  price_output = 50,
  price_nutrient = 15,
  verbose = TRUE
)

Arguments

dose

Numeric vector of nutrient doses (kg/ha).

yield

Numeric vector of yields (kg/ha), one per dose (replicated doses are allowed).

model

"quadratic" (default), "linear_plateau", "quadratic_plateau", "mitscherlich" or "all".

price_output

Price of the produce (Rs/kg).

price_nutrient

Price of the nutrient (Rs/kg of nutrient).

verbose

Logical. If TRUE (default), prints the results.

Value

For a single model, an object of class aridagri_response with model, coefficients, r_squared, adj_r_squared, rmse, aic, dose_max_yield, max_yield, dose_economic, yield_economic and fit. For model = "all", a list with comparison (one row per model), models and best (lowest AIC).

Details

  • Quadratic: \(Y = a + bX + cX^2\); maximum at \(-b/(2c)\); economic optimum \((P_n/P_y - b)/(2c)\).

  • Linear-plateau: \(Y = a + b \min(X, X_0)\); economic optimum at the join point \(X_0\) when \(b > P_n/P_y\).

  • Quadratic-plateau: quadratic up to its vertex \(X_0\), then flat.

  • Mitscherlich: \(Y = A - B e^{-cX}\); economic optimum \(\ln(B c P_y / P_n)/c\); also the dose giving 95% of the asymptote.

Plateau models are fitted by profile least squares over the join point; the Mitscherlich model by nls with a self-starting model. All four models have three parameters (the join point counts as one), so they are compared by R-squared, adjusted R-squared, RMSE and AIC on an equal footing (AIC differences equal those of stats::AIC()). An optimum beyond the highest dose tested is flagged as an extrapolation.

References

Cerrato, M.E. and Blackmer, A.M. (1990). Comparison of models for describing corn yield response to nitrogen fertilizer. Agronomy Journal, 82(1), 138-143. doi:10.2134/agronj1990.00021962008200010030x

Author

Lalit Kumar Rolaniya, ICAR-IIPR, Bikaner

Examples

dose <- c(0, 20, 40, 60, 80, 100)
yield <- c(1100, 1350, 1520, 1610, 1650, 1660)
nutrient_response(dose, yield, model = "quadratic",
                  price_output = 60, price_nutrient = 15)
#> 
#> ======================================================================
#>  NUTRIENT RESPONSE: QUADRATIC
#> ======================================================================
#>  Coefficients: a = 1108.6, b = 13.093, c = -0.076786 
#>  R-squared = 0.9973   Adjusted R-squared = 0.9954   RMSE = 10.51
#>  Dose for maximum yield = 85.3 kg/ha (yield 1667)
#>  Economic optimum dose = 83.6 kg/ha (yield 1666; price ratio 0.250)
nutrient_response(dose, yield, model = "all", price_output = 60,
                  price_nutrient = 15)
#> 
#> ======================================================================
#>  NUTRIENT RESPONSE: MODEL COMPARISON
#> ======================================================================
#>              Model    R2 Adj_R2   RMSE    AIC Dose_max Max_yield Dose_economic
#>          quadratic 0.997  0.995 10.511 36.229   85.256  1666.693        83.628
#>     linear_plateau 0.990  0.983 20.276 44.113   50.159  1640.000        50.159
#>  quadratic_plateau 0.999  0.999  5.750 28.990   79.580  1652.843        78.139
#>       mitscherlich 0.998  0.996  9.519 35.040       NA  1704.850       146.460
#>  Yield_economic
#>        1666.489
#>        1640.000
#>        1652.663
#>        1696.258
#> 
#>  Best model by AIC: quadratic_plateau