# Quadratic-plateau response

## Description

This tutorial demonstrates the quadratic_plateau() function for fitting a continuous response model and estimating a critical soil test value (CSTV). This function fits a segmented regression model that follows two phases: i) a curvilinear phase described as y = a + b * x + c * x^2, followed by ii) a plateau phase (Bullock and Bullock, 1994) were the ry response to increasing stv becomes NULL (flat), described as a plateau y = a + b*Xc + c*Xc, where y represents the fitted crop relative yield, x the soil test value, a the intercept (ry when stv = 0) , b the linear slope (as the change in ry per unit of soil nutrient supply or nutrient added), c the quadratic coefficient (giving the curve shape), and X_c the join point when the plateau phase starts (i.e. the CSTV).

This approach is a bit more complex than linear-plateau, but the curvature of the response brings more biological sense. Similar to linear-plateau, disadvantages are that: i) the user does not have control to estimate the CSTV (the Xc parameter) for an specific ry level; and ii) the default confidence interval estimation of the CSTV is generally unreliable (based on symmetric Wald’s intervals). We recommend the user to use a re-sampling technique (e.g. bootstrapping) for a more reliable confidence interval estimation for parameters and CSTV (for examples on bootstrapping, see nlraa package vignette. The quadratic_plateau() function works automatically with self-starting initial values to facilitate the model convergence.

## General Instructions

1. Load your dataframe with soil test value and relative yield data.

2. Specify the following arguments into the function -quadratic_plateau()-:

(a). data (optional),

(b). stv (soil test value) and ry (relative yield) columns or vectors,

(c). target (optional) if want to know stv level needed for a different ry than the plateau.

(d). tidy TRUE (produces a data.frame with results) or FALSE (store results as list),

(e). plot TRUE (produces a ggplot as main output) or FALSE (no plot, only results as data.frame),

(f). resid TRUE (produces plots with residuals analysis) or FALSE (no plot),

1. Run and check results.

2. Check residuals plot, and warnings related to potential limitations of this model.

3. Adjust curve plots as desired.

# Tutorial

library(soiltestcorr)

Suggested packages

# Install if needed
library(ggplot2) # Plots
library(dplyr) # Data wrangling
library(tidyr) # Data wrangling
library(utils) # Data wrangling
library(data.table) # Mapping
library(purrr) # Mapping

This is a basic example using three different datasets:

## Load datasets


# Example 1 dataset
# Fake dataset manually created
data_1 <- data.frame("RY"  = c(65,80,85,88,90,94,93,96,97,95,98,100,99,99,100),
"STV" = c(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15))

# Example 2. Native fake dataset from soiltestcorr package

data_2 <- soiltestcorr::data_test

# Example 3. Native dataset from soiltestcorr package, Freitas et al.  (1966), used by Cate & Nelson (1971)
data_3 <- soiltestcorr::freitas1966

# Fit quadratic_plateau()

## 1. Individual fits

### 1.1. tidy = FALSE

It returns a LIST (more efficient for multiple fits at once)


# Using dataframe argument, tidy = FALSE -> return a LIST
fit_1_tidy_false <-
soiltestcorr::quadratic_plateau(data = data_1,
ry = RY,
stv = STV,
tidy = FALSE)

utils::head(fit_1_tidy_false)
#> $intercept #> [1] 61.01 #> #>$slope
#> [1] 8.59
#>
#> $equation #> [1] "61 + 8.59x + -0.5x^2 if x<CSTV" #> #>$plateau
#> [1] 97.7
#>
#> $target #> [1] 97.7 #> #>$CSTV
#> [1] 8.6

### 1.2. tidy = TRUE

It returns a data.frame (more organized results)


# Using dataframe argument, tidy = FALSE -> return a LIST
fit_1_tidy_true <-
soiltestcorr::quadratic_plateau(data = data_1,
ry = RY,
stv = STV,
tidy = TRUE)

fit_1_tidy_true
#>   intercept slope                       equation plateau target CSTV LL_cstv
#> 1     61.01  8.59 61 + 8.59x + -0.5x^2 if x<CSTV    97.7   97.7  8.6     6.6
#>   UL_cstv             CI_type STVt AIC AICc   R2
#> 1    10.5 Wald Conf. Interval  8.6  75   79 0.94

### 1.3. Alternative using the vectors

You can call stv and ry vectors using the $. The tidy argument still applies for controlling the output type  fit_1_vectors_list <- soiltestcorr::quadratic_plateau(ry = data_1$RY,
stv = data_1$STV, tidy = FALSE) fit_1_vectors_tidy <- soiltestcorr::quadratic_plateau(ry = data_1$RY,
stv = data_1$STV, tidy = TRUE) ### 1.4. Data 2. Test dataset  fit_2 <- soiltestcorr::quadratic_plateau(data = data_2, ry = RY, stv = STV) utils::head(fit_2) #>$intercept
#> [1] 44.15
#>
#> $slope #> [1] 2.86 #> #>$equation
#> [1] "44.1 + 2.86x + -0.04x^2 if x<CSTV"
#>
#> $plateau #> [1] 96.4 #> #>$target
#> [1] 96.4
#>
#> $CSTV #> [1] 36.5 ### 1.5. Data 3. Freitas et al. 1966  fit_3 <- soiltestcorr::quadratic_plateau(data = data_3, ry = RY, stv = STK) utils::head(fit_3) #>$intercept
#> [1] 12.86
#>
#> $slope #> [1] 1.91 #> #>$equation
#> [1] "12.9 + 1.91x + -0.01x^2 if x<CSTV"
#>
#> $plateau #> [1] 95.3 #> #>$target
#> [1] 95.3
#>
stv = .$STV, tidy = TRUE))) utils::head(fit_multiple_map) #> # A tibble: 3 × 3 #> id data quadratic_plateau #> <chr> <list> <list> #> 1 1 <tibble [15 × 2]> <df [1 × 13]> #> 2 2 <tibble [137 × 2]> <df [1 × 13]> #> 3 3 <tibble [24 × 2]> <df [1 × 13]> ### 2.1. Using group_map Alternatively, with group_map, we do not require nested data. However, it requires to bind_rows and add an id column specifying the name of each dataset. This option return models as lists objects.  fit_multiple_group_map <- dplyr::bind_rows(data_1, data_2, .id = "id") %>% dplyr::group_by(id) %>% dplyr::group_map(~ soiltestcorr::quadratic_plateau(data = ., ry = RY, stv = STV, tidy = TRUE)) utils::head(fit_multiple_group_map) #> [[1]] #> intercept slope equation plateau target CSTV LL_cstv #> 1 61.01 8.59 61 + 8.59x + -0.5x^2 if x<CSTV 97.7 97.7 8.6 6.6 #> UL_cstv CI_type STVt AIC AICc R2 #> 1 10.5 Wald Conf. Interval 8.6 75 79 0.94 #> #> [[2]] #> intercept slope equation plateau target CSTV LL_cstv #> 1 44.15 2.86 44.1 + 2.86x + -0.04x^2 if x<CSTV 96.4 96.4 36.5 29.7 #> UL_cstv CI_type STVt AIC AICc R2 #> 1 43.4 Wald Conf. Interval 36.5 1023 1024 0.53 ## 3. Plots ### 3.1. Calibration Curve We can generate a ggplot with the same quadratic_plateau() function. We just need to specify the argument plot = TRUE.  quadratic_plateau_plot <- soiltestcorr::quadratic_plateau(data = data_3, ry = RY, stv = STK, plot = TRUE) quadratic_plateau_plot ### 3.1.2 Fine-tune the plots As ggplot object, plots can be adjusted in several ways. For example, modifying titles quadratic_plateau_plot_2 <- quadratic_plateau_plot + # Main title ggtitle("My own plot title")+ # Axis titles labs(x = "Soil Test K (ppm)", y = "Cotton RY(%)") quadratic_plateau_plot_2 Or modifying axis scales quadratic_plateau_plot_3 <- quadratic_plateau_plot_2 + # Axis scales scale_x_continuous(limits = c(20,220), breaks = seq(0,220, by = 20))+ # Axis limits scale_y_continuous(limits = c(30,100), breaks = seq(30,100, by = 10)) quadratic_plateau_plot_3 ### 3.3. Residuals We can generate a plot with the same quadratic_plateau() function. We just need to specify the argument resid = TRUE.  # Residuals plot soiltestcorr::quadratic_plateau(data = data_3, ry = RY, stv = STK, resid = TRUE) #>$intercept
#> [1] 12.86
#>
#> $slope #> [1] 1.91 #> #>$equation
#> [1] "12.9 + 1.91x + -0.01x^2 if x<CSTV"
#>
#> $plateau #> [1] 95.3 #> #>$target
#> [1] 95.3
#>
#> $CSTV #> [1] 86.2 #> #>$LL_cstv
#> [1] 45.5
#>
#> $UL_cstv #> [1] 126.9 #> #>$CI_type
#> [1] "Wald Conf. Interval"
#>
#> $STVt #> [1] 86.2 #> #>$AIC
#> [1] 187
#>
#> $AICc #> [1] 189 #> #>$R2
#> [1] 0.68`

References

Bullock, D.G. and Bullock, D.S. (1994), Quadratic and Quadratic-Plus-Plateau Models for Predicting Optimal Nitrogen Rate of Corn: A Comparison. Agron. J., 86: 191-195. 10.2134/agronj1994.00021962008600010033x