# Quadratic-plateau response

#### Adrian Correndo & Austin Pearce

## 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: a positive curvilinear response followed by a flat plateau phase. The join point is often interpreted as the CSTV. See Bullock and Bullock (1994) for example.\[ \begin{cases} x < j,\ y = a + bx + cx^2 \\ x > j,\ y = a + bj + cj^2 \end{cases} \]

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)
`j` the join point when the plateau phase starts (i.e., the CSTV).

This model is slightly more complex than the linear-plateau, but the curvature of the response is argued to be more biologically reasonably and economical useful. The `quadratic_plateau()` function works automatically with self-starting initial values to facilitate the model convergence.

Disadvantages are that:

- the default CSTV confidence interval (based on symmetric Wald’s intervals) is generally unreliable. We recommend the user try the `boot_quadratic_plateau()` function for a reliable confidence interval estimation of parameters via bootstrapping (resampling with replacement).

## General Instructions

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

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

   1. `data` (optional)

   2. `stv` (soil test value)

   3. `ry` (relative yield) columns or vectors

   4. `target` (optional) for calculating the soil test value at some RY level along the slope segment.

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

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

   7. `resid` `TRUE` (produces plots with residuals analysis) or `FALSE` (no plot)

3. Run and check results.

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

5. Adjust curve plots as desired with additional `ggplot2` functions.

# 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 dataset

    # Native fake dataset from soiltestcorr package
    corr_df <- soiltestcorr::data_test

# Fit quadratic_plateau()

## 1. Individual fits

### 1.1. `tidy` = TRUE (default)

It returns a tidy data frame (more organized results)

    quadratic_plateau(corr_df, STV, RY, tidy = TRUE)
    #> # A tibble: 1 × 17
    #>   intercept slope quadratic equation       plateau  CSTV lowerCL upperCL CI_type
    #>       <dbl> <dbl>     <dbl> <chr>            <dbl> <dbl>   <dbl>   <dbl> <chr>
    #> 1      44.1  2.86   -0.0392 44.1 + 2.86x …    96.4  36.5    29.7    43.4 Wald C…
    #> # ℹ 8 more variables: target <dbl>, STVt <dbl>, AIC <dbl>, AICc <dbl>,
    #> #   BIC <dbl>, R2 <dbl>, RMSE <dbl>, pvalue <dbl>

### 1.2. `tidy` = FALSE

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

    quadratic_plateau(corr_df, STV, RY, tidy = FALSE)
    #> $intercept
    #> [1] 44.14905
    #>
    #> $slope
    #> [1] 2.862648
    #>
    #> $quadratic
    #> [1] -0.03919089
    #>
    #> $equation
    #> [1] "44.1 + 2.86x + -0.04x^2 when x < 36.5"
    #>
    #> $plateau
    #> [1] 96.42367
    #>
    #> $CSTV
    #> [1] 36.52186
    #>
    #> $lowerCL
    #> [1] 29.7
    #>
    #> $upperCL
    #> [1] 43.4
    #>
    #> $CI_type
    #> [1] "Wald Conf. Interval"
    #>
    #> $target
    #> [1] 96.4
    #>
    #> $STVt
    #> [1] 36.5
    #>
    #> $AIC
    #> [1] 1023.43
    #>
    #> $AICc
    #> [1] 1023.73
    #>
    #> $BIC
    #> [1] 1035.11
    #>
    #> $R2
    #> [1] 0.53
    #>
    #> $RMSE
    #> [1] 9.85
    #>
    #> $pvalue
    #> [1] 0

### 1.3. Alternative using the vectors

You can use the `stv` and `ry` vectors from the data frame using the `$`.

    fit_vectors_tidy <- quadratic_plateau(stv = corr_df$STV, ry = corr_df$RY)

    fit_vectors_list <- quadratic_plateau(stv = corr_df$STV, ry = corr_df$RY, tidy = FALSE)

## 2. Multiple fits at once

    # Example 1. 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 %>%
      rename(STV = STK)

    data.all <- bind_rows(data_1, data_2, data_3, .id = "id")

Note: the `stv` column needs to have the same name for all datasets if binding rows.

### 2.1. Using `map()`

    # Run multiple examples at once with purrr::map()
    data.all %>%
      nest(data = c("STV", "RY")) %>%
      mutate(model = map(data, ~ quadratic_plateau(stv = .$STV, ry = .$RY))) %>%
      unnest(model)
    #> # A tibble: 3 × 19
    #>   id    data     intercept slope quadratic equation        plateau  CSTV lowerCL
    #>   <chr> <list>       <dbl> <dbl>     <dbl> <chr>             <dbl> <dbl>   <dbl>
    #> 1 1     <tibble>      61.0  8.59   -0.502  61 + 8.59x + -…    97.7  8.56     6.6
    #> 2 2     <tibble>      44.1  2.86   -0.0392 44.1 + 2.86x +…    96.4 36.5     29.7
    #> 3 3     <tibble>      12.9  1.91   -0.0111 12.9 + 1.91x +…    95.3 86.2     45.5
    #> # ℹ 10 more variables: upperCL <dbl>, CI_type <chr>, target <dbl>, STVt <dbl>,
    #> #   AIC <dbl>, AICc <dbl>, BIC <dbl>, R2 <dbl>, RMSE <dbl>, pvalue <dbl>

### 2.2. Using `group_modify()`

Alternatively, with `group_modify`, nested data is not required. However, it still requires a grouping variable (in this case, `id`) to identify each dataset. `group_map()` may also be used, though `list_rbind()` is required to return a tidy data frame of the model results instead of a list.

    data.all %>%
      group_by(id) %>%
      group_modify(~ quadratic_plateau(data = ., STV, RY))
    #> # A tibble: 3 × 18
    #> # Groups:   id [3]
    #>   id    intercept slope quadratic equation plateau  CSTV lowerCL upperCL CI_type
    #>   <chr>     <dbl> <dbl>     <dbl> <chr>      <dbl> <dbl>   <dbl>   <dbl> <chr>
    #> 1 1          61.0  8.59   -0.502  61 + 8.…    97.7  8.56     6.6    10.5 Wald C…
    #> 2 2          44.1  2.86   -0.0392 44.1 + …    96.4 36.5     29.7    43.4 Wald C…
    #> 3 3          12.9  1.91   -0.0111 12.9 + …    95.3 86.2     45.5   127.  Wald C…
    #> # ℹ 8 more variables: target <dbl>, STVt <dbl>, AIC <dbl>, AICc <dbl>,
    #> #   BIC <dbl>, R2 <dbl>, RMSE <dbl>, pvalue <dbl>

## 3. Bootstrapping

Bootstrapping is a suitable method for obtaining confidence intervals for parameters or derived quantities. Bootstrapping is a resampling technique (with replacement) that draws samples from the original data with the same size. If you have groups within your data, you can specify grouping variables as arguments in order to maintain, within each resample, the same proportion of observations than in the original dataset.

This function returns a table with as many rows as the resampling size (n) containing the results for each resample.

    boot_qp <- boot_quadratic_plateau(corr_df, STV, RY, n = 500) # only 500 for sake of speed

    boot_qp %>% head(n = 5)
    #> # A tibble: 5 × 14
    #>   boot_id intercept slope quadratic plateau  CSTV target  STVt   AIC  AICc   BIC
    #>     <dbl>     <dbl> <dbl>     <dbl>   <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
    #> 1       1      30.8  4.09   -0.0645    95.6  31.7   95.6  31.7 1063. 1064. 1075.
    #> 2       2      32.3  3.97   -0.0614    96.5  32.3   96.5  32.3 1024. 1024. 1035.
    #> 3       3      56.1  1.85   -0.0211    96.7  43.9   96.7  43.9  992.  992. 1003.
    #> 4       4      44.6  2.86   -0.0397    96.3  36.1   96.3  36.1 1004. 1004. 1015.
    #> 5       5      28.5  4.48   -0.0760    94.5  29.5   94.5  29.5 1091. 1091. 1103.
    #> # ℹ 3 more variables: R2 <dbl>, RMSE <dbl>, pvalue <dbl>

    # CSTV Confidence Interval
    quantile(boot_qp$CSTV, probs = c(0.025, 0.5, 0.975))
    #>     2.5%      50%    97.5%
    #> 28.54063 37.30023 50.58490

    # Plot
    boot_qp %>%
      ggplot2::ggplot(aes(x = CSTV))+
      geom_histogram(color = "grey25", fill = "#9de0bf", bins = 10)

## 4. Plots

### 4.1. Correlation Curve

We can generate a ggplot with the same `quadratic_plateau()` function.

We just need to specify the argument `plot = TRUE`.

    data_3 <- soiltestcorr::freitas1966

    plot_qp <- quadratic_plateau(data = data_3, STK, RY, plot = TRUE)

    plot_qp

### 4.2. Fine-tune the plots

As ggplot object, plots can be adjusted in several ways, such as modifying titles and axis scales.

    plot_qp +
      # Main title
      ggtitle("My own plot title")+
      # Axis titles
      labs(x = "Soil Test K (ppm)",
           y = "Cotton RY(%)") +
      # Axis scales
      scale_x_continuous(limits = c(20,220),
                         breaks = seq(0,220, by = 10))+
      # Axis limits
      scale_y_continuous(limits = c(30, 110),
                         breaks = seq(30, 110, by = 10))

### 4.3. Residuals

Set the argument `resid = TRUE`.

    # Residuals plot
    quadratic_plateau(data = data_3, STK, RY, resid = TRUE)

    #> # A tibble: 1 × 17
    #>   intercept slope quadratic equation       plateau  CSTV lowerCL upperCL CI_type
    #>       <dbl> <dbl>     <dbl> <chr>            <dbl> <dbl>   <dbl>   <dbl> <chr>
    #> 1      12.9  1.91   -0.0111 12.9 + 1.91x …    95.3  86.2    45.5    127. Wald C…
    #> # ℹ 8 more variables: target <dbl>, STVt <dbl>, AIC <dbl>, AICc <dbl>,
    #> #   BIC <dbl>, R2 <dbl>, RMSE <dbl>, pvalue <dbl>

#### 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