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:

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