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Description

This tutorial demonstrates the linear_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 linear phase described as y = a + b * x, followed by ii) a plateau phase (Anderson and Nelson, 1975) were the ry response to increasing stv becomes NULL (flat), described as plateau = y = a + b * Xc, where y represents the fitted crop relative yield, x the soil test value, a the intercept (ry when stv = 0) , b the slope (as the change in RY per unit of soil nutrient supply), and X_c the break point when the plateau phase starts (i.e. the CSTV).

The parameters of this regression model have a simple interpretation. Some disadvantages are that: i) the user does not have control to estimate the CSTV (the model 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 resampling technique (e.g. bootstrapping) for a more reliable confidence interval estimation for parameters and CSTV (for examples on bootstrapping, see nlraa package vignette. The linear_plateau() function works automatically with self-starting initial values to facilitate the model’s convergence.

General Instructions

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

  2. Specify the following arguments into the function -linear_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

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 linear_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::linear_plateau(data = data_1, 
                               ry = RY, 
                               stv = STV, 
                               tidy = FALSE)

utils::head(fit_1_tidy_false)
#> $intercept
#> [1] 65.87
#> 
#> $slope
#> [1] 5.09
#> 
#> $equation
#> [1] "65.9 + 5.09x if x<CSTV"
#> 
#> $plateau
#> [1] 97.4
#> 
#> $target
#> [1] 97.4
#> 
#> $CSTV
#> [1] 6.2

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::linear_plateau(data = data_1, 
                               ry = RY, 
                               stv = STV,
                               tidy = TRUE)

fit_1_tidy_true
#>   intercept slope               equation plateau target CSTV LL_cstv UL_cstv
#> 1     65.87  5.09 65.9 + 5.09x if x<CSTV    97.4   97.4  6.2     5.1     7.4
#>               CI_type STVt AIC AICc  R2
#> 1 Wald Conf. Interval  6.2  82   86 0.9

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::linear_plateau(ry = data_1$RY,
                               stv = data_1$STV,
                               tidy = FALSE)

fit_1_vectors_tidy <- 
  soiltestcorr::linear_plateau(ry = data_1$RY,
                               stv = data_1$STV,
                               tidy = TRUE)

1.4. Data 2. Test dataset


fit_2 <-
  soiltestcorr::linear_plateau(data = data_2, 
                               ry = RY,
                               stv = STV)

utils::head(fit_2)
#> $intercept
#> [1] 53.72
#> 
#> $slope
#> [1] 1.55
#> 
#> $equation
#> [1] "53.7 + 1.55x if x<CSTV"
#> 
#> $plateau
#> [1] 96.2
#> 
#> $target
#> [1] 96.2
#> 
#> $CSTV
#> [1] 27.4

1.5. Data 3. Freitas et al. 1966


fit_3 <-
  soiltestcorr::linear_plateau(data = data_3, 
                               ry = RY,
                               stv = STK)
utils::head(fit_3)
#> $intercept
#> [1] 39.24
#> 
#> $slope
#> [1] 0.75
#> 
#> $equation
#> [1] "39.2 + 0.75x if x<CSTV"
#> 
#> $plateau
#> [1] 95.6
#> 
#> $target
#> [1] 95.6
#> 
#> $CSTV
#> [1] 75

2. Multiple fits at once

2.1. Using map

Create nested data

Note: the stv column needs to have the same name for all datasets

# 
data.all <- dplyr::bind_rows(data_1, data_2,
                      data_3 %>% dplyr::rename(STV = STK),
                     .id = "id") %>% 
  tidyr::nest(data = c("STV", "RY"))
Fit

# Run multiple examples at once with map()
fit_multiple_map <-
  data.all %>%
  mutate(linear_plateau = purrr::map(data, 
                                     ~ soiltestcorr::linear_plateau(ry = .$RY,
                                                                    stv = .$STV,
                                                                    tidy = TRUE)))

utils::head(fit_multiple_map)
#> # A tibble: 3 × 3
#>   id    data               linear_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 dplyr::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::linear_plateau(data = ., 
                                           ry = RY,
                                           stv = STV,
                                           tidy = TRUE))

utils::head(fit_multiple_group_map)
#> [[1]]
#>   intercept slope               equation plateau target CSTV LL_cstv UL_cstv
#> 1     65.87  5.09 65.9 + 5.09x if x<CSTV    97.4   97.4  6.2     5.1     7.4
#>               CI_type STVt AIC AICc  R2
#> 1 Wald Conf. Interval  6.2  82   86 0.9
#> 
#> [[2]]
#>   intercept slope               equation plateau target CSTV LL_cstv UL_cstv
#> 1     53.72  1.55 53.7 + 1.55x if x<CSTV    96.2   96.2 27.4      24    30.7
#>               CI_type STVt  AIC AICc   R2
#> 1 Wald Conf. Interval 27.4 1026 1026 0.52

3. Plots

3.1. Calibration Curve

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

We just need to specify the argument plot = TRUE.


linear_plateau_plot <- 
  soiltestcorr::linear_plateau(data = data_3, 
                               ry = RY, 
                               stv = STK, 
                               plot = TRUE)

linear_plateau_plot

### 3.1.2 Fine-tune the plots

As ggplot object, plots can be adjusted in several ways.

For example, modifying titles

linear_plateau_plot_2 <- 
  linear_plateau_plot +
  # Main title
  ggtitle("My own plot title")+
  # Axis titles
  labs(x = "Soil Test K (ppm)",
       y = "Cotton RY(%)")

linear_plateau_plot_2

Or modifying axis scales

linear_plateau_plot_3 <-
linear_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))

linear_plateau_plot_3

3.3. Residuals

We can generate a plot with the same linear_plateau() function.

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


# Residuals plot

soiltestcorr::linear_plateau(data = data_3, 
                               ry = RY, 
                               stv = STK, 
                               resid = TRUE)

#> $intercept
#> [1] 39.24
#> 
#> $slope
#> [1] 0.75
#> 
#> $equation
#> [1] "39.2 + 0.75x if x<CSTV"
#> 
#> $plateau
#> [1] 95.6
#> 
#> $target
#> [1] 95.6
#> 
#> $CSTV
#> [1] 75
#> 
#> $LL_cstv
#> [1] 46.4
#> 
#> $UL_cstv
#> [1] 103.6
#> 
#> $CI_type
#> [1] "Wald Conf. Interval"
#> 
#> $STVt
#> [1] 75
#> 
#> $AIC
#> [1] 188
#> 
#> $AICc
#> [1] 190
#> 
#> $R2
#> [1] 0.66

References

Anderson, R. L., and Nelson, L. A. (1975). A Family of Models Involving Intersecting Straight Lines and Concomitant Experimental Designs Useful in Evaluating Response to Fertilizer Nutrients. Biometrics, 31(2), 303–318. 10.2307/2529422