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Description

This tutorial demonstrates the mitscherlich() function for fitting a continuous response model and estimating a critical soil test value. This function fits a Mitscherlich-type exponential regression model that follows a diminishing growth curve described as y = a * (1-exp(-c(x + b)), where a = asymptote, b = x-intercept, c = rate or curvature parameter. This approach is extensively used in agriculture to describe crops response to input since the biological meaning of its curved response. With 3 alternatives to fit the model, the mitscherlich() function brings the advantage of controlling the parameters quantity: i) type = 1 (DEFAULT), corresponding to the model without any restrictions to the parameters (y = a * (1-exp(-c(x + b))); ii) type = 2 (“asymptote 100”), corresponding to the model with only 2 parameters by setting the asymptote = 100 (y = 100 * (1-exp(-c(x + b))), and iii) type = 3 (“asymptote 100 from 0”), corresponding to the model with only 1 parameter by constraining the asymptote = 100 and xintercept = 0 (y = 100 * (1-exp(-c(x))).

Disadvantages this model might include: i) lacks a parameter corresponding directly with a critical soil test value (the model cannot be evaluated at the asymptote as it would result in a CSTV equal to Inf); and ii) there is no apparent confidence interval for an estimated CSTV. For this latter purpose, we recommend the user would to use a re-sampling technique (e.g. bootstrapping) for a reliable confidence interval estimation of parameters and CSTV (for examples on bootstrapping, see nlraa package vignette. The mitscherlich() function works automatically with self-starting initial values to facilitate the model’s convergence.

General Instructions

  1. Load your dataframe with soil test value (stv) and relative yield (ry) data.

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

(a). type select the type of parameterization of the model. i) type = 1 corresponds to the model without any restrictions to the parameters (y = a * (1-exp(-c(x + b))); ii) type = 2 (“asymptote 100”), corresponding to the model with only 2 parameters by setting the asymptote = 100 (y = 100 * (1-exp(-c(x + b))), and iii) type = 3 (“asymptote 100 from 0”), corresponding to the model with only 1 parameter by constraining the asymptote = 100 and xintercept = 0 (y = 100 * (1-exp(-c(x))).

(b). data (optional),

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

(d). target (optional) if want to know the stv level needed for a specific ry.

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

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

(g). 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 mitscherlich()

1. Individual fits

1.1.a type = 1

Type = “no restrictions” or Type = 1 for model with ‘no restrictions’


fit_1_type_1 <- 
  soiltestcorr::mitscherlich(data = data_1, 
                             ry = RY, 
                             stv = STV, 
                             type = 1, 
                             target = 90)

utils::head(fit_1_type_1)
#> $a
#> [1] 98.71
#> 
#> $b
#> [1] 2.07
#> 
#> $c
#> [1] 0.37
#> 
#> $equation
#> [1] "98.7(1-e^(-0.371(x+2.1)"
#> 
#> $target
#> [1] 90
#> 
#> $CSTV
#> [1] 4.5

1.1.b type = 2

Type = “asymptote 100” or Type = 2 for model with ‘asymptote = 100’


fit_1_type_2 <- 
  soiltestcorr::mitscherlich(data = data_1, 
                             ry = RY, 
                             stv = STV, 
                             type = 2, 
                             target = 90)

utils::head(fit_1_type_2)
#> $a
#> [1] 100
#> 
#> $b
#> [1] 2.56
#> 
#> $c
#> [1] 0.32
#> 
#> $equation
#> [1] "100(1-e^(-0.318(x+2.6)"
#> 
#> $target
#> [1] 90
#> 
#> $CSTV
#> [1] 4.7

1.1.c type = 3

Type = “asymptote 100 from 0” or Type = 3 for model with ‘asymptote = 100 and xintercept = 0’


fit_1_type_3 <- 
  soiltestcorr::mitscherlich(data = data_1, 
                             ry = RY, 
                             stv = STV, 
                             type = 3, 
                             target = 90)

utils::head(fit_1_type_3)
#> $a
#> [1] 100
#> 
#> $b
#> [1] 0
#> 
#> $c
#> [1] 0.81
#> 
#> $equation
#> [1] "100(1-e^(-0.811x)"
#> 
#> $target
#> [1] 90
#> 
#> $CSTV
#> [1] 2.8

RY type = 1, target = 90%, confidence level = 0.95, replace with your desired values

1.2. 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::mitscherlich(data = data_1, 
                               ry = RY, 
                               stv = STV, type = 1, target = 90, 
                               tidy = FALSE)

utils::head(fit_1_tidy_false)
#> $a
#> [1] 98.71
#> 
#> $b
#> [1] 2.07
#> 
#> $c
#> [1] 0.37
#> 
#> $equation
#> [1] "98.7(1-e^(-0.371(x+2.1)"
#> 
#> $target
#> [1] 90
#> 
#> $CSTV
#> [1] 4.5

1.3. tidy = TRUE

It returns a data.frame (more organized results)


# Using dataframe argument, tidy = FALSE -> return a LIST
fit_1_tidy_true <- 
  soiltestcorr::mitscherlich(data = data_1, 
                               ry = RY, 
                               stv = STV, type = 1, target = 90,
                               tidy = TRUE)

fit_1_tidy_true
#>       a    b    c                equation target CSTV AIC AICc   R2
#> 1 98.71 2.07 0.37 98.7(1-e^(-0.371(x+2.1)     90  4.5  64   68 0.97

1.4. 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::mitscherlich(ry = data_1$RY,
                             stv = data_1$STV,
                             type = 1,
                             tidy = FALSE)

fit_1_vectors_tidy <- 
  soiltestcorr::mitscherlich(ry = data_1$RY,
                             stv = data_1$STV,
                             type = 1,
                             tidy = TRUE)

1.5. Data 2. Test dataset


fit_2 <-
  soiltestcorr::mitscherlich(data = data_2, 
                             ry = RY,
                             stv = STV,
                             type = 1,
                             target = 90)

utils::head(fit_2)
#> $a
#> [1] 97.98
#> 
#> $b
#> [1] 3.91
#> 
#> $c
#> [1] 0.09
#> 
#> $equation
#> [1] "98(1-e^(-0.089(x+3.9)"
#> 
#> $target
#> [1] 90
#> 
#> $CSTV
#> [1] 24.4

1.6. Data 3. Freitas et al. 1966


fit_3 <-
  soiltestcorr::mitscherlich(data = data_3, 
                             ry = RY,
                             stv = STK, 
                             type = 1, 
                             target = 90)
utils::head(fit_3)
#> $a
#> [1] 96.38
#> 
#> $b
#> [1] -8.69
#> 
#> $c
#> [1] 0.05
#> 
#> $equation
#> [1] "96.4(1-e^(-0.046(x-8.7)"
#> 
#> $target
#> [1] 90
#> 
#> $CSTV
#> [1] 67.9

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 %>%
  dplyr::mutate(models = purrr::map(data, 
                                     ~ soiltestcorr::mitscherlich(ry = .$RY,
                                                                  stv = .$STV,
                                                                  type = 1,
                                                                  target = 90,
                                                                  tidy = TRUE)))

utils::head(fit_multiple_map)
#> # A tibble: 3 × 3
#>   id    data               models      
#>   <chr> <list>             <list>      
#> 1 1     <tibble [15 × 2]>  <df [1 × 9]>
#> 2 2     <tibble [137 × 2]> <df [1 × 9]>
#> 3 3     <tibble [24 × 2]>  <df [1 × 9]>

unnest(fit_multiple_map, models)
#> # A tibble: 3 × 11
#>   id    data         a     b     c equation       target  CSTV   AIC  AICc    R2
#>   <chr> <list>   <dbl> <dbl> <dbl> <chr>           <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1     <tibble>  98.7  2.07  0.37 98.7(1-e^(-0.…     90   4.5    64    68  0.97
#> 2 2     <tibble>  98.0  3.91  0.09 98(1-e^(-0.08…     90  24.4  1022  1022  0.54
#> 3 3     <tibble>  96.4 -8.69  0.05 96.4(1-e^(-0.…     90  67.9   187   189  0.67

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::mitscherlich(data = ., 
                                           ry = RY,
                                           stv = STV, type = 1, target = 90,
                                           tidy = TRUE))

utils::head(fit_multiple_group_map)
#> [[1]]
#>       a    b    c                equation target CSTV AIC AICc   R2
#> 1 98.71 2.07 0.37 98.7(1-e^(-0.371(x+2.1)     90  4.5  64   68 0.97
#> 
#> [[2]]
#>       a    b    c              equation target CSTV  AIC AICc   R2
#> 1 97.98 3.91 0.09 98(1-e^(-0.089(x+3.9)     90 24.4 1022 1022 0.54

3. Plots

3.1. Calibration Curve

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

We just need to specify the argument plot = TRUE.


mitscherlich_plot <- 
  soiltestcorr::mitscherlich(data = data_3, 
                               ry = RY, 
                               stv = STK, type = 1, target = 90, 
                               plot = TRUE)

mitscherlich_plot


3.1.2 Fine-tune the plots

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

For example, modifying titles

mitscherlich_plot_2 <- 
  mitscherlich_plot +
  labs(
    # Main title
    title = "My own plot title",
    # Axis titles
    x = "Soil Test K (ppm)",
    y = "Cotton RY(%)")

mitscherlich_plot_2

Or modifying axis scales

mitscherlich_plot_3 <-
mitscherlich_plot_2 +
  # Axis scales
  scale_x_continuous(breaks = seq(0,220, by = 20))+
  # Axis limits
  scale_y_continuous(breaks = seq(0,100, by = 10))

mitscherlich_plot_3

3.3. Residuals

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

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


# Residuals plot

soiltestcorr::mitscherlich(data = data_3, 
                               ry = RY, 
                               stv = STK, type = 1, target = 90, 
                               resid = TRUE)

#> $a
#> [1] 96.38
#> 
#> $b
#> [1] -8.69
#> 
#> $c
#> [1] 0.05
#> 
#> $equation
#> [1] "96.4(1-e^(-0.046(x-8.7)"
#> 
#> $target
#> [1] 90
#> 
#> $CSTV
#> [1] 67.9
#> 
#> $AIC
#> [1] 187
#> 
#> $AICc
#> [1] 189
#> 
#> $R2
#> [1] 0.67

References

Melsted, S.W. and Peck, T.R. (1977). The Mitscherlich-Bray Growth Function. In Soil Testing (eds T. Peck, J. Cope and D. Whitney). 10.2134/asaspecpub29.c1