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.
Load your dataframe with soil test value (stv) and relative yield
(ry) data.
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),
Run and check results.
Check residuals plot, and warnings related to potential
limitations of this model.
Adjust curve plots as desired.
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) # MappingThis is a basic example using three different 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::freitas1966type = 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] 4.5type = 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] 4.7type = 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] 2.8RY type = 1, target = 90%, confidence level = 0.95, replace with your
desired values
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] 4.5tidy = 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 0.97You 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)
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
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
#>
#> $CSTV
#> [1] 67.9 Note: the stv column needs to have the same name for
all datasets
#
data.all <- dplyr::bind_rows(data_1, data_2,
data_3 %>% dplyr::rename(STV = STK),
.id = "id") %>%
tidyr::nest(data = c("STV", "RY"))
# Run multiple examples at once with map()
fit_multiple_map <-
data.all %>%
dplyr::mutate(models = purrr::map(data,
~ soiltestcorr::mitscherlich(ry = .$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 0.67Alternatively, 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 0.54We 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
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_2Or 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_3We 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