Linear-plateau response
Adrian Correndo & Austin Pearce
Source:vignettes/linear_plateau_tutorial.Rmd
linear_plateau_tutorial.Rmd
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: a positive linear response and a flat plateau. The
join point or break point is often interpreted as the CSTV. See Anderson
and Nelson (1975) for examples.
wherey 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),j the join point (a.k.a, break point) when the plateau
phase starts (i.e., the CSTV).
The linear_plateau() function works automatically with
self-starting initial values to facilitate the model’s convergence. The
parameters of this regression model have simple interpretations.
Some disadvantages are that:
the crop relative yield response may be more curvilinear, such that a sharp break from linear response to plateau is unreasonable.
the default CSTV confidence interval (based on symmetric Wald’s intervals) is generally unreliable. We recommend the user try the
boot_linear_plateau()function for a reliable confidence interval estimation of parameters via bootstrapping (resampling with replacement).
General Instructions
Load your data frame with soil test value and relative yield data.
-
Specify the following arguments in
linear_plateau():data(optional)stv(soil test value)ry(relative yield) columns or vectorstarget(optional) for calculating the soil test value at some RY level along the slope segment.tidyTRUE(produces a data.frame with results) orFALSE(store results as list)plotTRUE(produces a ggplot as main output) orFALSE(no plot, only results as data.frame)residTRUE(produces plots with residuals analysis) orFALSE(no plot),
Run and check results.
Check residuals plot, and warnings related to potential limitations of this model.
Adjust curve plots as desired with additional
ggplot2functions.
Tutorial
library(soiltestcorr)
#> Registered S3 methods overwritten by 'ggpp':
#> method from
#> heightDetails.titleGrob ggplot2
#> widthDetails.titleGrob ggplot2Suggested packages
# Install if needed
library(ggplot2) # Plots
library(dplyr) # Data wrangling
library(tidyr) # Data wrangling
library(purrr) # MappingLoad dataset
# Native fake dataset from soiltestcorr package
corr_df <- soiltestcorr::data_testFit linear_plateau()
1. Individual fits
1.1. tidy = TRUE (default)
It returns a tidy data frame (more organized results)
linear_plateau(corr_df, STV, RY, tidy = TRUE)
#> # A tibble: 1 × 16
#> intercept slope equation plateau CSTV lowerCL upperCL CI_type target STVt
#> <dbl> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <dbl> <dbl>
#> 1 53.7 1.55 53.7 + 1.5… 96.2 27.4 24 30.7 Wald, … 96.2 27.4
#> # ℹ 6 more variables: 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)
linear_plateau(corr_df, STV, RY, tidy = FALSE)
#> $intercept
#> [1] 53.71661
#>
#> $slope
#> [1] 1.550917
#>
#> $equation
#> [1] "53.7 + 1.55x when x < 27"
#>
#> $plateau
#> [1] 96.18333
#>
#> $CSTV
#> [1] 27.38169
#>
#> $lowerCL
#> [1] 24
#>
#> $upperCL
#> [1] 30.7
#>
#> $CI_type
#> [1] "Wald, 95%"
#>
#> $target
#> [1] 96.2
#>
#> $STVt
#> [1] 27.4
#>
#> $AIC
#> [1] 1025.96
#>
#> $AICc
#> [1] 1026.26
#>
#> $BIC
#> [1] 1037.64
#>
#> $R2
#> [1] 0.52
#>
#> $RMSE
#> [1] 9.94
#>
#> $pvalue
#> [1] 01.3. Alternative using the vectors
You can use the stv and ry vectors from the
data frame using the $.
fit_vectors_tidy <- linear_plateau(stv = corr_df$STV, ry = corr_df$RY)
fit_vectors_list <- linear_plateau(stv = corr_df$STV, ry = corr_df$RY, tidy = FALSE)2. Multiple fits at once
# Example 1. Fake dataset manually created
data_1 <- data.frame("RY" = c(65,80,85,88,90,94,93,96,97,95,98,100,99,99,100),
"STV" = c(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15))
# Example 2. Native fake dataset from soiltestcorr package
data_2 <- soiltestcorr::data_test
# Example 3. Native dataset from soiltestcorr package, Freitas et al. (1966), used by Cate & Nelson (1971)
data_3 <- soiltestcorr::freitas1966 %>%
rename(STV = STK)
data.all <- bind_rows(data_1, data_2, data_3, .id = "id")Note: the stv column needs to have the same name for all
datasets if binding rows.
2.1. Using map()
# Run multiple examples at once with purrr::map()
data.all %>%
nest(data = c("STV", "RY")) %>%
mutate(model = map(data, ~ linear_plateau(stv = .$STV, ry = .$RY))) %>%
unnest(model)
#> # A tibble: 3 × 18
#> id data intercept slope equation plateau CSTV lowerCL upperCL CI_type
#> <chr> <list> <dbl> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <chr>
#> 1 1 <tibble> 65.9 5.09 65.9 + 5… 97.4 6.21 5.1 7.4 Wald, …
#> 2 2 <tibble> 53.7 1.55 53.7 + 1… 96.2 27.4 24 30.7 Wald, …
#> 3 3 <tibble> 39.2 0.752 39.2 + 0… 95.6 75.0 46.4 104. Wald, …
#> # ℹ 8 more variables: 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(~ linear_plateau(data = ., STV, RY))
#> # A tibble: 3 × 17
#> # Groups: id [3]
#> id intercept slope equation plateau CSTV lowerCL upperCL CI_type target
#> <chr> <dbl> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <dbl>
#> 1 1 65.9 5.09 65.9 + 5.0… 97.4 6.21 5.1 7.4 Wald, … 97.4
#> 2 2 53.7 1.55 53.7 + 1.5… 96.2 27.4 24 30.7 Wald, … 96.2
#> 3 3 39.2 0.752 39.2 + 0.7… 95.6 75.0 46.4 104. Wald, … 95.6
#> # ℹ 7 more variables: 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.
set.seed(123)
boot_lp <- boot_linear_plateau(corr_df, STV, RY, n = 500) # only 500 for sake of speed
boot_lp %>% head(n = 5)
#> # A tibble: 5 × 13
#> boot_id intercept slope plateau CSTV target STVt AIC AICc BIC R2
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 49.0 1.72 96.2 27.4 96.2 27.4 1028. 1029. 1040. 0.55
#> 2 2 56.4 1.38 95.9 28.5 95.9 28.5 1015. 1015. 1027. 0.49
#> 3 3 62.6 1.12 96.2 29.9 96.2 29.9 1004. 1004. 1016. 0.43
#> 4 4 60.0 1.22 97.1 30.4 97.1 30.4 992. 992. 1003. 0.56
#> 5 5 61.2 1.24 96.5 28.6 96.5 28.6 978. 979. 990. 0.51
#> # ℹ 2 more variables: RMSE <dbl>, pvalue <dbl>
# CSTV Confidence Interval
quantile(boot_lp$CSTV, probs = c(0.025, 0.5, 0.975))
#> 2.5% 50% 97.5%
#> 22.61863 27.60057 32.16080
# Plot
boot_lp %>%
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 linear_plateau()
function. We just need to specify the argument
plot = TRUE.
data_3 <- soiltestcorr::freitas1966
plot_lp <- linear_plateau(data = data_3, STK, RY, plot = TRUE)
plot_lp
4.2. Fine-tune the plots
As ggplot object, plots can be adjusted in several ways, such as modifying titles and axis scales.
plot_lp +
# 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(30,100),
breaks = seq(30,100, by = 10))
4.3. Residuals
Set argument resid = TRUE.
# Residuals plot
linear_plateau(data_3, STK, RY, resid = TRUE)
#> # A tibble: 1 × 16
#> intercept slope equation plateau CSTV lowerCL upperCL CI_type target STVt
#> <dbl> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <dbl> <dbl>
#> 1 39.2 0.752 39.2 + 0.7… 95.6 75.0 46.4 104. Wald, … 95.6 75
#> # ℹ 6 more variables: AIC <dbl>, AICc <dbl>, BIC <dbl>, R2 <dbl>, RMSE <dbl>,
#> # pvalue <dbl>