Author: Adrian Correndo

# Modified ALCC

#### Adrian Correndo

## Description

This tutorial is intended to show how to deploy the `mod_alcc()` function for estimating critical soil test values using the Modified Arcsine-Log Calibration Curve, originally introduced by Dyson & Conyers (2013) and modified by Correndo et al. (2017). This function produces the estimation of critical soil test values (CSTV) for a target relative yield (`ry`) with confidence intervals at adjustable confidence levels. Behind the scenes, the `mod_alcc()` is based on the standardized major axis (SMA), a bivariate regression model that assumes that both axis are random variables (Warton et al., 2006).

## General Instructions

1. Load your dataframe with soil test value (`stv`) and relative yield (`ry`) data.
2. Specify the following arguments into the function `mod_alcc()`

   1. `data` (optional)

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

   3. `target` of relative yield (e.g. 90%),

   4. desired `confidence` level (e.g. 0.95 for 1 - alpha(0.05)).

      - Used for the estimation of critical soil test value (CSTV) lower and upper limits.

   5. `tidy` TRUE (produces a data.frame with results) or FALSE-default- (store results as list),

   6. `plot` TRUE (produces a ggplot as main output) or FALSE (default; no plot, only results as data.frame or list)

3. Run and check results.
4. Check residuals plot, and warnings related to potential leverage points.
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(purrr) # Mapping

This is a basic example using three different datasets:

## Load datasets

    # Native fake dataset from soiltestcorr package
    corr_df <- soiltestcorr::freitas1966

# Fit `mod_alcc()`

## 1. Individual fits

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

### 1.1. `tidy` = FALSE

It returns a LIST (may be more efficient for multiple fits at once)

    # Using dataframe argument, tidy = FALSE -> return a LIST
    mod_alcc(data = corr_df, ry = RY, stv = STK, target=90, confidence = 0.95,
             tidy = TRUE)
    #> Warning: One or more original RY values exceeded 100%. All RY values greater
    #>           than 100% have been capped to 100%.
    #> Warning: 2 STV points exceeded the CSTV for 100% of RY.
    #>   Risk of leverage. You may consider a sensitivity analysis by removing extreme points,
    #>   re-run the mod_alcc(), and check results.
    #> Warning: 2 STV points exceeded two-times (2x)
    #>   the CSTV for 90% of RY. Risk of leverage. You may consider a sensitivity analysis by
    #>   removing extreme points, re-run the mod_alcc(), and check results.
    #> # A tibble: 1 × 18
    #>       n     r RMSE_alcc AIC_alcc AIC_sma BIC_sma p_value confidence target  CSTV
    #>   <int> <dbl>     <dbl>    <dbl>   <dbl>   <dbl>   <dbl>      <dbl>  <dbl> <dbl>
    #> 1    24 0.728      10.3     186.    17.0    22.9 5.57e-5       0.95     90  76.9
    #> # ℹ 8 more variables: LL <dbl>, UL <dbl>, CSTV90 <dbl>, n.90x2 <int>,
    #> #   CSTV100 <dbl>, n.100 <int>, Curve <list>, SMA <list>

### 1.2. `tidy` = TRUE

It returns a data.frame (more organized results)

    # Using dataframe argument, tidy = FALSE -> return a LIST
    mod_alcc(data = corr_df, ry = RY, stv = STK, target=90, confidence = 0.95, tidy = TRUE)
    #> Warning: One or more original RY values exceeded 100%. All RY values greater
    #>           than 100% have been capped to 100%.
    #> Warning: 2 STV points exceeded the CSTV for 100% of RY.
    #>   Risk of leverage. You may consider a sensitivity analysis by removing extreme points,
    #>   re-run the mod_alcc(), and check results.
    #> Warning: 2 STV points exceeded two-times (2x)
    #>   the CSTV for 90% of RY. Risk of leverage. You may consider a sensitivity analysis by
    #>   removing extreme points, re-run the mod_alcc(), and check results.
    #> # A tibble: 1 × 18
    #>       n     r RMSE_alcc AIC_alcc AIC_sma BIC_sma p_value confidence target  CSTV
    #>   <int> <dbl>     <dbl>    <dbl>   <dbl>   <dbl>   <dbl>      <dbl>  <dbl> <dbl>
    #> 1    24 0.728      10.3     186.    17.0    22.9 5.57e-5       0.95     90  76.9
    #> # ℹ 8 more variables: LL <dbl>, UL <dbl>, CSTV90 <dbl>, n.90x2 <int>,
    #> #   CSTV100 <dbl>, n.100 <int>, Curve <list>, SMA <list>

### 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_vectors_tidy <- mod_alcc(ry = corr_df$RY,
                                 stv = corr_df$STK,
                                 target = 90,
                                 confidence = 0.95)
    #> Warning: One or more original RY values exceeded 100%. All RY values greater
    #>           than 100% have been capped to 100%.
    #> Warning: 2 STV points exceeded the CSTV for 100% of RY.
    #>   Risk of leverage. You may consider a sensitivity analysis by removing extreme points,
    #>   re-run the mod_alcc(), and check results.
    #> Warning: 2 STV points exceeded two-times (2x)
    #>   the CSTV for 90% of RY. Risk of leverage. You may consider a sensitivity analysis by
    #>   removing extreme points, re-run the mod_alcc(), and check results.

    fit_vectors_list <- mod_alcc(ry = corr_df$RY,
                                 stv = corr_df$STK,
                                 target = 90,
                                 confidence = 0.95,
                                 tidy = FALSE)
    #> Warning: One or more original RY values exceeded 100%. All RY values greater
    #>           than 100% have been capped to 100%.
    #> Warning: 2 STV points exceeded the CSTV for 100% of RY.
    #>   Risk of leverage. You may consider a sensitivity analysis by removing extreme points,
    #>   re-run the mod_alcc(), and check results.
    #> Warning: 2 STV points exceeded two-times (2x)
    #>   the CSTV for 90% of RY. Risk of leverage. You may consider a sensitivity analysis by
    #>   removing extreme points, re-run the mod_alcc(), and check results.

## 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, ~ mod_alcc(stv = .$STV, ry = .$RY, target = 90))) %>%
      unnest(model)
    #> Warning: There were 9 warnings in `mutate()`.
    #> The first warning was:
    #> ℹ In argument: `model = map(data, ~mod_alcc(stv = .$STV, ry = .$RY, target =
    #>   90))`.
    #> Caused by warning:
    #> ! You have not specified the confidence level.
    #>                                 Please, modify if your desired confidence is different than the default (0.95)
    #> ℹ Run `dplyr::last_dplyr_warnings()` to see the 8 remaining warnings.
    #> # A tibble: 3 × 20
    #>   id    data         n     r RMSE_alcc AIC_alcc AIC_sma BIC_sma  p_value
    #>   <chr> <list>   <int> <dbl>     <dbl>    <dbl>   <dbl>   <dbl>    <dbl>
    #> 1 1     <tibble>    15 0.968      1.45     59.6  -23.5    -20.0 3.30e- 9
    #> 2 2     <tibble>   137 0.716     10.5    1038.    -1.04    13.6 7.31e-23
    #> 3 3     <tibble>    24 0.728     10.3     186.    17.0     22.9 5.57e- 5
    #> # ℹ 11 more variables: confidence <dbl>, target <dbl>, CSTV <dbl>, LL <dbl>,
    #> #   UL <dbl>, CSTV90 <dbl>, n.90x2 <int>, CSTV100 <dbl>, n.100 <int>,
    #> #   Curve <list>, SMA <list>

Note: the output table is still generated despite warnings regarding confidence level and leverage points.

### 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(~ mod_alcc(data = ., STV, RY, target = 90, confidence = 0.95))
    #> Warning: One or more original RY values exceeded 100%. All RY values greater
    #>           than 100% have been capped to 100%.
    #> # A tibble: 3 × 19
    #> # Groups:   id [3]
    #>   id        n     r RMSE_alcc AIC_alcc AIC_sma  BIC_sma  p_value confidence
    #>   <chr> <int> <dbl>     <dbl>    <dbl>   <dbl>    <dbl>    <dbl>      <dbl>
    #> 1 1        15 0.892      1.79     66.1  -82.7   -79.1   7.84e- 6       0.95
    #> 2 2       137 0.569     11.6    1067.  -290.   -275.    4.14e-13       0.95
    #> 3 3        24 0.708     20.0     218.    -5.01    0.885 1.10e- 4       0.95
    #> # ℹ 10 more variables: target <dbl>, CSTV <dbl>, LL <dbl>, UL <dbl>,
    #> #   CSTV90 <dbl>, n.90x2 <int>, CSTV100 <dbl>, n.100 <int>, Curve <list>,
    #> #   SMA <list>

## 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_alcc <- boot_mod_alcc(data = corr_df,
                               stv = STK, ry = RY,
                               target = 90, n = 500)
    #> Warning: There were 1332 warnings in `dplyr::mutate()`.
    #> The first warning was:
    #> ℹ In argument: `model = map(...)`.
    #> ℹ In group 1: `boot_id = 1`.
    #> Caused by warning:
    #> ! One or more original RY values exceeded 100%. All RY values greater
    #>           than 100% have been capped to 100%.
    #> ℹ Run `dplyr::last_dplyr_warnings()` to see the 1331 remaining warnings.

    boot_alcc %>% head(n = 5)
    #> # A tibble: 5 × 14
    #>   boot_id     n     r RMSE_alcc AIC_alcc AIC_sma BIC_sma   p_value target  CSTV
    #>     <dbl> <int> <dbl>     <dbl>    <dbl>   <dbl>   <dbl>     <dbl>  <dbl> <dbl>
    #> 1       1    24 0.508     12.4      195.   11.9     17.7 0.0112        90  81.9
    #> 2       2    24 0.750      9.50     182.    7.68    13.6 0.0000240     90  61.8
    #> 3       3    24 0.625     10.6      188.   10.6     16.5 0.00109       90  86.6
    #> 4       4    24 0.688      8.96     179.   16.1     22.0 0.000204      90  72.4
    #> 5       5    24 0.593      9.74     183.   13.3     19.2 0.00225       90  80.9
    #> # ℹ 4 more variables: CSTV90 <dbl>, n.90x2 <int>, CSTV100 <dbl>, n.100 <int>

    # CSTV Confidence Interval
    quantile(boot_alcc$CSTV, probs = c(0.025, 0.5, 0.975))
    #>     2.5%      50%    97.5%
    #> 63.70241 77.17436 93.90476

    # Plot
    boot_alcc %>%
      ggplot2::ggplot(aes(x = CSTV))+
      geom_histogram(color = "grey25", fill = "steelblue", bins = 10)

## 4. Plots

### 4.1. Calibration Curve

We can generate a ggplot with the same `mod_alcc()` function. We just need to specify the argument `plot = TRUE`.

    plt_alcc <- mod_alcc(data = corr_df,
                         ry = RY,
                         stv = STK,
                         target = 95,
                         plot = TRUE)

    plt_alcc

### 4.2. Fine-tune the plots

As ggplot object, plots can be adjusted in several ways, such as modifying titles and axis scales.

    plt_alcc +
      # 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 = 25)) +
      # Axis limits
      scale_y_continuous(limits = c(30,100),
                         breaks = seq(20,100, by = 20))

### 3.2. SMA regression

Behind the scenes, the `mod_alcc()` runs a Standardized Major Axis regression (SMA).

We can extract the SMA fit and also check the residuals of this model out as follows:

    fit_3 <- mod_alcc(data = corr_df, ry = RY, stv = STK, target = 90)
    # Extract SMA regression fit and residuals from fit_3 (data_3, (Freitas et al., 1966))
    SMA_freitas_1966 <- fit_3$SMA %>% as.data.frame()

    SMA_freitas_1966 %>%
      ggplot(aes(x = arc_RY, y = ln_STV))+
      ggtitle("SMA Regression. Dataset 3")+
      geom_point(shape=21, fill = "orange", size = 3, alpha = 0.75)+
      #SMA Line
      geom_path(aes(x=arc_RY, y = SMA_line, linetype = "SMA_fit"), linewidth = 1.5, col = "grey25")+
      scale_linetype_manual(name="", values = c("solid"))+
      #Critical value
      geom_vline(xintercept = 0, col = "grey10", size = 1.25, linetype = "dashed")+
      theme_bw()+
      # Axis titles
      labs(y = "ln_STV", y = "asin(sqrt(RY))-centered")

### 3.3. SMA residuals

    # Residuals plot
    SMA_freitas_1966 %>%
      ggplot(aes(x = fitted_axis, y = residuals))+
      ggtitle("Residuals SMA. Dataset 3")+
      geom_point(shape=21, fill = "orange", size = 3, alpha = 0.75)+
      geom_hline(yintercept = 0, col = "grey10", linewidth = 1.25, linetype = "dashed")+
      theme_bw()+
      # Axis titles
      labs(x = "Fitted Axis -SMA- (see Warton et al. 2006)", y = "Residuals (STV units)")

### References

Correndo, A.A., Salvagiotti, F., García, F.O. and Gutiérrez-Boem, F.H., 2017. A modification of the arcsine–log calibration curve for analysing soil test value–relative yield relationships. Crop and Pasture Science, 68(3), pp.297-304. 10.1071/CP16444

Dyson, C.B., Conyers, M.K., 2013. Methodology for online biometric analysis of soil test-crop response datasets. Crop & Pasture Science 64: 435–441. 10.1071/CP13009

Warton, D.I., Wright, I.J., Falster, D.S., Westoby, M., 2006. Bivariate line-fitting methods for allometry. Biol. Rev. Camb. Philos. Soc. 81, 259–291. 10.1017/S1464793106007007