# Bootstrapping

#### Adrian Correndo, Austin Pearce

## Description

This tutorial is intended to show how to implement bootstrapping to the soiltestcorr functions.

## What is bootstrapping?

Bootstrapping is a technique introduced in late 1970’s by Bradley Efron (Efron, 1979). It is a general purpose inferential approach that is useful for robust estimations, especially when the distribution of a statistic of quantity of interest is complicated or unknown (Faraway, 2014). It provides an alternative to perform confidence statements while relaxing the famous assumption of normality (Efron and Tibshirani, 1993). The underlying concept of bootstrapping is that the inference about a population parameter or quantity can be modeled by “resampling” the available data.

## How does it work?

In `soiltestcorr`, bootstrapping means resampling cases with replacement. That means that it draws random samples from the original data while maintaining the same original size (i.e, some observations may be included multiple times while others may not be sampled at all). Let’s suppose that we have a dataset with 50 observations (one per row). We first create bootstraps/replicates/samples by randomly sampling those 50 observations many times (e.g., 1,000 times). Each random sample has 50 total observations, conserving the same number of observations as the original data (50). Finally, we fit a chosen model to each of the 1000 samples, which, in the context of soiltestcorr, could result in a distribution of 1,000 estimates of a critical soil test value (CSTV), for example.

In general, the larger the number of observations and resamples, the more likely we’ll obtain a representative distribution of the population’s quantity of interest. The minimum number of iterations generally recommended is `n = 1000`. The number used is up to the user, and may represent a compromise between the quality of the outcomes and the necessary computational power or time for the iterations. If the final results from 1,000 samples is no different than 10,000 samples, the time savings may warrant fewer samples.

Once the distribution is obtained we may obtain a “summary” describing “moments” of the distribution such as the “mean”, “median”, “standard deviation”, “quantiles”, etc.. Very frequently, the general procedure is to extract the “median” (which is more robust to outliers than the mean), the 0.025- and 0.975-quantiles from the distribution (equivalent to 2.5th and 97.5th percentiles) in order to obtain the conventional 95% confidence interval. This interval in normally used to test hypotheses of interest (e.g. comparing groups, methods, etc.)

## The `soiltestcorr::boot_()` functions

You can implement bootstrapping to each of the six soiltestcorr available models, which count with their own `boot_…()` functions:

- `boot_mod_alcc()`

- `boot_cn_1965()`

- `boot_cn_1971()`

- `boot_linear_plateau()`

- `boot_quadratic_plateau()`

- `boot_mitscherlich()`

Each `boot_…()` function presents the same basic arguments than their respective original functions (i.e., `data`, `stv`, and `ry`), with the addition of two arguments related to the resampling:

- `n`, indicating the number of resamples to perform iterations.

- `...`, indicating open arguments that allow you to add grouping variable/s (factor or character) for running the bootstrapping by group. Be sure that these open arguments are placed by the end of each function.

## Examples

The following is a simple example using a simulated dataset that contains soil test values (STV) and relative yield (RY) for two different groups of soil: “clay_loam” and “sandy_loam”.

If you have groups in your data, you can specify the grouping variable/s as arguments in order to maintain within each bootstrap sample the same proportion of observations than in the original dataset. This will return a tibble for each sample with as many rows as the resampling size (n) containing the results.

The objective in this example is to obtain critical soil test values (CSTV) with confidence intervals at 90% RY for each of the different models that soiltestcorr offers.

    library(soiltestcorr)

Suggested packages

    # Install if needed
    library(ggplot2) # Plots
    library(dplyr) # Data wrangling
    library(tidyr) # Data wrangling
    library(purrr) # Mapping

    theme_set(theme_minimal())

### Data

    # Clay loam simulated dataset
    clay_loam <- dplyr::tibble(
      "soil" = "clay_loam",
      "RY"  = c(65,80,85,88,90,94,93,96,97,95,98,100,99,99,100),
      "STV" = seq(1, 15, by = 1) )

    # Sandy loam simulated dataset
    sandy_loam <- dplyr::tibble(
      "soil" = "sandy_loam",
      "RY" = c(55,70,80,84,89,92,90,94,99,92,96,100,100,99,99),
      "STV" = seq(2, 16, by = 1))

    # Merging soils datasets
    dataframe_soils <- bind_rows(clay_loam, sandy_loam)

#### 1. `boot_cn_1965()`

    set.seed(123)
    boot_cn_65 <- boot_cn_1965(data = dataframe_soils,
                               stv = STV, ry = RY,
                               target = 90,
                               n = 100, # only 100 replicates to save compute time on demo
                               # Group by soil
                               soil = soil)

    # CSTV Confidence Interval
    boot_cn_65_sum <- boot_cn_65 %>%
      group_by(soil) %>%
      # Obtain quantiles of interest
      summarise(q025 = quantile(CSTV, prob = 0.025, na.rm = TRUE),
                median = quantile(CSTV, prob = 0.500, na.rm = TRUE),
                q975 = quantile(CSTV, prob = 0.975, na.rm = TRUE)) %>%
      ungroup()

    # Plot CSTV
    boot_cn_65 %>%
      ggplot2::ggplot(aes(x = CSTV))+
      geom_density(aes(fill = soil), color = "grey40", alpha = 0.5)+
      # Draw lines
      geom_vline(data = boot_cn_65_sum, aes(xintercept = median, color = soil),
                 linetype = "dashed", linewidth = 1)+
      geom_vline(data = boot_cn_65_sum, aes(xintercept = q025, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      geom_vline(data = boot_cn_65_sum, aes(xintercept = q975, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      scale_x_continuous(limits = c(1,10), breaks = seq(1, 10, by = 1))+
      labs(title = "boot_cn_1965()", x = "CSTV for RY=90%")

#### 2. `boot_cn_1971()`

Note: for the Cate & Nelson (1971) approach, it is not possible to set a RY target for the CSTV estimation. It simply estimates the most efficient division of the data into the four quadrants in order to maximize the number of observations into positive quadrants.

    set.seed(123)
    boot_cn_71 <- boot_cn_1971(data = dataframe_soils,
                               stv = STV, ry = RY,
                               n = 100, # only 100 replicates to save compute time on demo
                               # Group by soil
                               soil = soil)

    # CSTV Confidence Interval
    boot_cn_71_sum <- boot_cn_71 %>%
                  group_by(soil) %>%
                # Obtain quantiles of interest
                  summarise(q025 = quantile(CSTV, prob = 0.025, na.rm = TRUE),
                            median = quantile(CSTV, prob = 0.500, na.rm = TRUE),
                            q975 = quantile(CSTV, prob = 0.975, na.rm = TRUE))

    # Plot CSTV
    boot_cn_71 %>%
      ungroup() %>%
      ggplot2::ggplot(aes(x = CSTV, y = CRYV))+
      geom_density_2d(aes(color = soil), shape = 21, alpha = 0.5)+
      # Draw lines
      geom_vline(data = boot_cn_71_sum, aes(xintercept = median, color = soil),
                 linetype = "dashed", linewidth = 1)+
      scale_x_continuous(limits = c(1,10), breaks = seq(1, 10, by = 1))+
      labs(title = "boot_cn_1971()", x = "CSTV for CRYV", y = "RV (%) at CSTV")

#### 3. `boot_mod_alcc()`

    set.seed(123)
    boot_alcc <- boot_mod_alcc(data = dataframe_soils,
                               stv = STV, ry = RY,
                               target = 90, n = 1000,
                               # Group by id
                               soil = soil)

    # CSTV Confidence Interval
    boot_alcc_sum <- boot_alcc %>%
                  group_by(soil) %>%
                # Obtain quantiles of interest
                  summarise(q025 = quantile(CSTV, prob = 0.025, na.rm = TRUE),
                            median = quantile(CSTV, prob = 0.500, na.rm = TRUE),
                            q975 = quantile(CSTV, prob = 0.975, na.rm = TRUE))

    # Plot CSTV
    boot_alcc %>%
      ungroup() %>%
      ggplot2::ggplot(aes(x = CSTV))+
      geom_density(aes(fill = soil), color = "grey40", alpha = 0.5)+
      # Draw lines
      geom_vline(data = boot_alcc_sum, aes(xintercept = median, color = soil),
                 linetype = "dashed", linewidth = 1)+
      geom_vline(data = boot_alcc_sum, aes(xintercept = q025, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      geom_vline(data = boot_alcc_sum, aes(xintercept = q975, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      scale_x_continuous(limits = c(1,10), breaks = seq(1, 10, by = 1))+
      labs(title = "boot_mod_alcc()", x = "CSTV for RY=90%")+
      theme_bw()

#### 2. `boot_linear_plateau()`

    set.seed(123)
    boot_lp <- boot_linear_plateau(data = dataframe_soils,
                                   stv = STV, ry = RY,
                                   target = 90, n = 1e3,
                                   # Group by soil
                                   soil = soil)

    # CSTV Confidence Interval
    boot_lp_sum <- boot_lp %>%
                  group_by(soil) %>%
                # Obtain quantiles of interest
                # Note: for linear_plateau, the CSTV for a specific RY target = STVt
                  summarise(q025 = quantile(STVt, prob = 0.025, na.rm = TRUE),
                            median = quantile(STVt, prob = 0.500, na.rm = TRUE),
                            q975 = quantile(STVt, prob = 0.975, na.rm = TRUE))

    # Plot STVt
    boot_lp %>%
      ungroup() %>%
      ggplot2::ggplot(aes(x = STVt))+
      geom_density(aes(fill = soil), color = "grey40", alpha = 0.5)+
      # Draw lines
      geom_vline(data = boot_lp_sum, aes(xintercept = median, color = soil),
                 linetype = "dashed", linewidth = 1)+
      geom_vline(data = boot_lp_sum, aes(xintercept = q025, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      geom_vline(data = boot_lp_sum, aes(xintercept = q975, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      scale_x_continuous(limits = c(1,10), breaks = seq(1, 10, by = 1))+
      labs(title = "boot_linear_plateau()", x = "STVt for RY=90%")+
      theme_bw()

#### 3. `boot_quadratic_plateau()`

    set.seed(123)
    boot_qp <- boot_quadratic_plateau(data = dataframe_soils,
                                   stv = STV, ry = RY,
                                   target = 90, n = 1000,
                                   # Group by soil
                                   soil = soil)
    #> Error in nlsModel(formula, mf, start, wts) :
    #>   singular gradient matrix at initial parameter estimates

    # CSTV Confidence Interval
    boot_qp_sum <- boot_qp %>%
                  group_by(soil) %>%
                # Obtain quantiles of interest
                # Note: for quad_plateau, the CSTV for a specific RY target = STVt
                  summarise(q025 = quantile(STVt, prob = 0.025, na.rm = TRUE),
                            median = quantile(STVt, prob = 0.500, na.rm = TRUE),
                            q975 = quantile(STVt, prob = 0.975, na.rm = TRUE))

    # Plot STVt
    boot_qp %>%
      ungroup() %>%
      ggplot2::ggplot(aes(x = STVt))+
      geom_density(aes(fill = soil), color = "grey40", alpha = 0.5)+
      # Draw lines
      geom_vline(data = boot_qp_sum, aes(xintercept = median, color = soil),
                 linetype = "dashed", linewidth = 1)+
      geom_vline(data = boot_qp_sum, aes(xintercept = q025, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      geom_vline(data = boot_qp_sum, aes(xintercept = q975, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      scale_x_continuous(limits = c(1,10), breaks = seq(2, 10, by = 1))+
      labs(title = "boot_quadratic_plateau()", x = "CSTV for RY=90%")+
      theme_bw()

#### 4. `boot_mitscherlich()`

    set.seed(123)
    boot_mits <- boot_mitscherlich(data = dataframe_soils,
                                   stv = STV, ry = RY, type = 1,
                                   target = 90, n = 1e3,
                                   # Group by soil
                                   soil = soil)

    # CSTV Confidence Interval
    boot_mits_sum <- boot_mits %>%
                  group_by(soil) %>%
                # Obtain quantiles of interest
                  summarise(q025 = quantile(CSTV, prob = 0.025, na.rm = TRUE),
                            median = quantile(CSTV, prob = 0.500, na.rm = TRUE),
                            q975 = quantile(CSTV, prob = 0.975, na.rm = TRUE))

    # Plot CSTV
    boot_mits %>%
      ungroup() %>%
      ggplot2::ggplot(aes(x = CSTV))+
      geom_density(aes(fill = soil), color = "grey40", alpha = 0.5)+
      # Draw lines
      geom_vline(data = boot_mits_sum, aes(xintercept = median, color = soil),
                 linetype = "dashed", linewidth = 1)+
      geom_vline(data = boot_mits_sum, aes(xintercept = q025, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      geom_vline(data = boot_mits_sum, aes(xintercept = q975, color = soil),
                  linetype = "dotted", linewidth = 0.5)+
      scale_x_continuous(limits = c(1,10), breaks = seq(1, 10, by = 1))+
      labs(title = "boot_mitscherlich()", x = "CSTV for RY=90%")+
      theme_bw()

### `set.seed()`

Note the use of `set.seed()` prior to bootstrapping ensures that results are reproducible. Even though random sampling is being used, `set.seed()` gives “an integer vector, containing the random number generator (RNG) state for random number generation in R.” This means that next time the code is run, the same bootstrap samples can be reproduced. Without `set.seed()`, your results will vary (even if just slightly) each time the code is run.

### References

Davison, A., Hinkley, D. (1997). Bootstrap Methods and their Application (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511802843

Efron, B. 1979. “Bootstrap Methods: Another Look at the Jackknife.” The Annals of Statistics 7 (1). https://doi.org/10.1214/aos/1176344552

Efron, B., Tibshirani, R.J. 1993. “Efficient Bootstrap Computations.” In, 338–57. Springer US. https://doi.org/10.1007/978-1-4899-4541-9_23

Faraway, J.J. 2014. Linear Models with r. 2nd ed. CRC press. NY. USA. 270p. ISBN 9781439887332. https://julianfaraway.github.io/faraway/LMR/

#### Additional webpages on bootstrapping in R

Bootstrap and Simulation for Nonlinear Models - nlraa https://femiguez.github.io/nlraa-docs/Bootstrapping.html (relevant to agriculture, and addresses resampling of residuals instead of cases)

Bootstrap confidence intervals - tidymodels. https://rsample.tidymodels.org/articles/Applications/Intervals.html