Beyond normality: the bootstrap method for hypothesis testing

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tl;dr: Parametric bootstrap methods can be used to test hypothesis and calculate p values while assuming any particular population distribution we may want. Non-parametric bootstrapping methods can be used to test hypotheses and calculate p values without having to assume any particular population as long as the sample can be assumed to be representative of the population and one can transform the data adequately to take into account the null hypothesis. The p values from bootstrap methods may differ from those from classical methods, especially when the assumptions of the classical methods do not hold. The different methods of calculation can push a p value beyond the 0.05 threshold which means that statements of statistical significance are sensitive to all the assumptions used in the test.

Introduction

In this article I show how to use parametric and non-parametric bootstrapping to test null hypotheses, with special emphasis on situations when the assumption of normality may not hold. To make it more relevant, I will use real data (from my own research) illustrate the application of these methods. If you get lost somewhere in this article, you may want to take a look at my previous post, where I introduced the basic concepts behind hypothesis testing and sampling distributions. As in the previous post, the analysis will be done in R, so before we get into the details, it is important to properly setup our R session:

for(name in c("ggplot2", "plotly","furrr", "distr6"))
  library(name, character.only = TRUE)
plan(multiprocess) # Turns on parallel computation
set.seed(2019) # Reproducible Monte Carlo simulation

The data I will use consists of measurements of individual plant biomass (i.e. the weight of a plant after we have remove all the water) exposed to a control treatment (C), drought (D), high temperature (HT) and high temperature and drought (HTD). First, let’s take a look at the data:

Biomass = data.frame(Treatment = rep(c("C", "D", "HT", "HTD"), each = 18),
                 Biomass = c(2.03,  4.49,   3.84,   2.66,   7.4,    3.04,   2.63,   7,  5.84,   6.99,   4.15,   5.74,   10.49,  23.3,   14.21,  16.97,  11.56,  17.94, 6.01,    6.94,   6.05,   5.23,   2.47,   6.24,   3.96,   4.47,   2.35,   4.37,   3.33,   6.04,   7.98,   11.44,  10.02,  9.64,   11.19,  12.71, 5.22,    4.61,   7.58,   4.7,    6.68,   4.88,   4.11,   4.28,   5.77,   1.54,   2.79,   7.64,   8.68,   7.68,   12, 7.06,   9.9,    17.94, 3.8, 3.8,    5.14,   6.06,   2.78,   2.63,   3.91,   4.65,   5.62,   4.5,    4.45,   5.44,   8.53,   5.59,   6.14,   4.92,   6.54,   7.01))

p = ggplot(data = Biomass, aes(x = Treatment, y = Biomass, colour = Treatment)) + geom_point() + geom_jitter(width = 0.25) + geom_violin(fill = NA) + theme(legend.position = "none")
ggplotly(p)

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