23-distributions.Rmd

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# Distributions

The underlying distribution of the variables in your sample population
makes a big difference in what inferential statistics you can use.
Different methods assume (or don't) different distributions. If you
violate assumptions, sometimes the statistical tests or estimates from
the methods may not be valid.

## Bounded

## Heteroscedascitity vs homoscedasicity

## Theoretical, existing, known

## Simulated, randomized, computational

## When to use either?

It seems like objections to bootstrapping linear models (and presumably
other complex models) fall into two categories: 1. Sampling design isn't
accounted for by complete randomization (ignoring stratification of
categories or other sampling vagaries) 2. It's less elegant (???).

Venables and Ripley 2002, pg 164, say "we see bootstrapping as having
little place in least-squares regression. If the errors are close to
normal, the standard theory suffices. If not, there are better methods
of fitting than least squares, or perhaps the data should be transformed
[...]" Hastie et al. 2008 (Elements of Statistical Learning) seem in
favor of bootstrapping

@johnston_bootstrap_2021 are enthusiastically in favor of the bootstrap
at least for their relatively simple design to replace a t-test. -
<https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8613103/#CR47> -- says do
all the other stuff like deal with random effects and autocorrelation
first. We have done this already.

Example: should we run a Redundancy Analysis (RDA) a la
<https://r.qcbs.ca/workshop10/book-en/redundancy-analysis.html> , which
I understand has multivariate normality assumptions. He has a small
sample size (around 35 I believe) and residuals are not coming out
normal in smaller linear models. -
<https://journals.sagepub.com/doi/10.1177/0049124189018002003> -
<https://statisticsbyjim.com/hypothesis-testing/bootstrapping/> -
<https://online.stat.psu.edu/stat555/node/119/> -
<https://www.sagepub.com/sites/default/files/upm-binaries/21122_Chapter_21.pdf> -
<https://link.springer.com/referenceworkentry/10.1007/978-1-4419-1153-7_84>