randomisation_theory.Rmd

---
title: "The logic of randomisation methods in statistics"
author: "Brad Duthie"
output:
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  beamer_presentation:
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header-includes:
- \usepackage{hyperref}
- \usepackage{caption}
- \usepackage{booktabs}
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colorlinks: true
linkcolor: blue
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---


## Two probability models

```{r, echo = FALSE}
library(knitr);
opts_chunk$set(echo = FALSE);
```

\LARGE

\begin{enumerate}
\item General idea of the key concept
\item Step by step example demonstrating the process
\item Related methods and tests
\end{enumerate}

## Assumptions on which statistical inference is made [1]

\normalsize


::: columns

:::: column

\underline{\textbf{Population model}}\vspace{2mm}
\begin{itemize}
\item Assumes random sampling from population
\item Assumes population distribution
\item Compares test statistic to distribution (e.g., $t$, $F$)
\item Can generalise to population of interest
\end{itemize} \pause

::::

:::: column

\underline{\textbf{Randomisation model}}\vspace{2mm}
\begin{itemize}
\item Does not assume random sampling form population
\item No assumption of population distribution
\item Randomises data to build a null distribution 
\item Statistical inference limited to sample$^{*}$ \pause 
\end{itemize}

::::

:::

\vspace{3mm}

*Note that we can generalise verbally [1, 2].


\footnotetext[1]{Ernst, M D. 2004. \textit{Stat. Sci.} \href{https://doi.org/10.1214/088342304000000396}{19:676-685}.}
\footnotetext[2]{Ludbrook, J, \& H Dudley. 1998. \textit{Am. Stat.}, \href{https://www.jstor.org/stable/2685470}{52:127-132}.}
\footnotetext[3]{Manly, B F J. 2007. \textit{Randomization, Bootstrap and Monte Carlo Methods in Biology} (3rd ed.). Chapman \& Hall/CRC.}


## The general idea of the randomisation approach


\begin{enumerate}
\item Randomly and repeatedly shuffle data in some way to generate a null distribution that is neutral with respect to our test statistic. \pause
\item Compare the observed test statistic with the null distribution of test statistics generated in step 1. \pause
\item Determine the \textit{probability of observing a test statistic at least as extreme as the one we actually observed given the null distribution.}
\end{enumerate}


\footnotetext[1]{Ernst, M D. 2004. \textit{Stat. Sci.} \href{https://doi.org/10.1214/088342304000000396}{19:676-685}.}
\footnotetext[2]{Manly, B F J. 2007. \textit{Randomization, Bootstrap and Monte Carlo Methods in Biology} (3rd ed.). Chapman \& Hall/CRC.}


## Empirical data from fig-fig wasp associations

```{r, out.width = "100%", echo = FALSE}
include_graphics("images/fig_tree.jpg");
```

Figs sampled from *Ficus petiolaris* in Baja, Mexico


## Do SO1 and SO2 fig wasps differ in ovipositor length?

```{r, out.width = "100%", echo = FALSE}
include_graphics("images/fig_wasps.jpg");
```

Cryptic species of *Idarnes* non-pollinating fig wasps

\footnotetext[1]{Duthie, A B, et al. 2015. \textit{Am. Nat.} \href{http://bradduthie.github.io/DuthieEtAl2015.pdf}{186:151-158}.}



## Ovipositor measurements for SO1 and SO2


```{r, echo = FALSE}
dat <- read.csv("data/wing_loadings.csv", sep = "\t");
SO1 <- dat[dat[,1] == "SO1", c(1, 11)];
SO2 <- dat[dat[,1] == "SO2", c(1, 11)];
rownames(SO1) <- NULL;
rownames(SO2) <- NULL;
colnames(SO1) <- c("Species", "Ovipositor (mm)");
colnames(SO2) <- c("Species", "Ovipositor (mm)");
```

\footnotesize

::: columns

:::: column


```{r, echo = FALSE}
kable(SO1);
```

::::

:::: column

```{r, echo = FALSE}
kable(SO2);
```

::::

:::

\footnotetext[1]{Duthie, A B, et al. 2015. \textit{Am. Nat.} \href{http://bradduthie.github.io/DuthieEtAl2015.pdf}{186:151-158}.}

\normalsize



## Measurement data for two fig wasp species

::: columns

:::: column

\textbf{Summary statistics for SO1}
\hrule\vspace{4mm}

```{r}
summary(SO1["Ovipositor (mm)"]);
```

$$\hat{\sigma}_{SO1} = `r sd(SO1[,2])`$$

::::

:::: column

\textbf{Summary statistics for SO2}
\hrule\vspace{4mm}

```{r}
summary(SO2["Ovipositor (mm)"]);
```

$$\hat{\sigma}_{SO2} = `r sd(SO2[,2])`$$

::::

:::

\vspace{10mm}

\textbf{Null hypothesis that ovipositor length does not differ with respect to species ($\mathbf{H_{0}: E[SO1] = E[SO2]}$)}

\footnotetext[1]{Duthie, A B, et al. 2015. \textit{Am. Nat.} \href{http://bradduthie.github.io/DuthieEtAl2015.pdf}{186:151-158}.}




## Calculating a p-value from randomised data


The difference between mean ovipositor lengths was:

$$E[SO1] - E[SO2] = `r mean(SO1[,2]) - mean(SO2[,2])`$$

\vspace{8mm}\pause

Assume that you randomly shuffle group identity (i.e., SO1 \& SO2) and recalculate the difference between means 9999 times, finding that 19 random resuffles are as or more extreme (i.e., larger absolute value) than the observed difference. \textbf{How do you calculate a p-value?}\pause



$$\hat{P} = \frac{19+1}{9999 + 1} = 0.002.$$

## Randomisation not restricted to categorical variables

\scriptsize

::: columns

:::: column


```{r, echo = FALSE}
SO1c <- dat[dat[,1] == "SO1", c(1, 11, 15)];
colnames(SO1c) <- c("Species", "Ovipositor (mm)", "Wing (mm)");
rownames(SO1c) <- NULL;
kable(SO1c);
```

::::

:::: column

\vspace{20mm}

```{r, echo = FALSE}
par(mar = c(5, 5, 1, 1));
plot(SO1c[,2], SO1c[,3], xlab = "Ovipositor length (mm)", pch = 20,
     ylab = "Wing length (mm)", cex.axis = 1.5, cex.lab = 2.75, cex = 2.5);
```


::::

:::

\vspace{2mm}
\normalsize

Compare estimated r = `r round(cor(SO1c[,2], SO1c[,3]), digits = 3)` with null distribution by randomly shuffling ovipositor length



## Null distribution not restricted to existing data


\scriptsize

::: columns

:::: column


```{r, echo = FALSE}
fpet <- read.csv("data/FpetTrees.csv");
fpet <- fpet[1:18,]
fptt <- fpet[,2:4];
kable(fptt);
```


::::

:::: column

\vspace{20mm}

```{r, echo = FALSE}
par(mar = c(5, 5, 1, 1));
plot(x = fpet[,4], y = fpet[,3], xlab = "Longitude", 
     ylab = "Latitude", cex = 2.5,
     pch = 20, cex.lab = 2.5, cex.axis = 1.25)
smpsze <- paste("N =", dim(fpet)[1]);
text(x = -113.1103, y = 28.2915, labels = smpsze, cex = 2.5);
```


::::

:::

\vspace{2mm}
\normalsize

Compare estimated n$^{th}$ nearest neighbour with null distribution by placing `r dim(fpet)[1]` trees in new locations.



## Assumptions on which statistical inference is made [1]

\normalsize


::: columns

:::: column

\underline{\textbf{Population model}}\vspace{2mm}
\begin{itemize}
\item Assumes random sampling from population
\item Assumes population distribution
\item Compares test statistic to distribution (e.g., $t$, $F$)
\item Can generalise to population of interest
\end{itemize}

::::

:::: column

\underline{\textbf{Randomisation model}}\vspace{2mm}
\begin{itemize}
\item Does not assume random sampling form population
\item No assumption of population distribution
\item Randomises data to build a null distribution 
\item Statistical inference limited to sample$^{*}$ 
\end{itemize}

::::

:::

\vspace{3mm}

*Note that we can generalise verbally [1, 2].


\footnotetext[1]{Ernst, M D. 2004. \textit{Stat. Sci.} \href{https://doi.org/10.1214/088342304000000396}{19:676-685}.}
\footnotetext[2]{Ludbrook, J, \& H Dudley. 1998. \textit{Am. Stat.}, \href{https://www.jstor.org/stable/2685470}{52:127-132}.}
\footnotetext[3]{Manly, B F J. 2007. \textit{Randomization, Bootstrap and Monte Carlo Methods in Biology} (3rd ed.). Chapman \& Hall/CRC.}