---
title: 'Formative Exercise 09: Functions'
output: 
  html_document:
    df_print: kable
---

```{r setup, include=FALSE}
# please do not alter this code chunk
knitr::opts_chunk$set(echo = TRUE,
                      message = FALSE, 
                      error = TRUE)

library(tidyverse)
library(broom)
```

Edit the code chunks below and knit the document. You can pipe your objects to `glimpse()` or `print()` to display them.

## Basic Iteration Functions

### Question 1

Set the vector `v1` equal to the following: 11, 13, 15, 17, 19, ..., 99, 101 (use a function; don't just type all the numbers).

```{r Q1}
v1 <- NULL
```

### Question 2

Set the vector `v2` equal to the following: "A" "A" "B" "B" "C" "C" "D" "D" "E" "E" (note the letters are all uppercase).

```{r Q2}
v2 <- NULL
```

### Question 3

Set the vector `v3` equal to the words "dog" 10 times, "cat" 9 times, "fish" 6 times, and "ferret" 1 time. 

```{r Q3}
v3 <- NULL
```

## map and apply functions

### Question 4a

Use `apply()` or `map()` functions to create a list of 11 vectors of 100 numbers sampled from 11 random normal distributions with means of 0 to 1.0 (in steps of 0.1) and SDs of 1. Assign this list to the object `samples`. Set the seed to `321` before you generate the random numbers to ensure reproducibility.


```{r Q4a}
samples <- NULL
```


### Question 4b

Use `apply()` or `map()` functions to create a vector of the sample means from the list `samples` in the previous question.

```{r Q4b}
sample_means <- NULL
```

## Custom functions

### Question 5a

Write a function called `my_add` that adds two numbers (`x` and `y`) together and returns the results.

```{r Q5a}
my_add <- NULL
```

### Question 5b

Create a vector testing your function `my_add`. Every item in the vector should evaluate to `TRUE` if your function is working correctly.

```{r Q5b}
my_add_test <- NULL
```

## Error handling

### Question 6

Copy the function `my_add` above and add an error message that returns "x and y must be numbers" if `x` or `y` are not both numbers.

```{r Q6}
my_add <- NULL
```

## Building up a custom function

### Question 7

Create a tibble called `dat` that contains 20 rows and three columns: `id` (integers 101 through 120), `pre` and `post` (both 20-item vectors of random numbers from a normal distribution with mean = 0 and sd = 1). Set seed to `90210` to ensure reproducible values.

```{r Q7}
dat <- NULL
```

### Question 8

Run a two-tailed, *paired-samples* t-test comparing `pre` and `post`. (check the help for `t.test`)

```{r Q8}
t <- NULL
```

### Question 9

Use `broom::tidy` to save the results of the t-test in question 8 in a table called `stats`.

```{r Q9}
stats <- NULL
```

### Question 10

Create a function called `report_t` that takes a data table as an argument and returns the result of a two-tailed, paired-samples t-test between the columns `pre` and `post` in the following format: 

"The mean increase from pre-test to post-test was #.###: t(#) = #.###, p = 0.###, 95% CI = [#.###, #.###]." 

Hint: look at the function `paste0()` (simpler) or `sprintf()` (complicated but more powerful).

NB: Make sure all numbers are reported to three decimal places (except degrees of freedom).

```{r Q10}
report_t <- NULL
```


### Question 11

Use inline R to include the results of `report_t()` on the `dat` data table in a paragraph below.



## Functions and GLM

### Question 12

Write a function to simulate data with the form.

$Y_i = \beta_0 + \beta_1 X_i + e_i$

The function should take arguments for the number of observations to return (`n`), the intercept (`b0`), the effect (`b1`), the mean and SD of the predictor variable X (`X_mu` and `X_sd`), and the SD of the residual error (`err_sd`). The function should return a tibble with `n` rows and the columns `id`, `X` and `Y`.

```{r Q12}

sim_lm_data <- function(n) {
  # add code here and define arguments above
}

dat12 <- sim_lm_data(n = 10) %>% print() # do not edit
```

### Question 13

Use the function from Question 12 to generate a data table with 10000 subjects, an intercept of 80, an effect of X of 0.5, where X has a mean of 0 and SD of 1, and residual error SD of 2.

```{r Q13a}
dat13 <- NULL
```

Analyse the data with `lm()`. Find where the analysis summary estimates the values of `b0` and `b1`. What happens if you change the simulation values?

```{r Q13b}
mod13 <- NULL
```


### Question 14

Use the function from Question 6 to calculate power by simulation for the effect of X on Y in a design with 50 subjects, an intercept of 80, an effect of X of 0.5, where X has a mean of 0 and SD of 1, residual error SD of 2, and alpha of 0.05.

Hint: use `broom::tidy()` to get the p-value for the effect of X.

```{r Q14}
power <- NULL
```

### Question 15

Calculate power (i.e., the false positive rate) for the effect of X on Y in a design with 50 subjects where there is no effect and alpha is 0.05.

```{r Q15}
false_pos <- NULL
```

Make a histogram of the p-values from the simulation above. Use geom_histogram with `binwidth=0.05` and `boundary=0`. What kind of distribution is this?

```{r Q15b}
ggplot()
```

