exercise_script_with_solutions.R

###############################################################################################################
# This R script contains                                                                                      #
#  - the examples shown in the step-by-step workshop pages                                                    #
#    (for you to run line-by-line to observe the outcome, or to modify and play with)                         #
#  - space called 'YOUR TURN' for you to write your own code to answer the exercises from the workshop pages  #
#  - a possible solution to those exercises                                                                   #
###############################################################################################################

#~~~~~~~~~ Random Numbers Generators and sampling theory -----

# sample 

## x is a sequence
x <- 1:10
x
?sample # default: replace = FALSE
sample(x)
sample(x, replace = TRUE)
sample(letters, size = 10)
sample(x, size = 100, replace = TRUE)

## x is a vector of combined values
x <- c(1,5,8)
x
sample(x, size = 6, replace = TRUE)




# YOUR TURN: generate random numbers
## Sample 100 values between 3 and 103 with replacement







### possible solutions ###
x <- 3:103
sample(x, 100, replace = TRUE)
##########################


# random number generator drawing from specific distributions
?runif  # runif(n, min, max) 
?rpois  # rpois(n, lambda) 
?rnorm  # rnorm(n, mean, sd) 
?rbinom # rbinom(n, prob)	



# YOUR TURN: generate random numbers
## Draw 100 values from a normal distribution with a mean of 0 and a sd of 1









### possible solutions ###
rnorm(n = 100, mean = 0, sd = 1)
rnorm(100) # if you sample from a normal distribution with a mean of 0 and a sd of 1, you do not need to provide them, they are the defaults
rnorm(100,0,1) # you do not need to label the arguments if you provide them in their default order, but you should provide names if they overwrite default values
##########################

## Draw 50 values from a normal distribution with a mean of 10 and sd of 5









### possible solutions ###
rnorm(sd = 5, mean = 10, n = 50) # if you label your arguments you can put them in whatever order you want!
rnrom(50, mean = 10, sd = 5) # using the rules above: non-default arguments in order, followed by named arguments using names
##########################

## Draw 1000 values from a poisson distribution with a lambda of 50









### possible solutions ###
rpois(n = 1000, lambda = 50)
rpois(1000,50)
##########################

## Draw 30 values from a uniform distribution between 0 and 10








### possible solutions ###
runif(n = 30, min = 0, max = 10)
runif(30, max = 10) # we can omit min = 0 because that's the default
##########################



# repeat 
?replicate # replicate(n, expression)
rnorm(10)
mean(rnorm(10))
replicate(10,rnorm(10))
replicate(10, mean(rnorm(100)))
hist(replicate(10, mean(rnorm(100))))

# YOUR TURN: generate random numbers, repeat, and plot
## Replicate 1000 times the mean of 10 values drawn from a uniform distribution between 0 and 10  









### possible solutions ###
replicate(1000, mean(runif(10, max = 10)))
hist(replicate(1000, mean(runif(10, max = 10))))
##########################

## Replicate 100 times the mean of 50 values drawn from a normal distribution of mean 10 and standard deviation 5  










### possible solutions ###
replicate(100, mean(rnorm(50, mean = 10, sd = 5)))
hist(replicate(100, mean(rnorm(50, mean = 10, sd = 5))))
##########################


# set seed
hist(replicate(100, mean(rnorm(10))))
hist(replicate(100, mean(rnorm(10))))
hist(replicate(100, mean(rnorm(10))))

set.seed(10)
hist(replicate(100, mean(rnorm(10))))

set.seed(10)
hist(replicate(100, mean(rnorm(10))))


# defining sample size within a replication (n) and the number of simulation/repeats/replication (nrep)
## single random sample of normal distribution N(0,1) with n = 10
set.seed(10)
x <- rnorm(10) # mean = 0, sd = 1 are the defaults
hist(x, breaks = 10, col = "grey", xlim = c(-4,4))
abline(v = 0, col = "red", lty = 2, lwd = 2)
abline(v = mean(x), col = "blue", lwd = 2)
par(xpd = TRUE) # turn off clipping of legend
# where a function has a long list of arguments, we can put them on a new line each
legend(
  0.9, 
  y = 1.5, 
  legend = c("mean(x)", "0"), 
  lty = c(1, 2), 
  col = c("blue","red")
)

## 24 sims of same distribution N(0,1) with n = 10
set.seed(10)
x24 <- replicate(24, rnorm(10))
par(mfrow = c(3,8), mar = c(0,0,0,0))
# apply is complicated because it takes a function as one of its arguments
x24Plot <- apply(
  x24,
  2,
  function(x) {
    # for fairly simple functions with lots of arguments, we sometimes just cram them all on the same line.
    # it's not great practice but it stops the script getting super long when dealing with graphical objects
    hist(x, col = "grey", xlim = c(-5,5), ylim = c(0,7), breaks = c(-5:5),
         main = "", ylab = "", xlab = "", xaxt = "n", yaxt = "n")
    abline(v = mean(x), col = "blue", lwd = 2)
    abline(v = 0, col = "red", lty = 2, lwd = 2)
  }
)

## distribution of mu and sd from 24 sims N(0,1) with n = 10
par(mfrow = c(1,2), mar = c(5,5,1,1))
hist(apply(x24, 2, mean), main = "Mean", col = "grey", xlim = c(-1,1))
abline(v = 0, col = "red", lty = 2, lwd = 2)
hist(apply(x24, 2, sd), main = "SD",col = "grey", xlim = c(0.6,1.4))
abline(v = 1, col = "red", lty = 2, lwd = 2)

## 24 sims of same distribution N(0,1) with n = 1000
set.seed(10)
x24b <- replicate(24, rnorm(1000))
par(mfrow = c(3,8), mar = c(0,0,0,0))
x24bPlot <- apply(
  x24b,
  2,
  function(x){
    hist(x, col = "grey", xlim = c(-5,5), ylim = c(0,500), breaks = c(-5:5),
         main = "", ylab = "", xlab = "", xaxt = "n", yaxt = "n")
    abline(v = mean(x), col = "blue", lwd = 2)
    abline(v = 0, col = "red", lty = 2, lwd = 2)
  }
)

## distribution of mu and sd from 24 sims N(0,1) with n = 1000
par(mfrow = c(1,2), mar = c(5,5,1,1))
hist(apply(x24b, 2, mean), main = "Mean",col = "grey", xlim = c(-1,1))
abline(v = 0, col = "red", lty = 2, lwd = 2)
hist(apply(x24b, 2, sd), main = "SD",col = "grey", xlim = c(0.6,1.4))
abline(v = 1, col = "red", lty = 2, lwd = 2)

## distribution of mu and sd from 1000 sims N(0,1) with n = 10
set.seed(10)
x1000 <- replicate(1000, rnorm(10))
par(mfrow = c(1,2), mar = c(5,5,1,1))
hist(apply(x1000, 2, mean), main = "Mean",col = "grey")
abline(v = 0, col = "red", lty = 2, lwd = 2)
hist(apply(x1000, 2, sd), main = "SD",col = "grey")
abline(v = 1, col = "red", lty = 2, lwd = 2)


#~~~~~~~~~ Functions -----

# writing a function
## function syntax:
## AwesomeFunctionName <- function(argument1, argument2,…){
##                                                         do stuff here
##                                                        }
## The last thing that appears in the 'do stuff here' section is the function's 
## "return value"

# YOUR TURN: write a function that takes input "nrep", replicates '(mean(rnorm(100)))'
# nrep times, and draws a histogram of the results












### possible solutions ###
#### step 1: the action
mean(rnorm(100))
#### step 2: replicate the action 1000 times
replicate(1000, mean(rnorm(100)))
#### step 3: plot the outcome of those simulations
hist(replicate(1000, mean(rnorm(100))))
#### step 4: replicate the action nrep time, with nrep defined outside the function
nrep <- 1000
replicate(nrep, mean(rnorm(100)))
#### step 5: wrap it in a function:
histrnorm100 <- function(nrep){
  hist(replicate(nrep, mean(rnorm(100))))
}
#### step 6: check that the function works
histrnorm100(9)
histrnorm100(1000)
##########################

# YOUR TURN: modify your function
## to draw a histogram of nrep mean(rnorm(n)), where n is another input
















### possible solutions ###
#### step 4: define parameters outside the function
nrep <- 100
n <- 10
replicate(nrep, mean(rnorm(n)))
#### step 5: wrap the action in a function:
histrnorm_n <- function(nrep, n){
  hist(replicate(nrep, mean(rnorm(n))))
}
#### step 6: check that the function works
histrnorm_n(10,10)
histrnorm_n(10,100)
histrnorm_n(100,100)
histrnorm_n(1000,100)
##########################


#~~~~~~~~~ Simulating no effect and check alpha -----

# YOUR TURN: draw from the same normal distribution twice 
## and see if the sample differ from each other
## will they differ significantly in 5% of the nrep?
### Figure out how to do a t.test in R  
### Generate two vectors of 10 values drawn from N(0,1) and compare them with a t test  
### Figure out how to extract the p-value from that object (HINT use `str` or `names`)
### Write a function simT that generates two vectors of n random normals, compare them with a t test and return the p-value  
### Repeat with nrep = 20 and draw a histogram for n = 10
### Repeat with nrep = 100 and draw a histogram for n = 10    
















### possible solutions ###
#### Figure out how to do a t.test in R
?t.test
#### Generate two vectors of 50  N(0,1) 
x1 <- rnorm(50,0,1)
x2 <- rnorm(50,0,1)
#### Compare them with a t test
t.test(x1,x2)
#### extract p value
str(t.test(x1,x2))
t.test(x1,x2)$p.value
#### write function 
simT <- function(n){
  x1 <- rnorm(n,0,1)
  x2 <- rnorm(n,0,1)
  t.test(x1, x2)$p.value 
}
#### test function
simT(50)

#### repeat function for n = 10 and for different nrep and plot
par(mfrow = c(1,2))
simTRep <- replicate(20, simT(10))
hist(simTRep, breaks = 21, col = c('red',rep('grey',20)), 
     main = "nrep = 20, n = 10", xlab = "pvalue")

simTRep2 <- replicate(100, simT(10))
hist(simTRep2, breaks = 21, col = c('red',rep('grey',20)), 
     main = "nrep = 100, n = 10", xlab = "pvalue")
##########################

#### repeat function for nrep = 1000 and various n
par(mfrow = c(1,2))
simTRep <- replicate(1000, simT(10))
hist(simTRep, breaks = 21, col = c('red',rep('grey',20)), 
     main = "nrep = 1000, n = 10", xlab = "pvalue")

simTRep2 <- replicate(1000, simT(100))
hist(simTRep2, breaks = 21, col = c('red',rep('grey',20)),  
     main = "nrep = 1000, n = 100", xlab = "pvalue")


#~~~~~~~~~ Simulating an effect and check power -----

# we can calculate the power of a t.test for a given sample size using:
power.t.test(n = NULL, delta = 0.5, sd = 1, sig.level = 0.05, power = 0.8)
# the required sample size is 64 per group.

# YOUR TURN: Use your simulation skills to work out the power of a t-test for a given sample size through simulation.
## Write a function which:
### 1. Draws from two random normal distributions with different means and a given sample size
### 2. Compares the means with a t.test and extracts the p.value
## Then, use that function to replicate the function 1000 times using the parameters used in the power calculation.
## Calculate the proportion of p-values that are <0.05











### possible solution ###
#### write new function 
simT2 <- function(n, m1, m2) {
  ##### n is sample size per group, m1 is mean of group 1, m2 is mean of group 2
  x1 <- rnorm(n, m1)
  x2 <- rnorm(n, m2)
  t.test(x1, x2)$p.value
}

##### repeat the function 1000 times 
##### note that we are using a difference of 0.5 between means to match the "delta" 
##### used in the power calculation 
set.seed(100)
p <- replicate(1000, simT2(n = 64, m1 = 0, m2 = 0.5))

#### plot the results
par(mfrow = c(1,1))
hist(p, breaks = 21, col = c('red',rep('grey',20)), 
     main = "nrep = 1000, n = 64, delta = 0.5", xlab = 'pvalue')

#### calculate the proportion "significant" 
prop.table(table(p < 0.05))
##########################

#### power is the probability that the test correctly rejects the null. Since we
#### know the population paramaters (as we set them in our simulation), we know
#### that there really is a difference, and the null should be rejected. The power
#### is therefore the proportion of p.values <0.05

# compare that to calculating the "power" parameter using the function below with all the other parameters provided (including n)
power.t.test(n = 64, delta = 0.5, sd = 1) # the results are similar


#~~~~~~~~~ Simulating for a preregistration  -----

# YOUR TURN: 
## Try to make a dataset that looks like this, using the 
## functions `data.frame()`, `sample()`, and `rnorm()`    

# smoking_status lung_cancer sex      age
# 1            Yes          No   M 12.67918
# 2            Yes         Yes   F 23.71397
# 3             No          No   M 28.87786
# 4            Yes          No   F 28.99327
# 5            Yes         Yes   F 30.41415
# 6             No          No   M 44.60615












#### possible solution ###
set.seed(1234)
N <- 10 # size of your dataset 
yes_no <- c("Yes", "No") # some options for the outcomes in the data

# the dataframe() function let's us create a data frame
# before the " = " is the column name. After " = " is the contents of the columm
df <- data.frame(
    smoking_status = sample(yes_no, size = N, replace = T),
    lung_cancer = sample(yes_no, size = N, replace = T), 
    sex = sample(c("M", "F"), size = N, replace = T),
    age = rnorm(N, 30, sd = 10)
)
# run the parts after the " = " alone to make sure you understand what's going on
sample(c("M", "F"), size = N, replace = T)

head(df)
##########################

# YOUR TURN:
## Run a logistic regression on the data with lung cancer as the outcome and
## adjusting for the other variables. 
## You could try something like:
## glm(lung_cancer ~ smoking_status, sex, age, family = binomial(link = "logit"), data = df)
## Why doesn't it work? Try to trouble shoot and get the code to work!

## HINT: are the variables the correct data type?
## HINT: once the model works, use summary() to look at the results











#### possible solution ###

# This mysterious error message appears because the categorical variables
# need to be factors for the model to run. So we are already learning
# something about how the data need to be that you might not have known
# before trying to run the code on a simulated dataset.

# You can convert the relevant variables to factors
df[1:3] <- lapply(df[1:3], as.factor)

# You could also do this one variable at a time:
# df$smoking_status <- as.factor(df$smoking_status)

# Now we can rerun the model:

m1 <- glm(
  lung_cancer ~ smoking_status + sex + age, 
  family = binomial(link = "logit"), 
  data = df
)

# and look at the results
summary(m1)
##########################