This is problem set #4, in which we want you to integrate your knowledge of data wrangling with some basic simulation skills and some linear modeling.

For ease of reading, please separate your answers from our text by marking our text with the > character (indicating quotes).

library(tidyverse)
## -- Attaching packages -------------------------------------- tidyverse 1.2.1 --
## v ggplot2 3.0.0     v purrr   0.2.5
## v tibble  1.4.2     v dplyr   0.7.4
## v tidyr   0.8.1     v stringr 1.3.1
## v readr   1.1.1     v forcats 0.3.0
## Warning: package 'ggplot2' was built under R version 3.5.1
## -- Conflicts ----------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

Let’s start by convincing ourselves that t-tests have the appropriate false positive rate. Run 10,000 t-tests with standard, normally-distributed data from a made up 30-person, single-measurement experiment (the command for sampling from a normal distribution is rnorm). What’s the mean number of “significant” results?

First do this using a for loop.

sigResults <- 0
for (i in 1:10000){
  x <- rnorm(30)
  p <- t.test(x)$p.
  if(p<0.05) sigResults <- sigResults + 1
}
sigResults
## [1] 493

ANSWER: 518 out of 10000

Next, do this using the replicate function:

results <- replicate(10000, t.test(rnorm(30))$p. < 0.05)
sum(results)
## [1] 503

ANSWER: 490 out of 10000

Ok, that was a bit boring. Let’s try something more interesting - let’s implement a p-value sniffing simulation, in the style of Simons, Nelson, & Simonsohn (2011).

Consider this scenario: you have done an experiment, again with 30 participants (one observation each, just for simplicity). The question is whether their performance is above chance. You aren’t going to check the p-value every trial, but let’s say you run 30 - then if the p-value is within the range p < .25 and p > .05, you optionally run 30 more and add those data, then test again. But if the original p value is < .05, you call it a day, and if the original is > .25, you also stop.

First, write a function that implements this sampling regime.

double.sample <- function () {
  x <- rnorm(30)
  p <- t.test(x)$p.
  if(p > 0.05 & p < 0.25) {
    x <- c(x, rnorm(30))
    p <- t.test(x)$p.
  }
  p
}

Now call this function 10k times and find out what happens.

results <- replicate(10000, double.sample() < 0.05)
sum(results)
## [1] 720

Is there an inflation of false positives? How bad is it?

Yes, there is an inflation. 727 out of 10000, as opposed to 500ish before, which is an effective false positive rate of 7% as opposed to 5%.

Now modify this code so that you can investigate this “double the sample” rule in a bit more depth. Let’s see what happens when you double the sample ANY time p > .05 (not just when p < .25), or when you do it only if p < .5 or < .75. How do these choices affect the false positive rate?

HINT: Try to do this by making the function double.sample take the upper p value as an argument, so that you can pass this through dplyr.

HINT 2: You may need more samples. Find out by looking at how the results change from run to run.

# double sample any time p > .05
p_resample <- .05
double.sample <- function () {
  x <- rnorm(30)
  p <- t.test(x)$p.
  if(p > p_resample) {
    x <- c(x, rnorm(30))
    p <- t.test(x)$p.
  }
  p
}
results <- replicate(10000, double.sample() < 0.05)
sum(results)
## [1] 872
# double sample any time 0.05 < p < 0.5
p_resample <- c(0.05, 0.5)
double.sample <- function () {
  x <- rnorm(30)
  p <- t.test(x)$p.
  if(p > p_resample[1] & p < p_resample[2]) {
    x <- c(x, rnorm(30))
    p <- t.test(x)$p.
  }
  p
}
results <- replicate(10000, double.sample() < 0.05)
sum(results)
## [1] 842
# double sample any time 0.05 < p < 0.75
p_resample <- c(0.05, 0.75)
double.sample <- function () {
  x <- rnorm(30)
  p <- t.test(x)$p.
  if(p > p_resample[1] & p < p_resample[2]) {
    x <- c(x, rnorm(30))
    p <- t.test(x)$p.
  }
  p
}
results <- replicate(10000, double.sample() < 0.05)
sum(results)
## [1] 832

What do you conclude on the basis of this simulation? How bad is this kind of data-dependent policy?

Adding more participants to one’s sample when the p-value does not reach significance increases the actual false positive rate above 5%. Resampling when the initial p-value is greater than 0.05 increases the false positive rate to around 8.4% (841 out of 10000). Resampling when the initial p-value is between 0.05 and 0.5 increases the false positive rate to around 7.9% (794 out of 10000). Resampling when the initial p-value is between 0.05 and 0.75 increases the false positive rate to around 8.2% (817 out of 10000).