This is problem set #3, 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)
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## ✔ readr   1.1.1     ✔ forcats 0.3.0
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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.

#normally-distributed data from a made up 30-person
data =data.frame(
  a=rnorm(n=30),
  group=rep(1:2, each=15))

#t-test for the sigle-measurement experiment
ttest_result=t.test(a~group, data=data)
p_value=(ttest_result$p.value)
significant=(p_value <.05)

#10,000 t-tests with standard, normally-distributed data
significant_vector=c()
for (i in 1:10000) {
  data=data.frame(
    a=rnorm(n=30),
    group=rep(1:2, each=15))

#t-test for the 10000 experiments
ttest_result=t.test(a~group, data=data)
p_value=(ttest_result$p.value)
significant=(p_value <=.05)
significant_vector[i]=significant
}

sum(significant_vector)/10000
## [1] 0.0479

Next, do this using the replicate function:

s=replicate(10000,t.test(rnorm(30))$p.value)
length(s[s <=.05])/10000
## [1] 0.0466

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_greses <- function(){
comienzo <- rnorm(n=30)
    p_comienzo=t.test(comienzo)$p.value
    if(p_comienzo>0.05 || p_comienzo< 0.25){
      final=c(rnorm(n=30), comienzo)
        p_final=t.test(final)$p.value
        return(p_final)
    } else {
      return(p_comienzo)
    } }
    #27 lec2

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

new_p <- replicate(10000,double.sample_greses())
significant_p <- new_p[new_p<= .05]
x=length(significant_p)/10000
print(x)
## [1] 0.0491
false= 1-(.05/x)
false
## [1] -0.01832994

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

There is an inflation of false positives. Of all the values calculated here, i found a false rate of .0825 or 825 false positives for 10,000 samples, which it is more than the 5% of posible p-values =<0.05.

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_greses1 <- function(){
comienzo1 <- rnorm(n=30)
    p_value=t.test(comienzo1)$p.value
    if(p_value<0.05) {
        return(p_value)
    }
    if(p_value>0.25){
      final1 <- c(rnorm(n=30), comienzo1)
      p_value=t.test(final1)$p.value
    }
    if(p_value<.5){
      final2 <- c(rnorm(n=30), comienzo1)
      p_value=t.test(final2)$p.value
    }
    if(p_value<.75){
      final3 <- c(rnorm(n=30), comienzo1)
      p_value=t.test(final3)$p.value
    }
    if(p_value>.05){
      final4 <- c(rnorm(n=30), comienzo1)
      p_value=t.test(final4)$p.value
    }
      return(p_value)
}
p_value1 <-replicate(10000, double.sample_greses1())
p_value_sign <-p_value1[p_value1<=.05]
length(p_value_sign)/10000
## [1] 0.1101

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

Based on the results, I can conclude that when I double the sample based on p-values, it increases the false positive rate. This is interesting because I have seen scholars doing something similar to evaluate wether or not they should get a bigger sample for an specific experiment. I wonder what other less bias methods are there out there to forecast how likely increasing the sampel will lead into the finidng of true positives.