This is problem set #4, in which we want you to integrate your knowledge of data wrangling with some basic simulation skills. It’s a short problem set to help consolidate your ggplot2 skills and then help you get your feet wet in testing statistical concepts through “making up data” rather than consulting a textbook or doing math.
For ease of reading, please separate your answers from our text by marking our text with the > character (indicating quotes).
Part 1: ggplot practice
This part is a warmup, it should be relatively straightforward ggplot2 practice.
Load data from Frank, Vul, Saxe (2011, Infancy), a study in which we measured infants’ looking to hands in moving scenes. There were infants from 3 months all the way to about two years, and there were two movie conditions (Faces_Medium, in which kids played on a white background, and Faces_Plus, in which the backgrounds were more complex and the people in the videos were both kids and adults). An eye-tracker measured children’s attention to faces. This version of the dataset only gives two conditions and only shows the amount of looking at hands (other variables were measured as well).
library(tidyverse)
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.3 ✔ readr 2.1.4
✔ forcats 1.0.0 ✔ stringr 1.5.0
✔ ggplot2 3.4.4 ✔ tibble 3.2.1
✔ lubridate 1.9.3 ✔ tidyr 1.3.0
✔ purrr 1.0.2
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
fvs <-read_csv("data/FVS2011-hands.csv")
Rows: 232 Columns: 4
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (1): condition
dbl (3): subid, age, hand.look
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
First, use ggplot to plot a histogram of the ages of children in the study. NOTE: this is a repeated measures design, so you can’t just take a histogram of every measurement.
# Assuming there's an identifier for each child, select distinct agesunique_ages <- fvs %>%distinct(subid, .keep_all =TRUE) %>%select(age)# Plot the histogram using ggplotggplot(unique_ages, aes(x = age)) +geom_histogram(binwidth =1, fill ="blue", color ="black") +labs(title ="Histogram of Ages", x ="Age (in months)", y ="Count of Children") +theme_minimal()
Second, make a scatter plot showing hand looking as a function of age and condition. Add appropriate smoothing lines. Take the time to fix the axis labels and make the plot look nice.
ggplot(fvs, aes(x = age, y = hand.look, color = condition)) +geom_point(alpha =0.6) +# Set transparency to see overlapping pointsgeom_smooth(method ="lm", se =TRUE) +# Add linear smoothing lines without confidence intervalslabs(title ="Hand Looking as a Function of Age and Condition",x ="Age (in months)",y ="Proportion of Time Looking at Hands",color ="Condition") +scale_color_brewer(palette ="Set1") +# Use a color palette that's distincttheme_minimal() +# Minimal theme for a nicer looktheme(legend.position ="bottom") # Move legend to the bottom
`geom_smooth()` using formula = 'y ~ x'
What do you conclude from this pattern of data?
Overall, children spend longer times looking at hands as they get older, and that this is especially pronounced when looking at hands in context.
What statistical analyses would you perform here to quantify these differences?
I would run an linear mixed effects model with age, condition, and their interaction predicting looking time.
Part 2: Simulation
library(tidyverse)
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).
The goal of these t-tests are to determine, based on 30 observations, whether the underlying distribution (in this case a normal distribution with mean 0 and standard deviation 1) has a mean that is different from 0. In reality, the mean is not different from 0 (we sampled it using rnorm), but sometimes the 30 observations we get in our experiment will suggest that the mean is higher or lower. In this case, we’ll get a “significant” result and incorrectly reject the null hypothesis of mean 0.
What’s the proportion of “significant” results (\(p < .05\)) that you see?
First do this using a for loop.
# Parameters for the simulationn_simulations <-10000sample_size <-30alpha <-0.05significant_results <-0for (i in1:n_simulations) {# Sample from a normal distribution with mean = 0 and sd = 1 sample_data <-rnorm(sample_size, mean =0, sd =1)# Perform a t-test against the null hypothesis (mean = 0) test_result <-t.test(sample_data, mu =0)# If the p-value is less than alpha, increment the counterif (test_result$p.value < alpha) { significant_results <- significant_results +1 }}# Calculate the proportion of significant resultsproportion_significant <- significant_results / n_simulationsproportion_significant
[1] 0.0532
Next, do this using the replicate function:
# Parameters for the simulationn_simulations <-10000sample_size <-30alpha <-0.05# Use replicate to run the t-testsset.seed(123) # Setting a seed for reproducibilityp_values <-replicate(n_simulations, { sample_data <-rnorm(sample_size, mean =0, sd =1) test_result <-t.test(sample_data, mu =0) test_result$p.value})# Calculate the proportion of significant resultsproportion_significant <-mean(p_values < alpha)proportion_significant
[1] 0.0465
How does this compare to the intended false-positive rate of \(\alpha=0.05\)?
It’s very close to it.
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 the true mean is different from 0. 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() {# Sample the first batch of 30 observations data <-rnorm(30, mean =0, sd =1)# Perform the initial t-test initial_test <-t.test(data)# Check the p-value and decide whether to continueif (initial_test$p.value <0.05|| initial_test$p.value >0.25) {# If the p-value is < .05 or > .25, stop and return the resultreturn(initial_test$p.value) } else {# If the p-value is between .05 and .25, sample another 30 observations data2 <-rnorm(30, mean =0, sd =1)# Combine the data and perform the final t-test final_test <-t.test(c(data, data2))return(final_test$p.value) }}
Now call this function 10k times and find out what happens.
# Set up a variable to store the resultsp.values <-replicate(10000, double.sample())# Calculate the proportion of significant results (p < .05)proportion_significant <-mean(p.values <0.05)# Print the resultproportion_significant
[1] 0.073
Is there an inflation of false positives? How bad is it?
Yes! 0.073/0.0465 ~= 1.5 times more!
Now modify this code so that you can investigate this “double the sample” rule in a bit more depth. In the previous question, the researcher doubles the sample only when they think they got “close” to a significant result, i.e. when their not-significant p is less than 0.25. What if the researcher was more optimistic? See what happens in these 3 other scenarios:
The researcher doubles the sample whenever their pvalue is not significant, but it’s less than 0.5.
The researcher doubles the sample whenever their pvalue is not significant, but it’s less than 0.75.
The research doubles their sample whenever they get ANY pvalue that is not significant.
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 <-function(p.upper.limit) {# Sample the first batch of 30 observations data <-rnorm(30, mean =0, sd =1)# Perform the initial t-test initial_test <-t.test(data)# Check the p-value against the upper limitif (initial_test$p.value <0.05|| initial_test$p.value > p.upper.limit) {# If the p-value is < .05 or > p.upper.limit, stop and return the resultreturn(initial_test$p.value) } else {# If the p-value is between .05 and p.upper.limit, sample another 30 observations data2 <-rnorm(30, mean =0, sd =1)# Combine the data and perform the final t-test final_test <-t.test(c(data, data2))return(final_test$p.value) }}# Run the simulation with different p-value upper limitsset.seed(123) # for reproducibilityp.upper.limits <-c(0.25, 0.5, 0.75, 1)results <-tibble(p.upper.limit = p.upper.limits) %>%mutate(p.values =map(p.upper.limit, ~replicate(100000, double.sample(.x))),proportion_significant =map_dbl(p.values, ~mean(.x <0.05)) )# Check the resultsresults
What do you conclude on the basis of this simulation? How bad is this kind of data-dependent policy?
These policies inflate false positives! Specifically, if you decrease your threshold to double the sample and rerun results, you’ll get more false positives!
