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 1.1.4 ✔ readr 2.1.5
✔ forcats 1.0.1 ✔ stringr 1.5.2
✔ ggplot2 4.0.0 ✔ tibble 3.3.0
✔ lubridate 1.9.4 ✔ tidyr 1.3.1
✔ purrr 1.1.0
── 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.
library(tidyverse)# Remove duplicate rows per childfvs_unique <- fvs %>%distinct(subid, .keep_all =TRUE)# Plot histogram of child agesggplot(fvs_unique, aes(x = age)) +geom_histogram(binwidth =1, color ="white") +labs(title ="Histogram of Child Ages",x ="Age (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 =0.5) +geom_smooth(method="lm") +labs(title ="Hand looking as a function of age and condition",x ="Age (months)",y ="Proportion of looking to hands",color ="Condition" ) +theme_minimal(base_size =14)
`geom_smooth()` using formula = 'y ~ x'
What do you conclude from this pattern of data?
The scatterplot suggests that older infants tend to look more at hands overall, regardless of condition. However, the Faces_Plus condition shows a noticeably stronger positive relationship between age and hand looking than the Faces_Medium condition. Younger infants look very little at hands in both conditions, but as children get older, looking to hands increases more steeply in the Faces_Plus videos—possibly because the scenes are more complex and contain more people, making hands more informative or salient. Overall, the pattern suggests both an age-related increase in attention to hands and a condition-specific difference in how strongly hand looking scales with age.
What statistical analyses would you perform here to quantify these differences?
To quantify these differences, I would run a regression model predicting hand looking from age (continuous predictor), condition (categorical predictor), and their interaction (age × condition). Because the dataset contains repeated observations per child, the appropriate model would be a linear mixed-effects regression with a random intercept for subid.
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.
set.seed(123) # for reproducibilityn_sim <-10000# number of simulationspvals <-numeric(n_sim) # empty vector to store p-valuesfor (i in1:n_sim) { x <-rnorm(30, mean =0, sd =1) # sample from null distribution test <-t.test(x, mu =0) # one-sample t-test pvals[i] <- test$p.value # store p-value}# proportion of p < .05 (false positives)mean(pvals <0.05)
[1] 0.0465
Next, do this using the replicate function:
set.seed(123)pvals_rep <-replicate(10000,t.test(rnorm(30, mean =0, sd =1), mu =0)$p.value)mean(pvals_rep <0.05)
[1] 0.0465
How does this compare to the intended false-positive rate of \(\alpha=0.05\)?
The proportion of “significant” results using replicate() ended up being right around 0.05, which is what we’d expect since the data are actually coming from a mean of 0. So basically, the t-test is doing what it’s supposed to—about 5% of the time it gives a false positive just by chance.
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() {# First sample of 30 x1 <-rnorm(30, mean =0, sd =1) p1 <-t.test(x1, mu =0)$p.value# If p < .05 or p > .25, stop and return the first p-valueif (p1 <0.05| p1 >0.25) {return(p1) }# Otherwise, collect 30 more points and test on all 60 x2 <-c(x1, rnorm(30, mean =0, sd =1)) p2 <-t.test(x2, mu =0)$p.valuereturn(p2)}
Now call this function 10k times and find out what happens.
Is there an inflation of false positives? How bad is it?
Yes, the false-positive rate is noticeably higher than .05. Which means this optional-stopping approach makes significant results more common even when the null is true.
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.
What do you conclude on the basis of this simulation? How bad is this kind of data-dependent policy?
As the cutoff gets bigger, the false-positive rate increases. When you double your sample for pretty much any nonsignificant p-value, the false positives get way higher than .05. So this kind of data-dependent stopping makes it much easier to get “significant” results even when nothing is going on.
