Threshold selection and normal distributions

Init

options(
  digits = 2
)

library(kirkegaard)

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library(patchwork)

theme_set(theme_bw())

Simulation

#make normal IQ data
set.seed(1)
d = tibble(
  IQ = rnorm(10e3, mean = 100, sd = 15)
) %>% mutate(
  SAT = (standardize(IQ) + rnorm(n(), sd = .7)) %>% standardize(),
  SAT_observed = (SAT + rnorm(n(), sd = .3)) %>% standardize()
)

#cors
cor(d)

##                IQ  SAT SAT_observed
## IQ           1.00 0.82         0.79
## SAT          0.82 1.00         0.96
## SAT_observed 0.79 0.96         1.00

#plots
ggplot(d, aes(IQ)) +
  geom_histogram() +
  ggtitle("True IQ before IQ selection") +
  ggplot(d %>% filter(IQ > 120), aes(IQ)) +
  geom_histogram() +
  ggtitle("True IQ after IQ threshold selection") +
  ggplot(d %>% filter(IQ > 120), aes(SAT)) +
  geom_histogram() +
  ggtitle("True SAT after IQ threshold selection") +
  ggplot(d %>% filter(IQ > 120), aes(SAT_observed)) +
  geom_histogram() +
  ggtitle("Observed SAT after IQ threshold selection")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.