Inputs not proxying for variance, but nevertheless well-correlated

Question

In a summary of this behavioral economics article, it says

Our third finding is that cross-consumer heterogeneity in biases is poorly explained by even a “kitchen sink” of other consumer characteristics, including classical decision inputs, demographics, and measures of survey effort. Most strikingly, we find more bias variance within classical sub-groups widely thought to proxy for behavioral biases than across them. E.g., we find more bias variation with the highest-education group than across the highest- and lowest-education groups.

And then:

Our fifth finding is that there are also some important correlations between biases and classical inputs. Classical inputs and demographics may not explain much of the variance in biases (per finding #3), but some of them are correlated with biases in patterns that align with prior work. Most notably, the average pairwise correlation between cognitive skills and biases is -0.25. Cognitive skills are strongly negatively correlated with most biases, but positively correlated with loss aversion and ambiguity aversion.

I'm having trouble coming up with a simple multivariate model where some input does not proxy for differences in a given statistic between samples, but nevertheless is well-correlated with that statistic. Evidently I need to sharpen up on my stats!

Could someone give me an illustrative toy example of this pattern, and/or perhaps a reading reference? Thanks

Answer 1

Graph it.

They say they get correlations around $0.25$. Let's see how that looks.

library(MASS)
set.seed(2021)
rho <- 0.25
X <- MASS::mvrnorm(1000, c(0, 0), matrix(c(1, rho, rho, 1), 2, 2))
plot(X)
cor(X)

library(MASS)
set.seed(2021)
rho <- 0.2
X <- MASS::mvrnorm(1000, c(0, 0), matrix(c(1, rho, rho, 1), 2, 2))
plot(X)
cor(X)
rho <- 0.9
Y <- MASS::mvrnorm(1000, c(0, 0), matrix(c(1, rho, rho, 1), 2, 2))
points(Y, col = 'red')
cor(X)

I believe they mean something like this, that there is a slight correlation, just not enough to explain much of the variability (tight fit to a diagonal line).