基于线性混合模型（lme4）计算重复测量相关系数的疑问

Alright, let's walk through this step by step using your Orange dataset example.

First, you're totally right to skip simple linear regression here—since each tree has repeated measurements of circumference and age, observations aren't independent (they're clustered by tree), which breaks the iid assumption. Grabbing lme4 for a linear mixed model is exactly the right call to account for that clustering.

Now, about that -0.471 value you're seeing: let's start with the most likely scenario. If you ran a random slope and intercept model (the kind that lets both a tree's starting circumference and growth rate vary by individual), your code probably looked like this:

library(lme4)
# Random slope + intercept model
mixed_model <- lmer(circumference ~ age + (age | Tree), data = Orange)
summary(mixed_model)

In this case, the -0.471 is the correlation between the random intercept and random slope for each tree. Here's what that means in plain terms:

• The random intercept represents a tree's extrapolated "starting" circumference at age 0
• The random slope represents how fast each tree's circumference grows with age
• A correlation of -0.471 tells us trees with a higher initial circumference tend to have slower growth rates, and vice versa. That makes biological sense—smaller trees might grow faster to catch up, while larger trees have already put on most of their size.

If you want to verify this calculation manually, you can extract the variance-covariance matrix from the model and compute it yourself:

# Extract random effect variances/covariances
var_corr <- VarCorr(mixed_model)
# Pull out the covariance between intercept and slope
covariance <- var_corr$Tree[1, 2]
# Calculate standard deviations of intercept and slope
sd_intercept <- sqrt(var_corr$Tree[1, 1])
sd_slope <- sqrt(var_corr$Tree[2, 2])
# Compute the correlation
calculated_corr <- covariance / (sd_intercept * sd_slope)

This should spit out the same -0.471 value you saw in the model output.

If you were actually hunting for the overall correlation between circumference and age (not the random effect correlation), you can calculate that while accounting for clustering. One simple way is to use the model's predicted values, but keep in mind the mixed model's fixed effect coefficient for age already tells you the average effect of age on circumference (which should be positive, since older trees are wider).

A quick sanity check: if you only ran a random intercept model (lmer(circumference ~ age + (1 | Tree), data = Orange)), there wouldn't be a correlation value in the output—so you must have included a random slope for age.

内容的提问来源于stack exchange，提问作者zesla