Regression_to_mean.qmd

---
title: "Regression to the mean in pre-post-designs"
---



Assumption: In a pre-post-treatment-control design, I would assume that the regression weight from Pre -> Post is 1: 

$$BDI_{post} = b_0 + 1*BDI_{pre} + e$$

$$BDI_{post} = b_0 + b_1*BDI_{pre} + b_2*treatment + e$$

Hence, the pre value simply is carried forward to the post value. 
We add random error, as participants of course go somewhat up and down, but there is no systematic trend.

```{r}
  library(Rfast)
  n <- 10000
  pre_post_cor <- 0.9
  mu <- c(23, 23)
  sigma <- matrix(c(117, pre_post_cor*sqrt(117)*sqrt(117), pre_post_cor*sqrt(117)*sqrt(117), 117), nrow=2, byrow=TRUE)
  BDI <- rmvnorm(n, mu, sigma) |> data.frame()
  
  
  xi <- rnorm(n, mean=23, sd=sqrt(117))
  pre <-  1*xi + rnorm(n, mean=0, sd=2)
  post <- 1*xi + rnorm(n, mean=0, sd=2)
  
  cor(pre, post)
  
  
  colMeans(BDI)
  var(BDI)
  
  names(BDI) <- c("pre", "post")
  BDI$id <- 1:nrow(BDI)
  summary(lm(post~pre, BDI))
  
  library(tidyr)
  library(ggplot2)
  BDI_long <- pivot_longer(BDI, c(pre, post), names_to = "time")
  ggplot(BDI_long, aes(x=time, y=value, group=id)) + geom_line()
  
  # typically RTM pattern: The pre-post difference is
  BDI$diff <- BDI$post-BDI$pre
  BDI$absdiff <- abs(BDI$post-BDI$pre)
  hist(BDI$diff)
  mean(BDI$diff)
  summary(lm(diff~pre, BDI))
  ggplot(BDI, aes(x=pre, y=absdiff)) + geom_point() + geom_smooth()
  
  ggplot(BDI, aes(x=pre, y=post)) + geom_point() + geom_smooth()
  
  BDI$predicted <- predict(lm(post~pre, BDI))
  mean(BDI$predicted)
  var(BDI$predicted)
  
  # The predicted values have the same mean (so no systematic treatment effect, as expected)
  # but much smaller variance:
  
  BDI_long2 <- pivot_longer(BDI, c(pre, post, predicted), names_to = "cat")
  
  library(ggplot2)
ggplot(BDI_long2, aes(x=as.factor(cat), y=value)) + 
  ggdist::stat_halfeye(adjust = .5, width = .3, .width = 0, justification = -.3, point_colour = NA) + 
  geom_boxplot(width = .1, outlier.shape = NA) +
  gghalves::geom_half_point(side = "l", range_scale = .4, alpha = .5)
```

Assumed that we use the predicted scores at t2 (post) and predict the next scores at t3 -- will it shrink ever more, until all data points are at the mean?
But this is not substantively reflected in the raw scores. Is it an artifact of regression?