Variance and interactions....

## The problem with (personal) non-ordinality:

n <- 30000 # 10k fictional people

# a pair of exposures, no measurmennt issues:
xa <- rnorm(n)
xb <- rnorm(n)

# Personal threshold 1 - 4 for each person, 
# reasonable scale design I think by which I mean the bins fill up sort of "normal" like this matters a lot!!
t1 <- runif(n,-2,0)
t2 <- t1 + runif(n,0,1.5)
t3 <- t2 + runif(n,0,2)
t4 <- t3 + runif(n,1,3)
  
  

# simulate simple interaction modle with true y:
y <- scale(xa + xb + xa*xb + rnorm(n))

# Make y ordinal with person specific thresholds
yord <-  (y > t1) + (y > t2) + (y > t3) + (y > t4)
hist(yord)

# scale
yord_s <- scale(yord)
hist(yord_s)

# rerun the interactions on true y before scaling issue:
summary(lm(y ~ xa + xb + xa:xb))
summary(lm(y^2 ~ xa))

# Run on ordinal data, loose some efficiency...
summary(lm(yord_s  ~ xa + xb + xa:xb))

# model the mean, the regress the variance on the predictor, which is the "variance regression model":
res <- lm(yord_s  ~ xa )$residuals
summary(lm(res^2 ~ xa))


### now see about false positives, here there is no interaction at all... :
y <- scale(xa + xb +  rnorm(n))

# make y ord again
yord <-  (y > t1) +(y > t2) + (y > t3) + (y > t4)

# sclae again
yord_s <- scale(yord)
hist(yord_s)

# the analyses with y, no interaction
summary(lm(y ~ xa + xb + xa:xb))
# variance regression with y no evidence for interaction:
res <- lm(y  ~ xa)$residuals
summary(lm(res^2 ~ xa))


# get some false positives if you use the ordinal scale:
summary(lm(yord_s  ~ xa + xb + xa:xb))

# model the mean, the regress the variance on the predictor, also false positives... 
res <- lm(yord_s  ~ xa)$residuals
summary(lm(res^2 ~ xa)) # potenially worse false positives?