How to unscale the coefficients from an lmer()-model fitted with a scaled response

Question

I fitted a model in R with the lmer()-function from the lme4 package. I scaled the dependent variable:

    mod <- l         


        

Answer 1

Updated: generalized to allow for scaling of the response as well as the predictors.

Here's a fairly crude implementation.

If our original (unscaled) regression is

Y = b0 + b1*x1 + b2*x2 ... 

Then our scaled regression is

(Y0-mu0)/s0 = b0' + (b1'*(1/s1*(x1-mu1))) + b2'*(1/s2*(x2-mu2))+ ...

This is equivalent to

Y0 = mu0 + s0((b0'-b1'/s1*mu1-b2'/s2*mu2 + ...) + b1'/s1*x1 + b2'/s2*x2 + ...)

So bi = s0*bi'/si for i>0 and

b0 = s0*b0'+mu0-sum(bi*mui)

Implement this:

 rescale.coefs <- function(beta,mu,sigma) {
    beta2 <- beta ## inherit names etc.
    beta2[-1] <- sigma[1]*beta[-1]/sigma[-1]
    beta2[1]  <- sigma[1]*beta[1]+mu[1]-sum(beta2[-1]*mu[-1])
    beta2
 }

Try it out for a linear model:

m1 <- lm(Illiteracy~.,as.data.frame(state.x77))
b1 <- coef(m1)

Make a scaled version of the data:

ss <- scale(state.x77)

Scaled coefficients:

m1S <- update(m1,data=as.data.frame(ss))
b1S <- coef(m1S)

Now try out rescaling:

icol <- which(colnames(state.x77)=="Illiteracy")
p.order <- c(icol,(1:ncol(state.x77))[-icol])
m <- colMeans(state.x77)[p.order]
s <- apply(state.x77,2,sd)[p.order]
all.equal(b1,rescale.coefs(b1S,m,s))  ## TRUE

This assumes that both the response and the predictors are scaled.

• If you scale only the response and not the predictors, then you should submit (c(mean(response),rep(0,...)) for m and c(sd(response),rep(1,...)) for s (i.e., m and s are the values by which the variables were shifted and scaled).
• If you scale only the predictors and not the response, then submit c(0,mean(predictors)) for m and c(1,sd(predictors)) for s.