lord.garbage - 5 months ago 21

R Question

I fitted a model in

`R`

`lmer()`

`lme4`

`mod <- lmer(scale(Y)`

~ X

+ (X | Z),

data = df,

REML = FALSE)

I look at the fixed-effect coefficients with

`fixef(mod)`

`> fixef(mod)`

(Intercept) X1 X2 X3 X4

0.08577525 -0.16450047 -0.15040043 -0.25380073 0.02350007

It is quite easy to calculate the means by hand from the fixed-effects coefficients. However, I want them to be unscaled and I am unsure how to do this exactly. I am aware that scaling means substracting the mean from every

`Y`

`lmer()`

Thanks for any help!

Update: The way I presented the model above seems to imply that the dependent variable is scaled by taking the mean over all responses and dividing by the standard deviation of all the responses. Usually, it is done differently. Rather than taking the overall mean and standard deviation the responses are standardized per subject by using the mean and standard deviation of the responses of that subject. (This is odd in an

`lmer()`

Answer

**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`

.