How to recover parameters of this simulated data set? I'm trying to get a handle on the R regression / mixed effects modeling tools by simulating some data using parameters for both population and random effects which I'd like to recover estimates for with the model. Below is my simulated data:
n <- 15
r <- 10
s <- 0.5
z <- 2
x <- sort(runif(n, min = 0, max = 100))
y <- log(2*x + 5*x^2)
sim <- data.frame(time = rep(x, r),
                  conv = rep(y, r) + rnorm(r*n, 0, s) + rep(rnorm(r, 0, z), each = n),
                  run =  factor(rep(1:r, each = n)))

I understand that $y$ is not linear so there needs to be a transformation to make the data linear.
$$exp(y) = 2x+5x^2$$
For these sorts of nonlinear transformations $g(x)$, I thought that $E(g(x)) \ne g(E(x))$, so I thought that the right approach would be to use a generalized model. However, its not clear to me how I should specify my link function. What is apparent from the simulation, is that the plots
plot(log(x), y)
plot(x, exp(y))

is linear. So, I would have thought that I could recover my model with:
fit.glmer <- glmer(data = sim,
                   formula = conv ~ 0 + time + I(time^2) + (1|run),
                   family = gaussian(link = log))

But I'm having trouble getting even close estimates from coef as well as getting confidence intervals around the mean. And how do I get estimates of s and z? 
Could some one give me some pointers? Certainly the parameters are recoverable for this simple of a model.
 A: The intercept suppression (-0) is egregiously violated in your code. While you generated data with no intercept on the linear scale, that is not the case for outcomes on the log scale. The line that intersects (1,1), (2,2) runs through the origin but the line that runs through (1, log(1)), (2, log(2)) does not.
See for instance:
library(lattice); xyplot(log(conv) ~ time | run, data=sim)

The warnings in R tell you how terrible the fit is. "Recovering" parameters is not well-defined. I assume you want to see the values of "s" and "r" that you used to generate the error terms and ICCs. Don't fit the model on the wrong scale for a start.
A: I think this question is more on programming using R, but not a really statistical question. May be out of topic here.
Try following code.
getME(fit.glmer,'X')
getME(fit.glmer,'Z')
getME(fit.glmer,'b')
getME(fit.glmer,'beta')

I assume you are using $ to extract fitting components. It will not work because lme4 is using S4 object (not S3). Details can be found here.
In S4, there are slots which can be accessed with @ operator. This is different from S3, which is essentially a list.
Following code will also work
fit.glmer@beta
fit.glmer@theta

