Fitting a Linear Mixed Effects Model

Suppose I have data on 4 units, $$X_i(t_{ij})$$, for $$i= 1,2,3,4$$ and $$j = 1,\dots,10$$. That is, I have 10 observations for each unit.

The observations for unit $$i$$ were recorded at times $$t_{i,1}, \dots, t_{i,10}$$, and the time between recordings is constant.

I do not know what $$t_{ij}$$ are, but I know the times between recordings is constant. I could choose $$t_{ij}$$ for $$i = 1,2,3,4$$ to be $$1,\dots,10$$ or $$0,\dots, 9$$, or any other equally spaced sequence of length 10, but which do I pick?

The model I am considering is

$$$$X_i(t_{ij}) = \eta + Z_i(t_{ij})w_i + \epsilon_{ij},$$$$

where $$\eta$$ is the mean, $$Z_i(t_{ij}) = [1, \log(t_{ij})]$$, $$w_i = (w_{0i}, w_{1i})' \sim N(0, \Sigma_w)$$, $$\epsilon_{ij} \sim N(0, \sigma^2)$$, and

$$$$\Sigma = \begin{bmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{bmatrix}.$$$$

In this example, I could not choose $$t = 0, \dots, 9$$ because of the $$\log(t)$$ term. In this example $$t$$ is chosen to be $$1, \dots, 10$$ for each unit. I can see why this was chosen, but could another sequence be chosen? If not why not?

If you consider a small dummy example; here $$x$$ is the data for each unit concatenated into one vector; unit represents which unit the data came from; and tt and tt2 are two different time sequences:

#Data
x = c(1.4806073,  1.2493809,  1.1612487,  1.1252604,  1.0057363,  0.9315882,  0.9091001,  0.8586410,  0.8297411,
0.7403425,  0.9725159,  1.1334336,  1.0876942,  1.0984994,  1.0607115,  1.0308212,  1.0824410,  1.0957307,
1.2664261,  1.1936734,  0.9573181,  1.2820057,  1.3489198,  1.4074968,  1.5751318,  1.6024103,  1.6203662,
1.6487979,  1.6294239,  1.7369618, -0.1207026, -0.1475024, -0.1562043, -0.2044265, -0.2118383, -0.2406047,
-0.1813440, -0.2671027, -0.3339175, -0.2192462)

tt = rep(1:10,4)
tt2 = tt + 20

unit = rep(1:4, each = 10)

DF1 = data.frame(unit, tt, x)
DF2 = data.frame(unit, tt2, x)

#Fits a linear mixed effects model
lme1=lme(x ~ 1, data = DF1, random = ~ I(log(tt))|unit)
lme2=lme(x ~ 1, data = DF2, random = ~ I(log(tt2))|unit)

mean((lme1$$fitted[,2] - lme2$$fitted[,2])^2)
#0.00165495

I was hoping that the two models would produce the same fitted values, but they do not. This means that the choice of sequence makes a difference when fitting the model.

One reason I ask is because I have seen similar models where the author has started counting from 0; and sometimes I have seen authors count from 1.