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
\begin{equation} X_i(t_{ij}) = \eta + Z_i(t_{ij})w_i + \epsilon_{ij}, \end{equation}
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
\begin{equation} \Sigma = \begin{bmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{bmatrix}. \end{equation}
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.
