Additional levels of variability
Modeling different levels of variability
Up to now, the distribution $\qpsii$ or equivalently the equation
\(
\psi_i = \model(\bbeta,c_i,\eta_i),
\)

(1) 
only describes the interindividual variability of the individual parameters $ (\psi_i)$. This model therefore assumes that:
 the individual parameter $\psi_i$ for individual $i$ remains constant during the whole study
 the $N$ individuals in the study are independent, i.e., the parameters $(\psi_i, 1\leq i \leq N)$ are mutually independent.
We will now see that these hypotheses can be weakened by considering additional levels of variability.
Let us look as the first assumption, and now consider introducing intraindividual variability of individual parameters in the model. A first simple model consists of splitting the study into $K$ time periods or "occasions", and assuming that individual parameters can vary from occasion to occasion but remain constant within each occasion. Then, we try to explain part of the intraindividual variability of the individual parameters by piecewiseconstant covariates, i.e., "occasiondependent" or "occasionvarying" (varying from occasion to occasion and constant within an occasion). The remaining part must then be described by random effects.
We will need some additional notation for describing this new statistical model. Let
 $\psi_{ik}$ be the vector of individual parameters of individual $i$ for occasion $k$, where $1\leq i \leq N$ and $1\leq k \leq K$.
 ${c}_{ik}$ be the vector of covariates of individual $i$ for occasion $k$. Some of these covariates remain constant (gender, group treatment, ethnicity, etc.) and others can vary (weight, treatment, etc.).
Let $\bpsi_i = (\psi_{i1}, \psi_{i2}, \ldots , \psi_{iK})$ be the sequence of $K$ individual parameters for individual $i$. The model for $\bpsi_i$ is now a joint distribution:
\(
\bpsi_i \sim \qpsii( \ \cdot \ ; c_{i1}, c_{i2}, \ldots, c_{iK}, \theta).
\)

(2) 
We also need to define:
 $\vari{\eta}{i}{0}$, the vector of random effects which describes the random interindividual variability of the individual parameters.
 $\vari{\eta}{ik}{1}$, the vector of random effects which describes the random intraindividual variability of the individual parameters in occasion $k$, for each $1\leq k \leq K$.
Here and in the following, superscript ${(0)}$ is used to represent interindividual variability, i.e., variability at the individual ("reference") level,
while superscript ${(1)}$ represents interoccasion variability, i.e., variability at the "occasion" level for each individual.
Then, for any individual $i$ and occasion $k$, model (1) becomes
\(
\psi_{ik} = \model(\bbeta,c_{ik},\vari{\eta}{i}{0},\vari{\eta}{ik}{1}).
\)

(3) 
As before, the prediction $\hpsi_{ik}$ of $\psi_{ik}$ is obtained in the absence of random effects:
If $\vari{\eta}{i}{0}\neq 0$, then the parameters $\psi_{ik}$ defined in (3) are no longer independent because they all depend on the same random effect $\vari{\eta}{i}{0}$. The joint distribution $\qpsii$ will therefore depend on the model $\model$ and in particular on the way in which the model integrates the random effects $\vari{\eta}{i}{0}$ and $\vari{\eta}{ik}{1}$. Let us now develop this further.
 Assume first an additive model for the random effects. Here, $\vari{\eta}{i}{0}$ and $\vari{\eta}{ik}{1}$ can be grouped into a random effect $\eta_{ik}$, where
 Assume now a Gaussian model of the form
 Assume furthermore a linear covariate model. For the sake of simplicity, we consider a unique covariate.Extension to multiple covariates, including categorical and continuous covariates, is straightforward. An initial covariate model deduced from our basic linear model proposed in (4) of The covariate model is written
\(
\eta_{ik} = \vari{\eta}{i}{0} + \vari{\eta}{ik}{1}.
\)

(4) 
Here is an example for an individual with three time periods, where the random effect is additive.
Possible decomposition of the random effects of a single subject over 3 time periods

If we also assume that $\vari{\eta}{i}{0}$ and $\vari{\eta}{ik}{1}$ are normally distributed with variancecovariance matrices $\vari{\Omega}{i}{0}$ and $\vari{\Omega}{ik}{1}$, then $\eta_{ik}$ is also normally distributed and the covariance between $\eta_{ik}$ and $\eta_{ik^\prime}$ is
\(
{\rm Cov} \left( \eta_{ik} , \eta_{ik^\prime} \right) = \left\{ \begin{array}{ll}
\vari{\Omega}{i}{0} + \vari{\Omega}{ik}{1} & {\rm if } \ k=k^\prime \\
\vari{\Omega}{i}{0} & {\rm otherwise} . \end{array} \right.
\)

(5) 
Model (3) then reduces to $\psi_{ik} = \model(\bbeta,c_{ik},\eta_{ik})$, where now the $\psi_{ik}$ are not independent.
Here, the $h(\psi_{i1}), \ldots , h(\psi_{iK})$ are correlated Gaussian vectors whose variancecovariance structure is that of the $(\eta_{ik})$ defined in (5).
\(
h(\psi_{ik}) = h(\psi_{\rm pop})+ \beta(c_{ik}  c_{\rm pop}) + \vari{\eta}{i}{0} + \vari{\eta}{ik}{1}.
\)

(6) 
We can then decompose the part of the variability explained by the covariate $c$ into interindividual and intraindividual components, exactly as we did with the random effects. Let $\cpop$ be the reference value of the covariate $c$ in the population as before, and also let $c_i$ be some reference (or typical) value for individual $i$. Then we can write
\(
\begin{array}{ccccc}
c_{ik}\cpop &= &(c_i  \cpop) & + &( c_{ik}  c_i) \\
&= &\vari{d}{i}{0} & + & \vari{d}{ik}{1} ,\\
\end{array}\)

(8) 
where $\vari{d}{i}{0}$ describes the variability of the reference individual value $c_i$ around the reference population value $\cpop$, and $\vari{d}{ik}{1}$ the fluctuations of the sequence of individual covariate values $(c_{ik})$ around $c_i$. Here is an illustration of this for one individual and three time periods.
Decomposition of a timevarying covariate of a single subject over three time periods

It is instructive to now write model (6) with the following decomposition:
\(
h(\psi_{ik})= h(\psi_{\rm pop})+ \left( \beta(c_{i}  c_{\rm pop}) + \vari{\eta}{i}{0} \right) +
\left( \beta(c_{ik}  c_{i}) + \vari{\eta}{ik}{1} \right) .
\)

(9) 
On the right hand side the first term gives the interindividual (or intraindividual) variability, whereas the second gives the interoccasion variability for this individual. If a covariate $c$ does not vary between occasions, it is the same as saying that for each $k$, $c_{ik} =c_i$. It may also be that some random effects do not exhibit interoccasion variability, i.e., $\vari{\eta}{ik}{1}=0$. Then, an individual parameter $\psi_i$ does not exhibit interoccasion variability if and only if both $c_{ik}c_i =0$ and $\vari{\eta}{ik}{1}=0$.
In general the goal is to construct a model based on the perceived variability of each of these two terms:
 The interindividual variability (IIV) model: choosing the model for the covariates that do not change from occasion to occasion, and a variancecovariance structure for the random effects $\vari{\eta}{i}{0}$.
 The interoccasion variability (IOV) model: choosing the model for the covariates that change from occasion to occasion, and a variancecovariance structure between the random effects $\vari{\eta}{ik}{1}$.
Model (9) assumes that interindividual and intraindividual variability of the covariate have the same magnitude of effect on the parameter, i.e., an increase of 1 unit of $c_{ik}$ with respect to $c_i$ has the same effect as an increase of 1 unit of $c_i$ with respect to $\cpop$. If we would rather not make this hypothesis, we can weight differently the covariates $(c_i  \cpop)$ and $(c_{ik}  c_i)$:
\(
h(\psi_{ik})= h(\psi_{\rm pop})+ \beta(c_{i}  c_{\rm pop}) + \gamma(c_{ik}  c_{i}) + \vari{\eta}{i}{0} + \vari{\eta}{ik}{1}.
\)

(10) 
Extensions to multilevel variability
Extension of the proposed approach to nested levels of variability is straightforward. We illustrate this with several examples.
 Suppose that an occasion can be split into several suboccasions. For instance, imagine that the same study (that lasts several days) is repeated each year. In this case, we might want to take into account year by year variability and day by day variability. To do this, we can introduce an additional level of intraindividual (or interoccasion) variability into the model:
 We can instead consider that the individuals are allocated to different centers or studies. Then, possible variability between centers or studies should also be taken into account by the statistical model.
 We can include in the model any combination of intergroup, interindividual and interoccasion variability with any combination of interactions.
 Center: $\vari{\eta}{\ell}{1}$
 Center and individual: $\vari{\eta}{\ell,i}{0}$
 Center, individual and occasion: $\vari{\eta}{\ell,i,k}{1}$.
\(
\psi_{i,k, l} = \model(\bbeta,c_{i,k,l},\vari{\eta}{i}{0},\vari{\eta}{i,k}{1},\vari{\eta}{i,k,l}{2}) .
\)

(11) 
Here, $c_{i,k,l}$ is the value of covariate $c$ for subject $i$ during suboccasion $l$ of occasion $k$, and $\vari{\eta}{i}{0}$, $\vari{\eta}{i,k}{1}$ and $\vari{\eta}{i,k,l}{2}$ describe different levels of random variability of the parameter. Like in (4), we can assume an additive model where the different levels of random effect can be grouped as a single one named $\eta_{i,k,l}$. Now both the explained and the unexplained parts of the variability can be decomposed into interindividual and two levels of intraindividual components:
\(
\begin{array}{ccccccc}
c_{i,k,l}\cpop &= &(c_{i}  \cpop) & + &( c_{i,k}  c_{i})& +& (c_{i,k,l}  c_{i,k}) \\
&= &\vari{d}{i}{0} & + & \vari{d}{i,k}{1} &+& \vari{d}{i,k,l}{2} \\
\eta_{i,k,l} &= &\vari{\eta}{i}{0} &+& \vari{\eta}{i,k}{1} &+& \vari{\eta}{i,k,l}{2} \,.
\end{array}\)

(12) 
Let $\ell=1,2,\ldots,L$ be the set of subgroups or studies. To keep things simple, first consider the case where there is only one occasion. Then, if individual $i$ is allocated to study $\ell$, its vector of individual parameters $\psi_{\ell,i}$ is described by a model that takes into account the interstudy variability:
\(
\psi_{\ell,i} = \model(\bbeta,c_{\ell,i},\vari{\eta}{\ell}{1},\vari{\eta}{\ell,i}{0}) ,
\)

(13) 
where $c_{\ell,i}$ is the vector of covariates of individual $i$ from group $\ell$, and $\vari{\eta}{\ell}{1}$ and $\vari{\eta}{i}{0}$ the random effects that describe the random components of the interstudy and interindividual variability (within the same study). Note that some components of $c_{\ell,i}$ might be specific to study $\ell$ and have no dependence on the given individual.
Consider for example a crossover study with $K$ occasions performed in $L$ centers, and assume the following levels of random variability:
Then, if we decide to assume an additive model for the random effects, all the random components of the variability can be combined into a unique vector of random effects:
For example, if we were considering an animal study, we might want to group animals with the same father (equivalent to "center"), and then try and characterize variability of some animal feature by a "father" effect, an "animal" effect, and an "occasion" effect.
$\mlxtran$ for multiple levels of variability models