## 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 inter-individual 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 intra-individual 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 intra-individual variability of the individual parameters by piecewise-constant covariates, i.e., "occasion-dependent" or "occasion-varying" (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 inter-individual variability of the individual parameters.
• $\vari{\eta}{ik}{1}$, the vector of random effects which describes the random intra-individual 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 inter-individual variability, i.e., variability at the individual ("reference") level, while superscript ${(1)}$ represents inter-occasion 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:

$$\begin{eqnarray} \hpsi_{ik} &=& \model(\bbeta,c_{ik},\vari{\eta}{i}{0}\equiv 0,\vari{\eta}{ik}{1} \equiv 0) \\ &=& \hmodel(\bbeta,c_{ik}). \end{eqnarray}$$

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.

1. 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
2.  $$\eta_{ik} = \vari{\eta}{i}{0} + \vari{\eta}{ik}{1}.$$ (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 variance-covariance 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.

3. Assume now a Gaussian model of the form
4. $$\begin{eqnarray} h(\psi_{ik})& = & h(\hpsi_{ik})+ \eta_{ik} \\ & = & h(\hpsi_{ik})+ \vari{\eta}{i}{0} + \vari{\eta}{ik}{1} . \end{eqnarray}$$

Here, the $h(\psi_{i1}), \ldots , h(\psi_{iK})$ are correlated Gaussian vectors whose variance-covariance structure is that of the $(\eta_{ik})$ defined in (5).

5. 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
6.  $$h(\psi_{ik}) = h(\psi_{\rm pop})+ \beta(c_{ik} - c_{\rm pop}) + \vari{\eta}{i}{0} + \vari{\eta}{ik}{1}.$$ (6)

Example:

Consider our model for the volume of distribution introduced in the covariate models Section, which assumes a linear relationship between the log-weight and the log-volume. If the weight varies from occasion to occasion, we can consider the following model:
 $$\log(V_{ik}) = \log(V_{\rm pop}) + \beta \, \log(w_{ik}/70) + \eta_{ik} .$$ (7)

We can then decompose the part of the variability explained by the covariate $c$ into inter-individual and intra-individual 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 time-varying 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 inter-individual (or intra-individual) variability, whereas the second gives the inter-occasion 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 inter-occasion variability, i.e., $\vari{\eta}{ik}{1}=0$. Then, an individual parameter $\psi_i$ does not exhibit inter-occasion 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 inter-individual variability (IIV) model: choosing the model for the covariates that do not change from occasion to occasion, and a variance-covariance structure for the random effects $\vari{\eta}{i}{0}$.
• The inter-occasion variability (IOV) model: choosing the model for the covariates that change from occasion to occasion, and a variance-covariance structure between the random effects $\vari{\eta}{ik}{1}$.

Model (9) assumes that inter-individual and intra-individual 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)

Example:

Consider a model that supposes a linear relationship between income and happiness. Denote $\psi_{ik}$ the happiness score (on some scale) for subject $i$ in year $k$, and $c_{ik}$ their income (in K€) in year $k$. Model (9) supposes that for each subject and year $k$, a difference in annual income of 1K€ with respect to the reference income in the population generates an increase of $\beta$ happiness.

There is no reason to expect here that the intra-individual variability (fluctuation in one individual's salary) has the same effect on happiness. Indeed, an increase in annual salary of 1K€ for some individuals might lead to more happiness than the fact of having a salary of 1K€, more than the reference salary.

Model (9) lets us take this into account, assuming for example that $\gamma>\beta$.

## Extensions to multi-level variability

Extension of the proposed approach to nested levels of variability is straightforward. We illustrate this with several examples.

1. Suppose that an occasion can be split into several sub-occasions. 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 intra-individual (or inter-occasion) variability into the model:
2.  $$\psi_{i,k, l} = \model(\bbeta,c_{i,k,l},\vari{\eta}{i}{0},\vari{\eta}{i,k}{1},\vari{\eta}{i,k,l}{2}) .$$ (11)

Here, $c_{i,k,l}$ is the value of covariate $c$ for subject $i$ during sub-occasion $l$ 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 inter-individual and two levels of intra-individual 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)

3. 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.
4. 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 inter-study 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 inter-study and inter-individual 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.

5. We can include in the model any combination of inter-group, inter-individual and inter-occasion variability with any combination of interactions.
6. Consider for example a cross-over study with $K$ occasions performed in $L$ centers, and assume the following levels of random variability:

• Center: $\vari{\eta}{\ell}{-1}$
• Center and individual: $\vari{\eta}{\ell,i}{0}$
• Center, individual and occasion: $\vari{\eta}{\ell,i,k}{1}$.

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:

$$\eta_{\ell,i,k} = \vari{\eta}{\ell}{-1} + \vari{\eta}{\ell,i}{0} + \vari{\eta}{\ell,i,k}{1} .$$

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.

Example 1:

TO DO