# Joint models

## Sommaire |

## Introduction

An important goal of longitudinal studies is to characterize relationships between different types of response data.

For instance, in a PKPD population study, we may be interested in the relationship between certain pharmacokinetics (absorption, distribution, metabolism and excretion) and pharmacodynamics (biochemical and physiological effects) of a drug. To do this, we need to measure some of both types of response data for several individuals from the same population, then try and characterize their relationship.

Alternatively, many clinical trials and reliability studies generate both longitudinal and survival (time-to-event) data. For example, in HIV clinical trials the viral load and the concentration of CD4 cells are widely used as biomarkers for progression to AIDS when studying the efficacy of drugs to treat HIV-infected patients. We might then be interested in the relationship between these variables and events such as seroconversion or death.

Therefore, in general a *joint model* is one that allows us to simultaneously describe the distribution of different types of observations made on the same individual. We consider this as usual in the population context.

Suppose that we have $L$ different types of observations for individual $i$: $y_i^{(1)}=(y_{ij}^{(1)},1\leq j \leq n_{i1})$, $y_i^{(2)}=(y_{ij}^{(2)},1\leq j \leq n_{i2})$, ..., $y_i^{(L)}=(y_{ij}^{(L)},1\leq j \leq n_{i,L})$, where $n_{i,\ell}$ is the number of observations of type $\ell$ made on individual $i$. Note that $n_{i,\ell}$ may be different for different $\ell$ for the same individual, and the observation times $(t_{ij}^{(\ell)})$ too.

Denote $y_i$ the set of observations for individual $i$: $ y_i = (y_i^{(1)},y_i^{(2)},\ldots,y_i^{(L)})$. For each individual, the joint probability distribution of the observations $y_i$ and the individual parameters $\psi_i$ can be decomposed as follows

We can then distinguish between different types of dependency between observations: independence, conditional independence and conditional dependence.

## Independent observations

Suppose first that the vector of individual parameters $\psi_i$ can be decomposed into $L$ independent sub-vectors $\psi_i^{(1)}$, $\psi_i^{(2)}$, ..., $\psi_i^{(L)}$ such that $y_i^{(\ell)}$ depends only on $\psi_i^{(\ell)}$:

Here, joint modeling does not bring anything new to the picture because all information on $\psi_i^{(\ell)}$ is contained in the related set of observations $y_i^{(\ell)}$. We can therefore model separately each set of observations.

In the same way that we jointly modeled these two types of independent continuous data, we can construct joint models using different types of data at the same time, i.e., various combinations of continuous, categorical, count and survival data, etc., if they are independent.

## Conditionally independent examples

In this case, the various observation types depend no longer only on disjoint (i.e., independent) individual parameters. We therefore write $\psi_i$ for the overall set of (partially or fully shared) individual parameters. Observations are nevertheless supposed independent when conditioning on $\psi_i$:

In such cases, each observation provides information on the individual parameter vector $\psi_i$.

This is the most common case when we are simultaneously modeling different types of longitudinal data of the form:

Here, the predictions $f_1$ and $f_2$ both depend on the same vector of individual parameters, which induces dependency between the observations $y_{i}^{(1)}$ and $y_{i}^{(2)}$. However, these observations are *conditionally independent* if the residual errors $\teps_{ij}^{(1)}$ and $\teps_{ij}^{(2)}$ are independent.

We can extend this framework to different types of data, considering for example categorical observations $y_i^{(2)}$ for which the probabilities $\prob{y_{ij}^{(2)} = k}$ depend on $f_1(t_{ij}^{(2)};\psi_i)$ and consequently $\psi_i$. We can also consider survival data for which the risk function depends on $f_1$.

## Conditionally dependent observations

In this case, there is a dependency structure between types of observation that no longer allows us to decompose the joint model into a product of models with only one type of observation in each.

This kind of dependency occurs when several types of longitudinal data are obtained at the same times, with correlated measurement errors. The joint conditional distribution $\qcyipsii$ of the observations is Gaussian if the residual errors are. The dependency structure between observations can then be characterized by a variance-covariance matrix for the errors.

We can also consider a natural decomposition of this joint distribution into a product of conditional distributions:

Here, the distribution of $y_i^{(2)}$ depends on the observation $y_i^{(1)}$, the distribution of $y_i^{(3)}$ depends on $y_i^{(1)}$ and $y_i^{(2)}$, etc.

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