3. Models¶
3.1. Introduction to Spatio-Temporal Models¶
3.1.1. Temporal Random Effects¶
Individual temporal variability for patient \(i\) is modeled with the latent disease age \(\psi_i(t)\) :
where:
\( \xi_i \) is the individual log speed factor
\( \tau_i \) is the estimated reference time
\( t_0 \) is the population reference time
The longitudinal \( \gamma_i(t)\) and survival \(S_i(t)\) processes are derived from \( \psi_i(t) \).
Key Hypothesis: Longitudinal and survival processes are linked by a shared latent disease age.
3.1.2. Spatial Random Effects¶
Disease presentation variability is captured by space-shifts : \(\mathbf{w}_i = \mathbf{A} \mathbf{s}_i\) where:
\(\mathbf{A}\): mixing matrix (dimension reduction with \(N_s \leq K-1 \) independent sources: \(N_s\) being the number of sources and \(K\) the number of outcomes).
\( \mathbf{s}_i\): Independent sources
For identifiability, \( \mathbf{A} \) is defined as a linear combination of an orthonormal basis \( (\mathbf{B}_k)_{1 \leq k \leq K} \) orthogonal to \( \text{Span}(\mathbf{v}_0) \) [36], so that:
with \(\beta\) the matrix of coefficients.
Each event ( \(l\) ) has a survival shift:
Interpretation: Space shifts (\(w_{i,k}\)) are more interpretable than sources (\(s_i \)), as they encapsulate total spatial variability effects.
3.2. Logistic Model¶
3.2.1. Definition¶
The logistic model is a statistical model used to describe sigmoidal trajectories of longitudinal outcomes over time, typically biomarkers or clinical scores. This model is particularly useful when outcomes exhibit a monotonic progression from one asymptote to another, also called as bounded outcome scores, which is common in disease progression [20]. The logistic model is governed by parameters, which control its speed, inflection point, and asymmetry.
In Leaspy, the logistic model is implemented using a non-linear mixed-effects framework, where the disease evolution is described through a latent time variable (latent disease age) shared across outcomes, and individual variations are modeled through subject-specific parameters [8].
This model provides interpretable components:
A population-level trajectory defined by a logistic function,
Individual deviations from this curve through time reparametrization and spatial shifts,
The ability to simulate, estimate, and personalize trajectories for unseen individuals.
3.2.2. Data¶
A logistic model is relevant when you have:
Longitudinal repeated measurements of one or more continuous outcomes (e.g., cognitive scores, clinical scores, biomarkers)
Monotonic or sigmoidal progression of outcomes over time (typically increasing or decreasing to a plateau)
No event or survival information required, unlike joint models
To fit a logistic model, you need a dataframe with the following columns:
ID: Patient identifierTIME: Time of measurementOne or more columns representing the longitudinal outcomes (e.g.,
OUTCOME_1,OUTCOME_2, …)
For the importation of dataframe:
dataset = dataframe.set_index(["ID", "TIME"]).sort_index()
print(dataset.head())
OUTCOME_1 OUTCOME_2
ID TIME
132-S2-0 81.661 0.44444 0.04000
82.136 0.60000 0.56000
82.682 0.39267 0.04000
83.139 0.58511 0.30000
83.691 0.57044 0.05040
data_logistic = Data.from_dataframe(dataset, "visit")
3.2.3. Mathematical background¶
The logistic trajectory for outcome \(k\), and from the latent disease age \(\psi_i(t)\) is defined by:
where:
\(\gamma_{i,k}(t)\) is the modeled outcome value,
\(t_0\) is the population reference time
\(v_{0,k}\) is the speed of progression for outcome \(k\) at reference time \(t_0\),
\(w_{i,k}\) is the individual space shift for outcome \(k\).
\(\frac{1}{1+g_k}\) is the value of the logistic curve at \(t_0\)
\(\psi_i(t)\) is the latent disease age, please have a look to part introduction to spatio-temporal models for more details.
3.3. Joint Model¶
3.3.1. Definition¶
Joint models are a class of statistical models that simultaneously analyze longitudinal data and survival data [2, 14]. Unlike traditional approaches that treat these processes separately, joint models integrate them into a unified framework, recognizing that they often share underlying biological mechanisms—for example, a slowly progressing biomarker may signal an increased risk of a clinical event. By linking the two submodels—typically through shared random effects [34] or latent processes [33], fraitly [35], models account for their interdependence, reducing biases from informative dropout or measurement error [38].
In Leaspy, the joint model [27] is implemented as a longitudinal spatio-temporal model, and a survival model, that are linked through a shared latent disease age, and, in the case of multiple longitudinal outcomes, spatial random effects (see description in first paragraph of this page). This approach allows for the incorporation of both temporal and spatial random effects, providing a more comprehensive understanding of the underlying disease process.
3.3.2. Data¶
A joint model is relevant when you have:
Longitudinal measurements as repeated biomarker readings, clinical scores
Time-to-event outcomes as survival, dropout, or failure events
A suspected association between the longitudinal process and event risk
You must have one dataframe with the following columns:
ID: Patient identifierTIME: Time of measurementEVENT_TIME: Time of eventEVENT_BOOL: Event indicator (1 if event occurred, 0 if censored and 2 if competing event)
For one patient, the event time and event bool are the same for each row.
For the importation of dataframe:
dataset = dataframe.set_index(["ID", "TIME"]).sort_index()
print(dataset.head())
OUTCOME_1 OUTCOME_2 EVENT_TIME EVENT_BOOL
ID TIME
132-S2-0 81.661 0.44444 0.04000 84.0 1
82.13600000000001 0.60000 0.00000 84.0 1
82.682 0.39267 0.04000 84.0 1
83.139 0.58511 0.00000 84.0 1
83.691 0.57044 0.00000 84.0 1
184-S2-0 80.994 0.32600 0.00000 83.4 0
81.47199999999998 0.35556 0.00000 83.4 0
82.08800000000001 0.40000 0.00000 83.4 0
82.488 0.31111 0.00000 83.4 0
data_joint = Data.from_dataframe(dataset, "joint")
3.3.3. Mathematical background¶
3.3.3.1. Longitudinal Submodel¶
The longitudinal submodel that can be used here is the logistic model, please have a look to part description logistic model for more details.
3.3.3.2. Survival Submodel¶
Cause-Specific Weibull Hazards (for competing risks):
This submodel captures how variations in the progression of longitudinal disease outcomes influence the probability and timing of multiple clinical events, while accounting for censoring and competing risks. To achieve this, the leaspy joint model uses a cause-specific hazard structure.
For each event \(l\) and patient \(i\), we model a cause-specific hazard \(h_{i,l}(t)\) [32]. This framework allows us to estimate the risk of each event separately and account for the presence of competing risks (where one event precludes others).
A Weibull distribution is used to model time-to-event data due to its flexibility in representing:
Increasing, decreasing, or constant hazard shapes,
Dependence on two interpretable parameters:
Scale parameter \(\nu_l\),
Shape parameter \(\rho_l\).
The hazard is modulated via a Cox-proportional hazard framework to incorporate the effect of longitudinal biomarkers using survival shifts.
Let \(u_{i} = \zeta s_{i}\), \(\zeta\) being the survival shift and \(s\) sources. The cause-specific hazard \(h_{i,l}(t)\) for event \(l\) and subject \(i\) is defined as:
Expanding the baseline hazard \(h_{0,i,l}(t)\), we get:
Where:
\(\nu_l\): Scale parameter of the Weibull distribution for event \(l\),
\(\rho_l\): Shape parameter of the Weibull distribution for event \(l\),
\(\xi_i\): Subject-specific parameter,
\(\tau_i\): Estimated reference time for individual \(i\),
Survival function \(S_{i,l}(t)\) is then defined as:
Finally, CIF for event \(l\) and subject \(i\) is defined as:
3.3.3.3. Model summary¶
For patient \(i\), outcome \(k\), and event \(l\):
In practice in leaspy, to use the joint model, you need to precise “joint” in Leaspy object creation, then you can use it to fit, personnalize, estimate and simulate.
leaspy_joint = JointModel(nb_events=2, source_dimension=3)
leaspy_joint.fit(data_joint, nb_iter=1000, nb_burnin=500)
For estimation, it is the CIF that is outputted by the model. Note that for prediction purposes, the CIF is corrected using the survival probability at the time of the last observed visit, following common practice in other packages [3].
3.4. Mixture Model¶
3.4.1. Definition¶
Mixture models are a class of statistical models that represent a population as a combination of underlying subpopulations, each described by its own probability distribution [23]. Standard mixed effects models assume a certain homogeneity, in the sense that inter-patient variability is considered as random variation around a fixed reference. Mixture models explicitly recognize that observed data may arise from distinct latent groups—for example, patients may cluster into different onset times, rates of disease progression or more advanced clinical scores. This formulation captures heterogeneity in the data, allowing for flexible modeling of complex, multimodal patterns. Estimation is typically carried out using an adaptation of MCMC-SAEM for mixture models as described in [23], which iteratively refine both the component parameters and the posterior probabilities of membership. By providing a probabilistic clustering framework, mixture models not only identify hidden structure but also quantify uncertainty in subgroup assignment, making them widely applicable in biomedical sciences.
In Leaspy the mixture model is implemented as an adaptation of the spatio-temporal logistic model where the individual parameters (tau, xiand sources) come from a mixture of gaussian distributions with a number of components defined by the user.
3.4.2. Data¶
The same rules apply as for the standard logistic model.
3.4.3. Mathematical background¶
To build a mixture model, we add another layer upon the hierarchical structure of the model. We suppose that each subgroup \(c\) can be described by a different set of \((\overline{\tau}^c, \overline{\xi}^c, \overline{s}^c)\) and has a respective probability of occurrence \(\pi^c\). We suppose that the population parameters come from normal distributions as in the standard model, while the individual parameters are sampled from a mixture of gaussians with mixing probabilities \(\pi^c\).
The individual parameters (random effects):
3.4.4. Model summary¶
To use the mixture model in Leaspy you need to choose the number of cluster you wish to estimate beforehand.
from leaspy.models import LogisticMultivariateMixtureModel
leaspy_mixture = LogisticMultivariateMixtureModel(source_dimension=1, n_clusters=2, dimension=3)
leaspy_mixture.fit(data_logistic, "mcmc_saem", n_iter=1000)
3.5. References¶
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