Longitudinal latent outcome modeling in “N-of-few”

Paired-samples longitudinal comparison

In a classic paired-samples longitudinal comparison, both the clinical assessments as well as the DMOs (lateral step deviation shown above again as an example) only begin to show significant changes after several years. The sample sizes required to show a meaningful treatment effect after a single year are estimated to be on the order of several hundreds.

Paired-samples longitudinal change correlation

Shown below is the 1-, 2-, and 3-year change of the lateral step deviation plotted against the change of the clinical score SARA. A correlation only starts to take hold after 3 years, further corroborating the temporal instability.

Outcome ranking

Due to the combinatorics of using a multitude of stride features, transformations thereof, context strata, and aggregation modalities, the number of available candidate DMOs is enormous. A rough assessment based on the cross-sectional and longitudinal analyses allows to screen and rank the DMOs for their suitability. The table below is an excerpt of such a ranking, sorted by the main metric ess, the pooled estimated 1-year sample size.

Preliminary results

Single-outcome trajectory modeling

A fixed-effects panel model allows to model the full trajectories of individual outcomes and to obtain relatively reliable future prediction intervals (using non-parametric estimators, as the data are ill-behaved in various ways). As single DMOs such as the lateral step deviation exhibit substantial variance, the resultung intervals are large. Furthermore, this particular choice of model lacks the partial pooling of a true mixed-effects model, and moreover does not allow to estimate the time point along the independent axis where the outcome starts to deviate from the healthy baseline, leading to misspecification in some cases.

Optimized composite outcome measures

Combining multiple outcomes into a single, more reliable measure of disease severity can be formulated as an optimization problem. As an initial attempt, several sets of top-ranking features were hand-picked. On each set, a regression of the change around the within-subject mean to the corresponding change in diseased time was performed. The resulting outcomes are listed in the ranking table (see below) as test_dw_instance_compound_… and do indeed show favorable characteristics.

Discussion & Outlook

With limited numbers of both subjects and time points, the reliability of the outcome measures becomes the key factor determining the statistical power. Considered as a whole, the rich pool of available outcomes has the potential to contain sufficient information to achieve the "N-of-few" goal.

Instead of combining outcomes into a single measure and modeling its trajectory, the logical next step would be a model that encompasses both simultaneously: A longitudinal model of latent disease severity. This model would consider as much information as possible when predicting a patient's future disease progression. Namely:

• The patient's own historical progression
• Other patients' and controls' progression patterns
• Multiple outcome measures

Key considerations:

• Few covariates other than time
• Generally sigmoidal shape with unknown asymptotes and offset
• Ill-behavedness:
• Heteroskedasticity
• Selection/attrition bias ("informative drop-out"?)
• Disease onset estimate has non-negligible error
• Disease onset estimate is not available for some patients
• Modeling on walking-bout level instead of visit level is an option
• Non-Bayesian approaches (e.g. structural equation modeling) will likely still require Monte Carlo methods due to ill-behavedness