Longitudinal Modeling in Clinical Trial Design: Methodological Advantages & Challenges

Adaptive designs have, rightfully, become pervasive in clinical trials. Early stopping for efficacy or futility, as in group sequential, goldilocks, and promising zone designs, aim to make decisions before the maximum sample size has been reached. These trials can decrease expected sample size without sacrificing power or type I error. Response adaptive randomization, arm dropping, and other randomization manipulation strategies have been researched thoroughly and implemented widely. Adaptive allocation can focus randomization so that more data is collected on arms of interest. These methods are used broadly and recognized as cutting edge methods that diverge from the classic paradigm of how information is used in a clinical trial.

A frontier that has similar goals as the previously mentioned adaptations, but has not been the focus of as much research is the use of early endpoint data in adaptive decision making. It’s a seemingly obvious statistical observation that a well-designed clinical trial should use all data available to it to make decisions. Despite that, it is common to ignore early data about subjects in a clinical trial, often to the extent that only subjects with complete information are included in statistical analyses.

Longitudinal modeling leverages data collected from individual subjects before the final endpoint time to improve estimation of the final endpoint. For the purposes of this article, this inclusion of early subject data in the primary analysis model is done through multiple imputation of the final endpoint given early subject data. The statistical model that links early endpoint data with the final endpoint to allow for the imputation is called the longitudinal model. This imputation through the longitudinal model allows for improvements in the estimation of the final endpoint, and, as a result, improves the efficiency in clinical trial decision making.

Importance in Clinical Trials

While in early stages of development, frequent assessments of outcomes like biomarker levels, symptom scores, or surrogate endpoints are more common, we argue that for confirmatory trials, too, we should assess whether the efficiency gain of longitudinal models justify more frequent repeat measures, even if it comes at an incremental operational cost.

By leveraging repeated measurements, longitudinal modeling enables the following:

1. Enhanced Statistical Estimation Efficiency

Multiple imputation of missing final endpoint data increases statistical efficiency by capturing currently enrolled subject trajectories and measuring within-subject variability across multiple time points. Imputing final data for subjects with only early endpoint data allows for the estimated treatment effect to incorporate likely trajectories of enrolled subjects, thereby estimating an effect closer to what will be observed once all subjects’ final data is observed. Additionally, the subjects without a final endpoint that are imputed contribute to the sample size, although not with a full subject weight, and improve the precision in the estimation of the final endpoint response.

2. Reduced Sample Size Requirements

Due to the increased statistical efficiency described above, clinical trials using longitudinal modeling often require fewer participants than traditional designs to achieve equivalent statistical power. This reduction in required sample size can lead to faster, more cost-effective trials and reduced overall patient exposure to potential risks. As always, this reduction in sample size can instead be realized as increased power if the sample size is not changed.

3. Improve Predictive Probability Calculations

Utilizing the longitudinal model’s knowledge of subjects’ standard progression through their follow-up period not only enhances the estimate of the treatment effect, but also allows for better predictions of the trial results at later time point. Predictive probabilities, often used in adaptive designs like the goldilocks design, can leverage longitudinal imputation models to improve the prediction of final endpoint values for subjects that have been enrolled, and have some available data, but do not have an observed final endpoint.

4. Improved Understanding of Disease Progression

Longitudinal approaches explicitly model likely values of the final endpoint conditional on the observed early endpoint data. This allows for direct analysis of disease trajectories for arms in the trial. These models characterize how diseases evolve naturally, and how interventions modify this progression, ultimately supporting more informative clinical conclusions.

5. Incorporation of Prior Information…or not

Many indications are well understood before the onset of a clinical trial, and a subject’s likely outcomes throughout their follow-up are known in advance. Longitudinal models can be structured so that they’re pre-dispositioned to predict subject outcomes based on past information. This can lead to benefits in trial decision making even before a single subject has complete data. On the other hand, longitudinal models can be given non-informative priors and be estimated only using subjects in the trial that have both intermediate and final endpoint data.

6. Flexibility in Handling Missing Data

Missing data, a frequent challenge in clinical trials, can be more robustly managed using longitudinal imputation methods. Subjects who are lost to follow-up (dropped out) can be imputed from the same longitudinal models as subjects who have not yet completed follow-up. This leads to an assumption about the dropped-out subjects that their endpoint is missing at random conditional on the early endpoint data, which is a less restrictive assumption than the common missing completely at random.

7. Individualized Patient Insights

Longitudinal analyses facilitate individual-level predictions and personalized medicine approaches. This individual-specific insight can help clinicians better understand the trajectory of the disease over time, informing tailored therapeutic decisions and enabling personalized healthcare strategies.

Challenges and Methodological Considerations

Longitudinal modeling often requires more advanced statistical methods than analysis models that only operate on the final endpoints for complete subjects. In the multiple imputation style of longitudinal modeling, this complication also results in added computational intensity. There are limited resources for designing and implementing a trial that incorporates longitudinal imputation in its primary analysis. In some cases, like when FACTS does not have the desired imputation model, the code must be written by hand to perform such analyses.

Additional challenges largely boil down to the model assumptions. The parametric form, correlation structure, and variance components of the longitudinal models must be specified in advance, and may not exactly match the collected data. Generally, the longitudinal model that was chosen was selected after careful consideration, so the potential risks were understood in advance. Either way, model checking and assumption evaluation checks are common when implementing an adaptive design with a longitudinal imputation model.

Applications of Longitudinal Modeling by Berry Consultants and FACTS

Berry Consultants has been instrumental in advocating and advancing the use of longitudinal modeling. Berry Consultants’ FACTS software (Fixed and Adaptive Clinical Trial Simulator) integrates longitudinal modeling techniques into a general adaptive design creation software to easily allow for explicitly leveraging longitudinal endpoint data. FACTS facilitates simulations and comprehensive analyses, enabling researchers to evaluate trial designs, predict outcomes, and refine methodological decisions for using intermediate endpoint data.

For continuous endpoints, a selection of the models are:

• Time-course hierarchical modeling (TCH): TCH models leverage Bayesian hierarchical modelling to estimate the final subject response by estimating the proportion of the final endpoint effect that will be observed at an early endpoint.

• Integrated Two-Parameter (ITP) models: ITP models integrate patient-level random effects with longitudinal observations to robustly estimate individual patient trajectories and population-level drug effects under a specific parametric shape.

• Linear Regression: The linear regression imputation model simply assigns a slope and intercept parameter that, when applied to the early endpoint observation, predict the final subject response. Each interim analysis may predict the final analysis with its own regression parameters.

• Kernel Density: Kernel Density models link early endpoint data to final endpoint data through a nonparametric estimate of the distribution of the final response conditional on early subject data.

For dichotomous endpoints, FACTS can multiply impute the final endpoint response from an early endpoint value through a beta binomial model or a logistic regression model. For time-to-event endpoints, the final event time for a subject can be multiply imputed based on a predictor observed before the event. The predictor can be continuous, dichotomous, or time-to-event itself, and a rich set of models for modelling the predictor values across arms are available.

Conclusion

Longitudinal modeling offers substantial efficiency gains in clinical trial design, enabling more informative studies. While incorporating these techniques requires careful attention to methodological complexities and computational demands, the potential benefits to clinical research—such as enhanced statistical efficiency, more insightful disease progression modeling, and individualized patient-level analyses—strongly advocate their broader adoption in clinical research frameworks.

Key References

1. Berry SM, Carlin BP, Lee JJ, Muller P. Bayesian Adaptive Methods for Clinical Trials. Chapman & Hall/CRC; 2010 (Provides a comprehensive Bayesian methodological framework, including longitudinal modeling and adaptive approaches.)

2. Saville BR, Berry SM. Efficiencies of platform clinical trials: A vision of the future. Clin Trials. 2016;13(3):358-366. doi:10.1177/1740774515626362 (Discusses longitudinal modeling strategies within the context of adaptive platform trials.)

3. Berry DA. Emerging innovations in clinical trial design. Clin Pharmacol Ther. 2016;99(1):82-91. doi:10.1002/cpt.285 (Highlights Bayesian longitudinal modeling as a key innovation improving early-phase trial efficiency.)

4. Quintana M, Saville BR, Vestrucci M, et al. Design and Statistical Innovations in a Platform Trial for Amyotrophic Lateral Sclerosis. Ann Neurol. 2023;94(3):547-560. doi:10.1002/ana.26714 (Illustrates longitudinal modeling methods, including hierarchical Bayesian approaches, implemented by Berry Consultants in an ALS platform trial.)

5. Berry SM, Petzold EA, Dull P, et al. A response-adaptive randomization platform trial for efficient evaluation of Ebola virus treatments: A model for pandemic response. Clin Trials. 2016;13(1):22-30. doi:10.1177/1740774515621721 (Demonstrates application of adaptive, Bayesian longitudinal modeling within urgent early-phase trials for infectious diseases.)

6. Diggle PJ, Liang KY, Zeger S L. Analysis of Longitudinal Data*. Oxford University Press 2002.

7. Berry, SM et al. Bayesian Adaptive Methods for Clinical Trials. CRC Press. 2010

8. FACTS Software. Berry Consultants. https://www.berryconsultants.com/software