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Bayesian Adaptive Clinical Trial Designs: INLA vs. MCMC

Bayesian methods have continuously played a key role in transforming clinical research in therapeutic areas such as oncology and rare diseases, and in addressing clinical development challenges for COVID-19 drugs, devices, and biologics. From early-phase trials to late-phase development, utilizing Bayesian tools can expedite and/or de-risk trials, even when used to augment a Frequentist framework.

Bayesian methods, with their ability to facilitate flexibility and learning, are often associated with early-phase clinical trials. Their benefits for dose-finding and unplanned stopping in Phase 1 and 2 trials are clearly documented. Recently, more Bayesian elements have appeared in some confirmatory trials, particularly those requiring historical borrowing. A 2018 FDA Guidance on the uses of Bayesian methods in medical device trials appeared to normalize the idea that Bayesian methods may be used in some confirmatory settings. Pediatric trials, which modify existing adult therapies for younger populations, can also benefit from Bayesian approaches.

How are Bayesian clinical trial designs different from Frequentist clinical trials?

The Bayesian approach allows every available piece of data to serve as evidence to update a hypothesis. Bayes’s rule consists of a hypothesis called a “prior” constructed either on evidence already collected, or scientific findings in the case of early-phase trials. A rule then explains how to update these priors in order to make sense of newly collected evidence. A “posterior” is then the result of the prior being updated in light of this new evidence. As the trial evolves with new in-trial insights, these Bayesian methods can enable statisticians and sponsors to create flexible trial designs and accelerate learning.

How are Bayesian design models computed and fit analytically?

Typically, in Bayesian models, the objective of interest is to compute the posterior distribution or a predictive distribution. The standard method statisticians use to estimate these is called MCMC, which stands for Markov Chain Monte Carlo. MCMC methods are a class of algorithms for sampling from a probability distribution by constructing a Markov chain that eventually converges to the desired distribution at equilibrium. MCMC techniques have gained in popularity and adoption over the last few decades and have vastly influenced the uptake of Bayesian methods.

What are some limitations of MCMC?

While MCMC methods are reliable and can compute the posterior distributions associated with any likelihood function, they may be slow to converge and can take a long time to execute. Depending upon the complexity of the hierarchical model under investigation (the dimension of the unknown parameters), this may result in a sub-optimal exploration of the design due to computational demands and limitations.

What is INLA?

The integrated nested Laplace approximation (INLA) is a method for approximate Bayesian inference. It can be an attractive alternative to MCMC methods due to its speed and ease of use. Unlike MCMC, which relies on the convergence of a Markov chain to the desired posterior distribution, INLA uses a Laplacian approximation to estimate the individual posterior marginals of the model parameters.

How does INLA work?

INLA uses a Laplacian approximation to estimate the probability distribution function of an unknown parameter by approximating it to a Gaussian distribution. The nesting occurs due to a posterior computation of the parameters conditional on the “data + hyperparameters” as a first step to computing the posterior distribution of the hyperparameters themselves. These two estimates are then used in a numerical integration calculation to compute the desired posterior marginal distribution. See the image below, for example, which shows the true posterior Gamma distribution in black and the INLA approximation to it in red. This example was based on a simple conjugate Poisson-Gamma example. The closer the posterior is to a normal-like curve, the more accurate the INLA approximation will be.

Krishna Padmanabhan_INLA vs MCMC_Fig1

What are the limitations associated with INLA?

INLA only works for Latent Gaussian Models, whose parameters form a Gaussian Markov Random Field. The former is typically achieved by setting normal priors to (some transformation of) the unknown parameters. INLA, although seemingly limited to a certain class of models, works for:

  • Linear Models
  • Generalized Linear Models
  • Linear Mixed Models
  • Generalized Linear Mixed Models
  • Generalized Additive Models
  • Time to Event (Survival) Models

Thankfully, these models cover a large majority of Phase 2 and 3 clinical trials. It can also work for Phase 1 studies such as the BLRM, which utilizes a Bayesian Logistic Regression model.

Typically, how much faster is INLA than MCMC?

In Cytel’s experience, we have noticed speed increases of >100x with INLA vs. MCMC in many instances, with virtually the exact same parameter estimates and confidence intervals. The graphic shows a comparison of compute times for different Bayesian clinical trial designs on a standard PC (Intel Core i7, 16GB RAM).

Comparison of Compute Times: JAGS vs. INLA
Trial Primary Endpoint Type MCMC (sec.) INLA (sec.)

MCMC:

50K iterations, 3 chains

INLA:

Standard INLA Simplified Laplace Approximation

Survival (Oncology) 187 1.1
Binary (Infectious Disease) 238 0.9
Repeated Measures (Nephrology) 153 2.7
Continuous (Rare Disease Biomarker) 36.2 1.0
Survival (CV)
(N=3000+)
>49K (13.7 hours) 27.35
Repeated Measures (Lipids) (N=7000+) >250K (~3 days) 396.2

How does this help you design a better Bayesian clinical trial?

Although the increase in computational speeds is impressive, the real advantage is in a deeper exploration of design parameters, thus optimizing your Bayesian design further. Frequently, an MCMC explores limited design parameters due to computational demands associated with the process. Using INLA, a more through exploration of the design parameters can take place in order to achieve a greater confidence in the resulting trial design.

How can I find out more about INLA and how Cytel can help with my Bayesian clinical trial design?

Cytel will continue to apply these and other cutting-edge methods in the design and analysis of Bayesian clinical trials for their clients. If interested, please contact us so we may begin a conversation on how we can be of assistance in designing your clinical trials.

 

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