Methods for non-ignorable missingness in longitudinal brain imaging studies.
Investigating the aging process of the human brain is an increasingly important endeavor, as the life expectancy of the world population is increasing. As a consequence, the number of individuals with cognitive impairments related to aging is expected to double within a 50-year period. Measuring the brain through different imaging techniques provides important insights on various aspects of brain structure and function, and their relation to cognitive aging. In order to follow the aging process on an individual basis, we must follow people over time and measure them repeatedly, i.e. do a longitudinal study. At Umeå Center for Functional Brain Imaging (UFBI), large longitudinal brain imaging studies are performed in collaboration with other world-leading aging research centers. These studies also collect extensive amounts of data through cognition tests, health tests, life style questionnaires, and genetic information, giving us unique longitudinal data covering most of the human adult life span. For various reasons including health problems, not all individuals entering our studies see them through to the end, and we have dropout. It is fair to assume that the dropout is related to the brain imaging outcome, in which case we have non-ignorable dropout. The analysis of longitudinal brain imaging data with non-ignorable dropout has, until now, been overlooked in the literature, and this project intends to fill this gap by developing necessary statistical models and methods.
Final report
Project aim and development.
The project aimed at developing statistical modeling techniques for longitudinal data with spatial structure, while having non-ignorable dropout. The plan was to use Bayesian Hierarchical Models together with Pattern-Mixture-Models. Although the models and methods are general, they were developed with a more specific application in mind, namely longitudinal brain imaging studies.
The project plan identified three sub-projects – to create models and methods that:
1. account for voxel-level spatial dependencies;
2. account for simultaneous space-time dependencies; and
3. model time-varying connections between brain regions.
The project was performed in accordance with the plan, although sub-project three was given more time than originally planned. This was mainly due to an increasing interest in connectivity modeling within the brain-imaging community, which became apparent during the first half of the project. The developed methods has been applied to data from both local collaborations (the Betula project) mentioned in the original application as well as open data from brain imaging consortia initiatives which became accessible during the course of the project.
Implementation.
The project has been carried out essentially by the members listed in the original application, with the addition of PhD student Tetiana Gorbach, who was funded by other sources but co-supervised by this project’s PI (Lundquist). For the later parts of sub-project 3, a cooperation was established with a research group at Linköping University headed by Prof. Mattias Villani. The Linköping group funded their participation in the project by themselves.
Most important results and discussion on conclusions.
In sub-projects 1 and 2, we propose a model for repeated observations of a multivariate outcome with spatial dependencies, using a Bayesian Hierarchical structure. This part of the model is similar to such models proposed previously for similar, but not identical, problems. We further propose using a Pattern-Mixture-Model (PMM) for performing the analysis along with sensitivity analysis when dropout is considered non-ignorable. For the sensitivity analysis we propose three ways of setting up the PMM sensitivity parameters such that observed spatial dependencies are taken into account, which is a significant novelty as this has received little to no attention at all previously. In a simulation study as well as using real data, we compare our proposals and conclude that one proposed method has a slight edge compared to the others – at least in our setting.
In sub-project 3, we propose cross-sectional and longitudinal models for analyses of effective and functional brain connectivity. Firstly, we consider functional brain connectivity, where we propose modeling observed correlations (or other association measures) using Bayesian Mixture Models (BMMs). We also extend this framework to encompass longitudinal data, and also introduce sensitivity analysis for non-ignorable dropout – as in sub-projects 1 & 2 we use PMM models for the latter. The use of BMMs in general occurs regularly in the literature, but quite rarely for longitudinal data, and the extension to nonignorable dropout has thus far received very little attention.
Also in sub-project 3, we propose to simultaneously model functional and effective brain connectivity using Bayesian Hierarchical Vector Autoregressive models (BH-VAR) in a cross-sectional setting, i.e. one brain scan session per subject. Due to the hierarchical nature, we can also obtain simultaneous inferences on subject- and group-level. Such extensive inferences come at a cost, in our case the cost is computational, currently our method is applicable when the number of connected regions is about 20-30. While this is an improvement of previously suggested models having similar computational complexity - results have commonly been presented for no more than five or six regions - the number of regions our model can address is unfortunately still quite low compared to the number of functional regions currently suggested – up 400 regions have been considered. If the estimation of functional connectivity is unimportant, we suggest a simplification of our model such that we are able to handle several hundreds of regions while still yielding valid inference on effective connectivity.
Further research questions.
The results from sub-projects 1 and 2 regarding the specification of PMM sensitivity analysis give initial insights into specifying such models while accounting for spatial associations. It is desirable to deepen those insights and investigate other potential areas of application.
Our results using the BH-VAR framework for connectivity analysis are quite interesting and promising, the largest current bottleneck being the computational issues. Thus, the immediate next step would be to make our full model computationally feasible for a tenfold increase in the number of regions the model can accommodate. Further – in the spirit of the other results in the projects - it is obviously quite interesting to extend this type of model to a longitudinal setting with repeated scanning sessions – and as an even further step to also consider nonrandom dropout in these types of models.
Dissemination of Results and cooperation with surrounding society.
Results have been or will be published in international peer-review journals and presented at international conferences. The journal publications are Open Access (OA), either by agreement between Umeå University and the publisher, or by payment of an OA-fee directly to the publisher.
The project aimed at developing statistical modeling techniques for longitudinal data with spatial structure, while having non-ignorable dropout. The plan was to use Bayesian Hierarchical Models together with Pattern-Mixture-Models. Although the models and methods are general, they were developed with a more specific application in mind, namely longitudinal brain imaging studies.
The project plan identified three sub-projects – to create models and methods that:
1. account for voxel-level spatial dependencies;
2. account for simultaneous space-time dependencies; and
3. model time-varying connections between brain regions.
The project was performed in accordance with the plan, although sub-project three was given more time than originally planned. This was mainly due to an increasing interest in connectivity modeling within the brain-imaging community, which became apparent during the first half of the project. The developed methods has been applied to data from both local collaborations (the Betula project) mentioned in the original application as well as open data from brain imaging consortia initiatives which became accessible during the course of the project.
Implementation.
The project has been carried out essentially by the members listed in the original application, with the addition of PhD student Tetiana Gorbach, who was funded by other sources but co-supervised by this project’s PI (Lundquist). For the later parts of sub-project 3, a cooperation was established with a research group at Linköping University headed by Prof. Mattias Villani. The Linköping group funded their participation in the project by themselves.
Most important results and discussion on conclusions.
In sub-projects 1 and 2, we propose a model for repeated observations of a multivariate outcome with spatial dependencies, using a Bayesian Hierarchical structure. This part of the model is similar to such models proposed previously for similar, but not identical, problems. We further propose using a Pattern-Mixture-Model (PMM) for performing the analysis along with sensitivity analysis when dropout is considered non-ignorable. For the sensitivity analysis we propose three ways of setting up the PMM sensitivity parameters such that observed spatial dependencies are taken into account, which is a significant novelty as this has received little to no attention at all previously. In a simulation study as well as using real data, we compare our proposals and conclude that one proposed method has a slight edge compared to the others – at least in our setting.
In sub-project 3, we propose cross-sectional and longitudinal models for analyses of effective and functional brain connectivity. Firstly, we consider functional brain connectivity, where we propose modeling observed correlations (or other association measures) using Bayesian Mixture Models (BMMs). We also extend this framework to encompass longitudinal data, and also introduce sensitivity analysis for non-ignorable dropout – as in sub-projects 1 & 2 we use PMM models for the latter. The use of BMMs in general occurs regularly in the literature, but quite rarely for longitudinal data, and the extension to nonignorable dropout has thus far received very little attention.
Also in sub-project 3, we propose to simultaneously model functional and effective brain connectivity using Bayesian Hierarchical Vector Autoregressive models (BH-VAR) in a cross-sectional setting, i.e. one brain scan session per subject. Due to the hierarchical nature, we can also obtain simultaneous inferences on subject- and group-level. Such extensive inferences come at a cost, in our case the cost is computational, currently our method is applicable when the number of connected regions is about 20-30. While this is an improvement of previously suggested models having similar computational complexity - results have commonly been presented for no more than five or six regions - the number of regions our model can address is unfortunately still quite low compared to the number of functional regions currently suggested – up 400 regions have been considered. If the estimation of functional connectivity is unimportant, we suggest a simplification of our model such that we are able to handle several hundreds of regions while still yielding valid inference on effective connectivity.
Further research questions.
The results from sub-projects 1 and 2 regarding the specification of PMM sensitivity analysis give initial insights into specifying such models while accounting for spatial associations. It is desirable to deepen those insights and investigate other potential areas of application.
Our results using the BH-VAR framework for connectivity analysis are quite interesting and promising, the largest current bottleneck being the computational issues. Thus, the immediate next step would be to make our full model computationally feasible for a tenfold increase in the number of regions the model can accommodate. Further – in the spirit of the other results in the projects - it is obviously quite interesting to extend this type of model to a longitudinal setting with repeated scanning sessions – and as an even further step to also consider nonrandom dropout in these types of models.
Dissemination of Results and cooperation with surrounding society.
Results have been or will be published in international peer-review journals and presented at international conferences. The journal publications are Open Access (OA), either by agreement between Umeå University and the publisher, or by payment of an OA-fee directly to the publisher.