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Cited by 276Functional Brain Connectivity StudiesAdvanced MRI Techniques and ApplicationsAdvanced Statistical Methods and Models

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Orthogonalization of Regressors in fMRI Models

Open access
Mumford, Jeanette A., Poline, Jean-Baptiste, Poldrack, Russell A.·2015·PLoS ONE
DOI 10.1371/journal.pone.0126255PMC PMC4412813
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Abstract

The occurrence of collinearity in fMRI-based GLMs (general linear models) may reduce power or produce unreliable parameter estimates. It is commonly believed that orthogonalizing collinear regressors in the model will solve this problem, and some software packages apply automatic orthogonalization. However, the effects of orthogonalization on the interpretation of the resulting parameter estimates is widely unappreciated or misunderstood. Here we discuss the nature and causes of collinearity in fMRI models, with a focus on the appropriate uses of orthogonalization. Special attention is given to how the two popular fMRI data analysis software packages, SPM and FSL, handle orthogonalization, and pitfalls that may be encountered in their usage. Strategies are discussed for reducing collinearity in fMRI designs and addressing their effects when they occur.

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Figures

Fig 1 How parametrically modulated regressors are constructed. The left graph shows the unconvolved regressors for 3 trials, where each trial for stimulus intensity and RT is modulated by the appropriate value. The right graph shows the convolved regressors and all pairwise correlations of regressors are above 0.80, indicating possible collinearity.
Fig 1 How parametrically modulated regressors are constructed. The left graph shows the unconvolved regressors for 3 trials, where each trial for stimulus intensity and RT is modulated by the appropriate value. The right graph shows the convolved regressors and all pairwise correlations of regressors are above 0.80, indicating possible collinearity.CC BY · PubMed Central
Fig 2 Illustration of parameter estimate behavior for 20 simulated subjects when model has low collinearity (top), high collinearity (middle) and high collinearity where the feedback regressor is orthogonalized with respect to the stimulus regressor (bottom). The Venn diagrams indicate how the variability is distributed across the 2 regressors where red is unique to stimulus, blue is unique to feedback and purple is shared. The rightmost column shows the parameter estimates for stimulus (red) and feedback (blue) across 20 subjects, where the solid lines indicate the true values. With low collinearity the variability of the parameter estimates is much smaller than the high collinearity case, although in both cases the estimates are unbiased on average. Orthogonalization does not change the feedback parameter estimates (middle and bottom blue data points are identical), but the stimulus feedback has lower variability and the estimate is much higher.
Fig 2 Illustration of parameter estimate behavior for 20 simulated subjects when model has low collinearity (top), high collinearity (middle) and high collinearity where the feedback regressor is orthogonalized with respect to the stimulus regressor (bottom). The Venn diagrams indicate how the variability is distributed across the 2 regressors where red is unique to stimulus, blue is unique to feedback and purple is shared. The rightmost column shows the parameter estimates for stimulus (red) and feedback (blue) across 20 subjects, where the solid lines indicate the true values. With low collinearity the variability of the parameter estimates is much smaller than the high collinearity case, although in both cases the estimates are unbiased on average. Orthogonalization does not change the feedback parameter estimates (middle and bottom blue data points are identical), but the stimulus feedback has lower variability and the estimate is much higher.CC BY · PubMed Central
Fig 3 Distribution of descriptive variability with 2 sets of parametric modulators, RT and stimulus intensity. The top row shows how descriptive variability is shared when there is no orthogonalization. In this case the unmodulated regressor’s interpretation is average BOLD when RT and intensity are 0. The second row shows the orthogonalization necessary to change the interpretation of the unmodulated regressor to the average activation. Note the shared variability between RT and intensity. The last two rows show how the variability is distributed when using the automated orthogonalization in SPM when RT is entered before intensity and vice versa. The modulation that is entered first is not adjusted for the second, which can lead to misleading results. SPM sequentially orthogonalizes a modulated regressor with the previous regressors, therefore two models may have to be run to obtain the specific variance of two modulations.
Fig 3 Distribution of descriptive variability with 2 sets of parametric modulators, RT and stimulus intensity. The top row shows how descriptive variability is shared when there is no orthogonalization. In this case the unmodulated regressor’s interpretation is average BOLD when RT and intensity are 0. The second row shows the orthogonalization necessary to change the interpretation of the unmodulated regressor to the average activation. Note the shared variability between RT and intensity. The last two rows show how the variability is distributed when using the automated orthogonalization in SPM when RT is entered before intensity and vice versa. The modulation that is entered first is not adjusted for the second, which can lead to misleading results. SPM sequentially orthogonalizes a modulated regressor with the previous regressors, therefore two models may have to be run to obtain the specific variance of two modulations.CC BY · PubMed Central
Fig 4 Efficiency as a function of correlation between two regressors for a single parameter (green), the sum of parameters (red) and their difference (blue). As collinearity increases, the efficiencies of all three of these contrasts of parameter estimates diminishes.
Fig 4 Efficiency as a function of correlation between two regressors for a single parameter (green), the sum of parameters (red) and their difference (blue). As collinearity increases, the efficiencies of all three of these contrasts of parameter estimates diminishes.CC BY · PubMed Central

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Metadata

Authors
Mumford, J. A., Poline, J.-B., & Poldrack, R. A.
Affiliations
  • University of Wisconsin–Madison (US)
  • University of California, Berkeley (US)
  • Stanford University (US)
Corresponding author
Mumford, Jeanette A.
ORCID
Venue
PLoS ONE
Publisher
Public Library of Science (PLoS)
Published
2015-04-28
Volume
10
Issue
4
Pages
e0126255
ISSN
1932-6203 (ISSN-L)
DOI
10.1371/journal.pone.0126255
PMC
PMC4412813
OpenAlex
W2013974533
Cited by
276
References
9
Citations by year
2026: 16 · 2025: 27 · 2024: 24 · 2023: 22 · 2022: 29 · 2021: 26 · 2020: 38 · 2019: 16 · 2018: 29 · 2017: 32 · 2016: 15 · 2015: 2
FWCI
15.63
Type
journal article
Language
en
Open access
Licence
Locations
  • PLoS ONE (publishedVersion, cc-by)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
Topics
  • Life Sciences › Neuroscience › Cognitive Neuroscience › Functional Brain Connectivity Studies
  • Health Sciences › Medicine › Radiology, Nuclear Medicine and Imaging › Advanced MRI Techniques and Applications
  • Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
Keywords
orthogonalization, multicollinearity, parameter estimation, statistical power, SPM, FSL, design matrix
MeSH terms
Algorithms, Humans, Magnetic Resonance Imaging, Models, Theoretical, Reaction Time, Regression Analysis, Software

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