Accurate and Computationally Efficient Analysis of
Longitudinal fMRI data
Thomas Nichols and Lourens Waldorp
University of Warwick, Coventry, England and University of Amsterdam, The Netherlands
Introduction
There are a growing number of longitudinal fMRI studies
collecting rich datasets. Standard software, SPM and FSL in
particular, cannot accurately model these data when there are
k > 2 visits (repeated measurements). Specifically, FSL can
only accomodate a single contrast (COPE) at the second level
(e.g. for visit 2 - visit 1) and SPM, though it models repeated
measures correlation, unrealistically assumes that the
correlation is equal over the whole brain. While proper
longitudinal modelling is available in standard statistics
software (e.g. SAS’s proc mix, or R’s lme), the iterative
estimation proceedures can be prohibiatively slow.
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Methods
It is common practice to fit longitudinal data via OLS by
including dummy variables for each subject to account for
subject-specific differences in the overall mean. While this
”poor man’s MFX” approach is not correct in general, it
happens to be valid when the data form a balanced 1-way
ANOVA and intra-visit correlation is the same for all pairs of
visit (the condition of ”compound symmetry”). When compound
symmetry doesn’t hold, the estimates will be suboptimal
(inefficient) and standard errors biased.
We compare three univariate models for MFX inference on
longitudinal data: OLS, GLS, and SW via Monte Carlo
simulation. For a range of subject sample sizes
(n = 12, 25, 50, 100, 200) and number of visits (k = 3, 5, 8), and
compound symmetric and non-compound symmetric
(autoregressive structure, AR(1)) intra-visit correlation
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OLS
GLS
SW
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Figure 1: With compound symmetry, in the rows are relative bias in the
standard error, FPR, and effeiciency of OLS (red), GLS (green) and SW
(blue) for k = 3, 5, 8 time points.
http://home.medewerker.uva.nl/l.j.waldorp
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Results
Fig. 1 shows the situation for compound symmetry. The
Relative Bias in standard errors, FPR of the three estimators
are approximately the same, indicating that the SW does not
perform poorly when compound symmetry holds. When
compound symmetry does not hold (Fig. 2), OLS is least
accurate. The FPRs in Fig. 2 show that SW is reasonably
robust against number of visits. All methods show stronger
errors for few number of subjects. Note that the variation of SW
is larger than both GLS and OLS.
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We use the R software to perform these evaluations in order to
have access to state of the art GLS MFX modelling tools and
sandwich estimators. Note that OLS is offered in all fMRI
software, and SW does not require iteration and is easily
implemented.
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In this work we propose the use of Ordinary Least Squares
(OLS) combined with the sandwich estimator (White, 1982),
which should provide fast and valid P-values (with possibly
sub-optimal sensitivity). We compare this approach to the
commonly used “poor man’s” OLS longitudinal model for
longitudinal data (which is fast, but possibly both inaccruate
P-values and poor sensitivity), and Generalized Least Square
(GLS) to fit a proper longintiudonal model that accounts for
intrasubject correlation (which is the gold standard, but may be
slow).
structure, we evaluate the properties of the MFX estimates of
the linear change in BOLD over time: (1) SE (Standard Error)
Bias, the average error in the estimated variance of BOLD
change, (2) FPR relative accuracy, the relative false positive
rate (FPR) compared to the nominal level, and (3) SE Stdev,
the standard deviation of the standard error estimates divided
by the Monte Carlo mean of the standard error. We use 1000
simulations for each setting.
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Figure 2: Without compound symmetry (AR(1)), in the rows are relative bias
in the standard error, FPR, and effeiciency of OLS (red), GLS (green) and
SW (blue) for k = 3, 5, 8 visits.
Discussion
We have shown that the widely used OLS approach to
modelling longitudinal data can be inaccurate in a variety of
settings, due to bias in the standard errors of the effect.
Although SW is not always significantly better than OLS, SW
can be more robust than OLS to more extreme covariance
structures than shown here. While GLS-based MFX provides
optimal inferences, the use of the Sandwich Estimator
Standard Errors with OLS provides a computationally efficient,
robust, flexible and accurate approach to longitudinal fMRI.
References
[1] Laird and Ware (1982). Biometrics, 38(4), 963-974.
[2] Verbeke and Molenberghs (2000). Linear mixed models. Springer.
[3] Beckmann (2004). IEEE Trans. Med. Im. 23, 137-152.
[4] White (1981). JASA, 76(374), 419-433.
waldorp@uva.nl