Fast and accurate modelling of longitudinal
neuroimaging data
Bryan Guillaume
1
, Thomas E. Nichols & Lourens Waldorp
2
University of Warwick, Coventry, United Kingdom, University of
3
Liège, Liège, Belgium, University of Amsterdam, Amsterdam,
4
Netherlands, GlaxoSmithKline
Introduction
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Figure 2: Same as figure 1, but comparing the unadjusted and adjusted SwE
methods; M is the number of subjects.
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Observation index
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Observation index
Figure 3: Absolute and relative age (with the first visit of each subject taken
as baseline) in the imbalanced real study linked to the study in [3].
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Real design
Compound Symmetry
F(1,M−2) at 0.05 for Hom. HC2
OTW F(1,N−p) at 0.05
1092 %
1079 %
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Hom. HC2 SwE
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Age.L
Gr1
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Real design
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F(1,M−2) at 0.05 for Hom. HC2
OTW F(1,N−p) at 0.05
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954 %
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Hom. HC2 SwE
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Relative FPR (%)
Age.Q
Gr2 vs Gr1
Gr1
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Age.Q
Gr1
Gr2 vs Gr1
Age.L
Gr2 vs Gr1
Age.Q
Gr2 vs Gr1
t.e.nichols@warwick.ac.uk
We have shown that the standard methods can lead to very inaccurate inferences when compound symmetry does not hold, or even when compound
symmetry does hold for between-subject covariates. In contrast, the SwE
method, particularly with the use of small sample adjustments, has been
shown to be very accurate in a large range of settings. By eliminating subjectindicator covariates, the SwE method is easy to specify, and is fast since it
does not require any iteration.
References
Figure 1: Relative FPR obtained on the difference between two groups in
terms of linear effect of visits; compound symmetry with constant correlation
ρ = 0.8 and Toeplitz with a linear decay of the correlation of ψ = 0.2 per visit;
N is the total number of observations and p is the number of parameters.
bryan.guillaume@doct.ulg.ac.be
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Figure 4: Performance with the imbalanced real study design on several
effects, compound symmetry with constant correlation ρ = 0.8 and Toeplitz
with a linear decay of the correlation of ψ = 0.1 per year.
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Discussion
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Het. HC0 SwE
N−OLS
LME
Relative FPR (%)
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Linear effect of visits
Group 2 versus group 1
Toeplitz
8 vis., F(1,N−p) at 0.05
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Relative FPR (%)
Linear effect of visits
Group 2 versus group 1
Compound symmetry
8 vis., F(1,N−p) at 0.05
Het. HC0 SwE with F(1,N−p)
Hom. HC2 SwE with F(1,M−2)
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Results
Standard methods (N-OLS and LME) can have vastly inflated false positives
when compound symmetry does not hold (see Toeplitz case in Fig. 1). With
enough subjects, the standard “Het. HC0” SwE method gives accurate inferences even when compound symmetry does not hold (Fig. 1). Nevertheless,
the “Het. HC0” SwE suffers of inaccuracy when the number of subjects is
small. Fortunately, the adjusted “Hom. HC2” SwE improves significantly the
behaviour of the method and gives accurate inferences even with 12 subjects
(Fig. 2). With the real study design (Fig. 3), we show that the adjusted “Hom.
HC2” SwE method is very accurate and outperforms the standard methods
(Fig. 4).
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Relative FPR (%)
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Het. HC0 SwE with F(1,N−p)
Hom. HC2 SwE with F(1,M−2)
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Relative FPR (%)
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We use a simple OLS model for the marginal model (i.e., no subject indicators) to create estimates of the parameters of interest; standard errors of
these estimates are obtained with the so-called Sandwich Estimator (SwE) in
order to account for the repeated measures correlation [1,2]. Standard SwE
methods assume large samples and make no assumption of homogeneous
subject variance-covariance. Here, we propose the use of small sample adjustments [5] and a homogeneous-subject assumption to improve P-value
accuracy. We simulate data to test performance in a range of settings consisting of N = 10, 20, 50, 100, 200 subjects with k = 3, 5, 8 repeated measures.
In addition, we also consider the case of a very imbalanced real study design
of longitudinal changes in a population of adolescents at risk from alcohol
abuse linked to the study in [3] with missing data and without clear common
time points (as represented in Fig. 3). For each setting, compound symmetric
and toeplitz (linearly decaying) correlations have been considered for 10, 000
Monte Carlo Gaussian null simulations. We compare different SwE variants
to the LME method implemented in R (lme4). We evaluate all methods in
terms of bias and variance of the standard error, and false positive rate (as
a percentage, 100% being nominal) for α = 0.05. While we have considered many different variants of SwE methods, for space considerations, we
only show the variants HC0 (computed by using the raw residuals) and HC2
(computed with small sample adjusted residuals), assuming Heterogeneous
(“Het.”) or Homogeneous (“Hom.”) covariance estimates over subjects.
Linear effect of visits
Group 2 versus group 1
Toeplitz
8 vis., F−test at 0.05
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Absolute age (years)
Methods
Linear effect of visits
Group 2 versus group 1
Compound symmetry
8 vis., F−test at 0.05
Relative FPR (%)
Longitudinal data analysis is of increasing importance in neuroimaging, particularly in structural and functional MRI studies. Unfortunately, the existing
neuroimaging analysis software packages do not model longitudinal data correctly when there are more than two time points. In particular, FSL cannot
accommodate more than 2 time points and SPM unrealistically assumes a
common correlation structure for the whole brain.
The gold standard method to deal with longitudinal data is the Linear Mixed
Effects (LME) modelling [4]. Unfortunately, it requires iterative algorithms that
can be prohibitively slow and may fail to converge. One possible alternative
used by FSL and SPM is the Naive Ordinary Least Squares (N-OLS) modelling, where subject indicator variables are used to fit subject-specific slopes.
For a balanced 1-way repeated measures ANOVA with compound symmetric
(all equal) correlations, this gives the same results as an iterative LME model.
However, real data analyses often have imbalanced designs and, for more
than 3 time points especially, a compound symmetric assumption is probably
not realistic. Here, we consider an alternate approach that is computationally
efficient and allows the use of either within- or between-subject covariates.
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Relative age (years)
1
1,2,4
[1] Eicker, F. (1963), The Annals of Mathematical Statistics, 34, 447-456.
[2] Eicker, F. (1967), Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability.
[3] Heitzeg, M.M. and Nigg, J.T. and Yau, W.Y.W. and Zucker, R.A. and Zubieta, J.K. (2010), Biological
psychiatry, 68(3), 287-295.
[4] Laird, N.M. and Ware, J.H. (1982), Biometrics, 38, 963-974.
[5] MacKinnon J.G., White H. (1985), Journal of Econometrics, 29, 305-325.
http://go.warwick.ac.uk/tenichols
waldorp@uva.nl
http://home.medewerker.uva.nl/l.j.waldorp