The SwE Toolbox: a Toolbox for the Analysis of Longitudinal and Repeated Measures Neuroimaging Data Bryan Guillaume1,2, Xue Hua3, Paul M. Thompson3, Lourens Waldorp4, Thomas E. Nichols1 and the Alzheimer’s Disease Neuroimaging Initiative 1University of Warwick, Coventry, United Kingdom, 2University of Li`ege, Li`ege, Belgium, 3University of California, Los Angeles, USA, 4University of Amsterdam, Amsterdam, Netherlands. Introduction Neuroimaging software packages like SPM and FSL currently model longi- tudinal and repeated measures neuroimaging data using restrictive assump- tions. In particular, SPM assumes a common covariance structure for all the voxels in the brain and FSL assumes Compound Symmetry (CS), the state of all equal variances and all equal covariances. While more accurate meth- ods have been recently proposed to analyse such data [1,2,4,5,7,8], there remain few easy-to-use implementations of these methods. Here, we present an SPM toolbox allowing the use of the Sandwich Estimator (SwE) method, a fast, non-iterative tool for longitudinal and repeated measures data [5], and illustrate its use on data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI). Methods The SwE toolbox estimates parameters of interest using an Ordinary Least Squares (OLS) model and their variances/covariances using the so-called Sandwich Estimator [5]. The toolbox consists of a set of Matlab scripts de- signed to work in conjunction with the most recent versions of SPM (i.e. SPM8 or SPM12). The toolbox offers a user interface (Figure 1) allowing an easy speciﬁcation of the design and data, and a display of results similar to stan- dard SPM analyses. The use of the toolbox can be divided into 3 stages: the model setup, the model estimation and the display of results. The setup stage calls the Matlab batch system (Figure 1, top left) with a dedicated module for the SwE toolbox, which can be used to easily specify the data and design of the analysis. The second stage estimates the model. Finally, the third stage allows the speciﬁcation of contrasts of interest to make inference and display results (Figure 1, middle and right) in a similar way as standard SPM analy- ses. Regarding the inferences, the current release version of the toolbox only allows for the use of parametric uncorrected and voxel-wise False Discovery Rate (FDR) inferences based on results in [5]; methods for non-parametric uncorrected, FDR, and Family-Wise Error (FWE) inferences based on a Wild Bootstrap [9] resampling method are forthcoming. Here, we demonstrate the toolbox on a highly unbalanced longitudinal dataset (i.e. with many missing visits) consisting of Tensor Base Morphometry images obtained from the ADNI project, where 229 healthy elderly Normal control, 400 Mild Cognitive Impairment (MCI) and 188 Alzheimer’s Disease (AD) sub- jects were scanned up to 6 times over a period of 3 years [6]. As comparison, we also analyse the same dataset using 2 alternative methods: the Naive- OLS (N-OLS) method which includes subject dummy variables in an OLS model and assumes, by construction, Compound Symmetry (CS), and the Summary Statistic OLS (SS-OLS) method which ﬁrst ﬁts a regression model for each subject to obtain subject-speciﬁc estimates of the parameters of in- terest, and then computes a simple model on these summary measures. In addition, we conducted a Box’s test of CS [3] on the largest subset of the ADNI dataset without missing data to check the validity of the CS assumption. Results Figure 2 shows the Box’s test F-score image (centred at the anterior com- missure) thresholded at 5% after using a Bonferroni correction. 56% of the in-mask voxels survived the thresholding, indicating a strong evidence of non- Compound Symmetry in the brain and challenging the validity of the N-OLS method. Figure 3 and 4 show a comparison of thresholded t-score images (centred at the anterior commissure) obtained with the N-OLS, SS-OLS and SwE meth- ods on the difference in (lon