[en] Despite the growing importance of longitudinal data in neuroimaging, the standard analysis methods make restrictive or unrealistic assumptions (e.g., assumption of Compound Symmetry--the state of all equal variances and equal correlations--or spatially homogeneous longitudinal correlations). While some new methods have been proposed to more accurately account for such data, these methods are based on iterative algorithms that are slow and failure-prone. In this article, we propose the use of the Sandwich Estimator method which first estimates the parameters of interest with a simple Ordinary Least Square model and second estimates variances/covariances with the "so-called" Sandwich Estimator (SwE) which accounts for the within-subject correlation existing in longitudinal data. Here, we introduce the SwE method in its classic form, and we review and propose several adjustments to improve its behaviour, specifically in small samples. We use intensive Monte Carlo simulations to compare all considered adjustments and isolate the best combination for neuroimaging data. We also compare the SwE method to other popular methods and demonstrate its strengths and weaknesses. Finally, we analyse a highly unbalanced longitudinal dataset from the Alzheimer's Disease Neuroimaging Initiative and demonstrate the flexibility of the SwE method to fit within- and between-subject effects in a single model. Software implementing this SwE method has been made freely available at http://warwick.ac.uk/tenichols/SwE.
Bates D., Maechler M., Bolker B. lme4: Linear mixed-effects models using S4 classes. R package version 0.999999-0 URL. http://CRAN.R-project.org/package=lme4.
Bell R.M., McCaffrey D.F. Bias reduction in standard errors for linear regression with multi-stage samples. Surv. Methodol. 2002, 28(2):169-182.
Bernal-Rusiel J.L., Greve D.N., Reuter M., Fischl B., Sabuncu M.R. Statistical analysis of longitudinal neuroimage data with linear mixed effects models. NeuroImage 2013, 66:249-260.
Bernal-Rusiel J.L., Reuter M., Greve D.N., Fischl B., Sabuncu M.R. Spatiotemporal linear mixed effects modeling for the mass-univariate analysis of longitudinal neuroimage data. NeuroImage 2013, 81:358-370.
Box G.E. Problems in the analysis of growth and wear curves. Biometrics 1950, 6(4):362-389.
Chen G., Saad Z.S., Britton J.C., Pine D.S., Cox R.W. Linear mixed-effects modeling approach to fMRI group analysis. NeuroImage 2013, 73:176-190.
Chesher A., Jewitt I. The bias of a heteroskedasticity consistent covariance matrix estimator. Econometrica 1987, 1217-1222.
Diggle P., Liang K., Zeger S. Analysis of longitudinal data oxford statistical science series 1994, 13.
Efron B., Tibshirani R.J. An introduction to the bootstrap (chapman & hall/crc monographs on statistics & applied probability) 1994.
Eicker F. Asymptotic normality and consistency of the least squares estimators for families of linear regressions. Ann. Math. Stat. 1963, 34(2):447-456.
Eicker F. Limit theorems for regressions with unequal and dependent errors. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1967, vol. 1:59-82. University of California Press, Berkeley.
Fay M., Graubard B. Small-sample adjustments for Wald-type tests using sandwich estimators. Biometrics 2001, 57(4):1198-1206.
Fitzmaurice G.M. A caveat concerning independence estimating equations with multivariate binary data. Biometrics 1995, 309-317.
Halekoh U., Højsgaard S. pbkrtest: Parametric bootstrap and Kenward Roger based methods for mixed model comparison. R package version 0.3-8 URL. http://CRAN.R-project.org/package=pbkrtest.
Hansen L.P. Large sample properties of generalized method of moments estimators. Econometrica 1982, 1029-1054.
Hardin J. Small sample adjustments to the sandwich estimate of variance http://www.stata.com/support/faqs/stat/sandwich.html.
Härdle W., Simar L. Applied Multivariate Statistical Analysis 2007, Springer Verlag.
Harville D. Maximum likelihood approaches to variance component estimation and to related problems. J. Am. Stat. Assoc. 1977, 320-338.
Hinkley D. Jackknifing in unbalanced situations. Technometrics 1977, 285-292.
Horn S., Horn R., Duncan D. Estimating heteroscedastic variances in linear models. J. Am. Stat. Assoc. 1975, 380-385.
Hua X., Hibar D.P., Ching C.R., Boyle C.P., Rajagopalan P., Gutman B.A., Leow A.D., Toga A.W., C.R.J., Harvey D., Weiner M.W., Thompson P.M. Unbiased tensor-based morphometry: improved robustness and sample size estimates for Alzheimer's disease clinical trials. NeuroImage 2013, 66(0):648-661.
Huber P. The behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the fifth Berkeley symposium on mathematical statistics and probability 1967, vol. 1:221-233.
Kauermann G., Carroll R. A note on the efficiency of sandwich covariance matrix estimation. J. Am. Stat. Assoc. 2001, 96(456):1387-1396.
Kenward M.G., Roger J.H. Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 1997, 983-997.
Lai T.L., Small D. Marginal regression analysis of longitudinal data with time-dependent covariates: a generalized method-of-moments approach. J. R. Stat. Soc. Ser. B (Stat Methodol.) 2007, 69(1):79-99.
Laird N., Ware J. Random-effects models for longitudinal data. Biometrics 1982, 963-974.
Li Y., Gilmore J.H., Shen D., Styner M., Lin W., Zhu H. Multiscale adaptive generalized estimating equations for longitudinal neuroimaging data. NeuroImage 2013, 72:91-105.
Liang K., Zeger S. Longitudinal data analysis using generalized linear models. Biometrika 1986, 73(1):13-22.
Lindquist M., Spicer J., Asllani I., Wager T. Estimating and testing variance components in a multi-level glm. NeuroImage 2012, 59(1):490-501.
Lipsitz S., Ibrahim J., Parzen M. A degrees-of-freedom approximation for a t-statistic with heterogeneous variance. J. R. Stat. Soc. Ser. D Stat. 1999, 48(4):495-506.
Long J., Ervin L. Using heteroscedasticity consistent standard errors in the linear regression model. Am. Stat. 2000, 217-224.
MacKinnon J., White H. Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. J. Econ. 1985, 29(3):305-325.
Mancl L., DeRouen T. A covariance estimator for gee with improved small-sample properties. Biometrics 2001, 57(1):126-134.
McDonald B.W. Estimating logistic regression parameters for bivariate binary data. J. R. Stat. Soc. Ser. B Methodol. 1993, 391-397.
Molenberghs G., Verbeke G. A note on a hierarchical interpretation for negative variance components. Stat. Model. 2011, 11(5):389-408.
Mueller S., Weiner M., Thal L., Petersen R., Jack C., Jagust W., Trojanowski J., Toga A., Beckett L. The Alzheimer's Disease Neuroimaging Initiative. Neuroimaging Clin. N. Am. 2005, 15(4):869.
Mumford J.A., Nichols T. Simple group fMRI modeling and inference. NeuroImage 2009, 47(4):1469-1475.
Nel D., Van der Merwe C. A solution to the multivariate Behrens-Fisher problem. Commun. Stat. Theory Meth. 1986, 15(12):3719-3735.
Neuhaus J., Kalbfleisch J. Between-and within-cluster covariate effects in the analysis of clustered data. Biometrics 1998, 638-645.
Nichols T.E., Holmes A.P. Nonparametric permutation tests for functional neuroimaging: a primer with examples. Hum. Brain Mapp. 2002, 15(1):1-25.
Pan W. On the robust variance estimator in generalised estimating equations. Biometrika 2001, 88(3):901-906.
Pan W., Wall M. Small-sample adjustments in using the sandwich variance estimator in generalized estimating equations. Stat. Med. 2002, 21(10):1429-1441.
Pepe M.S., Anderson G.L. A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data. Commun. Stat. Simul. Comput. 1994, 23(4):939-951.
Pinheiro J., Bates D. Mixed-effects Models in s and s-plus. Statistics and Computing 2000, Springer-Verlag, Berlin, D.
Pinheiro J., Bates D., DebRoy S., Sarkar D., R Core Team nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-113 2013.
R Core Team R: A Language and Environment for Statistical Computing 2013, R Foundation for Statistical Computing, Vienna, Austria, (URL http://www.R-project.org/).
Skup M., Zhu H., Zhang H. Multiscale adaptive marginal analysis of longitudinal neuroimaging data with time-varying covariates. Biometrics 2012, 68(4):1083-1092.
Thompson W.K., Hallmayer J., O'Hara R. Design considerations for characterizing psychiatric trajectories across the lifespan: application to effects of apoe-e4 on cerebral cortical thickness in Alzheimer's disease. Am. J. Psychiatr. 2011, 168(9):894-903.
Verbeke G., Molenberghs G. Linear Mixed Models for Longitudinal Data 2009, Springer.
Waldorp L. Robust and unbiased variance of glm coefficients for misspecified autocorrelation and hemodynamic response models in fmri. J. Biomed. Imaging 2009, 15.
West B., Welch K.B., Galecki A.T. Linear Mixed Models: A Practical Guide Using Statistical Software 2006, CRC Press.
White H. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 1980, 817-838.
Worsley K.J., Marrett S., Neelin P., Vandal A.C., Friston K.J., Evans A.C., et al. A unified statistical approach for determining significant signals in images of cerebral activation. Hum. Brain Mapp. 1996, 4(1):58-73.
Zhang H. Advances in Modeling and Inference of Neuroimaging Data 2008, (Ph.D. thesis), The University of Michigan.
Zhao L.P., Prentice R.L., Self S.G. Multivariate mean parameter estimation by using a partly exponential model. J. R. Stat. Soc. Ser. B Methodol. 1992, 805-811.
