The Annals of Applied Statistics

A lag functional linear model for prediction of magnetization transfer ratio in multiple sclerosis lesions

Gina-Maria Pomann, Ana-Maria Staicu, Edgar J. Lobaton, Amanda F. Mejia, Blake E. Dewey, Daniel S. Reich, Elizabeth M. Sweeney, and Russell T. Shinohara

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We propose a lag functional linear model to predict a response using multiple functional predictors observed at discrete grids with noise. Two procedures are proposed to estimate the regression parameter functions: (1) an approach that ensures smoothness for each value of time using generalized cross-validation; and (2) a global smoothing approach using a restricted maximum likelihood framework. Numerical studies are presented to analyze predictive accuracy in many realistic scenarios. The methods are employed to estimate a magnetic resonance imaging (MRI)-based measure of tissue damage (the magnetization transfer ratio, or MTR) in multiple sclerosis (MS) lesions, a disease that causes damage to the myelin sheaths around axons in the central nervous system. Our method of estimation of MTR within lesions is useful retrospectively in research applications where MTR was not acquired, as well as in clinical practice settings where acquiring MTR is not currently part of the standard of care. The model facilitates the use of commonly acquired imaging modalities to estimate MTR within lesions, and outperforms cross-sectional models that do not account for temporal patterns of lesion development and repair.

Article information

Source
Ann. Appl. Stat., Volume 10, Number 4 (2016), 2325-2348.

Dates
Received: May 2016
First available in Project Euclid: 5 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1483606862

Digital Object Identifier
doi:10.1214/16-AOAS981

Mathematical Reviews number (MathSciNet)
MR3592059

Zentralblatt MATH identifier
06688779

Keywords
Functional data analysis functional linear model magnetization transfer ratio image analysis

Citation

Pomann, Gina-Maria; Staicu, Ana-Maria; Lobaton, Edgar J.; Mejia, Amanda F.; Dewey, Blake E.; Reich, Daniel S.; Sweeney, Elizabeth M.; Shinohara, Russell T. A lag functional linear model for prediction of magnetization transfer ratio in multiple sclerosis lesions. Ann. Appl. Stat. 10 (2016), no. 4, 2325--2348. doi:10.1214/16-AOAS981. https://projecteuclid.org/euclid.aoas/1483606862


Export citation

References

  • Barkhof, F. (2002). The clinico-radiological paradox in multiple sclerosis revisited. Curr. Opin. Neurol. 15 239–245.
  • Besse, P. and Ramsay, J. O. (1986). Principal components analysis of sampled functions. Psychometrika 51 285–311.
  • Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics 149. Springer, New York.
  • Brex, P. A., Ciccarelli, O., O’Riordan, J. I., Sailer, M., Thompson, A. J. and Miller, D. H. (2002). A longitudinal study of abnormalities on MRI and disability from multiple sclerosis. N. Engl. J. Med. 346 158–164.
  • Chen, J. T., Kuhlmann, T., Jansen, G. H., Collins, D. L., Atkins, H. L., Freedman, M. S., O’Connor, P. W., Arnold, D. L., Group, C. M. S. et al. (2007). Voxel-based analysis of the evolution of magnetization transfer ratio to quantify remyelination and demyelination with histopathological validation in a multiple sclerosis lesion. NeuroImage 36 1152–1158.
  • Chen, J. T., Collins, D. L., Atkins, H. L., Freedman, M. S. and Arnold, D. L. (2008). Magnetization transfer ratio evolution with demyelination and remyelination in multiple sclerosis lesions. Ann. Neurol. 63 254–262.
  • Crainiceanu, C. M., Staicu, A.-M. and Di, C.-Z. (2009). Generalized multilevel functional regression. J. Amer. Statist. Assoc. 104 1550–1561.
  • Di, C.-Z., Crainiceanu, C. M., Caffo, B. S. and Punjabi, N. M. (2009). Multilevel functional principal component analysis. Ann. Appl. Stat. 3 458–488.
  • Ferraty, F., Vieu, P. and Viguier-Pla, S. (2007). Factor-based comparison of groups of curves. Comput. Statist. Data Anal. 51 4903–4910.
  • Goldsmith, J., Greven, S. and Crainiceanu, C. I. P. R. I. A. N. (2012). Corrected confidence bands for functional data using principal components. Biometrics.
  • Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493–1517.
  • Harezlak, J., Coull, B. A., Laird, N. M., Magari, S. R. and Christiani, D. C. (2007). Penalized solutions to functional regression problems. Comput. Statist. Data Anal. 51 4911–4925.
  • Hawkins, C. P., Munro, P. M. G., MacKenzie, F., Kesselring, J., Tofts, P. S., Du Boulay, E. P. G. H., Landon, D. N. and McDonald, W. I. (1990). Duration and selectivity of blood-brain barrier breakdown in chronic relapsing experimental allergic encephalomyelitis studied by gadolinium-DTPA and protein markers. Brain 113 365–378.
  • He, G., Müller, H.-G., Wang, J.-L. and Yang, W. (2010). Functional linear regression via canonical analysis. Bernoulli 16 705–729.
  • Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer, New York.
  • Ivanescu, A. E., Staicu, A.-M., Scheipl, F. and Greven, S. (2015). Penalized function-on-function regression. Comput. Statist. 30 539–568.
  • Jog, A., Roy, S., Carass, A. and Prince, J. L. (2013a). Pulse sequence based multi-acquisition MR intensity normalization. In SPIE Medical Imaging 86692H–86692H. International Society for Optics and Photonics.
  • Jog, A., Roy, S., Carass, A. and Prince, J. L. (2013b). Magnetic resonance image synthesis through patch regression. In IEEE 10th International Symposium on Biomedical Imaging (ISBI) 350–353. IEEE, New York.
  • Kim, K., Şentürk, D. and Li, R. (2011). Recent history functional linear models for sparse longitudinal data. J. Statist. Plann. Inference 141 1554–1566.
  • Krivobokova, T. and Kauermann, G. (2007). A note on penalized spline smoothing with correlated errors. J. Amer. Statist. Assoc. 102 1328–1337.
  • Malfait, N. and Ramsay, J. O. (2003). The historical functional linear model. Canad. J. Statist. 31 115–128.
  • McDonald, W. I., Compston, A., Edan, G., Goodkin, D., Hartung, H.-P., Lublin, F. D., McFarland, H. F., Paty, D. W., Polman, C. H., Reingold, S. C. et al. (2001). Recommended diagnostic criteria for multiple sclerosis: Guidelines from the international panel on the diagnosis of multiple sclerosis. Ann. Neurol. 50 121–127.
  • McLean, M. W., Hooker, G., Staicu, A.-M., Scheipl, F. and Ruppert, D. (2014). Functional generalized additive models. J. Comput. Graph. Statist. 23 249–269.
  • Meier, D. S., Weiner, H. L. and Guttmann, C. R. G. (2007). Time-series modeling of multiple sclerosis disease activity: A promising window on disease progression and repair potential? Neurotherapeutics 4 485–498.
  • Mejia, A., Sweeney, E. M., Dewey, B., Nair, G., Sati, P., Shea, C., Reich, D. S. and Shinohara, R. T. (2016). Statistical estimation of T1 relaxation times using conventional magnetic resonance imaging. NeuroImage 133 176–188.
  • Meyer, M. J., Coull, B. A., Versace, F., Cinciripini, P. and Morris, J. S. (2015). Bayesian function-on-function regression for multilevel functional data. Biometrics 71 563–574.
  • Morris, J. S. (2015). Functional regression. Annual Reviews of Statistics and Its Applications 2 321–359.
  • Polman, C. H., Reingold, S. C., Edan, G., Filippi, M., Hartung, H.-P., Kappos, L., Lublin, F. D., Metz, L. M., McFarland, H. F., O’Connor, P. W. et al. (2005). Diagnostic criteria for multiple sclerosis: 2005 revisions to the “McDonald criteria”. Ann. Neurol. 58 840–846.
  • Pomann, G.-M., Sweeney, E. M., Reich, D. S., Staicu, A.-M. and Shinohara, R. T. (2015). Scan-stratified case-control sampling for modeling blood-brain barrier integrity in multiple sclerosis. Stat. Med. 34 2872–2880.
  • Pomann, G.-M., Staicu, A., Lobaton, E. J., Mejia, A. F., Dewey, B. E., Reich, D. S., Sweeney, E. M.E. M. and Shinohara, R. T.R. T. (2016). Supplement to “A lag functional linear model for prediction of magnetization transfer ratio in multiple sclerosis lesions.” DOI:10.1214/16-AOAS981SUPP.
  • Ramsay, J. O. and Dalzell, C. J. (1991). Some tools for functional data analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol. 53 539–572.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Reich, D. S., White, R., Cortese, I. C., Vuolo, O., Shea, C. D., Collins, T. L. and Petkau, J. (2015). Sample-size calculations for short-term proof-of-concept studies of tissue protection and repair in multiple sclerosis lesions via conventional clinical imaging. Mult. Scler. 21 1693–1704.
  • Reiss, P. T. and Ogden, R. T. (2009). Smoothing parameter selection for a class of semiparametric linear models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 505–523.
  • Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B. Stat. Methodol. 53 233–243.
  • Roy, S., Carass, A. and Prince, J. (2011). A compressed sensing approach for MR tissue contrast synthesis. In Information Processing in Medical Imaging 371–383. Springer, Berlin.
  • Roy, S., Carass, A. and Prince, J. L. (2013). Magnetic resonance image example-based contrast synthesis. IEEE Trans. Med. Imag. 32 2348–2363.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics 12. Cambridge Univ. Press, Cambridge.
  • Scheipl, F. and Greven, S. (2015). Identifiability in penalized function-on-function regression models. Technical report, Univ. of Munich.
  • Scheipl, F., Staicu, A.-M. and Greven, S. (2015). Functional additive mixed models. J. Comput. Graph. Statist. 24 477–501.
  • Schmierer, K., Scaravilli, F., Altmann, D. R., Barker, G. J. and Miller, D. H. (2004). Magnetization transfer ratio and myelin in postmortem multiple sclerosis brain. Ann. Neurol. 56 407–415.
  • Staniswalis, J. G. and Lee, J. J. (1998). Nonparametric regression analysis of longitudinal data. J. Amer. Statist. Assoc. 93 1403–1418.
  • Suttner, L., Mejia, A., Dewey, B., Sati, P., Reich, D. S. and Shinohara, R. T. (2015). Statistical estimation of white matter microstructure from conventional MRI. UPenn Biostatistics Working Papers. Working Paper 44.
  • Sweeney, E. M., Shinohara, R. T., Shea, C. D., Reich, D. S. and Crainiceanu, C. M. (2013a). Automatic lesion incidence estimation and detection in multiple sclerosis using multisequence longitudinal MRI. Am. J. Neuroradiol. 34 68–73.
  • Sweeney, E. M., Shinohara, R. T., Shiee, N., Mateen, F. J., Chudgar, A. A., Cuzzocreo, J. L., Calabresi, P. A., Pham, D. L., Reich, D. S. and Crainiceanu, C. M. (2013b). OASIS is automated statistical inference for segmentation, with applications to multiple sclerosis lesion segmentation in MRI. NeuroImage: Clinical 2 402–413.
  • Sweeney, E. M., Shinohara, R. T., Dewey, B. E., Schindler, M. K., Muschelli, J., Reich, D. S., Crainiceanu, C. M. and Eloyan, A. (2015). Relating multi-sequence longitudinal intensity profiles and clinical covariates in new multiple sclerosis lesions. Preprint. Available at arXiv:1509.08359.
  • Van Den Elskamp, I. J., Lembcke, J., Dattola, V., Beckmann, K., Pohl, C., Hong, W., Sandbrink, R., Wagner, K., Knol, D. L., Uitdehaag, B. et al. (2008). Persistent T1 hypointensity as an MRI marker for treatment efficacy in multiple sclerosis. Mult. Scler. 14 764–769.
  • Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Chapman & Hall, Boca Raton, FL.
  • Wood, S. N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 3–36.
  • Wu, Y., Fan, J. and Müller, H.-G. (2010). Varying-coefficient functional linear regression. Bernoulli 16 730–758.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005a). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005b). Functional linear regression analysis for longitudinal data. Ann. Statist. 33 2873–2903.
  • Zhang, J.-T. and Chen, J. (2007). Statistical inferences for functional data. Ann. Statist. 35 1052–1079.

Supplemental materials

  • Algorithms and additional results. In this file, we include a sample sourcecode for estimation of the HFLM model, implementation details of the PW and GB approaches, figures for the simulated dense and sparse data, and additional results.