The Annals of Statistics

Random fields of multivariate test statistics, with applications to shape analysis

J. E. Taylor and K. J. Worsley

Full-text: Open access


Our data are random fields of multivariate Gaussian observations, and we fit a multivariate linear model with common design matrix at each point. We are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point. The problem is to find the P-value of the maximum of such a random field of test statistics. We approximate this by the expected Euler characteristic of the excursion set. Our main result is a very simple method for calculating this, which not only gives us the previous result of Cao and Worsley [Ann. Statist. 27 (1999) 925–942] for Hotelling’s T2, but also random fields of Roy’s maximum root, maximum canonical correlations [Ann. Appl. Probab. 9 (1999) 1021–1057], multilinear forms [Ann. Statist. 29 (2001) 328–371], χ̄2 [Statist. Probab. Lett 32 (1997) 367–376, Ann. Statist. 25 (1997) 2368–2387] and χ2 scale space [Adv. in Appl. Probab. 33 (2001) 773–793]. The trick involves approaching the problem from the point of view of Roy’s union-intersection principle. The results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma.

Article information

Ann. Statist., Volume 36, Number 1 (2008), 1-27.

First available in Project Euclid: 1 February 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 62H11: Directional data; spatial statistics

Integral geometry differential topology Euler characteristic Morse theory Roy’s maximum root Hotelling’s T^2 canonical correlation scale space


Taylor, J. E.; Worsley, K. J. Random fields of multivariate test statistics, with applications to shape analysis. Ann. Statist. 36 (2008), no. 1, 1--27. doi:10.1214/009053607000000406.

Export citation


  • Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
  • Adler, R. J. (2000). On excursion sets, tube formulae, and maxima of random fields. Ann. Appl. Probab. 10 1–74.
  • Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • Cao, J. and Worsley, K. J. (1999). The detection of local shape changes via the geometry of Hotelling’s T2 fields. Ann. Statist. 27 925–942.
  • Cao, J. and Worsley, K. J. (1999). The geometry of correlation fields with an application to functional connectivity of the brain. Ann. Appl. Probab. 9 1021–1057.
  • Collins, D. L., Holmes, C. J., Peters, T. M. and Evans, A. C. (1995). Automatic 3-D model-based neuroanatomical segmentation. Human Brain Mapping 3 190–208.
  • Friston, K. J., Büchel, C., Fink, G. R., Morris, J., Rolls, E. and Dolan, R. J. (1997). Psychophysiological and modulatory interactions in neuroimaging. NeuroImage 6 218–229.
  • Hadwiger, H. (1957). Vorlesüngen Über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin.
  • Johansen, S. and Johnstone, I. M. (1990). Hotelling’s theorem on the volume of tubes: Some illustrations in simultaneous inference and data analysis. Ann. Statist. 18 652–684.
  • Knowles, M. and Siegmund, D. (1989). On Hotelling’s approach to testing for a nonlinear parameter in a regression. Internat. Statist. Rev. 57 205–220.
  • Kuriki, S. and Takemura, A. (2001). Tail probabilities of the maxima of multilinear forms and their applications. Ann. Statist. 29 328–371.
  • Lin, Y. and Lindsay, B. G. (1997). Projections on cones, chi-bar squared distributions, and Weyl’s formula. Statist. Probab. Lett. 32 367–376.
  • Morse, M. and Cairns, S. S. (1969). Critical Point Theory in Global Analysis and Differential Topology. Academic Press, New York.
  • Siegmund, D. O. and Worsley, K. J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist. 23 608–639.
  • Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34–71.
  • Sun, J. and Loader, C. R. (1994). Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 1328–1345.
  • Sun, J., Loader, C. R. and McCormick, W. P. (2000). Confidence bands in generalized linear models. Ann. Statist. 28 429–460.
  • Takemura, A. and Kuriki, S. (1997). Weights of χ̅2 distribution for smooth or piecewise smooth cone alternatives. Ann. Statist. 25 2368–2387.
  • Takemura, A. and Kuriki, S. (2002). On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. Appl. Probab. 12 768–796.
  • Taylor, J. E. (2006). A Gaussian kinematic formula. Ann. Probab. 34 122–158.
  • Taylor, J. E. and Adler, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Probab. 31 533–563.
  • Taylor, J. E., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362–1396.
  • Taylor, J. E. and Worsley, K. J. (2007). Detecting sparse cone alternatives for Gaussian random fields, with an application to FMRI. Available at
  • Taylor, J. E. and Worsley, K. J. (2007). Detecting sparse signals in random fields, with an application to brain mapping. J. Amer. Statist. Assoc. 102 913–928.
  • Tomaiuolo, F., Worsley, K. J., Lerch, J., Di Paulo, M., Carlesimo, G. A., Bonanni, R., Caltagirone, C. and Paus, T. (2005). Changes in white matter in long-term survivors of severe non-missile traumatic brain injury: A computational analysis of magnetic resonance images. J. Neurotrauma 822 76–82.
  • Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of χ2, F and t fields. Adv. in Appl. Probab. 26 13–42.
  • Worsley, K. J. (1995). Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. in Appl. Probab. 27 943–959.
  • Worsley, K. J. (2001). Testing for signals with unknown location and scale in a χ2 random field, with an application to fMRI. Adv. in Appl. Probab. 33 773–793.
  • Worsley, K. J., Andermann, M., Koulis, T., MacDonald, D. and Evans, A. C. (1999). Detecting changes in nonisotropic images. Human Brain Mapping 8 98–101.
  • Worsley, K. J. and Friston, K. J. (2000). A test for a conjunction. Statist. Probab. Lett. 47 135–140.
  • Worsley, K. J., Taylor, J. E., Tomaiuolo, F. and Lerch, J. (2004). Unified univariate and multivariate random field theory. NeuroImage 23 S189–S196.