Roy's maximum root and maximum canonical correlation SPMs from multivariate multiple regression analysis of imaging data

Keith J. Worsley¹, Jonathan E. Taylor², Francesco Tomaiuolo,³
¹Department of Mathematics and Statistics, and Montreal Neurological Institute, McGill University, Montreal, Canada, ²Department of Statistics, Stanford University, ³IRCCS Fondazione 'Santa Lucia', Roma, Italy

This abstract fills a gap in the random field theory for the P-value of local maxima of SPMs from multivariate linear models. So far results are only available for:

- univariate images, single contrasts (T SPM, [1]),
- univariate images, multiple contrasts (F SPM, [1]),
- multivariate images, single contrasts (Hotelling's T² SPM, [2)].

We report results for

- multivariate images, multiple contrasts.

Examples of multivariate image data are: vector deformations to warp an MRI image to an atlas standard, diffusion tensors, and the HRF sampled at 1s intervals. Examples of multiple contrasts are: several polynomial effects, several performance measures, or differences between several groups.

For multivariate data and multiple contrasts, there are several different test statistics, all based on the eigen values r of W^-1B, where W is the error mean sum of squares matrix, and B is the regression mean sum of squares matrix. The natural choice is the likelihood ratio test Wilks' L, equivalent to the product of 1/(1+r), but the random field theory for this has so far proved intractable. However we have succeeded for an alternative, Roy's maximum root R=max r, reported here.

A simpler definition of Roy’s maximum root is the following. Take a linear combination of the multivariate image data, creating univariate image data. Work out the F SPM for relating the univariate image data to the multiple contrasts. Roy’s maximum root R is the maximum F SPM over all such linear combinations.

The maximum canonical correlation C can be defined analogously as the maximum univariate correlation between all linear combinations of the multivariate data and multiple covariates. The relation between the two is R=(C/b)/((1-C)/w), where b and w are the degrees of freedom of B and W. Applications are effective connectivity, detected by the maximum canonical correlation between multivariate image data at a single reference voxel, and that at all other voxels. If the reference voxel is varied as well, the search space is 6D rather than 3D, and the results of [3] have been extended in the same way.

The P-value of local maxima of a smooth SPM in D dimensions is well approximated by

P(local max > t) ~ S_d=0^D Resels_d EC_d(t),

where Resels_d and EC_d(t) are the resels and Euler characteristic density in d dimensions. Figure 1 gives a surprisingly simple expression for the EC density EC_d^R(t) of Roy’s maximum root for k components in terms of the EC density EC_d^F(t) of the F statistic with b,w df [1]. The EC density of Hotelling’s T² [2] now falls out as a special case when b=1.

Figure 1: EC density for Roy's maximum root SPM

These results are available in FMRISTAT (www.math.mcgill.ca/keith/fmristat/toolbox/stat_threshold.m) and applied to detecting regions of brain damage of n=17 non-missile brain trauma patients compared to n=19 age and sex matched controls, using vector deformations images, and correlating the deformations with 6 memory tests.

[1]. Worsley et al. (1996). Human Brain Mapping, 4:58-73
[2]. Cao, Worsley (1999). Annals of Statistics, 27:925-942.
[3]. Cao, Worsley (1999). Annals of Applied Probability, 9:1021-1057.