Inference on Individual Differences in Functional Imaging: A test for correlations among tasks Sumitra Purkayastha   , Tor D. Wager  and Thomas E. Nichols   Bayesian and Interdisciplinary Research Unit, Indian Statistical Institute, Kolkata 700108, INDIA,  Department of Psychology, Columbia University, New York, NY 10027, USA,  Department of Biostatistics, University of Michigan, Ann Arbor, MI 48109, USA Objective Complex experimental designs often yield multiple measurements of interest on each subject. For ex- ample, a   factorial design will yield contrasts for each of  subjects. When analyz- ing all  measurements together at the sec- ond level, the problem of inter-task dependence or “non-sphericity” must be accounted for. SPM ([1]) treats the inter-task correlations as nuisance param- eters, estimating an inter-task covariance and adjusting the inferences accordingly ([2]). However, correlations among tasks are of great theoretical interest in psychological studies, because they provide information about how tasks are related to one another. Hence the goal of this work is to ana- lyze inter-task correlations as parameters of interest, not simply as nuisance parameters. Speciﬁcally, we wish to detect pairs of effects where a high response in one effect predicts a high (or low) response in the other effect. This kind of information in human per- formance is the basis of arguments for general fac- tors underlying mental abilities (e.g., Spearman’s G) but has seldom been applied to brain imaging data. Working with a two-level model for fMRI data, we estimate the matrix of correlations between at each voxel and create statistic maps that detect the presence of non-zero correlation across differ- ent tasks. Then using scatterplots and the individual correlations, we assess and interpret speciﬁc patterns of dependence. We demonstrate our method with a fMRI study of attention switching. Methods We work with the following two-level model for fMRI data for  subjects:      N "!$# % '& )( +* , -)- -)    ./  /  N 01!2# 43 ( 5* , -)- -)    -)- -) 476 /  -)- -) 4/86 are independent - !9,:( In (1),   is of length ; , the number of scans. The ﬁrst level models the fMRI data for the * -the individ- ual and the second level models the population mean  (a length- < vector) while allowing for a nondiago- nal covariance 3 . We wish to test if the covariance matrix of a set of contrasts of the vectors of activation coefﬁcients is diagonal. We assume that the within- subject variability %  is known, which is reasonable since ; is generally large and consequently an accu- rate estimate of % can be obtained. Allowing some possible loss of information, we work with the following modiﬁed formulation of this problem. To begin with, we assume that    for all * . Denote by = , the < ma- trix whose rows give the contrast vectors. We pre-multiply both sides of the ﬁrst equation of (1) above by > ?9@BA =C!D T E(GF  T and both sides of the second equation by = and obtain a transformed two-level model. The observations in the ﬁrst-level are given by H  ?9@BA >   and the regression pa- rameters corresponding the two levels are given by I  ?9@BA =   I ?9@BA = J- The vectors H  I  I are all , . The covariance matrix in the second-level is given by KL = 3 = T - It is possible now to formulate our question in terms of K . We want to test the following: H M+NOK is diagonal against H NOK is non-diagonal. !  ( In this initial work, we will assume that we have an orthogonal design. That is, we assume that the < contrast estimators are independent, speciﬁcally that P ?Q@A =C!D T E(RF = T is diagonal. With this assump- tion, any correlation observed in H  at the second- level is attributable to population-level inter-task cor- relation. In this case we can use Bartlett’s modiﬁca- tion ([3]) of the relevant Likelihood Ratio Test (LRT) ([4], pp. 137-138) as an ad-hoc solution. However, when the d