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help > RE: Clarification on contrasts in CONN 2nd-level multivariate analysis
Oct 8, 2018 06:10 PM | Alfonso Nieto-Castanon - Boston University
RE: Clarification on contrasts in CONN 2nd-level multivariate analysis
Hi Martyn,
Yes, you are right the "univariate (SPM)" approach for multiple DVs is simply a univariate repeated measures model in SPM where both the conditions (pre- post-) and the components (MVPA) are treated as within-subject factors (with levels assumed non-independent with unequal variance). You are also right that, due to the approach chosen in CONN, a different model will be used/implemented for every different contrast that you may want to estimate.
Just for reference / additional info, in general CONN uses SPM's "two-stage" approach (described in more detail in ref attached), both for multivariate and repeated-measures univariate statistics. This means that, if you specify a contrast matrix M with K rows, then a separate image is computed for each subject and for each row of M, and the resulting NxK images are then entered into a new second-level model now with M=eye(K). In your example with 3-components (from MVPA) and 2-conditions (pre- and post-), and M=[1 -1 0 0 0 0;0 0 1 -1 0 0;0 0 0 0 1 -1] a new image will be computed separately for each subject and component estimating the difference pre- and post- in the corresponding MVPA component (so subject effects are already taken into account by this first-level contrast estimation). So, for the univariate repeated-measures approach, you would have something like:
y = reshape(Y*M', 3*N,1);
G = kron(eye(3),X);
y = G*beta + e;
with:
V = E{e*e'} = kron(Sigma,eye(N))
where Y is N-by-6 data matrix, V is a 3*N-by-3*N matrix modelling the repeated-measures structure of our data (non-independence between pairs of DVs within the same subject), the 3-by-3 matrix Sigma is estimated jointly across all voxels (note that this also allows DVs to have different scales, as long as the ratio between those scales is relatively constant across voxels), X in our example was a N-by-4 design matrix, the vector beta (in our example a 12 element vector) is estimated separately for each voxel, and the hypothesis tested is kron(eye(3),C)*beta = [0 0 0]' (M does not appear here as it has already been subsumed in the data y, and C in our original example was [1 -1 -1 1] modelling the interaction with the two between-subject effects)
In contrast, for the multivariate analyses on the same data, you would have something like:
y = Y*M';
y = X*beta + e;
with:
Sigma = E{e'*e}
where the 3-by-3 matrix Sigma, as well as beta (in our example now a 4-by-3 matrix), are estimated separately for each voxel, and the hypothesis tested now is C*beta = [0 0 0] (again with X, Y, M, and C just the same as before)
Hope this helps
Alfonso
Originally posted by Martyn McFarquhar:
Yes, you are right the "univariate (SPM)" approach for multiple DVs is simply a univariate repeated measures model in SPM where both the conditions (pre- post-) and the components (MVPA) are treated as within-subject factors (with levels assumed non-independent with unequal variance). You are also right that, due to the approach chosen in CONN, a different model will be used/implemented for every different contrast that you may want to estimate.
Just for reference / additional info, in general CONN uses SPM's "two-stage" approach (described in more detail in ref attached), both for multivariate and repeated-measures univariate statistics. This means that, if you specify a contrast matrix M with K rows, then a separate image is computed for each subject and for each row of M, and the resulting NxK images are then entered into a new second-level model now with M=eye(K). In your example with 3-components (from MVPA) and 2-conditions (pre- and post-), and M=[1 -1 0 0 0 0;0 0 1 -1 0 0;0 0 0 0 1 -1] a new image will be computed separately for each subject and component estimating the difference pre- and post- in the corresponding MVPA component (so subject effects are already taken into account by this first-level contrast estimation). So, for the univariate repeated-measures approach, you would have something like:
y = reshape(Y*M', 3*N,1);
G = kron(eye(3),X);
y = G*beta + e;
with:
V = E{e*e'} = kron(Sigma,eye(N))
where Y is N-by-6 data matrix, V is a 3*N-by-3*N matrix modelling the repeated-measures structure of our data (non-independence between pairs of DVs within the same subject), the 3-by-3 matrix Sigma is estimated jointly across all voxels (note that this also allows DVs to have different scales, as long as the ratio between those scales is relatively constant across voxels), X in our example was a N-by-4 design matrix, the vector beta (in our example a 12 element vector) is estimated separately for each voxel, and the hypothesis tested is kron(eye(3),C)*beta = [0 0 0]' (M does not appear here as it has already been subsumed in the data y, and C in our original example was [1 -1 -1 1] modelling the interaction with the two between-subject effects)
In contrast, for the multivariate analyses on the same data, you would have something like:
y = Y*M';
y = X*beta + e;
with:
Sigma = E{e'*e}
where the 3-by-3 matrix Sigma, as well as beta (in our example now a 4-by-3 matrix), are estimated separately for each voxel, and the hypothesis tested now is C*beta = [0 0 0] (again with X, Y, M, and C just the same as before)
Hope this helps
Alfonso
Originally posted by Martyn McFarquhar:
Hi Alfonso,
Thank you for the detailed response. So the approach is basically to form the univariate expression of the multivariate model? So for the example of 3 components with a pre a post you can have
y = reshape(Y', [n 1]);
G = kron(X, eye(6));
such that
y ~ N(G*beta, V)
and
V = blkdiag(Sigma(1), Sigma(2), Sigma(3), Sigma(4))
where each Sigma(i) is 6 x 6 covariance matrix of the dependancy structure between the original dependent variables (so the 3 components x 2 time-points)?
The SPM ReML procedure would then estimate a single covariance matrix from a pool of voxels from an initial model fit, which is then scaled by a voxel-specific noise term. So identical to a regular 2nd-level SPM analysis?
Presumably this is not suitable for DVs on different scales, but works fine for the case of the component?
The contrasts would then be something like?
C = kron(L,M)
Putting aside issues of having a global covariance estimate (which is par for the course in SPM), my question is then about the correct estimation of the variance-covariance matrix of the parameters. If you are using essentially a block-diagonal version of the original cell means design matrix, how are you able to derive the correct error terms for each contrast without inclusion of subject as a block in the design matrix?
For the regular multivariate statistics, this is implicit in the derivation of the SSCPh and SSCPe matrices, and for marginal models (such as the sandwich-estimator) this is formed from the design matrix and the estimate of V. For the regular SPM F-statistic, you have to force the correct error term due to the implicit use of the model residuals to form the singular variance estimate. This involves multiple models and the inclusion of subject (and all interactions) in the model. As you are essentially forming a univariate repeated-measures model in this approach, this is surely the only way to make sure the test statistic is correct. Is this what CONN is doing, or is there some other statistic being used that is able to accommodate the correlation structure in the derivation of the test statistic?
Best wishes,
- Martyn
Thank you for the detailed response. So the approach is basically to form the univariate expression of the multivariate model? So for the example of 3 components with a pre a post you can have
y = reshape(Y', [n 1]);
G = kron(X, eye(6));
such that
y ~ N(G*beta, V)
and
V = blkdiag(Sigma(1), Sigma(2), Sigma(3), Sigma(4))
where each Sigma(i) is 6 x 6 covariance matrix of the dependancy structure between the original dependent variables (so the 3 components x 2 time-points)?
The SPM ReML procedure would then estimate a single covariance matrix from a pool of voxels from an initial model fit, which is then scaled by a voxel-specific noise term. So identical to a regular 2nd-level SPM analysis?
Presumably this is not suitable for DVs on different scales, but works fine for the case of the component?
The contrasts would then be something like?
C = kron(L,M)
Putting aside issues of having a global covariance estimate (which is par for the course in SPM), my question is then about the correct estimation of the variance-covariance matrix of the parameters. If you are using essentially a block-diagonal version of the original cell means design matrix, how are you able to derive the correct error terms for each contrast without inclusion of subject as a block in the design matrix?
For the regular multivariate statistics, this is implicit in the derivation of the SSCPh and SSCPe matrices, and for marginal models (such as the sandwich-estimator) this is formed from the design matrix and the estimate of V. For the regular SPM F-statistic, you have to force the correct error term due to the implicit use of the model residuals to form the singular variance estimate. This involves multiple models and the inclusion of subject (and all interactions) in the model. As you are essentially forming a univariate repeated-measures model in this approach, this is surely the only way to make sure the test statistic is correct. Is this what CONN is doing, or is there some other statistic being used that is able to accommodate the correlation structure in the derivation of the test statistic?
Best wishes,
- Martyn
Threaded View
| Title | Author | Date |
|---|---|---|
| Martyn McFarquhar | Oct 4, 2018 | |
| Alfonso Nieto-Castanon | Oct 5, 2018 | |
| Martyn McFarquhar | Oct 5, 2018 | |
| Alfonso Nieto-Castanon | Oct 5, 2018 | |
| Martyn McFarquhar | Oct 8, 2018 | |
| Alfonso Nieto-Castanon | Oct 8, 2018 | |
| Martyn McFarquhar | Oct 9, 2018 | |
| Alfonso Nieto-Castanon | Oct 9, 2018 | |
| Ali Amad | Oct 11, 2018 | |
| Alfonso Nieto-Castanon | Oct 12, 2018 | |
| Ali Amad | Oct 19, 2018 | |
| Martyn McFarquhar | Oct 11, 2018 | |
| Alfonso Nieto-Castanon | Oct 12, 2018 | |
| Martyn McFarquhar | Oct 12, 2018 | |
| Alfonso Nieto-Castanon | Oct 12, 2018 | |
| Martyn McFarquhar | Oct 15, 2018 | |
| Martyn McFarquhar | Oct 5, 2018 | |