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**RE: Clarification on contrasts in CONN 2nd-level multivariate analysis**Oct 8, 2018 04:10 AM | Martyn McFarquhar

RE: Clarification on contrasts in CONN 2nd-level multivariate analysis

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 11, 2018 | |

Martyn McFarquhar |
Oct 12, 2018 | |

Alfonso Nieto-Castanon |
Oct 12, 2018 | |

Martyn McFarquhar |
Oct 15, 2018 | |

Martyn McFarquhar |
Oct 5, 2018 | |