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**RE: Clarification on contrasts in CONN 2nd-level multivariate analysis**Oct 5, 2018 10:10 PM | Alfonso Nieto-Castanon -

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

Hi Martyn,

You are actually on the right track, it is very much like having multiple design matrices. To be precise, if a standard multivariate test with three dependent variables uses a model of the form:

[Y1 Y2 Y3] = X * [B1 B2 B3] + E

with Y# your Nx1 dependent variables, X your NxM design matrix and E a Nx3 noise term with N independent samples, zero mean, and arbitrary 3x3 noise covariance matrix S, SPM's ReML approach would use instead a model of the form:

[Y1 = [X 0 0 * [B1 + E

Y2 0 X 0 B2

Y3] 0 0 X] B3]

with the same NxM design matrix X (now repeated in a block-diagonal manner three times into a new 3Nx3M design matrix) and a different 3Nx1 noise term E which is now univariate with zero mean, arbitrary scalar covariance s, and non-independent samples modeled by a 3Nx3N sample-to-sample covariance of the form kron(S,I) (to be precise in SPM the scalar covariance s is allowed to be different for each voxel, while the sample-to-sample covariance kron(S,I) is assumed constant -up to the previous scaling factor- across the entire brain)

In CONN both approaches are implemented (for seed-to-voxel or voxel-to-voxel analyses). You can see the differences in the corresponding design matrices and contrasts by clicking the "design" button in the GUI and then switching in the new window in the bottom dropdown menu between "univariate model (SPM)" and "multivariate model". In the "results explorer" window, CONN will use the "multivariate model" approach for non-parametric analyses, and the "univariate model (SPM)" approach for parametric analyses. Typically, if you have a single dependent variable (or even with multiple dependent variable if you are using a vector between-conditions and between-sources contrasts) then both approaches are actually identical and produce exactly the same statistics, but when you have multiple dependents (e.g. a between-conditions contrast matrix instead of a vector) then the two models will produce (slightly) different results (mostly due to the difference in the assumption regarding spatial homogeneity of the residual covariance matrix).

Hope this helps

Alfonso

You are actually on the right track, it is very much like having multiple design matrices. To be precise, if a standard multivariate test with three dependent variables uses a model of the form:

[Y1 Y2 Y3] = X * [B1 B2 B3] + E

with Y# your Nx1 dependent variables, X your NxM design matrix and E a Nx3 noise term with N independent samples, zero mean, and arbitrary 3x3 noise covariance matrix S, SPM's ReML approach would use instead a model of the form:

[Y1 = [X 0 0 * [B1 + E

Y2 0 X 0 B2

Y3] 0 0 X] B3]

with the same NxM design matrix X (now repeated in a block-diagonal manner three times into a new 3Nx3M design matrix) and a different 3Nx1 noise term E which is now univariate with zero mean, arbitrary scalar covariance s, and non-independent samples modeled by a 3Nx3N sample-to-sample covariance of the form kron(S,I) (to be precise in SPM the scalar covariance s is allowed to be different for each voxel, while the sample-to-sample covariance kron(S,I) is assumed constant -up to the previous scaling factor- across the entire brain)

In CONN both approaches are implemented (for seed-to-voxel or voxel-to-voxel analyses). You can see the differences in the corresponding design matrices and contrasts by clicking the "design" button in the GUI and then switching in the new window in the bottom dropdown menu between "univariate model (SPM)" and "multivariate model". In the "results explorer" window, CONN will use the "multivariate model" approach for non-parametric analyses, and the "univariate model (SPM)" approach for parametric analyses. Typically, if you have a single dependent variable (or even with multiple dependent variable if you are using a vector between-conditions and between-sources contrasts) then both approaches are actually identical and produce exactly the same statistics, but when you have multiple dependents (e.g. a between-conditions contrast matrix instead of a vector) then the two models will produce (slightly) different results (mostly due to the difference in the assumption regarding spatial homogeneity of the residual covariance matrix).

Hope this helps

Alfonso

*Originally posted by Martyn McFarquhar:*Hi Alfonso,

Just on the back of this, could you clarify what CONN would be using instead of Wilks' Lambda and how the SPM ReML approach (presumably for dealing with heterogeneity of variance) is implemented for multiple dependant variables? I can't think how this is possible given that you would get different heterogeneity estimates per-DV and so couldn't filter the design matrix in the same fashion (unless you used multiple design matrices)?

Best wishes,

- Martyn

Just on the back of this, could you clarify what CONN would be using instead of Wilks' Lambda and how the SPM ReML approach (presumably for dealing with heterogeneity of variance) is implemented for multiple dependant variables? I can't think how this is possible given that you would get different heterogeneity estimates per-DV and so couldn't filter the design matrix in the same fashion (unless you used multiple design matrices)?

Best wishes,

- Martyn

## Threaded View

Title | Author | Date |
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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 | |