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help > RE: Multivariate regression - assumptions?
Oct 14, 2015 12:10 AM | Alfonso Nieto-Castanon - Boston University
RE: Multivariate regression - assumptions?
Hi Jenna,
Some thoughts on your analyses first:
1) the choice of 'multivariate regression' in the first-level analysis tab is doing perhaps something different than what you intended. These measures pertain to the meaning of the connectivity measures estimated separately for each subject (subject-level / first-level analysis). In particular when choosing multivariate regression or semipartial correlation measures (e.g. instead of bivariate regression or bivariate correlation measures), CONN will be computing, for each seed ROI, the unique connectivity between this seed ROI and every target region after controlling for the contribution of all other seed ROIs. In other words, separately for each subject and for each target area/voxel, CONN is using a multiple regression model where it enters all of the seed/source ROI timeseries that you included in the 'sources' list as simultaneous regressors. Compared to that, when you use bivariate regression/correlation measures, CONN is using a separate regression model for each seed/source ROI (where only a single seed/source ROI timeseres is included as a regressor when fitting each target area/voxel). So, unless you have a reason to be interested in the unique connectivity associated with each seed ROI I would suggest to use bivariate correlation/regression measures instead in your first-level analysis model.
2) just to make sure: if your conn project includes both ASD and control subjects, and you wish to look at the association between any of your clinical score variables C1 to C4 with FC only within ASD subjects, you should:
a) make sure that the C1 to C4 second-level covariates contain zero values for all of the control subjects
b) have a 'ASD' and 'control' second-level covariates dummy-coding your subject groups (e.g. 'ASD' covariates contains 1's for your ASD subjects and 0's for controls)
and c) in the second-level results tab select 'ASD', 'C1' ... 'C4' in the subject-effects list and enter a between-subjects contrast [0 1 0 0 0; 0 0 1 0 0; 0 0 0 1 0; 0 0 0 0 1]
Coming back to your original question, the results of this test will be those areas where FC is associated with any of your C1-C4 covariates within the ASD subject group. This test is implemented as a second-level multivariate F-test (these analyses do not assume that the Fisher-transformed Z-values are homogeneous across your C1-C4 clinical variables; for seed-to-voxel analyses the variance/covariance between your C1-C4 regressors is estimated in an initial resML step, and for ROI-to-ROI analyses that same variance/covariance structure is explicitly estimated for each target ROI).
The residuals from these models are not explicitly saved anywhere, unfortunately. Perhaps a not terribly complicated way to obtain those residuals would be to:
a) first use 'extract values' to get the actual ROI-to-ROI or seed-to-voxel connectivity values of interest for each subject (e.g. into a new second-level covariate variable named 'FC')
and b) explicitly create a new second-level covariate (e.g. named 'residuals') and enter in its values field the following:
FC - FC / [ASD;C1;C2;C3;C4] * [ASD;C1;C2;C3;C4]
(change the 'FC', 'ASD', and 'C1' to 'C4' entries above to the actual names that you used in your CONN project for those second-level covariates). The new 'residuals' covariate will contain the residuals (one value per subject) from the second-level model used in your original multivariate test.
Hope this helps
Alfonso
Originally posted by Jenna Traynor:
Some thoughts on your analyses first:
1) the choice of 'multivariate regression' in the first-level analysis tab is doing perhaps something different than what you intended. These measures pertain to the meaning of the connectivity measures estimated separately for each subject (subject-level / first-level analysis). In particular when choosing multivariate regression or semipartial correlation measures (e.g. instead of bivariate regression or bivariate correlation measures), CONN will be computing, for each seed ROI, the unique connectivity between this seed ROI and every target region after controlling for the contribution of all other seed ROIs. In other words, separately for each subject and for each target area/voxel, CONN is using a multiple regression model where it enters all of the seed/source ROI timeseries that you included in the 'sources' list as simultaneous regressors. Compared to that, when you use bivariate regression/correlation measures, CONN is using a separate regression model for each seed/source ROI (where only a single seed/source ROI timeseres is included as a regressor when fitting each target area/voxel). So, unless you have a reason to be interested in the unique connectivity associated with each seed ROI I would suggest to use bivariate correlation/regression measures instead in your first-level analysis model.
2) just to make sure: if your conn project includes both ASD and control subjects, and you wish to look at the association between any of your clinical score variables C1 to C4 with FC only within ASD subjects, you should:
a) make sure that the C1 to C4 second-level covariates contain zero values for all of the control subjects
b) have a 'ASD' and 'control' second-level covariates dummy-coding your subject groups (e.g. 'ASD' covariates contains 1's for your ASD subjects and 0's for controls)
and c) in the second-level results tab select 'ASD', 'C1' ... 'C4' in the subject-effects list and enter a between-subjects contrast [0 1 0 0 0; 0 0 1 0 0; 0 0 0 1 0; 0 0 0 0 1]
Coming back to your original question, the results of this test will be those areas where FC is associated with any of your C1-C4 covariates within the ASD subject group. This test is implemented as a second-level multivariate F-test (these analyses do not assume that the Fisher-transformed Z-values are homogeneous across your C1-C4 clinical variables; for seed-to-voxel analyses the variance/covariance between your C1-C4 regressors is estimated in an initial resML step, and for ROI-to-ROI analyses that same variance/covariance structure is explicitly estimated for each target ROI).
The residuals from these models are not explicitly saved anywhere, unfortunately. Perhaps a not terribly complicated way to obtain those residuals would be to:
a) first use 'extract values' to get the actual ROI-to-ROI or seed-to-voxel connectivity values of interest for each subject (e.g. into a new second-level covariate variable named 'FC')
and b) explicitly create a new second-level covariate (e.g. named 'residuals') and enter in its values field the following:
FC - FC / [ASD;C1;C2;C3;C4] * [ASD;C1;C2;C3;C4]
(change the 'FC', 'ASD', and 'C1' to 'C4' entries above to the actual names that you used in your CONN project for those second-level covariates). The new 'residuals' covariate will contain the residuals (one value per subject) from the second-level model used in your original multivariate test.
Hope this helps
Alfonso
Originally posted by Jenna Traynor:
Hi Alfonso,
Thank you for your response. I am using 4 clinical scores of behaviour (labelled C1, C2, C3 and C4) as multiple predictor variables and FC is the outcome variable. However, I am a bit confused because I am looking at how each of these clinical scores is associated with FC in multiple source ROIs. So because I have more than one outcome variable (i.e., multiple ROIs) I thought it was a mutlivariate regression?
The way that I set this up is based on the fact that I initially had two groups (ASD and controls), and this was a within group analysis on ASD subjects only: I chose 'multivariate regression' in the 1st level analysis tab, and then when viewing in second level results I selected the following variables : ASD, C1, C2, C3, C4 and set up the contrast for example as [0, 1, 0, 0, 0] - to look at the association between FC and C1, (while controlling for the association between FC and all of the other clinical variables?). As I move through the source ROIs on the right, I can see the association between the selected predictor variable and connectivity between that source ROI and all target ROIS (brodmann areas).
So I guess my question was if there was any way to extract the residuals from this model in order to examine the validity of the assumptions. Is there reason to think that the homoscedasticity assumption does not apply here? Would I want the residual errors of observed - predicted Fisher-Z values to be homogenous across all clinical variables?
Thank you so much for your help, and please let me know if I have done something incorrectly.
Jenna
Thank you for your response. I am using 4 clinical scores of behaviour (labelled C1, C2, C3 and C4) as multiple predictor variables and FC is the outcome variable. However, I am a bit confused because I am looking at how each of these clinical scores is associated with FC in multiple source ROIs. So because I have more than one outcome variable (i.e., multiple ROIs) I thought it was a mutlivariate regression?
The way that I set this up is based on the fact that I initially had two groups (ASD and controls), and this was a within group analysis on ASD subjects only: I chose 'multivariate regression' in the 1st level analysis tab, and then when viewing in second level results I selected the following variables : ASD, C1, C2, C3, C4 and set up the contrast for example as [0, 1, 0, 0, 0] - to look at the association between FC and C1, (while controlling for the association between FC and all of the other clinical variables?). As I move through the source ROIs on the right, I can see the association between the selected predictor variable and connectivity between that source ROI and all target ROIS (brodmann areas).
So I guess my question was if there was any way to extract the residuals from this model in order to examine the validity of the assumptions. Is there reason to think that the homoscedasticity assumption does not apply here? Would I want the residual errors of observed - predicted Fisher-Z values to be homogenous across all clinical variables?
Thank you so much for your help, and please let me know if I have done something incorrectly.
Jenna
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| Title | Author | Date |
|---|---|---|
| Jenna Traynor | Oct 11, 2015 | |
| Alfonso Nieto-Castanon | Oct 11, 2015 | |
| Jenna Traynor | Oct 12, 2015 | |
| Alfonso Nieto-Castanon | Oct 14, 2015 | |
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