Distance

R2 plot: Without Exclusion

R2 plot: With Exclusion

R2 plot: With Exclusion and With 2 SD ceil

Corr: With Exclusion

  • For each ROI, we computed the relationship between various combinations of the dependent variables (euclidean distance and path length; path length and communicability).
  • We found that there was a strong correlation between euclidean distance and path length across several ROIs (and a p-value distribution suggested that many of these findings were significant), * Similarly, there was a strong [negative] correlation between path length and communicability across several ROIs.
  • These results suggest that the dependent variables for each constructed MLR model may be correlated with each other.
Euclidean Distance and Path Length

Path Length and Communicability

LMG for Individual R2 Contributions

  • From the correlational analysis, we conclude that the regressions are correlated with each other.

Problems with Existing Models of R2 Decomposition for Correlated Regressors

  • ANOVA in R: The difficulty in decomposing R2 for regression models with correlated regressors lies in the fact that the order in which regressors are entered into the model yields a different decomposition of the model sum of squares (i.e., the order in which regressors are entered into the model can have a very strong impact on their relative R2 contributions).

  • TYPE III SS/Stepwise-Regression: Type III Sum of Squares are often used to to compare what each regressor is able to explain in addition to all other regressors that are available. Here, we ascribe to each regressor the increase in R2 when including this regressor as the last of the p regressors. If regressors are correlated, these contributions do not add up to the overall R2, and typically add up to far less than the overall R2.Moreover, the direct effect of a regressor with the criterion cannot be calculated this way.

Current Approach * We evaluate R2 based on the proportionate contribution each predictor makes to R2, considering both its direct effect (i.e., its correlation with the criterion) and its effect when combined with the other variables in the regression equation.

  • We use LMG, which is based on sequential R2s (like ANOVA), but which takes care of the dependence on orderings by averaging over [permuted] orderings using simple unweighted averages. Johnson and Lebreton (2004) recommend lmg, since it clearly uses both direct effects (orders with xk first) and effects adjusted for other regressors in the model (orders with xk last). LMG decomposes R2 into non-negative contributions that automatically sum to the total R2. This is an advantage over simple metrics. (Grömping, U. (2007). Relative importance for linear regression in R: the package relaimpo. Journal of statistical software, 17, 1-27.)

(LMG: Lindemann, Merenda and Gold-1980)

R2 Individual: Without Exclusion

Sample Table (header rows: ROIs 1 to 10)
Individual R2 Contributions without Distance Exclusion
R2_ed R2_pl R2_co Total
0.0132972 0.0134925 0.0407130 0.0675026
0.0014548 0.0078898 0.0072842 0.0166288
0.0011824 0.0183753 0.0095648 0.0291225
0.0040214 0.0104478 0.0104103 0.0248795
0.0424400 0.0072535 0.0031434 0.0528369
0.0092011 0.0018007 0.0146924 0.0256943
0.0202613 0.0054078 0.0010898 0.0267590
0.0255592 0.0191125 0.0054265 0.0500982
0.0098052 0.0043447 0.0037019 0.0178518
0.0048110 0.0047099 0.0009763 0.0104972
Plot

R2 Individual: With Exclusion

Sample Table (header rows: ROIs 1 to 10)
Individual R2 Contributions with Distance Exclusion
R2_ed R2_pl R2_co Total
0.0050957 0.0079018 0.0337777 0.0467752
0.0079628 0.0007182 0.0000145 0.0086955
0.0026409 0.0073826 0.0015692 0.0115927
0.0006161 0.0039225 0.0022913 0.0068299
0.0170775 0.0072753 0.0007541 0.0251069
0.0036323 0.0020120 0.0059837 0.0116279
0.0153728 0.0037677 0.0037156 0.0228562
0.0090538 0.0160976 0.0015017 0.0266531
0.0029315 0.0026116 0.0013779 0.0069210
0.0004654 0.0043910 0.0002817 0.0051382
Plot

R2 Individual: With Exclusion and 2SD ceil

Sample Table (header rows: ROIs 1 to 10)
Individual R2 Contributions with Distance Exclusion and 2SD ceil
R2_ed R2_pl2SDceil R2_co2SDceil Total
0.0045098 0.0085365 0.0297464 0.0427927
0.0076761 0.0006020 0.0003276 0.0086057
0.0027062 0.0077401 0.0011110 0.0115573
0.0007387 0.0039420 0.0022163 0.0068970
0.0168337 0.0066026 0.0014437 0.0248799
0.0033952 0.0016574 0.0051411 0.0101937
0.0153807 0.0038641 0.0017099 0.0209547
0.0085277 0.0134875 0.0050121 0.0270273
0.0027787 0.0019248 0.0027577 0.0074612
0.0005586 0.0029915 0.0014265 0.0049766
Plot

R2-Brain Parcellation Map

  • These brain maps were generated from a Matlab code (and on workbench)

R2 Brain Parcellation Map Labeled Based on Gordon et al., 2016

R2 Brain Parcellation Map Labeled Based on BA areas

R2 Brain Parcellation Map: No labels

R2 Brain Parcellation thresholded at 0.05 R2 Brain Parcellation thresholded: 0.06 to 0.10 R2 Brain Parcellation thresholded: 0.10 to 0.20