code written: 2020-01-14
last ran: 2020-01-18
website: http://rpubs.com/navona/SPINS_normalityFA

Description. Here, we examine multivariate and univariate normality of the FA values in the n=72 Slicer tracts that met our segmentation threshold, from the n=390 participants who met eligibility, passed QC, and showed successful segmentations. We find that the FA values are not normally distributed, and there are too many outliers to trim or Winsor. We also find that power transformation via the Yeo-Johnson method does not produce much-improved multivariate or univariate normality. Thus, we decide not to transform the FA values, and instead, given our large N, invoke the properties of the CLT to meet the multivariate normality assumption of CCA (when N is large enough, the assumption of normality can be relaxed).

MVN tests. There exist several tests of multivariate normality. All tests were run at the 0.05 significance level. Though MVN tests can come up with different results, all suggest that the \(X\) set is not multivariate normal.

Mardia’s

##              Test        Statistic               p value Result
## 1 Mardia Skewness 77484.6635915663 1.27528873928695e-240     NO
## 2 Mardia Kurtosis 23.4805764761935                     0     NO
## 3             MVN             <NA>                  <NA>     NO

Henze-Zirkler’s

##            Test HZ p value MVN
## 1 Henze-Zirkler  1       0  NO

Royston’s

##      Test        H       p value MVN
## 1 Royston 836.8541 6.302094e-136  NO

Doornik-Hansen’s

##             Test       E  df       p value MVN
## 1 Doornik-Hansen 1380.39 144 8.638915e-201  NO

Energy

##          Test Statistic p value MVN
## 1 E-statistic  4.879344       0  NO

MVN plot. The Chi-square Q-Q plot is a widely used graphical approach to evaluate the agreement between two probability distributions. Each axis refers to the quantiles of probability distributions to be compared, where one of the axes indicates theoretical quantiles (hypothesized quantiles) and the other indicates the observed quantiles. If the observed data fit the hypothesized distribution, the points in the Q-Q plot will approximately lie on the reference line y = x. Below, we see large deviations from the reference line, which indicates departure from the multivariate normal distribution.

Multivariate outliers. A common cause of violating MVN is the presence of outliers. In some cases, these outliers can be addressed (“trimmed”, i.e., removed, or “Winsored”, i.e., recoded to less extreme values). Typically, multivariate outlier detection methods (two shown below) based on robust Mahalanobis distances are employed. However, thesee methods suggest so many outliers that it would not make sense to remove them. Moreover, we are confident that the observed FA values are ‘real’ (at least, relatively within site/scanner).

Mahalanobis distance

Adjusted Mahalanobis distance

Univariate tests.

Though MVN is not satisfied, we should still examine the univariate distributions of each contributing variable to the \(X\) set (i.e., all included Slicer FA values). (Note: while it is the case that if the data had a multivariate normal distribution, each of the variables would also have a univariate normal distribution, the converse does not hold.) Therefore, univariate tests and visualizations can assist in understanding the failure of MVN. Here, we see that many of the distributions of FA in the individual tracts are not normal.

Shapiro-Wilk’s

Test Variable Statistic p value Normality
Shapiro-Wilk AF_R 0.9930 0.0648 YES
Shapiro-Wilk AF_L 0.9913 0.0216 NO
Shapiro-Wilk CB_R 0.9918 0.0302 NO
Shapiro-Wilk CB_L 0.9885 0.0036 NO
Shapiro-Wilk CC1_C 0.9926 0.0492 NO
Shapiro-Wilk CC2_C 0.9900 0.009 NO
Shapiro-Wilk CC3_C 0.9862 9e-04 NO
Shapiro-Wilk CC4_C 0.9812 1e-04 NO
Shapiro-Wilk CC5_C 0.9745 <0.001 NO
Shapiro-Wilk CC6_C 0.9841 3e-04 NO
Shapiro-Wilk CC7_C 0.9848 4e-04 NO
Shapiro-Wilk CPC_L 0.9213 <0.001 NO
Shapiro-Wilk CPC_R 0.9832 2e-04 NO
Shapiro-Wilk CR.F_L 0.9881 0.0028 NO
Shapiro-Wilk CR.F_R 0.9953 0.2968 YES
Shapiro-Wilk CR.P_R 0.9867 0.0012 NO
Shapiro-Wilk CR.P_L 0.9822 1e-04 NO
Shapiro-Wilk CST_L 0.9714 <0.001 NO
Shapiro-Wilk CST_R 0.9693 <0.001 NO
Shapiro-Wilk EC_R 0.9821 1e-04 NO
Shapiro-Wilk EC_L 0.9762 <0.001 NO
Shapiro-Wilk EmC_L 0.9510 <0.001 NO
Shapiro-Wilk EmC_R 0.9447 <0.001 NO
Shapiro-Wilk ICP_L 0.9831 2e-04 NO
Shapiro-Wilk ICP_R 0.9914 0.0234 NO
Shapiro-Wilk ILF_R 0.9943 0.1569 YES
Shapiro-Wilk ILF_L 0.9921 0.036 NO
Shapiro-Wilk Intra.CBLM.I.P_R 0.9912 0.0197 NO
Shapiro-Wilk Intra.CBLM.I.P_L 0.9900 0.0091 NO
Shapiro-Wilk Intra.CBLM.PaT_L 0.9962 0.4665 YES
Shapiro-Wilk Intra.CBLM.PaT_R 0.9931 0.0719 YES
Shapiro-Wilk IOFF_L 0.8837 <0.001 NO
Shapiro-Wilk IOFF_R 0.9198 <0.001 NO
Shapiro-Wilk MCP_C 0.9837 2e-04 NO
Shapiro-Wilk MdLF_R 0.9975 0.8343 YES
Shapiro-Wilk MdLF_L 0.9969 0.6589 YES
Shapiro-Wilk PLIC_R 0.9919 0.0311 NO
Shapiro-Wilk PLIC_L 0.9947 0.2016 YES
Shapiro-Wilk SF_R 0.9971 0.7215 YES
Shapiro-Wilk SF_L 0.9937 0.105 YES
Shapiro-Wilk SLF.II_L 0.8713 <0.001 NO
Shapiro-Wilk SLF.II_R 0.9310 <0.001 NO
Shapiro-Wilk SLF.III_R 0.9952 0.2729 YES
Shapiro-Wilk SLF.III_L 0.9969 0.6603 YES
Shapiro-Wilk SO_L 0.9700 <0.001 NO
Shapiro-Wilk SO_R 0.9768 <0.001 NO
Shapiro-Wilk SP_R 0.9952 0.2701 YES
Shapiro-Wilk SP_L 0.9958 0.3937 YES
Shapiro-Wilk Sup.F_R 0.9953 0.2823 YES
Shapiro-Wilk Sup.F_L 0.9970 0.6997 YES
Shapiro-Wilk Sup.FP_L 0.9930 0.0656 YES
Shapiro-Wilk Sup.FP_R 0.9974 0.7988 YES
Shapiro-Wilk Sup.O_L 0.9562 <0.001 NO
Shapiro-Wilk Sup.O_R 0.9425 <0.001 NO
Shapiro-Wilk Sup.OT_R 0.9676 <0.001 NO
Shapiro-Wilk Sup.OT_L 0.9957 0.3559 YES
Shapiro-Wilk Sup.P_L 0.9957 0.3636 YES
Shapiro-Wilk Sup.P_R 0.9946 0.1818 YES
Shapiro-Wilk Sup.PO_L 0.9972 0.7498 YES
Shapiro-Wilk Sup.PO_R 0.9970 0.6943 YES
Shapiro-Wilk Sup.PT_R 0.9944 0.1678 YES
Shapiro-Wilk Sup.PT_L 0.9949 0.2193 YES
Shapiro-Wilk Sup.T_R 0.9969 0.671 YES
Shapiro-Wilk Sup.T_L 0.9970 0.7018 YES
Shapiro-Wilk TF_L 0.9906 0.0136 NO
Shapiro-Wilk TF_R 0.9959 0.4121 YES
Shapiro-Wilk TO_L 0.9833 2e-04 NO
Shapiro-Wilk TO_R 0.9688 <0.001 NO
Shapiro-Wilk TP_L 0.9894 0.0066 NO
Shapiro-Wilk TP_R 0.9929 0.0611 YES
Shapiro-Wilk UF_L 0.9791 <0.001 NO
Shapiro-Wilk UF_R 0.9807 <0.001 NO

Cramer-von Mises

Test Variable Statistic p value Normality
Cramer-von Mises AF_R 0.2814 5e-04 NO
Cramer-von Mises AF_L 0.2055 0.0046 NO
Cramer-von Mises CB_R 0.1410 0.0312 NO
Cramer-von Mises CB_L 0.3148 2e-04 NO
Cramer-von Mises CC1_C 0.0511 0.4955 YES
Cramer-von Mises CC2_C 0.1055 0.0941 YES
Cramer-von Mises CC3_C 0.1571 0.0191 NO
Cramer-von Mises CC4_C 0.1789 0.0099 NO
Cramer-von Mises CC5_C 0.3228 2e-04 NO
Cramer-von Mises CC6_C 0.2611 9e-04 NO
Cramer-von Mises CC7_C 0.1661 0.0146 NO
Cramer-von Mises CPC_L 3.2784 <0.001 NO
Cramer-von Mises CPC_R 0.3209 2e-04 NO
Cramer-von Mises CR.F_L 0.3029 3e-04 NO
Cramer-von Mises CR.F_R 0.1063 0.0917 YES
Cramer-von Mises CR.P_R 0.4478 <0.001 NO
Cramer-von Mises CR.P_L 0.2610 9e-04 NO
Cramer-von Mises CST_L 0.7141 <0.001 NO
Cramer-von Mises CST_R 0.7699 <0.001 NO
Cramer-von Mises EC_R 0.4302 <0.001 NO
Cramer-von Mises EC_L 0.4270 <0.001 NO
Cramer-von Mises EmC_L 1.4124 <0.001 NO
Cramer-von Mises EmC_R 1.8268 <0.001 NO
Cramer-von Mises ICP_L 0.5401 <0.001 NO
Cramer-von Mises ICP_R 0.3363 1e-04 NO
Cramer-von Mises ILF_R 0.0676 0.2998 YES
Cramer-von Mises ILF_L 0.1298 0.044 NO
Cramer-von Mises Intra.CBLM.I.P_R 0.1604 0.0173 NO
Cramer-von Mises Intra.CBLM.I.P_L 0.2064 0.0044 NO
Cramer-von Mises Intra.CBLM.PaT_L 0.0762 0.2314 YES
Cramer-von Mises Intra.CBLM.PaT_R 0.0222 0.9457 YES
Cramer-von Mises IOFF_L 4.0428 <0.001 NO
Cramer-von Mises IOFF_R 2.6477 <0.001 NO
Cramer-von Mises MCP_C 0.3831 <0.001 NO
Cramer-von Mises MdLF_R 0.0398 0.6839 YES
Cramer-von Mises MdLF_L 0.0448 0.5929 YES
Cramer-von Mises PLIC_R 0.1294 0.0446 NO
Cramer-von Mises PLIC_L 0.0901 0.1528 YES
Cramer-von Mises SF_R 0.0639 0.3355 YES
Cramer-von Mises SF_L 0.0582 0.3991 YES
Cramer-von Mises SLF.II_L 5.5627 <0.001 NO
Cramer-von Mises SLF.II_R 2.7907 <0.001 NO
Cramer-von Mises SLF.III_R 0.0991 0.1152 YES
Cramer-von Mises SLF.III_L 0.0686 0.2909 YES
Cramer-von Mises SO_L 0.6986 <0.001 NO
Cramer-von Mises SO_R 0.5017 <0.001 NO
Cramer-von Mises SP_R 0.0523 0.4778 YES
Cramer-von Mises SP_L 0.0963 0.1259 YES
Cramer-von Mises Sup.F_R 0.0750 0.2398 YES
Cramer-von Mises Sup.F_L 0.0503 0.506 YES
Cramer-von Mises Sup.FP_L 0.1500 0.0237 NO
Cramer-von Mises Sup.FP_R 0.0291 0.8586 YES
Cramer-von Mises Sup.O_L 0.5783 <0.001 NO
Cramer-von Mises Sup.O_R 0.7796 <0.001 NO
Cramer-von Mises Sup.OT_R 0.2548 0.0011 NO
Cramer-von Mises Sup.OT_L 0.0392 0.6964 YES
Cramer-von Mises Sup.P_L 0.0536 0.4583 YES
Cramer-von Mises Sup.P_R 0.0471 0.5524 YES
Cramer-von Mises Sup.PO_L 0.0423 0.6373 YES
Cramer-von Mises Sup.PO_R 0.0435 0.6165 YES
Cramer-von Mises Sup.PT_R 0.0700 0.2789 YES
Cramer-von Mises Sup.PT_L 0.0605 0.3716 YES
Cramer-von Mises Sup.T_R 0.0474 0.5487 YES
Cramer-von Mises Sup.T_L 0.0346 0.7765 YES
Cramer-von Mises TF_L 0.1723 0.0121 NO
Cramer-von Mises TF_R 0.0429 0.6264 YES
Cramer-von Mises TO_L 0.1432 0.0292 NO
Cramer-von Mises TO_R 0.3883 <0.001 NO
Cramer-von Mises TP_L 0.0449 0.5917 YES
Cramer-von Mises TP_R 0.0669 0.306 YES
Cramer-von Mises UF_L 0.4667 <0.001 NO
Cramer-von Mises UF_R 0.3880 <0.001 NO

Lilliefors

Test Variable Statistic p value Normality
Lilliefors (Kolmogorov-Smirnov) AF_R 0.0733 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) AF_L 0.0536 0.0091 NO
Lilliefors (Kolmogorov-Smirnov) CB_R 0.0449 0.0569 YES
Lilliefors (Kolmogorov-Smirnov) CB_L 0.0602 0.0017 NO
Lilliefors (Kolmogorov-Smirnov) CC1_C 0.0338 0.3431 YES
Lilliefors (Kolmogorov-Smirnov) CC2_C 0.0366 0.2321 YES
Lilliefors (Kolmogorov-Smirnov) CC3_C 0.0492 0.0245 NO
Lilliefors (Kolmogorov-Smirnov) CC4_C 0.0572 0.0038 NO
Lilliefors (Kolmogorov-Smirnov) CC5_C 0.0501 0.02 NO
Lilliefors (Kolmogorov-Smirnov) CC6_C 0.0583 0.0029 NO
Lilliefors (Kolmogorov-Smirnov) CC7_C 0.0535 0.0094 NO
Lilliefors (Kolmogorov-Smirnov) CPC_L 0.2187 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CPC_R 0.0801 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CR.F_L 0.0666 3e-04 NO
Lilliefors (Kolmogorov-Smirnov) CR.F_R 0.0341 0.3304 YES
Lilliefors (Kolmogorov-Smirnov) CR.P_R 0.0896 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CR.P_L 0.0662 3e-04 NO
Lilliefors (Kolmogorov-Smirnov) CST_L 0.0964 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CST_R 0.0809 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) EC_R 0.0792 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) EC_L 0.0838 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) EmC_L 0.1163 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) EmC_R 0.1350 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) ICP_L 0.0875 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) ICP_R 0.0901 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) ILF_R 0.0296 0.5548 YES
Lilliefors (Kolmogorov-Smirnov) ILF_L 0.0460 0.0464 NO
Lilliefors (Kolmogorov-Smirnov) Intra.CBLM.I.P_R 0.0509 0.0169 NO
Lilliefors (Kolmogorov-Smirnov) Intra.CBLM.I.P_L 0.0529 0.0108 NO
Lilliefors (Kolmogorov-Smirnov) Intra.CBLM.PaT_L 0.0310 0.4813 YES
Lilliefors (Kolmogorov-Smirnov) Intra.CBLM.PaT_R 0.0214 0.9349 YES
Lilliefors (Kolmogorov-Smirnov) IOFF_L 0.2298 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) IOFF_R 0.2027 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) MCP_C 0.0682 2e-04 NO
Lilliefors (Kolmogorov-Smirnov) MdLF_R 0.0253 0.7879 YES
Lilliefors (Kolmogorov-Smirnov) MdLF_L 0.0263 0.7384 YES
Lilliefors (Kolmogorov-Smirnov) PLIC_R 0.0410 0.1146 YES
Lilliefors (Kolmogorov-Smirnov) PLIC_L 0.0402 0.1297 YES
Lilliefors (Kolmogorov-Smirnov) SF_R 0.0341 0.3304 YES
Lilliefors (Kolmogorov-Smirnov) SF_L 0.0346 0.3103 YES
Lilliefors (Kolmogorov-Smirnov) SLF.II_L 0.2685 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) SLF.II_R 0.1977 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) SLF.III_R 0.0463 0.0433 NO
Lilliefors (Kolmogorov-Smirnov) SLF.III_L 0.0319 0.4341 YES
Lilliefors (Kolmogorov-Smirnov) SO_L 0.1205 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) SO_R 0.0731 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) SP_R 0.0341 0.3277 YES
Lilliefors (Kolmogorov-Smirnov) SP_L 0.0438 0.0695 YES
Lilliefors (Kolmogorov-Smirnov) Sup.F_R 0.0339 0.3381 YES
Lilliefors (Kolmogorov-Smirnov) Sup.F_L 0.0283 0.6291 YES
Lilliefors (Kolmogorov-Smirnov) Sup.FP_L 0.0508 0.0172 NO
Lilliefors (Kolmogorov-Smirnov) Sup.FP_R 0.0247 0.8164 YES
Lilliefors (Kolmogorov-Smirnov) Sup.O_L 0.0724 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) Sup.O_R 0.0870 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) Sup.OT_R 0.0576 0.0034 NO
Lilliefors (Kolmogorov-Smirnov) Sup.OT_L 0.0296 0.5567 YES
Lilliefors (Kolmogorov-Smirnov) Sup.P_L 0.0308 0.4893 YES
Lilliefors (Kolmogorov-Smirnov) Sup.P_R 0.0327 0.3953 YES
Lilliefors (Kolmogorov-Smirnov) Sup.PO_L 0.0287 0.6093 YES
Lilliefors (Kolmogorov-Smirnov) Sup.PO_R 0.0317 0.4436 YES
Lilliefors (Kolmogorov-Smirnov) Sup.PT_R 0.0363 0.2431 YES
Lilliefors (Kolmogorov-Smirnov) Sup.PT_L 0.0284 0.6205 YES
Lilliefors (Kolmogorov-Smirnov) Sup.T_R 0.0333 0.3663 YES
Lilliefors (Kolmogorov-Smirnov) Sup.T_L 0.0283 0.6275 YES
Lilliefors (Kolmogorov-Smirnov) TF_L 0.0443 0.0634 YES
Lilliefors (Kolmogorov-Smirnov) TF_R 0.0256 0.775 YES
Lilliefors (Kolmogorov-Smirnov) TO_L 0.0419 0.0978 YES
Lilliefors (Kolmogorov-Smirnov) TO_R 0.0590 0.0024 NO
Lilliefors (Kolmogorov-Smirnov) TP_L 0.0261 0.7471 YES
Lilliefors (Kolmogorov-Smirnov) TP_R 0.0299 0.541 YES
Lilliefors (Kolmogorov-Smirnov) UF_L 0.0678 2e-04 NO
Lilliefors (Kolmogorov-Smirnov) UF_R 0.0635 7e-04 NO

Shapiro-Francia

Test Variable Statistic p value Normality
Shapiro-Francia AF_R 0.9922 0.039 NO
Shapiro-Francia AF_L 0.9921 0.0361 NO
Shapiro-Francia CB_R 0.9925 0.0472 NO
Shapiro-Francia CB_L 0.9894 0.0076 NO
Shapiro-Francia CC1_C 0.9925 0.0479 NO
Shapiro-Francia CC2_C 0.9903 0.0127 NO
Shapiro-Francia CC3_C 0.9852 8e-04 NO
Shapiro-Francia CC4_C 0.9807 1e-04 NO
Shapiro-Francia CC5_C 0.9751 <0.001 NO
Shapiro-Francia CC6_C 0.9850 8e-04 NO
Shapiro-Francia CC7_C 0.9839 4e-04 NO
Shapiro-Francia CPC_L 0.9215 <0.001 NO
Shapiro-Francia CPC_R 0.9842 5e-04 NO
Shapiro-Francia CR.F_L 0.9891 0.0065 NO
Shapiro-Francia CR.F_R 0.9953 0.2509 YES
Shapiro-Francia CR.P_R 0.9862 0.0014 NO
Shapiro-Francia CR.P_L 0.9833 3e-04 NO
Shapiro-Francia CST_L 0.9730 <0.001 NO
Shapiro-Francia CST_R 0.9710 <0.001 NO
Shapiro-Francia EC_R 0.9804 1e-04 NO
Shapiro-Francia EC_L 0.9764 <0.001 NO
Shapiro-Francia EmC_L 0.9507 <0.001 NO
Shapiro-Francia EmC_R 0.9435 <0.001 NO
Shapiro-Francia ICP_L 0.9832 3e-04 NO
Shapiro-Francia ICP_R 0.9914 0.0242 NO
Shapiro-Francia ILF_R 0.9945 0.1594 YES
Shapiro-Francia ILF_L 0.9936 0.0886 YES
Shapiro-Francia Intra.CBLM.I.P_R 0.9920 0.0348 NO
Shapiro-Francia Intra.CBLM.I.P_L 0.9912 0.0219 NO
Shapiro-Francia Intra.CBLM.PaT_L 0.9952 0.235 YES
Shapiro-Francia Intra.CBLM.PaT_R 0.9934 0.079 YES
Shapiro-Francia IOFF_L 0.8837 <0.001 NO
Shapiro-Francia IOFF_R 0.9200 <0.001 NO
Shapiro-Francia MCP_C 0.9852 8e-04 NO
Shapiro-Francia MdLF_R 0.9979 0.8603 YES
Shapiro-Francia MdLF_L 0.9970 0.6121 YES
Shapiro-Francia PLIC_R 0.9931 0.0686 YES
Shapiro-Francia PLIC_L 0.9952 0.2385 YES
Shapiro-Francia SF_R 0.9978 0.8217 YES
Shapiro-Francia SF_L 0.9949 0.2036 YES
Shapiro-Francia SLF.II_L 0.8690 <0.001 NO
Shapiro-Francia SLF.II_R 0.9302 <0.001 NO
Shapiro-Francia SLF.III_R 0.9955 0.277 YES
Shapiro-Francia SLF.III_L 0.9974 0.7185 YES
Shapiro-Francia SO_L 0.9699 <0.001 NO
Shapiro-Francia SO_R 0.9774 <0.001 NO
Shapiro-Francia SP_R 0.9961 0.3861 YES
Shapiro-Francia SP_L 0.9961 0.4031 YES
Shapiro-Francia Sup.F_R 0.9947 0.1753 YES
Shapiro-Francia Sup.F_L 0.9970 0.6177 YES
Shapiro-Francia Sup.FP_L 0.9936 0.0903 YES
Shapiro-Francia Sup.FP_R 0.9980 0.8721 YES
Shapiro-Francia Sup.O_L 0.9544 <0.001 NO
Shapiro-Francia Sup.O_R 0.9404 <0.001 NO
Shapiro-Francia Sup.OT_R 0.9662 <0.001 NO
Shapiro-Francia Sup.OT_L 0.9948 0.188 YES
Shapiro-Francia Sup.P_L 0.9963 0.4493 YES
Shapiro-Francia Sup.P_R 0.9955 0.2796 YES
Shapiro-Francia Sup.PO_L 0.9969 0.5919 YES
Shapiro-Francia Sup.PO_R 0.9972 0.686 YES
Shapiro-Francia Sup.PT_R 0.9957 0.3168 YES
Shapiro-Francia Sup.PT_L 0.9957 0.3165 YES
Shapiro-Francia Sup.T_R 0.9975 0.7524 YES
Shapiro-Francia Sup.T_L 0.9977 0.8175 YES
Shapiro-Francia TF_L 0.9910 0.0188 NO
Shapiro-Francia TF_R 0.9958 0.338 YES
Shapiro-Francia TO_L 0.9839 4e-04 NO
Shapiro-Francia TO_R 0.9676 <0.001 NO
Shapiro-Francia TP_L 0.9890 0.0061 NO
Shapiro-Francia TP_R 0.9924 0.0447 NO
Shapiro-Francia UF_L 0.9810 1e-04 NO
Shapiro-Francia UF_R 0.9823 2e-04 NO

Anderson-Darling

Test Variable Statistic p value Normality
Anderson-Darling AF_R 1.2774 0.0025 NO
Anderson-Darling AF_L 1.1722 0.0046 NO
Anderson-Darling CB_R 0.9283 0.0183 NO
Anderson-Darling CB_L 1.7109 2e-04 NO
Anderson-Darling CC1_C 0.3693 0.4254 YES
Anderson-Darling CC2_C 0.7702 0.045 NO
Anderson-Darling CC3_C 1.0707 0.0082 NO
Anderson-Darling CC4_C 1.1396 0.0055 NO
Anderson-Darling CC5_C 2.2680 <0.001 NO
Anderson-Darling CC6_C 1.7374 2e-04 NO
Anderson-Darling CC7_C 0.9366 0.0175 NO
Anderson-Darling CPC_L 14.8956 <0.001 NO
Anderson-Darling CPC_R 1.8234 1e-04 NO
Anderson-Darling CR.F_L 1.6171 4e-04 NO
Anderson-Darling CR.F_R 0.5964 0.1186 YES
Anderson-Darling CR.P_R 2.0179 <0.001 NO
Anderson-Darling CR.P_L 1.7589 2e-04 NO
Anderson-Darling CST_L 3.9857 <0.001 NO
Anderson-Darling CST_R 4.3183 <0.001 NO
Anderson-Darling EC_R 2.0965 <0.001 NO
Anderson-Darling EC_L 2.3355 <0.001 NO
Anderson-Darling EmC_L 7.8793 <0.001 NO
Anderson-Darling EmC_R 9.2815 <0.001 NO
Anderson-Darling ICP_L 2.7717 <0.001 NO
Anderson-Darling ICP_R 1.5156 7e-04 NO
Anderson-Darling ILF_R 0.4459 0.281 YES
Anderson-Darling ILF_L 0.7397 0.0536 YES
Anderson-Darling Intra.CBLM.I.P_R 1.0220 0.0108 NO
Anderson-Darling Intra.CBLM.I.P_L 1.2148 0.0036 NO
Anderson-Darling Intra.CBLM.PaT_L 0.4662 0.2511 YES
Anderson-Darling Intra.CBLM.PaT_R 0.2399 0.7757 YES
Anderson-Darling IOFF_L 20.2036 <0.001 NO
Anderson-Darling IOFF_R 12.7692 <0.001 NO
Anderson-Darling MCP_C 2.2005 <0.001 NO
Anderson-Darling MdLF_R 0.2664 0.6881 YES
Anderson-Darling MdLF_L 0.2987 0.5847 YES
Anderson-Darling PLIC_R 0.8340 0.0313 NO
Anderson-Darling PLIC_L 0.6012 0.1178 YES
Anderson-Darling SF_R 0.3443 0.4854 YES
Anderson-Darling SF_L 0.5053 0.2014 YES
Anderson-Darling SLF.II_L 25.3267 <0.001 NO
Anderson-Darling SLF.II_R 13.3927 <0.001 NO
Anderson-Darling SLF.III_R 0.6019 0.1174 YES
Anderson-Darling SLF.III_L 0.3898 0.3813 YES
Anderson-Darling SO_L 3.5469 <0.001 NO
Anderson-Darling SO_R 2.9070 <0.001 NO
Anderson-Darling SP_R 0.4419 0.2872 YES
Anderson-Darling SP_L 0.5587 0.1481 YES
Anderson-Darling Sup.F_R 0.4954 0.213 YES
Anderson-Darling Sup.F_L 0.3006 0.5793 YES
Anderson-Darling Sup.FP_L 0.8923 0.0225 NO
Anderson-Darling Sup.FP_R 0.2031 0.8765 YES
Anderson-Darling Sup.O_L 3.3326 <0.001 NO
Anderson-Darling Sup.O_R 4.2916 <0.001 NO
Anderson-Darling Sup.OT_R 1.6029 4e-04 NO
Anderson-Darling Sup.OT_L 0.3173 0.5372 YES
Anderson-Darling Sup.P_L 0.4096 0.3428 YES
Anderson-Darling Sup.P_R 0.3708 0.4221 YES
Anderson-Darling Sup.PO_L 0.2975 0.5882 YES
Anderson-Darling Sup.PO_R 0.3134 0.546 YES
Anderson-Darling Sup.PT_R 0.4868 0.2237 YES
Anderson-Darling Sup.PT_L 0.4609 0.2586 YES
Anderson-Darling Sup.T_R 0.2893 0.6127 YES
Anderson-Darling Sup.T_L 0.2496 0.7444 YES
Anderson-Darling TF_L 1.0556 0.0089 NO
Anderson-Darling TF_R 0.3266 0.5191 YES
Anderson-Darling TO_L 1.0210 0.0108 NO
Anderson-Darling TO_R 2.3839 <0.001 NO
Anderson-Darling TP_L 0.4304 0.306 YES
Anderson-Darling TP_R 0.4968 0.2114 YES
Anderson-Darling UF_L 2.6753 <0.001 NO
Anderson-Darling UF_R 2.3245 <0.001 NO

Univariate visualizations. The same data is visualized as qq-plots and histograms below. In the qqplots, the coloured line is the reference line. In the histograms, the coloured line is the sample mean, and the shaded area is density.

qq-plots

histograms

Power transformation. We decided that the best way to try to overcome non-normality is via data transformation. Here, we transform the \(X\) set data with a Yeo-Johnson power transformation (from the car package). The Yeo-Johnson transformation identifies a lambda value and employs it as a power coefficient to transform values. Note that we use the Yeo-Johnson though there are no negative values in our FA data to be consistent with similar analyes applied to the clinical data, in which negative values are possible. In this situation, it is identical to the BoxCox method.

#function to perform transformation 
yjPowerTransformation <- function(x) {
  transformed <- x^powerTransform(x,family="yjPower")$lambda
  return(transformed)
}

#apply function to x set
xsetTransformed <- data.frame(apply(xset, 2, yjPowerTransformation))

Transformed MVN visualization. The transformed data continue to show substantial deviation from the reference line, which indicates departure from the multivariate normal distribution.

Transformed univariate tests. Wee note that the transformation does not result in more tests achieving normality.

Shapiro-Wilk’s

Test Variable Statistic p value Normality
Shapiro-Wilk AF_R 0.9927 0.0551 YES
Shapiro-Wilk AF_L 0.9783 <0.001 NO
Shapiro-Wilk CB_R 0.9903 0.0111 NO
Shapiro-Wilk CB_L 0.9879 0.0025 NO
Shapiro-Wilk CC1_C 0.9739 <0.001 NO
Shapiro-Wilk CC2_C 0.9862 9e-04 NO
Shapiro-Wilk CC3_C 0.9586 <0.001 NO
Shapiro-Wilk CC4_C 0.9534 <0.001 NO
Shapiro-Wilk CC5_C 0.9324 <0.001 NO
Shapiro-Wilk CC6_C 0.9720 <0.001 NO
Shapiro-Wilk CC7_C 0.9672 <0.001 NO
Shapiro-Wilk CPC_L 0.9014 <0.001 NO
Shapiro-Wilk CPC_R 0.9530 <0.001 NO
Shapiro-Wilk CR.F_L 0.9824 1e-04 NO
Shapiro-Wilk CR.F_R 0.9924 0.0454 NO
Shapiro-Wilk CR.P_R 0.9693 <0.001 NO
Shapiro-Wilk CR.P_L 0.9669 <0.001 NO
Shapiro-Wilk CST_L 0.9634 <0.001 NO
Shapiro-Wilk CST_R 0.9692 <0.001 NO
Shapiro-Wilk EC_R 0.9333 <0.001 NO
Shapiro-Wilk EC_L 0.9059 <0.001 NO
Shapiro-Wilk EmC_L 0.8454 <0.001 NO
Shapiro-Wilk EmC_R 0.8086 <0.001 NO
Shapiro-Wilk ICP_L 0.9366 <0.001 NO
Shapiro-Wilk ICP_R 0.9793 <0.001 NO
Shapiro-Wilk ILF_R 0.9834 2e-04 NO
Shapiro-Wilk ILF_L 0.9876 0.002 NO
Shapiro-Wilk Intra.CBLM.I.P_R 0.9759 <0.001 NO
Shapiro-Wilk Intra.CBLM.I.P_L 0.9848 4e-04 NO
Shapiro-Wilk Intra.CBLM.PaT_L 0.9792 <0.001 NO
Shapiro-Wilk Intra.CBLM.PaT_R 0.9295 <0.001 NO
Shapiro-Wilk IOFF_L 0.7263 <0.001 NO
Shapiro-Wilk IOFF_R 0.8167 <0.001 NO
Shapiro-Wilk MCP_C 0.9833 2e-04 NO
Shapiro-Wilk MdLF_R 0.9962 0.4804 YES
Shapiro-Wilk MdLF_L 0.9933 0.0805 YES
Shapiro-Wilk PLIC_R 0.9910 0.018 NO
Shapiro-Wilk PLIC_L 0.9930 0.068 YES
Shapiro-Wilk SF_R 0.9970 0.685 YES
Shapiro-Wilk SF_L 0.9928 0.0593 YES
Shapiro-Wilk SLF.II_L 0.7134 <0.001 NO
Shapiro-Wilk SLF.II_R 0.8379 <0.001 NO
Shapiro-Wilk SLF.III_R 0.9929 0.0614 YES
Shapiro-Wilk SLF.III_L 0.9967 0.6085 YES
Shapiro-Wilk SO_L 0.9067 <0.001 NO
Shapiro-Wilk SO_R 0.9278 <0.001 NO
Shapiro-Wilk SP_R 0.9939 0.1157 YES
Shapiro-Wilk SP_L 0.9898 0.0081 NO
Shapiro-Wilk Sup.F_R 0.9905 0.0126 NO
Shapiro-Wilk Sup.F_L 0.9892 0.0055 NO
Shapiro-Wilk Sup.FP_L 0.9780 <0.001 NO
Shapiro-Wilk Sup.FP_R 0.9957 0.3622 YES
Shapiro-Wilk Sup.O_L 0.8078 <0.001 NO
Shapiro-Wilk Sup.O_R 0.7734 <0.001 NO
Shapiro-Wilk Sup.OT_R 0.8947 <0.001 NO
Shapiro-Wilk Sup.OT_L 0.9824 1e-04 NO
Shapiro-Wilk Sup.P_L 0.9954 0.3089 YES
Shapiro-Wilk Sup.P_R 0.9903 0.0112 NO
Shapiro-Wilk Sup.PO_L 0.9951 0.255 YES
Shapiro-Wilk Sup.PO_R 0.9918 0.0296 NO
Shapiro-Wilk Sup.PT_R 0.9926 0.0501 YES
Shapiro-Wilk Sup.PT_L 0.9941 0.1394 YES
Shapiro-Wilk Sup.T_R 0.9952 0.2782 YES
Shapiro-Wilk Sup.T_L 0.9926 0.0524 YES
Shapiro-Wilk TF_L 0.9617 <0.001 NO
Shapiro-Wilk TF_R 0.9881 0.0028 NO
Shapiro-Wilk TO_L 0.9447 <0.001 NO
Shapiro-Wilk TO_R 0.9043 <0.001 NO
Shapiro-Wilk TP_L 0.9598 <0.001 NO
Shapiro-Wilk TP_R 0.9671 <0.001 NO
Shapiro-Wilk UF_L 0.9773 <0.001 NO
Shapiro-Wilk UF_R 0.9767 <0.001 NO

Cramer-von Mises

Test Variable Statistic p value Normality
Cramer-von Mises AF_R 0.2856 5e-04 NO
Cramer-von Mises AF_L 0.2161 0.0034 NO
Cramer-von Mises CB_R 0.1621 0.0164 NO
Cramer-von Mises CB_L 0.2982 3e-04 NO
Cramer-von Mises CC1_C 0.4045 <0.001 NO
Cramer-von Mises CC2_C 0.3143 2e-04 NO
Cramer-von Mises CC3_C 0.5409 <0.001 NO
Cramer-von Mises CC4_C 0.7348 <0.001 NO
Cramer-von Mises CC5_C 1.0294 <0.001 NO
Cramer-von Mises CC6_C 0.6287 <0.001 NO
Cramer-von Mises CC7_C 0.4302 <0.001 NO
Cramer-von Mises CPC_L 4.2242 <0.001 NO
Cramer-von Mises CPC_R 0.5503 <0.001 NO
Cramer-von Mises CR.F_L 0.3042 3e-04 NO
Cramer-von Mises CR.F_R 0.1626 0.0162 NO
Cramer-von Mises CR.P_R 0.7959 <0.001 NO
Cramer-von Mises CR.P_L 0.8075 <0.001 NO
Cramer-von Mises CST_L 0.8387 <0.001 NO
Cramer-von Mises CST_R 0.7744 <0.001 NO
Cramer-von Mises EC_R 0.5995 <0.001 NO
Cramer-von Mises EC_L 1.3660 <0.001 NO
Cramer-von Mises EmC_L 1.1177 <0.001 NO
Cramer-von Mises EmC_R 1.7822 <0.001 NO
Cramer-von Mises ICP_L 0.5333 <0.001 NO
Cramer-von Mises ICP_R 0.3075 3e-04 NO
Cramer-von Mises ILF_R 0.2943 4e-04 NO
Cramer-von Mises ILF_L 0.1579 0.0186 NO
Cramer-von Mises Intra.CBLM.I.P_R 0.2905 4e-04 NO
Cramer-von Mises Intra.CBLM.I.P_L 0.3025 3e-04 NO
Cramer-von Mises Intra.CBLM.PaT_L 0.1319 0.0413 NO
Cramer-von Mises Intra.CBLM.PaT_R 1.1666 <0.001 NO
Cramer-von Mises IOFF_L 3.8090 <0.001 NO
Cramer-von Mises IOFF_R 3.6565 <0.001 NO
Cramer-von Mises MCP_C 0.3948 <0.001 NO
Cramer-von Mises MdLF_R 0.0513 0.4925 YES
Cramer-von Mises MdLF_L 0.1024 0.1038 YES
Cramer-von Mises PLIC_R 0.1696 0.0131 NO
Cramer-von Mises PLIC_L 0.1737 0.0116 NO
Cramer-von Mises SF_R 0.0788 0.2144 YES
Cramer-von Mises SF_L 0.0888 0.1591 YES
Cramer-von Mises SLF.II_L 6.1464 <0.001 NO
Cramer-von Mises SLF.II_R 2.4866 <0.001 NO
Cramer-von Mises SLF.III_R 0.1587 0.0182 NO
Cramer-von Mises SLF.III_L 0.0816 0.1968 YES
Cramer-von Mises SO_L 1.4742 <0.001 NO
Cramer-von Mises SO_R 0.7802 <0.001 NO
Cramer-von Mises SP_R 0.1011 0.1081 YES
Cramer-von Mises SP_L 0.1662 0.0145 NO
Cramer-von Mises Sup.F_R 0.0648 0.3266 YES
Cramer-von Mises Sup.F_L 0.1370 0.0352 NO
Cramer-von Mises Sup.FP_L 0.1594 0.0178 NO
Cramer-von Mises Sup.FP_R 0.0588 0.3916 YES
Cramer-von Mises Sup.O_L 2.4683 <0.001 NO
Cramer-von Mises Sup.O_R 3.4793 <0.001 NO
Cramer-von Mises Sup.OT_R 1.5616 <0.001 NO
Cramer-von Mises Sup.OT_L 0.1683 0.0136 NO
Cramer-von Mises Sup.P_L 0.0489 0.5249 YES
Cramer-von Mises Sup.P_R 0.1870 0.0078 NO
Cramer-von Mises Sup.PO_L 0.0367 0.7418 YES
Cramer-von Mises Sup.PO_R 0.1110 0.079 YES
Cramer-von Mises Sup.PT_R 0.1305 0.0431 NO
Cramer-von Mises Sup.PT_L 0.0680 0.2962 YES
Cramer-von Mises Sup.T_R 0.0950 0.1313 YES
Cramer-von Mises Sup.T_L 0.0847 0.1794 YES
Cramer-von Mises TF_L 0.4369 <0.001 NO
Cramer-von Mises TF_R 0.2501 0.0013 NO
Cramer-von Mises TO_L 0.9458 <0.001 NO
Cramer-von Mises TO_R 1.3628 <0.001 NO
Cramer-von Mises TP_L 0.7490 <0.001 NO
Cramer-von Mises TP_R 0.4238 <0.001 NO
Cramer-von Mises UF_L 0.5119 <0.001 NO
Cramer-von Mises UF_R 0.4531 <0.001 NO

Lilliefors

Test Variable Statistic p value Normality
Lilliefors (Kolmogorov-Smirnov) AF_R 0.0754 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) AF_L 0.0528 0.011 NO
Lilliefors (Kolmogorov-Smirnov) CB_R 0.0469 0.0387 NO
Lilliefors (Kolmogorov-Smirnov) CB_L 0.0589 0.0024 NO
Lilliefors (Kolmogorov-Smirnov) CC1_C 0.0585 0.0027 NO
Lilliefors (Kolmogorov-Smirnov) CC2_C 0.0584 0.0028 NO
Lilliefors (Kolmogorov-Smirnov) CC3_C 0.0755 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CC4_C 0.0900 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CC5_C 0.0949 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CC6_C 0.0891 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CC7_C 0.0612 0.0013 NO
Lilliefors (Kolmogorov-Smirnov) CPC_L 0.2585 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CPC_R 0.0685 2e-04 NO
Lilliefors (Kolmogorov-Smirnov) CR.F_L 0.0653 4e-04 NO
Lilliefors (Kolmogorov-Smirnov) CR.F_R 0.0534 0.0095 NO
Lilliefors (Kolmogorov-Smirnov) CR.P_R 0.1460 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CR.P_L 0.1005 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CST_L 0.0966 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) CST_R 0.0803 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) EC_R 0.0811 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) EC_L 0.1080 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) EmC_L 0.1247 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) EmC_R 0.1634 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) ICP_L 0.0704 1e-04 NO
Lilliefors (Kolmogorov-Smirnov) ICP_R 0.0663 3e-04 NO
Lilliefors (Kolmogorov-Smirnov) ILF_R 0.0611 0.0013 NO
Lilliefors (Kolmogorov-Smirnov) ILF_L 0.0606 0.0016 NO
Lilliefors (Kolmogorov-Smirnov) Intra.CBLM.I.P_R 0.0557 0.0056 NO
Lilliefors (Kolmogorov-Smirnov) Intra.CBLM.I.P_L 0.0623 0.001 NO
Lilliefors (Kolmogorov-Smirnov) Intra.CBLM.PaT_L 0.0365 0.2348 YES
Lilliefors (Kolmogorov-Smirnov) Intra.CBLM.PaT_R 0.0953 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) IOFF_L 0.2177 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) IOFF_R 0.2200 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) MCP_C 0.0624 0.001 NO
Lilliefors (Kolmogorov-Smirnov) MdLF_R 0.0332 0.3708 YES
Lilliefors (Kolmogorov-Smirnov) MdLF_L 0.0415 0.106 YES
Lilliefors (Kolmogorov-Smirnov) PLIC_R 0.0368 0.2233 YES
Lilliefors (Kolmogorov-Smirnov) PLIC_L 0.0546 0.0072 NO
Lilliefors (Kolmogorov-Smirnov) SF_R 0.0393 0.1523 YES
Lilliefors (Kolmogorov-Smirnov) SF_L 0.0377 0.1959 YES
Lilliefors (Kolmogorov-Smirnov) SLF.II_L 0.3180 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) SLF.II_R 0.2244 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) SLF.III_R 0.0417 0.1 YES
Lilliefors (Kolmogorov-Smirnov) SLF.III_L 0.0367 0.227 YES
Lilliefors (Kolmogorov-Smirnov) SO_L 0.1297 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) SO_R 0.0820 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) SP_R 0.0355 0.2723 YES
Lilliefors (Kolmogorov-Smirnov) SP_L 0.0552 0.0063 NO
Lilliefors (Kolmogorov-Smirnov) Sup.F_R 0.0344 0.3181 YES
Lilliefors (Kolmogorov-Smirnov) Sup.F_L 0.0466 0.0414 NO
Lilliefors (Kolmogorov-Smirnov) Sup.FP_L 0.0427 0.0851 YES
Lilliefors (Kolmogorov-Smirnov) Sup.FP_R 0.0301 0.5313 YES
Lilliefors (Kolmogorov-Smirnov) Sup.O_L 0.1414 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) Sup.O_R 0.1838 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) Sup.OT_R 0.1171 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) Sup.OT_L 0.0454 0.052 YES
Lilliefors (Kolmogorov-Smirnov) Sup.P_L 0.0254 0.7812 YES
Lilliefors (Kolmogorov-Smirnov) Sup.P_R 0.0520 0.0132 NO
Lilliefors (Kolmogorov-Smirnov) Sup.PO_L 0.0283 0.6308 YES
Lilliefors (Kolmogorov-Smirnov) Sup.PO_R 0.0422 0.0923 YES
Lilliefors (Kolmogorov-Smirnov) Sup.PT_R 0.0428 0.0831 YES
Lilliefors (Kolmogorov-Smirnov) Sup.PT_L 0.0340 0.3363 YES
Lilliefors (Kolmogorov-Smirnov) Sup.T_R 0.0476 0.0336 NO
Lilliefors (Kolmogorov-Smirnov) Sup.T_L 0.0315 0.4564 YES
Lilliefors (Kolmogorov-Smirnov) TF_L 0.0630 8e-04 NO
Lilliefors (Kolmogorov-Smirnov) TF_R 0.0536 0.009 NO
Lilliefors (Kolmogorov-Smirnov) TO_L 0.0900 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TO_R 0.0992 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TP_L 0.0763 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TP_R 0.0635 7e-04 NO
Lilliefors (Kolmogorov-Smirnov) UF_L 0.0712 1e-04 NO
Lilliefors (Kolmogorov-Smirnov) UF_R 0.0729 <0.001 NO

Shapiro-Francia

Test Variable Statistic p value Normality
Shapiro-Francia AF_R 0.9920 0.034 NO
Shapiro-Francia AF_L 0.9790 <0.001 NO
Shapiro-Francia CB_R 0.9909 0.0176 NO
Shapiro-Francia CB_L 0.9888 0.0053 NO
Shapiro-Francia CC1_C 0.9747 <0.001 NO
Shapiro-Francia CC2_C 0.9872 0.0023 NO
Shapiro-Francia CC3_C 0.9574 <0.001 NO
Shapiro-Francia CC4_C 0.9546 <0.001 NO
Shapiro-Francia CC5_C 0.9321 <0.001 NO
Shapiro-Francia CC6_C 0.9729 <0.001 NO
Shapiro-Francia CC7_C 0.9680 <0.001 NO
Shapiro-Francia CPC_L 0.9023 <0.001 NO
Shapiro-Francia CPC_R 0.9544 <0.001 NO
Shapiro-Francia CR.F_L 0.9834 4e-04 NO
Shapiro-Francia CR.F_R 0.9927 0.0514 YES
Shapiro-Francia CR.P_R 0.9696 <0.001 NO
Shapiro-Francia CR.P_L 0.9684 <0.001 NO
Shapiro-Francia CST_L 0.9654 <0.001 NO
Shapiro-Francia CST_R 0.9709 <0.001 NO
Shapiro-Francia EC_R 0.9300 <0.001 NO
Shapiro-Francia EC_L 0.9055 <0.001 NO
Shapiro-Francia EmC_L 0.8411 <0.001 NO
Shapiro-Francia EmC_R 0.8035 <0.001 NO
Shapiro-Francia ICP_L 0.9349 <0.001 NO
Shapiro-Francia ICP_R 0.9798 1e-04 NO
Shapiro-Francia ILF_R 0.9838 4e-04 NO
Shapiro-Francia ILF_L 0.9891 0.0064 NO
Shapiro-Francia Intra.CBLM.I.P_R 0.9756 <0.001 NO
Shapiro-Francia Intra.CBLM.I.P_L 0.9859 0.0012 NO
Shapiro-Francia Intra.CBLM.PaT_L 0.9779 <0.001 NO
Shapiro-Francia Intra.CBLM.PaT_R 0.9297 <0.001 NO
Shapiro-Francia IOFF_L 0.7224 <0.001 NO
Shapiro-Francia IOFF_R 0.8152 <0.001 NO
Shapiro-Francia MCP_C 0.9849 7e-04 NO
Shapiro-Francia MdLF_R 0.9966 0.5205 YES
Shapiro-Francia MdLF_L 0.9935 0.087 YES
Shapiro-Francia PLIC_R 0.9923 0.0415 NO
Shapiro-Francia PLIC_L 0.9935 0.0831 YES
Shapiro-Francia SF_R 0.9976 0.7863 YES
Shapiro-Francia SF_L 0.9941 0.1252 YES
Shapiro-Francia SLF.II_L 0.7104 <0.001 NO
Shapiro-Francia SLF.II_R 0.8342 <0.001 NO
Shapiro-Francia SLF.III_R 0.9932 0.0732 YES
Shapiro-Francia SLF.III_L 0.9972 0.6734 YES
Shapiro-Francia SO_L 0.9071 <0.001 NO
Shapiro-Francia SO_R 0.9282 <0.001 NO
Shapiro-Francia SP_R 0.9946 0.1625 YES
Shapiro-Francia SP_L 0.9901 0.0113 NO
Shapiro-Francia Sup.F_R 0.9900 0.0105 NO
Shapiro-Francia Sup.F_L 0.9893 0.0071 NO
Shapiro-Francia Sup.FP_L 0.9784 <0.001 NO
Shapiro-Francia Sup.FP_R 0.9962 0.4139 YES
Shapiro-Francia Sup.O_L 0.8054 <0.001 NO
Shapiro-Francia Sup.O_R 0.7698 <0.001 NO
Shapiro-Francia Sup.OT_R 0.8950 <0.001 NO
Shapiro-Francia Sup.OT_L 0.9813 1e-04 NO
Shapiro-Francia Sup.P_L 0.9961 0.3886 YES
Shapiro-Francia Sup.P_R 0.9913 0.0226 NO
Shapiro-Francia Sup.PO_L 0.9947 0.1723 YES
Shapiro-Francia Sup.PO_R 0.9920 0.0337 NO
Shapiro-Francia Sup.PT_R 0.9938 0.1015 YES
Shapiro-Francia Sup.PT_L 0.9950 0.2157 YES
Shapiro-Francia Sup.T_R 0.9957 0.3142 YES
Shapiro-Francia Sup.T_L 0.9934 0.0809 YES
Shapiro-Francia TF_L 0.9613 <0.001 NO
Shapiro-Francia TF_R 0.9882 0.0039 NO
Shapiro-Francia TO_L 0.9460 <0.001 NO
Shapiro-Francia TO_R 0.9034 <0.001 NO
Shapiro-Francia TP_L 0.9596 <0.001 NO
Shapiro-Francia TP_R 0.9666 <0.001 NO
Shapiro-Francia UF_L 0.9793 1e-04 NO
Shapiro-Francia UF_R 0.9785 <0.001 NO

Anderson-Darling

Test Variable Statistic p value Normality
Anderson-Darling AF_R 1.2865 0.0024 NO
Anderson-Darling AF_L 1.5811 5e-04 NO
Anderson-Darling CB_R 1.0053 0.0118 NO
Anderson-Darling CB_L 1.6676 3e-04 NO
Anderson-Darling CC1_C 2.6512 <0.001 NO
Anderson-Darling CC2_C 1.8763 1e-04 NO
Anderson-Darling CC3_C 3.3359 <0.001 NO
Anderson-Darling CC4_C 4.7413 <0.001 NO
Anderson-Darling CC5_C 6.2437 <0.001 NO
Anderson-Darling CC6_C 3.5979 <0.001 NO
Anderson-Darling CC7_C 3.0263 <0.001 NO
Anderson-Darling CPC_L 19.3707 <0.001 NO
Anderson-Darling CPC_R 3.9460 <0.001 NO
Anderson-Darling CR.F_L 1.9332 1e-04 NO
Anderson-Darling CR.F_R 0.9720 0.0143 NO
Anderson-Darling CR.P_R 3.9159 <0.001 NO
Anderson-Darling CR.P_L 4.4266 <0.001 NO
Anderson-Darling CST_L 4.8392 <0.001 NO
Anderson-Darling CST_R 4.3377 <0.001 NO
Anderson-Darling EC_R 3.5266 <0.001 NO
Anderson-Darling EC_L 8.3941 <0.001 NO
Anderson-Darling EmC_L 8.1691 <0.001 NO
Anderson-Darling EmC_R 11.0190 <0.001 NO
Anderson-Darling ICP_L 3.9163 <0.001 NO
Anderson-Darling ICP_R 1.9824 <0.001 NO
Anderson-Darling ILF_R 1.8359 1e-04 NO
Anderson-Darling ILF_L 1.0239 0.0106 NO
Anderson-Darling Intra.CBLM.I.P_R 1.8641 1e-04 NO
Anderson-Darling Intra.CBLM.I.P_L 1.7730 2e-04 NO
Anderson-Darling Intra.CBLM.PaT_L 1.0303 0.0103 NO
Anderson-Darling Intra.CBLM.PaT_R 7.0188 <0.001 NO
Anderson-Darling IOFF_L 22.2533 <0.001 NO
Anderson-Darling IOFF_R 19.3437 <0.001 NO
Anderson-Darling MCP_C 2.2570 <0.001 NO
Anderson-Darling MdLF_R 0.3482 0.4756 YES
Anderson-Darling MdLF_L 0.7214 0.0594 YES
Anderson-Darling PLIC_R 1.0022 0.012 NO
Anderson-Darling PLIC_L 0.9421 0.0169 NO
Anderson-Darling SF_R 0.4065 0.3486 YES
Anderson-Darling SF_L 0.6783 0.0759 YES
Anderson-Darling SLF.II_L 30.6464 <0.001 NO
Anderson-Darling SLF.II_R 13.0643 <0.001 NO
Anderson-Darling SLF.III_R 0.8632 0.0265 NO
Anderson-Darling SLF.III_L 0.4480 0.2778 YES
Anderson-Darling SO_L 8.9507 <0.001 NO
Anderson-Darling SO_R 5.5689 <0.001 NO
Anderson-Darling SP_R 0.6105 0.1118 YES
Anderson-Darling SP_L 0.9459 0.0166 NO
Anderson-Darling Sup.F_R 0.5350 0.1699 YES
Anderson-Darling Sup.F_L 0.9244 0.0187 NO
Anderson-Darling Sup.FP_L 1.2322 0.0033 NO
Anderson-Darling Sup.FP_R 0.3676 0.4293 YES
Anderson-Darling Sup.O_L 15.3361 <0.001 NO
Anderson-Darling Sup.O_R 20.0749 <0.001 NO
Anderson-Darling Sup.OT_R 9.6132 <0.001 NO
Anderson-Darling Sup.OT_L 1.0937 0.0072 NO
Anderson-Darling Sup.P_L 0.3785 0.4052 YES
Anderson-Darling Sup.P_R 1.1557 0.005 NO
Anderson-Darling Sup.PO_L 0.2861 0.6229 YES
Anderson-Darling Sup.PO_R 0.6458 0.0914 YES
Anderson-Darling Sup.PT_R 0.7396 0.0536 YES
Anderson-Darling Sup.PT_L 0.4884 0.2217 YES
Anderson-Darling Sup.T_R 0.4860 0.2247 YES
Anderson-Darling Sup.T_L 0.6163 0.1081 YES
Anderson-Darling TF_L 2.8771 <0.001 NO
Anderson-Darling TF_R 1.4375 0.001 NO
Anderson-Darling TO_L 5.8828 <0.001 NO
Anderson-Darling TO_R 8.5358 <0.001 NO
Anderson-Darling TP_L 4.4680 <0.001 NO
Anderson-Darling TP_R 2.6381 <0.001 NO
Anderson-Darling UF_L 2.9118 <0.001 NO
Anderson-Darling UF_R 2.7574 <0.001 NO

Transformed univariate visualizations. Likewise, the qq-plots and histograms don’t appear to be particularly improved.

qq-plots

histograms

Decision. Because the power transformation doesn’t appear to net large benefits, we elect not to transform the variables (transformation can often obfuscate interpretation, so it should only be applied if helpful). Given our large sample (n=390), we will instead assume the CLT is in effect and the assumption of normality is inessential. Moreover, we are hopeful that the distributions will appear more normal after RISH harmonization.