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

MVN tests. All MVN tests suggest that the \(Y\) set is not multivariate normal.

Mardia’s

##              Test        Statistic              p value Result
## 1 Mardia Skewness 1394.10367987404 9.75220120854385e-73     NO
## 2 Mardia Kurtosis 14.1409556484297                    0     NO
## 3             MVN             <NA>                 <NA>     NO

Henze-Zirkler’s

##            Test       HZ p value MVN
## 1 Henze-Zirkler 1.046297       0  NO

Royston’s

##      Test        H      p value MVN
## 1 Royston 481.3512 7.778447e-94  NO

Doornik-Hansen’s

##             Test        E df p value MVN
## 1 Doornik-Hansen 18105.83 28       0  NO

Energy

##          Test Statistic p value MVN
## 1 E-statistic  3.524195       0  NO

MVN plot. The Chi-square Q-Q plot shows large deviations from the reference line, which indicates departure from the multivariate normal distribution.

Multivariate outliers. Multivariate outlier detection methods suggest so many outliers that it would not make sense to remove them.

Mahalanobis distance

Adjusted Mahalanobis distance

Univariate tests. Here, we see that many of the distributions of many of the variables in the \(Y\) set are not normal.

Shapiro-Wilk’s

Test Variable Statistic p value Normality
Shapiro-Wilk Processing speed 0.9897 0.0076 NO
Shapiro-Wilk Attention & vigilance 0.9665 <0.001 NO
Shapiro-Wilk Working memory 0.9951 0.2501 YES
Shapiro-Wilk Verbal learning 0.9847 4e-04 NO
Shapiro-Wilk Visual learning 0.9829 1e-04 NO
Shapiro-Wilk Problem solving 0.9849 4e-04 NO
Shapiro-Wilk RMET 0.9433 <0.001 NO
Shapiro-Wilk RAD 0.9277 <0.001 NO
Shapiro-Wilk ER_40 0.8693 <0.001 NO
Shapiro-Wilk TASIT_1 0.8926 <0.001 NO
Shapiro-Wilk TASIT_2 0.8823 <0.001 NO
Shapiro-Wilk TASIT_3 0.9284 <0.001 NO
Shapiro-Wilk IRI 0.9914 0.0227 NO
Shapiro-Wilk EA 0.9332 <0.001 NO

Cramer-von Mises

Test Variable Statistic p value Normality
Cramer-von Mises Processing speed 0.1380 0.0342 NO
Cramer-von Mises Attention & vigilance 0.2051 0.0046 NO
Cramer-von Mises Working memory 0.0739 0.2478 YES
Cramer-von Mises Verbal learning 0.1379 0.0343 NO
Cramer-von Mises Visual learning 0.3089 3e-04 NO
Cramer-von Mises Problem solving 0.2073 0.0043 NO
Cramer-von Mises RMET 0.9903 <0.001 NO
Cramer-von Mises RAD 1.5912 <0.001 NO
Cramer-von Mises ER_40 2.3562 <0.001 NO
Cramer-von Mises TASIT_1 1.7538 <0.001 NO
Cramer-von Mises TASIT_2 2.2039 <0.001 NO
Cramer-von Mises TASIT_3 0.8612 <0.001 NO
Cramer-von Mises IRI 0.0736 0.2505 YES
Cramer-von Mises EA 0.9933 <0.001 NO

Lilliefors

Test Variable Statistic p value Normality
Lilliefors (Kolmogorov-Smirnov) Processing speed 0.0449 0.0564 YES
Lilliefors (Kolmogorov-Smirnov) Attention & vigilance 0.0599 0.0019 NO
Lilliefors (Kolmogorov-Smirnov) Working memory 0.0446 0.0605 YES
Lilliefors (Kolmogorov-Smirnov) Verbal learning 0.0600 0.0018 NO
Lilliefors (Kolmogorov-Smirnov) Visual learning 0.0684 2e-04 NO
Lilliefors (Kolmogorov-Smirnov) Problem solving 0.0598 0.0019 NO
Lilliefors (Kolmogorov-Smirnov) RMET 0.1275 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) RAD 0.1392 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) ER_40 0.1405 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TASIT_1 0.1590 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TASIT_2 0.1461 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TASIT_3 0.1051 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) IRI 0.0394 0.1487 YES
Lilliefors (Kolmogorov-Smirnov) EA 0.0957 <0.001 NO

Shapiro-Francia

Test Variable Statistic p value Normality
Shapiro-Francia Processing speed 0.9906 0.0151 NO
Shapiro-Francia Attention & vigilance 0.9665 <0.001 NO
Shapiro-Francia Working memory 0.9960 0.3841 YES
Shapiro-Francia Verbal learning 0.9854 9e-04 NO
Shapiro-Francia Visual learning 0.9838 4e-04 NO
Shapiro-Francia Problem solving 0.9866 0.0017 NO
Shapiro-Francia RMET 0.9445 <0.001 NO
Shapiro-Francia RAD 0.9291 <0.001 NO
Shapiro-Francia ER_40 0.8664 <0.001 NO
Shapiro-Francia TASIT_1 0.8918 <0.001 NO
Shapiro-Francia TASIT_2 0.8831 <0.001 NO
Shapiro-Francia TASIT_3 0.9269 <0.001 NO
Shapiro-Francia IRI 0.9917 0.0296 NO
Shapiro-Francia EA 0.9330 <0.001 NO

Anderson-Darling

Test Variable Statistic p value Normality
Anderson-Darling Processing speed 0.8823 0.0238 NO
Anderson-Darling Attention & vigilance 1.7529 2e-04 NO
Anderson-Darling Working memory 0.4609 0.2586 YES
Anderson-Darling Verbal learning 1.0538 0.009 NO
Anderson-Darling Visual learning 1.9351 1e-04 NO
Anderson-Darling Problem solving 1.4463 0.001 NO
Anderson-Darling RMET 6.1082 <0.001 NO
Anderson-Darling RAD 9.3415 <0.001 NO
Anderson-Darling ER_40 13.4772 <0.001 NO
Anderson-Darling TASIT_1 10.5432 <0.001 NO
Anderson-Darling TASIT_2 13.2858 <0.001 NO
Anderson-Darling TASIT_3 5.3011 <0.001 NO
Anderson-Darling IRI 0.5848 0.1271 YES
Anderson-Darling EA 6.0253 <0.001 NO

Univariate visualizations. The same data is visualized as qq-plots and histograms below. In the qq-plots, 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 transformed the \(Y\) set data with a Yeo-Johnson power transformation (from the car package). Yeo-Johnson is employed as there are negative values in our \(Y\) set (namely, in the EA task). However, we observer that the transformation does not result in more tests achieving normality.

Shapiro-Wilk’s

Test Variable Statistic p value Normality
Shapiro-Wilk Processing speed 0.9956 0.3562 YES
Shapiro-Wilk Attention & vigilance 0.9870 0.0017 NO
Shapiro-Wilk Working memory 0.9944 0.1832 YES
Shapiro-Wilk Verbal learning 0.9943 0.1684 YES
Shapiro-Wilk Visual learning 0.9890 0.0059 NO
Shapiro-Wilk Problem solving 0.9864 0.0012 NO
Shapiro-Wilk RMET 0.9855 7e-04 NO
Shapiro-Wilk RAD 0.9765 <0.001 NO
Shapiro-Wilk ER_40 0.9014 <0.001 NO
Shapiro-Wilk TASIT_1 0.9729 <0.001 NO
Shapiro-Wilk TASIT_2 0.9672 <0.001 NO
Shapiro-Wilk TASIT_3 0.9852 6e-04 NO
Shapiro-Wilk IRI 0.9969 0.6795 YES
Shapiro-Wilk EA 0.8988 <0.001 NO

Cramer-von Mises

Test Variable Statistic p value Normality
Cramer-von Mises Processing speed 0.0486 0.5294 YES
Cramer-von Mises Attention & vigilance 0.2233 0.0027 NO
Cramer-von Mises Working memory 0.0780 0.2196 YES
Cramer-von Mises Verbal learning 0.0873 0.1661 YES
Cramer-von Mises Visual learning 0.1514 0.0227 NO
Cramer-von Mises Problem solving 0.2072 0.0043 NO
Cramer-von Mises RMET 0.2594 0.001 NO
Cramer-von Mises RAD 0.4581 <0.001 NO
Cramer-von Mises ER_40 1.3911 <0.001 NO
Cramer-von Mises TASIT_1 0.5705 <0.001 NO
Cramer-von Mises TASIT_2 0.4264 <0.001 NO
Cramer-von Mises TASIT_3 0.1953 0.0061 NO
Cramer-von Mises IRI 0.0466 0.5614 YES
Cramer-von Mises EA 1.3836 <0.001 NO

Lilliefors

Test Variable Statistic p value Normality
Lilliefors (Kolmogorov-Smirnov) Processing speed 0.0335 0.3775 YES
Lilliefors (Kolmogorov-Smirnov) Attention & vigilance 0.0637 8e-04 NO
Lilliefors (Kolmogorov-Smirnov) Working memory 0.0390 0.1735 YES
Lilliefors (Kolmogorov-Smirnov) Verbal learning 0.0413 0.1185 YES
Lilliefors (Kolmogorov-Smirnov) Visual learning 0.0554 0.007 NO
Lilliefors (Kolmogorov-Smirnov) Problem solving 0.0550 0.0077 NO
Lilliefors (Kolmogorov-Smirnov) RMET 0.0817 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) RAD 0.0929 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) ER_40 0.1103 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TASIT_1 0.1035 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TASIT_2 0.0875 <0.001 NO
Lilliefors (Kolmogorov-Smirnov) TASIT_3 0.0630 0.001 NO
Lilliefors (Kolmogorov-Smirnov) IRI 0.0370 0.2355 YES
Lilliefors (Kolmogorov-Smirnov) EA 0.1216 <0.001 NO

Shapiro-Francia

Test Variable Statistic p value Normality
Shapiro-Francia Processing speed 0.9968 0.5801 YES
Shapiro-Francia Attention & vigilance 0.9887 0.0059 NO
Shapiro-Francia Working memory 0.9955 0.3058 YES
Shapiro-Francia Verbal learning 0.9951 0.2382 YES
Shapiro-Francia Visual learning 0.9908 0.0194 NO
Shapiro-Francia Problem solving 0.9881 0.0044 NO
Shapiro-Francia RMET 0.9875 0.0032 NO
Shapiro-Francia RAD 0.9783 <0.001 NO
Shapiro-Francia ER_40 0.8961 <0.001 NO
Shapiro-Francia TASIT_1 0.9751 <0.001 NO
Shapiro-Francia TASIT_2 0.9700 <0.001 NO
Shapiro-Francia TASIT_3 0.9869 0.0024 NO
Shapiro-Francia IRI 0.9975 0.7588 YES
Shapiro-Francia EA 0.8991 <0.001 NO

Anderson-Darling

Test Variable Statistic p value Normality
Anderson-Darling Processing speed 0.3309 0.5121 YES
Anderson-Darling Attention & vigilance 1.3347 0.0018 NO
Anderson-Darling Working memory 0.5180 0.1873 YES
Anderson-Darling Verbal learning 0.6074 0.1137 YES
Anderson-Darling Visual learning 0.9685 0.0146 NO
Anderson-Darling Problem solving 1.3568 0.0016 NO
Anderson-Darling RMET 1.4943 7e-04 NO
Anderson-Darling RAD 2.7768 <0.001 NO
Anderson-Darling ER_40 8.4578 <0.001 NO
Anderson-Darling TASIT_1 3.4135 <0.001 NO
Anderson-Darling TASIT_2 2.9878 <0.001 NO
Anderson-Darling TASIT_3 1.3557 0.0016 NO
Anderson-Darling IRI 0.3059 0.5646 YES
Anderson-Darling EA 8.8642 <0.001 NO