We will give a clear example where no univariate test detects a difference between 2 populations but the multivariate test does. We will summarize the purposes of multivariate techniques, their assumptions, and the importance of balanced designs. We may begin discussing one-way MANOVA theory.
The following data are the clear example. The data are originally due to Kevin E. Bonine and can be found in Applied Multivariate Statistical Analysis by Johnson and Wichern. The two species are Cnemidophorus and Sceloporus. The measurements are snout-vent length in millimeters and mass in grams.
Species SVL mass log_SVL log_mass C 74.0 7.513 4.30407 2.01663 C 69.5 5.032 4.24133 1.61582 C 72.0 5.867 4.27667 1.76934 C 80.0 11.088 4.38203 2.40586 C 56.0 2.419 4.02535 0.88335 C 94.0 13.610 4.54329 2.61080 C 95.5 18.247 4.55913 2.90400 C 99.5 16.832 4.60016 2.82328 C 97.0 15.910 4.57471 2.76695 C 90.5 17.035 4.50535 2.83527 C 91.0 16.526 4.51086 2.80493 C 67.0 4.530 4.20469 1.51072 C 75.0 7.230 4.31749 1.97824 C 69.5 5.200 4.24133 1.64866 C 91.5 13.450 4.51634 2.59898 C 91.0 14.080 4.51086 2.64476 C 90.0 14.665 4.49981 2.68546 C 73.0 6.092 4.29046 1.80698 C 69.5 5.264 4.24133 1.66089 C 94.0 16.902 4.54329 2.82743 S 77.0 13.911 4.34381 2.63268 S 62.0 5.236 4.12713 1.65556 S 108.0 37.331 4.68213 3.61982 S 115.0 41.781 4.74493 3.73244 S 106.0 31.995 4.66344 3.46558 S 56.0 3.962 4.02535 1.37675 S 60.5 4.367 4.10264 1.47408 S 52.0 3.048 3.95124 1.11449 S 60.0 4.838 4.09434 1.57650 S 64.0 6.535 4.15888 1.87717 S 96.0 22.610 4.56435 3.11839 S 79.5 13.342 4.37576 2.59092 S 55.5 4.109 4.01638 1.41318 S 75.0 12.369 4.31749 2.51519 S 64.5 7.120 4.16667 1.96291 S 87.5 21.077 4.47164 3.04818 S 109.0 42.989 4.69135 3.76094 S 96.0 27.201 4.56435 3.30325 S 111.0 38.901 4.70953 3.66102 S 84.5 19.747 4.43675 2.98300 S 80.0 14.666 4.38203 2.68553 S 62.0 4.790 4.12713 1.56653 S 61.5 5.020 4.11904 1.61343 S 62.0 5.220 4.12713 1.65250 S 64.0 5.690 4.15888 1.73871 S 63.0 6.763 4.14313 1.91147 S 71.0 9.977 4.26268 2.30028 S 69.5 8.831 4.24133 2.17827 S 67.5 9.493 4.21213 2.25055 S 66.0 7.811 4.18965 2.05553 S 64.5 6.685 4.16667 1.89987 S 79.0 11.980 4.36945 2.48324 S 84.0 16.520 4.43082 2.80457 S 81.0 13.630 4.39445 2.61227 S 82.5 13.700 4.41280 2.61740 S 74.0 10.350 4.30407 2.33699 S 68.5 7.900 4.22683 2.06686 S 70.0 9.103 4.24850 2.20860 S 77.5 13.216 4.35028 2.58143 S 70.0 9.787 4.24850 2.28105