It is believed by some that most tests are multidimensional, meaning that they measure more than one underlying construct. The primary objective of this study is to illustrate how variations in the secondary ability distribution affect the statistical detection of dimensionality and to demonstrate the difference between substantive and statistical dimensionality. Given dichotomous data simulated to be multidimensional, this study shows how varying the ability distributions affects the results obtained from DIMTEST, a nonparametric statistical procedure based on the theory of essential unidimensionality. Results indicate that the power of DIMTEST decreased as the mean of the secondary ability distribution approached the extremes and/or as the standard deviation of the secondary ability distribution approached zero. This has important implications for both researchers and practitioners because although a test may measure additional dimensions from a substantive viewpoint, these dimensions may not be detected statistically. (Contains 3 tables and 3 figures.)