Bootstrap Test of Distribution Shape 1 Running head: BOOTSTRAP TEST OF DISTRIBUTION SHAPE A Bootstrap Test of Shape Invariance Across Distributions

The shape of a response time distribution provides valuable clues about the underlying mental processing. If a manipulation affects the shape of an RT distribution, then it is reasonable to suspect that the manipulation has done more than simply speed or slow the rate of processing. We develop a nonparametric bootstrap test of shape invariance. Simulations reveal the test is sufficiently powered to detect small shape changes in reasonably sized experiments while maintaining appropriate Type I error control. The test is simple and can be applied broadly in cognitive psychology. An application to a number priming experiment provides a demonstration of how shape changes may be detected. Bootstrap Test of Distribution Shape 3 A Bootstrap Test of Shape Invariance Across Distributions Response time (RT), the time taken to complete a task, is a common dependent variable that has been used to draw inferences about the nature of mental processing (e.g., Luce, 1986). Although many researhcers draw conclusions from analysis of mean RT, it is increasingly common to see more sophisticated analyses of entire RT distributions. These more sophistcated analyses often provide additional constraint on cognitive and perceptual theories. Selective examples include Ashby, Tien, and Balakrishnan (1993); Dzhafarov (1992); Hockley (1984); Logan (1992); Ratcliff (1978); Ratcliff and Rouder (1998, 2000); Rouder (2000); Rouder, Ratcliff, and McKoon (2000); Spieler, Faust, and Balota, (1996); Theeuwes (1992, 1994); Townsend & Nozawa (1995); Van Zandt, Colonius, & Proctor (2000); and Vickers (1980). Although there are several methods of analyzing distributions, we advocate that researchers consider how properties of location, scale, and shape change across conditions or populations (Rouder, Sun, Speckman, Lu, & Zhou, 2003; Rouder, Lu, Speckman, Sun, & Jiang, 2005). Figure 1 provides an example of these properties. The left panel shows the case when only location differs between distributions; the center and right panels show the same for scale and shape, respectively. Location and scale are formally defined as follows: Let the density of a continuous random variable exist everywhere and be expressed as f(t | θ1, . . . , θp), where θ1, . . . , θp are parameters. Let z = (t− θ1)/θ2. We refer to the density as being in location-scale form if there exists a density g such that f(t | θ1, . . . , θp) = 1 θ2 g ( t− θ1 θ2 | θ3, . . . , θp )