A dominant theme in modeling human perceptual judgments is that sensory neural activity is summed or integrated until a critical bound is reached. Such models predict that, in general, the shape of response time distributions change across conditions, although in practice, this shape change may be subtle. An alternative view is that response time distributions are shape invariant across conditions or groups. Shape invariance is predicted by some race models in which the first of several parallel fibers to communicate the signal determines the response. We competitively assess a specific gradual growth model, the one-bound diffusion model, against a natural shape-invariant competitor: shape invariance in an inverse Gaussian distribution. Assessment of subtle shape change versus shape invariance of response time distributions is aided by a Bayesian approach that allows the pooling of information across multiple participants. We find, conditional on reasonable distributional assumptions, subtle shape changes in response time that are highly concordant with a simple diffusion gradual growth model and discordant with shape invariance. (Contains 7 figures and 3 footnotes.)