Supplement to “From ROC Curves to Psychological Theory”

Distributions have several different types of associated functions. The most familiar is the density function. Figure 1A and 1D show density functions for the normal distribution and gamma distribution, respectively. Density functions are related to histograms in that if enough data are collected then the histogram and density share the same shape. Hence, density functions serve as a first-order graphical representation of a distribution. An alternative view of a distribution is provided by the cumulative distribution function (CDF). CDFs describe the probability of an observation being below a specified value, and Figure 1B and 1E provides the CDFs for the normal and gamma distribution, respectively. The highlighted point in Figure 1B corresponds to a latent strength of 1.0 and a cumulative probability of .84, indicating that 84% of the mass of distribution is less than 1.0. Quantile functions are the inverse of the CDF. They describe the value (latent strength values in this case) associated with a cumulative probability, and the corresponding examples are shown in Figure 1C and 1F. The highlighted point in Figure 1C shows that the cumulative probability of .84 corresponds to a latent strength of 1.0. These three graphical representations, densities, CDFs, and quantile functions are alternative graphical representations of a distribution and may be used interchangeably.