Delta Plot

A delta plot is a graphical display for comparing two distributions that is closely related to quantile-quantile (QQ) plots. A common application in education is to inspect whether there is differential item functioning by plotting distributions of item discriminability in one population versus that of another. The setup is one with two populations (or conditions), which are generically called Population A and Population B. Let X(p) and Y(p) denote the pth percentile for the dependent measure for Populations A and B, respectively; let d(p) = Y(p) − X(p) be the difference, or the effect, at the pth percentile; and let m(p) be the average effect at the pth percentile. The delta plot is a plot of the effect, d(p), as a function of the average, m(p).

Figure 1 provides an example for item discriminability parameters. The distributions are in Panel A; the corresponding delta plot is the solid line in Panel B. The open circles show the 10th percentile for each distribution, and the values are respectively .68 and .74 for the control and treatment conditions. The difference at the 10th percentile is .06, and the average is .71, which is the open circle in Panel B. The filled circles show the same for the 90th percentile. The line is the plot for the remaining percentiles, and it has positive slope.

Figure 1 Delta Plots: (A) Two distributions of IRT discriminability parameters that are to be compared. (B) Corresponding delta plot shows that these distributions are not equal. (C) Delta plot patterns are sometimes used to document the time course of processing.

The key question in education is often whether two distributions differ. If they are the same—that is, if items are functioning without differentials across populations—then the delta plot line [Page 481]should be a horizontal line at the value of 0, which in the figure is indicated with the dashed line labeled “equality.” The delta plot line is not on this equality line, and the deviation indicates differential item functioning. Items tend to have a larger standard deviation in Population B than in Population A.

Delta plots are rotated versions of quantile-quantile plots, and the rotation allows the analyst to choose a smaller range for the y-axis. With this smaller range, deviations from horizontal lines at 0 are visually emphasized.

In experimental psychology, delta plots are used with response time distributions to interpret the time course of effects. Panel C shows two delta plots that have opposite patterns: The one labeled “typical” shows a rising pattern (i.e., the effect is a slowing of the slower responses) and the other plot, labeled “alternative,” starts high and decreases. The effect is a slowing of the faster responses. The typical plot is indeed typical in many applications; for example, it describes the slowing from cognitive aging. The fastest of the elderly are about as fast as the fastest college-age individuals, but the slowest of the elderly are quite a bit slower than the slowest of the college-age individuals. The alternative pattern occurs for just a handful of phenomena, and it indicates a fast effect that decays quickly in time.

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