tween A and the probability that the test is correct. Note that, for these curves, when the test is correct with the same probability for deceptive as for nondeceptive examinees, this shared probability is between 5 percent and 10 percent lower than the value of A for each ROC. Elsewhere on the ROC, percent correct depends heavily on the base rate and, in some circumstances, may not be lower than the value of A.

Under this model, the method of maximum likelihood estimation is commonly used to estimate the ROC, and hence A. However, this method fails without at least one observation in each of the categories used to determine the ROC points. When some categories are only sparsely filled, it also can produce unstable and inadmissible results: that is, ROC curves that idiosyncratically dip below the 45-degree diagonal instead of increasing steadily from the lower left to the upper right-hand corners of the graph. In either of these instances, we estimated A directly from the empirical ROC data, by connecting points from the same study to each other, the leftmost point to the lower left-hand corner, and the rightmost point to the upper right-hand corner, and determining A as the area within the polygon generated by those lines and the lower and right-hand plotting axes. In the signal detection theory literature, this is known as the “trapezoidal estimate.” For our data, where one-point ROCs with equivariance binormal maximum likelihood estimates exist, the resulting estimates of A tended to be higher than the trapezoidal estimates by about 0.1; for two-point ROCs, the discrepancy between the trapezoidal and binormal (possibly with unequal variances) estimates of A was much smaller, generally 0.01-0.03. Had sample sizes been large enough to allow the use of a binormal estimate in all cases, we conjecture that the median values of A reported in Chapter 5 and below would have increased by 0.02-0.03 for laboratory studies and perhaps 0.01 for field studies.

Accuracy in Laboratory Studies Figure H-3 plots values of A from the extrapolated ROCs from our 52 laboratory datasets, in descending order of A from left to right. Below each point is suspended a line of length equal to the estimated standard error of the associated A, to give an indication of the inherent variability in these numbers given the sizes of the various studies. From the lengths of most of these lines, it is clear that few of these studies estimate A precisely. Furthermore, the apparent precision of the high estimates at the upper left may well be due to the fact that values of A that are near the maximum due to chance necessarily produce unduly low estimates of variability. We note, in any event, that the large majority of A values are between 0.70 and 0.95, and that half the studies fall between the lower and upper quartiles of A = 0.813 and 0.910, represented by the horizontal lines.

One might suspect that the highest and lowest values of A would



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