Receiver Operating Characteristic - Z-transformation

Z-transformation

If a z-transformation is applied to the ROC curve, the curve will be transformed into a straight line. This z-transformation is based on a normal distribution with a mean of zero and a standard deviation of one. In memory strength theory, one must assume that the zROC is not only linear, but has a slope of 1.0. The normal distributions of targets (studied objects that the subjects need to recall) and lures (non studied objects that the subjects attempt to recall) is the factor causing the zROC to be linear.

The linearity of the zROC curve depends on the standard deviations of the target and lure strength distributions. If the standard deviations are equal, the slope will be 1.0. If the standard deviation of the target strength distribution is larger than the standard deviation of the lure strength distribution, then the slope will be smaller than 1.0. In most studies, it has been found that the zROC curve slopes constantly fall below 1, usually between 0.5 and 0.9. Many experiments yielded a zROC slope of 0.8. A slope of 0.8 implies that the variability of the target strength distribution is 25% larger than the variability of the lure strength distribution.

Another variable used is d'. d' is a measure of sensitivity for yes-no recognition that can easily be expressed in terms of z-values. d' measures sensitivity, in that it measures the degree of overlap between target and lure distributions. It is calculated as the mean of the target distribution minus the mean of the lure distribution, expressed in standard deviation units. For a given hit rate and false alarm rate, d' can be calculated with the following equation: d'=z(hit rate)- z(false alarm rate). Although d' is a commonly used parameter, it must be recognized that it is only relevant when strictly adhering to the very strong assumptions of strength theory made above.

The z-transformation of a ROC curve is always linear, as assumed, except in special situations. The Yonelinas Familiarity-Recollection model is a two-dimensional account of recognition memory. Instead of the subject simply answering yes or no to a specific input, the subject gives the input a feeling of familiarity, which operates like the original ROC curve. What changes, though, is a parameter for Recollection (R). Recollection is assumed to be all-or-none, and it trumps familiarity. If there were no recollection component, zROC would have a predicted slope of 1. However, when adding the recollection component, the zROC curve will be concave up, with a decreased slope. This difference in shape and slope result from an added element of variability due to some items being recollected. Patients with anterograde amnesia are unable to recollect, so their Yonelinas zROC curve would have a slope close to 1.0.

Read more about this topic:  Receiver Operating Characteristic