Likelihood Ratio Decisions in Memory: Three Implied Regularities
Glanzer, Murray, Hilford, Andrew, Maloney, Laurence T., Psychonomic Bulletin & Review
We analyze four general signal detection models for recognition memory that differ in their distributional assumptions. Our analyses show that a basic assumption of signal detection theory, the likelihood ratio decision axis, implies three regularities in recognition memory: (1) the mirror effect, (2) the variance effect, and (3) the z-ROC length effect. For each model, we present the equations that produce the three regularities and show, in computed examples, how they do so. We then show that the regularities appear in data from a range of recognition studies. The analyses and data in our study support the following generalization: Individuals make efficient recognition decisions on the basis of likelihood ratios.
(ProQuest: ... denotes formulae omitted.)
In a typical recognition memory test, individuals consider a series of test items presented in random order. Some of the test items have been seen previously (old), others are new, and the prior probability that an item is old is π. In the simplest case, the individuals are asked to classify each item as "old" or "new," and their performance is measured by the proportion of correct classifications.
Signal detection models of the recognition process assume that the information available on a single trial can be represented by a random variable X. The distribution of this variable is fO(x) when the item is old (O) and fN(x) when it is new (N). If X is a continuous random variable, fO(x) and fN(x) are probability density functions, whereas if X is discrete, they are probability mass functions.1
Given X, the likelihood ratio (LR) for "old" over "new" responses is
This ratio is a measure of the evidence in the data favoring "old" over "new" (Royall, 1999). The likelihood ratio decision rule compares the likelihood ratio in favor of "old" with a fixed criterion,
L(X) > β, (2)
and returns an "old" response if the likelihood ratio exceeds β, or otherwise returns a "new" response. If the criterion β is set to (1 - π)/π (the prior odds in favor of "new"), the resulting decision rule has the highest expected proportion of correct responses (Duda, Hart, & Stork, 2001, p. 26; Green & Swets, 1966/1974, p. 23) of any decision rule.
Even when the item information X is multivariate, the LR rule converts the item information to a univariate measure of evidence in favor of "old" over "new," and a simple comparison of the prior probabilities of old and new items determines whether the evidence justifies an "old" response. If there are more than two response categories, the LR rule can be easily generalized (Duda et al., 2001, chap. 2). If responses are allowed to be graded (e.g., individuals give a confidence rating for each choice), the LR rule is also easily generalized by assuming that there are multiple criteria (Green & Swets, 1966/1974, pp. 40-43).
A more convenient form of the LR rule, which we will use, replaces the comparison in Equation 1 with a comparison of log likelihoods. The resulting log-likelihood ratio (Λ) rule leads to exactly the same decisions as the LR rule:
Λ = λ(X) > log(β), (3)
where, for convenience, we define Λ = λ(X) as the random likelihood corresponding to the random strength variable X and
We refer to the latter function as the transfer function. It maps from the evidence axis to the log-likelihood axis. We emphasize that Λ = λ(X) is a random variable-the evidence available on each trial-whereas λ(x) is a function that will prove useful in what follows.
The LR rule can be applied for any choice of the two distributions fN(x) and fO(x) (Wickens, 2002, p. 165). In work on recognition memory, these two distributions are typically assumed to be normal, differing in their means and possibly their standard deviations:
When σO = σN = σ, we refer to the model as equalvariance normal. …