Likelihood Ratio Decisions in Memory: Three Implied Regularities

By Glanzer, Murray; Hilford, Andrew et al. | Psychonomic Bulletin & Review, June 2009 | Go to article overview

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

... (1)

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

... (4)

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:

... (5)

and

... (6)

When σO = σN = σ, we refer to the model as equalvariance normal. …

The rest of this article is only available to active members of Questia

Sign up now for a free, 1-day trial and receive full access to:

  • Questia's entire collection
  • Automatic bibliography creation
  • More helpful research tools like notes, citations, and highlights
  • Ad-free environment

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

Likelihood Ratio Decisions in Memory: Three Implied Regularities
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Full screen

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.