The Core of Latent-Strength Theory ◮ CORE: Decisions are made by placing criteria on a latent strength axis. ◮ SIDE ASSUMPTION: Parametric form of latent strength (e.g., normal). ◮ These side assumptions have no psychological content. Made for convenience. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Perfectly Good Latent-Strength Models 0.4 0.3 Density 0.2 0.1 0.0 −2 0 2 4 6 Latent Strength Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Perfectly Good Latent-Strength Models 0.4 0.3 Density 0.2 0.1 0.0 −2 0 2 4 6 Latent Strength Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Perfectly Good Latent-Strength Models 0.4 0.3 Density 0.2 0.1 0.0 −2 0 2 4 6 Latent Strength Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Perfectly Good Latent-Strength Models 0.3 Density 0.2 0.1 0.0 0 5 10 15 Latent Strength Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Perfectly Good Latent-Strength Models 0.6 Density 0.4 0.2 0.0 0 5 10 15 Latent Strength Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Latent Strength Is Too Flexible ◮ CORE: Decisions are made by placing criteria on a latent strength axis. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Latent Strength Is Too Flexible ◮ CORE: Decisions are made by placing criteria on a latent strength axis. ◮ THEOREM: The core is unfalsifiable. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Latent Strength Is Too Flexible ◮ CORE: Decisions are made by placing criteria on a latent strength axis. ◮ THEOREM: The core is unfalsifiable. ◮ For any family of ROC curves, there exists some set of latent strength distributions N , S 1 , S 2 , . . . that can perfectly predict the family. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Latent Strength Is Too Flexible ◮ CORE: Decisions are made by placing criteria on a latent strength axis. ◮ THEOREM: The core is unfalsifiable. ◮ For any family of ROC curves, there exists some set of latent strength distributions N , S 1 , S 2 , . . . that can perfectly predict the family. ◮ CONSEQUENCE: Goodness of fit tests are tests of the contentless side assumptions rather than the core. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Latent Strength Is Too Flexible ◮ CORE: Decisions are made by placing criteria on a latent strength axis. ◮ THEOREM: The core is unfalsifiable. ◮ For any family of ROC curves, there exists some set of latent strength distributions N , S 1 , S 2 , . . . that can perfectly predict the family. ◮ CONSEQUENCE: Goodness of fit tests are tests of the contentless side assumptions rather than the core. ◮ MY VIEW: The debate in the ROC literature on multiple vs. single process is misguided because in every case it is critically dependent on contentless parametric assumptions. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Example of Flexibility 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Latent Mnemonic Strength Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Example of Flexibility Fam Recollection 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Latent Mnemonic Strength Jeffrey N. Rouder Exploring the Structure of Recognition Memory
NEEDED: New conceptualization of single process. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
NEEDED: New conceptualization of single process. ◮ More constrained than the core of signal detection Jeffrey N. Rouder Exploring the Structure of Recognition Memory
NEEDED: New conceptualization of single process. ◮ More constrained than the core of signal detection ◮ Less arbitrary than the side parametric assumptions Jeffrey N. Rouder Exploring the Structure of Recognition Memory
NEEDED: New conceptualization of single process. ◮ More constrained than the core of signal detection ◮ Less arbitrary than the side parametric assumptions ◮ Behavior of a family of ROC curves rather than any one curve. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
My Approach SINGLE PROCESS IMPLIES: Jeffrey N. Rouder Exploring the Structure of Recognition Memory
My Approach SINGLE PROCESS IMPLIES: ◮ All ROC curves differ from one another in just one way. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
My Approach SINGLE PROCESS IMPLIES: ◮ All ROC curves differ from one another in just one way. ◮ ROC curve that differ in just one way are said to be Single Process Representable (SPR) Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Examples of Single-Process ROC Families 1.0 0.8 0.6 Hit Rate Normal- 0.4 Distribution Signal Detection 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False−Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Examples of Single-Process ROC Families 1.0 0.8 0.6 Hit Rate High Threshold 0.4 Model 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False−Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Examples of Single-Process ROC Families 1.0 0.8 0.6 Hit Rate Gamma- 0.4 Distribution Signal Detection 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False−Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Not Single Process Representable 1.0 0.8 0.6 Hit Rate Yonelinas Dual 0.4 Process Model 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False−Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Not Single Process Representable 1.0 0.8 0.6 Hit Rate Unequal-Variance Normal- 0.4 Distribution Signal Detection (UVSD) 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False−Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
One Formalization 1.0 4 0.8 2 0.6 Hit Rate Φ − 1 ( h ) 0 0.4 −2 −4 0.2 −6 0.0 0.0 0.2 0.4 0.6 0.8 1.0 −6 −4 −2 0 2 4 Φ − 1 ( f ) False−Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
One Formalization 1.0 0 0.8 −1 0.6 log ( 1 − h ) Hit Rate −2 0.4 −3 0.2 −4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 −4 −3 −2 −1 0 log ( 1 − f ) False−Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Linear Single Process Representability Linear Single Process Representability: ◮ Does there exist a linearizing transform? ◮ Do all the transformed lines have the same slope? Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Linear Single Process Representability ◮ Let ω i be the function that describes the ROC curve for the i th condition; e.g., h = ω i ( f ). ◮ Let Ω be the collection of all such curves under consideration. ◮ Collection Ω is linearly single-process representable (lSPR) if there exists a strictly increasing function function θ such that for any ω i ∈ Ω θ ( ω i ( f )) = θ ( f ) + c i , 0 ≤ f ≤ 1 , where c i is constant across f Jeffrey N. Rouder Exploring the Structure of Recognition Memory
SPR Is General ROCs: ◮ Working Memory ◮ Auditory/Visual Perception ◮ Speech ◮ Tumor Detection Jeffrey N. Rouder Exploring the Structure of Recognition Memory
SPR Across Domains 1.0 1.0 0.8 0.8 0.6 0.6 Hit Rate Hit Rate 0.4 0.4 0.2 0.2 Tone Detection Tone Detection Recognition Memory Recognition Memory 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 False−Alarm Rate False−Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
SPR Across Domains ◮ How can tone recognition rely on the same single process as recognition memory? Jeffrey N. Rouder Exploring the Structure of Recognition Memory
SPR Across Domains ◮ How can tone recognition rely on the same single process as recognition memory? ◮ Abstracted information, common decision system (much like a statistician who uses a t -test across multiple domains) Jeffrey N. Rouder Exploring the Structure of Recognition Memory
SPR Across Domains ◮ How can tone recognition rely on the same single process as recognition memory? ◮ Abstracted information, common decision system (much like a statistician who uses a t -test across multiple domains) ◮ SPECULATION: ROCs tell us more about common decision processing than mnemonic or other upstream processing. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing Linear SPR How does one know if a suitable linearizing function θ exists? Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing SPR ◮ Functional Approximation: θ ( x ) ≈ g ( x , β ) + β 0 Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing SPR ◮ Functional Approximation: θ ( x ) ≈ g ( x , β ) + β 0 ◮ Check whether g ( ω i ( f ) , β ) = g ( f , β ) + c i for all ω i ∈ Ω. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing SPR In Practice �� 1 − 2 3 β 1 − 1 � Φ − 1 g ( x , β 1 , β 2 ) = 3 β 2 x + � x − 1 � � x − 2 � � β 1 I ( x − 1 / 3) + β 2 I ( x − 2 / 3) . 3 3 Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing SPR In Practice �� 1 − 2 3 β 1 − 1 � Φ − 1 g ( x , β 1 , β 2 ) = 3 β 2 x + � x − 1 � � x − 2 � � β 1 I ( x − 1 / 3) + β 2 I ( x − 2 / 3) . 3 3 ◮ Two-piece linear spline on z-ROC function. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing SPR In Practice �� 1 − 2 3 β 1 − 1 � Φ − 1 g ( x , β 1 , β 2 ) = 3 β 2 x + � x − 1 � � x − 2 � � β 1 I ( x − 1 / 3) + β 2 I ( x − 2 / 3) . 3 3 ◮ Two-piece linear spline on z-ROC function. ◮ g = Φ − 1 if β 1 = β 2 = 0 Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing SPR In Practice �� 1 − 2 3 β 1 − 1 � Φ − 1 g ( x , β 1 , β 2 ) = 3 β 2 x + � x − 1 � � x − 2 � � β 1 I ( x − 1 / 3) + β 2 I ( x − 2 / 3) . 3 3 ◮ Two-piece linear spline on z-ROC function. ◮ g = Φ − 1 if β 1 = β 2 = 0 ◮ Estimate separate parameters ( β 1 i , β 2 i ) for each ω i . Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing SPR In Practice �� 1 − 2 3 β 1 − 1 � Φ − 1 g ( x , β 1 , β 2 ) = 3 β 2 x + � x − 1 � � x − 2 � � β 1 I ( x − 1 / 3) + β 2 I ( x − 2 / 3) . 3 3 ◮ Two-piece linear spline on z-ROC function. ◮ g = Φ − 1 if β 1 = β 2 = 0 ◮ Estimate separate parameters ( β 1 i , β 2 i ) for each ω i . ◮ SPR holds if β 1 and β 2 do not vary across curves, i.e., β 1 = β 1 i and β 2 = β 2 i for all i . Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Conclusions 1. The core of single-process signal-detection models is too general. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Conclusions 1. The core of single-process signal-detection models is too general. 2. Single-process representability (SPR) implies that ROCs differ in one way. May be formally defined and tested (Linear SPR). Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Conclusions 1. The core of single-process signal-detection models is too general. 2. Single-process representability (SPR) implies that ROCs differ in one way. May be formally defined and tested (Linear SPR). 3. SPR seems especially well-suited to explore behavior across several domains. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
PART II: And Then We Began Collecting Recognition Memory and Perception Data Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Consider a Simple Discrete-State Model ◮ Three States: Detect Old Item, Detect New Item, Guess ◮ Conditional Independence: Responses are determined only by which state one is in. ◮ No false detection. Participant can enter one of two states on any trial (correct detect state, guess state). ◮ Certainty: Detection leads to a “certainty” response. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Discrete States With Certainty Assumption 1.0 ● 6 1 ● d s d n ● θ 1 θ 1 1 1 ● Hit Rate θ 2 θ 2 2 2 ● 0.5 1 − d s 1 − d n θ 3 θ 3 3 3 θ 4 θ 4 4 4 θ 5 θ 5 5 5 6 6 ∑ ∑ θ i = 1 θ i = 1 i = 1 θ 6 i = 1 θ 6 0.0 6 6 0.0 0.5 1.0 Old Item Presented New Item Presented False Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Discrete States W/O Certainty Assumption 6 3 ∑ φ 6 ∑ φ 1 φ i = 1 φ i = 1 6 1 i = 4 i = 1 1.0 ● φ 5 φ 2 5 2 ● d s d n φ 4 φ 3 4 3 ● θ 1 θ 1 1 1 Hit Rate θ 2 θ 2 2 2 0.5 ● 1 − d s 1 − d n θ 3 θ 3 3 3 θ 4 θ 4 4 4 ● θ 5 θ 5 5 5 6 6 ∑ ∑ θ i = 1 θ i = 1 i = 1 θ 6 i = 1 θ 6 0.0 6 6 0.0 0.5 1.0 Old Item Presented New Item Presented False Alarm Rate Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing Discrete State Models ◮ W/O Certainty Assumption, Discrete-State Models predict that the ROC points are connected by straight lines. ◮ Can account for any single ROC curve, no constraint. ◮ Constraint across the comparison of several ROC curves Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Testing Discrete State Models ◮ i. Show you the paradigm ◮ ii. Then the predictions for discrete-state and typical latent-strength models ◮ iii. Finally, the data Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Paradigm ◮ Study A List of Words (e.g., “FROG”) ◮ Test: 2AFC, Move Slider TABLE FROG Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Paradigm STUDY: ◮ 1 repetition ◮ 2 repetitions ◮ 4 repetitions TEST: Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Paradigm STUDY: ◮ 1 repetition ◮ 2 repetitions ◮ 4 repetitions TEST: ◮ Normal : Old Word + New Word Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Paradigm STUDY: ◮ 1 repetition ◮ 2 repetitions ◮ 4 repetitions TEST: ◮ Normal : Old Word + New Word ◮ Sneaky : Two New Words (0 repetitions). Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Paradigm STUDY: ◮ 1 repetition ◮ 2 repetitions ◮ 4 repetitions TEST: ◮ Normal : Old Word + New Word ◮ Sneaky : Two New Words (0 repetitions). ◮ These sneaky trials allow us to isolate responses under guessing. Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Differing Predictions Latent Strength Theory Discrete State Theory No Repetitions No Repetitions Sure Left Sure Right Sure Left Sure Right Word Was Word Was Word Was Word Was Studied Studied Studied Studied Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Differing Predictions Latent Strength Theory Discrete State Theory Few Repetitions Few Repetitions Sure Left Sure Right Sure Left Sure Right Word Was Word Was Word Was Word Was Studied Studied Studied Studied Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Differing Predictions Latent Strength Theory Discrete State Theory Many Repetitions Many Repetitions Sure Left Sure Right Sure Left Sure Right Word Was Word Was Word Was Word Was Studied Studied Studied Studied Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Differing Predictions Latent Strength Theory Discrete State Theory Many Many Few Few None None Sure Left Sure Right Sure Left Sure Right Word Was Word Was Word Was Word Was Studied Studied Studied Studied Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Differing Predictions Latent Strength Theory Discrete State Theory No Repetitions No Repetitions Few Repetitions Few Repetitions Many Repetitions Many Repetitions Sure "New" Sure "Old" Sure "New" Sure "Old" Word Was Word Was Word Was Word Was Studied Studied Studied Studied Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Differing Predictions Latent Strength Theory Discrete State Theory No Repetitions No Repetitions Few Repetitions Few Repetitions Many Repetitions Many Repetitions Sure "New" Sure "Old" Sure "New" Sure "Old" Word Was Word Was Word Was Word Was Studied Studied Studied Studied Jeffrey N. Rouder Exploring the Structure of Recognition Memory
Data: Select Subjects Probability 1.00 Study Condition 0 Repetitions 0.75 1−2 Repetitions 4 Repetitions 0.50 0.25 0.25 "Target was on List" "Lure was on List" (Incorrect Response) (Correct Response) 0.50 0.75 [High Confidence] [Low Confidence] [Low Confidence] [High Confidence] 1.00 Jeffrey N. Rouder Exploring the Structure of Recognition Memory
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