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Analysis of Interaction Effects James Jaccard New York University

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Will cover the basics of interaction analysis, highlighting multiple regression based strategies Overview Will discuss advanced issues and complications in interaction analysis. This treatment will be somewhat superficial but hopefully informative

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Conceptual Foundations of Interaction Analysis

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Causal Theories Most (but not all) theories rely heavily on the concept of causality, i.e., we seek to identify the determinants of a behavior or mental state and/or the consequences of a behavior or environmental/mental state I am going to ground interaction analysis in a causal framework

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Causal Theories Causal theories can be complicated, but at their core, there are five types of causal relationships in causal theories Direct Causal Relationships A direct causal relationship is when a variable, X, has a direct causal influence on another variable, Y:

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Direct Causal Relationships

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Indirect Causal Relationships An indirect causal relationship is when a variable, X, has a causal influence on another variable, Y, through an intermediary variable, M:

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Indirect Causal Relationships

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Spurious Relationship A spurious relationship is one where two variables that are not causally related share a common cause:

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Bidirectional Causal Relationships A bidirectional causal relationship is when a variable, X, has a causal influence on another variable, Y, and that effect, Y, has a “simultaneous” impact on X:

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Bidirectional Causal Relationships

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Moderated Causal Relationships A moderated causal relationship is when the impact of a variable, X, on another variable, Y, differs depending on the value of a third variable, Z

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Moderated Causal Relationships

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The variable that “moderates” the relationship is called a moderator variable.

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Causal Theories We put all these ideas together to build complex theories of phenomena. Here is one example:

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Interaction Analysis Interactions, when translated into causal analysis, focus on moderated relationships When I encounter an interaction effect, I think:

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Key step in interaction analysis is to identify the focal independent variable and the moderator variable. Sometimes it is obvious – such as with the analysis of a treatment for depression on depression as moderated by gender Interaction Analysis

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Sometimes it is not obvious – such as an analysis of the effects of gender and ethnicity on the amount of time an adolescent spends with his or her mother Interaction Analysis Statistically, it matters not which variables take on which role. Conceptually, it does.

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The Statistical Analysis of Interactions

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Omnibus tests – I do not use these Some Common Practices Hierarchical regression – I use sparingly Focus on unstandardized coefficients - we tend to stay away from standardized coefficients in interaction analysis because they can be misleading and they do not have “clean” mathematical properties

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Y = a + b 1 X + e A “Trick” We Will Use: Linear Transformations Satisfaction = a + b 1 Grade + e Satisfaction = 12 + -.50 Grade + e

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Y = a + b 1 X + e A “Trick” We Will Use: Linear Transformations Satisfaction = a + b 1 Grade + e Satisfaction = 12 + -.50 Grade + e Satisfaction = 9 + -.50 (Grade – 6) + e

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Y = a + b 1 X + e A “Trick” We Will Use: Linear Transformations Satisfaction = a + b 1 Grade + e Satisfaction = 12 + -.50 Grade + e Satisfaction = 9 + -.50 (Grade – 6) + e “Mean centering” is when we subtract the mean

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Will focus on four cases: Categorical IV and Categorical MV Assume you know the basics of multiple regression and dummy variables in multiple regression Interaction Analysis Continuous IV and Categorical MV Categorical IV and Continuous MV Continuous IV and Continuous MV

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Categorical IV and Categorical MV

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Y = Relationship satisfaction (0 to 10) X = Gender (female = 1, male = 0) Z = Grade (6 th = 1, 7 th = 0) 6th7th Female8.07.0 Male7.04.0

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Categorical IV and Categorical MV 6th7th Female8.07.0 Male7.04.0 Three questions: Is there a gender difference for 6 th graders? Is there a gender difference for 7 th graders? Are these gender effects different?

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Categorical IV and Categorical MV 6th7th Female8.07.0 Male7.04.0 Gender effect for 6 th grade: 8 – 7 = 1

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Categorical IV and Categorical MV 6th7th Female8.07.0 Male7.04.0 Gender effect for 6 th grade: 8 – 7 = 1 Gender effect for 7 th grade: 7 – 4 = 3

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Categorical IV and Categorical MV 6th7th Female8.07.0 Male7.04.0 Gender effect for 6 th grade: 8 – 7 = 1 Gender effect for 7 th grade: 7 – 4 = 3 Interaction contrast: (8-7) – (7– 4) = -2

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Y = a + b 1 Gender + b 2 Grade + b 3 (Gender)(Grade) Categorical IV and Categorical MV 6th7th Female8.07.0 Male7.04.0 Y = 4.0 + 3.0 Gender + b 2 Grade + -2.0 (Gender)(Grade)

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Y = a + b 1 Gender + b 2 Grade + b 3 (Gender)(Grade) Categorical IV and Categorical MV 6th7th Female8.07.0 Male7.04.0 Y = 4.0 + 3.0 Gender + b 2 Grade + -2.0 (Gender)(Grade) Flipped: Y = 7.0 + 1.0 Gender + b 2 Grade + 2.0 (Gender)(Grade)

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Extend to groups > 2 (add 8 th grade) Categorical IV and Categorical MV 6th7th Female8.07.0 Male7.04.0 Inclusion of covariates How to generate means and tables

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Continuous IV and Categorical MV

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Y = Relationship satisfaction (0 to 10) X = Time spent together (in hours) Z = Gender (female = 1, male = 0)

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Continuous IV and Categorical MV Y = Relationship satisfaction (0 to 10) X = Time spent together (in hours) Z = Gender (female = 1, male = 0) For females: b = 0.33 For males: b = 0.20 Three questions: Are the effects different: 0.33 – 0.20

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Continuous IV and Categorical MV Y = Relationship satisfaction (0 to 10) X = Time spent together (in hours) Z = Gender (female = 1, male = 0) For females: b = 0.33 For males: b = 0.20 Y = a + b 1 Gender + 0.20 Time + 0.13 (Gender)(Time)

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Continuous IV and Categorical MV Y = Relationship satisfaction (0 to 10) X = Time spent together (in hours) Z = Gender (female = 1, male = 0) For females: b = 0.33 For males: b = 0.20 Y = a + b 1 Gender + 0.20 Time + 0.13 (Gender)(Time) Flipped: Y = a + b 1 Gender + 0.33 Time + -0.13 (Gender)(Time)

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Continuous IV and Categorical MV Do not estimate slopes separately; use flipped reference group strategy Extend to groups > 2 (use grade as example)

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Categorical IV and Continuous MV

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Study conducted in Miami with bi-lingual Latinos

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Categorical IV and Continuous MV Study conducted in Miami with bi-lingual Latinos Ad language: Half shown ad in Spanish (0) and half in English (1)

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Categorical IV and Continuous MV Study conducted in Miami with bi-lingual Latinos Ad language: Half shown ad in Spanish (0) and half in English (1) Latino identity: 1 = not at all, 7 = strong identify

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Categorical IV and Continuous MV Study conducted in Miami with bi-lingual Latinos Ad language: Half shown ad in Spanish (0) and half in English (1) Latino identity: 1 = not at all, 7 = strong identify Outcome = Attitude toward product (1 = unfavorable, 7 = unfavorable) Hypothesized moderated relationship

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Common Analysis Form: Median Split Many researchers not sure how to analyze this, so use median split for continuous moderator variable and conduct ANOVA Why this is bad practice….

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Categorical IV and Continuous MV Identity Mean English – Mean Spanish 1 1.50 2 1.00 3 0.50 4 0.00 5 -0.50 6 -1.00 7 -1.50

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Categorical IV and Continuous MV Identity Mean English – Mean Spanish 1 1.50 2 1.00 3 0.50 4 0.00 5 -0.50 6 -1.00 7 -1.50 Y = a + b 1 Ad language + b 2 Identity + b 3 Ad X Identity

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Categorical IV and Continuous MV In order to make intercept meaningful, subtracted 1 from Latino Identity measure, so ranged from 0 to 6 Y = a + b 1 Ad language + b 2 Identity + b 3 Ad X Identity

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Categorical IV and Continuous MV

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Mean attitude for Spanish ad for Latino ID = 1 is 3.215

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Categorical IV and Continuous MV Mean attitude for Spanish ad for Latino ID = 1 is 3.215 Mean difference for Latino ID = 1 is 1.707 (p < 0.05)

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Categorical IV and Continuous MV Mean attitude for Spanish ad for Latino ID = 1 is 3.215 Mean attitude for English ad for Latino ID = 1 is 4.922 Mean difference for Latino ID = 1 is 1.707 (p < 0.05)

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Categorical IV and Continuous MV Identity Mean English Mean Spanish Difference 1 4.922 3.215 1.707* 2 3 4 5 6 7

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Categorical IV and Continuous MV Identity Mean English Mean Spanish Difference 1 4.922 3.215 1.707* 2 4.915 3.662 1.253* 3 4 5 6 7

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Categorical IV and Continuous MV Identity Mean English Mean Spanish Difference 1 4.922 3.215 1.707* 2 4.915 3.662 1.253* 3 4.908 4.108 0.800* 4 5 6 7

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Categorical IV and Continuous MV Identity Mean English Mean Spanish Difference 1 4.922 3.215 1.707* 2 4.915 3.662 1.253* 3 4.908 4.108 0.800* 4 4.901 4.555 0.346* 5 4.895 5.002 -0.107 6 4.888 5.449 -0.561* 7 4.882 5.896 -1.014* (Common practice, Mean = 3, SD = 1.2; Show R program)

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Continuous IV and Continuous MV Y: Child anxiety (0 to 20) X: Parent anxiety (0 to 20) Z: Parenting behavior: Control (0 to 20)

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Continuous IV and Continuous MV Y: Child anxiety (0 to 20) X: Parent anxiety (0 to 20) Z: Parenting behavior: Control (0 to 20) Control b for Y onto X 7.10 8.20 9.30 10.40 11.50 12.60 13.70

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Continuous IV and Continuous MV Control b for Y onto X 7.10 8.20 9.30 10.40 11.50 12.60 13.70 Y = a + b 1 Control + 0.10 PA + 0.10 (Control)(PA) (Common practice versus regions of significance) (Why we include component parts)

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Advanced Topics

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Three Way Interactions

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Identify focal independent variable Identify first order moderator variable Identify second order moderator variable

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Three Way Interactions Grade 7Grade 8 Female6.0 Male5.04.0 IC 1 = (6-5) - (6-4) = -1 IC = (6-5) – (6-4) = -1 European American

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Three Way Interactions Grade 7Grade 8 Female6.0 Male5.04.0 IC 1 = (6-5) - (6-4) = -1 Grade 7Grade 8 Female6.0 Male6.0 IC = (6-5) – (6-4) = -1IC = (6-6) – (6-6) = 0 European AmericanLatinos

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Three Way Interactions Grade 7Grade 8 Female6.0 Male5.04.0 IC 1 = (6-5) - (6-4) = -1 Grade 7Grade 8 Female6.0 Male6.0 IC = (6-5) – (6-4) = -1IC = (6-6) – (6-6) = 0 European AmericanLatinos TW = [(6-5) – (6-4)] - [(6-6) – (6-6)] = -1

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Three Way Interactions G7 (1)G8 (0) Female (1)6.0 Male (0)5.04.0 IC 1 = (6-5) - (6-4) = -1 G7 (1)G8 (0) Female (1)6.0 Male (0)6.0 IC = (6-5) – (6-4) = -1IC = (6-6) – (6-6) = 0 European American (1)Latinos (0) Y = 6.0 + 0 Gender + b2 Grade + b3 Ethnic + 0 (Gender)(Grade) + b5 (Gender)(Ethnic) + b6 (Grade)(Ethnic) + -1 (Gender)(Grade)(Ethnic) TW = [(6-5) – (6-4)] - [(6-6) – (6-6)] = -1

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Modeling Non-Linear Interactions

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Y = α + β 1 X + β 2 Z + ε Modeling Non-Linear Interactions β 1 = α’ + β 3 Z + β 4 Z 2

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Y = α + β 1 X + β 2 Z + ε Substitute right hand side for β 1 : Modeling Non-Linear Interactions β 1 = α’ + β 3 Z + β 4 Z 2 Y = α + (α’ + β 3 Z + β 4 Z 2 ) X + β 2 Z + ε

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Y = α + β 1 X + β 2 Z + ε Substitute right hand side for β 1 : Modeling Non-Linear Interactions β 1 = α’ + β 3 Z + β 4 Z 2 Y = α + (α’ + β 3 Z + β 4 Z 2 ) X + β 2 Z + ε Expand: Y = α + α’X + β 3 XZ + β 4 XZ 2 + β 2 Z + ε

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Modeling Non-Linear Interactions Re-arrange terms: Y = α + α’X + β 3 XZ + β 4 XZ 2 + β 2 Z + ε Y = α + α’X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε

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Modeling Non-Linear Interactions Re-arrange terms: Y = α + α’X + β 3 XZ + β 4 XZ 2 + β 2 Z + ε Y = α + α’X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε Re-label and you have your model: Y = α + β 1 X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε

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Modeling Non-Linear Interactions Re-arrange terms: Y = α + α’X + β 3 XZ + β 4 XZ 2 + β 2 Z + ε Y = α + α’X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε Re-label and you have your model: Y = α + β 1 X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε Use centering strategy to isolate effect of X on Y (β 1 ) at any given value of Z; also consider modeling intercept

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Exploratory Interaction Analysis

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Use program in R Y = Tenured or not (using MLPM) X = Number of articles published Y = α + β 1 X + ε Z = Number of years since hired

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X COEFFICENT AND M VALUES N M Value X Slope 478 1.000.000 475 2.000.002 457 3.000.007 408 4.000.007 330 5.000.009 246 6.000.008 166 7.000.005 115 8.000.009 74 9.000.011 48 10.000.001

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Regression Mixture Modeling

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BI = α + β 1 Aact + β 2 PN + β 3 PBC + ε Mixture Regression But, in reality, we probably are mixing heterogeneous population segments with different coefficients characterizing the segments When we regress Y onto a set of predictors, we assume that people are drawn from a single population with common linear coefficients

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With “mixed” populations, the overall regression analysis can characterize neither segment very well and lead to sub-optimal inferences and intervention strategies Mixture Regression

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Another Example of Aggregation Bias

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Mixture Regression

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A four class model fits data best (entries are linear coefficients) Segment 1 (42%):.33.02.01 -.01 Mixture Model for Heavy Episodic Drinking Segment 2 (17%):.10.29.30.01 Segment 3 (21%):.30.29.05.04 Segment 4 (20%):.48.09.25 -.03 Aact SN DN PBC

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Interaction Analysis and Establishing Generalizability

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It is common for people to conclude that an effect “generalizes” in the absence of a statistically significant interaction effect Problem is that we can never accept the null hypothesis of a zero interaction contrast Generalizability Example with RCT of obesity treatment and gender Solution: Adopt the framework of equivalence testing

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Step 1: Specify a threshold value that will be used to define functional equivalence Step 2: Specify the range of functional equivalence Generalizability Step 3: Calculate the 95% CI for the interaction contrast Step 4: Determine if the CI is completely within the range of functional equivalence

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Measurement Error

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It is well known that measurement error can bias parameter estimates in multiple regression. This holds with vigor for interaction analysis One approach to dealing with measurement error in general is to use latent variable modeling Measurement Error

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Latent Variable Regression

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There are a about a half a dozen approaches to how best to model latent variable interactions (e.g., quasi-maximum likelihood; Bayesian). I recommend the approach developed by Herbert Marsh as a good balance between utility and complexity, coupled with Huber-White sandwich estimators for robustness Latent Variable Regression Latent variable regression using multiple group analysis

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Multi-Group Modeling in SEM

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Assumption Violations

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If assumptions of normality or variance homogeneity are suspect Huber-White sandwich estimators Assumption Violations Use approaches with robust standard errors Be careful of outlier resistant robust methods Bootstrapping Rand Wilcox work with smoothers

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