LATENT DIFFERENCE SCORE MODELS Bob Vandenberg Terry College of Business Department of Management...
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LATENT DIFFERENCE SCORE MODELSBob Vandenberg
Terry College of BusinessDepartment of ManagementUniversity of Georgia
OverviewFramework for workshopIntroductionChangeWhy longitudinal?Other longitudinal statistical procedures and shortcomingsLatent difference: Only a T1 and T2 measureNonlatent methods and shortcomingLatent methodsType DR and Type modelsEmpirical examplesObserved variable onlySingle-Indicator ApproachMultiple-Indicator Approach
My Primary SourcesMcArdle, J.J., & Nesselroade, J.R. (2014). Longitudinal Data Analysis Using Structural Equation Models. American Psychological Association: Washington, D.C.Particularly Chapter 9 Newsom, J.T. (2015). Longitudinal Structural Equation Modeling: A Comprenhensive Introduction. Routledge: New York.Particularly Chapter 9
AssumptionsNeed an understanding of CFA and structural modelsExtension of basic modelsUsed Mplus 7.x (Muthn & Muthn, 1998-2013)Simply my personal choice can also be completed in LISREL, EQS, AMOS, or Mx.Some focus on equations . . .but I am very application oriented and thus, will be presenting lots of empirical examples with syntax
IntroductionChangeWhy is it so important?#1 = because the theoretical frameworks of interest to us explicitly or implicitly imply itUnderlying the use of terms such as related to or associated with is a true desire to say as X changes so does Y; or a change in X causes a change in YReality is we neither operationalize change nor use a design that permits true causal inferencesConcern here is with change, not cause
IntroductionComplexities in Changes Over TimeMultiple Levelse.g., changes in newcomer information seeking over time.Cross-Levelschanges in P-G fit, P-O fit over time.Multivariaterelationships linking changes in different focal variables.
IntroductionResearch DesignsCross-sectional research designCollects data on a variable at one time point.Longitudinal research designCollects data on a variable at multiple time points.Note:Longitudinal research designs are meant to examine changes over time. A prospective study that uses variable A at T1 to predict variable B at T2, with A measured only at T1 and B measured only at T2, is NOT a longitudinal research design.
IntroductionWhy longitudinal?McArdle & Nesselroade (2014) give 5 reasons1 = direct identification of intraindividual changeComes in many forms such as level, frequency and amplitude2 = direct identification of interindividual differences in intraindividual changeChange patterns differ meaningfully between individuals
IntroductionWhy longitudinal?5 reasons (continued)3 = analysis of interrelations in or among changesE.g., change in commitment impact change in performance?4 = Analysis of determinants of intraindividual changeWhat causes the changes in levels, amplitude and frequency?5 = Analysis of determinants of interindividual differences in intraindividual changeWhat causes differences in the pattern
IntroductionLongitudinal Research DesignsMy webcast in 2007 focused on latent growth models where you have 3 or more repeated measuresTodays webcast focuses on using latent models with only two points in timeLatent Difference Score Models
Latent DifferenceNonlatent method 1 Subtract T1 from T2 to get a simple difference scoreIf positive spoke of gain; if negative of lossWhile widely used prior to 70s, Chronbach & Furby (1970) heavily criticized it and use dropped off dramatically.Saw a parallel phenomenon in mid-90s with congruence score and Jeff Edwards severe critique
Latent DifferenceNonlatent method 1C&F (1970) criticism based on . . .Y1n = yn + e1n and Y2n = yn + e2n where y is same unobserved true scores and e is two random errors then . . . Dyn = Y2n Y1n= (yn + e2n) (yn + e1n)= (yn yn) + (e2n e1n)= e2n e1n= etn ; that is, change is completely composed of random error
Latent DifferenceNonlatent method 2Autoregression or Type A modelRegress T2 onto T1 in Step 1 and residual regressed onto control and substantive variables in subsequent steps
Latent DifferenceNonlatent method 2Criticism 1: Means assumed to be zeroHowever, in models depicting change, sample means carry useful statistical informationInformation needed to estimate both inter- and intraindividual differences in change or growthCriticism 2: By controlling T1, model eliminates all predictors except those that predict changes in rank order of observations over time.
Latent DifferenceGroundwork for latent approachesLain by Nesselroade (1972; 1974)Dont assume exact same true scoreY1n = y1n + e1n and Y2n = y2n + e2n where y1 is unobserved true at T1 and y2 is unobserved true score at T2, then . . .Dyn = Y2n Y1n= (y2n + e2n) (y1n + e1n)= (y2n y1n) + (e2n e1n)= y + (e2n e1n)= y + etn
Latent DifferenceFramework for Remainder of SectionUse affective commitmentIllustrate two latent methods conceptuallyType DR: Regression of ChangesType : Regression Change ModelVery simple difference between themWill only use Type DR for empirical examplesEmpirical examplesObserved variable onlySingle-indicator approachMultiple-indicator approach
Latent DifferenceType DR: Regression of Changes 1 21 AC1 1 AC4 1 AC 2 24 1 1McArdle & Nesselroade (2014) notation
Latent DifferenceType : Regression Change Model 1 21 AC1 1 AC4 1 AC 2 1 0 1McArdle & Nesselroade (2014) notation
Latent DifferenceType DR vs. Type Substantively drive decisionIf you feel that T1 variable is a major factor in the change, then Type E.g. Weight loss greater for those heaviest to begin with; resource rich firms can change faster, then firms with lessIf you feel that T1 variable is not a major factor, then Type DRE.g., Most attitude research, doesnt consider T1 to be a factor in the change
Latent DifferenceObserved Variables ExampleType DR followed by Type Level 1 is there change?Averages of affective commitment at T1 and T4After this, build on the Type DRLevel 2 theory testinglatent difference in turnover intention as an outcome affective commitment diferenceorg fit as an antecedent to affective commitment change
Latent DifferenceType DR Syntax for Level 1Model:CH by AffAvgT4@1;AffAvgT4 on AffAvgT1@1;AffAvgT4@0;CH with AffAvgT1;[CH AffAvgT1 AffAvgT4@0];CH AffAvgT1;
Latent DifferenceCH by AffAvgT4@1; 1 21 AC1 1 AC4 1 AC 2 24 1 1
Latent DifferenceAffAvgT4 on AffAvgT1@1; 1 21 AC1 1 AC4 1 AC 2 24 1 1
Latent DifferenceAffAvgT4@0; 1 21 AC1 1 AC4 1 AC 2 24 1 1
Latent DifferenceCH with AffAvgT1; 1 21 AC1 1 AC4 1 AC 2 24 1 1
Latent Difference[CH AffAvgT1 AffAvgT4@0]; 1 21 AC1 1 AC4 1 AC 2 24 1 1Not shown but mean of AC4 fixed to 0
Latent DifferenceCH AffAvgT1; 1 21 AC1 1 AC4 1 AC 2 24 1 1
Latent DifferenceModel OutcomesJust Identified so perfect fitCH BY AFFAVGT4 1.000 0.000 999.000 999.000 AFFAVGT4 ON AFFAVGT1 1.000 0.000 999.000 999.000 CH WITH AFFAVGT1 1 -0.438 0.073 -5.975 0.000 Means AFFAVGT1 1 1.071 0.061 17.431 0.000 CH 2.017 0.065 31.210 0.000
Latent DifferenceModel Outcomes (cont.)
Intercepts AFFAVGT4 0.000 0.000 999.000 999.000 Variances AFFAVGT1 21 1.133 0.093 12.247 0.000 CH 2 1.254 0.102 12.247 0.000 Residual Variances AFFAVGT4 0.000 0.000 999.000 999.000
Latent DifferenceType Syntax for Level 1 Model: CH by AffAvgT4@1; AffAvgT4 on AffAvgT1@1; AffAvgT4@0; CH on AffAvgT1; [CH AffAvgT1 AffAvgT4@0]; CH AffAvgT1;
Latent DifferenceType : Regression Change Model 1 21 AC1 1 AC4 1 AC 2 1 0 1
Latent DifferencesType Outcomes CH BY AFFAVGT4 1.000 0.000 999.000 999.000 CH ON AFFAVGT1 1 -0.387 0.056 -6.845 0.000 AFFAVGT4 ON AFFAVGT1 1.000 0.000 999.000 999.000 Means AFFAVGT1