Post on 12-Jan-2016
LATENT DIFFERENCE SCORE MODELS
Bob Vandenberg
Terry College of BusinessDepartment of Management
University of Georgia
Overview• Framework for workshop
– Introduction• Change
• Why longitudinal?
• “Other” longitudinal statistical procedures and shortcomings
– Latent difference: Only a T1 and T2 measure• Nonlatent methods and shortcoming
• Latent methods
– Type DR and Type Δ models
– Empirical examples
» Observed variable only
» Single-Indicator Approach
» Multiple-Indicator Approach
My Primary Sources
• McArdle, 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
Assumptions
• Need an understanding of CFA and structural models– Extension of basic models
• Used Mplus 7.x (Muthén & Muthén, 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
Introduction
• Change– Why is it so important?
• #1 = because the theoretical frameworks of interest to us explicitly or implicitly imply it
– Underlying 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 Y
» Reality is we neither operationalize change nor use a design that permits true causal inferences
– Concern here is with “change,” not cause
Introduction
• Complexities in Changes Over Time– Multiple Levels
• e.g., changes in newcomer information seeking over time.
– Cross-Levels• changes in P-G fit, P-O fit over time.
– Multivariate• relationships linking changes in different focal
variables.
Introduction
• Research Designs– Cross-sectional research design
• Collects data on a variable at one time point.
– Longitudinal research design• Collects 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.
Introduction
• Why longitudinal?– McArdle & Nesselroade (2014) give 5
reasons• 1 = direct identification of intraindividual
change– Comes in many forms such as level, frequency and
amplitude
• 2 = direct identification of interindividual differences in intraindividual change
– Change patterns differ meaningfully between individuals
Introduction• Why longitudinal?
– 5 reasons (continued)• 3 = analysis of interrelations in or among
changes– E.g., change in commitment impact change in
performance?
• 4 = Analysis of determinants of intraindividual change
– What “causes” the changes in levels, amplitude and frequency?
• 5 = Analysis of determinants of interindividual differences in intraindividual change
– What causes differences in the pattern
Introduction• Longitudinal Research Designs
– My webcast in 2007 focused on latent growth models where you have 3 or more repeated measures
– Today’s webcast focuses on using latent models with only two points in time
• Latent Difference Score Models
Latent Difference
• Nonlatent method 1 – Subtract T1 from T2 to get a simple
difference score• If positive spoke of gain; if negative of loss• While 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 Difference• Nonlatent method 1
– C&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
= Δetn ; that is, change is completely composed of random error
Latent Difference
• Nonlatent method 2– Autoregression or Type A model
• Regress T2 onto T1 in Step 1 and residual regressed onto control and substantive variables in subsequent steps
Latent Difference
• Nonlatent method 2– Criticism 1: Means assumed to be zero
• However, in models depicting change, sample means carry useful statistical information
– Information needed to estimate both inter- and intraindividual differences in change or growth
– Criticism 2: By controlling T1, model eliminates all predictors except those that predict changes in rank order of observations over time.
Latent Difference• Groundwork for latent approaches
– Lain by Nesselroade (1972; 1974)• Don’t assume exact same true score
• Y1n = 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)
= Δy + (e2n – e1n)
= Δy + Δetn
Latent Difference• Framework for Remainder of Section
– Use affective commitment– Illustrate two latent methods conceptually
• Type DR: Regression of Changes• Type Δ: Regression Change Model• Very simple difference between them
– Will only use Type DR for empirical examples
– Empirical examples• Observed variable only• Single-indicator approach• Multiple-indicator approach
Latent Difference• Type DR: Regression of Changes
σ1Δ
σ21 AC1 1 AC4 1 ΔAC σ2
Δ
σ24
μ1 μΔ
1
McArdle & Nesselroade (2014) notation
Latent Difference• Type Δ: Regression Change Model
α1
σ21 AC1 1 AC4 1 ΔAC ζ2
μ1 α0
1
McArdle & Nesselroade (2014) notation
Latent Difference
• Type DR vs. Type Δ– Substantively drive decision
• If you feel that T1 variable is a major factor in the change, then Type Δ
– E.g. Weight loss greater for those heaviest to begin with; resource rich firms can change faster, then firms with less
• If you feel that T1 variable is not a major factor, then Type DR
– E.g., Most attitude research, doesn’t consider T1 to be a factor in the change
Latent Difference• Observed Variables Example
– Type DR followed by Type Δ• Level 1 – is there change?
– Averages of affective commitment at T1 and T4
– After this, build on the Type DR• Level 2 – theory testing• latent difference in turnover intention as an
outcome affective commitment diference• org fit as an antecedent to affective
commitment change
Latent Difference
• Type DR Syntax for Level 1– Model:– CH by AffAvgT4@1;– AffAvgT4 on AffAvgT1@1;– AffAvgT4@0;– CH with AffAvgT1;– [CH AffAvgT1 AffAvgT4@0];– CH AffAvgT1;
Latent Difference– CH by AffAvgT4@1;
σ1Δ
σ21 AC1 1 AC4 1 ΔAC σ2
Δ
σ24
μ1 μΔ
1
Latent Difference– AffAvgT4 on AffAvgT1@1;
σ1Δ
σ21 AC1 1 AC4 1 ΔAC σ2
Δ
σ24
μ1 μΔ
1
Latent Difference– AffAvgT4@0;
σ1Δ
σ21 AC1 1 AC4 1 ΔAC σ2
Δ
σ24
μ1 μΔ
1
Latent Difference– CH with AffAvgT1;
σ1Δ
σ21 AC1 1 AC4 1 ΔAC σ2
Δ
σ24
μ1 μΔ
1
Latent Difference– [CH AffAvgT1 AffAvgT4@0];
σ1Δ
σ21 AC1 1 AC4 1 ΔAC σ2
Δ
σ24
μ1 μΔ
1
Not shown but mean of AC4 fixed to 0
Latent Difference– CH AffAvgT1;
σ1Δ
σ21 AC1 1 AC4 1 ΔAC σ2
Δ
σ24
μ1 μΔ
1
Latent Difference
• Model Outcomes– Just Identified so perfect fit
• CH BY• AFFAVGT4 1.000 0.000 999.000 999.000• • AFFAVGT4 ON• AFFAVGT1 1.000 0.000 999.000 999.000• • CH WITH
• AFFAVGT1 σ1Δ -0.438 0.073 -5.975 0.000
• • Means
• AFFAVGT1 μ1 1.071 0.061 17.431 0.000
• CH μΔ 2.017 0.065 31.210 0.000
Latent Difference
• Model Outcomes (cont.)
• Intercepts• AFFAVGT4 0.000 0.000 999.000 999.000• • Variances
• AFFAVGT1 σ21 1.133 0.093 12.247 0.000
• CH σ2Δ 1.254 0.102 12.247 0.000
• • Residual Variances• AFFAVGT4 0.000 0.000 999.000 999.000
Latent Difference
• Type Δ 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 Difference• Type Δ: Regression Change Model
α1
σ21 AC1 1 AC4 1 ΔAC ζ2
μ1 α0
1
Latent Differences
• Type Δ Outcomes • CH BY
• AFFAVGT4 1.000 0.000 999.000 999.000
•
• CH ON
• AFFAVGT1 α1 -0.387 0.056 -6.845 0.000
•
• AFFAVGT4 ON
• AFFAVGT1 1.000 0.000 999.000 999.000
•
• Means
• AFFAVGT1 μ1 1.071 0.061 17.430 0.000
•
Latent Differences
• Type Δ Outcomes (cont.)• Intercepts
• AFFAVGT4 0.000 0.000 999.000 999.000
• CH α0 2.432 0.085 28.511 0.000
•
• Variances
• AFFAVGT1 σ21 1.133 0.093 12.247 0.000
•
• Residual Variances
• AFFAVGT4 0.000 0.000 999.000 999.000
• CH ζ2 1.084 0.089 12.249 0.000
Latent Difference
• Level 2 or Conditional Model – Type DR
Aff – T1
σ2iaf
f
ΔAC ΔTI
σ2iturn
TI – T1
saff sturn siaf
f
siturn
βsta
Org. Fit
βAF
Latent Difference• Level 2 Model Syntax
– Model:
– CHAff by AffAvgT4@1;
– AffAvgT4 on AffAvgT1@1;
– AffAvgT4@0;
– CHAff with AffAvgT1;
– [CHAff AffAvgT1 AffAvgT4@0];
– CHAff AffAvgT1;
–
– CHTI by TrnAvgT4@1;
– TrnAvgT4 on TrnAvgT1@1;
– TrnAvgT4@0;
– CHTI with TrnAvgT1;
– [CHTI TrnAvgT1 TrnAvgT4@0];
– CHTI TrnAvgT1;
Latent Difference• Level 2 Model Syntax (cont.)
– Next two lines are the conditional hypotheses
– CHAff on AvgFit;– CHTI on CHAff;
Latent Difference
• Level 2 or Conditional Model – Type DR
Aff – T1
σ2iaf
f
ΔAC ΔTI
σ2iturn
TI – T1
saff sturn siaf
f
siturn
βsta
Org. Fit
βAF
Latent Difference• Level 2 Outcomes
– Model fit – now appropriate• Chi-Square Test of Model Fit
• Value 126.151
• Degrees of Freedom 10
• P-Value 0.0000
• RMSEA (Root Mean Square Error Of Approximation)
• Estimate 0.197
• 90 Percent C.I. 0.167 0.228
• Probability RMSEA <= .05 0.000
• CFI/TLI
• CFI 0.574
• TLI 0.404
• PRETEND MODEL FIT WAS ABSOLUTELY GREAT
Latent Difference
• Level 2 Outcomes– CHTI ON
CHAFF -0.163 0.054 -3.013 0.003•
– CHAFF ON AVGFIT 0.309 0.083 3.703 0.000
Latent Difference
• Revisit 5 reasons for longitudinal– 1 = Is there direct identification of
intraindividual change– Yes
– 2 = Is there direct identification of interindividual differences in intraindividual change
– Yes
Latent Difference
• 5 Reasons (continued)– 3 = Are there interrelations in or among changes
– Yes commitment clearly linked to TI– 4 = Are there determinants of intraindividual change
– Yes for No group; No for Yes group; Classic interaction
– 5 = Are there determinants of interindividual differences in intraindividual change
– Yes, because links between commitment and turnover intention directly consistent with conceptual expectations
Latent Difference• My skeptical musings
– Basically “tricking” the program• By incorporating the parameterization as
specified, basically “tricking” program into creating a so-called latent change variable from the observed scores (e.g., fixing variance to zero, mean to zero, etc.).
– Why not actually use a true latent variable?
• Next two methods, do this.• Still based on DR model, but easily can
become delta model
Latent Difference
si
Aff –T1)
λ1= .964
Aff –T4)
λ4= .969
1 =0 4 =0
τ1 = 0 τ4 = 0
Latent Difference
• Empirical examples; Single Indicator Approach– Using the averages of affective
commitment at T1 and T4 in this particular database
Latent Difference
• Single Indicator Syntax• Model:
• AffT1 by AffAvgT1@.964; !Square root of composite reliability index• AffAvgT1@.08; !(1- CRI) * Varianc of AffAvgT1• AffT4 by AffAvgT4@.969;• AffAvgT4@.071;• [AffAvgT1@0 AffAvgT4@0];• Base Change | AffT1@0 AffT4@1;• AffT4@0;• AffT1@0;• Base Change;• [Base Change];• Base with Change;
Latent Difference BASE WITH CHANGE -0.389 0.079 -4.942 0.000 Means BASE 1.111 0.064 17.430 0.000 CHANGE 2.076 0.067 31.067 0.000
σ2iaf
f
ChangeAC ChangeTI
σ2iturn
saff sturn siaf
f
siturn
βsta
Org. Fit
βAF
Latent Difference
Base Aff Base TI
Latent Difference
• Level 2 Syntax• Model:
• AffT1 by AffAvgT1@.964; !Square root of composite reliability index• AffAvgT1@.08; !(1- CRI) * Varianc of AffAvgT1• AffT4 by AffAvgT4@.969;• AffAvgT4@.071;• [AffAvgT1@0 AffAvgT4@0];• Base Change | AffT1@0 AffT4@1;• AffT4@0;• AffT1@0;• Base Change;• [Base Change];• Base with Change;
Latent Difference• Single Indicator Syntax• [AffAvgT1@0 AffAvgT4@0];• [TrnAvgT1@0 TrnAvgT4@0];• [OrgFit@0];• BaseAf ChangeAf | AffT1@0 AffT4@1;• BaseTI ChangeTI | TrnT1@0 TrnT4@1;• AffT1@0 AffT4@0;• TrnT1@0 TrnT4@0;• BaseAf ChangeAF BaseTI ChangeTI;• [BaseAF ChangeAF BaseTI ChangeTI OrgFitLV];• BaseAF with ChangeAf;• BaseTI with ChangeTI;• BaseAF with OrgFitLV@0;• BaseTI with OrgFitLV@0;• ChangeTI on ChangeAF (p1);• ChangeAF on OrgfitLV (p2);
Latent Difference• Single Indicator Syntax• ANALYSIS: Type = General;• Bootstrap = 1000;• Model Constraint:• • New (p2p1);• p2p1 = p2*p1;• • OUTPUT: SAMPSTAT STAND(STDYX) RESIDUAL CINTERVAL
(BCBOOTSTRAP) TECH1;
Latent Difference
• Select Output– CHANGETI ON
• CHANGEAF -0.190 0.059 -3.231 0.001
• CHANGEAF ON
• ORGFITLV 0.347 0.102 3.397 0.001
• New/Additional Parameters
• P2P1 -0.066 0.028 -2.356 0.018
• The estimates (i.e., -.19, etc.) are the center values from the bootstrap distribution created on each one.
σ2iaf
f
ChangeAC ChangeTI
σ2iturn
saff sturn siaf
f
siturn
βsta
Org. Fit
βAF
Latent Difference
Base Aff Base TI
The -.066 is the product of these 2 paths.
Latent Difference
si
1 =0 4 =0
y21 y61
Aff –T1)
y11 y14 y24 y64
14 24 64
Aff –T4)
11 21 61
All tau values (item intercepts) and latent means of T1 and T2 set to 0
Latent Difference
• Multiple Indicator Approach– Advantages over other approaches
• Measurement model in truest sense– Better accounting for of measurement error
• Metric (factor loading) invariance between groups
– McArdle & Nesselroade (2014) stated that stability of factor loadings necessary to make meaningful comparisons and to accurately access change
– Invariance tested before creating model and constraints carried into the model
• Can account for autoregressiveness
Latent Difference• Selected Syntax - Invariance• Model:• • AffT1 by aff1t1@1• aff2t1 (1)• aff3t1 (2)• aff4t1(3)• aff5t1(4)• aff6t1 (5);• • AffT4 by aff1t4@1• aff2t4 (1)• aff3t4 (2)• aff4t4(3)• aff5t4 (4)• aff6t4 (5);
Latent Difference• Selected Syntax - Autoregressive• aff1t1 with aff1t4;• aff2t1 with aff2t4;• aff3t1 with aff3t4;• aff4t1 with aff4t4;• aff5t1 with aff5t4;• aff6t1 with aff6t4
Latent Difference• Selected Syntax – Latent Variables• Base Change | AffT1@0 AffT4@1;• AffT4@0;• AffT1@0;• Base;• Change;• [Base Change];• Base with Change;
Latent Difference
si
1 =0 4 =0
y21 y61
Aff –T1)
y11 y14 y24 y64
14 24 64
Aff –T4)
11 21 61
All tau values (item intercepts) and latent means of T1 and T2 set to 0
Latent Difference• Selected Output – Model Fit• Chi-Square Test of Model Fit• • Value 119.706• Degrees of Freedom 124• P-Value 0.5923 • RMSEA (Root Mean Square Error Of Approximation)• • Estimate 0.000• 90 Percent C.I. 0.000 0.026• Probability RMSEA <= .05 1.000• • CFI/TLI• • CFI 1.000• TLI 1.001
Latent Difference• Selected Output – Latent Variables• BASE WITH• CHANGE -0.348 0.074 -4.718 0.000
• Means• BASE 1.061 0.062 17.121 0.000• CHANGE 1.990 0.073 27.323 0.000
• Variances• BASE 1.031 0.094 11.015 0.000• CHANGE 1.058 0.101 10.444 0.000
Latent Difference
• Level 2 Model– As with other examples above ran a model
with OrgFit as Antecdent to Change in Affective Commit. And the latter as antecedent to Change in turnover intention.
• Model Fit well and as in the others all paths significant
• Tested mediation– P2P1 -0.076 0.031 -2.441 0.015
Latent Difference
• The End
• Thank you for listening