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