1. Misleading Results from Combining Residualized and Simple Gain Scores in Longitudinal Analyses
Robert E. Larzelere Isaac J. Washburn Mwarumba Mwavita Ronald B. Cox, Jr.
Taren M. Swindle
Oklahoma State U. & U. of Arkansas for Medical Science
2016 Modern Modeling Methods Conference

2. Combining Predictors of Two Types of Gain: Outline
Combining in same regression equation
Combining in complex longitudinal model
Bidirectional latent change model
Hybrid of Autoregressive latent trajectory model
Misleading results can occur in predicting one type of change controlling for other type of change

3. Predicting Change: Central Goal of Developmental Science
2 types of gain scores
Simple gain: y2 - y1
Residualized gain: y2 | y1
Inconsistent results
Lord’s (1967) paradox
Larzelere, Ferrer, et al. (2010)
Why not combine both?
In same regression equation
In more complex longitudinal model

4. Lord’s Paradox
160
Men
Wave-2 Weight
130
Women
130
160
Wave-1 Weight

5. History Residualized gains, not simple gains (d)
Cronbach & Furby (1970)
Reliability of d < Y1 or Y2
Predicting simple gain makes a comeback
Rogosa, Willett, Allison, Johnson, et al.
Latent growth & multilevel models

6. Combine in the Same Regression Equation?
Y2 – Y1 = b00 + b10X1 + e (simple gain)
Y2 – Y1 = b01+ b11X1 + b2Y1 + e (control for Y1 to add residualized gain)
[Adding Y1 to both sides:]
Y2 = b01 + b11X1 + (b2 +1)Y1 + e
b11 controls for Y1, thus b11 is same as for residualized gain, NOT a distinct result
b2 +1 = b21 in standard residualized gain equation

7. Estimate Both Types of Gain in Same Model?
E.g. #1: Bidirectional dual change score model: McArdle & Hamagame (2001)
E.g. #2: Hybrid of autoregressive latent trajectory model: Bollen & Curran (2004), hybrid: Smith, Dishion et al. (2014)
Conclusion: Interpretations on b for one gain type must account for other gain type

8. Bidirectional Dual Change Model: Residualized Change
Y[0]
Y[1]
Y[t-1]
Y[t]
ry0xs
ry0x0 1
rx0ys
X[0]
X[1]
X[t-1]
X[t]

9. Model Equivalent to BDCM
YY[0]
Y[1]
Y[t-1]
Y[t]
by* * * *
X[0]
X[1]
X[t-1]
X[t]

10. Bidirectional Dual Change Model
Y[0]
Y[1]
Y[t-1]
Y[t]
ry0xs
ry0x0 1
rx0ys
X[0]
X[1]
X[t-1]
X[t]

11. Method
Testing bidirectional associations between parenting variables and child outcomes
1st 4 waves of NLSCY
Ages 2-3, 4-5, 6-7, 8-9
N = 1456: complete data, except <20% scale items
Parenting variables (NLSCY):
positive interaction, consistency
Child outcomes (NLSCY; proxies at Wave 1):
Aggression, hyperactivity

12. Counter-Intuitive & Unstable Cross-Lagged g’s in BDCM
Initial results: Positive parental interactions predicted increasing aggression & hyperactivity (e.g., gx = .23**)
But r(y0*,xsl*) = -.30**
Note: Cross-lagged gx is + only when y0 has -r with xsl over all 4 waves
Improving model fit and minimizing irrelevant aspects often reversed signs
Unstable due to adjusting for each other?

13. Fixing growth r’s to 0: Reduces puzzling results
Positive parenting & aggression: Table 1
Cross-lagged effect can reverse sign
gx = -.24*** instead of +.23** (PosInter  Aggr)
gy = -.02* instead of +.09*** (Aggr  PosInter)
Parental consistency then matched literature
gx = -.25*** instead of +.11 (Consistency Aggr)
gy = .01 instead of .00 (Aggr  Consistency)

14. Fixing growth r’s to 0: Reduces puzzling results & helps power
Positive parenting & hyperactivity: Table 2
One lagged effect shrunk to p < .10
gx = .04^ instead of +.24*** (PosInter  Hyper)
gy = -.05*** instead of -.13 (Hyper PosInter)
Parental consistency (one g lost; one g found)
gx = -.03 instead of -.28** (Consistency Hyper)
gy = -.08* instead of +.09 (Hyper Consistency)
Note: More statistical power when not juggling two change scores

15. Cross-lagged g’s unstable across improved fit attempts
Modification indices can improve fit, but g’s change dramatically (Table 2 only)
Freeing e4 reverses signs of 3 of 4 g’s
gx = -.20^ instead of +.24*** (PosInter  Hyper)
gy = -.18 instead of -.13 (Hyper PosInter)
(Parental consistency )
gx = +.54* instead of -.28**(Consistency Hyper)
gy = -.10 instead of +.09 (Hyper Consistency)
Effect of PosInter now reduces hyperactivity, but Consistency now increases it

16. Conclusions about BDCM
Predicts residualized latent gain scores, not simple latent gain scores
Juggling two gain scores creates problems
Counter-intuitive results in +gx
Positive interact  Aggression or Hyperactivit y
Growth parameter then opposite r(y0*,xsl*) < 0
Becomes n.s. or reverses size, when r 0
Cross-lagged g ‘s unstable across minor model changes

17. #2: Hybrid Autoregressive Latent Trajectory Model
It also combines simple and residualized gain scores in model

18. Hybrid ALT (Smith et al., ’14)
Coercive
Interaction
Coercive
Interaction
Coercive
Interaction
Coercive
Interaction
.08**
.04
.16***
Non-
compliance
Non-
compliance
Non-
compliance
Non-
compliance
Intercept
Slope

19. Robustness of Cross-Lagged Path from Wave 1 to 2
Robustness needed in developmental, like in econometrics -- Duncan et al. (2014)
Robustness across simple & residualized gain scores – Larzelere et al. (submitted)
Contrasting biases for simple vs. residualized gain scores
Residualized biased against corrective action
Simple gains biased for corrective actions

20. Mean r’s & b’s for Antisocial
From Larzelere, Ferrer et al. (2010
Residualized-gain b’s always closer to W1 differences [r (y1,x)]
Most simple-gain b’s have opposite sign
Hostile-ineff
.49
.35
.56
.09
-.15
Disc tactics
.27
.20
.05
-.07
Tx & Ritalin
.11
.13
.07
.02

21. r’s & b’s for Smith et al. (W1&2)
yt is Noncomply at W1 & W2
Both estimated bs negative, vs. .08**
Only other predictor of Noncomply at W2:
Intercept and slope of Noncomply over waves
Thus discrepancy due to cross-lagged b controlling for growth curve slope
Coercive Interactions
.19
.01
.23
-.03
-.15

22. b predicting change + only when controlling for growth b
Growth curve across all 4 waves:
b = -.49* for Coercive predicting slope of Noncomply over all 4 waves
Cross-lagged across waves
b = -.04 (n.s) predicting W2 Noncomply from W1 Coercive
Published b = .08** only because its b is less negative than growth slope prediction
Other cross-lagged bs not so far off

23. Conclusion from Hybrid ALT
Misleading predictors of change can occur controlling for predictor of other change
Having 2 change predictions violates a causal assumption: not to control for an other effect
Checking complex analyses with simpler analyses for robustness can detect this
Check cross-lagged b using 3 r’s
b(y2-y1)x = ry2x – ry2
Can test intermediate complexity from R matrix

24. General Conclusions
Combining predictors of 2 types of change can yield misleading conclusions
Combining in same equation yields predictor of residualized change
Combining in complex longitudinal models
Yield each effect controlling for the other effect
b may be + or – only when controlling for other type of change
Coefficients can be less stable than single gain predictor

25. Thank You Funding by NICHD grant R03 HD044679
Endowed Parenting Professorship at Oklahoma State University

26. Lord’s Paradox
160
Men
Wave-2 Weight
130
Women
130
160
Wave-1 Weight

27. Differing Conclusions
Simple change: (solid line)
No mean change for either sex
Thus no sex differences in change
Residualized change (dashed lines)
For any W1 weight, predicted W2 weight has men > women
Bias in direction of pre-existing differences
“Under-adjustment bias” – Campbell (1975)
“Residual confounding” -- epidemiologists

28. Tx
Wave-2 Antisocial
Comparison
Wave-1 Antisocial

29. Counterfactuals: Simple (S) & Residualized (R) ChangeTx
Wave-2 Antisocial M
Control
Wave-1 Antisocial

30. Counterfactuals for 3 Analyses

31. Counterfactuals for Two Types of Change and for r (if bX = 0)
Simple change: Y2 = 0X + Y1
Counterfactual = no change
Residualized change: Y2 = 0X + b1Y1
Counterfactual = regression toward grand mean
Longitudinal correlation Y2 = 0X
Counterfactual = subsequent grand mean