Misleading Results from Combining Residualized and Simple Gain Scores in Longitudinal Analyses

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 • b11controls for Y1, thus b11 is same as forresidualized gain, NOT a distinct result • b2 +1 = b21in 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[1] Y[t-1] Y[0] Y[t] ry0xs 1 ry0x0 rx0ys X[0] X[1] X[t-1] X[t]

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

10. Bidirectional Dual Change Model Y[1] Y[t-1] Y[0] Y[t] ry0xs 1 ry0x0 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’sin BDCM • Initial results: Positive parental interactions predicted increasing aggression & hyperactivity (e.g., gx= .23**) • But r(y0*,xsl*) = -.30** • Note: Cross-lagged gxis + only when y0 has -r with xslover 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 fixed @ 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 .16*** .04 .08** 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

21. r’s &b’s for Smith et al. (W1&2) • ytis 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

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) Change S Tx R 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