Growth Curve Modeling Approach to Moderated Mediation for Longitudinal Data

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1 Growth Curve Modeling Approach to Moderated Mediation for Longitudinal Data JeeWon Cheong Department of Health Education & Behavior University of Florida This research was supported in part by NIH grants (R01 AA A1; R01 DA09757)

2 Overview of Presentation Basic Models for Mediation and Moderated Mediation Longitudinal Mediation: Two-wave vs. Multiple-wave Mediation Growth Curve Modeling Approach to: 1) Mediation 2) Moderated Mediation

3 Mediation Model M M = b 0 + ax + e 1 Y = b 0 + cx + bm + e 2 a b X c Y Mediated (Indirect) Effect = ab Example: BMI Alcohol Demand Drinking Quantity & Problems (Murphy et al., 2015)

4 Moderation Model Z X c Y Y = b 0 + cx + b 1 Z + b 2 XZ + e 1 = (b 0 + b 1 Z) + (c + b 2 Z)X + e 1 The effect of X on Y depends on the level of the moderator Z. Example: The effect of Peer Influence on Adolescent Alcohol use was weaker for high Parental Monitoring, compared to low monitoring (Barnes et al., 2006).

5 Moderated Mediation Model

6 Moderated Mediation Model e.g., Z moderates X M path M = b 0 +b 1 X + b 2 Z + b 3 XZ + e 1 = (b 0 + b 2 Z) + (b 1 + b 3 Z)X + e 1 Y = b 0 + b 4 X + b 5 M + e 2 Effect of X on M, a path in ab Mediated Effect = (b 1 + b 3 Z)b 5 Mediated effect at high level of Z: (b 1 + b 3 Z high )b 5 Mediated effect at low level of Z: (b 1 + b 3 Z low )b 5

7 Longitudinal Mediation Using two-wave data M 1 X a M 2 b c Y 1 Y 2 M 2 = b 0 + b 1 M 1 + ax + e 1 Y 2 = b 0 + b 2 M 1 + b 3 Y 1 + cx + bm + e 2

8 Longitudinal Mediation Two waves of data may not give us a complete picture of longitudinal change. Outcome T 1 T 2 T 3 T 4 T 5 T 6

9 Longitudinal Mediation Depending on the time of wave 2 measurement, effect size may vary. Outcome T 1 T 2 T 3 T 4 T 5 T 6

10 Longitudinal Mediation Challenges with two wave data: Are the mediated effects at post-test real? Will we observe similar results at later waves? How can we include repeated measures in mediation analysis? What would significant mediation mean in such cases?

11 Latent Growth Modeling (LGM) Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y t = IS + t*sl +e t (t = 0, 1,, 5) IS IS: Initial Status SL: Slope 2 3 SL 4 5 Y 1 = IS + 0xSL +e 1 Y 2 = IS + 1xSL +e 2 Y 3 = IS + 2xSL +e 3 Y 4 = IS + 3xSL +e 4 Y 5 = IS + 4xSL +e 5 Y 6 = IS + 5xSL +e 6

12 LGM Mediation Cheong, MacKinnon, & Khoo (2003). Structural Equation Modeling, 10,

13 LGM Mediation a Trajectory M b X c Trajectory Y Mediated effect = ab a: the extent to which X affects the slope of trajectory (growth rate) of M b: the extent to which the slope of trajectory of M is related to the slope of trajectory of Y

14 Data Illustration: ATLAS Adolescents Teaching and Learning to Avoid Steroids (Goldberg, et al., 2000; MacKinnon et al., 2001) A longitudinal, randomized school-based prevention trial for high school football players in OR and WA (31 schools) Goals: Reduce adolescent athletes anabolic steroid (AS) use Enhance health behaviors alternative to AS use: Nutrition behaviors & Strength Training

15 Data Illustration: ATLAS ATLAS Program Implementation T 1 T 2 T 3 T 4 T 5 T 6 Full Program Booster Program Booster Program

16 LGM Mediation in ATLAS Outcome: Nutrition Behaviors 5 Nutrition Behaviors Control Treatment T1 T2 T3 T4 T5 T6 χ 2 = , df = 39 RMSEA =.057 [90% CI (.046,.069)] Source: Cheong, et al. (2003)

17 LGM Mediation in ATLAS Mediator: Peer as an Information Source 6 Peer as an Information Source Control 4 T1 T2 T3 T4 T5 T6 χ 2 = , df = 39 RMSEA =.067 [90% CI (.056,.078)] Source: Cheong, et al. (2003)

18 LGM Mediation in ATLAS χ 2 = , df = 72 RMSEA =.050 [90% CI (.045,.056)] Mediated Effect =.765 SE αβ =.210 z αβ = 3.643, p<.001; 95% CI (.372, 1.196) Source: Cheong, et al. (2003)

19 LGM Mediation LGM mediation approach has been applied to various studies (e.g., Dekovic et al., 2010; Kouros et al. 2010; Pantin et al., 2009; Jagers, et al, 2007; Prado et al., 2007, etc.). However, little is known about the accuracy of the estimated mediated effects and statistical power when mediation is tested in LGM framework.

20 Monte Carlo Simulation Study on LGM Mediation Cheong (2011). Structural Equation Modeling, 18,

21 Simulation Conditions Simulation Conditions Measurement occasions: 3 & 5 occasions R 2 of repeated measures (Proportion of variance explained by growth factors):.50 &.80 Effect size of mediated effect: Proportion mediated Med effect = = Total effect ab ab + c Small:.10 (a =.18; b =.16; c =.25) Medium:.30 (a =.35; b =.31; c =.25) Large:.50 (a =.52; b =.49; c =.25) Sample size: 100, 200, 500, 1000, 2000, & 5000

22 Bias of Mediated effect (ab): R 2 =.50 R 2 = Measurements Measurements Rel. Bias ab L ES M ES S ES Measurements Measurements Rel. Bias ab Sample Size Sample Size

23 Statistical Power Three methods for testing mediated effects examined in the simulation study

24 Statistical Power: 3 Measurement Occasions R 2 =.50 Empirical Power S ES (Prop Med=.10) M ES (Prop Med=.30) L ES (Prop Med=.50) R 2 =.80 Empirical Power Sample Size Sample Size Sobel Test Joint Significance Asymmetric CI Sample Size

25 Statistical Power: 5 Measurement Occasions R 2 =.50 Empirical Power S ES (Prop Med =.10) M ES (Prop Med =.30) L ES (Prop Med =.50) R 2 =.80 Empirical Power Sample Size Sample Size Sobel Test Joint Significance Asymmetric CI Sample Size

26 LGM Mediation Summary: Monte Carlo simulation study In general, growth curve modeling approach to mediation requires relatively large samples. However, other conditions may improve accuracy of estimates and statistical power: Effect size Number of measurement time points R 2 of repeated measures Methods for testing mediation

27 Longitudinal Moderated Mediation When the study design involves multilevel data structure, mediation analyses may need to take into account the dependencies among participants. e.g., students nested within schools, patients nested within therapists Variables at different levels (e.g., individual characteristics, school characteristics) may affect trajectories of M and Y. Growth curve modeling approach to mediation can be carried out in multilevel modeling framework. Moderated mediation can be investigated as cross-level interaction.

28 Longitudinal Moderated Mediation Repeated measures of individuals nested within schools Growth model in 3 levels Level 1: Within-individual Level Growth of the mediator and the outcome Level 2: Between-individual Individual characteristics that may affect the growth of the mediator and the outcome Level 3: Between-school Level School level characteristics Cheong & Khoo (in prep)

29 Longitudinal Moderated Mediation Level 2: Between Individual Level Z Trajectory M Level 1: Within Individual Level a b X Level 3: Between School Level c Trajectory Y Level 1: Within Individual Level

30 Data Illustration: MPP Midwestern Prevention Project (Pentz, et al., 1986; 1989) A randomized school-based prevention program for adolescents gateway drug use 57 schools randomly assigned to Treatment vs. Control groups Mediator: Number of friends who use drugs (FrnUse) Measured from T1 to T4 Outcome: Alcohol drinking in the past month (AlcUse) Measured from T1 to T7

31 Longitudinal Moderated Mediation Level 1 (Within-Individual): Level 2 (Between Individual): Level 3 (Between School): FrnUse tij = IS m0ij + SL m1ij *T tij + ε mtij IS m0ij = γ m00j + γ m01j *Impuls ij + υ m0ij SL m1ij = γ m10j + γ m11j *Impuls ij + υ m1ij γ m00j = c m000 + δ m00j γ m01j = c m010 + δ m01j γ m10j = c m100 + a 1 *Group j + δ m10j γ m11j = c m110 + a 2 *Group j + δ m11j Cheong & Khoo (in prep)

32 Longitudinal Moderated Mediation FrnUse tij = [c m000 + c m010 *Impuls ij + c m100 *T tij + c m110 *Implus ij *T tij + (a 1 + a 2 *Impuls ij ) *Group j *T tij ] + [(δ m10j + υ m1ij )*T tij + δ m11j *Impuls ij *T tij + υ m0ij + δ m00j + ε mtij ] Group (T/C) effect on the slope of trajectory of M (FrnUse) moderated by impulsivity Cross-level Interaction: Interaction between level-2 (Impulsivity) and level-3 (Group) characteristics

33 Estimated Growth Curve of Mediator Moderated by Impulsivity Number of Friends who use drugs Estimated Means of Number of Drug Using Friends Control Low Impuls Control High Impuls Treatment Low Impuls Treatment High Impuls T1 T2 T3 T4 Measurement Time Points Low Impulsivity: 1 sd below the mean; High Impulsivity : 1 sd above the mean

34 Longitudinal Moderated Mediation Mediated effect: Low Impulsivity = a low b High Impulsivity = a high b Low Impulsivity High Impulsivity a low =.011 (.226) a high = (.177) b =.157 (.020)*** Mediated effect Low Impulsivity: a low b =.002 (.036); 95%CI (-.069,.072) High Impulsivity: a high b = (.028); 95% CI (-.087,.024)

35 Longitudinal Moderated Mediation Summary: Illustration with MPP data The findings suggest that the MPP program may be more effective for the adolescents with higher level of impulsivity: For highly impulsive adolescents, the program showed a trend of slowing down the increased affiliation with friends who use drugs, which, in turn, was associated with less accelerated alcohol use. This trend was not clear among adolescents with lower levels of impulsivity.

36 Overall Summary When growth curve modeling is applied to mediation analysis, relatively large samples are needed to estimate the mediated effects accurately and to have reasonable statistical power. Alternatives to large samples: Increase the number of measurements Choose reliable measures and model accurate trajectory shapes Increase R 2 Use mediation tests that take into account the non-normal distribution of the mediated effects

37 Overall Summary Multilevel growth curve modeling approach to mediation analysis allows researchers: To take into account the dependency of individual participants nested within higher level units; To explicitly model the effects of predictors at different levels; and To investigate moderated mediation as cross-level interaction.

38 Thank you!