1. A Random Coefficient Model Example: Observations Nested Within Individuals Academy of Management, 2010 Montreal, Canada Jeffrey B. Vancouver Ohio University

2. Within-Person Level of Analysis Longitudinal Models Growth curve modeling Time series Repeated-measures – Multiple decisions, tasks, etc. – Where, time is often a nuisance variable (practice/fatigue effects) to be controlled – Lagged effects models 2

3. Sample Phenomena Dynamic criteria Learning Socialization Treatment/intervention evaluation Stress, attitude, turnover research Predicting slopes and intercepts Testing causal hypotheses 3

4. Training Effects Example Finding best-fitting trajectory by ignoring individual violates independence of observation assumption More importantly, – Might want to find out what determines Intercept Slope E.g., – Training (e.g., self-paced training for new Plant A employees) – Individual differences (predictor scores) 4

5. 01234567 Time (months) Individual Trajectories Performance High low 5

6. Unconditional Models Unconditional (null) Means model – L1: y = π 0 + e – L2: π 0 = β 00 + r Unconditional growth model – L1: y = π 0 + π 1 time + e – L2: π 0 = β 00 + r π 1 = β 10 + r 6 Note changes in notation (pi, e for within-person)

7. Conditional Models Conditional growth model – L1: y = π 0 + π 1 time + e – L2: π 0 = β 00 + β 01 TRAIN + β 02 COGA + r π 1 = β 10 + β 11 TRAIN + β 11 COGA + r 7 Dummy code for in training group (1) or not (0) cognitive ability score

8. The outcome variable is PERFORM The model specified for the fixed effects was: ---------------------------------------------------- Level-1 Level-2 Coefficients Predictors ---------------------- --------------- INTRCPT1, P0 INTRCPT2, B00 TRAIN, B01 COGA, B02 TIME slope, P1 INTRCPT2, B10 TRAIN, B11 COGA, B12 Summary of the model specified (in equation format) --------------------------------------------------- Level-1 Model Y = P0 + P1*(TIME) + E Level-2 Model P0 = B00 + B01*(TRAIN) + B02*(COGA) + R0 P1 = B10 + B11*(TRAIN) + B12*(COGA) + R1 HLM Output 8

9. Sigma_squared = 0.25585 Tau INTRCPT1,P0 0.21291 0.00825 TIME,P1 0.00825 0.00369 Tau (as correlations) INTRCPT1,P0 1.000 0.295 TIME,P1 0.295 1.000 ---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, P0 0.932 TIME, P1 0.352 ---------------------------------------------------- Variance/Covariances/ICC(2) 9

10. The outcome variable is PERFORM Final estimation of fixed effects (with robust standard errors) ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, P0 INTRCPT2, B00 4.252554 0.056756 74.926 129 0.000 TRAIN, B01 -0.684429 0.080463 -8.506 129 0.000 COGA, B02 0.019250 0.116900 0.165 129 0.870 For TIME slope, P1 INTRCPT2, B10 0.191658 0.012714 15.075 129 0.000 TRAIN, B11 0.045514 0.019159 2.332 129 0.005 COGA, B12 0.038086 0.021712 1.754 129 0.081 ---------------------------------------------------------------------------- Fixed Effects Output 10

11. The outcome variable is PERFORM Final estimation of fixed effects (with robust standard errors) ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, P0 INTRCPT2, B00 4.252554 0.056756 74.926 129 0.000 TRAIN, B01 0.264429 0.090463 3.506 129 0.000 COGA, B02 0.019250 0.116900 0.165 129 0.870 For TIME slope, P1 INTRCPT2, B10 0.191658 0.012714 15.075 129 0.000 TRAIN, B11 0.025514 0.019159 1.332 129 0.185 COGA, B12 0.038086 0.021712 1.754 129 0.081 ---------------------------------------------------------------------------- Fixed Effects Output (re “centered” time) 11

12. Interpretations a Training group – Groups were not equivalent on performance initially – Training effect made up for initial deficit (-.68) and then some (+.26). Cognitive ability (g) – Not much of a player; only marginally predicted growth in performance over time – Might be useful to test interaction of training condition and g to see if g led to faster improvement for training group 12 a Caveat: this is a made up example

13. Final estimation of variance components: ----------------------------------------------------------------------------- Random Effect Standard Variance df Chi-square P-value Deviation Component ----------------------------------------------------------------------------- INTRCPT1, R0 0.46142 0.21291 129 4451.18764 0.000 TIME slope, R1 0.06074 0.00369 129 220.46760 0.000 level-1, E 0.50582 0.25585 ----------------------------------------------------------------------------- Statistics for current covariance components model -------------------------------------------------- Deviance = 4758.579866 Number of estimated parameters = 4 Random Effects Output 13

14. Time Series (internally valid designs) Interrupted: – O O O O O X O O O O O Control series – O O O O X O O O O O O O O O O – O O O O O O O O O O O X O O O Non-equivalent dependent variables – O A O A O A O A X O A O A O A – O B O B O B O B X O B O B O B Removal – O O O X O O O O X O O O Switching Replications (bringing waitlisted on-line) 14

15. Examining Effect of Interventions/Events Using HLM y = π 0 + π 1 t + π 2 d + π 3 td + e y = outcome t = time d = dummy t = 0 1 2 3 4 5 6 7 8 9... k d = 0 0 0 0 0 1 1 1 1 1 1 1 1 d = 0 0 0 0 0 1 2 3 4 5 6 7 8 z = 1 if in treatment/exposed to event 0 if control/not exposed Level 1: occasion (within person) Level 2: individual π 0 = β 00 + β 01 z + u π 1 = β 10 + β 11 z + u π 2 = β 20 + β 21 z + u π 3 = β 30 + β 31 z + u t-k =......... -5 -4 -3 -2 -1 0 15

16. Other issues Nonlinear effects – Growth CURVE implies time effects are not linear Adding polynomials (e.g., time 2 ; time 3 ) provides simple, preliminary (and often final) test for curves – Singer & Willett (2003) Applied Longitudinal Data Analysis Graph results – Plug variable values into equations, minding centering decisions – Helpful to self and audience Dichotomous DVs (e.g., choice) – Bernoulli distribution available in HLM (output includes odds ratios) – E.g., in Vancouver, More, & Yoder (2008) JAP Examining relationships among time-varying variables using lagged RCM, manipulations – E.g., Vancouver, Thompson, Tischner, & Putka (2002) JAP 16

17. Q & A Dave Hofmann Mark Gavin Jeff Vancouver 17

18. Graph Results 18

19. Predicting Effects of Time-Varying Passively Observed Variables If X and Y are measured simultaneously, reciprocal and 3 rd variable effects confound interpretations Lagged RCM can test reciprocal issue: – provided lags are properly specified – provided trend effects controlled (only issue if reciprocity) Control for time Control for y (t-1) – 3 rd variable problem still possible 19

20. Y tA = π A X (t - 1)A Y tj = π j X (t – 1)j : Lagged RCM Y tB = π B X (t - 1)B Individual A B : n 1234 xxxx yyy xxxx yyy :::: xxxx yyy Time Y ij = π 0j + π 1j (X ij ) + r ij Or more generally: 20