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

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

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

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

Time (months) Individual Trajectories Performance High low 5

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)

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

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

Sigma_squared = Tau INTRCPT1,P TIME,P Tau (as correlations) INTRCPT1,P TIME,P Random level-1 coefficient Reliability estimate INTRCPT1, P TIME, P Variance/Covariances/ICC(2) 9

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, B TRAIN, B COGA, B For TIME slope, P1 INTRCPT2, B TRAIN, B COGA, B Fixed Effects Output 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, B TRAIN, B COGA, B For TIME slope, P1 INTRCPT2, B TRAIN, B COGA, B Fixed Effects Output (re “centered” time) 11

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

Final estimation of variance components: Random Effect Standard Variance df Chi-square P-value Deviation Component INTRCPT1, R TIME slope, R level-1, E Statistics for current covariance components model Deviance = Number of estimated parameters = 4 Random Effects Output 13

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

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 = k d = d = 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 =

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

Q & A Dave Hofmann Mark Gavin Jeff Vancouver 17

Graph Results 18

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

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