A GENTLE INTRODUCTION TO DYNAMIC STRUCTURAL EQUATION MODELING

A GENTLE INTRODUCTION TO DYNAMIC STRUCTURAL EQUATION MODELING (DSEM) FOR INTENSIVE LONGITUDINAL DATA ANGELA B. BRADFORD SCHOOL OF FAMILY LIFE

DSEM • USED TO ANALYZE INTENSIVE LONGITUDINAL DATA (ILD) • COMBINES MULTILEVEL MODELING, TIME SERIES MODELING, STRUCTURAL EQUATION MODELING, AND TIME VARYING EFFECTS MODELING • GOAL IS TO PARSE OUT AND MODEL THESE TYPES OF CORRELATIONS, GIVING A FULLER PICTURE OF THE DYNAMICS OF ILD • UNLIKE CROSS-CLASSIFIED MODELING (I. E. , LONG FORMAT GROWTH MODEL), IT ALLOWS YOU TO REGRESS A VARIABLE ON: • OTHER VARIABLES AT THE SAME TIMEPOINT • ITSELF AT A PREVIOUS TIMEPOINT • OTHER VARIABLES AT PREVIOUS TIMEPOINT Asparouhov, T. , Hamaker, E. L. & Muthén, B. (2017). Dynamic structural equation models. Technical Report. Version 3. Submitted for publication. Retrieved from https: //www. statmodel. com/Time. Series. shtml.

DSEM (CONT. ) • MODELS INTRA-INDIVIDUAL CHANGES OVER TIME ON LEVEL 1 AND ALLOWS THE PARAMETERS OF THESE CHANGES TO VARY ACROSS INDIVIDUALS ON LEVEL 2 USING RANDOM EFFECTS • SAMPLES WITH MANY SUBJECTS (E. G. , 200) AND FEW TIME POINTS (E. G. , 20 -50) PERFORM BETTER THAN THOSE WITH FEW SUBJECTS AND MANY TIME POINTS (IN TERMS OF BIASED SE AND POWER)

LAG • CREATE A LAG VARIABLE Observation Variable (T) Lagged Variable (T 1) 1 2. 68 2 2. 90 2. 68 3 3. 12 2. 90 4 3. 11 3. 12 5 3. 67 3. 11 6 3. 84 3. 67 • CORRELATION BETWEEN THESE IS THE AUTOCORRELATION (REGRESS T ON T 1)

BETWEEN VS. WITHIN • BETWEEN PART: INDIVIDUALS’ MEAN OVER TIME ON A VARIABLE (I. E. , BASELINE); IN OTHER WORDS, THE OVER-TIME MEAN FOR EACH PERSON • WITHIN PART: AN ESTIMATE OF INDIVIDUALS’ SCORE OVER TIME; WITHINPERSON CENTERED OR CLUSTER-MEAN CENTERED SCORE (DEVIATION FROM THE MEAN OVER TIME)

WITHIN- AND BETWEEN- PERSON PARTS • WITHIN-PERSON/CLUSTER PART: • MODELED WITH FIRST-ORDER AUTOREGRESSIVE MODEL • T REGRESSED ON T-1 • PHI IS THE ESTIMATED PARAMETER, RANGING BETWEEN 0 AND 1, AND IS CALLED “INERTIA. ” THE CLOSER PHI IS TO 1, THE LONGER IT TAKES TO RECOVER FROM A CHANGE FROM THE INDIVIDUAL’S MEAN. • BETWEEN-PERSON/CLUSTER PART: • AVERAGE ACROSS ALL INDIVIDUALS • YOU HAVE THE MEAN AND THE PHI, MODELED AS A RANDOM EFFECTS

EXAMPLE • INTERPERSONAL NEUROBIOLOGY SUGGESTS THAT ONE’S PHYSIOLOGY “CATCHES” • THERAPISTS SHOULD HAVE THE MOST REGULATORY INFLUENCE IN THE ROOM, THEREBY HELPING CLIENTS REGULATE • RESEARCH ON EMPATHY IN THE MEDICAL PROFESSION SUGGEST THAT THE MOST EMPATHETIC PHYSICIANS HAVE PHYSIOLOGY THAT SYNCHRONIZES WITH THEIR CLIENTS • SO, DOES THERAPIST AND CLIENT LAGGED PHYSIOLOGY PREDICT THERAPIST AND CLIENT PHYSIOLOGY?

T physio (t-1) T physio (t) W physio (t-1) W physio (t) H physio (t-1) H physio (t)

MPLUS INPUT (FOR MULTIVARIATE MODEL) These are the autocorrelations of t regressed on t-1.

These are the cross-lagged effects. These are correlations of variables at time t-1. These are the random slopes modeled above with the | symbol, correlated with the random mean at time t.

INTRACLASS CORRELATION • (TYPE= TWOLEVEL BASIC), CALCULATED AS: • BETWEEN VARIANCE/(BETWEEN+WITHIN VARIANCE) • HOW MUCH VARIABILITY IS BETWEEN CLUSTERS (IN MEANS) VS. WITHIN Variable Correlation Variable Correlatio n THRV 7_23 0. 27 TEDA 4_23 0. 07 TIMP 6_23 0. 29 WHRV 7_23 0. 51 WEDA 4_23 0. 02 WIMP 6_23 0. 38 HHRV 7_23 0. 56 HEDA 4_23 0. 05 HIMP 6_23 0. 46

CONVERGENCE AND MODEL FIT • POTENTIAL SCALE REDUCTION (PSR) AS CLOSE TO 1. 00 AS POSSIBLE • NORMAL POSTERIOR PARAMETER DISTRIBUTION • TRACE PLOTS SHOW CONVERGENCE • MODEL FIT IS ASSESSED WITH DIC (AND RELATIVE FIT WITH ΔDIC). LOWER IS BETTER. • UNSTABLE/DIFFICULT TO COMPUTE IF LATENT VARIABLES ARE TREATED AS PARAMETERS • COMPARING SAMPLE STATISTICS TO MODEL-ESTIMATED QUANTITIES

PARAMETER TRACE PLOTS • SHOULD LOOK LIKE THIS:

MINE LOOK LIKE THIS

POSTERIOR PARAMETER DISTRIBUTIONS • SHOULD LOOK LIKE THIS:

MINE LOOK LIKE THIS

SO • I SHOULD BE RUNNING MORE ITERATIONS UNTIL I HAVE BETTER EVIDENCE OF CONVERGENCE. • NOTE: MY PSR IS 1. 002, WHICH IS GOOD, BUT MPLUS ALWAYS GIVE THIS WARNING: • TECH 8 TELLS YOU WHAT THE PSR IS AT EACH 100 ITERATIONS

RESULTS

STANDARDIZED RESULTS • CAN REQUEST STANDARDIZED RESULTS • MPLUS USES WITHIN-PERSON STANDARDIZATION (AS YOU WOULD IF IT WERE TIMESERIES ANALYSIS) • ADD THIS LINE TO THE INPUT: • OUTPUT: • STANDARDIZED(CLUSTER); • RESIDUAL(CLUSTER); • STANDARDIZES BY CLUSTER AND THEN GIVES AVERAGE OF THOSE STANDARDIZED PARAMETERS • R-SQUARE TELLS YOU AVERAGE ACROSS CLUSTERS • ALSO GIVES WITHIN-CLUSTER RESULTS

THANK YOU!