1. What is a “Launch” relationship? Definition: Individual differences in the rate of change of a target variable is predicted from individual differences in the initial level of an antecedent variable. Target: Rate of change of the outcome Predictor: Initial level of antecedent

2. Example: Individual differences in attachment at 18 months of age LAUNCHES the trajectory of the quality of relationships with peers throughout life. Quality of peer relationships Secure attachment Insecure attachment AGE/TIME

3. What is an “Ambient Level” relationship? Definition: Individual differences in the average level of an antecedent variable present across time is important in shaping individual differences in the rate of change in target variable. Target: Rate of change of the outcome Predictor: Average level of antecedent

4. Example: A certain ambient level of parental warmth is necessary for optimal development of the psychosocial functioning of a child. Increases in parental warmth are not necessary, but decreasing parental warmth may be detrimental. Parental Warmth Psychosocial functioning AGE/TIME Parental Warmth Psychosocial functioning AGE/TIME

5. What is a “Change-to-Change” relationship? Definition: Individual differences in the pattern of change in an antecedent variable predicts individual differences in the trajectory of a target variable. Target: Rate of change of the outcome Predictor: Rate of change of the antecedent

6. Example: As self-esteem improves, positive affect also improves. As self-esteem decays, positive affect also decays. It is the pattern of change in self-esteem that predicts positive affect. Self-esteem Positive Affect Self-esteem Positive Affect AGE/TIME

7. What do you need to test Launch, Change-to-Change, and Ambient Level Hypotheses? 1. Longitudinal data 2. At least 3 times of measurement (although you can have missing data) 3. Estimates of the Intercept, Slope, and Ambient Level of the trajectory of antecedent and/or target variables FOR EACH STUDY PARTICIPANT, where a) The Intercept of the Developmental Trajectory of an Antecedent Variable is an indicator of LAUNCH. Intercept of the Trajectory Level of an antecedent variable AGE/TIME

8. b) The Slope of the Developmental Trajectory of an Antecedent or Target Variable is the indicator of CHANGE Run Rise Level of the antecedent or target variable (Slope = Rise/Run) Slope is the rate of change Slope and c) the Mean Level of the Developmental Trajectory of an Antecedent Variable is the indicator of AMBIENT LEVEL. AGE/TIME

9. How do you estimate the Intercept and Slope of a Developmental Trajectory of a Variable for each individual study participant? Use hierarchical linear modeling. Hierarchical linear modeling (HLM), which is also called a random effects model or the general linear mixed model, can be used to estimate both fixed and random effects simultaneously. Fixed effects are variables that are assumed to represent all possibilities in the entire population of interest. Random effects are those that are assumed to have been randomly selected from all possibilities. In this case, times of measurement and study participants are assumed to have been randomly selected from the entire possible collection of times and participants. When intercept and time of measurement are specified as random effects in HLM, and there are repeated measures of variables of interest for each subject, estimates of the intercept and slope of the developmental trajectory of each participant can be made based upon information available about the average study population trajectory and the individual’s own repeated measures of the variable of interest.

10. In the following example the MIXED procedure available in SAS/STAT software was used to 1) assess the population trajectory of Perceived Control in children in 3rd to 7th grade and to determine if there were group differences in trajectories. 2) output estimates of the intercept and slope of each individual’s developmental trajectory of perceived control, and 3) output values of perceived control predicted by the HLM model. This example is a portion of a cross-sectional sequential study that had LAUNCH, AMBIENT LEVEL, and CHANGE-TO-CHANGE hypotheses. In this study, two waves of children, initially in grades 3 to 6, participated in a study of perceived control and motivation in the classroom. Children completed a measure of perceived control (as well as other measures) in the fall and spring of the school year for up to three consecutive years. The numbers of children participating at each time of measurement are shown in Figure 1.

11. 1) Assessing the Population Trajectory of Perceived Control and Between Group Differences in Developmental Trajectories SUBNUM SEX WAVE GRADE TIME CON 4410 2 1 6 7 30.11 4410 2 1 6 8 31.39 4410 2 1 7 10 34.42 6340 2 1 6 7 32.14 6340 2 1 6 8 23.58 6340 2 1 7 9 19.10 6340 2 1 7 10 22.58 7598 2 1 6 8 29.33 7598 2 1 7 10 9.75 8360 1 2 3 1 28.25 8360 1 2 3 2 22.46 8360 1 2 4 3 39.00 8360 1 2 4 4 21.67 ETC... TIME indicates the time of measurement and is coded as follows: 1 = Measurement of a 3rd grade student in the fall of the school year 2 = 3rd grade spring measure 3 = 4th grade fall measure... 9 = 7th grade fall measure 10 = 7th grade spring measurement. CON is the measure of Perceived Control. In order to use HLM to estimate the population trajectory of perceived control (or subsample trajectories) the data are structured as in the following example. As you can see, there is a record for each subject at each time of measurement that an assessment was completed. Also, not all participants had data at all times of measurement.

12. The following is an example SAS program using PROC MIXED. Lines were numbered on the right to provide points of reference for the following section. PROC MIXED DATA=SASUSER.GW36; CLASS SUBNUM SEX WAVE;#1 MODEL CON=TIME SEX WAVE TIME*SEX TIME*WAVE / SOLUTION CHISQ;#2 RANDOM INTERCEPT TIME / TYPE=UN SUBJECT=SUBNUM;#3 RUN; In line #1, SUBNUM, SEX, and WAVE were specified as CLASS variables. This indicates that these are nominal variables. Line #2 contains the MODEL statement. CON (perceived control) is the dependent variable. CON is measured at multiple points in TIME. Therefore, the first independent variable specified is TIME which specifies the time of measurement. SEX and WAVE are additional independent variables. These effects will test whether the level of the trajectories of perceived control differed between participants grouped by sex or wave. TIME*SEX and TIME*WAVE are interaction effects which test whether changes in perceived control over time differ by sex or by wave. SOLUTION requests a solution for the fixed effect parameters be printed and a CHISQ requests of Chi-square test of these effects in addition to the F test. Higher level effects may be added to test whether a development trajectory has a quadratic, cubic, etc. shape. For example, to test a quadratic shape, TIME*TIME would be included as an additional model effect. Line #3 contains the RANDOM statement which indicates that the intercept and time are random effects. TYPE allows for selection of the covariance structure of random effects. TYPE=UN requests an unstructured covariance matrix. This is recommended for a correlated random coefficient model. SUBJECT=SUBNUM identifies the subjects in you database.

13. The SAS output can be interpreted as follows: Since this is an iterative method, the iterations are first printed. The Covariance Parameter Estimates are computed using the default residual maximum likelihood method (REML). UN(1,1) is the test of whether the variance in the intercepts of perceived control trajectories is significantly different from 0. The test performed is the Wald Z. Since, Pr > |Z| is less than.05, the variance of the intercepts of the developmental trajectories of perceived control it is significantly different from 0. UN(2,2) tests the variation among the slopes of the developmental trajectories of perceived control. Since, Pr > |Z| is less than.05, there is also significant variation in the slope of the trajectories of perceived control. Model Fitting Information for CON gives various pieces of information on the fit of the model including the number of observations, the residual variance estimate and the square root of this estimate. The Solution for Fixed Effects table is next printed. This information indicates that the intercept of the average perceived control trajectory of SEX 2 is 37.83 (the Intercept Estimate) while the intercept for SEX 1 is 37.83 +.81 = 38.64. In addition, the estimate of the slope of the perceived control trajectory of SEX 2 is -2.02 (the Time Estimate), while the slope for SEX 1 is 1.06 less (-2.02-1.06=-3.08). Thus, group SEX 1 starts with a higher initial level of perceived control than group SEX 2, but declines more rapidly. (However, in the next section it is shown that these differences are not significant.)

14. The Tests of Fixed Effects table report s tests of the unique effect of each fixed effect specified. Neither the TIME*SEX interaction or the TIME*WAVE interaction are significant at the p F). These results indicate that there was no significant differences in the slopes of perceived control between participants grouped by sex or grouped by wave. The overall effects of SEX and WAVE are also not significant (although, the interactions should be removed and the model re-fit to draw this conclusion). Therefore, in this model only the effect of TIME is significant indicating that perceived control significantly changed from 3rd grade to 7th grade. In addition, since the parameter estimate of TIME equals -2.02, there is a significant downward slope of perceived control from 3rd to 7th grade for this population.

15. 2) Output Estimates of the Intercept and Slope of Each Individual’s Developmental Trajectory of Perceived Control An additional option is added to the PROC MIXED program to request a solution to the random effects designated. This solution provides estimates of the intercepts and slopes of the trajectory of the dependent variable for each study participant. The option is underlined in the following example SAS program : PROC MIXED DATA=SASUSER.GW36; CLASS SUBNUM SEX WAVE; MODEL CON=TIME SEX WAVE TIME*SEX TIME*WAVE / SOLUTION CHISQ; RANDOM INTERCEPT TIME / TYPE=UN SUBJECT=SUBNUM SOLUTION; RUN; The following is an example of the output received by including the SOLUTION option (as well as some additional programming to actually output the information to a file): SUBNUM INTERCEPT INTSE TIME TIMESE 4410 4.93949332 6.80477868 2.00358954 2.38626018 6340 4.44533301 6.76336746 -0.60975877 2.33977441 7598 6.79776566 9.50030688 -2.52123912 2.83436687 8360 10.50907960 6.76349884 1.14906584 2.33977923 INTERCEPT is the estimated intercept of each subject’s perceived control trajectory. TIME is the estimated slope of the trajectory of perceived control for each subject. For example, the developmental trajectory of perceived control for SUBNUM 4410 had an estimated intercept of 4.94 and an estimated slope of 2.00. INTSE and TIMESE are the standard errors of the intercept and slope estimates, respectively, for each subject. Figure 2 illustrates the trajectories of perceived control of a few study participants and the estimates of the intercepts and slopes of their trajectories.

16. The intercepts and slopes can then be used as “usual” individual difference variables in more traditional correlation or regression analyses. for example, in Table 1 two multiple regressions examining relationships between the intercept and/or slope of the trajectories of perceived control and the slope parameter of the developmental trajectory of engagement in the classroom are shown. These regression results indicate that there was no “LAUNCH” relationship between student perceptions of perceived control in the academic domain (CON) and engagement in the classroom (  =.01). However, there was a “CHANGE-TO-CHANGE” relationship (  =.22, p<.001). Note, that in the 2nd regression model, the intercept of the trajectory of engagement in the classroom was included as an independent variable. This is to account for possible ceiling effects and to determine the unique LAUNCH and CHANGE-TO-CHANGE relationships between perceived control and engagement after partialling out the intercept of engagement.

17. Finally, HLM can be used to output values of a variable of interest at each time of measurement for all participants. PROC MIXED DATA=SASUSER.GW36; CLASS SUBNUM SEX WAVE; MODEL CON=TIME SEX WAVE TIME*SEX TIME*WAVE / SOLUTION CHISQ PREDICTED; RANDOM INTERCEPT TIME / TYPE=UN SUBJECT=SUBNUM; ID SUBNUM TIME; MAKE PREDICTED OUT=SASUSER.CONPRED; RUN; The PREDICTED option requests predicted values for the dependent variable for each subject at each time of measurement. The input dataset must include a record for each subject and time for which you want to output predicted values. For example: SUBNUM SEX WAVE GRADE TIME CON 4410 2 1 3 1. 4410 2 1 3 2. 4410 2 1 4 3 34.42 4410 2 1 4 4 32.14 4410 2 1 5 5 23.58 4410 2 1 5 6 19.10 4410 2 1 6 7. 4410 2 1 6 8 18.00 4410 2 1 7 9. 4410 2 1 7 10 18.50 ETC... 3) Output Values of Perceived Control Predicted by the HLM Model.

18. The ID statement asks that each record in the output dataset contain these variables for identification. The MAKE statement asks SAS to output the predicted values of the dependent variable to a dataset name CONPRED. The following is an example of the output received by including the P option and MAKE statement: SUBNUM TIME CONOBS CONPREDCONV_PR CONSE_P CONL95M CONU95M CONRES 4410 1.35.01 2.05 4410 2.35.06 1.54 4410 3 34.4234.32 2.34 ETC... 4410 4 32.1432.25 2.21 4410 5 23.5826.45 2.10 4410 6 19.1019.13 1.78 4410 7.18.67 1.79 4410 8 18.0018.10 1.87 4410 9.18.25 1.76 4410 10 18.5018.43 1.87 ETC... CONOBS is the actual score for perceived control, CONPRED is the value predicted by HLM. In addition, the output table includes the variance of the predicted value, the standard error of the predicted value, the lower and upper 95% confidence interval of the predicted value, and the residual. These predicted values can be used to plot complete trajectories of perceived control from fall of the 3rd grade to spring of the 7th grade whether or not a participant had missing data (see Figure 2). These could also be used to compute ambient levels. The ambient level for each participant would be the average of his/her 10 predicted values of CON.

19. For further information see: Burchinal, M., & Appelbaum, M.I. (1991). Estimating individual developmental function: Methods and their assumptions. Child Development, 62, 23-43. Bryk, A.S., & Raudenbush, S.W. (1992). Hierarchical linear models: Application and data analysis methods. Newbury Park, CA: Sage. Francis, D.J., Fletcher, J.M., Steubing, K.K., Davidson, K.C., & Thompson, N.M. (1991). Analysis of change: Modeling individual growth. Journal of consulting and Clinical Psychology, 59, 27-37. Bailey, Jr., D.B., Burchinal, M.R., & McWilliam, R.A. (1993). Age of peers and early childhood development. Child Development, 64, 848-862. Skinner, E.A. (1995). Perceived control, motivation, and coping. Newbury Park, CA: Sage Publications. Wellborn, J.G., Connell, J.P., & Skinner, E.A. (1989). The Student’s Perceptions of Control Questionnaire (SPOCQ): Academic domain. Technical report, University of Rochester, New York. Willet, J.B., Ayoub, C.C. & Robinson, D. (1991). Using growth modeling to examine systematic differences in growth: An example of change in the functioning of families at risk of maladaptive parenting, child abuse, and neglect. Journal of Consulting and Clinical Psychology, 59, 38-47.