1. © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 1 S077: Applied Longitudinal Data Analysis Week #3: What Topics Are Covered In Today’s Overview?

2. Sample: 254 people identified at unemployment offices. Design: Goal was to collect 3 waves of data per person at 1, 5 &11 months after job loss. In reality, though, data set is not time-structured:  Interview 1 was within 1 day & 2 months of job loss.  Interview 2 was between 3 & 8 months of job loss.  Interview 3 was between 10 & 16 months of job loss.  In addition, not everyone completed 2nd & 3rd interview. Outcome: Depression measured on the CES-D scale:  Twenty 4-point items (possible score ranged from 0 to 80). Time-Varying Predictor: Unemployment status (UNEMP)  132 were unemployed at every interview.  61 were always working after first interview.  41 were still unemployed at second interview, but working by third.  19 were working at second interview, but were unemployed again by third. Research Question:  How does unemployment affect the symptomatology of depression, over time? Sample: 254 people identified at unemployment offices. Design: Goal was to collect 3 waves of data per person at 1, 5 &11 months after job loss. In reality, though, data set is not time-structured:  Interview 1 was within 1 day & 2 months of job loss.  Interview 2 was between 3 & 8 months of job loss.  Interview 3 was between 10 & 16 months of job loss.  In addition, not everyone completed 2nd & 3rd interview. Outcome: Depression measured on the CES-D scale:  Twenty 4-point items (possible score ranged from 0 to 80). Time-Varying Predictor: Unemployment status (UNEMP)  132 were unemployed at every interview.  61 were always working after first interview.  41 were still unemployed at second interview, but working by third.  19 were working at second interview, but were unemployed again by third. Research Question:  How does unemployment affect the symptomatology of depression, over time? Source: Liz Ginexi and colleagues (2000), Journal of Occupational Health Psychology. (ALDA, Section 5.3..1, pp160-161) S077: Applied Longitudinal Data Analysis I. Time-Varying Predictors: Illustrative Example © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 2

3. ID 7589 has 3 waves, all unemployed ID 65641 has 3 waves, re-employed after 1 st wave ID 65641 has 3 waves, re-employed after 1 st wave ID 53782 has 3 waves, re- employed at 2 nd, unemployed again at 3 rd TIME = MONTHS since job loss (ALDA, Table 5.6, p161) UNEMP (by design, must be 1 at wave 1) S077: Applied Longitudinal Data Analysis I. What Does A Person-Period Dataset Look Like, When It Contains Time-Varying Predictors? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 3 How Shall We Proceed? What Multilevel Models for Change Must We Fit, To Address the Research Question? How Shall We Proceed? What Multilevel Models for Change Must We Fit, To Address the Research Question?

4. Level-1 Model: Level-2 Model: Composite Model: (ALDA, Section 5.3.1, pp 159-164) Let’s just get a sense of the data by ignoring UNEMP, and first fitting the usual unconditional growth model Where do we add time-varying predictor UNEMP?  How can it go in at Level-2?  But, it looks like we could just stick it in here? On the first day of job loss, the average person has an estimated CES-D of 17.7 On average, CES-D declines by 0.42/mo There’s statistically significant within- person residual variation. There’s statistically significant variation in initial status and rates of change. S077: Applied Longitudinal Data Analysis I. Model A – Ignoring the Time-Varying Predictor &Fitting an Unconditional Growth Model © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 4

5. © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 5 (ALDA, Section 5.3.1, pp 159-164) Population average rate of change in CES-D, controlling for UNEMP. Population average difference, over time, in CES-D by UNEMP status. Logical impossibility? 02468101214 Months since job loss 5 10 15 20 CES-D Remains unemployed   02468101214 Months since job loss 5 10 15 20 CES-D Reemployed at 5 months How Can We Understand This Graphically? Although the magnitude of the time-varying predictor’s effect remains constant, the time-varying nature of UNEMP’s values implies the existence of many possible true individual trajectories, such as: How Can We Understand This Graphically? Although the magnitude of the time-varying predictor’s effect remains constant, the time-varying nature of UNEMP’s values implies the existence of many possible true individual trajectories, such as: What Happens When We Fit This Model To Our Data? 02468101214 Months since job loss 5 10 15 20 CES-D Reemployed at 10 months   02468101214 Months since job loss 5 10 15 20 CES-D Reemployed at 5 months Unemployed again at 10   S077: Applied Longitudinal Data Analysis I. Model B -- Add the Time-Varying Predictor into the Composite Specification Directly?

6. © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 6 (ALDA, Section 5.3.1, pp. 162-167) What about the variance components? Monthly rate of decline is cut in half by controlling for UNEMP (still stat. sig.) UNEMP has a large and stat. sig. effect. Model B is a better fit (difference in deviance = 25.5, 1 df, p<.001) Consistently Employed (UNEMP=0): Consistently Unemployed (UNEMP=1): 02468101214 Months since job loss 5 10 15 20 CES-D UNEMP = 0 UNEMP = 1 Look what happens to folk who get a job!!! S077: Applied Longitudinal Data Analysis I. Fitting & Interpreting Model B, Which Now Contains Time-Varying Predictor UNEMP

7. (ALDA, Section 5.3.1, pp. 162-167) When including time-invariant predictors in the model, we already understand which variance components will differ and how:  Level-1 Variance Components will remain relatively stable because time-invariant predictors cannot predict (much of) the within-person variation.  Level-2 Variance Components will decline if the time-invariant predictors successfully predict some of the between-person variation. When including time-varying predictors, all the variance components can change, but  Although you can continue to interpret a model-to-model decrease in the magnitude of the level-1 variance components meaningfully, …  Changes in the level-2 variance components may not appear to make sense!!!  Look what happened to the Level-2 VC’s In this example, they’ve increased! Why? Because including a time-varying predictor changes the meaning of the individual growth parameters (e.g., intercept now refers to value of the outcome when all level-1 predictors, including UNEMP are 0).  Look what happened to the Level-2 VC’s In this example, they’ve increased! Why? Because including a time-varying predictor changes the meaning of the individual growth parameters (e.g., intercept now refers to value of the outcome when all level-1 predictors, including UNEMP are 0). Level-1 VC, Adding UNEMP to the unconditional growth model (A) reduces its magnitude 68.85 to 62.39. UNEMP predicts 9.4% of the variation in CES-D scores. Level-1 VC, Adding UNEMP to the unconditional growth model (A) reduces its magnitude 68.85 to 62.39. UNEMP predicts 9.4% of the variation in CES-D scores. We can clarify what’s happened by decomposing the composite specification back into a level 1/level-2 representation … S077: Applied Longitudinal Data Analysis I. Careful -- Variance Components Can Be Weird When You Add Time-Varying Predictors © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 7

8. (ALDA, Section 5.3.1, pp. 168-169) Level-2 Models: Level-1 Model: But, notice that … in Model B: The level-2 model for  2i has no residual! The value of  2i has effectively been “fixed” – to the same hypothesized value – for everyone in the population. So, the model is implicitly hypothesizing that there is no stochastic variation across people in this parameter. But, notice that … in Model B: The level-2 model for  2i has no residual! The value of  2i has effectively been “fixed” – to the same hypothesized value – for everyone in the population. So, the model is implicitly hypothesizing that there is no stochastic variation across people in this parameter. But, Is this reasonable? Should we really assume that the effect of the person-specific predictor is identical across people? No, with a little thought, we can try to relax the assumption  But, Is this reasonable? Should we really assume that the effect of the person-specific predictor is identical across people? No, with a little thought, we can try to relax the assumption  S077: Applied Longitudinal Data Analysis I. A Clue! Decomposing the Composite Specification of Model B Into Its L1/L2 Specification © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 8

9. (ALDA, Section 5.3.1, pp. 169-171) Level-1 Model: Level-2 Models: You can allow the effect of UNEMP to differ randomly across people in the pop by adding in a level-2 residual. Check your software to make sure you know what you’re doing … You can allow the effect of UNEMP to differ randomly across people in the pop by adding in a level-2 residual. Check your software to make sure you know what you’re doing … But, you may not be able to afford the cost:  Adding this residual adds 3 new variance components.  If you have only a few waves of data, you may be straining the resources of your data.  Here, we can’t actually fit this model!! But, you may not be able to afford the cost:  Adding this residual adds 3 new variance components.  If you have only a few waves of data, you may be straining the resources of your data.  Here, we can’t actually fit this model!! Moral: The multilevel model for change can easily handle time-varying predictors, but… Think carefully about the consequences for both the structural and stochastic parts of the model. Don’t just “buy” the default specification in your software.  Until you’re sure you know what you’re doing, always write out your model in both the composite and L1/L2 specifications before specifying code to a computer package. Moral: The multilevel model for change can easily handle time-varying predictors, but… Think carefully about the consequences for both the structural and stochastic parts of the model. Don’t just “buy” the default specification in your software.  Until you’re sure you know what you’re doing, always write out your model in both the composite and L1/L2 specifications before specifying code to a computer package. So … Are we happy with Model B as the “final” model??? Is there any other way to allow the effect of UNEMP to vary – if not across people, across TIME? So … Are we happy with Model B as the “final” model??? Is there any other way to allow the effect of UNEMP to vary – if not across people, across TIME? S077: Applied Longitudinal Data Analysis I. Adding the “Omitted” Level-2 Stochastic Variation in the Effect of UNEMP © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 9

10. What happens when we fit this model to data? (ALDA, Section 5.3.2, pp. 171-172) To allow the effect of a time-varying predictor to differ over time, just add its interaction with TIME … Two possible (equivalent) interpretations: The effect of UNEMP differs across occasions. The rate of change in depression differs by unemployment status. Two possible (equivalent) interpretations: The effect of UNEMP differs across occasions. The rate of change in depression differs by unemployment status.  But you need to think very carefully about the hypothesized error structure: We’ve added another level-1 parameter to represent the interaction. Just like we asked about the main effect of time-varying predictor UNEMP, perhaps we should allow the interaction effect also to differ stochastically across people in the pop? We won’t right now, but soon...  But you need to think very carefully about the hypothesized error structure: We’ve added another level-1 parameter to represent the interaction. Just like we asked about the main effect of time-varying predictor UNEMP, perhaps we should allow the interaction effect also to differ stochastically across people in the pop? We won’t right now, but soon... Because of the way in which we’ve constructed the models with time-varying predictors, we’ve automatically constrained UNEMP to have only a “main effect” on the trajectory’s elevation. When analyzing the effects of time-invariant predictors, we automatically allowed predictors to affect the trajectory’s slope... S077: Applied Longitudinal Data Analysis I. Model C: What If the Effect of a Time-Varying Predictor Differs Over Time? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 10

11. (ALDA, Section 5.3.2, pp. 171-172) Consistently unemployed (UNEMP=1) Perhaps we should constrain the slope of the trajectory for the reemployed to be zero? What about people who get a job? Main effect of TIME is now positive (!) & not stat sig ?!?!?!?!?!?!?!?! UNEMP×TIME interaction is stat sig (p<.05). Model C is a better fit than Model B (  Deviance = 4.6, 1 df, p<.05). Model C is a better fit than Model B (  Deviance = 4.6, 1 df, p<.05). S077: Applied Longitudinal Data Analysis I. Model C: What If the Effect of a Time-Varying Predictor Differs Over Time? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 11

12. Model D: What happens when we fit this model to data? (ALDA, Section 5.3.2, pp. 172-173) Should we remove the main effect of TIME? (which is the slope when UNEMP=0)  Yes, but this creates a lack of congruence between the model’s fixed and stochastic parts So, let’s better align the parts by having UNEMP*TIME be both fixed and random  If we’re allowing the UNEMP*TIME slope to differ randomly across people, might we also need to allow the effect of UNEMP itself to do the same? UNEMP*TIME has both a fixed & random effect UNEMP has both a fixed & random effect S077: Applied Longitudinal Data Analysis I. Model D: Constraining the Slope of the Individual Growth Trajectory for the Re-Employed? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 12

13. (ALDA, Section 5.3.2, pp. 172-173) What about people who get a job? Consistently unemployed Consistently employed Best fitting model S077: Applied Longitudinal Data Analysis I: Model D: Constraining Slope of the Individual Growth Trajectory for the Re-Employed? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 13

14. Sample: 888 male high school dropouts, from an earlier analysis. Research Design: –Each was interviewed between 1 and 13 times after dropping out. –34.6% (n=307) earned a GED at some point during data collection. OLD Research Questions: –How do log(WAGES) change over time? –Do wage trajectories differ by ethnicity and highest grade completed? NEW Research Questions: What is the effect of GED attainment? Does earning a GED: –affect the wage trajectory’s elevation? –affect the wage trajectory’s slope? –create a discontinuity in the wage trajectory? Sample: 888 male high school dropouts, from an earlier analysis. Research Design: –Each was interviewed between 1 and 13 times after dropping out. –34.6% (n=307) earned a GED at some point during data collection. OLD Research Questions: –How do log(WAGES) change over time? –Do wage trajectories differ by ethnicity and highest grade completed? NEW Research Questions: What is the effect of GED attainment? Does earning a GED: –affect the wage trajectory’s elevation? –affect the wage trajectory’s slope? –create a discontinuity in the wage trajectory? Data source: Murnane, Boudett and Willett (1999), Evaluation Review (ALDA, Section 6.1.1, pp 190-193) S077: Applied Longitudinal Data Analysis II. Modeling Discontinuous Individual Change: Illustrative Example © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 14

15. (ALDA, Figure 6.1, p 193) Let’s start by considering four plausible effects of GED receipt by imagining what the wage trajectory might look like for someone who got a GED 3 years after labor force entry (post dropout) How do we represent trajectories like these within the context of a linear growth model???  GED 0246810 EXPER 1.5 2.0 2.5 LNW A: No effect of GED whatsoever B: An immediate shift in elevation; no difference in rate of change D: An immediate shift in rate of change; no difference in elevation F: Immediate shifts in both elevation & rate of change S077: Applied Longitudinal Data Analysis II. Theory: How Might GED Receipt Affect an Individual’s Wage Trajectory? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 15

16. (ALDA, Section 6.1.1, pp 194-195) Key Idea: It’s easy; simply include GED as a time-varying effect at level-1 … Pre-GED (GED=0): Post-GED (GED=1): S077: Applied Longitudinal Data Analysis II. Trajectory B: Including a Discontinuity in Elevation, and Not In Slope © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 16

17. (ALDA, Section 6.1.1, pp 195-198) Pre-GED (POSTEXP=0): Post-GED (POSTEXP clocked in same cadence as EXPER): POSTEXP ij  A new time-varying predictor that clocks “TIME since GED receipt” (in the same cadence as EXPER). Equals 0 prior to GED. After GED is received, it counts “Post GED experience.” POSTEXP ij  A new time-varying predictor that clocks “TIME since GED receipt” (in the same cadence as EXPER). Equals 0 prior to GED. After GED is received, it counts “Post GED experience.” S077: Applied Longitudinal Data Analysis II. Trajectory D: Including a Discontinuity in Slope, and Not in Elevation © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 17

18. (ALDA, Section 6.1.1, pp 195-198) Pre-GED Post-GED S077: Applied Longitudinal Data Analysis II. Trajectory F: Including Discontinuities In Both Elevation and Slope © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 18 Now, let’s try fitting all these models …

19. (ALDA, Section 6.1.2, pp 201-202) (UERATE-7) is the local area unemployment rate (a time-varying control predictor), centered around 7% for interpretability -7 To compare this deviance statistic to more complex models appropriately, we need to know how many parameters have been estimated to achieve this value of deviance. Benchmark against which we will compare the discontinuous models 5 fixed effects 4 random effects S077: Applied Longitudinal Data Analysis II. Let’s Start By Fitting a “Baseline Model” In Which There Are No Discontinuities © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 19

20. (ALDA, Section 6.1.2, pp 202-203) Instead of constructing tables of (seemingly endless) parameter estimates, we’re going to construct a summary table that presents the… Baseline just shown specific terms in the model specific terms in the model n parameters (for d.f.) n parameters (for d.f.) deviance statistic (for model comparison) deviance statistic (for model comparison) S077: Applied Longitudinal Data Analysis II. Now, What’s the Best Way to Proceed … ? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 20

21. (ALDA, Section 6.1.2, pp 202-203) B: Add GED as both a fixed and random effect (1 extra fixed parameter; 3 extra random)  Deviance=25.0, 4 df, p<.001—keep GED effect B: Add GED as both a fixed and random effect (1 extra fixed parameter; 3 extra random)  Deviance=25.0, 4 df, p<.001—keep GED effect C: But, does the GED discontinuity differ across people? (do we need to keep the extra variance components for the effect of GED?)  Deviance=12.8, 3 df, p<.01— keep variance components. C: But, does the GED discontinuity differ across people? (do we need to keep the extra variance components for the effect of GED?)  Deviance=12.8, 3 df, p<.01— keep variance components. What about adding the discontinuity in slope? S077: Applied Longitudinal Data Analysis II. First Steps: Investigating A Potential Discontinuity in Elevation by Adding Effect of GED © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 21

22. D: Adding POSTEXP as both a fixed and random effect (1 extra fixed parameter; 3 extra random)  Deviance=13.1, 4 df, p<.05— keep POSTEXP effect D: Adding POSTEXP as both a fixed and random effect (1 extra fixed parameter; 3 extra random)  Deviance=13.1, 4 df, p<.05— keep POSTEXP effect E: But does the POSTEXP slope differ across people? (do we need to keep the extra variance components for the effect of POSTEXP?)  Deviance=3.3, 3 df, ns—don’t need the POSTEXP random effects (but in comparison with A still need POSTEXP fixed effect) E: But does the POSTEXP slope differ across people? (do we need to keep the extra variance components for the effect of POSTEXP?)  Deviance=3.3, 3 df, ns—don’t need the POSTEXP random effects (but in comparison with A still need POSTEXP fixed effect) What if we include both types of discontinuity? (ALDA, Section 6.1.2, pp 203-204) S077: Applied Longitudinal Data Analysis II. Next Steps: Investigating a Discontinuity in Slope by Adding the Effect of POSTEXP © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 22

23. (ALDA, Section 6.1.2, pp 204-205) F: Add GED and POSTEXP simultaneously (each as both fixed and random effects) F: Add GED and POSTEXP simultaneously (each as both fixed and random effects) comp. with D shows Importance of GED comp. with D shows Importance of GED comp. with B shows importance of POSTEXP comp. with B shows importance of POSTEXP S077: Applied Longitudinal Data Analysis II. Examining The Addition of Elevation and Slope Discontinuities Simultaneously © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 23

24. We actually fit several other possible models (see ALDA) but Model F was the best alternative— so…how do we display its results? (ALDA, Section 6.1.2, pp 204-205) Each results in a worse fit, suggesting that Model F (which includes both random effects) is better (even though Model E suggested we might be able to eliminate the variance component for POSTEXP) Each results in a worse fit, suggesting that Model F (which includes both random effects) is better (even though Model E suggested we might be able to eliminate the variance component for POSTEXP) S077: Applied Longitudinal Data Analysis II. Can We Simplify by Eliminating theVariance Components for POSTEXP or GED? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 24

25. (ALDA, Section 6.1.2, pp 204-206) LNW 12 th grade dropouts 9 th grade dropouts Highest grade completed Those who stay longer have higher initial wages. This differential remains constant over time. Highest grade completed Those who stay longer have higher initial wages. This differential remains constant over time. White/ Latino Black Race At dropout, no racial differences in wages. Racial disparities increase over time because wages for Blacks increase at a slower rate. Race At dropout, no racial differences in wages. Racial disparities increase over time because wages for Blacks increase at a slower rate. earned a GED GED receipt has two effects Upon GED receipt, wages rise immediately by 4.2%. Post-GED receipt, wages rise annually by 5.2% (vs. 4.2% pre-receipt). GED receipt has two effects Upon GED receipt, wages rise immediately by 4.2%. Post-GED receipt, wages rise annually by 5.2% (vs. 4.2% pre-receipt). S077: Applied Longitudinal Data Analysis II. Displaying Fitted Growth Trajectories for Prototypical HS Dropouts © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 25

26. (ALDA, Section 6.2, pp 208-210) When facing obviously non-linear trajectories, we begin by trying transformation: –A straight line—even on a transformed scale—is a simple form with easily interpretable parameters. –Since many outcome metrics are ad hoc, transformation to another ad hoc scale may sacrifice little. When facing obviously non-linear trajectories, we begin by trying transformation: –A straight line—even on a transformed scale—is a simple form with easily interpretable parameters. –Since many outcome metrics are ad hoc, transformation to another ad hoc scale may sacrifice little. Earlier, we modeled ALCUSE, an outcome that we formed by taking the square root of the researchers’ original alcohol use measurement We can ‘de-transform’ the findings and return to the original scale, by squaring the predicted values of ALCUSE and re- plotting Prototypical Individual Growth Trajectories Are Now Non-linear: By transforming the outcome before analysis, we have effectively modeled non-linear change over time. Prototypical Individual Growth Trajectories Are Now Non-linear: By transforming the outcome before analysis, we have effectively modeled non-linear change over time. So … How do we know which variable to transform, using which transformation … ? S077: Applied Longitudinal Data Analysis III: Modeling Non-Linear Change Using Transformations © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 26

27. (ALDA, Section 6.2.1, pp. 210-212) Step 1: What kinds of transformations do we consider? Generic variable V expand scale compress scale Step 2: How do we know when to use which transformation? 1.Plot empirical growth trajectories. 2.Find linearizing transformations by moving “up” or “down” in the direction of the “Bulge.” Step 2: How do we know when to use which transformation? 1.Plot empirical growth trajectories. 2.Find linearizing transformations by moving “up” or “down” in the direction of the “Bulge.” S077: Applied Longitudinal Data Analysis III. The “Rule of the Bulge” and the “Ladder of Transformations” © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 27

28. (ALDA, Section 6.2.1, pp. 211-213) Down in TIME Up in IQ How Else Might We Model Non-linear Change? S077: Applied Longitudinal Data Analysis III. Transforming the Growth Trajectory of a Single Child in the Berkeley Growth Study © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 28

29. (ALDA, Section 6.3.1, pp. 213-217) Polynomial of the “zero order” (because TIME 0 =1): Like including a constant predictor 1 in the level-1 model. Intercept represents vertical elevation. Different people can have different elevations. Polynomial of the “zero order” (because TIME 0 =1): Like including a constant predictor 1 in the level-1 model. Intercept represents vertical elevation. Different people can have different elevations. Polynomial of the “first order” (because TIME 1 =TIME): Familiar individual growth model. Varying intercepts and slopes yield criss-crossing lines. Polynomial of the “first order” (because TIME 1 =TIME): Familiar individual growth model. Varying intercepts and slopes yield criss-crossing lines. Second order polynomial for quadratic change: Includes both TIME and TIME 2.  0i = intercept (but now both TIME & TIME 2 are 0).  1i = instantaneous rate of change when TIME=0 (there is no longer a constant slope).  2i = curvature parameter; larger its value, more dramatic its effect. Peak is called a “stationary point”— a quadratic has only one. Second order polynomial for quadratic change: Includes both TIME and TIME 2.  0i = intercept (but now both TIME & TIME 2 are 0).  1i = instantaneous rate of change when TIME=0 (there is no longer a constant slope).  2i = curvature parameter; larger its value, more dramatic its effect. Peak is called a “stationary point”— a quadratic has only one. Third order polynomial for cubic change: Includes TIME, TIME 2 and TIME 3. Can keep on adding other powers of TIME. Each extra term adds another stationary point—cubic has two. Third order polynomial for cubic change: Includes TIME, TIME 2 and TIME 3. Can keep on adding other powers of TIME. Each extra term adds another stationary point—cubic has two. S077: Applied Longitudinal Data Analysis III. Representing Non-Linear Individual Change by a Polynomial Function of Time © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 29

30. Sample: 45 boys & girls identified in 1 st grade. Goal was to study behavior changes over time (until 6 th grade) Research Design:  At the end of every school year, teachers rated each child’s level of externalizing behavior using Achenbach’s Child Behavior Checklist: 3 point scale (0=rarely/never; 1=sometimes; 2=often). 24 aggressive, disruptive, or delinquent behaviors. Outcome :  EXTERNAL — ranges from 0 to 68 (simple sum of scores). Question Predictor:  FEMALE. Research Question:  How does children’s level of externalizing behavior change over time?  Do trajectories of change differ for boys and girls? Sample: 45 boys & girls identified in 1 st grade. Goal was to study behavior changes over time (until 6 th grade) Research Design:  At the end of every school year, teachers rated each child’s level of externalizing behavior using Achenbach’s Child Behavior Checklist: 3 point scale (0=rarely/never; 1=sometimes; 2=often). 24 aggressive, disruptive, or delinquent behaviors. Outcome :  EXTERNAL — ranges from 0 to 68 (simple sum of scores). Question Predictor:  FEMALE. Research Question:  How does children’s level of externalizing behavior change over time?  Do trajectories of change differ for boys and girls? Source: Margaret Keiley et al. (2000) J. of Abnormal Child Psychology (ALDA, Section 6.3.2, p. 217) S077: Applied Longitudinal Data Analysis III. Change as a Polynomial Function of Time: Illustrative Example © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 30

31. (ALDA, Section 6.3.2, pp 217-220) Little change over time (flat line?) Linear decline (at least until 4 th grade) Quadratic change (but with varying curvatures) Two stationary points? (suggests a cubic) Two stationary points? (suggests a cubic) Three stationary points? (suggests a quartic!!!) Three stationary points? (suggests a quartic!!!) When faced with so many different patterns, how do you select a common polynomial for analysis? S077: Applied Longitudinal Data Analysis III. Selecting a Suitable Polynomial Trajectory for Individual Change? © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 31

32. (ALDA, Section 6.3.2, pp 217-220) First impression: Most fitted trajectories provide a reasonable summary for each child’s data Second impression: Maybe these ad hoc decisions aren’t the best? Quadratic? Would a quadratic do? Third realization: We need a common polynomial across all cases (and might the quartic be just too complex)? Using sample data to draw conclusions about the shape of the underlying true trajectories is tricky— let’s compare alternative models  S077: Applied Longitudinal Data Analysis III. Examining Possible Polynomial Trajectories, Using Exploratory OLS Methods © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 32

33. (ALDA, Section 6.3.3, pp 220-223) Add polynomial functions of TIME to person period data set Compare goodness of fit (accounting for all the extra parameters that get estimated) A: stat significant between- and within-child variation B: no fixed effect of TIME but stat sig var comps  Deviance=18.5, 3df, p<.01 B: no fixed effect of TIME but stat sig var comps  Deviance=18.5, 3df, p<.01 C: no fixed effects of TIME & TIME 2 but stat sig var comps  Deviance=16.0, 4df, p<.01 C: no fixed effects of TIME & TIME 2 but stat sig var comps  Deviance=16.0, 4df, p<.01 D: still no fixed effects for TIME terms, but now var comps are ns also  Deviance=11.1, 5df, ns Quadratic (C) is best choice—and it turns out there are no gender differentials at all. S077: Applied Longitudinal Data Analysis III. Using Model-to-Model Comparisons To Test Higher-Order Polynomial Terms © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 33

34. Sample: 17 1st and 2nd graders –During a three week period, Terry repeatedly played a two- person checkerboard game called Fox ‘n Geese, with participants (hopefully) learning from experience: Fox is controlled by the experimenter, at one end of the board. Children have four geese, that they use to try to trap the fox. –Great for studying cognitive development because: There exists a strategy that children can learn that will guarantee victory. This strategy is not immediately obvious to children. Many children can deduce the strategy over time. Research Design: –Each child played up to 27 games (each game is a “wave”). –Outcome, NMOVES is the number of moves made by the child before making a catastrophic error (guaranteeing defeat)—ranges from 1 to 20. Research Question: –How does the child’s success change over time? –What is the impact of a child’s reading (or cognitive) ability — READ (score on a standardized reading test) – on the change trajectory. Sample: 17 1st and 2nd graders –During a three week period, Terry repeatedly played a two- person checkerboard game called Fox ‘n Geese, with participants (hopefully) learning from experience: Fox is controlled by the experimenter, at one end of the board. Children have four geese, that they use to try to trap the fox. –Great for studying cognitive development because: There exists a strategy that children can learn that will guarantee victory. This strategy is not immediately obvious to children. Many children can deduce the strategy over time. Research Design: –Each child played up to 27 games (each game is a “wave”). –Outcome, NMOVES is the number of moves made by the child before making a catastrophic error (guaranteeing defeat)—ranges from 1 to 20. Research Question: –How does the child’s success change over time? –What is the impact of a child’s reading (or cognitive) ability — READ (score on a standardized reading test) – on the change trajectory. Data source: Terry Tivnan (1980) Doctoral Dissertation, HGSE (ALDA, Section 6.4.1, pp. 224-225) S077: Applied Longitudinal Data Analysis III. Truly Non-Linear Individual Change: Illustrative Example © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 34

35. (ALDA, Section 6.4.2, pp. 225-228) A lower asymptote, because everyone makes at least 1 move and it takes a while to figure out what’s going on A lower asymptote, because everyone makes at least 1 move and it takes a while to figure out what’s going on An upper asymptote, because a child can make only a finite # moves each game A smooth curve joining the asymptotes, that initially accelerates and then decelerates These three features suggest a level-1 logistic change trajectory which, unlike our previous growth models, is non-linear in the individual growth parameters. S077: Applied Longitudinal Data Analysis III. Selecting a Truly Non-Linear Trajectory for Individual Change © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 35

36. (ALDA, Section 6.4.2, pp 226-230)  0i is related to, and determines, the intercept  1i determines the rapidity with which the trajectory approaches the upper asymptote Upper asymptote in this particular model is constrained to be 20 (1+19) Higher the value of  0i, the lower the intercept When  1i is small, the trajectory rises slowly (often not reaching an asymptote) When  1i is large, the trajectory rises more rapidly Models can be fit in usual way using provided your software can do it  S077: Applied Longitudinal Data Analysis III. Understanding the Logistic Individual Change Trajectory © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 36

37. (ALDA, Section 6.4.2, pp 229-232) Begins low and rises smoothly and non-linearly Not statistically significant (note small n’s), but better READers approach asymptote more rapidly S077: Applied Longitudinal Data Analysis III. Investigating Systematic Inter-Individual Differences in Logistic Change © Willett & Singer, Harvard University Graduate School of EducationS077/Week #3– Slide 37