1. Growth Curve Modeling
Day 1 Foundations of Growth Curve Models June 28 & 29, Michael Bader

2. Objectives
Describe how latent growth, latent growth trajectory analysis, and growth mixture models are related to, and different from, one another
Interpret results all three types of growth models
Identify the proper modeling technique for analytical questions regarding change
Clearly articulate the benefits, assumptions, and shortcomings of different models of change

3. Organization of Class Discuss the components of model
Simulate data corresponding to the model
Analyze real-life data

4. Free (as in beer and as in speech)
Increasingly used for research
Excellent graphics through the ggplot library

5. Integrated Development Environment for R
Free (as in beer, not as in speech)

6. Agenda Day 1 Models (means & fancy means)
Linear Trends & Modeling Time
Growth Models
Day 2
Modeling time
Covariates
Latent growth trajectory/growth mixture models

7. Structure of Topics For each model Description of model
Simulation example
Analytical example (using Zillow)

8. Models

9. A simplified or idealized description or conception of a particular system, situation, or process, often in mathematical terms, that is put forward as a basis for theoretical or empirical understanding, or for calculations, predictions, etc.; a conceptual or mental representation of something.
Oxford English Dictionary

10. Models
Estimate
Uncertainty

11. The mean

12. Why do we take the mean?

13. Take the mean of the following numbers:
{1,4,7,9,8,2,5,3,10,6}
Calculate the error for each number

14. The Mean

15. As an estimate, the mean minimizes the error

16. The Mean as Model

17. The Mean as Model
Outcome (yi): the value observed on y for datum (observation) i
Estimate (µ): is the best guess (by minimizing the error) of the data as a whole
Error (ei): is the deviation of the mean from the observed outcome for datum (observation) i

18. Estimates describe the data while the error describes a datum

19. The Mean as Model Simulation Example Plan the Population
Population size
Set mean
Set standard deviation
Conjure the population into existence
Analyze the population we created

20. The Mean as Model Analysis Example Gather our data Describe our data
Analyze our data
Interpret the data

21. QUESTIONS?

22. Linear Regression of a trend

23. Linear Trend

24. Linear Trend
Best estimate: ∑(et) = 0 Uncertainty: σt

25. Linear regression is nothing more than a fancy mean

26. Linear Trend Simulation Example Plan the Population
Number of repeated observations
Initial level of outcome
Change per unit time
Conjure the population into existence
Analyze the population we created
File: 01b_lineartrend_simulation.R

27. Plan Population

28. Linear Trend Analysis Example Gather our data Describe our data
Analyze our data
Interpret the data
File: 01b_lineartrend_analysis.R

29. Linear Trend
Let’s interpret the results:

30. QUESTIONS?

31. Nonlinear Change
Quadratic change

32. Multiple linear trends

33. Two linear trends

34. Multiple Linear Trends
Simulation Example
Plan the Population
Number of repeated observations
Initial level of outcome (for each city)
Change per unit time (for each city)
Conjure the population into existence
Analyze the population we created

35. Two Linear Trends
New York: Philadelphia:

40. Multiple Linear Trends

41. Multiple Linear Trends

42. Random intercept Model

43. Random Intercept Model

44. Multiple Linear Trends
Simulation Example
Plan the Population
Number of repeated observations
Initial level of outcome
Variation in initial level of outcome across cities
Change per unit time
Conjure the population into existence
Analyze the population we created

45. Random Intercepts Analysis Example Gather our data Describe our data
Analyze our data
Interpret the data

46. Interpreting the Results

47. QUESTIONS?

48. Random slopes

49. Random Slopes

50. Random Slopes Simulation Example Plan the Population
Conjure the population into existence
Analyze the population we created

51. Random Slopes

52. Random Intercepts & slopes

53. Random Intercepts & Slopes

54. Random Intercepts & Slopes

55. Random Intercepts & Slopes
Simulation Example
Plan the Population
Conjure the population into existence
Analyze the population we created

56. Random Intercepts & Slopes

57. Random Intercepts & Slopes
Uncreative, straightforward interpretation The average metropolitan area started the past year with (logged) home values per square foot of 4.763, with a standard of (Logged) values increased, on average, by per month, with a standard deviation of

58. Meaningful interpretation Home values twelve months ago were distributed a\-round a mean of $117 and a standard deviation of 9.2% in these metropolitan areas. The average metropolitan area saw home values increase by 0.48% per month, with a standard deviation of 0.5%..

59. Random Intercepts & Slopes
Analysis Example
Gather our data
Describe our data
Analyze our data
Interpret the data

62. Interpreting the results Across the largest 150 metropolitan areas, the median home price per square foot in last April was $123, but prices varied by a standard deviation of 46%. Since last April, median home prices increased, on average, by 0.55% per month, or 6.6% over the year. Increases in home prices varied around the average increase by 0.02% percent per month. The modest correlation between the random terms of the intercept and slope means that increases were larger in metropolitan areas that started with higher median values.

63. Random Intercepts & Slopes

65. QUESTIONS?

66. Review

67. Models distill important information about the world and have
An outcome
Deterministic component (data-level)
Stochastic component (datum-level)
All of our models attempt to minimize error (everything we do creates fancy means)
Determine what varies in the model and then analyze it

68. QUESTIONS?