1. Lecture 1 (Chapter 1)

2. Introduction This course describes statistical methods for the analysis of longitudinal data, with a strong emphasis on applications in the biological and health sciences –Univariate statistics: each subject gives rise to a single measurement, termed response –Multivariate statistics: each subject gives rise to a vector of measurements, or different responses –Longitudinal data: each subject gives rise to a vector of measurements, but these represent the same response measured at a sequence of observation times Repeated responses over time on independent units (persons)

3. Basic issues and exploratory analyses –Definition and examples of LDA –Approaches to LDA –Exploring correlation Statistical methods for continuous measurements –General Linear Model with correlated data Weighted Least Squares estimation Maximum Likelihood estimation Parametric models for covariance structure Generalized linear models for continuous/discrete responses –Marginal Models –Log Linear Model and Poisson Model for count responses –Logistic model for binary responses –GEE estimation methods –Estimation techniques –Random Effects Models (Multi-level models) –Transition Models Topics

4. Longitudinal study: people are measured repeatedly over time Cross-sectional study: a single outcome is measured for each individual In LDA, we can investigate: –changes over time within individuals (ageing effects) –differences among people in their baseline levels (cohort effects) LDA requires special statistical methods because the set of observations on one subject tends to be inter-correlated Introduction

5. Reading ability appears to be poorer among older children.

6. Let’s assume that the reading ability of each child has been measured twice.

7. LDA can distinguish between changes over time within individual & differences among people in their baseline levels

8. Repeated observations are likely to be correlated, so assumption of independence is violated What if we used standard regression methods anyway (ignore correlation)? –Correlation may be of scientific focus –Incorrect inference –Inefficient estimates of β Why special methods?

9. Opportunities Distinguish “longitudinal” from “cross- sectional” effects Choose several targets of estimation Challenges Repeated observations tend to be auto- correlated Correlation must be modeled What is special about longitudinal data?

10. Repeated observations on individuals Scientific questions regression methods Response = f(predictors) Discrete/continuous responses and predictors Types of Studies Time series studies Panel studies (Sociology & Economics) Prospective studies (Clinical Trials) Characteristics

11. 1.CD4+ cell numbers (continuous) 2.Growth of Sitka spruce – tree size (continuous) 3.Protein contents of milk (continuous) 4.Indonesian Children’s health study (binary) 5.Analgesic Crossover trial (binary) 6.Epileptic seizures (count) 7.Health Effects of Air Pollution (count) Note: All these datasets are posted on the course website Examples

12. HIV attacks CD4+ cells, which regular the body’s immunoresponse to infectious agents 2376 values of CD4+ cell number plotted against time since seroconversion for 369 infected men enrolled in the MACS Q: What is the impact of HIV infection on CD4 counts over time? Ex: CD4+ cell numbers

13. Goals: 1.Characterize the typical time course of CD4+ cell depletion 2.Identify factors which predict CD4+ cell changes 3.Estimate the average time course of CD4+ cell depletion 4.Characterize the degree of heterogeneity across men in the rate of progression Ex: CD4+ cell numbers (cont’d)

14. This figure displays 2376 values of CD4+ cell numbers plotted against time since sero-conversion for 369 infected men enrolled in the MACS study.

18. Ex: Growth of trees Data for 79 trees over two growing seasons –54 trees were grown with ozone exposure at 70 ppb –25 trees were grown under control conditions Goal: Compare the growth patterns of trees under the two conditions

19. Goal: To assess the effect of ozone pollution on tree growth Figure 1.3. Log-size of 79 sitka spruce trees over 2 growing seasons

20. Ex: Protein content of milk Milk was collected weekly for 79 Australian cows and analyzed for their protein content Cows were maintained on one of three diets Goal: To determine how diet affects the protein content in milk Problem: About half of the 79 sequences are incomplete (i.e. missing data)

21. Barley (25 cows) Mixed (27 cows) Lupins (27 cows)

22. Ex: Indonesian Children’s Health Study (ICHS) Dr. Sommer conducted a study to determine effects of vitamin A deficiency in pre-school children Over 3000 children were examined for up to 6 visits to assess whether they suffered from respiratory infection or xerophthalmia (an ocular manifestation of vit A deficiency). Weight and height are also measured. Q: What are the predictors of infection?

23. Ex: ICHS (cont’d) Goals: 1.Estimate the increase in risk of respiratory infection for children who are vitamin A deficient while controlling for other demographic factors 2.Estimate the degree of heterogeneity in the risk of disease among children

24. Ex: ICHS (cont’d) Respiratory Infection (RI) i (0, 1, 0, 0, 1, 1) Xerophthalmia i (1, 1, 0, 0, 1, 1)

25. Ex: Analgesic Crossover trial 3-period crossover trial of an analgesic drug for relieving pain for primary dysmenorrhea 3 levels of analgesic (control, low, and high) were given to each of the 86 women Women were randomized to one of the 6 possible orders for administering the 3 treatment levels Pain was relieved for 26% with placebo, 71% with low dose, and 80% with high dose Q: Treatment effect? Q: How about carry-over effects?

26. Ex: Analgesic Crossover trial (cont’d) Ignoring the order of treatments, pain was relieved for 22 women under placebo

27. Ex: Epileptic seizures Clinical trial of 59 epileptics For each patient, the number of epileptic seizures was recorded during a baseline period of 8 weeks Patients were randomized to treatment with the antiepileptic drug progabide or placebo Number of seizures was then recorded in 4 consecutive 2-week intervals Q: Does progabide reduce the epileptic seizure rate?

29. Placebo Progabide

30. What do these examples have in common? These are repeated observations on each experimental unit Units can be assumed independent of one another Multiple responses within each unit are likely to be correlated The objectives can be formulated as regression problems whose purpose is to describe the dependence of the response on explanatory variables The choice of the statistical model must depend on the type of the outcome variable

31. Course Overview Scientific objectives include: –Characterize change –Component of variation –Hypothesis testing We will focus on regression models We will consider (up to 6) case studies in detail Computing using Stata will be introduced (SAS macros will be provided)

32. Notation

33. Notation (cont’d) Regression Model (n 1 x 1) (n 1 x p) (p x 1) (n 1 x 1) Suppose:

34. Cross-sectional versus Longitudinal Study with respect to its baseline value

35. Cross-sectional versus Longitudinal Study (cont’d)

38. Example

40. Approaches to LDA If we have one observation on each experimental unit, we are confined to modeling the population average of Y, called marginal mean response If we have repeated measurements, there are several approaches we can use: 1.Reduce the repeated values into one or two summary variables 2.Analyze each summary variable as a function of covariates x i (two-stage analysis)

41. Approaches to LDA (cont’d) For example, in the Sitka spruce data, we can: 1.Estimate the growth rate of each tree 2.Compare the rates across ozone groups x ij = x i Explanatory variables do not change over time.

42. Where is the correlation coming from?

43. Approaches to LDA

44. ICHS Example To illustrate the three approaches, we consider the ICHS example. Let:

45. Compare with Standard logistic regression model (only one measurement for each subject) Marginal Models

46. Random Effects Models

47. Transition Models

48. Appropriate EDA techniques Lines plots (spaghetti plots) Average and distribution plots (boxplot, quantiles) Empirical covariance Residual “pairs” plots Variograms (for unequally spaced observations) Scatterplot of a response against time Smoothing to highlight the typical response as a function of an explanatory variable

49. Why LDA?

51. Visualize the Independence Case:

52. Visualize the Correlation Case:

55. Compare with Independence Case:

56. LDA & Correlation:

57. …LDA takes care of R

58. Visualizing Correlation Structures:

60. Uniform, Exchangeable, Compound Symmetry…

61. Visualizing Correlation Structures: Uniform, Exchangeable, Compound Symmetry…

62. Visualizing Correlation Structures: Uniform, Exchangeable, Compound Symmetry…

63. Visualizing Correlation Structures: Stationary (Toeplitz): many forms…

64. Visualizing Correlation Structures: Stationary (Toeplitz): decreasing

65. Visualizing Correlation Structures: Stationary (Toeplitz): “mdependent” (ha) (m=2)

66. Visualizing Correlation Structures: Stationary (Toeplitz): “mdependent” (ha) (m=3)

67. Visualizing Correlation Structures: Stationary (Toeplitz): Autoregressive (AR)

68. Visualizing Correlation Structures: Stationary (Toeplitz): Autoregressive (AR)

69. Visualizing Correlation Structures: Stationary (Toeplitz): Autoregressive (AR)