Announcement

1. Announcement • AMS Conversation on Non-Academic Employment: Friday, January 15, 9:30am to 11am, Moscone West Room 2008 • Also you can drop by Booth No. 812 on Friday, January 15, 6-7pm., to talk to the people there about the Certificate in Quantitative Finance (CQF), a 6-month, part-time course for those interested in derivatives, development, quantitative trading or risk management.

2. Structural Equation Modeling with New Development for Mixed DesignsPlus Introductions toTime Series Modeling, Bootstrap Resampling, & Partial Correlation Network Analysis (PCNA) Kathryn Sharpe Wei Zhu

3. Outline • Part I: SEM & Introduction to Time Series Models • A brief introduction to SEM with an eating disorder example • A fMRI time series study + AR and MV models in SEM notation • SEM for Mixed Designs with a PET study example • Part II: PCNA and Bootstrap Resampling • Partial Correlation Network Analysis • Bootstrap Resampling

4. Part I: SEM & Introduction to Time Series Models 1. A brief introduction to SEM with an eating disorder example

5. SEM Basics • SEM is a set of usually inter-related linear regression equations. • SEM without latent variables is called • Path Analysis. • SEM is a confirmatory procedure. We cannot create hypotheses by way of SEM, only test a particular hypothesized model against a data set.

6. Assumptions of SEM Large samples: SEM researchers suggest a sample size of at least ten times the number of parameters we will be estimating. The variables follow a multivariate normal distribution. Now, we present a simple example of SEM.

7. The Hypothesis We received a hypothesis and data from a psychologist interested in determining what factors in a young woman’s life influence her risk for developing an eating disorder. “The following path is proposed as a representation of the relationship and inter-relationships among age of menarche, press for thinness, body image, and self-concept. Age at onset of menarche will lead to a more negative body image and self-concept which will lead to an increased press for thinness and an increase in eating disordered symptomatology.” We can test the validity of her hypothesis using SEM.

8. The Data Summary of variables incorporated into the model: • Age of first menstrual period • Body Image score • Self Concept score: measured differently for adolescents • and for adults, so we have two separate models • Drive for thinness • Risk for developing an eating disorder The psychologist evaluated many aspects of each participant’s life and combined the results to create a score for each variable.

9. Listwise/Pairwise Deletion What do we do when we have missing measurements? There are several ways a researcher can deal with missing data values. Two of them are listed here. In both cases, we assume that any missing data is missing completely at random. If much data is lost in deletion, imputation should be considered. Listwise Deletion: We delete a subject from the study if any of its measurement values are missing. Pairwise Deletion: We omit those subjects from a particular calculation who do not have the corresponding measurement value. The subjects are present in any calculation for which their value exists. In our study, we use listwise deletion to delete those subjects who do not have the age of first menstrual period variable. We lose less than 10% of our data in deletion.

10. Creating Two Models We have two groups of subjects as classified by the grade of the participant. Grade is a variable included in the data but not used in the SEM model, as it was not present in the hypothesis. • If a subject has grade ≤ 12, she is an adolescent. • If a subject has grade >12, she is an adult. The psychologist in this study first requested two separate models — one for adults and one for adolescents.

11. Path Diagrams Age of Menstruation (AM) Age of Menstruation (AM) Body Image (BI) Adolescent Self Worth (SW) Body Image (BI) Adult Self Worth (SW) Drive for Thinness (DT) Drive for Thinness (DT) Risk for Disorder (RD) Risk for Disorder (RD) Directional Arrows indicate cause and effect

12. The Equations Because there are no latent variables, we can write basic regression equations for each variable. Our system is as follows:

13. SEM Programs The most popular software packages for SEM are: • LISREL • EQS (Peter Bentler) • AMOS • PROC CALIS (and PROC TCALIS) in SAS For this example, we will use PROC CALIS. It takes our linear equations (previous slide) and estimates the parameters for the model. Then it evaluates the goodness of fit of the model.

14. SAS Code SAS correlation procedure Suppress Pearson correlations Specifies output Dataset with type Covariance matrix Proc calis: use cov rather than corr proccorr cov nocorr data=eddata outp=edcova(type=cov); run; proccalis cov mod data=edcova; Lineqs bi = b1 am + b2 sw + E1, sw = b3 am + b4 bi + E2, dt = b5 bi + b6 sw + E3, rd = b7 dt + E4 ; Std E1-E4 = The1-The4 ; Cov E1 E2 = Ps1; Run; Give the linear equations describing the system Give variances of exogenous variables Error variances We must set the error equal to a parameter value; otherwise it is assumed to be 0 BI and SW are correlated so we must estimate the correlation of their error terms’ variances

15. Results of SAS Analysis: Adolescents Age of Menstruation (AM) SAS gives us parameter estimates, error estimates, and t-values for each path included in the model. We use a t-test to determine which paths are significant. In addition, we can calculate the confidence interval: -.1232 (-.77,.52) -1.9912 (-4.02,.04) -.0525 (-2.02,1.92) Body Image (BI) Adolescent Self Worth (SW) -.1040 (-.27,.06) If zero is contained in the interval, then the path is not significant. -.2699 (-1.33,.79) .3341 (.04,.63) Drive for Thinness (DT) .8292 (.72,.94) Risk for Disorder (RD)

16. Results of SAS Analysis: Adults Age of Menstruation (AM) Here again, we use a t-test and/or the confidence intervals to evaluate which paths are significant. -.4157 (-2.89,2.06) .2262 (-.46,.99) -.2998 (-2.19,1.59) Body Image (BI) Adolescent Self Worth (SW) -.1289 (-.33,.07) SAS Output: .4255 (-.43,1.28) .8285 (.55,1.11) Drive for Thinness (DT) .8960 (.77,1.03) Risk for Disorder (RD)

17. Goodness of Fit After reporting the parameter estimates, SAS reports many different measures of fit so we can evaluate it in any way we choose. The more measures we use to evaluate our model, the better. Adolescent: A good fit does not necessarily mean a perfect model. We can still have unnecessary variables or be missing important ones. By convention, a model is “good” if: Adult: • GFI > .90/.95, • Small Chi-Square value, • large p-value, • RMSEA Estimate should • be close to zero.

18. Part I: SEM & Introduction to Time Series Models 2. A fMRI time series study + AR and MV models in SEM notation

19. SEM for Brain Functional Pathway Analysis SEM can be utilized to study brain functional pathways based on brain image studies including the positron emission tomography (PET), & the functional magnetic resonance imaging (fMRI) studies. Each PET scan generates only one 3D image, while each fMRI scan generates a time series of images tabulating continuous brain functional activities.

20. Experiment Study of Visual Attention fMRI Data Visual Attention Task : - Subjects : 28 volunteers ( 14 males and 14 females) - Methods : three-ball tracking task. 6 identified cortical regions for visual attention network : cerebellum (CEREB), left posterior parietal cortex (PPC), left anterior parietal cortex (APC), thalamus (THAL), supplementary motor area (SMA), left lateral prefrontal cortex (LPFC). 5 times cue task cue target balls attentive tracking response show target balls 1.25 s 1.5 s 7.75 s 1 s 1 s static balls cue task static balls passive viewing 5 times 20

21. Study Design Data : fMRI data at coordinates corresponding the 6 regions from each individual data set.  Extracting the activation (onset) conditions.  Hemodynamic delay : The first two time points ( 2 TR, 6 sec) at each onset ON OFF ON OFF ON OFF PASSIVE ACTIVE PASSIVE ACTIVE PASSIVE ACTIVE 60 sec 60 sec 60 sec 60 sec 60 sec 60 sec 20 Images 20 Images 20 Images 20 Images 20 Images 20 Images 21

22. Structure of fMRI Data 22

23. Contemporaneous Pathway The initial path Model : Consideration of the prior literatures and inspection of experts identified 7 anatomically possible directional paths in the left brain hemisphere . LPFC SMA APC THAL PPC CEREB 23

24. Longitudinal Pathways Modeled by MAR(1) time t-1 time t 24

25. - The original model with the longitudinal relationsand the contemporaneous relations in the same path diagram. - This model is used as the original model through all approaches to visual attention fMRI data. SMA t-1 LPFC t-1 t t t-1 t THAL t-1 t APC t-1 t t-1 t PPC CEREB Unified SEM of Visual Attention fMRI data 25

26. Autoregressive & Moving Average Models • AR(1) Residuals • MV(1) Residuals • AR(1) Response

27. Autoregressive & Moving Average Models in SEM Notations AR(1) Residuals, MV(1) Residuals, AR(1) Response

28. Autoregressive & Moving Average Models • AR(k) Residuals • MV(k) Residuals • AR(k) Response

30. We’ll guess same as last month

32. We’ll guess same as last month plus a little more for a possible trend

33. This is easy, who needs forecasting

34. Continue with our successful method: guess the same as last month plus a little more for a possible trend

36. Definitely looks like a trend

38. Trend might be a tad steeper than I thought

39. Opps

40. Momentary deviation, trend will continue

41. See, I told you this was easy!

42. Trend will continue

43. Opps, another momentary fluctuation:

44. Trend should continue

45. Oh oh!

46. Sales has leveled off: Lets average last few points

47. Oh oh, maybe things are going down hill

48. Let’s be conservative and Assume a negative trend

49. Thank goodness, we are still basically level

50. We’ll guess same as last month