1. Unit 5c: Adding Predictors to the Discrete Time Hazard Model © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 1 http://xkcd.com/893/

2. Reviewing Life Table Analysis, Hazard Functions, Survival Functions Building the Discrete Time Hazard Model Comparing Nested Models © Andrew Ho, Harvard Graduate School of Education Unit 5c– Slide 2 Multiple Regression Analysis (MRA) Multiple Regression Analysis (MRA) Do your residuals meet the required assumptions? Test for residual normality Use influence statistics to detect atypical datapoints If your residuals are not independent, replace OLS by GLS regression analysis Use Individual growth modeling Specify a Multi-level Model If time is a predictor, you need discrete- time survival analysis… If your outcome is categorical, you need to use… Binomial logistic regression analysis (dichotomous outcome) Multinomial logistic regression analysis (polytomous outcome) If you have more predictors than you can deal with, Create taxonomies of fitted models and compare them. Form composites of the indicators of any common construct. Conduct a Principal Components Analysis Use Cluster Analysis Use non-linear regression analysis. Transform the outcome or predictor If your outcome vs. predictor relationship is non-linear, Use Factor Analysis: EFA or CFA? Course Roadmap: Unit 5c Today’s Topic Area

3. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 3 New data example … data described in FIRSTSEX_info.html ….FIRSTSEX_info.html New data example … data described in FIRSTSEX_info.html ….FIRSTSEX_info.html 822 person-period records.Sample size Singer & Willett, 2003Singer & Willett, 2003, Chapter 11.More Info Capaldi, D. M., Crosby, L., & Stoolmiller, M. (1996). Predicting The Timing Of First Sexual Intercourse For At-Risk Adolescent Males. Child Development, 67, 344-359. Source Person-level dataset that records he high-school grade (7 th – 12 th ) in which at-risk adolescent boys reported experiencing heterosexual sex for first time, with data on: 1.Whether the boy had suffered a parental transition during early childhood (eg., a parental divorce and/or a parental death prior to 7 th grade). 2.The parents’ level of antisocial behavior during the boy’s early childhood. Overview FIRSTSEX.txtDataset Research Questions 1.Whether, and if so in which grade, at-risk adolescent boys report first experiencing heterosexual sex? 2.How the risk of reported first heterosexual sex depends on the boy’s experiences with parental death and divorce during early childhood? Research Questions 1.Whether, and if so in which grade, at-risk adolescent boys report first experiencing heterosexual sex? 2.How the risk of reported first heterosexual sex depends on the boy’s experiences with parental death and divorce during early childhood? New Data Example In DTSA terms, first sex is death, and survival is virginity.

4. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 4 Getting the Data in Shape Adolescent Boy 1 first reported sexual intercourse in Grade 9, had no early parenting transitions and a high level of parent antisocial behavior during the boy’s childhood. Adolescent Boy 2 has a censored record (never had intercourse through Grade 12), had early parenting transitions, and an average level of parent antisocial behavior. Adolescent Boy 1 first reported sexual intercourse in Grade 9, had no early parenting transitions and a high level of parent antisocial behavior during the boy’s childhood. Adolescent Boy 2 has a censored record (never had intercourse through Grade 12), had early parenting transitions, and an average level of parent antisocial behavior.

5. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 5 A Histogram of GRADE This tells us how many adolescent boys report first sex in any given grade. In red, we have the number of boys whose records are censored, who never report having sex through Grade 12. Why doesn’t this give us a good sense of how Hazard Probabilities differ over time?

6. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 6 The Hazard Function The sample probability of reporting the loss of your virginity in Grade 12 (conditional on never doing so before entering the risk set) is 32.5%.

7. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 7 The Survival Function How to find the median survival time…

8. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 8 Conditional Hazard Probabilities and Functions What are the hazard probabilities conditional on whether the boy had an early parental transition? The sample probability of reporting the loss of your virginity in Grade 12 (conditional on being a reported virgin entering the risk set) is 47% if you had an early parental transition and 19% if you did not.

9. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 9 Conditional Survival Probabilities and Functions What are the survival probabilities conditional on whether the boy had an early parental transition? The sample probability of maintaining one’s reported virginity past Grade 12 is 19% if you had an early parental transition and 47% if you did not. How to find the median survival time conditional on PT!

10. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 10 Conditional Hazard Logits as the Target of Modeling

11. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 11 From Person-Level to Person-Period Hazard Probabilities. tabulate EVENT GRADE, column Remember how to generate hazard probabilities in a person-period dataset, by tabulate or by egen. From before…

12. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 12 Fitting the Discrete Time Hazard Model These z-tests test the null hypothesis that the logit is 0 (the probability is 50%) in the population. The fitted logits/probabilities reproduce the sample logits/probabilities exactly and incorporates them into a statistical framework.

13. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 13 Now, Conditional Hazard Probabilities in a Person-Period Dataset Remember that we don’t model probabilities directly. Instead, we model their logits. By adding the categorical predictor variable, PT, to the “by” statement, we create conditional hazard probabilities in a person-period dataset. (It’s straightforward for person-level data with the “ ltable, by ” approach).

14. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 14 Fitting the Discrete Time Hazard Model with a Predictor  As always, the interpretation of the main effect is easier to see on the logit scale than it is on the probability scale.

15. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 15 Building the Discrete Time Hazard Model with esttab For each model, we store using eststo:, save the deviance (-2LL), save the predicted probabilities, and label the predicted probability variable. Instead of adding dummy variables for each grade, we plot a model that is linear in the logits. And then add the question predictor… We can see if the above model is more parsimonious than a model that uses all the grade dummies without sacrificing prediction.

16. © Andrew Ho, Harvard Graduate School of EducationUnit 5c– Slide 16 Building the Discrete Time Hazard Model with esttab