© Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 1 S077: Applied Longitudinal Data Analysis Week #5: What Are The Topics Covered In Today’s Overview?

(ALDA, Section 12.1, pp ) General Specification For TIME Is: Completely general, placing no constraints on the shape of the baseline (logit) hazard function. Easily interpretable—each associated parameter represents logit hazard in time period j for the baseline group. Provides estimates identical to life-table estimates. General Specification For TIME Is: Completely general, placing no constraints on the shape of the baseline (logit) hazard function. Easily interpretable—each associated parameter represents logit hazard in time period j for the baseline group. Provides estimates identical to life-table estimates. PRO Three Reasons For Considering An Alternative Specification Your study involves many discrete time periods (because data collection has been lengthy or time is less coarsely discretized). Hazard probability is expected to be near 0 in some time periods (causing convergence problems) Some time periods have small risk sets (because either the initial sample is small or hazard and censoring dramatically diminish the risk set over time). Three Reasons For Considering An Alternative Specification Your study involves many discrete time periods (because data collection has been lengthy or time is less coarsely discretized). Hazard probability is expected to be near 0 in some time periods (causing convergence problems) Some time periods have small risk sets (because either the initial sample is small or hazard and censoring dramatically diminish the risk set over time). The variable PERIOD in the person-period data set can be treated as continuous TIME. © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 2 S077: Applied Longitudinal Data Analysis I.1 Pros And Cons Of The Completely General Specification For “Main Effect Of TIME”? General Specification For TIME Is Also: Nothing more than an analytic decision, not a requirement of the discrete-time hazard model. Completely lacking in parsimony. If J is large, it requires the inclusion of many unknown parameters. A problem when it yields fitted functions that fluctuate erratically across time periods because of nothing more than sampling variation. General Specification For TIME Is Also: Nothing more than an analytic decision, not a requirement of the discrete-time hazard model. Completely lacking in parsimony. If J is large, it requires the inclusion of many unknown parameters. A problem when it yields fitted functions that fluctuate erratically across time periods because of nothing more than sampling variation. CON

(ALDA, Section , pp ) Completely General Specification: always the “best fitting” model (lowest Deviance) Completely General Specification: always the “best fitting” model (lowest Deviance) Constant Specification: always the “worst fitting” model (highest Deviance) Use of ONE facilitates programming Constant Specification: always the “worst fitting” model (highest Deviance) Use of ONE facilitates programming Polynomial Specifications: As in growth modeling, a systematic set of choices. Choose centering constant “c” to ease interpretation. Because each lower order model is nested within each higher order model, deviance statistics can be directly compared to help make analytic decisions. 4 th and 5 th order polynomials are rarely adopted, but help give you a sense of whether you should stick with the completely general specification. Polynomial Specifications: As in growth modeling, a systematic set of choices. Choose centering constant “c” to ease interpretation. Because each lower order model is nested within each higher order model, deviance statistics can be directly compared to help make analytic decisions. 4 th and 5 th order polynomials are rarely adopted, but help give you a sense of whether you should stick with the completely general specification. © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 3 S077: Applied Longitudinal Data Analysis I.1 Replace the General Specification by a Polynomial Representations For TIME

Sample: 260 faculty members (who had received a National Academy of Education/Spencer Foundation Post-Doctoral Fellowship) Research design: Each was tracked for up to 9 years after taking his/her first academic job. By the end of data collection, n=166 (63.8%) had received tenure; the other 36.2% were censored (because they might eventually receive tenure somewhere). For simplicity, we don’t include any substantive predictors (although the study itself obviously did). Sample: 260 faculty members (who had received a National Academy of Education/Spencer Foundation Post-Doctoral Fellowship) Research design: Each was tracked for up to 9 years after taking his/her first academic job. By the end of data collection, n=166 (63.8%) had received tenure; the other 36.2% were censored (because they might eventually receive tenure somewhere). For simplicity, we don’t include any substantive predictors (although the study itself obviously did). Data Source: Beth Gamse & Dylan Conger (1997) Abt Associates Report (ALDA, Section p 412) © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 4 S077: Applied Longitudinal Data Analysis I.1 Illustrative Example: Time To Tenure In Colleges And Universities The Quadratic Looks Reasonably Good, But Can We Test Whether It’s “Good Enough”? As expected, deviance declines as model becomes more general General Constant Linear Quadratic Cubic

(ALDA, Section , pp ) Two comparisons always worth making Is the added polynomial term necessary? Is this polynomial as good as the general spec? Lousy Better, but not as good as general As good as general, better than linear Clear preference for quadratic (although cubic has some appeal) © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 5 S077: Applied Longitudinal Data Analysis I.1 Testing Alternative Polynomial Specifications For TIME No better than quadratic

Sample: 1,393 adults ages 17 to 57 (drawn randomly through a phone survey in metropolitan Toronto) Research Design: Each asked whether and, if so, when (age in years) s/he had first experienced a depressive episode? 387 (27.8%) reported first onset between 4 and 39. Time-Varying Question Predictor: PD, first parental divorce: 145 (10.4%) had experienced a parental divorce while still at risk of first depression onset. PD is time-varying, indicating whether the parents of individual i divorced during time period j. PD ij = 0, in periods before the divorce PD ij = 1 in periods coincident with, or subsequent to, the divorce Additional Time-Invariant Predictors: FEMALE – which we’ll include in the models, now. NSIBS (total number of siblings)—which we’ll use soon. Sample: 1,393 adults ages 17 to 57 (drawn randomly through a phone survey in metropolitan Toronto) Research Design: Each asked whether and, if so, when (age in years) s/he had first experienced a depressive episode? 387 (27.8%) reported first onset between 4 and 39. Time-Varying Question Predictor: PD, first parental divorce: 145 (10.4%) had experienced a parental divorce while still at risk of first depression onset. PD is time-varying, indicating whether the parents of individual i divorced during time period j. PD ij = 0, in periods before the divorce PD ij = 1 in periods coincident with, or subsequent to, the divorce Additional Time-Invariant Predictors: FEMALE – which we’ll include in the models, now. NSIBS (total number of siblings)—which we’ll use soon. Data Source: Blair Wheaton and colleagues (1997) (ALDA, Section 12.3, p 428) © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 6 S077: Applied Longitudinal Data Analysis I.2 Including Time-Varying Predictors: Illustrative Example What happened to my Mommy?

(ALDA, Section 12.3, p 428) ID AGE PD FEMALE NSIBS EVENT ID 40: Reported first depression onset at 23; first parental divorce at age 9 Many Periods per Person (because we have retrospective annual data from age 4 to respondent’s current age, up to age 39). In fact, there are 36,997 records in this pp data set, and only 387 events—would we really want to include 36 TIME dummies? Many Periods per Person (because we have retrospective annual data from age 4 to respondent’s current age, up to age 39). In fact, there are 36,997 records in this pp data set, and only 387 events—would we really want to include 36 TIME dummies? First depression onset at age 23 PD is time-varying: Her parents divorced when she was 9. Cubic function of TIME fits nearly as well as completely general spec (  2 =34.51, 32 df, p>.25) & measurably better than quadratic (  2 =5.83, 1 df, p<.05) FEMALE and NSIBS are time-invariant predictors. © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 7 S077: Applied Longitudinal Data Analysis I.2 Including A Time-varying Predictor In The Person-Period Data Set

(ALDA, Section , p ) What does  1 tell us? Contrasts the population logit hazard for people who have experienced a parental divorce with those who have not. But, because PD ij is time-varying, membership in the parental divorce group changes over time so we’re not always comparing the same people. The predictor effectively compares different groups of people at different times! But, we’re still assuming that the effect of the time-varying predictor is constant over time. What does  1 tell us? Contrasts the population logit hazard for people who have experienced a parental divorce with those who have not. But, because PD ij is time-varying, membership in the parental divorce group changes over time so we’re not always comparing the same people. The predictor effectively compares different groups of people at different times! But, we’re still assuming that the effect of the time-varying predictor is constant over time. Sample logit(proportions) of people experiencing first depression onset at each age, by PD status at that age Hypothesized population model (note constant effect of PD) Implicit particular realization of population model (for those whose parents divorce when they’re age 20) © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 8 S077: Applied Longitudinal Data Analysis I.2 Including A Time-Varying Predictor In The Discrete-Time Hazard Model

(ALDA, Section , pp ) e =1.51  Controlling for gender, at every age from 4 to 39, the estimated odds of first depression onset are about 50% higher for individuals who experienced a concurrent, or previous, parental divorce. e =1.73  Controlling for parental divorce, the estimated odds of first depression onset are 73% higher for women. What about a woman whose parents divorced when she was 20? © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 9 S077: Applied Longitudinal Data Analysis I.2 Interpreting A Fitted DT Hazard Model That Includes A Time-Varying Predictor

(ALDA, Section 12.4, pp 443) Linear Additivity Assumption Unit differences in a predictor—time- invariant or time-varying—correspond to fixed differences in logit-hazard. Linear Additivity Assumption Unit differences in a predictor—time- invariant or time-varying—correspond to fixed differences in logit-hazard. Data source: Nina Martin & Margaret Keiley (2002) Sample: 1,553 adolescents (n=887, 57.1% had been abused as children). Research design: Incarceration history from age 8 to 18. n=342 (22.0.8%) had been arrested. RQs: What’s the effect of abuse on the risk of first arrest? What’s the effect of race? Does the effect of abuse differ by race (or conversely, does the effect of race differ by abuse status)? Data source: Nina Martin & Margaret Keiley (2002) Sample: 1,553 adolescents (n=887, 57.1% had been abused as children). Research design: Incarceration history from age 8 to 18. n=342 (22.0.8%) had been arrested. RQs: What’s the effect of abuse on the risk of first arrest? What’s the effect of race? Does the effect of abuse differ by race (or conversely, does the effect of race differ by abuse status)? Violated by interactions among substantive predictors? © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 10 S077: Applied Longitudinal Data Analysis I.3 Linear Additivity Assumption: Uncovering Violations And Simple Solutions

(ALDA, Section , pp ) What Is The Shape Of The Logit Hazard Functions? For all groups, the risk of 1 st arrest is low during childhood, accelerates during the teen years, and peaks between ages 14 and 17. What Is The Shape Of The Logit Hazard Functions? For all groups, the risk of 1 st arrest is low during childhood, accelerates during the teen years, and peaks between ages 14 and 17. How Does The Level of the Logit hazard Functions Differ Across Groups? While abused children appear to be at greater risk of 1 st arrest consistently, the differential is especially pronounced among African- American children How Does The Level of the Logit hazard Functions Differ Across Groups? While abused children appear to be at greater risk of 1 st arrest consistently, the differential is especially pronounced among African- American children As in regular regression, when the effect of one predictor differs by the levels of another, we need to include their interaction as a predictor! © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 11 S077: Applied Longitudinal Data Analysis I.3 Evidence Of An Interaction Between ABUSE And RACE

© Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 12 (ALDA, Section , pp ) Estimated odds ratios for the 4 possible prototypical individuals In comparison to a White child who had not been abused, the odds of 1 st arrest are: 28% higher for a Black child who had not been abused (note: this is not stat sig.) 43% higher for a White child who had been abused (this is stat sig.) Nearly 3 times higher for a Black child who had been abused. In comparison to a White child who had not been abused, the odds of 1 st arrest are: 28% higher for a Black child who had not been abused (note: this is not stat sig.) 43% higher for a White child who had been abused (this is stat sig.) Nearly 3 times higher for a Black child who had been abused. This Is Not The Only Way To Violate The Linear Additivity Assumption … S077: Applied Longitudinal Data Analysis I.3 Including and Interpreting The Interaction Between ABUSE And RACE

(ALDA, Section , pp ) Predictor’s effect is constant over time Predictor’s effect increases over time Predictor’s effect decreases over time Predictor’s effect is particularly pronounced in certain time periods © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 13 S077: Applied Longitudinal Data Analysis I.4 Proportionality Assumption: Is A Predictor’s Effect Constant Over Time?

© Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 14 (ALDA, Section , pp ) A Completely General Representation: The predictor has a unique effect in each period A Completely General Representation: The predictor has a unique effect in each period A More Parsimonious Representation: The predictor’s effect changes linearly with time A More Parsimonious Representation: The predictor’s effect changes linearly with time  1 assesses the effect of X 1 in time period c  2 describes how this effect linearly increases (if positive) or decreases (if negative) Another Parsimonious Representation: The predictor’s effect differs across epochs Another Parsimonious Representation: The predictor’s effect differs across epochs  2 assesses the additional effect of X 1 during those time periods declared to be “later” in time S077: Applied Longitudinal Data Analysis I.4 Discrete-Time Hazard Models That Do Not Invoke The Proportionality Assumption

(ALDA, Section 12.4, pp 443) Sample: 3,790 high school students who participated in the Longitudinal Survey of American Youth (LSAY) Research design: Tracked from 10 th grade through 3 rd semester of college—a total of 5 periods Only n=132 (3.5%) took a math class for all of the 5 periods! RQs: When are students most at risk of first dropping out of math? What’s the effect of gender? Does the gender differential vary over time? Sample: 3,790 high school students who participated in the Longitudinal Survey of American Youth (LSAY) Research design: Tracked from 10 th grade through 3 rd semester of college—a total of 5 periods Only n=132 (3.5%) took a math class for all of the 5 periods! RQs: When are students most at risk of first dropping out of math? What’s the effect of gender? Does the gender differential vary over time? © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 15 S077: Applied Longitudinal Data Analysis I.4 The Proportionality Assumption: Uncovering Violations And Simple Solutions Data source: Suzanne Graham (1997) HGSE Dissertation

(ALDA, Section , pp ) All models include a completely general specification for TIME that makes use of 5 time dummies: HS11, HS12, COLL1, COLL2, and COLL3 All models include a completely general specification for TIME that makes use of 5 time dummies: HS11, HS12, COLL1, COLL2, and COLL (4) ns 6.50 (1) p= © Willett & Singer, Harvard University Graduate School of Education S077/Week #5– Slide 16 S077: Applied Longitudinal Data Analysis I.4 Checking The Proportionality Assumption: Is Effect Of FEMALE Constant Over Time?