1. John B. Willett & Judith D. Singer Harvard Graduate School of Education Introducing discrete-time survival analysis ALDA, Chapter Eleven “To exist is to change, to change is to mature” Henri Bergson

2. Chapter 11: Fitting basic discrete-time hazard models Review basic descriptive statistics for discrete-time survival data (Ch 10) Life table Hazard function Survivor function Median lifetime Specifying a suitable discrete-time hazard model (§11.1 & 11.2)—both heuristic and formal representations Fitting the discrete-time hazard model to data (§11.3)—it turns out that it’s very easy to fit the model Interpreting parameter estimates (§11.4)—very different from growth modeling, but more similar to logistic regression Displaying fitted hazard and survivor functions (§11.5)—as in growth modeling, we’ll display fitted functions at prototypical predictor values Comparing (nested) discrete-time hazard models using goodness-of-fit statistics (§11.5)—methods for data analysis and model comparison Review basic descriptive statistics for discrete-time survival data (Ch 10) Life table Hazard function Survivor function Median lifetime Specifying a suitable discrete-time hazard model (§11.1 & 11.2)—both heuristic and formal representations Fitting the discrete-time hazard model to data (§11.3)—it turns out that it’s very easy to fit the model Interpreting parameter estimates (§11.4)—very different from growth modeling, but more similar to logistic regression Displaying fitted hazard and survivor functions (§11.5)—as in growth modeling, we’ll display fitted functions at prototypical predictor values Comparing (nested) discrete-time hazard models using goodness-of-fit statistics (§11.5)—methods for data analysis and model comparison

3. Illustrative example: Grade at first heterosexual intercourse Sample: 180 middle school boys (all considered “at risk”) Research design: Large panel study in which each boy was tracked from 7 th through 12 th grades By the end of data collection (at the end of 12 th grade), n=126 (70.0%) had had sex The remaining n=54 (30%) were still virgins. These censored observations pose a challenge for data analysis. Question predictor: PT, for parenting transition, a dichotomy indicating whether the boy lived with his biological parents during his early formative years (before 7 th grade when data collection began) 72 boys (40%) lived with both biological parents (PT=0) 108 boys (60%) experienced at least one parenting transition before 7 th grade (PT=1) Ultimately, we’ll also examine a continuous predictor, PAS, which assesses the parents’ level of antisocial behavior during the child’s formative years (also time-invariant—behavior before the study started). Because the original scale is totally arbitrary, scores have been standardized to a mean of 0 and sd of 1 Sample: 180 middle school boys (all considered “at risk”) Research design: Large panel study in which each boy was tracked from 7 th through 12 th grades By the end of data collection (at the end of 12 th grade), n=126 (70.0%) had had sex The remaining n=54 (30%) were still virgins. These censored observations pose a challenge for data analysis. Question predictor: PT, for parenting transition, a dichotomy indicating whether the boy lived with his biological parents during his early formative years (before 7 th grade when data collection began) 72 boys (40%) lived with both biological parents (PT=0) 108 boys (60%) experienced at least one parenting transition before 7 th grade (PT=1) Ultimately, we’ll also examine a continuous predictor, PAS, which assesses the parents’ level of antisocial behavior during the child’s formative years (also time-invariant—behavior before the study started). Because the original scale is totally arbitrary, scores have been standardized to a mean of 0 and sd of 1 Data source: Deborah Capaldi & colleagues (1996) Child Development (ALDA, Section 11.1, pp 358-360)

4. The life table: Summarizing the distribution of event occurrence over time (ALDA, Section 10.1, pp 326-329) Risk set n censored in interval j n experiencing target event in interval j J intervals, T=7, 8, …, 12 How might we summarize the distribution of event occurrence?

5. Assessing the conditional risk of event occurrence: The discrete-time hazard function (ALDA, Section 10.2.1, pp 330-339) Discrete-time hazard Conditional probability that individual i will experience the target event in time period j (T i = j) given that s/he didn’t experience it in any earlier time period (T i  j) h(t ij )=Pr{T i = j|T i  j} As a probability (only in discrete time), hazard is bounded by 0 and 1. This is an issue for modeling that we’ll need to address Estimation is easy because each value of hazard is based on that interval’s risk set. Discrete-time hazard Conditional probability that individual i will experience the target event in time period j (T i = j) given that s/he didn’t experience it in any earlier time period (T i  j) h(t ij )=Pr{T i = j|T i  j} As a probability (only in discrete time), hazard is bounded by 0 and 1. This is an issue for modeling that we’ll need to address Estimation is easy because each value of hazard is based on that interval’s risk set.

6. Cumulating risk over time: The survivor function (and median lifetime) (ALDA, Section 10.2, pp 330-339) 6789101112 Grade 0.00 0.25 0.50 0.75 1.00 S(t) Discrete-time survival probability Probability that individual i will “survive” beyond time period j (T i > j) (i.e.,will not experience the event until after time period j). S(t ij )=Pr{T i > j} Also a probability bounded by 0 and 1. At the beginning of time, S(t i0 )=1.0 Strategy for estimation: Since h(t ij ) tells us about the probability of event occurrence, 1-h(t ij ) tells us about the probability of non-occurrence (i.e., about survival) Discrete-time survival probability Probability that individual i will “survive” beyond time period j (T i > j) (i.e.,will not experience the event until after time period j). S(t ij )=Pr{T i > j} Also a probability bounded by 0 and 1. At the beginning of time, S(t i0 )=1.0 Strategy for estimation: Since h(t ij ) tells us about the probability of event occurrence, 1-h(t ij ) tells us about the probability of non-occurrence (i.e., about survival) ML = 10.6 Estimated median lifetime

7. Person-period data set: one row for every person-period until event occurrence or censoring—different from growth modeling EVENT indicates either event occurrence or censoring Converting a person-level data set into a person-period data set (ALDA, Section 10.5.1, pp 351-354) Person-level data set: one row per person IDTCENSORPT 193901 1261201 4071210 ID 407 was censored, remaining a virgin through 12 th grade ID 126 had sex in the 12 th grade ID 193 had sex in the 9 th grade

8. Contemplating a DTSA model: Inspecting sample plots of within-group hazard and survivor functions (ALDA, Section 11.1.1, pp 358-361) Q’s to ask when examining sample hazard f n s: What is the shape of each hazard function?—here, their shape is similar—both beginning low and climbing steadily over time. Does the relative level of hazard differ across groups?—here, hazard for boys with a parenting transition is consistently higher Suggests partitioning variation in risk into: A baseline profile of risk A shift in risk corresponding to variation in the predictor Q’s to ask when examining sample hazard f n s: What is the shape of each hazard function?—here, their shape is similar—both beginning low and climbing steadily over time. Does the relative level of hazard differ across groups?—here, hazard for boys with a parenting transition is consistently higher Suggests partitioning variation in risk into: A baseline profile of risk A shift in risk corresponding to variation in the predictor Q’s to ask when examining sample survivor f n s: They tend to be less useful because they assess the predictor’s cumulative effect—here, telling us that the ML for boys with a PT is 10.0 vs. 11.7 when PT=0.  Note: reversal of relative rankings Q’s to ask when examining sample survivor f n s: They tend to be less useful because they assess the predictor’s cumulative effect—here, telling us that the ML for boys with a PT is 10.0 vs. 11.7 when PT=0.  Note: reversal of relative rankings We’re almost ready to go, but back to the bounded nature of hazard

9. As in regular regression, we use transformation to deal with hazard’s bounds: Understanding the effects of taking odds and logits (ALDA, Section 11.1.2, pp 362-365) Facts about odds scale Symmetric about 1 (50/50) Effect most prominent when hazard is larger Easy to get back to raw hazard: But it’s still bounded below by 0 and it’s asymmetric (raw differences have different meanings depending upon value of odds) Facts about odds scale Symmetric about 1 (50/50) Effect most prominent when hazard is larger Easy to get back to raw hazard: But it’s still bounded below by 0 and it’s asymmetric (raw differences have different meanings depending upon value of odds) Facts about logit scale Not bounded at all, although you need to get used to negative values (whenever hazard<.50) Usually regularizes distance betw hazard f n s Stretches distance between small values Compresses distance between large values It’s easy to get back to raw hazard Facts about logit scale Not bounded at all, although you need to get used to negative values (whenever hazard<.50) Usually regularizes distance betw hazard f n s Stretches distance between small values Compresses distance between large values It’s easy to get back to raw hazard

10. " " " " " " ! ! ! ! ! ! 6789101112 Grade 0.0 -2.0 -3.0 -4.0 Logit(hazard) PT=0 PT=1 What population model might have generated these sample data? Plotting sample hazard estimates and overlaying alternative hypothesized models (ALDA, Section 11.1.1, pp 366-369) Flat population logit hazard, shifted when PT switches from 0 to 1 Linear population logit hazard, shifted when PT switches from 0 to 1 General population logit hazard, shifted when PT switches from 0 to 1 Three reasonable features of a population discrete-time hazard model 1.For each predictor value, there is a population logit-hazard function. When the predictor(s)=0, we call it the “baseline” logit-hazard function. 2.Each population logit-hazard function is constrained to have the identical shape, regardless of predictor value. This is an assumption, and it can—and will—be relaxed later. 3.The distance between each of these logit hazard functions is identical in every time period. Differences in predictor value only “shift” the logit-hazard function “vertically.” This assumption can—and will—be relaxed later In the meantime, the magnitude of this shift is the magnitude of the predictor’s effect Three reasonable features of a population discrete-time hazard model 1.For each predictor value, there is a population logit-hazard function. When the predictor(s)=0, we call it the “baseline” logit-hazard function. 2.Each population logit-hazard function is constrained to have the identical shape, regardless of predictor value. This is an assumption, and it can—and will—be relaxed later. 3.The distance between each of these logit hazard functions is identical in every time period. Differences in predictor value only “shift” the logit-hazard function “vertically.” This assumption can—and will—be relaxed later In the meantime, the magnitude of this shift is the magnitude of the predictor’s effect

11. How do we specify a discrete-time hazard model that has these 3 features? (ALDA, Section 11.2, pp369-372) Recode PERIOD into a set of TIME indicators Constant vertical shift in logit hazard associated with variation in PT How does this model relate to the previous graph?

12. Carefully unpacking the discrete-time hazard model (ALDA, Section 11.2.1, pp 372-376) When PT=0, you get the baseline logit hazard function When PT=1, you shift this entire baseline vertically by  1 And we can add predictors just as in regular (logistic) regression How does this model behave when hazard is expressed in the other scales?

13. What does the DT hazard model look like when expressed on the other scales? (ALDA, Section 11.2.2, pp 376-379) On the logit scale, the distances between functions is identical in every time period (assumption built into our model) On the odds scale, one function is a constant magnification (or dimunition) of the other —they are proportional On the odds scale, one function is a constant magnification (or dimunition) of the other —they are proportional On the hazard scale, the functions have no constant relationship (Would need to use a complementary log-log transformation to get a proportional hazards model) On the hazard scale, the functions have no constant relationship (Would need to use a complementary log-log transformation to get a proportional hazards model) The “standard” DTSA model is a proportional odds model!

14. Fitting the model to data: Use logistic regression in the person-period data set (ALDA, Section 11.3, pp 378-386) All parameter estimates, standard errors, t- and z- statistics, goodness-of-fit statistics, and tests will be correct for the discrete-time hazard model OutcomeTIME indicatorsSubstantive predictors  ’s estimate the baseline logit hazard function  ’s assess the effects of substantive predictors

15. Strategies for interpreting the  ’s: ML estimates of the baseline hazard function (ALDA, Section 11.4.1, pp 386-388) Because there are no predictors in Model A, this baseline is for the entire sample If est’s are approx equal, baseline is flat If est’s decline, hazard declines If est’s increase (as they do here), hazard increases Simplifying interpretation by transforming back to odds and hazard ^ Because there are no substantive predictors, Model A’s estimates are the full sample estimates

16. Strategies for interpreting the  ’s: ML estimates of the substantive predictors’ effects (ALDA, Section 11.4.2 & 11.4.3, pp 388-390) ^ Dichotomous predictors As in regular logistic regression, antilogging a parameter estimate yields the estimated odds-ratio associated with a 1-unit difference in the predictor: The estimated odds of first intercourse for boys who have experienced a parenting transition are 2.4 times higher than the odds for boys who did not experience such a transition. Continuous predictors Antilogging still yields a estimated odds-ratio associated with a 1-unit difference in the predictor: The estimated odds of first intercourse for boys whose parents exhibited “1 unit more” of antisocial behavior are 1.56 times the odds for boys whose parental antisocial behavior was one unit lower. Because odds ratios are symmetric about 1, you can also invert the odds ratios and change the reference group Estimated odds of first intercourse for boys who did not experience a parenting transition are 1/(2.40)=.42 or approximately 40% the odds for boys who did Estimated odds of first intercourse for boys who parents have “1 unit less” of antisocial behavior are 1/(1.56)=.641 or approximately 2/3 rd s the odds for boys whose parents were 1 unit higher

17. Displaying fitted hazard and survivor functions Illustrating the general idea using Model B for a single dichotomous predictor (ALDA, Section 11.5.1, pp 392-394) With a single dichotomous predictor, there are only 2 possible prototypical functions: PT=0 (for boys from stable homes with no parenting transitions before 7 th grade) PT=1 (for boys who experienced one of more early parenting transitions) With a single dichotomous predictor, there are only 2 possible prototypical functions: PT=0 (for boys from stable homes with no parenting transitions before 7 th grade) PT=1 (for boys who experienced one of more early parenting transitions)

18. Displaying fitted hazard and survivor functions (ALDA, Section 11.5.1, pp 392-394) Constant vertical separation of 0.8736 (the parameter estimate for PT) Easy to see the effect of PT  Non-constant vertical separation (no simple interpretation because the model is proportional in odds, not hazard) Effect of PT cumulates into a large difference in estimated median lifetimes (9.9 vs. 11.8  2 years)

19. Displaying fitted hazard and survivor functions when some predictors are continuous (ALDA, Section 11.5.1, pp 392-394) As in growth modeling, select substantively interesting prototypical values and proceed in just as you did for dichotomous predictors here, we’ll choose +/- 1 sd PAS (lo=- 1, medium=0, and high=+1) As in growth modeling, select substantively interesting prototypical values and proceed in just as you did for dichotomous predictors here, we’ll choose +/- 1 sd PAS (lo=- 1, medium=0, and high=+1) PASPT=0PT=1 Low (-1)>12.010.7 Medium (0)11.510.1 High (+1)10.99.6 Estimated Median Lifetimes

20. Comparing goodness of fit using deviance statistics and information criteria : The strategies are generally the same as in growth modeling (ALDA, Section 11.6, pp 397-402) TIME dummies Deviance smaller value, better fit,  2 dist., compare nested models AIC, BIC smaller value, better fit, compare non- nested models Model B vs. Model A provides an uncontrolled test of H 0 :  PT =0  Deviance=17.30(1), p<.001 Model C vs. Model A provides an uncontrolled test of H 0 :  PAS =0  Deviance=14.79(1), p<.001 Model D vs. Models B&C provide controlled tests [Both rejected as well]