John B. Willett & Judith D. SingerHarvard Graduate School of

Education

Introducing discrete-time survival analysis ALDA, Chapter Eleven

“To exist is to change, to change is to mature”Henri Bergson

Chapter 11: Fitting basic discrete-time hazard modelsChapter 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

Illustrative example: Grade at first heterosexual intercourseIllustrative 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 12th gradesBy the end of data collection (at the end of 12th grade), n=126 (70.0%) had had sexThe 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 7th 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 12th gradesBy the end of data collection (at the end of 12th grade), n=126 (70.0%) had had sexThe 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 7th 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)

The life table: Summarizing the distribution of event occurrence over timeThe life table: Summarizing the distribution of event occurrence over time

(ALDA, Section 10.1, pp 326-329)

Risk set n censored in interval jn experiencing target

event in interval j

J intervals,T=7, 8, …, 12

How might we summarize the distribution of event occurrence?

How might we summarize the distribution of event occurrence?

Assessing the conditional risk of event occurrence: The discrete-time hazard functionAssessing the conditional risk of event occurrence: The discrete-time hazard function

(ALDA, Section 10.2.1, pp 330-339)

6 7 8 9 10 11 12

Grade

0.00

0.10

0.20

0.30

h(t)

riskatn

eventsnth

j

jj ,)(ˆ

0833.0180

15)(ˆ 7 th

1519.0158

24)(ˆ 9 th

3250.080

26)(ˆ 12 th

Discrete-time hazardConditional probability that individual i will experience the

target event in time period j (Ti = j) given that s/he didn’t experience it in any earlier time period (Ti j)

h(tij)=Pr{Ti= j|Ti 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 addressEstimation is easy because each value of hazard is based on that interval’s risk set.

Discrete-time hazardConditional probability that individual i will experience the

target event in time period j (Ti = j) given that s/he didn’t experience it in any earlier time period (Ti j)

h(tij)=Pr{Ti= j|Ti 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 addressEstimation is easy because each value of hazard is based on that interval’s risk set.

Cumulating risk over time: The survivor function (and median lifetime)Cumulating risk over time: The survivor function (and median lifetime)

(ALDA, Section 10.2, pp 330-339)

6 7 8 9 10 11 12

Grade

0.00

0.25

0.50

0.75

1.00

S(t)

7444.0]1519.01[8778.0)(ˆ9 tS

Discrete-time survival probabilityProbability that individual i will “survive”

beyond time period j (Ti > j)

(i.e.,will not experience the event until after time period j).

S(tij)=Pr{Ti > j}

Also a probability bounded by 0 and 1.

At the beginning of time, S(ti0)=1.0

Strategy for estimation: Since h(tij) tells us about the probability of event occurrence, 1-h(tij) tells us about the probability of non-occurrence (i.e., about survival)

Discrete-time survival probabilityProbability that individual i will “survive”

beyond time period j (Ti > j)

(i.e.,will not experience the event until after time period j).

S(tij)=Pr{Ti > j}

Also a probability bounded by 0 and 1.

At the beginning of time, S(ti0)=1.0

Strategy for estimation: Since h(tij) tells us about the probability of event occurrence, 1-h(tij) tells us about the probability of non-occurrence (i.e., about survival)

9167.0]0833.01[0.1)(ˆ7 tS

)](ˆ1)[(ˆ)(ˆ1 jjj thtStS

ML = 10.6

Estimated median lifetime

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

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 setConverting 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

Person-level data set:one row per person

ID T CENSOR PT

193 9 0 1

126 12 0 1

407 12 1 0

ID 407 was censored,

remaining a virgin through 12th grade

ID 126 had sex in the 12th grade

ID 193 had sex in the 9th grade

Contemplating a DTSA model: Inspecting sample plots of within-group hazard and survivor functions

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 ns:• 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 ns:• 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 ns:• 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 ns:• 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

We’re almost ready to go, but back to the bounded nature of hazard

As in regular regression, we use transformation to deal with hazard’s bounds: Understanding the effects of taking odds and logits

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)

6 7 8 9 10 11 12

Grade

0.0

0.2

0.4

0.6

0.8

1.0Estimated hazard

No early parenting transitions

One or more early parenting transitions

6 7 8 9 10 11 12

Grade

0.0

0.3

0.5

0.8

1.0Estimated odds

No early parenting transitions

One or more early parenting transitions

hazard

hazardodds

1

odds

6 7 8 9 10 11 12

Grade

0.0

-1.0

-2.0

-3.0

-4.0

Estimated logit(hazard)

No early parenting transitions

One or more early parenting transitions

hazard

hazardodds

1log)log(

logit

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)

odds

oddshazard

1

Facts about logit scaleNot bounded at all, although you need to get used to negative values (whenever hazard<.50)Usually regularizes distance betw hazard fns

Stretches distance between small valuesCompresses distance between large values

It’s easy to get back to raw hazard

Facts about logit scaleNot bounded at all, although you need to get used to negative values (whenever hazard<.50)Usually regularizes distance betw hazard fns

Stretches distance between small valuesCompresses distance between large values

It’s easy to get back to raw hazard

logit

ehazard

1

1

6 7 8 9 10 11 12

Grade

0.0

-1.0

-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

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 model1. 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 model1. 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

How do we specify a discrete-time hazard model that has these 3 features?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

ijjij PTDDDthlogit 1121277 ][)(

Constant vertical shift in logit hazard associated

with variation in PT

How does this model relate to the previous graph?

How does this model relate to the previous graph?

Carefully unpacking the discrete-time hazard modelCarefully unpacking the discrete-time hazard model

(ALDA, Section 11.2.1, pp 372-376)

ijjij PTDDDthlogit 1121277 ][)(

6 7 8 9 10 11 12

Grade

0.0

-1.0

-2.0

-3.0

-4.0

Logit(hazard)

PT = 0

PT = 1

(D7=1) (D8=1) (D12=1).........

12

8

9

1011

7

1

When PT=0, you get the baseline logit hazard function

When PT=1, you shift this entire baseline vertically by 1

1

iijjij PASPTDDDt 21121277 ][)( hlogitAnd we can add predictors just as in regular (logistic) regression

How does this model behave when hazard is expressed in the other scales?

How does this model behave when hazard is expressed in the other scales?

What does the DT hazard model look like when expressed on the other scales?What does the DT hazard model look like when expressed on the other scales?

(ALDA, Section 11.2.2, pp 376-379)

6 7 8 9 10 11 12

Grade

0.0

-1.0

-2.0

-3.0

-4.0

Logit(hazard)

PT = 0

PT = 1

1

logit

ehazard

1

1

6 7 8 9 10 11 12

Grade

0.0

0.1

0.2

0.3

0.4

0.5Hazard

PT = 0

PT = 1

logiteodds

6 7 8 9 10 11 12

Grade

0.0

0.2

0.4

0.6

0.8Odds

PT = 0

PT = 1

exp(1)

On the logit scale, the distances between functions is identical in every time period

(assumption built into our model)

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! The “standard” DTSA model is a proportional odds model!

Fitting the model to data: Use logistic regression in the person-period data setFitting 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

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 indicators Substantive predictors

PASPTDDD

PASDDD

PTDDD

DDD

2112128877

212128877

112128877

12128877

...)

...)

...)

...)

j

j

j

j

h(t logit :D Model

h(t logit :C Model

h(t logit :B Model

h(t logit :A Model

’s estimate the baseline logit hazard function

’s assess the effects of substantive predictors

Strategies for interpreting the ’s: ML estimates of the baseline hazard functionStrategies 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

Strategies for interpreting the ’s: ML estimates of the substantive predictors’ effectsStrategies for interpreting the ’s: ML estimates of the substantive predictors’ effects

(ALDA, Section 11.4.2 & 11.4.3, pp 388-390)

^

4.28736.0ˆee PT

Dichotomous predictorsAs 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 predictorsAntilogging still yields a

estimated odds-ratio associated with a 1-unit

difference in the predictor:

56.14428.0ˆee PAS

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 2/3rds the odds for boys whose

parents were 1 unit higher

Displaying fitted hazard and survivor functionsIllustrating the general idea using Model B for a single dichotomous predictor

Displaying fitted hazard and survivor functionsIllustrating the general idea using Model B for a single dichotomous predictor

(ALDA, Section 11.5.1, pp 392-394)

)](ˆ1)[(ˆ)(ˆ1 jjj thtStS PTth logit jj 1

ˆˆ)(ˆ

)(ˆ1

1)(ˆ

jth logitje

th

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)

Displaying fitted hazard and survivor functionsDisplaying 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 PTNon-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)

Displaying fitted hazard and survivor functions when some predictors are continuousDisplaying 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)

6 7 8 9 10 11 12

Grade

0.0

0.1

0.2

0.3

0.4

0.5Fitted hazard

6 7 8 9 10 11 12

Grade

0.0

0.5

1.0Fitted survival probability

One or more early parenting transitions

One or more early parenting transitions

No early parenting transitions

No early parenting transitions

PAS=+1

PAS= -1

PAS= 0

PAS=+1

PAS= -1

PAS= 0

PAS = -1

PAS = +1PAS = 0

PAS = -1

PAS = +1PAS = 0

PAS PT=0 PT=1

Low (-1) >12.0 10.7

Medium (0) 11.5 10.1

High (+1) 10.9 9.6

Estimated Median Lifetimes

Comparing goodness of fit using deviance statistics and information criteria:The strategies are generally the same as in growth modeling

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 H0: PT=0Deviance=17.30(1), p<.001

Model C vs. Model A provides an uncontrolled test of H0: PAS=0Deviance=14.79(1), p<.001

Model D vs. Models B&C provide controlled tests

[Both rejected as well]