Session 3: Event History Analysis: Basic Models Karl Ulrich Mayer Life Course Research: Theoretical...

Session 3:Event History Analysis: Basic Models

Karl Ulrich Mayer

Life Course Research:Theoretical Issues, Empirical Applications and Methodological Problems

Sociological Methodology Workshop Series, Academia Sinica, Taipei, TaiwanSeptember 20-24, 2004

1. Research Designs and Time-Continuous Data

2. Terminology on Time-Continuous Data

3. Censoring and Patterns of Censoring

4. Transition Probability

5. Mean Transition Rate in Interval [t,t‘)

6. Examples and Exercises

7. (Instantaneous) Transition Rate at Time t: r(t)

8. Probability Distribution of T and Survival Function

9. Methods of Survival Analysis: Mortality Table Method

10. Methods of Survival Analysis: Product-Limit Estimator

11. Methods of Survival Analysis: Comparison of Survival Functions

12. Methods of Survival Analysis: Analysis of Local Interdependence

13. Exponential Model: Basics

14. Exponential Model: Time-Constant Covariates

15. Exponential Model: Duration-Dependent Rates (Employment Transition Rates)

16. Exponential Model: Time-Dependent Covariates

17. Event History Analysis with Logistic Regression

Cox – Partial Likelihood Models

Cox‘s Proportional Hazards Regression Model

Functions (graphics)

Event History Analysis: Literature

Software for Event History Analysis

Outline

Event History Analysis 2004 /1-1

1. Research Designs and Time-Continuous Data

Cross-sectional data:

Measurement only at time t2

Panel data: Measurements at a sequence of discrete points in time t1, t2, ...

Event History Data:

Continuous measurement in time up to the time of the survey t4

Measurement mostly retrospective:

Advantage: relatively cost-efficient

Disadvantage: Potential recall errort2

Married

Single

t4 Time t

Consensual union

t3

Event data

t2

Married

Single

t4Time t

Consensual union

t3t1

Panel Data

°°

° °

t2

Married

Single

State space Y

Time t

Consensual union

Cross-sectional data

°


2a. Terminology on Time-Continuous Data: 1-Process Model

Episode (Spell): time interval, in which a person i stays i in a state: here the third measured episode

si(3) is the starting time; ti(3) is the ending time

ti(3) - si(3) is equal to the duration of the episode

oi(3) is the initial state in episode 3 (Origin); d i(3) is the new state after the end of episode 3 (Destination)

Time axis (clock: Specification of time dimension, in which process is being measured)

Examples: Age (Duration since birth), Duration since age 15, Duration since occurrence of an event

si(3) ti(3)

di(3)

oi(3)Married

Single

t4

State space Y

Time t

Consensual union

Family state of a person i across time

State space: set of possible outcomes of a processual variable Y („family situations“).

The state space of the process is discrete.


2b. Terminology on Time-Continuous Data

Event/transition: change in a state of the processual variable Y

State and ending time of a state is usually defined by an event.

Censoring: the end of episode 5 is unknown.

Reason: limitation of the observation window (e.g. time of interview)

si(3) ti(3)

di(3)di(2)

oi(3)Married

Single

t4

State space Y

Time t

Consensual union

Family situations of person i across time


2c. Terminology on Time-Continuous Data: Special Models

One-Episode-Model with a state space of two states : Origin and Destination

Variant: There are several different destination states: multi-state model

si ti

oi

di Ever Married

Never Married

t4

State space Y

Time t


2d. Terminology on Time-Continuous Data: Special Models

Multiple-Episodes-Multi-State-Model

Repeatable vs. absorbing states

Married

Single

t4

State space Y

Time t

Consensual union


2e. Terminology on Time-Continuous Data

Describing a multiple-episode-multi-state-process of person i

{(ui, mi, oi, di, si, ti, xi); mi,=1,..., Mi}

ui is the identification number of person i,

mi is the number of the episode,

xi is a vector with additional time-constant or time-variant attributes


3. Censoring and Patterns of Censoring

Left Censoring Right Censoring


3a. Dual-Process-Model or Parallel Processes

Process 2

Process 1

t3 t4

Sub-episodes in process 2


4. Transition Probability

(conditional) transition probability in interval [t, t‘)

q[t, t‘) = Pr ( t T < t‘ | t T ), for t < t‘

or

T is a random variable, which represents the timing of the event

q[t, t‘) = number of persons i with t ti < t‘ / number of persons i with t ti

q[t, t‘) = number of events in [t, t‘) / number of persons i with yti = oi

q[t, t‘) = number of events in [t, t‘) / number of persons „at risk“ at time t

si ti

oi

di Ever Married

Never Married

t4

State space Y

Time t

One-Episode-Two-States-Model


5a. Mean Transition Rate in Interval [t, t‘)

Mean transition rate in time interval [t, t‘)

r[t, t‘) = number of persons i with t ti < t‘ / total of durations, which persons i

with t ti spent in time interval [t,t‘) in state oi

r[t, t‘) = number of events in [t, t‘) / total of durations, which persons i with t ti spend in interval [t,t‘) in state oi

r[t, t‘) = number of events in [t, t‘) / total of durations, which persons i are in time interval [t,t‘) „at risk“

si ti

oi

di Ever Married

Never Married

t4

State space Y

Time t

One-Episode-Two-States Model


5b. Mean Transition Rate in Interval [t, t‘)

Mean Transition Rate in Interval[t, t‘)

„measures“ the average event flow in interval [t, t‘) per time unit (month)

Analogy: average speed!

si ti

oi

di Ever Married

Never Married

t4

State space Y

Time t

One-Episode-Two-States-Model


6a. Examples and Exercises

One-Episode-Multi-State-Model:

The state space of Y is {„1“(„Single“), 2 („consensual union“), 3 („Married“)}

We observe only the first transition out of state „1“ into state „2“ or „3“

Respondent 20 21 22 23 24 25 26 27 28 29 301 1 1 2 2 2 3 3 3 1 2 2 21 Y 4 M2 1 1 2 2 2 2 2 2 1 1 1 21 Y 8 M3 1 1 1 1 1 1 1 1 3 3 3 27 Y 11 M4 1 1 1 1 1 1 1 1 1 1 2 29 Y 1 M5 1 2 2 1 1 2 3 3 2 2 2 20 Y 3 M6 1 1 1 3 3 3 3 1 1 1 1 22 Y 7 M7 1 1 1 2 2 2 2 2 3 3 3 22 Y 7 M8 1 1 1 1 2 2 2 2 2 2 2 23 Y 2 M9 1 1 1 1 1 1 1 1 3 3 3 27 Y 5 M10 1 1 1 1 1 1 1 1 1 1 1 censored

Exact age at time of change

"Survey sample" (10 respondents)State at Age t


6b. Examples and Exercises

Respondent 20 21 22 23 24 25 26 27 28 29 301 1 1 2 2 2 3 3 3 1 2 2 21 Y 4 M2 1 1 2 2 2 2 2 2 1 1 1 21 Y 8 M3 1 1 1 1 1 1 1 1 3 3 3 27 Y 11 M4 1 1 1 1 1 1 1 1 1 1 2 29 Y 1 M5 1 2 2 1 1 2 3 3 2 2 2 20 Y 3 M6 1 1 1 3 3 3 3 1 1 1 1 22 Y 7 M7 1 1 1 2 2 2 2 2 3 3 3 22 7 M8 1 1 1 1 2 2 2 2 2 2 2 23 Y 2 M9 1 1 1 1 1 1 1 1 3 3 3 27 Y 5 M10 1 1 1 1 1 1 1 1 1 1 1 censored

"Survey sample" (10 respondents)State at Age t Exact age at time of

change

Example


6c. Examples and Exercises

Respondent 20 21 22 23 24 25 26 27 28 29 301 1 1 2 2 2 3 3 3 1 2 2 21 Y 4 M2 1 1 2 2 2 2 2 2 1 1 1 21 Y 8 M3 1 1 1 1 1 1 1 1 3 3 3 27 Y 11 M4 1 1 1 1 1 1 1 1 1 1 2 29 Y 1 M5 1 2 2 1 1 2 3 3 2 2 2 20 Y 3 M6 1 1 1 3 3 3 3 1 1 1 1 22 Y 7 M7 1 1 1 2 2 2 2 2 3 3 3 22 Y 7 M8 1 1 1 1 2 2 2 2 2 2 2 23 Y 2 M9 1 1 1 1 1 1 1 1 3 3 3 27 Y 5 M10 1 1 1 1 1 1 1 1 1 1 1 censored

"Survey sample" (10 respondents)State at Age t Exact age at time of

change

r[20,30)1j

20 21 22 23 24 25 26 27 28 29 20-30

r[t,t+1)12 1/111

r[t,t+1)13 0/120

Age-specific Average Transition Rates

Age t

r[t,t+1)1j

= Number of episodes with change to j (events) in the interval [t,t+1) / Cumulative episode time (length of stay) in the interval [t,t+1) in state 1 measured in months ("person months at risk")

The estimates amount to: 1/110 = 1/(9*12+2); 0/114 = 0/(9*12+6)


7a. (Instantaneous) Transition Rate at Time t: r(t)

r(t) = l i m r[t, t‘) = l i m q[t, t‘) / ( t‘ -t) t‘ t t‘ t

r(t) = l i m (transition probability per time unit) t‘ t

si ti

oi

di Ever Married

Never Married

t4

State space Y

Time t

One-Episode-Two-State-Model


7b. (Instantaneous) Transition Rate at Time t: r(t)

Inversely, it holds:

r[t, t‘) = [ r() d ] / ( t‘ -t) = : R(t) / (t‘-t)

If r(t) in [t,t‘) is constantly equal to r, it follows that

r[t, t‘) = r

t

t‘


8a. Probability Distribution of T and Survival Function

For the random variable T (event time point) the probability distribution F(t) is defined as follows:

F(t) = Pr (T t)

The corresponding probability density is f(t):

f(t) = l i m (F(t‘) - F(t)) / (t‘ - t)=dF(t)/d(t)t‘ t

= l i m Pr ( t T < t‘) / (t‘ - t) t‘ t

= F´(t) as the first moment where differentiable.and the „survival function“ G(t)

G(t) = 1 - F(t) = Pr (T > t)


8b. Probability Distribution of T and Survival Function

r(t), f(t) and G(t) are standing in close relationship to each other:

r(t) = l i m q[t, t‘) / (t‘ - t) = l i m Pr ( t T < t‘ | T t ) / ( t‘ -t) t‘ t t‘ t

= l i m [Pr ( t T < t‘ ) / (t‘ - t)] * 1 / Pr (T t) t‘ t

= f(t) / G(t)

Recall and note: q[t, t‘) = Pr ( t T < t‘ | T t)


8c. Probability Distribution of T and Survival Function

It follows

G(t) = exp ( - r() d ) =: exp ( - H(t))

and

q[t, t‘) = Pr ( t T < t‘ | t T ) = [G(t) - G(t‘)] / G(t)

= 1 - G(t‘) / G(t)

= 1 - exp ( - r() d )

(t‘ - t) * r(t)

since 1 - exp (-x) x for small x.

0

t

t'

t


8d. Probability Distribution of T and Survival Function

As illustration the following figure for a model with a constant Rate r

The Functions F(t), G(t), r(t) and f(t)

0

0,2

0,4

0,6

0,8

1

0 5 10 15

r(t)

F(t)

G(t)

f(t)


8e. Probability Distribution of T and Survival Function

As illustration for a model with time variable Rate r


0

0,2

0,4

0,6

0,8

1

0 5 10 15

r(t)

F(t)

G(t)

f(t)


9a. Methods of Survival Analysis: Mortality Table Method

Step 1: Cut the time axis in L time intervals of equal length Il, l =1,...L:Il = [l, l+1); l =1,...L

and one interval with no upper limit IL+1 = [L, )

oi

di Ever Married

Never Married

t4

State space Y

Time t1 l l+1

Il


9b. Methods of Survival Analysis: Mortality Table Method

Step 2: Estimate the transition probabilities ql for the intervals Il, l =1,...L taking the censored spells into account:

given:El the number of events (transitions) in Il .

Rl = ( Nl - 0,5 * Zl ) and Nl the persons „at risk“ at time l Zl the number of censored events in Il

Step 3: Estimate the „transition probabilities“ pl for theintervals Il, l =1,...L :

l

l1lll R

E)),([qq

ll q1p


9c. Methods of Survival Analysis: Mortality Table Method

Step 4: Estimate the values of the survival function G(t) at the points l , l =1,...L taking censored cases into account:

Step 5: Estimate approximatively the values of the density function f(t) for the midpoints of the intervals Il, l =1,...L :

l1l

1ll

l1l

l1ll1ll

GG

FF

)2

(f f

lll

1-l1-lll

11

G1)(τFF :follows it and

pG)(τGG

1)(τGG


9d. Methods of Survival Analysis: Mortality Table Method

Step 6: Compute approximatively the values of the rate function r(t) for the midpoints of the intervals Il, l =1,...L :

Then it follows:

2/)GG(

fr

1ll

ll

l

l

l1l1ll

1ll

l1ll q2

q21

2/)GG(

GG1r


9e. Methods of Survival Analysis: Mortality Table Method

Step 7: Compute the standard deviations of the estimates of Gl , fl , rl for l =1,...L :

21

2l1ll

21

il

ll

21

il

l1l

1i il

l

l1l

lll

21

1l

1i il

lll

))(r(411)

Rq

r()rSE(

)Rq

p

Rp

q(

Gq)fSE(

Rp

q/G)GSE(


9f. Methods of Survival Analysis: Mortality Table Method

Step 1: Cut the time axis in L time intervals of equal length Il, l =1,...LStep 2: Estimate the transition probabilities ql Step 3: Estimate the „transition probabilities“ pl

Step 4: Estimate the values of the survival function G(t)Step 5: Estimate approximatively the values of the density function f(t) Step 6: Compute approximatively the values of the rate function r(t)Step 7: Compute the standard deviations of the estimates of Gl , fl , rl

oi

di Ever Married

Never Married

t4

State space Y

Time t1 l l+1

Il


10a. Methods of Survival Analysis: Product-Limit Estimator

This estimator also goes under the name of Kaplan-Meier-Estimator.

The survival function G(t) is being estimated, without cutting the time axis in discrete intervals. The estimates are, therefore, „close to the events“.

si ti

oi

di Ever Married

Never Married

t4

State space Y

Time t

One-Episode-Two-State-Model


10b. Methods of Survival Analysis: Product-Limit Estimator

Step 1: Sort the episodes of the observation units i=1, ... N according to length viz. ti ( if the common starting point is 0)

i = 5211452

t

t5 = 1

t14 = 4

t2 = 2


10c. Methods of Survival Analysis: Product-Limit Estimator

Step 2: If there are no censored events, estimate:

i = 5211452

t

t5 = 1

t14 = 4

t2 = 2

1lll

l

ll

l

t for )(G)t(G

to according

risk“ „at number the is R ,N

R)(G


10d. Methods of Survival Analysis: Product-Limit Estimator

Step 2: If there are no cases of censoring, estimate:

Case iti = taul Ri+ G(taul)0 60 1

5 2 59 0,98332 4 58 0,96671 5 57 0,95

14 9 55 0,916752 9 55 0,9167

.

.

.

G(t)

0

0,2

0,4

0,6

0,8

1

0 5 10 15 20 25 30

t


10e. Methods of Survival Analysis: Product-Limit Estimator

Step 2: If there are no cases of censoring:

Or let it be:

El = number of events at l Rl = number „at risk“ at l („risk set“)

i = 5211452

t

t5 = 1

t14 = 4

t2 = 2

)R

E(1...)

R

E(1)

R

E(11)(G

l

l

2

2

1

1l

1)-(l

l

1

21l R

R...

R

R

N

R1)(G


10f. Methods of Survival Analysis: Product-Limit Estimator

Step 3: In case of censored cases:

Let b: El = number of events at l Zl = number of censored events in [l-1, l)Rl = (number „at risk“ at l) = Rl-1 - El-1 - Zl

(Rl contains the cases censored exactly atl)

i = 5211452

t

t5 = 1

t14 = 3

t2 (censored)

1ltl for )l(G tl:l

)lRlE

(1 (t)G

or

)lRlE

(1...)2R2E

(1)1R1E

(11)l(G


10g. Methods of Survival Analysis: Product-Limit Estimator

Step 3: In case of censoring:

Case iti = taul Ri G(taul)0 60 1

5 2 60 0,98332 4 Z 0,98331 5 58 0,9664

14 9 57 0,932552 9 57 0,9325

.

.

.

G(t)

0

0,2

0,4

0,6

0,8

1

0 5 10 15 20 25 30

t


10h. Methods of Survival Analysis: Product-Limit Estimator

Step 4: Estimate the standard deviation for the survival function

21

:l lll

l

l)ER(R

E)(G))(GSE(

t

tt


10h. Methods of Survival Analysis: Product-Limit Estimator

Step 5: Estimate the cumulative Rate R(t).We know it follows:

R(t))exp())dr(exp(G(t)t

0

From that it follows:

(t))G( log -(t)H


10i. Methods of Survival Analysis: Product-Limit Estimator

Case i ti Ri G(taul) H(t)0 60 1 0

5 2 60 0,983 0,0172 4 Z 0,983 0,0171 5 58 0,966 0,034

14 9 57 0,932 0,0752 9 57 0,932 0,07

.

.

G(t); H(t)

0

0,2

0,4

0,6

0,8

1

0 10 20 30

t


10j. Methods of Survival Analysis: Product-Limit Estimator

This estimator also goes under the name of Kaplan-Meier

Step 1: Sort the episodes of the observation units i=1, ... N according to length viz. ti ( if the common starting point is 0)

Step 2: Estimate

1lll t:l l

l t for )(G)R

E(1 (t)G

l

Step 3: Estimate the standard deviation for G(t)

21

t :l lll

l

l)ER(R

E)t(G))t(GSE(

Step 4: Estimate the cumulative rate H(t).

(t))G( log -(t)H


11a. Methods of Survival Analysis: Comparison of Survival Functions

A simple but powerful additional method:

Estimate the survival function in regard to an event for different sub-populations und compare them to each other.

Example: One-Episode-Two-States-Model

Event: Birth of first child

Compare: Men and women

East- und West Germans

Old and young cohorts


11b. Methods of Survival Analysis: Comparison of Survival Functions

Figure 1: Transition to first birth, cohorts 1959-1961 and 1971, Kaplan-Meier survival curve

Panel 1: East German cohort 1959/61 and 1971 Panel 2: East and West German cohort 1971

0%

50%

100%

16 18 20 22 24 26

Age of woman

Cohort 1971

Cohort 1959/61

0%

50%

100%

16 18 20 22 24 26

Age of woman

East Germany

West Germany

Source: GLHS, East German cohorts 1959-61 and East and West German cohort 1971


11c. Methods of Survival Analysis: Comparison of Survival Functions

Estimate for survival functions for one event in different sub-populations (subsets of episodes!)

Is the difference between the survival functions important? Is it significant

Statistical testing

Tests on the difference of survival functions in sub-populations:

Comparison between the survival function expected under the equality assumption and the observed distribution of events in sub-population

!! Assumptions: The survival functions do not cross-over


11d. Methods of Survival Analysis: Comparison of Survival Functions

Basic principle of tests

Step 1: Sort all episodes according to their respective length/duration 1, 2, 3, ....

Step 2: Determine the number of events and of the risk sets for each time point l and each sub-population g=1,..., m: Elg und Rlg

Step 3: Determine the difference between observed and expected number of events for time point l and each sub-population g:

Dlg = Elg - Rlg * (El/ Rl)


11e. Methods of Survival Analysis: Comparison of Survival Functions

Step 4: Calculate for each sub-population the sum of these differences weighted with a factor Wl

l

lglg D*W U

Compute the vector u = (U1 , ..., Um )

Step 5: Calculate the variances of Wl *Dlg and the cova-riances of these terms for different sub-populations and sum them across all l. Compute the Matrix V

m1,...,gm;1,...,gg,g 2121)(V

Step 6: Compute the test statistic 2m-1 :

S = u‘ * V-1 * u


11f. Methods of Survival Analysis: Comparison of Survival Functions

Example 1: The weighting factors Wl are all equal to 1

log rank - Test

Then we can compute the test statistic as follows:

m

1g g

2g

E

U S

where Eg is the expected number of events in the sub-population g .


11g. Methods of Survival Analysis: Comparison of Survival Functions

Example 2: The weighting factors Wl are equal to Rl

Wilcoxon - Test (Breslow)

There are other specifications of Wl, which are leading to other versions of the Wilcoxon-Tests .

The Wilcoxon-Tests are sensible especially for differences of the survival functions at the beginning of the process.

The log rank-Test is sensible especially for differences of the survival functions at the end of the process


12a. Methods of Survival Analysis: Analysis of Local Interdependence

With the explorative method of Survival Analysis one can also determine simple interdependencies between different processes

Starting point: One- Episode-Two State Prozesses A and B.

si ti(B)

oi

di(B) One child

Childless

B

si ti(A)

oi

di(A) Ever Married

Never Married

A


12b. Methods of Survival Analysis: Analysis of Local Interdependence

Interrelation of First Birth and First Marriage Panel 3: East German cohorts 1959/61 and 1971 Panel 4: East and West German cohort 1971

0%

50%

100%

-5 -4 -3 -2 -1 0 1 2 3 4 5

Age of first child

Cohort 1971

Cohort 1959/61

0%

50%

100%

-5 -4 -3 -2 -1 0 1 2 3 4 5

Age of fitst child

East GermanyWest Germany

Notes: In Panel 3 and 4, we only selected women who had a first child at the time of interview.

Source: GLHS, East German cohorts 1959-61 and East and West German cohort 1971


12c. Methods of Survival Analysis: Analysis of Local Interdependence

Compute a new time scale:a) - Select all cases with an event in process A

- Transform for case i the time t for process B in t - ti(A)

si ti(B)

oi

di(B) One child

Childless

B

si ti(A)

oi

di(A) Ever Married

Never Married

A

ti(B) - ti(A)0

New time scale


12d. Methods of Survival Analysis: Analysis of Local Interdependence

Compute a new time scale:b) - Select all cases with an event in process B

- Transform for case i the time t for process A in t - ti(B)

si ti(B)

oi

di(B) One child

Childless

B

si ti(A)

oi

di(A) Ever Married

Never Married

A

ti(A) - ti(B) 0

New time scale


Parametric models of event history analysis fix a specific probability distribution F(t) for a distribution of waiting times or event time points T .

The parameters of the probability distribution can be modelled conditional of attributes of the observation units (covariates) and then be estimated given sample observations.

This equivalent to estimating the rate function r(t) which corresponds to the probability distribution of T conditional on covariates.

Rate regression

13a. Exponential Model: Basics


The parametric model, which will fulfill our specifications of the rate function is quite simple:

The Exponential Model

T is taken to be distributed according to an exponential function t:

F(t) = 1- exp (- a t ), a > 0

and

f(t) = a exp (- a t)G(t) = exp (- a t)

r(t) = f(t) / G(t) = a

13b. Exponential Model: Basics


The exponential model

T is distributed according to the exponential distribution:

F(t) = 1- exp (- a t ), a > 0

and

E(T) = 1/a = 1/r („mean waiting time“)

Var(T) = 1/ a

13c. Exponential Model: Basics


As illustration the following figure for an exponential model with the constant rate r(t) = a = 0,2


0

0,2

0,4

0,6

0,8

1

0 5 10 15

r(t)

F(t)

G(t)

f(t)

13d. Exponential Model: Basics


How can one estimate on the basis of an observed sample the parameter a of the exponential distribution which is equal to the time constant rate?

The commonly used estimator is the Maximum-Likelihood Method (Remember: for linear regression one uses the „Minimum-Distance-Estimator“.)

The Maximum-Likelihood-Method selects out of all possible values those values of a parameter a of the probability distribution F for which the observed sample is „maximally likely“.

More precisely:...for which the density of the probability for the realization of the sample (Likelihood) is highest as computed according to the postulated probability distribution

13e. Exponential Model: The Estimation of a


The value of the density dependent on the parameter a is given by the Likelihood function L(a| ti , i S), where S is the observed sample.

Let us assume again the One-episode-two-state-model.

E is to be the set of observation units, for which events were observed, and Z is to be the set of censored cases.Then L will be calculated as follows:

13f. Exponential Model: The Estimation of a

i

i

i

ii )G(t)r(t)G(t)f(t S)i ,t|L(aSiEiZiEi


Then we derive:

13g. Exponential Model: The Estimation of a

S)i,t|L(aln max a

or

S)i,t|L(a max a

ia

ia

ln (L) is the Log-Likelihood.


13h. Exponential Model: The Estimation of a

i ti Censored a LL1 0,25 1 0,02 -23,7371382 0,5 0 0,04 -19,84325493 0,75 1 0,06 -17,67546434 1 1 0,08 -16,21437195 1,25 1 0,1 -15,14051066 1,5 1 0,12 -14,31158127 2 0 0,14 -13,65167718 2,5 1 0,16 -13,11548889 3,5 0 0,18 -12,6737906

0,2 -12,3066275Sum ti Events 0,22 -11,9997664

13,25 6 0,24 -11,7426981Avg. Rate 0,26 -11,5274419

0,45283019 0,28 -11,34779410,3 -11,1988368

0,32 -11,07660570,34 -10,977858

Log-Likelihood-Estimate for the Parameter a of the Exponential Distribution

-20-19-18-17-16-15-14-13-12-11-10

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1

a

LL


Rate regression with time constant covariates:

Estimating the effects of time constant attributes of the observation units on the time constant transition rate a which in the exponential model is identical to the parameter a.

The estimation equation is:

r = exp (0 + 1 X 1 + ... + m X m)

X1 ,..., Xm are time constant attributes of the observation units,

1, ... , m, are regression coefficients, 0 is the constant

14a. Exponential Model: Time-Constant Covariates


The coefficients 0, ... , m are estimated using the Maximum-Likelihood Method. For our model then follows:

14b. Exponential Model: Time-Constant Covariates

SiEi

irt-rln

S)i,t|,...,L(ln maxS)i,t|L(rln max r i0,...,

ir 0

mm

SiEi

)G(tln)(trln S)i ,t|L(rln iii

Si

m

jjij

Ei

m

jjij i

10

10 t)x(exp-)x(


Interpretation of the coefficients of the covariates

14c. Exponential Model: Time-Constant Covariates

m1 Xm

X10

mm110

)exp(...)exp()exp(

)X...Xexp( r

It follows:If the covariate Xj increases by one unit, Then the estimate for r changes by the factor exp(j)

Or by (exp(j) - 1) * 100 %


Several possibilities exist to test the significance of the coefficients.

1. t - Test (just as in linear regression)

)ˆ(ˆ

ˆ t

j

j

14d. Exponential Model: Time-Constant Covariates

is approximatively normally distributed, if the sample is sufficiently large.


Several possibilities exist to test the significance of the coeffcients.

2. Likelihood-Ratio Test

14e. Exponential Model: Time-Constant Covariates

Example: Test, whether the covariates X1 und X2 contribute a significant part in explaining the estimation of rate r:

reference model: r = exp(0 )

enlarged model: r = exp (0 + 1 X 1 + 2 X2).

We get the maximal Log-Likelihood-values for the two models: ln L[Enl. model] and ln L[reference model]. The latter is smaller than the former.


Several possibilities exist to test the significance of the coefficients.

2. Likelihood-Ratio Test

14f. Exponential Model: Time-Constant Covariates

Then compute the following term

LR = 2 (ln L[Enl. model] - ln L[reference model])

This test statistic is 2-distributed. The degrees of freedom are 2. It is equal to the additional number of parameters in the enlarged model.


Rate regression with time constant covariates:

r(t) = r = exp (0 + 1 X 1 + ... + m X m)

X1 ,..., Xm are time constant of the observation units,

1, ... , m, are the regressions coefficients, 0 is the constant.

These models estimate effects on a time constant rate, i.e. it is assumed that the rate does not depend on the spell duration, but also that the effects of the variables do not depend on the spell durations

15a. Exponential Model: Duration-Dependent Rates

„Piecewise Constant Exponential Model“:

Here we assume that the rate is not constant over the entire duration of the episode, but rather is only „piecewise“ constant.


15b. Exponential Model: Duration-Dependent Rates

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

1 4 7 10

13

16

19

22

25

28

31

34

37

40

Constant Rate

Piecewise constant rate

Months

„Piecewise Constant Exponential Model“:

One dissects the observation interval into intervals, which do not need to be equal in length: Divide the time axis in L time intervals Il, l =1,...L:

Il = [l, l+1); l =1,...L The last interval is open at the upper limit.


15c. Exponential Model: Duration-Dependent Rates

oi

di Ever Married

Never Married

t4

State space Y

Time t1 l l+1

Il

The rate regression is conducted as follows:

ll 0, I für t ),exp( r(t)


15d. Exponential Model: Duration-Dependent Rates

In this case the constant varies with the duration of the episode.

For the estimation the Maximum-Likelihood-Method is being used. The rates vary with the duration intervals and require a decomposition of the integral for computing the survival function into subset integrals.

In addition, one can include time constant covariates.

In the model with proportional effects the coefficients of the covariates are independent of the duration (the duration interval Il).

lmm11l 0, I t for),X...Xexp( r(t)


15e. Exponential Model: Duration-Dependent Rates

Thus in this case the constant varies discretely with the duration of the episode, the Coefficients of the covariates are constant.

indicates the values of the interval specific constants for the rate („basis rate“) in interval+ Il.

)exp( l 0,

Finally, one can estimate the coefficients of the covariates depending on the duration (of the duration interval Il).

lmlm,1l,1l 0, I t for),X...Xexp( r(t)


15f. Exponential Model: Duration-Dependent Rates

In this case both the constant and the coefficients of the covariates vary discretely with the duration of the episode.


1. Rate regression with time constant covariates

r(t) = r = exp (0 + 1 X 1 + ... + m X m)

2. Rate regression with duration dependent effects

16a. Exponential Model: Time-Dependent Covariates

lmlm,1l,1l 0, I t for),X...Xexp( r(t)

3. Rate regression with time dependent covariates

(t))X...(t)Xexp( r(t) mm110


16b. Exponential Model: Time-Dependent Covariates

Rate regression with time-dependent covariates allow to estimate transition rates conditional on time-varying conditions in one or more parallel processes.

Y: unmarried

X: living with parents

Y: unmarried

X: not living with parents

Y: married

X: living with parents

Y: married

X: not living with parents

„Marriage process“

„leaving home“


16c. Exponential Model: Time-Dependent Covariates-Example

The state space of Y is {"1" ("Single"), 2 ("Consensual Union"), 3 ("Married")} Co-variate Ht: yellow: living with parents; white: not living with parents (change in month given as a subscript)

Respondent 20 21 22 23 24 25 26 27 28 29 301 1 1 2Mo 2 2 2 3 3 3 1 2 2 21 Y 4 M2 1 1 2 2 2 2 2 2 1 1 1 21 Y 8 M3 1 1 6Mo 1 1 1 1 6Mo 1 1 3 3 3 27 Y 11 M4 1 1 1 1 6Mo 1 6Mo 1 1 6Mo 1 1 1 2 29 Y 1 M5 1 2 2 1 1 2 3 3 2 2 2 20 Y 3 M6 1 1 1 6Mo 3 3 3 3 1 1 1 1 22 Y 7 M7 1 6Mo 1 1 2 2 2 2 2 3 3 3 22 Y 7 M8 1 1 1 1 2 6Mo 2 2 2 2 2 2 23 Y 2 M9 1 1 1 1 1 1 1 1 6Mo 3 3 3 27 Y 5 M10 1 1 1 1 1 6Mo 1 1 1 6Mo 1 1 1 censored

"Survey sample" (10 respondents)

Status at Age t Exact age at time of change

20 21 22 23 24 25 26 27 28 29

r[t,t+1)12

(living with parents) 0/90

r[t,t+1)12

(not living with parents) 1/15

Age t

r[t,t+1)1j (Ht)= Number of episodes with change to j in the interval [t,t+1) in household type Ht /

Person months "at risk" in the interval [t,t+1) in state 1 in household type Ht

Average Transition Rate according to Age and Household Type

Age-independent

The estimates amount to: 0/90 = 0/(7*12+6); 1/15 = 1(1*12+3)


16d. Exponential Model: Time-Dependent Covariates

Non-Parental HH

Parental. HH

t4

State space X

Time t

Episode split 2 sub-episodes (L=2)

oi

di Married

Single

t4

State space Y

Time tsi ti


16e. Exponential Model: Time-Dependent Covariates

Estimation on the basis of episode splittings

Preconditions: the time-dependent covariates change discretely across time and have discrete values, i.e. are dichotomous or polytomous.

Episode splitting: Dissect – in analogy to what we did when introducing the duration-dependent rate – the episodes in sub-episodes Il, l=1,...,L, in which the covariates are constant.

The survival function is then computed according to

L

1lil,il,i )s|G(t )G(t

where tl,i is the end- and sl,i the starting time of the episode Il and it holds that sl,i = si and tL,i = ti.


16f. Exponential Model: Time-Dependent Covariates

Estimating on the basis of the episode splitting

It holds:

where r(sl) = r(t) is being modelled according to

lmm110 I t for,(t))X...(t)Xexp( r(t)

))r(s)s(texp(}τd )τr(exp{)G(s

)G(t )s|G(t lll

t

sl

lll

l

l


16g. Exponential Model: Time-Dependent Covariates

Estimating on the basis of episode splitting

Then compute the Log-Likelihood according to

Si lEiSiEi

iL

1il,il,iii )s|G(tln)(trln )G(tln)(trln L(r)ln

The modelling of the duration dependency of a rate can be considered as a special case of a model with time-dependent covariates, which indicates in which duration interval one is at the moment.


Logistic Regression

Let Y be a dichotomous variable with values 1 and 0 (married vs. non-married) and is binomially distributed with parameters p = P(Y=1).

Then for the following model the parameters 0, 1,..., m are being estimated using Maximum-Likelihood:

17a. Event History Analysis with Logistic Regression

mm110

mm110

X...X )1)P(Y-1

1)P(Yln(

)X...Xexp( 1)P(Y-1

1)P(Y


Logistic regression

17b. Event History Analysis with Logistic Regression

Logit. or odd"-log" means 1)P(Y-1

1)P(Yln

Chance". relative" or odd"" means 1)P(Y-1

1)P(Y

These terms like transition rates cannot be observed directly.


Logistic regression

17c. Event History Analysis with Logistic Regression

For case i one estimates the probability as follows:

)xˆ...xˆêxp(-1

1

)xˆ...xˆêxp(1

)xˆ...xˆêxp(1)(yP

im,mi1,10

im,mi1,10

im,mi1,10

P(Y=1) is distributed like a logistic distribution.


oi

di Married(1)

Unmarried (0)

t4

State space Y

Time tsi ti

17d. Event History Analysis with Logistic Regression

Step 1: Changing to discrete time:

Divide the episode (si, ti] for the observation unit i in monthly intervals Mil ,

l=1,..., L. The last month MiL marks the end of the episode (si, ti]. We

assume here that for non-censored events the event takes place at the end of

the month MiL .


17e. Event History Analysis with Logistic Regression

Step 2: Estimate for all i the conditional probability that a month for the first

time ends with an event: P(Y(Mil)=1|0), l=1,...L; i=1,...,N.

This estimator corresponds to the relation of the number of events to the number of months „at risk“, i.e. the average monthly transition rate r.

Step 3: Estimate the probability that a month ends for the first time with an

event (P(Y(Mil) = 1|0), l=1,...L; i=1,...,N, conditional on time constant or

time variant covariates X1, ..., Xm, the values of which are given for each

observation unit and each episode month l .

One can estimate this using Logistic Regression .What is being estimated is

then the „odd“ of P(Y(Mil)=1|0).


17f. Event History Analysis with Logistic Regression

Step 4: Estimate the transition rates according to:

)xˆ...xˆêxp(-1

1

)xˆ...xˆêxp(1

)xˆ...xˆêxp(0)|1(yP r

li,m,mli,1,10

li,m,mli,1,10

li,m,mli,1,10li,li,




Event History AnalysisLiterature

Software for Event History AnalysisBMDP (1L, 2L)

GLIM Generalized Linear Interactive Modeling

RATE Invoking RATE

SAS (LIFEREG)

SIR Scientific Information Retrieval

SPSS Statistical Package for the Social Science

LIMDEP 5.1

RATE C Supplement to BMDP

P3FUN

FORTRAN Program for episode splitting given discrete time-dependent covariatesIn: Blossfeld, Hans-Peter, Alfred Hamerle, and Karl Ulrich Mayer (1989): Event History Analysis. Statistical Theory and Application in the Social Sciences. Hillsdale, N.J.: Lawrence Erlbaum Associates Publishers, (pp.283-284)

FORTRAN Program for episode splitting given continuous time-dependent covariatesIn: Blossfeld, Hamerle, Mayer, 1989 (pp. 285)

GLIM Macros to estimate the Weibull and Log-Logistic models of Roger and PeacockIn: Blossfeld, Hamerle, Mayer, 1989 (pp. 286-287)

PARAT Hillmar Schneider 81991): Verweildauer mit GAUSS. Frankfurt am Main/New York: Campus Verlag.

TDA Blossfeld, Hans-Peter and Götz Rohwer (2002): Techniques of Event History Analysis. Mahwah, NJ: Lawrence Erlbaum Associates, 310 pp.

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Session 3: Event History Analysis: Basic Models Karl Ulrich Mayer Life Course Research: Theoretical...

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