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Session 3: Event History Analysis: Basic Models Karl Ulrich Mayer Life Course Research: Theoretical Issues, Empirical Applications and Methodological Problems.

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Presentation on theme: "Session 3: Event History Analysis: Basic Models Karl Ulrich Mayer Life Course Research: Theoretical Issues, Empirical Applications and Methodological Problems."— Presentation transcript:

1 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, Taiwan September 20-24, 2004

2 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

3 Event History Analysis 2004 /1-1 1. Research Designs and Time-Continuous Data Cross-sectional data: Measurement only at time t 2 Panel data: Measurements at a sequence of discrete points in time t 1, t 2,... Event History Data: Continuous measurement in time up to the time of the survey t 4 Measurement mostly retrospective: Advantage: relatively cost-efficient Disadvantage: Potential recall error t2t2 Married Single t4t4 Time t Consensual union t3t3 Event data t2t2 Married Single t4t4 Time t Consensual union t3t3 t1t1 Panel Data ° ° °° t2t2 Married Single State space Y Time t Consensual union Cross-sectional data °

4 Event History Analysis 2004 /1-2 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 s i (3) is the starting time; t i (3) is the ending time t i (3) - s i (3) is equal to the duration of the episode o i (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 s i (3)t i (3) d i (3) o i (3) Married Single t4t4 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.

5 Event History Analysis 2004 /1-3 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) s i (3)t i (3) d i (3) d i (2) o i (3) Married Single t4t4 State space Y Time t Consensual union Family situations of person i across time

6 Event History Analysis 2004 /1-4 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 sisi titi oioi didi Ever Married Never Married t4t4 State space Y Time t

7 Event History Analysis 2004 /1-5 2d. Terminology on Time-Continuous Data: Special Models Multiple-Episodes-Multi-State-Model Repeatable vs. absorbing states Married Single t4t4 State space Y Time t Consensual union

8 Event History Analysis 2004 /1-6 2e. Terminology on Time-Continuous Data Describing a multiple-episode-multi-state-process of person i {(u i, m i, o i, d i, s i, t i, x i ); m i,=1,..., M i } u i is the identification number of person i, m i is the number of the episode, x i is a vector with additional time-constant or time-variant attributes

9 Event History Analysis 2004 /1-7 3. Censoring and Patterns of Censoring Left Censoring Right Censoring

10 Event History Analysis 2004 /1-8 3a. Dual-Process-Model or Parallel Processes Process 2 Process 1 t 3 t 4 Sub-episodes in process 2

11 Event History Analysis 2004 /2-1 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  t i < t‘ / number of persons i with t  t i q [t, t‘) = number of events in [t, t‘) / number of persons i with y ti = o i q [t, t‘) = number of events in [t, t‘) / number of persons „at risk“ at time t sisi titi oioi didi Ever Married Never Married t4t4 State space Y Time t One-Episode-Two-States-Model

12 Event History Analysis 2004 /2-2 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  t i < t‘ / total of durations, which persons i with t  t i spent in time interval [t,t‘) in state o i r [t, t‘) = number of events in [t, t‘) / total of durations, which persons i with t  t i spend in interval [t,t‘) in state o i r [t, t‘) = number of events in [t, t‘) / total of durations, which persons i are in time interval [t,t‘) „at risk“ sisi titi oioi didi Ever Married Never Married t4t4 State space Y Time t One-Episode-Two-States Model

13 Event History Analysis 2004 /2-3 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! sisi titi oioi didi Ever Married Never Married t4t4 State space Y Time t One-Episode-Two-States-Model

14 Event History Analysis 2004 /2-4 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“

15 Event History Analysis2004 /2-5 6b. Examples and Exercises Example

16 Event History Analysis 2004 /2-6 6c. Examples and Exercises The estimates amount to: 1/110 = 1/(9*12+2); 0/114 = 0/(9*12+6)

17 Event History Analysis 2004 /3-1 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 sisi titi oioi didi Ever Married Never Married t4t4 State space Y Time t One-Episode-Two-State-Model

18 Event History Analysis 2004 /3-2 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‘

19 Event History Analysis 2004 /3-3 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)

20 Event History Analysis 2004 /3-4 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)

21 Event History Analysis 2004 /3-5 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

22 Event History Analysis 2004 /3-6 8d. Probability Distribution of T and Survival Function As illustration the following figure for a model with a constant Rate r

23 Event History Analysis 2004 /3-7 8e. Probability Distribution of T and Survival Function As illustration for a model with time variable Rate r

24 Event History Analysis 2004 /4-1 9a. Methods of Survival Analysis: Mortality Table Method Step 1: Cut the time axis in L time intervals of equal length I l, l =1,...L: I l = [  l,  l+1 ); l =1,...L and one interval with no upper limit I L+1 = [  L,  ) oioi didi Ever Married Never Married t4t4 State space Y Time t 11 ll  l+1 IlIl

25 Event History Analysis 2004 /4-2 9b. Methods of Survival Analysis: Mortality Table Method Step 2: Estimate the transition probabilities q l for the intervals I l, l =1,...L taking the censored spells into account: given: E l the number of events (transitions) in I l. R l = ( N l - 0,5 * Z l ) and N l the persons „at risk“ at time  l Z l the number of censored events in I l Step 3: Estimate the „transition probabilities“ p l for the intervals I l, l =1,...L :

26 Event History Analysis 2004 /4-3 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 I l, l =1,...L :

27 Event History Analysis 2004 /4-4 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 I l, l =1,...L : Then it follows:

28 Event History Analysis 2004 /4-5 9e. Methods of Survival Analysis: Mortality Table Method Step 7: Compute the standard deviations of the estimates of G l, f l, r l for l =1,...L :

29 Event History Analysis 2004 /4-6 9f. Methods of Survival Analysis: Mortality Table Method Step 1: Cut the time axis in L time intervals of equal length I l, l =1,...L Step 2: Estimate the transition probabilities q l Step 3: Estimate the „transition probabilities“ p l 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 G l, f l, r l oioi didi Ever Married Never Married t4t4 State space Y Time t 11 ll  l+1 IlIl

30 Event History Analysis 2004 /4-7 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“. sisi titi oioi didi Ever Married Never Married t4t4 State space Y Time t One-Episode-Two-State-Model

31 Event History Analysis 2004 /4-8 10b. Methods of Survival Analysis: Product-Limit Estimator Step 1: Sort the episodes of the observation units i=1,... N according to length viz. t i ( if the common starting point is 0) i =5 2 1 14 52 t t 5 =  1 t 14 =  4 t 2 =  2

32 Event History Analysis 2004 /4-9 10c. Methods of Survival Analysis: Product-Limit Estimator Step 2: If there are no censored events, estimate: i =5 2 1 14 52 t t 5 =  1 t 14 =  4 t 2 =  2

33 Event History Analysis 2004 /4-10 10d. Methods of Survival Analysis: Product-Limit Estimator Step 2: If there are no cases of censoring, estimate:

34 Event History Analysis 2004 /4-11 10e. Methods of Survival Analysis: Product-Limit Estimator Step 2: If there are no cases of censoring: Or let it be: E l = number of events at  l R l = number „at risk“ at  l („risk set“) i =5 2 1 14 52 t t 5 =  1 t 14 =  4 t 2 =  2

35 Event History Analysis 2004 /4-12 10f. Methods of Survival Analysis: Product-Limit Estimator Step 3: In case of censored cases: Let b: E l = number of events at  l Z l = number of censored events in [  l-1,  l ) R l = (number „at risk“ at  l ) = R l-1 - E l-1 - Z l (R l contains the cases censored exactly at  l ) i =5 2 1 14 52 t t 5 =  1 t 14 =  3 t 2 (censored)

36 Event History Analysis 2004 /4-13 10g. Methods of Survival Analysis: Product-Limit Estimator Step 3: In case of censoring:

37 Event History Analysis 2004 /4-14 10h. Methods of Survival Analysis: Product-Limit Estimator Step 4: Estimate the standard deviation for the survival function

38 Event History Analysis 2004 /4-14 10h. Methods of Survival Analysis: Product-Limit Estimator Step 5: Estimate the cumulative Rate R(t). We know it follows: From that it follows:

39 Event History Analysis 2004 /4-15 10i. Methods of Survival Analysis: Product-Limit Estimator

40 Event History Analysis 2004 /4-16 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. t i ( if the common starting point is 0) Step 2: Estimate Step 3: Estimate the standard deviation for G(t) Step 4: Estimate the cumulative rate H(t).

41 Event History Analysis 2004 /5-1 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

42 Event History Analysis 2004 /5-2 11b. Methods of Survival Analysis: Comparison of Survival Functions

43 Event History Analysis 2004 /5-3 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

44 Event History Analysis 2004 /5-4 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: E lg und R lg Step 3: Determine the difference between observed and expected number of events for time point  l and each sub- population g: D lg = E lg - R lg * (E l / R l )

45 Event History Analysis 2004 /5-5 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 W l Compute the vector u = (U 1,..., U m ) Step 5: Calculate the variances of W l *D lg and the cova- riances of these terms for different sub-populations and sum them across all l. Compute the Matrix V Step 6: Compute the test statistic  2 m-1 : S = u‘ * V -1 * u

46 Event History Analysis 2004 /5-6 11f. Methods of Survival Analysis: Comparison of Survival Functions Example 1: The weighting factors W l are all equal to 1  log rank - Test Then we can compute the test statistic as follows: where E g is the expected number of events in the sub-population g.

47 Event History Analysis 2004 /5-7 11g. Methods of Survival Analysis: Comparison of Survival Functions Example 2: The weighting factors W l are equal to R l  Wilcoxon - Test (Breslow) There are other specifications of W l, 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

48 Event History Analysis 2004 /5-8 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. sisi t i (B) oioi d i (B) One child Childless B sisi t i (A) oioi d i (A) Ever Married Never Married A

49 Event History Analysis 2004 /5-9 12b. Methods of Survival Analysis: Analysis of Local Interdependence

50 Event History Analysis 2004 /5-10 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 - t i (A) sisi t i (B) oioi d i (B) One child Childless B sisi t i (A) oioi d i (A) Ever Married Never Married A t i (B) - t i (A)0 New time scale

51 Event History Analysis 2004 /5-11 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 - t i (B) sisi t i (B) oioi d i (B) One child Childless B sisi t i (A) oioi d i (A) Ever Married Never Married A t i (A) - t i (B)0 New time scale

52 Event History Analysis 2004 /6-1 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

53 Event History Analysis 2004 /6-2 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

54 Event History Analysis 2004 /6-3 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

55 Event History Analysis 2004 /6-4 As illustration the following figure for an exponential model with the constant rate r(t) = a = 0,2 13d. Exponential Model: Basics

56 Event History Analysis 2004 /6-5 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

57 Event History Analysis 2004 /6-6 The value of the density dependent on the parameter a is given by the Likelihood function L(a| t i, 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

58 Event History Analysis 2004 /6-7 Then we derive: 13g. Exponential Model: The Estimation of a ln (L) is the Log-Likelihood.

59 Event History Analysis 2004 /6-8 13h. Exponential Model: The Estimation of a

60 Event History Analysis 2004 /7-1 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 ) X 1,..., X m are time constant attributes of the observation units,  1,...,  m, are regression coefficients,  0 is the constant 14a. Exponential Model: Time-Constant Covariates

61 Event History Analysis 2004 /7-2 The coefficients  0,...,  m are estimated using the Maximum- Likelihood Method. For our model then follows: 14b. Exponential Model: Time-Constant Covariates

62 Event History Analysis 2004 /7-3 Interpretation of the coefficients of the covariates 14c. Exponential Model: Time-Constant Covariates It follows: If the covariate X j increases by one unit, Then the estimate for r changes by the factor exp(  j ) Or by (exp(  j ) - 1) * 100 %

63 Event History Analysis 2004 /7-4 Several possibilities exist to test the significance of the coefficients. 1. t - Test (just as in linear regression) 14d. Exponential Model: Time-Constant Covariates is approximatively normally distributed, if the sample is sufficiently large.

64 Event History Analysis 2004 /7-5 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 X 1 und X 2 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 X 2 ). 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.

65 Event History Analysis 2004 /7-6 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.

66 Event History Analysis 2004 /8-1 Rate regression with time constant covariates: r(t) = r = exp (  0 +  1 X 1 +... +  m X m ) X 1,..., X m 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

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72 „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. Event History Analysis2004 /8-2 15b. Exponential Model: Duration-Dependent Rates Months

73 „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 I l, l =1,...L: I l = [  l,  l+1 ); l =1,...L The last interval is open at the upper limit. Event History Analysis2004 /8-3 15c. Exponential Model: Duration-Dependent Rates oioi didi Ever Married Never Married t4t4 State space Y Time t 11 ll  l+1 IlIl

74 The rate regression is conducted as follows: Event History Analysis2004 /8-4 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.

75 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 I l ). Event History Analysis2004 /8-5 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+ I l.

76 Finally, one can estimate the coefficients of the covariates depending on the duration (of the duration interval I l ). Event History Analysis2004 /8-6 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.

77 Event History Analysis2004 /9-1 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 3. Rate regression with time dependent covariates

78 Event History Analysis2004 /9-2 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“

79 Event History Analysis2004 /9-3 16c. Exponential Model: Time-Dependent Covariates-Example The estimates amount to: 0/90 = 0/(7*12+6); 1/15 = 1(1*12+3)

80 Event History Analysis2004 /9-4 16d. Exponential Model: Time-Dependent Covariates Non-Parental HH Parental. HH t4t4 State space X Time t Episode split  2 sub-episodes (L=2) oioi didi Married Single t4t4 State space Y Time t sisi titi

81 Event History Analysis2004 /9-5 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 I l, l=1,...,L, in which the covariates are constant. The survival function is then computed according to where t l,i is the end- and s l,i the starting time of the episode I l and it holds that s l,i = s i and t L,i = t i.

82 Event History Analysis2004 /9-6 16f. Exponential Model: Time-Dependent Covariates Estimating on the basis of the episode splitting It holds: where r(s l ) = r(t) is being modelled according to

83 Event History Analysis2004 /9-7 16g. Exponential Model: Time-Dependent Covariates Estimating on the basis of episode splitting Then compute the Log-Likelihood according to 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.

84 Event History Analysis2004 /10-1 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

85 Event History Analysis2004 /10-2 Logistic regression 17b. Event History Analysis with Logistic Regression These terms like transition rates cannot be observed directly.

86 Event History Analysis2004 /10-3 Logistic regression 17c. Event History Analysis with Logistic Regression For case i one estimates the probability as follows: P(Y=1) is distributed like a logistic distribution.

87 Event History Analysis2004 /10-4 oioi didi Married(1) Unmarried (0) t4t4 State space Y Time t sisi titi 17d. Event History Analysis with Logistic Regression Step 1: Changing to discrete time: Divide the episode (s i, t i ] for the observation unit i in monthly intervals M il, l=1,..., L. The last month M iL marks the end of the episode (s i, t i ]. We assume here that for non-censored events the event takes place at the end of the month M iL.

88 Event History Analysis2004 /10-5 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(M il )=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(M il ) = 1|0), l=1,...L; i=1,...,N, conditional on time constant or time variant covariates X 1,..., X m, 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(M il )=1|0).

89 Event History Analysis2004 /10-6 17f. Event History Analysis with Logistic Regression Step 4: Estimate the transition rates according to:

90 Cox – Partial Likelihood Models

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105 Event History Analysis Literature

106 Software for Event History Analysis BMDP(1L, 2L) GLIMGeneralized Linear Interactive Modeling RATEInvoking RATE SAS(LIFEREG) SIRScientific Information Retrieval SPSSStatistical Package for the Social Science LIMDEP 5.1 RATE CSupplement to BMDP P3FUN FORTRANProgram for episode splitting given discrete time-dependent covariates In: 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) FORTRANProgram for episode splitting given continuous time-dependent covariates In: Blossfeld, Hamerle, Mayer, 1989 (pp. 285) GLIMMacros to estimate the Weibull and Log-Logistic models of Roger and Peacock In: Blossfeld, Hamerle, Mayer, 1989 (pp. 286-287) PARATHillmar Schneider 81991): Verweildauer mit GAUSS. Frankfurt am Main/New York: Campus Verlag. TDABlossfeld, Hans-Peter and Götz Rohwer (2002): Techniques of Event History Analysis. Mahwah, NJ: Lawrence Erlbaum Associates, 310 pp.


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