Event History Models 1

Sociology 229A: Event History Analysis

Class 3

Copyright © 2008 by Evan Schofer

Do not copy or distribute without permission

Announcements

• Assignment #1 due!

• Assignment #2 handed out• Due in 1 week

• Agenda• The idea of an EHA model• Interpreting coefficients• Types of EHA models

– Break• EHA data structures.

Survivor: Marriage

• Compare survivor for women, men:Kaplan-Meier survival estimates, by dfem

analysis time0 50 100

0.00

0.25

0.50

0.75

1.00

dfem 0

dfem 1

Survivor plot for Men

(declines later)

Survivor plot for Women

(declines earlier)

Integrated Hazard: Marriage

• Compare Integrated Hazard for women, men:Nelson-Aalen cumulative hazard estimates, by dfem

analysis time0 50 100

0.00

1.00

2.00

3.00

dfem 0

dfem 1

Integrated Hazard for men increases slower (and remains lower)

than women

Figure 3. Estimated hazard rateof entry into first marriage for entire sample

Est

ima

ted

Ha

zard

Ra

te

Age in Years12 20 30 40 50 60 70 80

12 20 30 40 50 60 70 80

0

.05

.1

.15

.2

0

.05

.1

.15

.2

Hazard Plot: Marriage• Hazard Rate: Full Sample

Hazard Plot: Marriage• Smoothed Hazard Rate: Full Sample

0.0

2.0

4.0

6.0

8.1

0 20 40 60 80analysis time

Smoothed hazard estimate

From Plots to Tests to Models

• It appears from the plots that women get married faster than men

• Issue:• How do we test hypotheses about the difference in

rates?• Can we be confident that the observed difference

between men and women is not merely due to sampling variability?

• One crude solution: Compute confidence intervals around plots

• Ex: sts graph, surv ci

Tests of Equality for Survivor Fns

• Idea: Conduct a hypothesis test to see if survivor functions differ across groups

• Like a t-test for difference in means…

• Example: Log-Rank Test• Based on calculating the expected # failures at each

point in time if there were no difference between groups

• Then, compute difference between observed failures and expected value for each group

• Analogous to a chi-square test of independence for a crosstab.

Log-rank Test

• Example: Do women marry earlier than men?. sts test sex, logrank

failure _d: married == 1 analysis time _t: endtime

Log-rank test for equality of survivor functions

| Events Eventssex | observed expected------+-------------------------1 | 10118 12820.672 | 13990 11287.33------+-------------------------Total | 24108 24108.00

chi2(1) = 1389.65 Pr>chi2 = 0.0000

Significant Chi-square (p<.05) indicates that survivor plots differ

Tests of Equality for Survivor Fns

• Stata offers a variety of tests• They mainly differ by how they weight cases

– Some place greater weight on early failures, others on later

– Tests available in Stata• Log rank, Wilcoxon, Tarone-Ware, Peto-Peto-Prentice• See Cleves et al for advice about which to use• Usually, the results are similar across tests

– But they differ in sensitivity to early vs. late events

– Also: Cox test• Based on a different principle• Can be used with weighted data (“pweights”).

EHA Models

• Strategy: Model the hazard rate as a function of covariates

• Much like regression analysis

• Determine coefficients• The extent to which change in independent variables

results in a change in the hazard rate

• Use information from sample to compute t-values (and p-values)

• Test hypotheses about coefficients

EHA Models

• Issue: In standard regression, we must choose a proper “functional form” relating X’s to Y’s

• OLS is a “linear” model – assumes a liner relationship– e.g.: Y = a + b1X1 + b2X2 … + bnXn + e

• Logistic regression for discrete dependent variables – assumes an ‘S-curve’ relationship between variables

• When modeling the hazard rate h(t) over time, what relationship should we assume?

• There are many options: assume a flat hazard, or various S-shaped, U-shaped, or J-shaped curves

• We’ll discuss details later…

Constant Rate Models

• The simplest parametric EHA model assumes that the base hazard rate is generally “flat” over time

• Any observed changes are due to changed covariates• Called a “Constant Rate” or “Exponential” model• Note: assumption of constant rate isn’t always tenable

• Formula: nnXbXbXbath 2211)(ln

• Usually rewritten as:

)()( 2211)( βXXbXbXba eeth nn

Figure 3. Estimated hazard rateof entry into first marriage for entire sample

Est

ima

ted

Ha

zard

Ra

te

Age in Years12 20 30 40 50 60 70 80

12 20 30 40 50 60 70 80

0

.05

.1

.15

.2

0

.05

.1

.15

.2

Constant Rate Models• Is the constant rate assumption tenable?

Constant Rate Models• Question: Is the constant rate assumption

tenable?

• Answer: Harder question than it seems…• The hazard rate goes up and down over time

– Not constant at all – even if smoothed

• However, if the change was merely the result of independent variables, then the underlying (base) rate might, in fact, be constant

• If your model doesn’t include variables that account for time variation in h(t), then a constant-rate model isn’t suitable.

Constant Rate Models

• Let’s run an analysis anyway…

• Ignore possible violation of assumptions regarding the functional form of h(t)

• Recall -- Constant rate model is:

)()( 2211)( βxXbXbXba nnn eeth

• In this case, we’ll only specify one X var:• DFEMALE – dummy variable indicating women• Coefficient reflects difference in hazard rate for women

versus men.

Constant Rate Model: Marriage

• A simple one-variable model comparing gender. streg sex, dist(exponential) nohr

No. of subjects = 29269 Number of obs = 29269No. of failures = 24108Time at risk = 693938 LR chi2(1) = 213.53Log likelihood = -30891.849 Prob > chi2 = 0.0000

------------------------------------------------------------------------------ _t | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- Dfemale | .1898716 .0130504 14.55 0.000 .1642933 .2154499 _cons | -3.655465 .0216059 -169.19 0.000 -3.697812 -3.613119------------------------------------------------------------------------------

• The positive coefficient for DFemale indicates a higher hazard rate for women

Constant Rate Coefficients

• Interpreting the EHA coefficient: b = .19

• Coefficients reflect change in log of the hazard– Recall one of the ways to write the formula:

nnXbXbXbath 2211)(ln

• But – we aren’t interested in log rates• We’re interested in change in the actual rate

• Solution: Exponentiate the coefficient• i.e., use “inverse-log” function on calculator• Result reflects the impact on the actual rate.

Constant Rate Coefficients

• Exponentiate the coefficient to generate the “hazard ratio”

Ratio Hazard21.1)19(.)( ee coef

• Multiplying by the hazard ratio indicates the increase in hazard rate for each unit increase in the independent variable

• Multiplying by 1.21 results in a 21% increase• A hazard ratio of 2.00 = a 100% increase• A hazard ratio of .25 = a decreased rate by 75%.

Constant Rate Coefficients

• The variable FEMALE is a dummy variable• Women = 1, Men = 0• Increase from 0 to 1 (men to women) reflects a 21%

increase in the hazard rate

– Continuous measures, however can change by many points (e.g., Firm size, age, etc.)

• To determine effects of multiple point increases (e.g., firm size of 10 vs. 7) multiply repeatedly

• Ex: Hazard Ratio = .95, increase = 3 units:• .95 x .95 x .95 = .86 – indicating a 14% decrease.

Hypothesis Tests: Marriage

• Final issue: Is the 21% higher hazard rate for women significantly different than men?

• Or is the observed difference likely due to chance?

• Solution: Hazard rate models calculate standard errors for coefficient estimates

• Allowing calculation of T-values, P-values

--------------------------------------------------

_t | Coef. Std. Err. t P>|t|

--------+---------------------------------------

Female | .1898716 .0130504 14.55 0.000

_cons | -3.465594 .0099415 -348.60 0.000

--------------------------------------------------

Types of EHA Models

• Two main types of proportional EHA Models

• 1. Parametric Models• specify a functional form of h(t)• Constant rate; Also: Gompertz, Weibull,etc.

• 2. Cox Models• Also called “semi-parametric”• Doesn’t specify a particular form for h(t)

• Each makes assumptions• Like OLS assumptions regarding functional form, error

variance, normality, etc• If assumptions are violated, results can’t be trusted.

Parametric Models

• Parametric models make assumptions about the shape of the hazard rate over time

– Conditional on X

• Much like OLS regression assumes a linear relationship between X and Y, logit assumes s-curve

• Options: constant, Gompertz, Weibull• There is a piecewise exponential option, too

• Note: They also make standard statistical assumptions:

• Independent random sample• Properly specified model, etc, etc…

Cox Models

• The basic Cox model:)(

02211)()( nnXbXbXbethth

• Where h(t) is the hazard rate

• h0(t) is some baseline hazard function (to be inferred from the data)• This obviates the need for building a specific

functional form into the model

• bX’s are coefficients and covariates

Cox Model: Example

• Marriage example:

No. of subjects = 29269 Number of obs = 29269No. of failures = 24108 Time at risk = 693938

LR chi2(1) = 1225.71Log likelihood = -229548.82 Prob > chi2 =

0.0000

-------------------------------------------------- _t | Coef. Std. Err. z P>|z| --------+-----------------------------------------Female | .4551652 .0131031 34.74 0.000 --------------------------------------------------

Cox vs. Parametric: Differences• Cox Models do not make assumptions about

the time-dependence of the hazard rate– Cox models focus on time-ordering of observed

events ONLY• They do not draw information from periods in which no

events occur– After all, to do this you’d need to make some assumption

about what rate you’d expect in that interval…

– Benefit: One less assumption to be violated– Cost: Cox model is less efficient than a properly

specified parametric model• Standard errors = bigger; more data needed to get

statistically significant results.

Cox vs. Parametric: Similarities• Models discussed so far are all “proportional

hazard” models• Assumption: covariates (X’s) raise or lower the

hazard rate in a proportional manner across time• Ex: If women have higher risk of marriage than men,

that elevated risk will be consistent over all time…

• Another way of putting it:– Cox Models assume that independent variables

don’t interact with time • At least, not in ways you haven’t controlled for• i.e., that the hazard rate at different values of X are

proportional (parallel) to each other over time

Proportional Hazard Models

• Proportionality: X variables shift h(t) up or down in a proportional manner

h(t

)

time

Proportional

Women

Men

h(t

)

Not Proportional

Women

Men

Proportional Hazard Models

• Issue: Does the hazard rate for women diverge or converge with men over time?

• If so, the proportion (or ratio) of the rate changes. • The proportional hazard assumption is violated

• Upcoming classes:• We’ll discuss how to check the proportional hazard

assumption and address violations…