Event History Models Sociology 229: Advanced Regression Class 5 Copyright © 2008 by Evan Schofer Do not copy or distribute without permission

Announcements Assignment 3 due Agenda EHA models Discrete time models More details on Cox models & other fully parametric Proportional Hazard models Break Discussion of paper: Allison and McGinnis

Review Event history analysis focuses attention on rates of events/failures over time Descriptive approaches include: Survivor Plots Hazard plots Integrated / cumulative hazard plots Also, we can conduct non-parametric tests to see if rates differ across groups Example: Log rank test

Hazard Plot: Marriage Smoothed Hazard Rate: Full Sample

EHA Models Strategy: Model the hazard rate as a function of covariates Goal: Estimate coefficients that show impact of independent variables on the hazard rate Also, we can 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: Usually rewritten as:

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: 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 = 29269 No. of failures = 24108 Time at risk = 693938 LR chi2(1) = 213.53 Log 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: 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” 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: 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 = 29269 No. of failures = 24108 Time at risk = 693938 LR chi2(1) = 1225.71 Log 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…

Reading Discussion Hironaka, Ann M.  2005.  “World Patterns in Civil War Duration.”  Chapter 2 in Neverending Wars.  Cambridge, MA:  Harvard University Press. How are the models set up? What were the outcomes? Findings? Empirical Example:  Soule, Sarah A and Susan Olzak.  2004.  “When Do Movements Matter? The Politics of Contingency and the Equal Rights Amendment.”  American Sociological Review, Vol. 69, No. 4. (Aug., 2004), pp. 473-497.