Event History Models: Cox & Discrete Time Models Sociology 229: Advanced Regression Class 6 Copyright © 2010 by Evan Schofer Do not copy or distribute without permission

Announcements Assignment 4 Handed out Today’s agenda More complex EHA assignment Today’s agenda Cox models Parametric Models Reading Discussion

Cox Models Where h(t) is the hazard rate 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 Also written as:

Cox Model: Example Mostly similar to exponential model… Cox regression -- Breslow method for ties No. of subjects = 92 Number of obs = 1938 No. of failures = 77 Time at risk = 1938 Wald chi2(6) = 65.49 Log pseudolikelihood = -287.27209 Prob > chi2 = 0.0000 (Std. Err. adjusted for 92 clusters in newid3) ------------------------------------------------------------------------------ | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp | .4572288 .2025104 2.26 0.024 .0603157 .8541419 degradation | -.4311475 .1131853 -3.81 0.000 -.6529867 -.2093083 education | .0027517 .0136965 0.20 0.841 -.024093 .0295964 democracy | .2836321 .0911985 3.11 0.002 .1048862 .4623779 ngo | .2874221 .1614045 1.78 0.075 -.0289248 .603769 ingo | -.026845 .2391101 -0.11 0.911 -.4954922 .4418021 Most effects = similar… though education effect loses significance…

Cox Model: Baseline Hazard Cox models involve a “baseline hazard” Note: baseline = when all covariates are zero Question: What does the baseline hazard look like? Or baseline survivor & integrated hazard? Stata can estimate the baseline survivor, hazard, integrated hazard. Two steps: 1. You must ask stata to save the info when you run the Cox model Ex: stcox gdp degradation education democracy ngo ingo, robust nohr basehc(h0) 2. Use “stcurve” command to plot the baseline curves Ex: stcurve, hazard OR stcurve, survival

Cox Model: Baseline Hazard Baseline rate: Adoption of environmental law

Cox Model: Baseline Hazard Note: It may not always make sense to plot the baseline hazard Baseline shows hazard when X variables are zero Sometimes zero values aren’t very useful/interesting Example: Does it make sense to plot hazard of countries adopting laws, if X vars = zero? Hazard rate might be quite low In some cases, you’ll just get a flat zero curve Or extremely high values Solutions: 1. Rescale indep vars before running cox model 2. Use stcurve to choose relevant values of vars.

Cox Model: Estimated Hazards You can also use stcurve to plot estimated hazard rates based on values of indep vars Ex: What is hazard curve if democracy = 1, 5, 10? Strategy: use “at” subcommand: stcurve , hazard at(democ=1) at2(democ=10) NOTE: All other variables are pegged at the mean…

Cox: Estimated Hazard Rate Hazard rate for adoption of environmental law

Cox Model Diagnostics Issues that you must deal with: 1. How to estimate results with “ties” in your data Ties = cases that fail at the exact same time 2. How to identify violations of the proportional hazard assumption 3. Dealing with outliers/influential cases 4. Assessing model fit Most of this applies to parametric models Ties are not a concern But, additional issues come up: choosing the right functional form (shape) to model the hazard.

Cox Model Issues: Ties How to handle ties in data It is mathematically complex to estimate models when there are tied failures That is: two cases that have events at the exact same time Several mathematical approaches: Breslow approximation – simplest approach Stata default, but not the best choice! Efron approximation – generally better More computationally intensive, but given the power of modern computers it is not an issue stcox var1 var2 var3, efron

Cox Model Issues: Ties Exact marginal – “continuous time approximation” Box-Steffensmeier & Jones: “Averaged Likelihood” Assumes ties didn’t happen EXACTLY at the same time… and considers all possible orderings Exact partial – “discrete” Box-Steffensmeier & Jones: “exact discrete method” Assumes ties happened EXACTLY at the same time Advice: Use Efron at a minimum Exact methods are often more accurate Exact marginal often makes most sense… events rarely occur at the EXACT same time… unless you have discrete data But, exact methods can take a LONG time. For big datasets with many ties, Efron is OK.

Proportional Hazard Assumption Key assumption: Proportional hazards Estimated Hazard ratios are proportional over time i.e., Estimates of a hazard ratio do NOT vary over time Example: Effect of “abstinence” program on sexual behavior Issue: Do abstinence programs lower the rate in a consistent manner across time? Or, perhaps the rate is lower initially… but then the rate jumps back up (maybe even exceeds the control group). Groups are assumed to have “parallel” hazards Rather than rates that diverge, converge (or cross).

Proportional Hazard Assumption Strategies: 1. Visually examine raw hazard plots for sub-groups in your data Watch for non-parallel trends A crude method… not the best approach… but often identifies big violations

Proportional Hazard Assumption Visual examination of raw hazard rate You want them to change proportionally If one doubles, so does the other…

Proportional Hazard Assumption 2. Plot –ln(-ln(survival plot)) versus ln(time) across values of X variables What stata calls “stphplot” Parallel lines indicate proportional hazards Again, convergence and divergence (or crossing) indicates violation A less-common approach: compare observed survivor plot to predicted values (for different values of X) What stata calls “stcoxkm” If observed are similar to predicted, assumption is not likely to be violated.

Proportional Hazard Assumption -ln(-ln(survivor)) vs. ln(time) – “stphplot” Parallel=good Convergence suggests violation of proportional hazard assumption (But, I’ve seen worse!)

Proportional Hazard Assumption Cox estimate vs. observed KM – “stcoxkm” Predicted differs from observed for countries in West

Proportional Hazard Assumption 3. Piecewise Models Piecewise = break model up into pieces (by time) Ex: Split analysis in to “early” vs “late” time If coefficients vary in different time periods, hazards are not proportional Example: stcox var1 var2 var3 if _t < 10 stcox var1 var2 var3 if _t >= 10 Look for large changes in coefficients!

Proportional Hazard Assumption In a piecewise model, coefficients would differ in non-proportional models Non-Proportional Proportional Early Late Here, the effect is the same in both time periods Here, the effect is negative in the early period and positive in the late period

Piecewise Models Look at coefficients at 2 (or more) spans of time EARLY . stcox gdp degradation education democracy ngo ingo if year < 1985, robust nohr           _t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------          gdp |   .4465818   .4255587     1.05   0.294    -.3874979    1.280661  degradation |   -.282548   .1572746    -1.80   0.072    -.5908005    .0257045    education |  -.0195118   .0328195    -0.59   0.552    -.0838368    .0448131    democracy |   .2295673   .2625205     0.87   0.382    -.2849634     .744098          ngo |   .6792462   .3110294     2.18   0.029     .0696399    1.288853         ingo |   .6664661   .4804229     1.39   0.165    -.2751456    1.608078 ------------------------------------------------------------------------------ Note: Effect of ngo is larger in early period LATE . stcox gdp degradation education democracy ngo ingo if year >= 1985, robust nohr           _t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------          gdp |   .4963942    .357739     1.39   0.165    -.2047613     1.19755  degradation |  -.5702894   .2395257    -2.38   0.017    -1.039751   -.1008277    education |   .0142118   .0143762     0.99   0.323    -.0139649    .0423886    democracy |   .2541799   .0981386     2.59   0.010     .0618317    .4465281          ngo |   .1742862   .1448187     1.20   0.229    -.1095532    .4581256         ingo |  -.1134661   .2104308    -0.54   0.590    -.5259028    .2989707 ------------------------------------------------------------------------------

Proportional Hazard Assumption 4. Tests based on re-estimating model Try including time interactions in your model Recall: Interactions – effect of A on C varies with B If effect of variable X on hazard rate (or ratio) varies with time, then hazards aren’t proportional Recall example: Abstinence programs Perhaps abstinence programs have a big effect initially, but the effect diminishes (or reverses) later on

Proportional Hazard Assumption Red = Abstinence group; green = control Positive time interaction No time interaction In non-proportional case, the effect of abstinence programs varies across time

Proportional Hazard Assumption Strategy: Create variables that reflect the interaction of X variables with time Significant effects of time interactions indicate non-proportional hazard Fortunately, inclusion of the interaction term in the model corrects the problem. Issue: X variables can interact with time in multiple ways… Linearly With “log time” or time squared With time dummies You may have to try a range of things…

Proportional Hazard Assumption Red = Abstinence group; green = control Linear time interaction Effect grows consistently over time Try “Abstinence*time” Interaction with time-period… Effect differs early vs. late Try “Abstinence*DLate”

Proportional Hazard Assumption 5. Grambsch & Therneau test Ex: Stata “estat phtest” Test for non-zero slope of Schoenfeld residuals vs time Implies log hazard ratio function = proportional Can be applied to general model, or for each variable stcox gdp degradation education democracy ngo ingo, robust nohr scaledsch(sca*) schoenfeld(sch*) . estat phtest Test of proportional hazards assumption Time: Time ---------------------------------------------------------------- | chi2 df Prob>chi2 ------------+--------------------------------------------------- global test | 18.14 6 0.0059 Significant chi-square indicates violation of proportional hazard assumption

Proportional Hazard Assumption Variable-by-variable test “estat phtest”: Note: Certain variables are especially problematic… . estat phtest, detail Test of proportional hazards assumption Time: Time ---------------------------------------------------------------- | rho chi2 df Prob>chi2 ------------+--------------------------------------------------- gdp | 0.09035 0.63 1 0.4277 degradation | -0.22735 3.41 1 0.0646 education | 0.06915 0.47 1 0.4950 democracy | -0.04929 0.20 1 0.6560 ngo | -0.18691 4.56 1 0.0327 ingo | -0.03759 0.34 1 0.5609 global test | 18.14 6 0.0059

Proportional Hazard Assumption Notes on estat phtest : 1. STATA 9/10: Requires that you calculate “schoenfeld residuals” when you run the original cox model And, if you want a test for each variable, you must also request scaled schoenfeld residuals 2. Test is based on identifying non-zero time trend… but how should we characterize time? Options: normal/linear time, log time, time dummies, etc Results may differ depending on your choice Ex: estat phtest, log – specifies “log time” Plot of smoothed Schoenfeld residuals can indicate best way to characterize time Linear trend (not a curve) indicates that time is characterized OK Ex: estat phtest, plot(ngo) OR estat phtest, log plot(ngo)

Proportional Hazard Assumption What if the assumption is violated? 1. Improve model specification Add time interactions to address nonproportionality Ex: If high democracies are not proportional to low democracies, try adding “highdemoc*time” Variables can be interacted with linear time, log time, time dummies, etc., to address the issue 2. Model groups separately Split sample along variables that are non-proportional.

Proportional Hazard Assumption What if the assumption is violated? 3. Use a stratified Cox model Allows a different baseline hazard for each group But, you can’t estimate effect of stratifying variable! Ex: stcox var1 var2 var3, strata(Dhighdemoc) 4. Use a piecewise model Split time into chunks… in which PH assumption is met Requires sufficient sample size in all time periods!

Proportional Hazard Assumption What if the assumption is violated? 5. Live with it (but temper your conclusions) Violation of proportional hazard assumption tends to: Overestimate the effect of variables whose hazard ratios are increasing over time And, underestimate those whose hazard ratios are decreasing However, Allison points out: Cox model is reasonably robust Other issues (e.g., model misspecification) are bigger issues

Discrete Time EHA Models Distinction: Continuous vs. Discrete EHA “Discrete time”: time divided into integer chunks Years, decades, months Spell start & end times are essentially “rounded off” Continuous time: time conceptualized as an unbroken continuum Times need not be rounded off High levels of precision are possible Not just integers, but decimals.

Discrete Time EHA Models Issue: Discrete vs. continuous time gives rise to different EHA models Example: The hazard rate is defined for continuous time: The hazard rate over discrete (identical-sized) chunks of time is (ti):

Discrete Time EHA Models Issue: If the hazard rate in discrete time is a probability, maybe we can model it as such… Standard options for modeling probabilities: Logistic regression (logit) model Probit model Complementary log/log model (cloglog) An asymmetric function Starts slowly from p=0, but accelerates more rapidly toward p=1 at the end Often used when predicted probabilities are very low or high.

Discrete Time EHA Models Example: Discrete time logit model Where p is the probability of an event (Y=1) for a discrete chunk of time Complementary log log model looks like this:

Discrete Time EHA Models Basic logit/probit/cloglog models are like constant-rate/exponential models They assume a constant baseline hazard, represented by constant in the model Discrete EHA models are are proportional hazard models Logit output reports coefficients and odds ratios… But, it is appropriate to refer to them as hazard ratios Coefficient interpretation is the same Raw coeficientss require exponentiation to interpret…

Discrete Time EHA: Data Discrete time models require split-spell data where each spell has constant length Example: every record in your data represents 1 year Number of cases represents total time at risk Ex: If caseid 1 has 10 records, it was at risk for 10 years… This differs from continuous models, where records can represent variable amounts of time E.g., by providing specific start and end times…

Discrete Time EHA Data Discrete time data looks like other examples of split spell data But, each record MUST be the same length Example: Country data over time: Logit/probit/cloglog simply models outcome of 1 newname2 newid3 year law eventnum start end ss es pop INDIA 1119 1978 0 1 1978 1979 0 0 656941 INDIA 1119 1979 0 1 1979 1980 0 0 672021 INDIA 1119 1980 0 1 1980 1981 0 0 687332 INDIA 1119 1981 0 1 1981 1982 0 0 702821 INDIA 1119 1982 0 1 1982 1983 0 0 718426 INDIA 1119 1983 0 1 1983 1984 0 0 734072 INDIA 1119 1984 0 1 1984 1985 0 0 749677 INDIA 1119 1985 0 1 1985 1986 0 0 765147 INDIA 1119 1986 1 1 1986 1987 0 1 781893 Event (Y=1)

Discrete Time Logit Model Logit model for discrete time EHA It is a constant rate model In fact, results are almost the same as streg… . logit es gdp degradation education democracy ngo ingo Logistic regression Number of obs = 1938 LR chi2(6) = 47.83 Prob > chi2 = 0.0000 Log likelihood = -299.90676 Pseudo R2 = 0.0739 ------------------------------------------------------------------------------ es | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp | -.02752 .2274919 -0.12 0.904 -.473396 .4183559 degradation | -.5264763 .1404763 -3.75 0.000 -.8018049 -.2511477 education | .0415878 .0141799 2.93 0.003 .0137957 .0693799 democracy | .2429383 .0981245 2.48 0.013 .0506179 .4352587 ngo | .4534059 .177047 2.56 0.010 .1064001 .8004117 ingo | .3298737 .2341225 1.41 0.159 -.128998 .7887455 _cons | -4.724106 1.916741 -2.46 0.014 -8.48085 -.9673627

Discrete Time and Cox Models A Cox model can also be estimated in the discrete time context Indeed, the discrete time example helps illustrate what a Cox model really is (even in continuous time) Idea: Use a conditional logit model Conditioned on the cases in the risk set at each point in time … rather than a traditional logit model

Discrete Time and Cox Models A conditional logit model estimates common coefficients across models for many groups Looks at within-group factors, net of overall rate within each group… sorta like a fixed-effects model… Box-Steffensmeier & Jones, p. 80 Thus, effects are modeled net of the “baseline hazard” Interpretation: A Cox model is like pooling a large set of logit results In the continuous time context, the group is the current risk set at the time of any failure

Discrete Time and Cox Models A conditional logit model on discrete time EHA yields identical results to a Cox Model; If you specify the “exact partial” method for handling ties in the continuous time Cox model We’ll cover this later

Discrete Time Cox Model Conditional logit model – a cox model Yields identical results to cox when using discrete data . clogit es gdp degradation education democracy ngo ingo, group(year) Conditional (fixed-effects) logistic regression Number of obs = 1472 LR chi2(6) = 25.49 Prob > chi2 = 0.0003 Log likelihood = -224.89587 Pseudo R2 = 0.0536 ------------------------------------------------------------------------------ es | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp | .4806954 .2499626 1.92 0.054 -.0092223 .9706132 degradation | -.4624672 .1502146 -3.08 0.002 -.7568824 -.168052 education | .0036883 .0148541 0.25 0.804 -.0254251 .0328017 democracy | .3066401 .0971026 3.16 0.002 .1163225 .4969578 ngo | .314372 .1715222 1.83 0.067 -.0218052 .6505493 ingo | -.0329307 .2009382 -0.16 0.870 -.4267624 .360901

Discrete vs. Continuous EHA In practice, we can often use either discrete or continuous methods Even though time is theoretically continuous, our measures are usually limited to discrete time intervals Ex: year, month, day… For yearly spell data (or any other consistent interval) the data sets are pretty much identical If time resolution is extremely poor, there can be advantages to using discrete time models Otherwise, continuous time models provide greater flexibility And more modeling options.

EHA Example In-class group activity: Let’s design a study Outcome of interest: Students dropping a course What is the risk set? How would you set up the data? What are key independent variables? What kind of model would you use? Work in groups of 2-4, and be prepared to discuss your thoughts…

Reading Discussion 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. Long, J. Scott, Paul D. Allison, and Robert McGinnis.  1993.  “Rank Advancement in Academic Careers:  Sex Differences and the Effects of Productivity.”  American Sociological Review, 58, 5:703-722.