Event History Models 2 Sociology 229A: Event History Analysis Class 4 Copyright © 2008 by Evan Schofer Do not copy or distribute without permission

Announcements Assignment 2 due Assignment # handed out Agenda More EHA models Discrete time models More details on Cox models & other fully parametric Proportional Hazard models Break Discussion of paper: Allison and McGinnis

Event History Example What factors affect how soon a country passes an environmental protection law? Event: Passing an environmental law in a given year Risk set: All countries that have not yet passed an environmental protection law –We decided that risk begins at 1970 (when such laws were invented) Countries independent after 1970 are treated as entering the analysis “late” Option #2: Duration since independence (age) –But, that was less appropriate for the research question.

Example: Environmental Laws Cross-national time series dataset of nearly 100 countries Event: when a country writes its first comprehensive environmental law (e.g., EPA) Data taken from various sources Independent variables: GDP, population, democracy, degradation, education, domestic and international NGOs Time duration: analyses are from In other words, countries enter the “risk set” in 1970, or when they become independent Total sample of 97 countries 73 countries have an event between 1970 and 1998.

Time-Varying Data Structure newname2newid3yearlaweventnumstartendssespop INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA Example: Law written SpellState Population

Time-Varying Data Structure newname2newid3yearlaweventnumstartendssespop INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA Stset command: stset end, failure(es==1) time0(start) Note: It is common to drop cases that are not at risk (ex: if start state = 1) BUT, it is not necessary… Stata drops cases after the event by default…unless you specify exit(time.)

Time-Varying Data Structure What if countries pass multiple laws? Called “repeated events 1. start state could be reset to zero 2. We can override the stata default of removing cases after the first event occurs: exit(time.) newname2newid3yearlaweventnumstartendssespop INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA

Smoothed Hazard Function West vs. non-West

EHA Models in Stata Cox Models: stcox indep1 indep2 indep3 Default output shows hazard ratios Useful options: nohr – requests raw coefs (not hazard ratios) vce(robust) – specifies robust standard errors vce(cluster varname) – better SEs for non- independent (clustered) data.

EHA Models in Stata Parametric Models: streg streg ind1 ind2 ind3, dist(exponential) You must specify a functional form (distribution) Ex: Exponential, weibull, gompertz, etc. We’ll discuss choices later Streg shares many options with stcox: nohr vce(robust), vce(cluster)

Constant Rate Model: Example Simple one-variable model comparing west vs. non-west streg west, dist(exponential) nohr Exponential regression -- log relative-hazard form No. of subjects = 97 Number of obs = 2047 No. of failures = 81 Time at risk = 2047 Wald chi2(1) = Log pseudolikelihood = Prob > chi2 = (Std. Err. adjusted for 97 clusters in newid3) | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] west | _cons |

Constant Rate Model: Example Model with time-varying covariates No. of subjects = 92 Number of obs = 1938 No. of failures = 77 Time at risk = 1938 Wald chi2(6) = Log pseudolikelihood = Prob > chi2 = (Std. Err. adjusted for 92 clusters in newid3) | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] gdp | degradation | education | democracy | ngo | ingo | _cons | Democratic countries enact laws at a higher rate than less-democratic countries

Constant Rate Model: Example Same model – with Hazard Ratios No. of subjects = 92 Number of obs = 1938 No. of failures = 77 Time at risk = 1938 Wald chi2(6) = Log pseudolikelihood = Prob > chi2 = (Std. Err. adjusted for 92 clusters in newid3) | Robust _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] gdp | degradation | education | democracy | ngo | ingo | A 1-point increase in democracy increases the hazard rate by 25.8%!

Constant Rate Model : Example What if we expect global civil society to have a particularly strong effect in the non-West? Option #1: Create an interaction term No. of subjects = 92 Number of obs = 1938 No. of failures = 77 Time at risk = 1938 Wald chi2(8) = Log pseudolikelihood = Prob > chi2 = (Std. Err. adjusted for 92 clusters in newid3) | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] gdp | degradation | education | democracy | ngo | ingo | nonwest | ingoXnonwest | _cons |

Constant Rate Model : Example What if we expect global civil society to have a particularly strong effect in the non-West? Option #2: Include only non-Western countries in the analysis No. of subjects = 76 Number of obs = 1720 No. of failures = 61 Time at risk = 1720 Wald chi2(6) = Log pseudolikelihood = Prob > chi2 = (Std. Err. adjusted for 76 clusters in newid3) | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] gdp | degradation | education | democracy | ngo | ingo | _cons |

Cox Models The basic Cox model: Where h(t) is the hazard rate h 0 (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) = Log pseudolikelihood = Prob > chi2 = (Std. Err. adjusted for 92 clusters in newid3) | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] gdp | degradation | education | democracy | ngo | ingo | Most effects = similar… though education effect loses significance…

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 (t i ):

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 newname2newid3yearlaweventnumstartendssespop INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA INDIA 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) = Prob > chi2 = Log likelihood = Pseudo R2 = es | Coef. Std. Err. z P>|z| [95% Conf. Interval] gdp | degradation | education | democracy | ngo | ingo | _cons |

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) = Prob > chi2 = Log likelihood = Pseudo R2 = es | Coef. Std. Err. z P>|z| [95% Conf. Interval] gdp | degradation | education | democracy | ngo | ingo |

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