Event History Analysis 3 Sociology 8811 Lecture 17 Copyright © 2007 by Evan Schofer Do not copy or distribute without permission

Announcements Topic: More Event History Analysis Models, data structures, etc Note: We have fallen a bit behind schedule Lectures = slightly behind the reading assignments I’ll try to catch up But, you can adjust your reading efforts accordingly.

Review: EHA In essence, EHA models a dependent variable that reflects both: 1. Whether or not a patient experiences mortality 2. When it occurs (like a OLS regression of duration Dependent variable is best conceptualized as a rate of some occurrence EHA involves both descriptive and parametric analysis of data

Survivor: Marriage Compare survivor for women, men: Survivor plot for Men (declines later) Survivor plot for Women (declines earlier)

Integrated Hazard: Marriage Compare Integrated Hazard for women, men: Integrated Hazard for men increases slower (and remains lower) than women

Hazard Plot: Marriage Hazard Rate: Full Sample

Hazard Plot: Marriage Smoothed Hazard Rate: Full Sample

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?

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 Events sex | observed expected ------+------------------------- 1 | 10118 12820.67 2 | 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 ex: some place greater weight on early failures Tests available in Stata Log rank, Wilcoxon, Tarone-Ware, Peto-Peto-Prentice See Stata manual “Survivor Analysis & Epidemiological Tables” for advice about which to use In many cases, the results are similar across tests 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: Usually rewritten as:

Constant Rate Models Question: Is the constant rate assumption tenable?

Constant Rate Models Question: Is the constant rate assumption tenable? Answer: Probably not The hazard rate goes up and down over time Not constant at all – even if smoothed 2. The change over time isn’t likely the result of changing covariates (X’s) in our model However, if the change was merely the result of some independent variable, then the underlying (unobserved) rate might, in fact, be constant.

Constant Rate Models Let’s run an analysis anyway… Ignore the violation of assumptions regarding the functional form of the hazard rate 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 is one example Also: Piecewise Exponential, Gompertz, Weibull,etc. 2. Cox Models 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, models can’t be trusted.

Parametric Models These models make assumptions about the overall shape of the hazard rate over time 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 Assumptions Cox Models assume that independent variables don’t interact with time At lease, 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 Example: Marriage rate – women vs. men Women have a higher rate at all points in time Question: Does the hazard rate for women diverge or converge with men over time? If so, the proportion (or ratio) of the rate changes. The assumption is violated. Use a different model

Cox Model Assumptions: Proportionality: Look for parallel h(t)’s for different sub-groups (values of X’s) h(t) time Good Women Men h(t) Bad Women Men

Cox Model Assumptions: Hazard rates are often too spiky to discern trends Options: 1. Smooth the hazard plots OR 2. Check the integrated hazard rate Look for differences in the overall shape of the curve Note: divergence is OK on an integrated hazard

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

Reading Discussion Frank, David J., Ann M. Hironaka, and Evan Schofer. 2000. “The Nation State and the Natural Environment, 1900-1995.” American Sociological Review, 65 (Feb): 96-116.