1. Parametric EHA Models Sociology 229A: Event History Analysis Class 6 Copyright © 2008 by Evan Schofer Do not copy or distribute without permission

2. Announcements Assignment #4 due Assignment #5 handed out Class topic: Parametric EHA models More diagnostics: Outliers

3. Parametric Proportional Hazard Models Cox models do not specify a functional form for the hazard curve, h(t) Rather, they examine effects of variables net of a baseline hazard trend (to be inferred from the data) h(t) = h 0 (t)e  X = h 0 (t)exp(  X) Parametric models specify the general shape of the hazard curve Approach is more familiar – more like regression –We can model Y as a constant, a linear function, a logit function, a binomial function (poisson), etc For instance, we could assume h(t) was a linear –Then solve for values of a hazard slope that best fit the data (plus effects of other covariates on hazard rate).

4. Parametric Proportional Hazard Models Parametric models work best when you choose a curve that fits the data Just like OLS regression – which works best when the relationship between two variables is roughly linear If the actual relationship between two variables is non-linear, coefficient estimates may be incorrect –Though sometimes one can transform variables (e.g., logging them) to get a good fit… –Parametric models are more efficient than Cox models They can generate more precise estimates for a given sample size But, they can also be more wildly incorrect if you mis-specify h(t)! –Note: These are proportional hazard models – like Cox! You must still check the proportional hazard assumption.

5. Exponential (Constant Rate) Model Exponential models are simplest: Note that there is no “t” in the equation… no coefficient that specifies time dependence of the hazard rate –Rather, there are just exponentiated BXs –PLUS: a, the constant Note 2: Box-Steffensmeier & Jones: h(t)=e -(  X) An exponential model solves for the constant value (a) that best fits the data… Along with values of Bs, which reflect effects of X vars In effect, the model assumes a constant hazard rate.

6. Exponential (Constant Rate) Model Another way of looking at it: An exponential model is a lot like a cox model But, with the assumption that the baseline hazard is a constant! Cox Exponential

7. Exponential (Constant Rate) Model Basic Model. Constant reflects base rate. streg gdp degradation education democracy ngo ingo, dist(exponential) nohr Exponential regression -- log relative-hazard form No. of subjects = 92 Number of obs = 1938 No. of failures = 77 Time at risk = 1938 Wald chi2(6) = 94.29 Log pseudolikelihood = 282.11796 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp | -.044568.1842564 -0.24 0.809 -.4057039.3165679 degradation | -.4766958.1044108 -4.57 0.000 -.6813372 -.2720543 education |.0377531.0130314 2.90 0.004.0122121.0632942 democracy |.2295392.0959669 2.39 0.017.0414475.417631 ngo |.4258148.1576803 2.70 0.007.1167671.7348624 ingo |.3114173.365112 0.85 0.394 -.4041891 1.027024 _cons | -4.565513 1.864396 -2.45 0.014 -8.219663 -.9113642 ------------------------------------------------------------------------------ Constant shows base hazard rate estimated from data: exp(-4.57) =.01

8. Exponential (Constant Rate) Model Suppose we plotted the baseline hazard rate estimated from our exponential model It would be a flat line: h(t) =.01 –This is the estimated hazard if all X vars are zero If we plotted the estimated hazard for some values of X (ex: democracy = 10), we would get a higher value –Since democracy has a positive effect, Democ = 10 would yield a higher hazard than democ = 0 –But, again, the estimated hazard rate trend would be a flat line over time…

9. Exponential Model: Baseline Hazard Ex: stcurve, hazard See, the estimated baseline hazard really is flat!

10. Exponential Model: Estimated Hazard stcurve, hazard at1(democ=1) at2(democ=10) Here are estimated hazards for 2 groups Other vars pegged at mean

11. Exponential Model: Baseline Hazard Issue: Actual hazard is rising. A problem? Is an exponential model appropriate? Answer: It can be, IF we have X variables that account for increasing hazard If not, fit will be poor!

12. Exponential (Constant Rate) Model Cleves et al. 2004, p. 216: In the exponential model, h(t) being constant means that the failure rate is independent of time, and thus the failure process is said to lack memory. You may be tempted to view exponential regression as suitable for use only in the simplest of cases. This would be unfair. There is another sense in which the exponential model is the basis for all other models. The baseline hazard… is constant … the way in which the overall hazard varies is purely a function of  X. The overall hazard need not be constant with time; it is just that every bit of how the hazard varies must be specified in BX. If you fully understand a process, you should be able to do that. When you do not understand a process, you are forced to assign a role to time, and in that way, you hope, put to the side your ignorance and still describe the part of the process that you do understand. In addition, exponential models can be used to model the overall hazard as a function of time, if they include t or functions of t as covariates.

13. Exponential (Constant Rate) Model The exponential model is extremely flexible… You specify substantive covariates (X variables) to explain failures –It is probably not due to some inherent feature of time, but rather due to some variable that you hope to control for –If you do a great job, you will fully explain why hazard rate appears to go up (or down) over time And, you can include functions of time as independent variables to address temporal variation –Independent (X) variable scan include time dummies, log time, linear time, time interactions, etc –That is, if you can’t explain time variation with substantive X variables, you can add time variables to model it But, if you mis-specify your model, results will be biased –In that case, you might be better off with a Cox model…

14. Piecewise Exponential Model If you have a lot of cases, you can estimate a piecewise model –Essentially a separate model for different chunks of time Model will yield different coefficients and base rate (constant) for multiple chunks of time Even if hazard is not constant over time, it may be more or less constant in each period –This allows you to effectively model any hazard trend –A related approach: Put in time-period dummies This gives a single set of bX coefficient estimates But, allows you to specify changes in the hazard rate over different periods –NOTE: Don’t forget to omit one of the time dummies!

15. Parametric Models Let’s try a more complex parametric model Example: Let’s specify a linear time trend Linear Exponential In this case, we estimate a constant (a) and slope (  0 ) which best summarize the time dependence of the hazard rate Note: this isn’t common – we have better options…

16. Gompertz Models Another option: an exponentiated line Rather than a linear function of time and exponentiated function of  X, we’ll exponentiate everything: Slope coefficient is often represented by gamma:  Note: Exponentiation alters the line… it isn’t a simple linear function anymore. –It is flat if gamma = 0 –It is monotonically increasing if gamma > 0 –It is monotonically decreasing if gamma < 0 Exponentiated Linear: Gompertz

17. Gompertz Models Exponentiating a linear function generates a curve defined by the value of gamma (  ) Model estimates value of  that best fits the data  = 0  < 0  > 0  >> 0

18. Gompertz Model Example: streg gdp degradation education democracy ngo ingo, robust nohr dist(gompertz) Gompertz regression -- log relative-hazard form No. of subjects = 92 Number of obs = 1938 No. of failures = 77 Time at risk = 1938 Wald chi2(6) = 46.48 Log pseudolikelihood = 307.64758 Prob > chi2 = 0.0000 _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.4633559.2104244 2.20 0.028.0509316.8757802 degradation | -.4394712.1434178 -3.06 0.002 -.720565 -.1583775 education |.0026837.0145341 0.18 0.854 -.0258026.03117 democracy |.2890106.092612 3.12 0.002.1074943.4705268 ngo |.2522894.1658275 1.52 0.128 -.0727265.5773054 ingo |.0037688.2275176 0.02 0.987 -.4421575.4496952 _cons | -253.035 45.28363 -5.59 0.000 -341.7892 -164.2807 -------------+---------------------------------------------------------------- gamma |.124117.0224506 5.53 0.000.0801146.1681195 ------------------------------------------------------------------------------ Model estimates gamma to be positive, significant. Implies increasing baseline hazard

19. Gompertz Model: Estimated Hazard stcurve, hazard at1(democ=1) at2(democ=10) Estimated hazards for 2 groups Other vars pegged at mean Note: curves are actually proportional – hard to see because bottom curve is nearly zero…

20. Weibull Models Another option: the Weibull curve Another curve that can fit monatonic hazards Model estimates p to best fit the model –Hazard is flat if p = 1 –Hazard is monotonically increasing if p > 1 –Hazard is monotonically decreasing if p < 1. Weibull

21. Weibull: Visually The Weibull family: Monotonic increasing or decreasing, depending on p Time Hazard Rate p = 1 p = 4 p =.5 p = 2

22. Weibull Model Example: streg gdp degradation education democracy ngo ingo, robust nohr dist(weibull) Weibull regression -- log relative-hazard form No. of subjects = 92 Number of obs = 1938 No. of failures = 77 Time at risk = 1938 LR chi2(6) = 23.71 Log likelihood = 307.6045 Prob > chi2 = 0.0006 _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.4631871.2360589 1.96 0.050.0005202.9258541 degradation | -.4396978.1486662 -2.96 0.003 -.7310781 -.1483175 education |.0027319.0141652 0.19 0.847 -.0250314.0304953 democracy |.288927.0913855 3.16 0.002.1098147.4680394 ngo |.2522595.1610192 1.57 0.117 -.0633324.5678514 ingo |.004058.1835743 0.02 0.982 -.355741.363857 _cons | -1884.071 280.0398 -6.73 0.000 -2432.939 -1335.203 -------------+---------------------------------------------------------------- /ln_p | 5.511481.1486542 37.08 0.000 5.220124 5.802837 -------------+---------------------------------------------------------------- p | 247.5173 36.79449 184.9571 331.2381 1/p |.0040401.0006006.003019.0054067 ------------------------------------------------------------------------------

23. Parametric: Model Fit Parametric models use maximum likelihood estimation (MLE) Comparisons among nested models can be made using a likelihood ratio test (LR test) Just like logit: Addition of groups of variables can be tested with lrtest –Some parametric models are themselves nested Ex: A Weibull model simplifies to an exponential model if p = 1 –Thus, exponential is nested within Wiebull LR tests can be used to see if Weibull is preferable to exponential.

24. Parametric Model Fit: AIC Non-nested parametric models can be compared via the Akaike Information Criterion k = # independent variables in the model c = # shape parameters in model (ex: p in Weibull) –Exponential has one parameter (a); Weibull has 2. AIC compares likelihoods, but corrects for parameters in the model – rewarding simpler models… Low values = better model fit –Even for negative values… -100 is better than -50.

25. Parametric: Model Fit How do you know which model fits best? 1. Look at the shape parameter Weibull: p, Gompertz: gamma If gamma is near zero or p near 1, they aren’t improving on fit compared to an exponential model 2. Conduct a likelihood ratio test For nested models only 3. Compare fit statistics: AIC Run models, then request “estat ic” Lower values = better.

26. Likelihood Ratio Test Ex: Compare Gompertz to exponential –Likelihood ratio test Run full model (weibull or gompertz) estimates store fullmodel Run base model estimates store basemodel lrtest fullmodel basemodel, force.. lrtest gompertz exponential, force Likelihood-ratio test LR chi2(1) = 51.06 (Assumption: exponential nested in gompertz) Prob > chi2 = 0.0000 Significant effect indicates that full model (Gompertz) fits better than exponential

27. Parametric: Model Fit AIC: Weibull, Gompertz, Exponential Request “estat ic” after each model is run ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- weibull | 1938 295.7504 307.6045 8 -599.209 -554.6537 ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- gompertz | 1938 295.7926 307.6476 8 -599.2952 -554.7399 ----------------------------------------------------------------------------- Model | Obs ll(null) ll(model) df AIC BIC -------------+--------------------------------------------------------------- Exponential | 1938 259.5519 282.118 7 -550.2359 -511.25 ----------------------------------------------------------------------------- AIC Results: Lower = better. Gompertz & Weibull fit better than Exponential; Little difference between Gompertz/Weibull.

28. Ancillary Parameters Gompertz & Weibull models have parameters that determine the shape of the curve Gamma (  ), p Ex: Bigger  = greater increase of h(t) over time –You can actually specify covariate effects on those parameters Effectively allowing a different curve shape across values of X variables Ex: If you think that hazard increases more for men than women, you can look to see if Dmale affects  –streg male educ, dist(gompertz) ancillary(male) –Model estimates effect of male on hazard AND on gamma…

29. Choosing a Hazard Model A Cox model is a good starting point Less problems due to accidental mis-specification of the time-dependence of the hazard rate Box-Steffensmeier & Jones point to cites: Cox models are 95% as efficient as parametric models under many circumstances –Cox models treat time dependence as a “nuisance”, put the focus on substantive covariates Which is often desirable.

30. Choosing a Hazard Model Parametric models are good when 1. You have strong theoretical expectations about the hazard rate 2. You are confident that you can fit the time dependence well with a parametric model 3. You need the most efficient estimates possible AGAIN: Substantive model specification is typically more important Biases due to omitted variables are often greater than biases due to poor model choice (e.g., Cox vs. Weibull) Also: In small samples, outliers are likely to be more important.

31. PH Assumption Models discussed today are proportional hazard models… Require the same assumption as Cox models But, most of the “tests” of proportionality are only available in Cox models But: You can still use piecewise models and interaction terms to check the assumption.

32. Residuals in EHA OLS regression: Residuals = difference between predicted value of Y and observed Y-hat – Y i EHA: Residuals are more complicated You could compute predicted failure minus observed… But, what about censored cases? What is observed? There are a number of different ways to calculate residuals… each with different properties.

33. Residuals – Summary From Cleves et al. (2004) An Introduction to Survival Analysis Using Stata, p. 184: 1. Cox-Snell residuals … are useful for assessing overall model fit 2. Martingale residuals Are useful in determining the functional form of the covariates to be included in the model 3. Schoenfeld residuals (scaled & unscaled), score residuals, and efficient score residuals Are useful for checking & testing the proportional hazard assumption, examining leverage points, and identifying outliers NOTE: A residual is produced for each independent variable… 4. Deviance residuals Are useful fin examining model accuracy and identifying outliers.

34. Martingale/deviance Residuals Outliers Martingale residuals: difference over time of observed failures minus expected failures Feature: range from +1 to –infinity –Deviance residuals = martingale residuals that are rescaled to be symmetric around zero Easier to interpret Extreme martingale or deviance residuals may indicate outliers Plot residuals vs. time, case number, IVs, etc. Or simply sort data by residuals & list the cases.

35. Martingale & Deviance Residuals: Outliers Stata code to identify outliers: *run Cox Model, calculate martingale residuals stcox var1 var2 var3, robust nohr mgale(mg) * Creates variable “mg” which contains martingale residuals * Next, compute deviance residuals using “predict” predict dev, deviance gen caseid = _n * create plots of various types scatter mg caseid * Deviance residual plots are generally easier to interpret scatter dev caseid, mlabel(newname2)

36. Deviance Residuals Plot Extreme values may be outliers Here, no obvious outliers are visible

37. Scaled Schoenfeld Residuals: Outliers Stata code to identify outliers: *run Cox Model, calculate residuals stcox var1 var2 var3, nohr schoenfeld(sch*) scaledsch(sca*) *Creates variables containing schoenfeld & scaled schoenfeld * residuals… labeled sch1, sch2, sch3… respectively gen caseid = _n * create plots of various types scatter sca1 caseid, mlabel(caselabel) scatter sca2 caseid, mlabel(caselabel) scatter sca3 caseid, mlabel(caselabel) … -- repeat for as many X variables as you have in the model

38. Scaled Schoenfeld Residuals: Plot A set of residuals is created for each X var Not too bad, but Latvia is a bit suspicious…

39. Scaled Schoenfeld Residuals: Plot This can’t be good! Here is a plot for a different X var: INGOs…

40. Efficient Score Residuals: Influential Cases Procedure for identifying outliers using ESRs It is possible to compute DFBETAs based on ESRs DFBETA: Change in coefficient a variable’s coefficient due to a particular case in the analysis –Cases with big DFBETAS may be overly influential –Issue: Stata cannot automatically compute DFBETAS… You have to compute them manually Also, computation = limited to 800 cases (for “intercooled stata”) Hopefully stata will improve this in the future.

41. ESRs: Influential Cases Stata code to estimate DFBETAs: * Run Cox model, request efficient score residuals * Creates vars: esr1 to esr5 corresponding to vars listed in model stcox gdp var1 var2 var3 var4, robust nohr esr(esr*) * Create room for a matrix of up to 800 rows (for your cases) set matsize 800 * Create esr matrix mkmat esr1 esr2 esr3 esr4, matrix(esr) * Multiply ESRs and Var/Cov matrix to estimate DFBETAs, save results mat V=e(V) mat Inf = esr*V svmat Inf, names(s) * Label estimates for subsequent plots label var s1 "dfbeta – var 1" label var s2 "dfbeta – var 2" label var s3 "dfbeta – var 3" label var s4 "dfbeta – var 4" * Plot DFBETAs for each variable vs. time or case number scatter s1 _t, yline(0) mlab(caseID) s(i) scatter s1 casenumber, yline(0) mlab(caseID) s(i) * Look for extreme values (for each IV – s1 to s4)

42. DFBETA Example DFBETA for NGOs (plotted by casenumber) DFBETA value indicates that presence of Latvia changes NGO coefficient by +.075 standard deviations

43. Outliers Cox Model: change due to removal of outlier ------------------------------------------------------------------------------ | 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 ------------------------------------------------------------------------------. RESULTS WITH LATVIA REMOVED: ------------------------------------------------------------------------------ | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.3654458.2031124 1.80 0.072 -.0326471.7635388 degradation | -.4472621.1110395 -4.03 0.000 -.6648956 -.2296286 education | -.0002829.0141668 -0.02 0.984 -.0280494.0274837 democracy |.2715732.0904942 3.00 0.003.0942078.4489385 ngo |.2245402.1644891 1.37 0.172 -.0978526.546933 ingo |.2735146.200823 1.36 0.173 -.1200912.6671204 ------------------------------------------------------------------------------ Removing Latvia changes things…

44. Reading Discussion Empirical Example: Schofer, Evan. 2003. “The Global Institutionalization of Geological Science, 1800-1990.” American Sociological Review, 68 (Dec): 730-759.