Event History Analysis 6 Sociology 8811 Lecture 20 Copyright © 2007 by Evan Schofer Do not copy or distribute without permission.

Event History Analysis 6

Sociology 8811 Lecture 20

Copyright © 2007 by Evan SchoferDo not copy or distribute without permission

Announcements

• Paper Assignment #2 handed out previously• Due April 26

• Class topic: • More details on Cox model diagnostics• Next class: parametric models, AFT models (if time)…• Then – new topics!

Review: Cox Model Basics

• Choosing how to deal with ties• Several algorithms available• Efron method is better than Breslow (default)• Exact marginal is even better – but time consuming

• The baseline hazard rate• Can be computed in stata…• Plus, you can estimate the hazard rates at particular

values of variables

• The proportional hazard assumption• We’ll continue discussing several methods…

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).


• Strategies:

• 1. Visually examine raw hazard plots for sub-groups in your data

• Watch for non-parallel trends• A simple, crude method… but often identifies big

violations

Proportional Hazard Assumption• Visual examination of raw hazard rate

0.0

5.1

.15

1970 1980 1990 2000analysis time

west = 0 west = 1

Smoothed hazard estimates, by west

Parallel trends in hazard rate indicate proportionality


• 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”

Convergence suggests violation of proportional hazard assumption

(But, I’ve seen worse!)

-10

12

34

-ln[-

ln(S

urv

ival

Pro

babi

lity)

]

7.585 7.59 7.595 7.6 7.605ln(analysis time)

west = 0 west = 1

Proportional Hazard Assumption• Cox estimate vs. observed KM – “stcoxkm”

0.0

00.

20

0.4

00.

60

0.8

01.

00

Sur

viva

l Pro

bab

ility

1970 1980 1990 2000analysis time

Observed: west = 0 Observed: west = 1Predicted: west = 0 Predicted: west = 1

Predicted differs from observed for countries in West


• 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!


• In a piecewise model, coefficients would differ in non-proportional models

Proportional Non-Proportional

Here, the effect is the same in both time periods

Early Late Early Late

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 timeEARLY. 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------------------------------------------------------------------------------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------------------------------------------------------------------------------

Note: Effect of ngo is larger in early period


• 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


• Red = Abstinence group; green = control

No time interaction Positive timeinteraction

In non-proportional case, the effect of abstinence programs varies across time


• 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…


• Red = Abstinence group; green = control

Linear time interactionEffect grows consistently over timeTry “Abstinence*time”

Interaction with time-period… Effect differs early vs. late Try “Abstinence*DLate”


• 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


• Variable-by-variable test “estat phtest”:

. 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 ----------------------------------------------------------------

Note: Certain variables are especially problematic…

Proportional Hazard Assumption• Notes on estat phtest :

– 1. 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)


• 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.



• 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!



• 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

Cox Model: Residuals

• OLS regression: Residuals = difference between predicted value of Y and observed

• Y-hat – Yi

• 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.

Cox Model: 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

• 4. Deviance residuals• Are useful fin examining model accuracy and identifying outliers.

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.

Martingale & Deviance Residuals: Outliers

• Stata code to identify outliers:

*run Cox Model, calculate martingale residualsstcox var1 var2 var3, robust nohr mgale(mg)* Creates variable “mg” which contains martingale residuals* Next, compute deviance residuals using “predict”predict dev, deviancegen caseid = _n* create plots of various typesscatter mg caseid* Deviance residual plots are generally easier to interpretscatter dev caseidscatter dev caseid, mlabel(newname2)scatter dev _t

Deviance Residuals Plot• Extreme values may be outliers

CROATIA

LATVIA

MACEDONIA

SLOVAKIA

SLOVENIA

ALGERIA

ANGOLA

BENIN

BUR-FASO

BURUNDI

CAMEROON

CHADCOMOROS

CONGO

EGYPT

ETHIOPIA w e

GAMBIA

GHANA

GUINEA

IVORY-CO

KENYA

MADAGASC

MALAWI

MALI

MAURITANMAURITIUS

MOROCCO

MOZAMBIQ

NIGER

NIGERIA

RWANDA

SENEGAL

SIERRA-L

SO-AFRICA

TANZANIA

TOGO

UGANDA

ZAMBIA

ZIMBABWE

CANADA

COSTA-RICUBA

DOM-REP

EL-SALVA

GUATEMA

HONDURAS

JAMAICA

MEXICO

NICARAGPANAMATRIN&TOB

USA

ARGENTIN

BOLIVIA

BRAZIL

CHILE

COLOMBIA

ECUADORGUYANA

PARAGUAY

PERU

URUGUAY

BANGLAD

CYPRUS

KAMPUCH

INDIA

INDONES

IRAN

ISRAEL

JAPAN

JORDAN

KOREA-R(S

LEBANON

MALAYSIA

NEPAL

PAKISTAN

PHILIPPI

SINGAPOR

SRI-LAN

SYRIA

THAILAND

TURKEY

BELGIUM

DENMARKFINLAND

ICELAND

IRELAND

LUXEMB

NETHERL

NORWAY

PORTUGAL

SWEDEN

SWITZERL

AUSTRAL

NEW-ZEAL

-2-1

01

2de

via

nce

re

sidu

al

0 1000 2000 3000caseid

Here, no obvious outliers are visible

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.

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 modelstcox 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 matrixmkmat esr1 esr2 esr3 esr4, matrix(esr)* Multiply ESRs and Var/Cov matrix to estimate DFBETAs, save resultsmat V=e(V)mat Inf = esr*Vsvmat Inf, names(s)* Label estimates for subsequent plotslabel 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 numberscatter 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)

DFBETA Example• DFBETA for NGOs (plotted by casenumber)

LATVIA

MACEDONIA

MACEDONIAMACEDONIAMACEDONIA

MACEDONIA

SLOVAKIASLOVAKIASLOVAKIASLOVAKIASLOVAKIASLOVAKIASLOVAKIASLOVAKIASLOVENIASLOVENIASLOVENIASLOVENIASLOVENIASLOVENIASLOVENIASLOVENIASLOVENIAALGERIAALGERIAALGERIAALGERIAALGERIAALGERIAALGERIAALGERIAALGERIAALGERIAALGERIAALGERIAALGERIA

ALGERIA

ANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLAANGOLA

ANGOLA

BENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENINBENIN

BENIN

BUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASOBUR-FASO

BUR-FASO

BURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDIBURUNDI

BURUNDI

CAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROONCAMEROON

CAMEROON

CHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHADCHAD

CHAD

COMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCOMOROSCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGOCONGO

CONGO

EGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPTEGYPT

EGYPT

ETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIAETHIOPIA w eETHIOPIA w eETHIOPIA w eETHIOPIA w eETHIOPIA w eETHIOPIA w eETHIOPIA w eETHIOPIA w eGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIAGAMBIA

GAMBIA

GHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGHANAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEAGUINEA

GUINEAIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-COIVORY-CO

KENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYAKENYA

KENYA

MADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASCMADAGASC

MADAGASC

MALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWIMALAWI

MALAWI

MALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALIMALI

MALI

MAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITANMAURITAN

MAURITAN

MAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUSMAURITIUS

MAURITIUS

MOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOROCCOMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQMOZAMBIQNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGERNIGER

NIGER

NIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIANIGERIA

NIGERIA

RWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDARWANDASENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSENEGALSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-LSIERRA-L

SO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICASO-AFRICA

SO-AFRICA

TANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATANZANIATOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOTOGOUGANDAUGANDA

-.05

0.0

5.1

dfb

eta

- ng

o

0 200 400 600 800casenumber

DFBETA value indicates that presence of Latvia changes NGO coefficient by +.075 standard deviations

Cox-Snell residuals: Model Fit

• Cox-Snell residuals can be plotted to assess model fit

• If model fits well, graph of integrated (cumulative) hazard conditional on Cox-Snell residuals vs. Cox-Snell residuals will fall on a line

– Strategy in stata:• Run Cox model, request martingale residuals• Use “predict” to compute Cox-Snell residuals• Stset your data again, with Cox-Snell as time variable• Compute integrated hazard• Graph integrated hazard versus residuals.

Cox-Snell residuals: Model Fit

• Cox-Snell residuals can be plotted to assess model fit

• If model fits well, graph of integrated (cumulative) hazard conditional on Cox-Snell residuals vs. Cox-Snell residuals will fall on a line

– Strategy in stata:• Run Cox model, request martingale residuals• Use “predict” to compute Cox-Snell residuals• Stset your data again, with Cox-Snell as time variable• Compute integrated hazard• Graph integrated hazard versus residuals.

Cox-Snell Model Fit Example• Cox-Snell Plot for Environmental Law data

0.5

11.

52

0 .2 .4 .6partial Cox-Snell residual

Nelson-Aalen cumulative hazardpartial Cox-Snell residual

This looks quite bad. Cumulative hazard should fall on the line… Instead, there is a sizable gap.

Note: Don’t worry about small deviations from the line at the right edge of the plot. There are typically few cases there…

Stratified Cox Models

• Stratified models allow different baseline hazards for sub-groups in your data

• But constrain all model coefficients to be the same across all groups

• Useful if we think that some sub-groups have very different hazard curves over time

– AND we aren’t really interested in differences across those groups – it is just a nuisance

• Another option is to simply analyze groups separately– But, we lose sample size. Stratifying avoids that.

Cox Models for Grouped Data

• Sometimes cases are not independent• Ex: Students in same class; people in same family

• Two useful options:– 1. Stata’s “cluster” command: Adjusts standard

errors based on group membership•stcox var1 var2 var3, cluster(FamilyID)

– 2. Cox model with shared frailty • Another name for a random effects model

– We’ll discuss this later in the course– Don’t confuse with non-shared frailty models!

•stcox var1 var2 var3, shared(FamilyID)

Stata Note: TVC

• Note: Stata has a special option for “time varying covariates”

• You DO NOT need to use this!!!– It is designed for cases where you wish to specify the

character of time variation (e.g., a rate of decay)

• You can simply create time-varying variables in your dataset…

– STATA will analyze it properly using the stcox command.

Event History Analysis 6 Sociology 8811 Lecture 20 Copyright © 2007 by Evan Schofer Do not copy or distribute without permission.

Documents

proportional hazards

proportional example

hazard rates

hazard ratios

baseline hazard rate

raw hazard plots

proportionality slide

west slide