Parametric EHA Models

Sociology 229A: Event History AnalysisClass 6

Copyright © 2008 by Evan SchoferDo not copy or distribute without permission

Announcements

• Assignment #4 due• Assignment #5 handed out

• Class topic: • Parametric EHA models• More diagnostics: Outliers

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) = h0(t)eX = h0(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).

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.

Exponential (Constant Rate) Model

• Exponential models are simplest:)()( 2211)( βXaXbXbXba eeth nn

• 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 .

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!

)(0 )()( βXethth

Cox

)()()( βXaXa eeeth Exponential

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 = 1938No. of failures = 77Time at risk = 1938 Wald chi2(6) = 94.29Log 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

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…

Exponential Model: Baseline Hazard• Ex: stcurve, hazard

-.96

9705

91.

030

294

Ha

zard

func

tion

1970 1980 1990 2000analysis time

Exponential regression

See, the estimated baseline hazard really is flat!

Exponential Model: Estimated Hazard• stcurve, hazard at1(democ=1) at2(democ=10)

.05

.1.1

5.2

.25

.3H

aza

rd fu

nctio

n

1970 1980 1990 2000analysis time

democracy=1 democracy=10

Exponential regression

Here are estimated hazards for 2 groups

Other vars pegged at mean

Exponential Model: Baseline Hazard• Issue: Actual hazard is rising. A problem?

0.0

2.0

4.0

6.0

8.1

1970 1980 1990 2000analysis time

Smoothed hazard estimateIs an exponential model appropriate?

Answer:

It can be, IF we have X variables that account for increasing hazard

If not, fit will be poor!

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.

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…

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!

Parametric Models

• Let’s try a more complex parametric model• Example: Let’s specify a linear time trend

)(0 )()( βXetβath

Linear

)()()( βXaXa eeeth 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…

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

)()()( 0)( βXtaβXtβa eeeth Exponentiated Linear: Gompertz

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

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 = 1938No. of failures = 77Time at risk = 1938 Wald chi2(6) = 46.48Log 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

Gompertz Model: Estimated Hazard• stcurve, hazard at1(democ=1) at2(democ=10)

Estimated hazards for 2 groups

Other vars pegged at mean

01

23

4H

aza

rd fu

nctio

n

1970 1980 1990 2000analysis time

democracy=1 democracy=10

Gompertz regression

Note: curves are actually proportional – hard to see because bottom curve is nearly zero…

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.

)(1)( βXap eptth Weibull

Weibull: Visually

• The Weibull family: Monotonic increasing or decreasing, depending on p

Time

Haz

ard

Rat

e

p = 1

p = 4

p = .5

p = 2

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 = 1938No. of failures = 77Time at risk = 1938 LR chi2(6) = 23.71Log 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------------------------------------------------------------------------------

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.

Parametric Model Fit: AIC

• Non-nested parametric models can be compared via the Akaike Information Criterion

)(2)ln(2 ckLAIC • 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.

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.

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

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.

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…

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.

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.

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.

Residuals in EHA

• 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.

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.

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 caseid, mlabel(newname2)

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

Scaled Schoenfeld Residuals: Outliers

• Stata code to identify outliers:

*run Cox Model, calculate residualsstcox var1 var2 var3, nohr schoenfeld(sch*) scaledsch(sca*)*Creates variables containing schoenfeld & scaled schoenfeld* residuals… labeled sch1, sch2, sch3… respectivelygen caseid = _n* create plots of various typesscatter 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

Scaled Schoenfeld Residuals: Plot

• A set of residuals is created for each X var

LATVIA

MACEDONIA

SLOVENIA

ALGERIA

ANGOLA

BENIN

BUR-FASOBURUNDI

CAMEROON

CHAD

COMOROS

CONGOEGYPTGAMBIA

GHANA

GUINEA

KENYA

MADAGASC

MALAWI

MALI

MAURITAN

MAURITIUS

MOZAMBIQ

NIGER

NIGERIASENEGAL

SO-AFRICA

TOGO

UGANDA

ZAMBIA

CANADACOSTA-RI

DOM-REPEL-SALVAGUATEMA

HONDURAS

MEXICO

NICARAG

PANAMA

TRIN&TOB

USA

ARGENTIN

BOLIVIA

CHILE

COLOMBIA

ECUADORGUYANA

PARAGUAY

PERU

URUGUAY

BANGLAD

KAMPUCH

INDIA

INDONES

JAPAN

KOREA-R(S

MALAYSIANEPALPAKISTAN

PHILIPPISRI-LAN

SYRIA

THAILAND

TURKEYBELGIUM

DENMARK

FINLAND

ICELANDIRELAND

LUXEMB

NETHERLNORWAY

PORTUGALSWEDEN

SWITZERL

AUSTRALNEW-ZEAL

-50

510

scal

ed S

choe

nfel

d -

gdp

0 1000 2000 3000caseid

Not too bad, but Latvia is a bit suspicious…

Scaled Schoenfeld Residuals: Plot

LATVIA

MACEDONIASLOVENIA

ALGERIA

ANGOLA

BENINBUR-FASOBURUNDICAMEROON

CHAD

COMOROS

CONGOEGYPT

GAMBIAGHANA

GUINEA

KENYAMADAGASCMALAWIMALIMAURITANMAURITIUSMOZAMBIQ

NIGER

NIGERIA

SENEGALSO-AFRICATOGOUGANDAZAMBIA

CANADACOSTA-RI

DOM-REP

EL-SALVA

GUATEMAHONDURAS

MEXICO

NICARAGPANAMA

TRIN&TOB

USA

ARGENTINBOLIVIACHILECOLOMBIA

ECUADORGUYANAPARAGUAYPERU

URUGUAYBANGLAD

KAMPUCH

INDIAINDONES

JAPANKOREA-R(SMALAYSIA

NEPAL

PAKISTANPHILIPPISRI-LAN

SYRIATHAILANDTURKEYBELGIUM

DENMARKFINLANDICELAND

IRELAND

LUXEMBNETHERLNORWAY

PORTUGAL

SWEDENSWITZERL

AUSTRALNEW-ZEAL

-30

-20

-10

010

scal

ed S

choe

nfel

d -

ingo

0 1000 2000 3000caseid

This can’t be good!

• Here is a plot for a different X var: INGOs…

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

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…

Reading Discussion

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