csss 569 visualizing data 0.35 in visualizing...
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CSSS 569 · Visualizing Data
VISUALIZING INFERENCE
Christopher Adolph
Department of Political Science
and
Center for Statistics and the Social Sciences
University of Washington, Seattle
CENTER for STATISTICS and the SOCIAL SCIENCES
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.Lessons so far…..
......
Principles for effective visual display
How to avoid cognitive pitfalls in designing visuals
Basics of R graphics
Application of the above to Exploratory Data Analysis (EDA)
Most resources on scientific visualization stop with EDA,or using graphics to understand the data
But quantitative social science emphasizes modeling data
Next step: Designing effective visuals for understanding models
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Roadmap for Visualizing Inference
...1 Obtaining Quantities of Interest from models
...2 Introduction to the tile package
...3 Graphical approaches to model inference
...4 Graphical approaches to model robustness
...5 Visualizing interactive models
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What determines cross-national inflation performance?
Source: Adolph, BBC, Ch. 3
Method: Time series cross-section regression with compositional covariates
Who votes in American elections?
Source: King, Tomz, and Wittenberg (2000)
Method: Logistic regression
How do Chinese leaders gain power?
Source: Shih, Adolph, and Liu (2012)
Method: Bayesian model of partially observed ranks
When do governments choose liberal or conservative central bankers?
Source: Adolph, BBC, Ch. 8
Method: Zero-inflated compositional data model
How long do central bankers stay in office?
Source: Adolph, BBC, Ch. 9
Method: Cox proportional hazards model
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Presenting Estimated Models in Social Science
Most empirical work in social science is regression model-driven,with a focus on conditional expectation
Our regression models are
full of covariates
often non-linear
usually involve interactions and transformations
If there is anything we need to visualize well, it is our models
Yet we often just print off tables of parameter estimates
Limits readersʼ and analystsʼ understanding of the results
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Presenting Estimated (Causal) Models in Social Science
What if you work in a causal inference framework?
Still a great need for visualization:
to show robustness across different techniques
to show differences across quantities of interest(e.g., ATE vs. ATT)
to show variation across different kinds of subjects(variation in LATEs/LATTs)
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What Most Discussions of Statistical Graphics Leave Out
Tufteʼs books have had a huge impact on information visualization
However, they have two important limits:
Modeling Most examples are either exploratory or very simple models;
Social scientists want cutting edge applications
Tools Need to translate aesthetic guidelines into software
Social scientists are unlikely to do this on their own –and shouldnʼt have to!
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Goals for Visualizing Model Inference
Key need Ready-to-use techniques to visually present model results:
for many variables
for many robustness checks
showing uncertainty
without accidental extrapolation
for an audience without deep statistical knowledge
“Tufte without Tears”
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Goals for Visualizing Model Inference
Key need Ready-to-use techniques to visually present model results:
for many variables
for many robustness checks
showing uncertainty
without accidental extrapolation
for an audience without deep statistical knowledge
“Tufte without Tears”
Chris Adolph (University of Washington) VISUALIZING INFERENCE 8 / 134
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Goals for Visualizing Model Inference
Key need Ready-to-use techniques to visually present model results:
for many variables
for many robustness checks
showing uncertainty
without accidental extrapolation
for an audience without deep statistical knowledge
“Tufte without Tears”
Chris Adolph (University of Washington) VISUALIZING INFERENCE 8 / 134
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Goals for Visualizing Model Inference
Key need Ready-to-use techniques to visually present model results:
for many variables
for many robustness checks
showing uncertainty
without accidental extrapolation
for an audience without deep statistical knowledge
“Tufte without Tears”
Chris Adolph (University of Washington) VISUALIZING INFERENCE 8 / 134
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Goals for Visualizing Model Inference
Key need Ready-to-use techniques to visually present model results:
for many variables
for many robustness checks
showing uncertainty
without accidental extrapolation
for an audience without deep statistical knowledge
“Tufte without Tears”
Chris Adolph (University of Washington) VISUALIZING INFERENCE 8 / 134
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Voting Example (Logit Model)
We will explore a simple dataset using asimple model of voting
People either vote (Votei = 1),or they donʼt (Votei = 0)
Many factors could influence turn-out;we focus on age and education
National Election Survey (2000) data
“Did you vote in 2000 election?”
vote00 age hsdeg coldeg1 49 1 00 35 1 01 57 1 01 63 1 01 40 1 01 77 0 00 43 1 01 47 1 11 26 1 11 48 1 0
…
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.Logit of Decision to Vote, 2000 Presidential NES..
......
est. s.e. p-value
Age 0.074 0.017 0.000Age2 -0.0004 0.0002 0.009High School Grad 1.168 0.178 0.000College Grad 1.085 0.131 0.000Constant -3.05 0.418 0.000
Age enters as a quadratic to allow the probability of votingto first rise and eventually fall over the life course
Results look sensible, but what do they mean?
Which has the bigger effect, age or education?
What is the probability a specific person will vote?
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An Alternative to Eye-glazing Tables
...1 Estimate your model as normal; treat the output as an intermediate step
...2 Translate your model results back into the scale of the response variable
▶ Modeling war? Show the change in probability of war associated with x
▶ Modeling counts of crimes committed? Show how those counts vary with x
▶ Unemployment rate time series? Show how a change in x shifts the unemploymentrate over the following t years
...3 Calculate or simulate the uncertainty in these final quantities of interest
...4 Present visually as many scenarios calculated from the model as needed
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An Alternative to Eye-glazing Tables
...1 Estimate your model as normal; treat the output as an intermediate step
...2 Translate your model results back into the scale of the response variable
▶ Modeling war? Show the change in probability of war associated with x
▶ Modeling counts of crimes committed? Show how those counts vary with x
▶ Unemployment rate time series? Show how a change in x shifts the unemploymentrate over the following t years
...3 Calculate or simulate the uncertainty in these final quantities of interest
...4 Present visually as many scenarios calculated from the model as needed
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An Alternative to Eye-glazing Tables
...1 Estimate your model as normal; treat the output as an intermediate step
...2 Translate your model results back into the scale of the response variable
▶ Modeling war? Show the change in probability of war associated with x
▶ Modeling counts of crimes committed? Show how those counts vary with x
▶ Unemployment rate time series? Show how a change in x shifts the unemploymentrate over the following t years
...3 Calculate or simulate the uncertainty in these final quantities of interest
...4 Present visually as many scenarios calculated from the model as needed
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An Alternative to Eye-glazing Tables
...1 Estimate your model as normal; treat the output as an intermediate step
...2 Translate your model results back into the scale of the response variable
▶ Modeling war? Show the change in probability of war associated with x
▶ Modeling counts of crimes committed? Show how those counts vary with x
▶ Unemployment rate time series? Show how a change in x shifts the unemploymentrate over the following t years
...3 Calculate or simulate the uncertainty in these final quantities of interest
...4 Present visually as many scenarios calculated from the model as needed
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A bit more formally… (KTW, 2000, AJPS)
We want to know the behavior of E(y|x) as we vary x
In non-linear models with multiple regressors, this gets tricky
The effect of x1 depends on all the other xʼs and β̂ʼs
Generally, we will need to make a set of “counterfactual” assumptions:x1 = a, x2 = b, x3 = c, …
Choose a, b, c, . . . to match a particular counterfactual case of interest or
Hold all but one of the xʼs at their mean values (or other baseline, such as thefactual values by case), then systematically vary the remaining x
The same trick works if we are after differences in y related to changes in x, such asE(yscen2 − yscen1|xscen1, xscen2) or E(yscen2/yscen1|xscen1, xscen2)
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Calculating quantities of interest
Our goal to obtain “quantities of interest,” like
Expected Values: E(y|xc)
Differences: E(yc2 − yc1|xc1, xc2)
Risk Ratios: E(yc2/yc1|xc1, xc2)
or any other function of the above
for some counterfactual xcʼs.
For our Voting example, thatʼs easy – just plug xc into the systematic component:
E(y|xc) = logit−1(xcβ)
=1
1+ exp(−xcβ)= Pr(y|xc)
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Calculating quantities of interest
Our goal to obtain “quantities of interest,” like
Expected Values: E(y|xc)
Differences: E(yc2 − yc1|xc1, xc2)
Risk Ratios: E(yc2/yc1|xc1, xc2)
or any other function of the above
for some counterfactual xcʼs.
For our Voting example, thatʼs easy – just plug xc into the systematic component:
E(y|xc) = logit−1(xcβ) =1
1+ exp(−xcβ)
= Pr(y|xc)
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Calculating quantities of interest
Our goal to obtain “quantities of interest,” like
Expected Values: E(y|xc)
Differences: E(yc2 − yc1|xc1, xc2)
Risk Ratios: E(yc2/yc1|xc1, xc2)
or any other function of the above
for some counterfactual xcʼs.
For our Voting example, thatʼs easy – just plug xc into the systematic component:
E(y|xc) = logit−1(xcβ) =1
1+ exp(−xcβ)= Pr(y|xc)
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Getting confidence intervals is harder, but there are several options:
For maximum likelihood models,simulate the response conditional on the regressors
These simulations can easily be summarized as CIs: sort them and takepercentiles
See King, Tomz, and Wittenberg, 2000, American Journal of Political Science,and the Zelig or simcf packages for R or Clarify for Stata.
For Bayesian models, usual model output is a set of posterior draws
See Andrew Gelman and Jennifer Hill, 2006, Data Analysis Using Hierarchical/Multilevel Models, Cambridge UP.
Once we have the quantities of interest and confidence intervals,weʼre ready to make some graphs…but how?
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Here is the graphthat King, Tomz, andWittenberg createdfor this model
How would we makethis ourselves?
Could use the defaultgraphics in Zelig orClarify (limiting, notas nice as the above)
Could do it by hand(tedious)
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Here is the graphthat King, Tomz, andWittenberg createdfor this model
How would we makethis ourselves?
Could use the defaultgraphics in Zelig orClarify (limiting, notas nice as the above)
Could do it by hand(tedious)
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Weʼll return to thisexample later anddevelop tools formaking plots like this
Note we donʼt haveto always presentmodel inference inthis format
Beginners often fixateon plots like this, witha continuouscovariate on thex-axis, but there areother options…
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American Interest Rate Policy
Recall this example from my own work on central banking (Bankers, Bureaucrats, andCentral Bank Politics, Cambridge U.P., 2013, Ch. 4)
Federal Reserve Open MarketCommittee (FOMC) sets interest rates10×/year
Members of the FOMC vote on theChairʼs proposed interest rate
Dissenting voters signal whether theywould like a higher or lower rate
Dissents are rare but may besymptomatic of how the actual rate getschosen
Many factors could influenceinterest rate votes:
Individual Career backgroundAppointing partyInteractions of above
Economy Expected inflationExpected unemployment
Politics Election cycles
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American Interest Rate Policy
Recall this example from my own work on central banking (Bankers, Bureaucrats, andCentral Bank Politics, Cambridge U.P., 2013, Ch. 4)
Federal Reserve Open MarketCommittee (FOMC) sets interest rates10×/year
Members of the FOMC vote on theChairʼs proposed interest rate
Dissenting voters signal whether theywould like a higher or lower rate
Dissents are rare but may besymptomatic of how the actual rate getschosen
Many factors could influenceinterest rate votes:
Individual Career backgroundAppointing partyInteractions of above
Economy Expected inflationExpected unemployment
Politics Election cycles
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Response variable: FOMC Votes (1 = ease, 2 = accept, 3 = tighten)
EVs param. s.e. EVs param. s.e.FinExp −0.021 (0.146) E(Inflation) 0.019 (0.015)GovExp −0.753 (0.188) E(Unemployment) −0.035 (0.022)FMExp −1.039 (0.324) In-Party, election year −0.182 (0.103)CBExp −0.142 (0.141) Republican −0.485 (0.102)EcoExp × Repub 0.934 (0.281) Constant 2.490 (0.148)EcoExp × Dem −0.826 (0.202) Cutpoint (τ) 3.745 (0.067)
N 2957 ln likelihood −871.68
Table 1: Problematic presentation: FOMC member dissenting votes—Ordered probit parameters.
Estimated ordered probit parameters, with standard errors in parentheses, from the regression of a j = 3 categoryvariable on a set of explanatory variables (EVs). Although such nonlinear models are often summarized by tableslike this one, especially in the social sciences, it is difficult to discern the effects of the EVs listed at right onthe probability of each of the j outcomes. Because the career variables XXXExp are logically constrained to aunit sum, even some of the signs are misleading. The usual quantities of interest for an ordered probit modelare not the parameters (β and τ), but estimates of Pr(yj |xc, β, τ) for hypothetical levels of the EVs xc, whichI plot in Figure 1.
And the hopelessness of directly interpretingordered probit on compositional covariates
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Instead, we used adotplot of probabilitiessimulated for a series ofinteresting scenarios
FMExp
GovExp
EcoExp × Dem
Republican
In-Party & Election
E(Unemployment)
E( Inflation)
CBExp
FinExp
EcoExp × Repub
Response to anIncrease in …
Probability of hawkish dissent
Change in P(hawkish dissent)
0.03x 0.1x 0.2x 0.5x 1x 2x 5x 10x
0.1% 0.4% 0.8% 2% 4% 8% 20% 40%
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Instead, we used adotplot of probabilitiessimulated for a series ofinteresting scenarios
…which we carefullysorted to produce a morereadable diagonalizedpresentation
FMExp
GovExp
EcoExp × Dem
Republican
In-Party & Election
E(Unemployment)
E( Inflation)
CBExp
FinExp
EcoExp × Repub
Response to anIncrease in …
Probability of hawkish dissent
Change in P(hawkish dissent)
0.03x 0.1x 0.2x 0.5x 1x 2x 5x 10x
0.1% 0.4% 0.8% 2% 4% 8% 20% 40%
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And explained the resultsin terms of thosescenarios, uniting the textof our report with thefigure:
“The average centralbanker dissents in favorof tighter interest rates4% of the time. Incontrast, former treasuryofficials in the FOMCdissent 0.6% of the time,with a 95% CI from0.05% to 2%.”
FMExp
GovExp
EcoExp × Dem
Republican
In-Party & Election
E(Unemployment)
E( Inflation)
CBExp
FinExp
EcoExp × Repub
Response to anIncrease in …
Probability of hawkish dissent
Change in P(hawkish dissent)
0.03x 0.1x 0.2x 0.5x 1x 2x 5x 10x
0.1% 0.4% 0.8% 2% 4% 8% 20% 40%
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And explained the resultsin terms of thosescenarios, uniting the textof our report with thefigure:
“Other formerbureaucrats issue hawkishdissents 1% of the time[95% CI: 0.5 to 2.0], allelse equal.”
FMExp
GovExp
EcoExp × Dem
Republican
In-Party & Election
E(Unemployment)
E( Inflation)
CBExp
FinExp
EcoExp × Repub
Response to anIncrease in …
Probability of hawkish dissent
Change in P(hawkish dissent)
0.03x 0.1x 0.2x 0.5x 1x 2x 5x 10x
0.1% 0.4% 0.8% 2% 4% 8% 20% 40%
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Comparative Inflation Performance
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
Now instead of studying individual central bankers in the United States, we study apanel of 20 central banks across the industrialized world (pre-Euro data)
We ask what effect the average career composition of the central bank policy boardhas on inflation
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Comparative Inflation Performance
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
Now instead of studying individual central bankers in the United States, we study apanel of 20 central banks across the industrialized world (pre-Euro data)
We ask what effect the average career composition of the central bank policy boardhas on inflation
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Comparative Inflation Performance
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
We imagine a central bank that initially has central bankers with typical careerexperience (i.e., the global average in each category)
Then, we imagine raising experience in one category (say finance, or FinExp),and use the model to predict how inflation will change over the next 5 years
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Comparative Inflation Performance
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
Why not just show a coefficient for each career category?Two reasons to show the first difference in inflation over time:
1. Raising FinExp means lowering the other categories,so effects are blended across coefficients
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Comparative Inflation Performance
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
Why not just show a coefficient for each career category?Two reasons to show the first difference in inflation over time:
2. Effects in time series models build over time;coefficients show (somewhat arbitrary) first period effects
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Comparative Inflation Performance
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
We simply iterate the KTW simulation algorithm over 5 periods,computing for each period the difference from inflation under the average board
I used ldvsimfd() in the simcf package for R;see my course on Panel Data Analysis offered at Essex Summer School
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Comparative Inflation Performance
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
In the plot above, I show two different scenarios iterated over time:increasing finance experience, or increasing finance ministry experience
Both produce significant reductions in inflation compared to the baseline,and mostly converge to new equilibria after 5 years
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0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd EcoExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd BusExp
−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd GovExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd CBExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
Once weʼve explainedthe model, simulationmethod, and a single plotin our graphic, we canexpand to multipledisplays
The plot at left replacesan eye-glazing, opaque,and (because ofcompositional constraints)misleading table ofregression coefficients
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“bbc” — / / — : — page —
Table 3.7. Log inflation regressed on central banker characteristics, twenty countries, 1973to 2000, quarterly.
Expected DV: lnInflationVariable Sign
FinExpj,t−2 −
FMExpj,t−2 −/+
CBExpj,t−2 +/−
GovExpj,t−2 +
CBIj,t−2 −
CBCCmedj,t−2 −
CBIj,t−2 × CBCCmed
j,t−2 −
(Imports/GDP)j,t−2 −
%EcDegreej,t−2 −
ln πj,t−1
ln πj,t−2
Fixed effects x x x xNser R2 LM test critical =
Least squares estimates with panelcorrected standard errors in parenthesesECB members excluded after LM test refers to a Lagrange Multipliertest for serial correlation
People often ask, “Whatif the journal insists on atable instead of thefigure?”
In my experience, no oneprefers this table to thegraph
Give them both, focusyour write-up on thegraphic, and make surethe graphic explainseverything you wanted toget from the table
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Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd CBCC−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd CBI
cbi−c
cbi−3
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp+ FMExp
−GovExp− CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
No tradeoffs: The small multiple graphs are more accessible to a broad audience andmore useful to specialists than a table
You can always include the table as an appendix for those who want to “look underthe hood,” but cast your argument in terms of the graphics
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Wanted: an easy-to-use R package that
...1 takes as input the output of estimated statistical models
...2 makes a variety of plots for model interpretation
...3 plots “triples” (lower, estimate, upper) from estimated models well
...4 lays out these plots in a tiled arrangement (small multiples)
...5 takes care of axes, titles, and other fussy details
With considerable work, one could
coerce Rʼs basic graphics to do this badly
or get ggplot2 or lattice to do this fairly well for a specific case
But an easy-to-use, general solution is lacking
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Wanted: an easy-to-use R package that
...1 takes as input the output of estimated statistical models
...2 makes a variety of plots for model interpretation
...3 plots “triples” (lower, estimate, upper) from estimated models well
...4 lays out these plots in a tiled arrangement (small multiples)
...5 takes care of axes, titles, and other fussy details
With considerable work, one could
coerce Rʼs basic graphics to do this badly
or get ggplot2 or lattice to do this fairly well for a specific case
But an easy-to-use, general solution is lacking
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The tile package
My answer is the tile package, written using Rʼs grid graphics
Some basic tile graphic types:
scatter Scatterplots with fits, CIs, and extrapolation checkinglineplot Line plots with fits, CIs, and extrapolation checkingropeladder Dot plots with CIs and extrapolation checking
Each can take as input draws from the posterior of a regression model
A call to a tile function makes a multiplot layout:
ideal for small multiples of model parameters
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An example tile layout, minus traces
1 5 10 20 350
0.2
0.4
0.6
0.8
1
Row Title 1
Column Title 1Plot Title 1
Under Title 1
X Axis 1
Y A
xis
1
1 5 10 20 35
0
0.2
0.4
0.6
0.8
1
Column Title 2Plot Title 2
Under Title 2
Top Axis 2
Right A
xis 2
1 5 10 20 350
0.2
0.4
0.6
0.8
1
Column Title 3Plot Title 3
Under Title 3
X Axis 3
1 5 10 20 35
0
0.2
0.4
0.6
0.8
1
Row Title 2
Plot Title 4
Under Title 4
Top Axis 4
Right A
xis 4
1 5 10 20 350
0.2
0.4
0.6
0.8
1
Plot Title 5
Under Title 5
X Axis 5
1 5 10 20 35
0
0.2
0.4
0.6
0.8
1
Plot Title 6
Under Title 6
Top Axis 6
Right A
xis 6
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
▶ Could be a set of points, or text labels, or lines, or a polygon
▶ Could be a set of points and symbols, colors, labels, fit line, CIs, and/orextrapolation limits
▶ Could be the data for a dotchart, with labels for each line
▶ Could be the marginal data for a rug
▶ All annotation must happen in this step
▶ Primitive traces: linesTile(), pointsile(), polygonTile(), polylinesTile(),textTile(), and circleTile()
▶ Complex traces: lineplot(), scatter(), ropeladder(), and rugTile()
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
▶ Could be a set of points, or text labels, or lines, or a polygon
▶ Could be a set of points and symbols, colors, labels, fit line, CIs, and/orextrapolation limits
▶ Could be the data for a dotchart, with labels for each line
▶ Could be the marginal data for a rug
▶ All annotation must happen in this step
▶ Primitive traces: linesTile(), pointsile(), polygonTile(), polylinesTile(),textTile(), and circleTile()
▶ Complex traces: lineplot(), scatter(), ropeladder(), and rugTile()
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
▶ Could be a set of points, or text labels, or lines, or a polygon
▶ Could be a set of points and symbols, colors, labels, fit line, CIs, and/orextrapolation limits
▶ Could be the data for a dotchart, with labels for each line
▶ Could be the marginal data for a rug
▶ All annotation must happen in this step
▶ Primitive traces: linesTile(), pointsile(), polygonTile(), polylinesTile(),textTile(), and circleTile()
▶ Complex traces: lineplot(), scatter(), ropeladder(), and rugTile()
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
▶ Could be a set of points, or text labels, or lines, or a polygon
▶ Could be a set of points and symbols, colors, labels, fit line, CIs, and/orextrapolation limits
▶ Could be the data for a dotchart, with labels for each line
▶ Could be the marginal data for a rug
▶ All annotation must happen in this step
▶ Primitive traces: linesTile(), pointsile(), polygonTile(), polylinesTile(),textTile(), and circleTile()
▶ Complex traces: lineplot(), scatter(), ropeladder(), and rugTile()
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
▶ Could be a set of points, or text labels, or lines, or a polygon
▶ Could be a set of points and symbols, colors, labels, fit line, CIs, and/orextrapolation limits
▶ Could be the data for a dotchart, with labels for each line
▶ Could be the marginal data for a rug
▶ All annotation must happen in this step
▶ Primitive traces: linesTile(), pointsile(), polygonTile(), polylinesTile(),textTile(), and circleTile()
▶ Complex traces: lineplot(), scatter(), ropeladder(), and rugTile()
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
▶ Could be a set of points, or text labels, or lines, or a polygon
▶ Could be a set of points and symbols, colors, labels, fit line, CIs, and/orextrapolation limits
▶ Could be the data for a dotchart, with labels for each line
▶ Could be the marginal data for a rug
▶ All annotation must happen in this step
▶ Primitive traces: linesTile(), pointsile(), polygonTile(), polylinesTile(),textTile(), and circleTile()
▶ Complex traces: lineplot(), scatter(), ropeladder(), and rugTile()
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Trace functions in tile
Primitive trace functions:
linesTile Plot a set of connected line segmentspointsTile Plot a set of pointspolygonTile Plot a shaded regionpolylinesTile Plot a set of unconnected line segmentscircleTile Plot a set of circlestextTile Plot text labels
Complex traces for model or data exploration:
lineplot Plot lines with confidence intervals, extrapolation warningsropeladder Plot dotplots with confidence intervals, extrapolation warnings,
and shaded rangesrugTile Plot marginal data rugs to axes of plotsscatter Plot scatterplots with text and symbol markers,
fit lines, and confidence intervals
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
...2 Plot the data traces. Using the tile() function, simultaneously plot all traces to allplots.
▶ This is the step where the scaffolding gets made: axes and titles
▶ Set up the rows and columns of plots
▶ Titles of plots, axes, rows of plots, columns of plots, etc.
▶ Set up axis limits, ticks, tick labels, logging of axes
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
...2 Plot the data traces. Using the tile() function, simultaneously plot all traces to allplots.
▶ This is the step where the scaffolding gets made: axes and titles
▶ Set up the rows and columns of plots
▶ Titles of plots, axes, rows of plots, columns of plots, etc.
▶ Set up axis limits, ticks, tick labels, logging of axes
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Three steps to make tile plots
...1 Create data traces. Each trace contains the data and graphical parameters neededto plot a single set of graphical elements to one or more plots.
...2 Plot the data traces. Using the tile() function, simultaneously plot all traces to allplots.
...3 Examine output and revise. Look at the graph made in step 2, and tweak the inputparameters for steps 1 and 2 to make a better graph.
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2 3 4 5 6 7
0
20
40
60
80
Australia
Belgium
Canada
Denmark
Finland
France
GermanyItaly
NetherlandsNorway
Sweden
Switzerland
United Kingdom
United States
MajoritarianProportionalUnanimity
Effective number of parties
% li
fted
from
pov
erty
by
taxe
s &
trans
fers
Letʼs make this plot
CODE EXAMPLE
inequalityScatter.R
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Plot simulations of QoI
Generally, we want to plot triples: lower, estimate, upperWe could do this for specific discrete scenarios, e.g.
Pr(Voting) given five distinct sets of xʼs
Recommended plot: Dotplot with confidence interval lines
Or for a continuous stream of scenarios, e.g.,
Hold all but Age constant, then calculate Pr(Voting) at every level of Age
Recommended plot: Lineplot with shaded confidence intervals
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Plot simulations of QoI
Generally, we want to plot triples: lower, estimate, upperWe could do this for specific discrete scenarios, e.g.
Pr(Voting) given five distinct sets of xʼs
Recommended plot: Dotplot with confidence interval lines
Or for a continuous stream of scenarios, e.g.,
Hold all but Age constant, then calculate Pr(Voting) at every level of Age
Recommended plot: Lineplot with shaded confidence intervals
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This example isobviously superior tothe table of logitcoefficients
But is there anythingwrong or missinghere?
18 year old collegegrads?!
And what about highschool dropouts?
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This example isobviously superior tothe table of logitcoefficients
But is there anythingwrong or missinghere?
18 year old collegegrads?!
And what about highschool dropouts?
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20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
Age of Respondent
Pro
babi
lity
of V
otin
g
Less than HS
High School
College
Logit estimates:95% confidence
interval is shaded
Here is the graphicredrawn in tile
tile helps ussystematize plottingmodel results,and helps avoidunwantedextrapolation bylimiting results to theconvex hull
CODE EXAMPLE
votingLineplots.R
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Voting Example Redux
Next step: learn to simulate and plot first differences and relative risks
We could do this with our current example.E.g., hold age fixed and compute the change in Pr(Vote)given an increase in education
But for pedagogical reasons, it will be more useful to add an additional covariate
We now add to our voting model whether the respondent was married
Theory: Marriage should increase voting by increasing concernfor a variety of public goods, or by forming ties to a local community, etc.
How would this competing model normally be presented?
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.Logit of Decision to Vote, 2000 Presidential NES..
......
M1 M2
Age 0.075 0.061(0.017) (0.017)
Age2 -0.0004 -0.0003(0.0002) (0.0002)
High School Grad 1.124 1.099(0.180) (0.181)
College Grad 1.080 1.053(0.131) (0.132)
Married 0.373(0.110)
Constant -3.019 -2.866(0.418) (0.421)
log likelihood -1101.370 -1099.283N 1783 1783
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Comparing Logistic Regression Models
But we can also compare our results in an intelligible way.
20 30 40 50 60 70 80 900
0.2
0.4
0.6
0.8
1
Age of Respondent
Prob
abilit
y of
Vot
ing
Less than HS
High School
College
Logit estimates:95% confidence
interval is shaded
Model 2 (control for Married)
Effects of Age and Education havenʼt discernably changedChris Adolph (University of Washington) VISUALIZING INFERENCE 45 / 134
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Comparing Logistic Regression Models
But we can also compare our results in an intelligible way.
20 30 40 50 60 70 80 900
0.2
0.4
0.6
0.8
1
Age of Respondent
Prob
abilit
y of
Vot
ing
Less than HS
High School
College
Logit estimates:95% confidence
interval is shaded
Model 2 (control for Married)
Our first attempt to show model robustness – weʼll find more efficient waysChris Adolph (University of Washington) VISUALIZING INFERENCE 46 / 134
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A common misconception about confidence intervals
20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
Age of Respondent
Pro
babi
lity
of V
otin
g
Currently Married
Not Married
Logit estimates:95% confidence
interval is shaded
Ceteris paribus,marriage has a moderateeffect.
Is this effect statisticallysignificant?
Overlapping CIs ofexpected values doesnʼtalways imply aninsignificant difference
This is not necessarily thecase in “close calls”
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A common misconception about confidence intervals
20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
Age of Respondent
Pro
babi
lity
of V
otin
g
Currently Married
Not Married
Logit estimates:95% confidence
interval is shaded
Ceteris paribus,marriage has a moderateeffect.
Is this effect statisticallysignificant?
Overlapping CIs ofexpected values doesnʼtalways imply aninsignificant difference
This is not necessarily thecase in “close calls”
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A common misconception about confidence intervals
20 30 40 50 60 70 80 90
−0.1
0
0.1
0.2
0.3
0.4
0.5
Age of Respondent
Diff
eren
ce in
Pro
babi
lity
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
The right way to assessstatistical significance:simulate the CI of the firstdifference (or relativerisk) directly
This first difference isalways bounded awayfrom zero, hence alwayssignificant
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Avoid mistakenly rejecting significant first differences
20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
Age of Respondent
Pro
babi
lity
of V
otin
g
Currently Married
Not Married
Logit estimates:95% confidence
interval is shaded
20 30 40 50 60 70 80 90
−0.1
0
0.1
0.2
0.3
0.4
0.5
Age of Respondent
Diff
eren
ce in
Pro
babi
lity
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
Expected values estimate both difference & location;demanding a more detailed estimate from the model increases uncertainty
First differences and relative risks estimate the difference only, so they have slightlytighter confidence intervals
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Relative risk plots
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0.9
1
1.1
1.2
1.3
1.4
1.5
Age of Respondent
Rel
ativ
e R
isk
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
Consider showing relativerisks instead of (or insupport of) firstdifferences
Relative risks show “howmany times more likely” acategorical outcome isunder the counterfactual
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Relative risk plots
20 30 40 50 60 70 80 90
0.9
1
1.1
1.2
1.3
1.4
1.5
Age of Respondent
Rel
ativ
e R
isk
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
For continuous outcomes,RR shows how many timesbigger the outcome isunder the counterfactual
As with first differences,relative risk should besimulated directly tocalculate CIs correctly
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Setting up before-and-after scenarios
20 30 40 50 60 70 80 90
−0.1
0
0.1
0.2
0.3
0.4
0.5
Age of Respondent
Diff
eren
ce in
Pro
babi
lity
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
20 30 40 50 60 70 80 90
0.9
1
1.1
1.2
1.3
1.4
1.5
Age of Respondent
Rel
ativ
e R
isk
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
Setting up counterfactuals for FDs or RRs is tricky, as we will see in the code
Here I set before and after age to the same value (which varied across the plot)but I set Married to different values (0 before, 1 after)
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Setting up before-and-after scenarios
20 30 40 50 60 70 80 90
−0.1
0
0.1
0.2
0.3
0.4
0.5
Age of Respondent
Diff
eren
ce in
Pro
babi
lity
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
20 30 40 50 60 70 80 90
0.9
1
1.1
1.2
1.3
1.4
1.5
Age of Respondent
Rel
ativ
e R
isk
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
Take care in selecting the before and after values of all covariates
Most common place to make mistakes, with huge substantive consequencesChris Adolph (University of Washington) VISUALIZING INFERENCE 53 / 134
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Setting up before-and-after scenarios
20 30 40 50 60 70 80 90
−0.1
0
0.1
0.2
0.3
0.4
0.5
Age of Respondent
Diff
eren
ce in
Pro
babi
lity
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
20 30 40 50 60 70 80 90
0.9
1
1.1
1.2
1.3
1.4
1.5
Age of Respondent
Rel
ativ
e R
isk
of V
otin
g
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
RETURN TO CODE EXAMPLE
votingLineplots.R
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Robustness Checks
So far, weʼve presenting conditional expectations & differences from regressions
But are we confident that these were the “right” estimates?
The language of inference usually assumes we
correctly specified our model
correctly measured our variables
chose the right probability model
donʼt have influential outliers, etc.
Weʼre never completely sure these assumptions hold.
Most people present one model, and argue it was the best choice
Sometimes, a few alternatives are displayed
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Robustness Checks
So far, weʼve presenting conditional expectations & differences from regressions
But are we confident that these were the “right” estimates?
The language of inference usually assumes we
correctly specified our model
correctly measured our variables
chose the right probability model
donʼt have influential outliers, etc.
Weʼre never completely sure these assumptions hold.
Most people present one model, and argue it was the best choice
Sometimes, a few alternatives are displayed
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.The race of the variables..
......
Model 1 Model 2 Model 3 Model 4 Model 5
My variable X.XX X.XX X.XX X.XXof interest, x1 (X.XX) (X.XX) (X.XX) (X.XX)
A control X.XX X.XX X.XX X.XX X.XXI ”need” (X.XX) (X.XX) (X.XX) (X.XX) (X.XX)
A control X.XX X.XX X.XX X.XX X.XXI ”need” (X.XX) (X.XX) (X.XX) (X.XX) (X.XX)
A candidate X.XX X.XXcontrol (X.XX) (X.XX)
A candidate X.XX X.XXcontrol (X.XX) (X.XX)
Alternate X.XXmeasure of x1 (X.XX)
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Robustness Checks
Problems with the approach above?
...1 Lots of space to show a few permutations of the model
Most space wasted or devoted to ancillary info
...2 What if weʼre really interested in E(y|X), not β̂?
E.g., because of nonlinearities, interactions, scale differences, etc.
...3 The selection of permutations is ad hoc.
Weʼll try to fix 1 & 2.
Objection 3 is harder, but worth thinking about.
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Robustness Checks
Problems with the approach above?
...1 Lots of space to show a few permutations of the model
Most space wasted or devoted to ancillary info
...2 What if weʼre really interested in E(y|X), not β̂?
E.g., because of nonlinearities, interactions, scale differences, etc.
...3 The selection of permutations is ad hoc.
Weʼll try to fix 1 & 2.
Objection 3 is harder, but worth thinking about.
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Robustness Checks
Problems with the approach above?
...1 Lots of space to show a few permutations of the model
Most space wasted or devoted to ancillary info
...2 What if weʼre really interested in E(y|X), not β̂?
E.g., because of nonlinearities, interactions, scale differences, etc.
...3 The selection of permutations is ad hoc.
Weʼll try to fix 1 & 2.
Objection 3 is harder, but worth thinking about.
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Robustness Checks
Problems with the approach above?
...1 Lots of space to show a few permutations of the model
Most space wasted or devoted to ancillary info
...2 What if weʼre really interested in E(y|X), not β̂?
E.g., because of nonlinearities, interactions, scale differences, etc.
...3 The selection of permutations is ad hoc.
Weʼll try to fix 1 & 2.
Objection 3 is harder, but worth thinking about.
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Robustness Checks: An algorithm
...1 Identify a relation of interest between a concept X and a concept Y
...2 Choose:
▶ a measure of X , denoted X,
▶ a measure of Y , denoted y,
▶ a set of confounders, Z,
▶ a functional form, g(·)▶ a probability model of y, f(·)
...3 Estimate the probability model y ∼ f(μ, α), μ = g(vec(X, Z), β).
...4 Simulate the quantity of interest such as E(y|X), E(y2 − y1|X1, X2), orE(y2/y1|X1, X2) to obtain a point estimate and confidence interval.
...5 Repeat 2–4, changing at each iteration one of the choices in step 2.
...6 Compile the results in a variant of the dot plot called a ropeladder.
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liberal conservative
other
0.2
0.8
0.2
0.4
0.6
0.4
0.6
0.4
0.6
0.8
0.2
0.8
L R+
(.6,.1,.3)
(.4,.3,.3)
(.5,.2,.3)
In Gallery 7, we revieweda compositional datamodel from Ch. 8 ofBankers, Bureaucrats,and Central Bank Politics
We used a ternary plot toshow the careercomposition of appointedcentral bankers dependson the partisanship of theappointing government
How would we showrobustness underalternative specifications?
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One tool for robustness: Small Multiples & Backup Dancers
FixedEffects
PreviousLevels
OtherPCoG
Measures
liberal
other
L R
(.6,.1,.3)(.4,.3,.3)
liberal
other
LR
(.5,.2,.3)
(.4,.3,.3)
liberal
other
L
R
L
R
L
R
conservativeconservativeconservative
Once people understand ternary plots,they will immediately absorb a small, simplified version
Each of these small multiples shows our result under a different model
The similarity of each plot is immediately obvious here
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One tool for robustness: Small Multiples & Backup Dancers
FixedEffects
PreviousLevels
OtherPCoG
Measures
liberal
other
L R
(.6,.1,.3)(.4,.3,.3)
liberal
other
LR
(.5,.2,.3)
(.4,.3,.3)
liberal
other
L
R
L
R
L
R
conservativeconservativeconservative
Once people understand ternary plots,they will immediately absorb a small, simplified version
Each of these small multiples shows our result under a different model
The similarity of each plot is immediately obvious here
If not, putting the original plot in gray in the background helps:Amanda Cox call these “backup dancers”
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One tool for robustness: Small Multiples & Backup Dancers
FixedEffects
PreviousLevels
OtherPCoG
Measures
liberal
other
L R
(.6,.1,.3)(.4,.3,.3)
liberal
other
LR
(.5,.2,.3)
(.4,.3,.3)
liberal
other
L
R
L
R
L
R
conservativeconservativeconservative
But in this case, I need lots of robustness checks
Because of the multiple equations, my statistical model is so demandingitʼs hard to include many regressors at once
If I try them one at a time, I would fill pages with triangle plots
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One tool for robustness: Small Multiples & Backup Dancers
FixedEffects
PreviousLevels
OtherPCoG
Measures
liberal
other
L R
(.6,.1,.3)(.4,.3,.3)
liberal
other
LR
(.5,.2,.3)
(.4,.3,.3)
liberal
other
L
R
L
R
L
R
conservativeconservativeconservative
However, the horizontal dimension is the substantively important one:the one that affects affects economic outcomes
So I create a new QoI: Central Banker Career Conservatism (CBCC)
CBCC = Conservative Experience - Liberal Experience
And use my model to predict changes in CBCC and plot them on a ropeladder
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Robustness Ropeladder: Partisan central banker appointment
Control added
[None]
Office appointed to
CBI (3 index avg.)
CBI (Cukierman)
Lagged inflation
Lagged unemployment
Trade openness
Endebtedness
Financial Sector Employment
Financial Sector Score
Time trend
Central bank staff size
-0.4 -0.2 0.0 0.4 0.60.2 -0.4 -0.2 0.0 0.40.2 0.6
Estimated increase in Central Bank Conservatism (CBCC) resulting from …
Shifting control from
low (μ-1.5 sd)
to high (μ+1.5)
Shifting PCoG from
From left gov (μ-1.5 sd)
to right gov (μ+1.5 sd)
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Anatomy of a ropeladder plot
I call this a ropeladder plot.
The column of dots shows the relationship between y and a specific X under differentmodel assumptions
Each entry corresponds to a different assumption about the specification, or themeasures, or the estimation method, etc.
If all the dots line up, with narrow, similar CIs, we say the finding is robust, and reflectsthe data under a range of reasonable assumptions
If the ropeladder is “blowing in the wind”, we may be skeptical of the finding.It depends on model assumptions that may be controversial
The shaded gray box shows the full range of the point estimates for the QoI.
Narrow is better.
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Anatomy of a ropeladder plot
I call this a ropeladder plot.
The column of dots shows the relationship between y and a specific X under differentmodel assumptions
Each entry corresponds to a different assumption about the specification, or themeasures, or the estimation method, etc.
If all the dots line up, with narrow, similar CIs, we say the finding is robust, and reflectsthe data under a range of reasonable assumptions
If the ropeladder is “blowing in the wind”, we may be skeptical of the finding.It depends on model assumptions that may be controversial
The shaded gray box shows the full range of the point estimates for the QoI.
Narrow is better.
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Anatomy of a ropeladder plot
I call this a ropeladder plot.
The column of dots shows the relationship between y and a specific X under differentmodel assumptions
Each entry corresponds to a different assumption about the specification, or themeasures, or the estimation method, etc.
If all the dots line up, with narrow, similar CIs, we say the finding is robust, and reflectsthe data under a range of reasonable assumptions
If the ropeladder is “blowing in the wind”, we may be skeptical of the finding.It depends on model assumptions that may be controversial
The shaded gray box shows the full range of the point estimates for the QoI.
Narrow is better.Chris Adolph (University of Washington) VISUALIZING INFERENCE 65 / 134
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Why ropeladders?
...1 Anticipate objections on model assumptions, and have concrete answers.
Avoid: “I ran it that other way, and it came out the ʻsame.”ʼ
Instead: “I ran it that other way, and look – it made no substantive or statisticaldifference worth speaking of.”
Or: “…it makes this much difference.”
...2 Investigate robustness more thoroughly
Traditional tabular presentation would have run to 7 pages, making comparisonhard and discouraging a thorough search
...3 Find patterns of model sensitivity
Two seemingly unrelated changes in specification had the same effect.(Unemployment and Financial Sector Size)
Turned out to be a missing third covariate (time trend)
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Why ropeladders?
...1 Anticipate objections on model assumptions, and have concrete answers.
Avoid: “I ran it that other way, and it came out the ʻsame.”ʼ
Instead: “I ran it that other way, and look – it made no substantive or statisticaldifference worth speaking of.”
Or: “…it makes this much difference.”
...2 Investigate robustness more thoroughly
Traditional tabular presentation would have run to 7 pages, making comparisonhard and discouraging a thorough search
...3 Find patterns of model sensitivity
Two seemingly unrelated changes in specification had the same effect.(Unemployment and Financial Sector Size)
Turned out to be a missing third covariate (time trend)
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Why ropeladders?
...1 Anticipate objections on model assumptions, and have concrete answers.
Avoid: “I ran it that other way, and it came out the ʻsame.”ʼ
Instead: “I ran it that other way, and look – it made no substantive or statisticaldifference worth speaking of.”
Or: “…it makes this much difference.”
...2 Investigate robustness more thoroughly
Traditional tabular presentation would have run to 7 pages, making comparisonhard and discouraging a thorough search
...3 Find patterns of model sensitivity
Two seemingly unrelated changes in specification had the same effect.(Unemployment and Financial Sector Size)
Turned out to be a missing third covariate (time trend)Chris Adolph (University of Washington) VISUALIZING INFERENCE 66 / 134
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0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd EcoExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd BusExp
−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd GovExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd CBExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
Recall the TSCS model ofinflation performance
How would we showrobustness here?
Once we understand thedynamics over time, wecan simplify ourpresentation
What if we isolate the 5year mark, and comparethe estimated effects ofcovariate on inflation atthat point under differentmodels?
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0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FinExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd FMExp
−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd EcoExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd BusExp
−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd GovExp−4
−2
0
2
4
0.0 2.5 5.0
Years after ...
+1 sd CBExp
Change in inflation, over time, from changing career composition of the central bank
Inflation-reducing
career types*
(Finance, Finance Ministry)
Neutral
career types†
(Economics, Business)
Inflation-increasing
career types*
(Government,
Central Bank Staff)
Career Index
(CBCC)‡
FinExp FMExp
GovExp CBExp
CentralBank
Independence‡
cbi-c =
Cukierman index
cbi-3 =avg of 3 indexes
Recall the TSCS model ofinflation performance
How would we showrobustness here?
Once we understand thedynamics over time, wecan simplify ourpresentation
What if we isolate the 5year mark, and comparethe estimated effects ofcovariate on inflation atthat point under differentmodels?
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Robustness for several QoIs at once
−3 −1 1 −3 −1 1 −1 1 3 −1 1 3 −3 −1 1
FinExp FMExp CBExp GovExp CBCC
Baseline
Robust Estimation
Add πworld
Use Cukierman CBI
Omit Imports/GDP
Add Exchange Regime
Add % Left-Appointees
Add Partisan CoG of Aptees
Add % with Econ PhDs
Change in inflation, five years after +1 s.d. in …Specification
Each ropeladder, or column, shows the effect of a different variable on the response
That is, reading across shows the results from a single model
Reading down shows the results for a single question across different modelsChris Adolph (University of Washington) VISUALIZING INFERENCE 68 / 134
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Robustness for several QoIs at once
−3 −1 1 −3 −1 1 −1 1 3 −1 1 3 −3 −1 1
FinExp FMExp CBExp GovExp CBCC
Baseline
Robust Estimation
Add πworld
Use Cukierman CBI
Omit Imports/GDP
Add Exchange Regime
Add % Left-Appointees
Add Partisan CoG of Aptees
Add % with Econ PhDs
Change in inflation, five years after +1 s.d. in …Specification
Arrows indicate confidence intervals that extend outside the plot
Choosing our own plotting area using limits= is critical for ropeladders
Focus on the area with the point estimates and on any problematic CIsChris Adolph (University of Washington) VISUALIZING INFERENCE 69 / 134
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Robustness for several QoIs at once
−3 −1 1 −3 −1 1 −1 1 3 −1 1 3 −3 −1 1
FinExp FMExp CBExp GovExp CBCC
Baseline
Robust Estimation
Add πworld
Use Cukierman CBI
Omit Imports/GDP
Add Exchange Regime
Add % Left-Appointees
Add Partisan CoG of Aptees
Add % with Econ PhDs
Change in inflation, five years after +1 s.d. in …Specification
To write up robustness, show this graphic and relegate tables to the appendix
You can be specific about the nature of robustness (no “hand-waving”),
yet still write up 8 robustness checks on 5 covariates in 2 pages totalChris Adolph (University of Washington) VISUALIZING INFERENCE 70 / 134
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Our tools are flexible
From the first few examples,you might think lineplots are for model inferenceand dotplots (ropeladders) are for model robustness
But these tools are flexible and reward creativity
In the following examples,I use dotplots made with ropeladder() to explore models,then use lineplots to explore robustness
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When simulation is the only option: Chinese leadership
Shih, Adolph, and Liu investigate the advancement of elite Chinese leaders in theReform Period (1982–2002)
Explain (partially observed) ranks of the top 300 to 500 Chinese Communist Partyleaders as a function of:
Demographics age, sex, ethnicityEducation level of degreePerformance provincial growth, revenueFaction birth, school, career, and family ties to top leaders
Bayesian model of partially observed ranks of CCP officials
Model parameters difficult to interpret: on a latent scaleand individual effects are conditioned on all other ranked members
Only solution:Simulate ranks of hypothetical officials as if placed in the observed hierarchy
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356 285 214 142 71 0
0 20 40 60 80 100
Expected rank
Expected percentile
< High School
Female
High School
Han
Non−Princeling
Average Member
Graduate School
College
Jiang Zemin Faction
Male
Relative GDP Growth +1 sd
Princeling
Party Exp +1 sd
Relative Fiscal Growth +1 sd
Minority
Hu Jintao Faction
Deng Faction
Age +1 sdAlt Central Cmte Central Cmte SC
Alt Central Cmte Central Cmte SC
Black circles showexpected ranks forotherwise averageChinese officialswith thecharacteristic listedat left
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356 285 214 142 71 0
0 20 40 60 80 100
Expected rank
Expected percentile
< High School
Female
High School
Han
Non−Princeling
Average Member
Graduate School
College
Jiang Zemin Faction
Male
Relative GDP Growth +1 sd
Princeling
Party Exp +1 sd
Relative Fiscal Growth +1 sd
Minority
Hu Jintao Faction
Deng Faction
Age +1 sdAlt Central Cmte Central Cmte SC
Alt Central Cmte Central Cmte SC
Thick blackhorizontal linesare 1 std errorbars, and thin linesare 95% CIs
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356 285 214 142 71 0
0 20 40 60 80 100
Expected rank
Expected percentile
< High School
Female
High School
Han
Non−Princeling
Average Member
Graduate School
College
Jiang Zemin Faction
Male
Relative GDP Growth +1 sd
Princeling
Party Exp +1 sd
Relative Fiscal Growth +1 sd
Minority
Hu Jintao Faction
Deng Faction
Age +1 sdAlt Central Cmte Central Cmte SC
Alt Central Cmte Central Cmte SC
Gray triangles areofficials withrandom effects at±1 sd; how muchunmeasuredfactors matter
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356 285 214 142 71 0
0 20 40 60 80 100
Expected rank
Expected percentile
< High School
Female
High School
Han
Non−Princeling
Average Member
Graduate School
College
Jiang Zemin Faction
Male
Relative GDP Growth +1 sd
Princeling
Party Exp +1 sd
Relative Fiscal Growth +1 sd
Minority
Hu Jintao Faction
Deng Faction
Age +1 sdAlt Central Cmte Central Cmte SC
Alt Central Cmte Central Cmte SC
It helps to sortrows of the plotfrom smallest tolargest effect(diagonalization)
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'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
−20
0
20
40C
hang
e in
R
ank
Perc
entil
e
'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
−20
0
20
40
Cha
nge
in
Ran
k Pe
rcen
tile
GDP Growth +1 sd Fiscal Revenue +1 sd
College Degree
Ethnic Minority
Female
Graduate Degree
We re-estimate the model separately for each year,leading to a large number of results with varying sets of covariates
A complex lineplot helps organize these results and facilitate comparisons
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'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
−20
0
20
40C
hang
e in
R
ank
Perc
entil
e
'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
−20
0
20
40
Cha
nge
in
Ran
k Pe
rcen
tile
GDP Growth +1 sd Fiscal Revenue +1 sd
College Degree
Ethnic Minority
Female
Graduate Degree
Note that these results are now first differences:
the expected percentile change in rank for an otherwise average official who gainsthe characteristic noted
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'82 '87 '82 '87 '82 '87 '82 '87 '92 '97 '02 '82 '87
−20
0
20
40C
hang
e in
R
ank
Perc
entil
e
'87 '92 '92 '97 '02 '97 '02 '82 '87 '92 '97 '02
−20
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20
40
Cha
nge
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tile
Mao Faction
Long Marcher 2nd Field Army Vet Deng Faction Hu Yaobang Fact'n
Zhao Ziyang Faction
Jiang Zemin Faction
Hu Jintao Faction Princeling
Over time, officialsʼ economic performance never matters, but factions often do
Runs counter to the conventional wisdom that meritocratic selection of officials liesbehind Chinese economic success
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Chinese Officials Robustness
Our findings were controversial: countered the widely accepted belief that Chineseofficials are rewarded for economic performance
Critics asked for lots of alternative specifications to probe our results
We used tile to show exactly what difference these robustness checks made usingoverlapping lineplots
We provide detailed one-to-one comparisons of our model with each alternative,for a lengthy appendix…
…And a single page summary for the printed article collecting all robustness checks
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'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
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Cha
nge
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tile
'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
−20
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20
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Cha
nge
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Ran
k Pe
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tile
GDP Growth +1 sd Fiscal Revenue +1 sd
College Degree
Ethnic Minority
Female
Graduate Degree
Original Model Factional Tie Requires Job Overlap
Some critics worried that our measures of faction were too sensitive,so we considered a more specific alternative
This didnʼt salvage the conventional wisdom on growth…
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'82 '87 '82 '87 '82 '87 '82 '87 '92 '97 '02 '82 '87
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Cha
nge
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'87 '92 '92 '97 '02 '97 '02 '82 '87 '92 '97 '02
−20
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20
40
Cha
nge
in
Ran
k Pe
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tile
Mao Faction
Long Marcher 2nd Field Army Vet Deng Faction Hu Yaobang Fact'n
Zhao Ziyang Faction
Jiang Zemin Faction
Hu Jintao Faction Princeling
Original Model Factional Tie Requires Job Overlap
But did (unsuprisingly) strengthen our factional results
(Specific measures pick up the strongest ties)
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'82 '87 '82 '87 '82 '87 '82 '87 '92 '97 '02 '82 '87
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Cha
nge
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tile
'87 '92 '92 '97 '02 '97 '02 '82 '87 '92 '97 '02
−20
0
20
40
Cha
nge
in
Ran
k Pe
rcen
tile
Mao Faction
Long Marcher 2nd Field Army Vet Deng Faction Hu Yaobang Fact'n
Zhao Ziyang Faction
Jiang Zemin Faction
Hu Jintao Faction Princeling
Original Model Surprise Performance, ARMA(p,q)
Other critics worried about endogeneity or selection effects flowing from politicalpower to economic performance
We used measures of unexpected growth to zero in on an officialʼs own performancein office – which still nets zero political benefit
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'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
−20
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Cha
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tile
'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
−20
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Cha
nge
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GDP Growth +1 sd Fiscal Revenue +1 sd
College Degree
Ethnic Minority
Female
Graduate Degree
Original Model Surprise Performance, ARMA(p,q)
The above summarizes results combined from 2 versions of a model applied over 5periods, each with 5 multiply imputed datasets (50 models)
But it still takes many pages to show all our robustness checks.Is there a more efficient way to show that our results stay essentially the same?
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'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
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'82 '87 '92 '97 '02 '82 '87 '92 '97 '02 '82 '87 '92 '97 '02
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Cha
nge
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GDP Growth +1 sd Fiscal Revenue +1 sd
College Degree
Ethnic Minority
Female
Graduate Degree
Original Model Ignore exact ACC ranks Job Overlap Only7 models controlling Surprise Performance, Five Factor Perform. & Schools
In our printed article, we show only this plot,which overlays the full array of robustness checks
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'82 '87 '82 '87 '82 '87 '82 '87 '92 '97 '02 '82 '87
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nge
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'87 '92 '92 '97 '02 '97 '02 '82 '87 '92 '97 '02
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Cha
nge
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Mao Faction
Long Marcher 2nd Field Army Vet Deng Faction Hu Yaobang Fact'n
Zhao Ziyang Faction
Jiang Zemin Faction
Hu Jintao Faction Princeling
Original Model Ignore exact ACC ranks Job Overlap Only7 models controlling Surprise Performance, Five Factor Perform. & Schools
Conveys hundreds of separate findings in a compact, readable form
No knowledge of Bayesian methods or partial rank coefficients required!
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More uses of ropeladder traces for model exploration
We will discuss implementation of ropeladders – for robustness and general modelinference – shortly
But first, letʼs explore three more uses of ropeladder dotplots that show off the fullrange of features of these traces:
Exploring interactive models using differences-in-differences
Grouping variables and interactions for easier comprehension and explanation
Grouping categorical responses to multinomial models
Remember, ropeladders are flexible – surely the most flexible way to present models
Be willing to experiment to make your model easier to explain
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
Recall ourcomparativeinflationexample
Central bankercareers →Inflationperformance
But is this aresult ofsocialization orincentives?
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...Recall ourcomparativeinflationexample
Central bankercareers →Inflationperformance
But is this aresult ofsocialization orincentives?
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...Interactionsmay reveal thesources ofcareer effects
I interact“conservative”careers with:
Age
Future JobAwarded
Public Voting
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
Focus on the“Age ofCentralBanker” trio ofresults
This modelinteractsCareerConservatism(CBCC) withcentral bankerage
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...We simulatethe effect of +1sd CBCC giveneither 65 yearold officials or45 year oldofficials
We areespeciallyinterested inthe differenceof the firstdifferencesacross thesescenarios
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
We use theshape ofsymbols tosuggest the“building up”of the full effectfor 65 yearolds
While open vs.filled indicatessignificance
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
Looking at thewhole plot, wefindconservatismhas biggerinflation-fighting effectswhen centralbankers end uptaking jobs inthe financialsector
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
Conservatismmay bestronger whenvotes arepublic, but thisis not quitesignificant
Age is a wash
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
Ropeladderscan exploreinteractiveeffects byworkingthrough eachcombination ofvalues for theinteractedcovariates
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−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
2 keys tosuccess:
Find theminimumnumber ofcombinationsneeded to tellthe story
Clearorganization –designedaround theclearestwrite-up of theresult
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Chapter 9 of BBC explorescorrelates of central bankertenure in 20 industrializedcountries using a Coxproportional hazards model
Covariate
AgeCareer typesEconomic performanceChange in governmentPerformance × Party
Last is most interesting: arecentral bankers graded on apartisan curve, with the Leftpenalizing unemploymentand the Right inflation?
“bbc” — / / — : — page —
Table 9.1. Cox proportional hazards estimates of central banker tenure.
Hazard 95% CICovariate ratio lower upper
Age> 75 70 < Age≤ 75 65 < Age≤ 70 Other Government Experience Abs diff in PCoG appt party vs current Financial Experience Finance Ministry Experience Current PCoG× Inflation Unemployment Inflation Current PCoG×Unemployment Central Bank Staff Experience Economics Experience Current Partisan Center of Gravity PCoG
N individualslog likelihood − L test p < 10−9
Entries are hazard ratios exponentiated coefficients and their associated percent confidence intervals Hazard ratios greater than one indicate factors makingretirement/dismissal more likely Confidence intervals are calculated using standard errors clustered by country; significant results are those with lower and upperbounds on the same side of
Results
The estimated Cox proportional hazards model is presented in Table withcovariates sorted so that the greatest hazards are listed first and the variablesmost likely to lengthen tenure at the central bank are listed last Coefficientshave been exponentiated to form hazard ratios so that values greater than oneindicate increases in the hazard rate and hence shorter expected tenures whilevalues less than one show reductions in the hazard rate longer tenures Forexample the estimated hazard ratio of for being older than years meansthat crossing the year threshold raises the probability of departure by
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Chapter 9 of BBC explorescorrelates of central bankertenure in 20 industrializedcountries using a Coxproportional hazards model
Covariate
AgeCareer typesEconomic performanceChange in governmentPerformance × Party
Last is most interesting: arecentral bankers graded on apartisan curve, with the Leftpenalizing unemploymentand the Right inflation?
“bbc” — / / — : — page —
Table 9.1. Cox proportional hazards estimates of central banker tenure.
Hazard 95% CICovariate ratio lower upper
Age> 75 70 < Age≤ 75 65 < Age≤ 70 Other Government Experience Abs diff in PCoG appt party vs current Financial Experience Finance Ministry Experience Current PCoG× Inflation Unemployment Inflation Current PCoG×Unemployment Central Bank Staff Experience Economics Experience Current Partisan Center of Gravity PCoG
N individualslog likelihood − L test p < 10−9
Entries are hazard ratios exponentiated coefficients and their associated percent confidence intervals Hazard ratios greater than one indicate factors makingretirement/dismissal more likely Confidence intervals are calculated using standard errors clustered by country; significant results are those with lower and upperbounds on the same side of
Results
The estimated Cox proportional hazards model is presented in Table withcovariates sorted so that the greatest hazards are listed first and the variablesmost likely to lengthen tenure at the central bank are listed last Coefficientshave been exponentiated to form hazard ratios so that values greater than oneindicate increases in the hazard rate and hence shorter expected tenures whilevalues less than one show reductions in the hazard rate longer tenures Forexample the estimated hazard ratio of for being older than years meansthat crossing the year threshold raises the probability of departure by
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The table is actually fairlyinterpretable, except:
The career covariates arecompositional, so their effectsare blended
The interaction terms are hardto mentally combine, and itʼsimpossible to get CIs withouta computer to help
…so maybe this isnʼt thatinterpretable
“bbc” — / / — : — page —
Table 9.1. Cox proportional hazards estimates of central banker tenure.
Hazard 95% CICovariate ratio lower upper
Age> 75 70 < Age≤ 75 65 < Age≤ 70 Other Government Experience Abs diff in PCoG appt party vs current Financial Experience Finance Ministry Experience Current PCoG× Inflation Unemployment Inflation Current PCoG×Unemployment Central Bank Staff Experience Economics Experience Current Partisan Center of Gravity PCoG
N individualslog likelihood − L test p < 10−9
Entries are hazard ratios exponentiated coefficients and their associated percent confidence intervals Hazard ratios greater than one indicate factors makingretirement/dismissal more likely Confidence intervals are calculated using standard errors clustered by country; significant results are those with lower and upperbounds on the same side of
Results
The estimated Cox proportional hazards model is presented in Table withcovariates sorted so that the greatest hazards are listed first and the variablesmost likely to lengthen tenure at the central bank are listed last Coefficientshave been exponentiated to form hazard ratios so that values greater than oneindicate increases in the hazard rate and hence shorter expected tenures whilevalues less than one show reductions in the hazard rate longer tenures Forexample the estimated hazard ratio of for being older than years meansthat crossing the year threshold raises the probability of departure by
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4 6 8 10 12 14 16
0.5x 0.75x 1x 1.25x 1.5x
Conditional median central banker tenure, in years
Conditional median central banker tenure, relative to baseline
Inflation is
Unemployment is
Inflation is
Unemployment is
Inflation is
Unemployment is
Left Appointee
Right Appointee
100% of CareerBackground in
●
low ●
low ●
low ●
low ●
●low
low ●
left stays
right stays
gov
fm
eco
high
high ●
high ●
●high
high ●
high ●
●right
●
fin
●cb
Under average government
Under Left government
Under Right government
Under changing governments
By career type
left
●
●
●
●
●
●
We can replace the entiretable with a complex dotplot
(Aside: Itʼs okay to providehandouts of really large plots– they donʼt display on LCDprojectors well)
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4 6 8 10 12 14 16
0.5x 0.75x 1x 1.25x 1.5x
Conditional median central banker tenure, in years
Conditional median central banker tenure, relative to baseline
Inflation is
Unemployment is
Inflation is
Unemployment is
Inflation is
Unemployment is
Left Appointee
Right Appointee
100% of CareerBackground in
●
low ●
low ●
low ●
low ●
●low
low ●
left stays
right stays
gov
fm
eco
high
high ●
high ●
●high
high ●
high ●
●right
●
fin
●cb
Under average government
Under Left government
Under Right government
Under changing governments
By career type
left
●
●
●
●
●
●
Instead of thinking, “Whatcovariates do I plot,” ask:
“What is the minimum set ofscenarios that will explore thefull model space”
The key is picking outcounterfactuals that exploreeffects of both inflation andunemployment under eachtype of government andunder each possible changein government
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Ropeladder Robustness Example: 1960 US crime rates
My apologies – this example isnʼt particularly substantively interesting or sharp
We have data from each of the 50 US states on crime rates in 1960
And a variety of covariates as seen on the next slide
We will fit a set of models with the same specification but different estimators
We will then consider several ropeladder-based presentations of robustness
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Kitchen sink models of 1960 US crime rates
Linear Robust Poisson Neg Bin
Constant -28820.91 -17784.56 -19.08 -15.43(10199.82 ) (8158.71 ) (1.77) (7.81)
% males aged 14–24 1156.49 2480.55 1.1 1.53(522.98 ) (418.32 ) (0.1) (0.4)
Southern state 0.97 138.11 0.06 0.06(141.49 ) (113.18 ) (0.02) (0.11)
Mean education (yrs) 1802.64 1413.62 1.84 2.11(590.84 ) (472.61 ) (0.11) (0.45)
Police spending 1960 897.54 422.45 0.81 0.76(813.8 ) (650.95 ) (0.15) (0.62)
Police spending 1959 6.66 651.14 0.01 0.01(823.35 ) (658.59 ) (0.15) (0.63)
Labor participation 143.91 2235.29 0.63 0.62(727.79 ) (582.15 ) (0.13) (0.56)
Males per 1000 94.71 -3469.7 -1.46 -2.3(1943.8 ) (1554.82 ) (0.36) (1.49)
State population -79.39 -138.58 -0.08 -0.07(51.4 ) (41.12 ) (0.01) (0.04)Chris Adolph (University of Washington) VISUALIZING INFERENCE 102 / 134
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Kitchen sink models of 1960 US crime rates, continued
continuedLinear Robust Poisson Neg Bin
Nonwhites per 1000 61.25 32.47 0.11 0.11(47.85 ) (38.28) (0.01) (0.04)
Unem, males 14–24 -325.65 -444.95 -0.18 -0.18(336.46 ) (269.13 ) (0.06) (0.26)
Unem, males 35–39 475.14 895.28 0.39 0.46(239.62 ) (191.67 ) (0.04) (0.18)
Gross state product, pc 282.31 -196.44 0.69 0.64(420.2 ) (336.11 ) (0.08) (0.32)
Income inequality 1461.68 943.27 1.68 1.56(386.64 ) (309.27 ) (0.07) (0.3)
Pr(imprisonment) -226.39 -443.28 -0.29 -0.31(103.39 ) (82.7 ) (0.02) (0.08)
E(time in prison) -69.91 -294.41 -0.16 -0.27(184.13 ) (147.29 ) (0.03) (0.14)
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Ropeladder Robustness Example: US Crime
−100 0 100200300
1x 1.2x 1.4x
E(crime rate per 100,000)
E(crime rate) / average
Pr(Prison) +0.5 sd
Police Spending +0.5 sd
Unemployment (t−2) +0.5 sd
Non−White Pop +0.5 sd
Male Pop +0.5 sd
Education +0.5 sd
Inequality +0.5 sd
●
●
●
●
●
●
●
A simple inference dotplot with an extra axis showing relative risk
CODE EXAMPLE
crimeRopeladders.r
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Ropeladder Robustness Example: US Crime
−500 0 500 1000
0.5x 1x 1.5x 2x
E(crime rate per 100,000)
E(crime rate) / average
Pr(Prison) +0.5 sd
Police Spending +0.5 sd
Unemployment (t−2) +0.5 sd
Non−White Pop +0.5 sd
Male Pop +0.5 sd
Education +0.5 sd
Inequality +0.5 sd
●
●
●
●
●
●
●
An inference dotplot with a marginal plot of the data
The data vary more widely than the first differences, stretching the plot
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Ropeladder Robustness Example: US Crime
−500 0 500 1000
0.5x 1x 1.5x 2x
Linear
E(crime rate per 100,000)
E(crime rate) / average
Pr(Prison) +0.5 sd
Police Spending +0.5 sd
Unemployment (t−2) +0.5 sd
Non−White Pop +0.5 sd
Male Pop +0.5 sd
Education +0.5 sd
Inequality +0.5 sd
−500 0 500 1000
0.5x 1x 1.5x 2x
Robust
E(crime rate per 100,000)
E(crime rate) / average
−500 0 500 1000
0.5x 1x 1.5x 2x
Poisson
E(crime rate per 100,000)
E(crime rate) / average
−500 0 500 1000
0.5x 1x 1.5x 2x
Neg Bin
E(crime rate per 100,000)
E(crime rate) / average
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Side-by-side inference dotplots
The focus here is comparisons across covariates within models
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Ropeladder Robustness Example: US Crime
−200 −100 0 100 200 300 400
0.75x 1x 1.25x 1.5x
E(crime rate per 100,000)
E(crime rate) / average
Pr(Prison) +0.5 sd
Police Spending +0.5 sd
Unemployment (t−2) +0.5 sd
Non−White Pop +0.5 sd
Male Pop +0.5 sd
Education +0.5 sd
Inequality +0.5 sd
●
●
●
●
●
●
●
linear
linear
linear
linear
linear
linear
linear
robust
robust
robust
robust
robust
robust
robust
poisson
poisson
poisson
poisson
poisson
poisson
poisson
negbin
negbin
negbin
negbin
negbin
negbin
negbin
A superplot ofropeladders
Equal focus oncomparisons acrosscovariates and acrossmodels
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Ropeladder Robustness Example: US Crime
−200 0 200 400
.75x 1x 1.25x 1.5x
Pr(Prison)
E(crime rate per 100,000)
E(crime rate) / average
Linear
Robust & Resistant
Poisson
Negative Binomial
−200 0 200 400
.75x 1x 1.25x 1.5x
Police Spending
E(crime rate per 100,000)
E(crime rate) / average
−200 0 200 400
.75x 1x 1.25x 1.5x
Unemployment
E(crime rate per 100,000)
E(crime rate) / average
−200 0 200 400
.75x 1x 1.25x 1.5x
Non−White Pop
E(crime rate per 100,000)
E(crime rate) / average
−200 0 200 400
.75x 1x 1.25x 1.5x
Male Pop
E(crime rate per 100,000)
E(crime rate) / average
Linear
Robust & Resistant
Poisson
Negative Binomial
−200 0 200 400
.75x 1x 1.25x 1.5x
Education
E(crime rate per 100,000)
E(crime rate) / average
−200 0 200 400
.75x 1x 1.25x 1.5x
Inequality
E(crime rate per 100,000)
E(crime rate) / average
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Side-by-side robustness ropeladders –focus is now on comparisons across models, not variables
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How Do I Visualize Interactions of Covariates?
To effectively visualize interactive specifications, you need:
1. A strategy for constructing counterfactuals that survey the model space
2. An algorithm that assembles logically coherent counterfactualsand correctly computes QoIs and their CIs
What you donʼt need: special machinery to calculate “marginal effects”
A generic counterfactual and simulation package that can use model formulas willcorrectly compute EVs, FDs, and RRs of the outcome variable
simcf does this – thatʼs basically why it exists
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.Strategies for Visualizing Interactive Covariates..
......
Interaction Counterfactual Strategy Plot
discrete with one cf for each combination ropeladder
discrete of values (full factorial)
continuous with choose combinations ropeladder
discrete of representative valuesor
combine a continuum with lineplot
each discrete value
continuous with choose combinations ropeladder
continuous of represented valuesor
combine a continuum with lineplot
each discrete valueor
combine a continuum with 3D functionala continuum boxplots
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Discrete × Discrete Interactions: Ropeladders
State Region
Local
0.8
2.08.
0
0.6
4.0
6.0
0.4
6.0
4.0
0.2
8.0
2.0
Public HealthPrimary CareSecondary/TertiaryPharmacueticals
•(Pha
rma-
ceut
ical
s)
•
•
•
•
Finance
Framework
Implementation
Provision
Stat
e
Loca
l
●
●●
●
Prov
ision
Fina
nce
Impl
emen
tatio
n
Fram
ewor
k
Letʼs convert this to a ropeladder ON THE WHITEBOARDChris Adolph (University of Washington) VISUALIZING INFERENCE 111 / 134
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Continuous × Discrete Interactions: Ropeladders
−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
In this model ofinflation, acontinuousvariable –central bankerconservatism,CBCC – isinteracted withbinaryvariables likewhether publicvotes are taken
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Continuous × Discrete Interactions: Ropeladders
−3 −2 −1 0
−3 −2 −1 0
Cumulative change in inflation after 5 years
Cumulative change in inflation after 5 years
Age ofCentral Banker
Future JobMatches Policy
Monetary PolicyVotes
Future Jobs andPublic Votes
65 years
yes
public
both
45 years
no
secret
neither
diff-in-diffs
diff-in-diffs
diff-in-diffs
diff-in-diffs
Increase CBCC by +1 sd, given...
I selected aspecific changein mycontinuousvariable andcomputedcounterfactualsunder eachvalue of mydiscretevariable
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Continuous × Discrete Interactions: Lineplots
20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
Age of Respondent
Pro
babi
lity
of V
otin
g
Currently Married
Not Married
Logit estimates:95% confidence
interval is shaded
Letʼs return to our votingexample, where votingwas a function of Age,Age2, Education, andMarriage Status
Suppose we add theinteractions Age ×Married and Age2 ×Married
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Continuous × Discrete Interactions: Lineplots
20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
Age of Respondent
Pro
babi
lity
of V
otin
g
Currently Married
Not Married
Logit estimates:95% confidence
interval is shaded
Warning: I have notheoretical reason to doso, and the model fitsuggests this is anoverspecified model. Wejust want an example ofhow to do this
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Continuous × Discrete Interactions: Lineplots
20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
Age of Respondent
Pro
babi
lity
of V
otin
g
Currently Married
Not Married
Logit estimates:95% confidence
interval is shaded
To get the new plots fromvotingLineplots.R,I need to change onlyone line:
Continuous Discrete Interactions: Lineplots
The only change I needto make to my codevotingLineplots.R is tochange one line:
model2 <-
vote00 ~ age + I(age^2)
+ hsdeg + coldeg
+ marriedo
changes to
model2 <-
vote00 ~ age*marriedo
+ I(age^2)*marriedo
+ hsdeg + coldeg
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Continuous × Discrete Interactions: Lineplots
20 30 40 50 60 70 80 90
−0.1
0
0.1
0.2
0.3
0.4
0.5
Age of Respondent
Diff
eren
ce in
Pro
babi
lity
of V
otin
g
20 30 40 50 60 70 80 90
0.9
1
1.1
1.2
1.3
1.4
1.5
Age of RespondentR
elat
ive
Ris
k of
Vot
ing
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
simcf takes care of the rest –it will correctly set up interactions and combines their uncertainty into the QoIs
The first differences and relative risks above are your marginal effect plots
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Continuous × Discrete Interactions: Lineplots
20 30 40 50 60 70 80 90
−0.1
0
0.1
0.2
0.3
0.4
0.5
Age of Respondent
Diff
eren
ce in
Pro
babi
lity
of V
otin
g
20 30 40 50 60 70 80 90
0.9
1
1.1
1.2
1.3
1.4
1.5
Age of RespondentR
elat
ive
Ris
k of
Vot
ing
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
Married compared to Not Married
Logit estimates:95% confidence
interval is shaded
The results suggest this interaction wasnʼt a great idea…
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Continuous × Continuous Interactions: Ropeladders
0.5x 0.75x 1x 1.25x 1.5x
Conditional median central banker tenure, relative to baseline
Inflation is
Unemployment is
Inflation is
Unemployment is
Inflation is
Unemployment is
●
low ●
low ●
low ●
low ●
●low
low ●
high
high ●
high ●
●high
high ●
high ●
Under average government
Under Left government
Under Right government
ernments
eer type
Weʼve already dealtwith a continuous ×continuous interaction:
Central banker tenuredepended on:
Inflation × Party CoG
and
Unemployment× Party CoG
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Continuous × Continuous Interactions: Ropeladders
0.5x 0.75x 1x 1.25x 1.5x
Conditional median central banker tenure, relative to baseline
Inflation is
Unemployment is
Inflation is
Unemployment is
Inflation is
Unemployment is
●
low ●
low ●
low ●
low ●
●low
low ●
high
high ●
high ●
●high
high ●
high ●
Under average government
Under Left government
Under Right government
ernments
eer type
I simply examined everycombination of high,low, and averagepartisanship with highand low inflation orunemployment
Grouping and labelingthe dotplot helpscatalog thecombinations
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Continuous × Continuous Interactions: Lineplots
CBNA
Measure
Long-run Unemployment Under …
Low CBNA High CBNA
5 Year Difference,
Low High CBNA
Unem Unem ΔUnem
CBI
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−8
−4
0
4
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−2
−2
−2
In Chapter 6 of Bankers, Bureaucrats, and Central Bank Politics, I consider theinteractive effects of central bank “nonaccommodation” (autonomous conservatism)and wage bargaining centralization on unemployment
I build on and test a complex literature positing interactive, nonlinear effectsChris Adolph (University of Washington) VISUALIZING INFERENCE 121 / 134
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Continuous × Continuous Interactions: Lineplots
CBNA
Measure
Long-run Unemployment Under …
Low CBNA High CBNA
5 Year Difference,
Low High CBNA
Unem Unem ΔUnem
CBI
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−8
−4
0
4
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−2
−2
−2
I investigate how different measures of nonaccommodation affect the results
I start with a crude “independence only” measure
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Continuous × Continuous Interactions: Lineplots
CBNA
Measure
Long-run Unemployment Under …
Low CBNA High CBNA
5 Year Difference,
Low High CBNA
Unem Unem ΔUnem
CBI
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−8
−4
0
4
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−2
−2
−2
The left and middle show expected unemploymentacross the continuum of CWB for two different levels of CBNA
The right plot shows the first difference in unemploymentgiven a change in CBNA at each level of CWB
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Continuous × Continuous Interactions: Lineplots
CBNA
Measure
Long-run Unemployment Under …
Low CBNA High CBNA
5 Year Difference,
Low High CBNA
Unem Unem ΔUnem
CBI
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−8
−4
0
4
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−2
−2
−2
This is an intuitive measure of the wage-bargaining-conditional effect ofnonaccommodation
simcf can produce this, with the right syntaxNote weʼve also iterated over time, so you would use ldvsimfd()
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Continuous × Continuous Interactions: Lineplots
CBNA
Measure
Long-run Unemployment Under …
Low CBNA High CBNA
5 Year Difference,
Low High CBNA
Unem Unem ΔUnem
CBI not significant
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
not significant
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
not significant
−8
−4
0
4
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
−2
−2
−2
Why no CIs? Because they would fill the whole plot!
I could make the plot area bigger, but that would make comparison hard
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Continuous × Continuous Interactions: Lineplots
CBNAMeasure
Long-run Unemployment Under …
Low CBNA High CBNA5 Year Difference,
Low → High CBNA
Unem Unem ΔUnem
CBI not significant
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
not significant
0
5
10
15
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
not significant
−8
−4
0
4
0.1 0.2 0.3 0.4 0.5
Centralization of Wage Bargaining
CBCC
0
5
10
15
0.1 0.2 0.3 0.4 0.5
0
5
10
15
0.1 0.2 0.3 0.4 0.5
−2
0
2
4
6
0.1 0.2 0.3 0.4 0.5
CBI ×CBCC
0
5
10
15
0.1 0.2 0.3 0.4 0.5
0
5
10
15
0.1 0.2 0.3 0.4 0.5
−2
0
2
4
6
0.1 0.2 0.3 0.4 0.5
MPA ×CBCC
0
5
10
15
0.1 0.2 0.3 0.4 0.5
0
5
10
15
0.1 0.2 0.3 0.4 0.5
−2
0
2
4
6
0.1 0.2 0.3 0.4 0.5
The real goal here is arobustness exercise
Measures of CBNAincorporating careerconservatism produce similarand generally more preciseresults, alone or incombination with differentmeasures of autonomy
Yet another approach toshowing robustness – one thatemphasizes similarity of fitsand CIs for conditionalrelationships
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Continuous × Continuous Interactions: 3D Boxplots?
I was long a skeptic ofincluding confidence volumesin 3D plots
This example made me abeliever
If it is really important to seesmooth variation in 2interacting continuouscovariates at the same time,investigate functionalboxplots
Source: Ying Sun and MarcG. Genton. 2011. “Functionalboxplots.” JCGS 20:2)
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Strategies for Interactions of 3 Continuous Variables
Any set of interactions involving 2 or fewer continuous variables can be addressedwith the above methods
What if you have 3 continuous variables? Some strategies:
.Strategy 1 – Ropeladder..
......
X1 X2 X3
high high highhigh high low
values high low highused high low low
in low high highcounterfactuals low high low
low low highlow low low
Descriptive names for these combinations essential for presentation in this case
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Strategies for Interactions of 3 Continuous Variables
Any set of interactions involving 2 or fewer continuous variables can be addressedwith the above methods
What if you have 3 continuous variables? Some strategies:
.Strategy 2 – Lineplots (overlapping and/or side-by-side)..
......
X1 X2 X3
values high highused contin- high low
in uum low highcounterfactuals low low
Use tiling of lineplots to your advantage in this case
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Strategies for Interactions of 3 Continuous Variables
0 0.1 0.2 0.3 0.4−10
−5
0
5
10
15
20
25
Left Government
Centralization of Wage Bargaining
0 0.1 0.2 0.3 0.4−10
−5
0
5
10
15
20
25
Right Government
Centralization of Wage Bargaining
Unem-ploy-ment
LowCBNA
LowCBNA
HighCBNA
HighCBNA
U
In Chapter 7 of BBC, I add a third interactive term to wage bargaining centralizationand central bank nonaccommodation: partisanship of government
The theory and model is complex, but graphically, I just plot 4 traces instead of 2Chris Adolph (University of Washington) VISUALIZING INFERENCE 130 / 134
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Strategies for Interactions of 3 Continuous Variables
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1Theoretical Prediction
Centralization of Wage Bargaining(theoretical scale)
Centralization of Wage Bargaining(empirical scale)
0 0.1 0.2 0.3 0.4
−4
−2
0
2
4
Statistical Model
HighCBNA
LowCBNA
HighCBNA
LowCBNA
Right Left Gov Right Left Gov
Uafter
3years
Uarbitrary
scale
If we plot first differences across partisanship, weʼre back to a 2 trace plot,but with a separate continuum of first difference for each level of CBNA
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Strategies for Interactions of 3 Continuous Variables
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1Theoretical Prediction
Centralization of Wage Bargaining(theoretical scale)
Centralization of Wage Bargaining(empirical scale)
0 0.1 0.2 0.3 0.4
−4
−2
0
2
4
Statistical Model
HighCBNA
LowCBNA
HighCBNA
LowCBNA
Right Left Gov Right Left Gov
Uafter
3years
Uarbitrary
scale
For complex models it helps to have a theory and to show it in the same format– both to justify and to explain the empirical result
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Strategies for Interactions of 3 Continuous Variables
Any set of interactions involving 2 or fewer continuous variables can be addressedwith the above methods
What if you have 3 continuous variables? Some strategies:
.Strategy 3 – Side-by-side 2D boxplots?..
......
X1 X2 X3
valuesused contin- contin- high
in uum uum lowcounterfactuals
Iʼve never tackled this problem – but this is the strategy Iʼd use
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Lessons for practice of data analysis
Simulation + Graphics can summarize complex models for a broad audience
You might even find something you missed as an analyst
And even for fancy or complex models, we can and should show uncertainty
Payoff to programming: this is hard the first few times, but gets easier, especially withsimcf+tile
Your code is re-usable and encourages more ambitious modeling
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