day10-AdjustmentBalance.Rmd

---
title: |
 | Statistical Adjustment and Assessment of Adjustment in Observational Studies
date: '`r format(Sys.Date(), "%B %d, %Y")`'
author: |
  | ICPSR 2023 Session 1
  | Jake Bowers \& Tom Leavitt
bibliography:
 - 'BIB/MasterBibliography.bib'
fontsize: 10pt
geometry: margin=1in
graphics: yes
biblio-style: authoryear-comp
biblatexoptions:
  - natbib=true
output:
  beamer_presentation:
    slide_level: 2
    keep_tex: true
    latex_engine: xelatex
    citation_package: biblatex
    template: icpsr.beamer
    incremental: true
    includes:
        in_header:
           - defs-all.sty
    md_extensions: +raw_attribute-tex_math_single_backslash+autolink_bare_uris+ascii_identifiers+tex_math_dollars
    pandoc_args: [ "--csl", "chicago-author-date.csl" ]
---


<!-- To show notes  -->
<!-- https://stackoverflow.com/questions/44906264/add-speaker-notes-to-beamer-presentations-using-rmarkdown -->

```{r setup1_env, echo=FALSE, include=FALSE}
library(here)
source(here::here("rmd_setup.R"))
```

```{r setup2_loadlibs, echo=FALSE, include=FALSE}
## Load all of the libraries that we will use when we compile this file
## We are using the renv system. So these will all be loaded from a local library directory
library(dplyr)
library(ggplot2)
library(coin)
library(RItools)
library(optmatch)
```
## Today

  1. Agenda: The problem of covariance adjustment to reduce "bias"/
     confounding. **How can we answer the question about whether we have
     adjusted enough.** A simple approach: stratification on one categorical
     variable (and interaction effects). A more complex approach: find sets
     that are as similar as possible in terms of a continuous variable
     (bipartite matching). Balance assessment after stratification.
  2. Recap:
  2. We are now moving into the "Observational Studies" or Matching part of the course.
  3. Questions arising from the reading or assignments or life?


## So far/Randomized experiments

 - Randomization plus choice of tests statistic tells us which distribution
   characterizes our hypothesis (plus Central Limit Theorem for tests of the
   weak null of no average effects).
 - Randomization tells us that estimators of average causal effects are
   unbiased. (and tells us how to summarize how the values of those estimates
   would vary across different runs of the same experiment with the same
   experimental pool).
 - Randomization does not guarantee strict equality of covariate distributions
   but does guarantee "closeness on average" of those distributions. We can
   compare the covariate distributions in actual experiments to those of
   theoretical / hypothesized experiments (in this class using `balanceTest` and
   the `overall` or $d^2$ omnibus test of overall balance).
 - Randomization thus guarantees that comparisons between groups as-randomized
   do not have systematic confounding with any observed or unobserved
   covariates: outcome comparisons tell us only about the causal effect of the
   treatment (which can have complex theoretical mechanisms, or even unclear or
   contested mechanisms of course).

## So far/Non-random compliance {.fragile}

 - We can learn about the causal effects of non-randomized variables, $D$, if we have an instrument, $Z$ under certain assumptions:

\begin{center}
\begin{tikzcd}[column sep=large]
Z  \arrow[from=1-1,to=1-2, "\text{ITT}_D \ne 0"] \arrow[from=1-1, to=1-4, bend left, "\text{0 (exclusion)}"] & d  \arrow[from=1-2,to=1-4, "CACE"] & & y \\
(x_1 \ldots x_p) \arrow[from=2-1,to=1-1, "\text{0 (as if randomized)}"]  \arrow[from=2-1,to=1-2] \arrow[from=2-1,to=1-4]
\end{tikzcd}
\end{center}

 - The @angrist1996 assumptions for estimating the CACE/LATE are: (1) sutva, (2) ignorable $Z$ (see above), (3) exclusion (see above), (4) no defiers (or they might call this "monotonicity"), (5) non-zero causal effect of $Z$ on $D$.

 - This enables **Encouragement Designs**
   
 - We can test the sharp null of no complier average effects directly (under
   those assumptions): this helps us sidestep the problem of estimating $\Var\left[\cfrac{\widehat{\text{ITT}}_Y}{\widehat{\text{ITT}}_D}\right]$.

# How to assess the randomization process in an experiment.

## The Neyman-Rubin Model for (simple) experiments

This is what randomization ensures:
$$ ({y_t, y_c},{X}) \perp {Z} $$

i.e., each of $X$, $y_{c}$ and $y_{t}$ is balanced between treatment and control groups (in expectation; given variability from randomization).

\begin{itemize}
\item In controlled experiments, random assignment justifies this argument.
\item In natural experiments, justified otherwise, this is an article of faith.
\item In an experiment, the $x$es aren't necessary for inference (although
they can be used, carefully, to increase precision in both the design and
analysis phases of a project).
\item \textbf{However, the part with the $x$es has testable consequences.}
\end{itemize}

# Did we control for enough?

```{r cache=TRUE}
load(url("http://jakebowers.org/Data/meddat.rda"))
```

##  Introducing the Medellin Data

Cerdá et al. collected data on about roughly `r nrow(meddat)`
neighborhoods in Medellin, Colombia. About  `r signif(sum(meddat$nhTrt),2)` had
access to the new Metrocable line and `r signif(sum(1-meddat$nhTrt),2)` did not.


\centering
\includegraphics[width=.7\textwidth]{medellin-gondola.jpg}

<!-- For more on the Metrocable project see <https://www.medellincolombia.co/where-to-stay-in-medellin/medellin-orientation/> and <https://archleague.org/article/connective-spaces-and-social-capital-in-medellin-by-jeff-geisinger/> -->


##  Introducing the Medellin Data

Cerdá et al. collected data on about roughly `r nrow(meddat)`
neighborhoods in Medellin, Colombia. About  `r signif(sum(meddat$nhTrt),2)` had
access to the new Metrocable line and `r signif(sum(1-meddat$nhTrt),2)` did not.

\centering
\includegraphics[width=.8\textwidth]{medellin-conc-pov.jpg}

##  Introducing the Medellin Data: Variables Collected

\scriptsize
```{r eval=FALSE}
## The Intervention
nhTrt        Intervention neighborhood (0=no Metrocable station, 1=Metrocable station)

## Some Covariates (there are others, see the paper itself)
nh03         Neighborhood id
nhGroup      Treatment (T) or Control (C)
nhTrt        Treatment (1) or Control (0)
nhHom        Mean homicide rate per 100,000 population in 2003
nhDistCenter Distance to city center (km)
nhLogHom     Log Homicide (i.e. log(nhHom))

## Outcomes (BE03,CE03,PV03,QP03,TP03 are baseline versions)
BE      Neighborhood amenities Score 2008
CE      Collective Efficacy Score 2008
PV      Perceived Violence Score 2008
QP      Trust in local agencies Score 2008
TP      Reliance on police Score 2008
hom     Homicide rate per 100,000 population Score 2008-2003 (in log odds)

HomCount2003 Number of homicides in 2003
Pop2003      Population in 2003
HomCount2008 Number of homicides in 2008
Pop2008      Population in 2008
```

Get rates from counts:

```{r rates, echo=TRUE}
meddat<- mutate(meddat, HomRate03=(HomCount2003/Pop2003)*1000,
                HomRate08=(HomCount2008/Pop2008)*1000)
```

## What is the effect of the Metrocable on Homicides?

One approach:  Estimate the average treatment effect of Metrocable on
Homicides after the stations were built.

```{r lmone, echo=TRUE}
## code here
themeans<-group_by(meddat,nhTrt) %>% summarise(ybar=mean(HomRate08))
themeans
diff(themeans$ybar)
lmOne <- lm(HomRate08~nhTrt,meddat)
coef(lmOne)["nhTrt"]
library(estimatr)
difference_in_means(HomRate08~nhTrt,meddat)
```

Another approach, test the null of no effects: `balanceTest` and `oneway_test`
(with `distribution=asymptotic()`) use a large sample Central Limit Theorem
based approximation to a randomization distribution (which here is "as-if
randomized").

```{r initialtest, echo=TRUE}
balanceTest(nhTrt~HomRate08,data=meddat)
meddat$nhTrtF <- factor(meddat$nhTrt)
test2 <- oneway_test(HomRate08~nhTrtF,data=meddat,distribution=asymptotic())
## This next uses a permutation approach to approximate the as-if-randomized
## randomization distribution.
test3 <- oneway_test(HomRate08~nhTrtF,data=meddat,distribution=approximate(nresample=10000))
pvalue(test2)
pvalue(test3)
```

## Do we have any concerns about confounding?

Sometimes people ask about "bias from observed confounding" or "bias from selection on observables".

How would we interpret the following results where we look at the relationship
between one potential confounder ("proportion having more than a high school
degree") and receipt of a new Metrocable station (`nhTrt==1`)? (Recall how we
justified the use of `balanceTest` in terms of randomization above.)

```{r xbmed, echo=TRUE}
xbMed1 <- balanceTest(nhTrt~nhAboveHS,data=meddat)
xbMed1$overall
xbMed1$results
```


## One approach to this problem: model-based adjustment

Let's try to just adjust for this covariate in a very common manner:

```{r echo=FALSE}
lm1 <- lm(HomRate08~nhTrt+nhAboveHS,data=meddat)
```

```{r echo=FALSE}
preddat <- expand.grid(nhTrt=c(0,1),nhAboveHS=range(meddat$nhAboveHS))
preddat$fit <- predict(lm1,newdata=preddat)
```

\centering
```{r, out.width=".9\\textwidth", echo=FALSE}
par(oma=rep(0,4),mgp=c(1.5,.5,0),mar=c(3,3,0,0))
with(meddat, plot(nhAboveHS,HomRate08,pch=c(1,2)[nhTrt+1]))
with(subset(preddat,subset=nhTrt==0),lines(nhAboveHS,fit,lty=1))
with(subset(preddat,subset=nhTrt==1),lines(nhAboveHS,fit,lty=2))
## locator()
text(c(0.111807,0.001629), c(1.871,2.204), labels=c("Treat","Control"),pos=1)
text(c(.3,.5),c( coef(lm1)[1]+coef(lm1)[3]*.3 , coef(lm1)[1]+coef(lm1)[2]+coef(lm1)[3]*.5),
     labels=c("Control","Treat"))
```

## Exactly what does this kind of adjustment do?

Notice that I can get the same coefficient (the effect of Metrocable on
Homicides adjusted for HS-Education in the neighborhood) either directly (as
earlier) or via **residualization**:

```{r echo=TRUE}
coef(lm1)["nhTrt"]
eYX <- residuals(lm(HomRate08~nhAboveHS,data=meddat))
eZX <- residuals(lm(nhTrt ~ nhAboveHS, data=meddat))
lm1a <- lm(eYX~eZX)
coef(lm1a)[2]
```


## Did we adjust enough?

Maybe adding some more information to the plot can help us decide whether, and to what extend, we effectively "controlled for" the proportion of the neighborhood with more than High School education. Specifically, we might be interested in assessing extrapolation/interpolation problems arising from our linear assumptions.

\centering
```{r, out.width=".7\\textwidth", echo=FALSE, warning=FALSE, message=FALSE}
par(oma=rep(0,4),mgp=c(1.5,.5,0),mar=c(3,3,0,0))
with(meddat, plot(nhAboveHS,HomRate08,pch=c(1,2)[nhTrt+1]))
with(subset(preddat,subset=nhTrt==0), lines(nhAboveHS,fit,lty=1))
with(subset(preddat,subset=nhTrt==1),lines(nhAboveHS,fit,lty=2))
with(subset(meddat,subset=nhTrt==0),lines(loess.smooth(nhAboveHS,HomRate08,deg=1,span=2/3),lty=1))
with(subset(meddat,subset=nhTrt==1),lines(loess.smooth(nhAboveHS,HomRate08,deg=1,span=.8),lty=2))
## locator()
text(c(0.111807,0.001629), c(1.871,2.204), labels=c("Treat","Control"),pos=1)
text(c(.3,.5),c( coef(lm1)[1]+coef(lm1)[3]*.3 , coef(lm1)[1]+coef(lm1)[2]+coef(lm1)[3]*.5),
     labels=c("Control","Treat"))
with(subset(meddat,subset=nhTrt==0),rug(nhAboveHS))
with(subset(meddat,subset=nhTrt==1),rug(nhAboveHS,line=-.5))
```

How should we interpret this adjustment? How should we judge the improvement that we made? What concerns might we have?

```{r echo=FALSE, eval=FALSE}
thecovs <- unique(c(names(meddat)[c(5:7,9:24)],"HomRate03"))
balfmla<-reformulate(thecovs,response="nhTrt")

xbMed<-balanceTest(balfmla,
	      data=meddat,
        p.adjust.method="none")
xbMed$overall
xbMed$results["nhAboveHS",,]
```


```{r echo=FALSE, eval=FALSE}
outcomefmla <- reformulate(c("nhTrt",thecovs),response="HomRate08")
lmbig <- lm_robust(outcomefmla,data=meddat)
lmbig
```


## How would you adjust for Proportion Above HS Degree?

So, part of the Metrocable effect might not reflect the causal effect of
Metrocable per se, but rather the education of people in the
neighborhood. How should we remove `nhAboveHS` from our estimate or test? What
strategies can you think of?

Features of a good adjustment process:

  - Blind to outcome analysis (to preserve false positive rate and deter critics). Able to be pre-registered. Perhaps even reviewed by stakeholders.
  - Easy to interpret ("controlling for"  versus "holding constant")
  - Easy to diagnoses (Easy to answere the question "Did we adjust enough?")

## Stratification V 1.0

```{r strat1, echo=TRUE}
lm1a <- lm(HomRate08~nhTrt,data=meddat,subset=nhAboveHS>=.1)
lm1b <- lm(HomRate08~nhTrt,data=meddat,subset=nhAboveHS<.1)
res_strat <- c(hiEd_Effect=coef(lm1a)["nhTrt"],loEd_Effect= coef(lm1b)["nhTrt"])
res_strat
n_strat <- table(meddat$nhAboveHS>=.1)
n_strat
stopifnot(sum(n_strat)==nrow(meddat)) ## A test of code
sum(res_strat * rev(n_strat)/45) ## What is happening here? A weighted average.
```

*But, standard errors? p-values? confidence intervals?*

## Stratified adjustment: One-step V 2.0

```{r strat2, echo=TRUE}
## Weight by block size
ate1c <- difference_in_means(HomRate08~nhTrt, blocks = I(nhAboveHS>=.1),data=meddat)
ate1c
## Weight by both block size and  proportion in treatment vs control ("harmonic weight")
lm1c <- lm_robust(HomRate08~nhTrt, fixed_effects = ~I(nhAboveHS>=.1),data=meddat)
coef(lm1c)["nhTrt"]
lm1d <- lm_robust(HomRate08~nhTrt+I(nhAboveHS>=.1),data=meddat)
coef(lm1d)["nhTrt"]
## This next weighted by number of treated observations in each strata
xbate1 <- balanceTest(nhTrt~HomRate08+strata(I(nhAboveHS>=.1)),data=meddat)
xbate1$results[,c("adj.diff","p"),]
strat_by_trt <- with(meddat,table(nhAboveHS>=.1,nhTrt))
nt_strat <- strat_by_trt[,"1"]
sum(res_strat * rev(nt_strat)/sum(nt_strat))
```

## Balance assessment after stratification

Did we adjust enough? What would *enough* mean?

```{r xbHS1, echo=TRUE}
xbHS1 <- balanceTest(nhTrt~nhAboveHS+strata(I(nhAboveHS>=.1)),data=meddat)
xbHS1$overall
xbHS1$results[1,,]  ## the covariate specific z-test
```
## Disadvantages and Advantages of Simple Stratification

  -  (+) Easy to explain what  "controlling for" or "adjustment" means.
  -  (-) Hard to justify any particular cut-point
  -  (-) We could probably adjust *more* --- comparing neighborhoods similar in education rather than just  within  big   strata


## The Curse of Dimensionality and linear adjustment for one more variable.

What about more than one variable? Have we controlled for both population
density and educational attainment enough? How would we know?

```{r lm2x, echo=TRUE}
lm2x <- lm(HomRate08 ~ nhTrt + nhPopD + nhAboveHS, data=meddat)
coef(lm2x)["nhTrt"]
```

Maybe another plot?

```{r eval=FALSE}
meddat$nhTrtF <- factor(meddat$nhTrt)
library(car)
scatter3d(HomRate08~nhAboveHS+nhPopD,
	  groups=meddat$nhTrtF,
	  data=meddat,surface=TRUE,
    fit=c("linear")) #additive"))
```

```{r echo=FALSE, eval=FALSE}
scatter3d(HomRate08~nhAboveHS+nhPopD,
	  groups=meddat$nhTrtF,
	  data=meddat,surface=TRUE,
    fit=c("additive"))

```

## The Problem of Using  the Linear Model for  Adjustment {.allowframebreaks}

 - \bh{Problem of Interepretability:} "Controlling for" is  "removing (additive) linear relationships" it is  not "holding constant"
 - \bh{Problem of Diagnosis and Assessment:} What is the  standard against which we can compare a given linear covariance adjustment specification?
 - \bh{Problem of extrapolation and interpolation:} Often known as "common support", too.
 - \bh{Problems of overly influential points and curse of  dimensionality:} As dimensions increase, odds of influential  point increase (ex. bell curve in one dimension, one very influential point in 2 dimensions); also real limits on number of covariates (roughly $\sqrt{n}$ for OLS).
 - \bh{Problems of bias:} We know that the difference of means is an unbiased
   estimator of the ATE (or ITT aka ITT$_Y$) or ITT$_D$ in randomized
   experiments, but that the covariance adjusted difference of means is not
   unbiased (although the bias may be small).

\begin{equation}
Y_i = \beta_0 + \beta_1 Z_i + e_i
\end{equation}

This is a common practice because, we know that the formula to estimate $\beta_1$ above is the same as the difference of means in $Y$ between treatment and control groups:

\begin{equation}
\hat{\beta}_1 = \overline{Y|Z=1} - \overline{Y|Z=0} = \frac{\cov(Y,Z)}{\var(Z)}.
\end{equation}

Now what about adjusting ro $X$, what does OLS give us for $\beta_1$?

\begin{equation}
Y_i = \beta_0 + \beta_1 Z_i + \beta_2 X_i + e_i
\end{equation}

What is $\beta_1$ in this case? We know the matrix representation here $(\bX^{T}\bX)^{-1}\bX^{T}\by$, but here is the scalar formula for this particular case:

$$ \hat{\beta}_1 = \frac{\var(X)\cov(Z,Y) - \cov(X,Z)\cov(X,Y)}{\var(Z)\var(X) - \cov(Z,X)^2} $$

## Can we improve stratified adjustment?

Rather than two strata, why not three?

```{r lm1cut3, echo=TRUE}
lm1cut3 <- lm(HomRate08~nhTrt+cut(nhAboveHS,3),data=meddat)
coef(lm1cut3)["nhTrt"]
```
But why those cuts? And why not 4? Why not...?

\medskip

**One idea to side-step the problem of choosing cut-points**: collect observations into strata such that the sum of the
differences in means of nhAboveHS within strata is smallest? This is the idea
behind `optmatch` and other matching approaches.

## The optmatch workflow: The distance matrix

Introduction to `optmatch` workflow. To minimize differences requires a matrix
of those differences (in general terms, a matrix of distances between the
treated and control units)

```{r optm1, echo=TRUE}
tmp <- meddat$nhAboveHS
names(tmp) <- rownames(meddat)
absdist <- match_on(tmp, z = meddat$nhTrt,data=meddat)
absdist[1:3,1:3]
abs(meddat$nhAboveHS[meddat$nhTrt==1][1] - meddat$nhAboveHS[meddat$nhTrt==0][1] )
```

## Created a Stratified Research Design

```{r fm1, echo=TRUE}
fm1 <- fullmatch(absdist,data=meddat)
summary(fm1, min.controls=0, max.controls=Inf )
table(meddat$nhTrt,fm1)

pm1 <- pairmatch(absdist,data=meddat)
summary(pm1, min.controls=0, max.controls=Inf )
table(meddat$nhTrt,pm1,exclude=c())
```


## Evaluate the design: Within set differences

```{r echo=FALSE}
meddat$fm1 <- fm1
meddat$pm1 <- pm1
```

Differences within sets versus raw differences.

```{r echo=FALSE, out.width=".9\\textwidth"}
library(gridExtra)
bpfm1 <- ggplot(meddat,aes(x=fm1,y=nhAboveHS)) +
	geom_boxplot() +
	 stat_summary(fun=mean, geom="point", shape=20, size=3, color="red", fill="red")
meddat$nostrata <- rep(1,45)
bporig <- ggplot(meddat,aes(x=nostrata,y=nhAboveHS))+
	 geom_boxplot()+
	 stat_summary(fun=mean, geom="point",
		      shape=20, size=3, color="red", fill="red")

grid.arrange(bpfm1,bporig,ncol=2,layout_matrix=matrix(c(1,1,1,1,2),nrow=1))
```

## Evaluate the design: Within set differences

```{r echo=FALSE, out.width=".9\\textwidth"}
bppm1 <- ggplot(meddat,aes(x=pm1,y=nhAboveHS)) +
	geom_boxplot() +
	 stat_summary(fun=mean, geom="point", shape=20, size=3, color="red", fill="red")

grid.arrange(bppm1,bporig,ncol=2,layout_matrix=matrix(c(1,1,1,1,2),nrow=1))
```

## Evaluate the design: Inspect within set differences (Full Match)

```{r sdiffs, echo=FALSE}
rawmndiffs <- with(meddat, mean(nhAboveHS[nhTrt==1]) - mean(nhAboveHS[nhTrt==0]))
setdiffsfm1 <- meddat %>% group_by(fm1) %>% summarize(mneddiffs =
						   mean(nhAboveHS[nhTrt==1]) -
						   mean(nhAboveHS[nhTrt==0]),
					   mnAboveHS = mean(nhAboveHS),
					   minAboveHS = min(nhAboveHS),
					   maxAboveHS = max(nhAboveHS))

setdiffsfm1
#summary(setdiffs$mneddiffs)
#quantile(setdiffs$mneddiffs, seq(0,1,.1))
```



## Evaluate the design: Inspect within set differences (Pair Match)


```{r echo=FALSE, warnings=FALSE}
setdiffspm1 <- meddat %>% group_by(pm1) %>% summarize(mneddiffs =
						   mean(nhAboveHS[nhTrt==1]) -
						   mean(nhAboveHS[nhTrt==0]),
					   mnAboveHS = mean(nhAboveHS),
					   minAboveHS = min(nhAboveHS),
					   maxAboveHS = max(nhAboveHS))

setdiffspm1
```


## Evaluate the design: Compare to a randomized experiment.
The within-set differences look different from those that would be expected
from a randomized experiment.

```{r xbhs2, echo=TRUE}
xbfm1 <- balanceTest(nhTrt~nhAboveHS+strata(fm1),
                  data=meddat)
xbfm1$results[,,]
xbfm1$overall
```



## What is balanceTest doing?

```{r xbagain, echo=TRUE}
setmeanDiffs <- meddat %>% group_by(fm1) %>%
  summarise(diffAboveHS=mean(nhAboveHS[nhTrt==1])-mean(nhAboveHS[nhTrt==0]),
            nb=n(),
            nTb = sum(nhTrt),
            nCb = sum(1-nhTrt),
            hwt = ( 2*( nCb * nTb ) / (nTb + nCb))
            )
setmeanDiffs
```

## What is balanceTest doing with multiple sets/blocks?

The test statistic is a weighted average of the set-specific differences (same
approach as we would use to test the null in a block-randomized experiment)

```{r wtmns, echo=TRUE}
## The descriptive mean difference using block-size weights (focusing on the
## treated observations)
with(setmeanDiffs, sum(diffAboveHS*nTb/sum(nTb)))
## The mean diff used as the observed value in the testing
with(setmeanDiffs, sum(diffAboveHS*hwt/sum(hwt)))
## Compare to balanceTest output
xbfm1$results[,,"fm1"]
```

## Can we improve the design?

Here we (1) require sets to contain neighborhoods that differ by no more than
.08 on proportion educated more than high-school and (2) require than no set
have more than 2 treated neighborhoods.

```{r}
## Look at the kinds of differences between units that are possible.
quantile(as.vector(absdist),seq(0,1,.1))

absdist_cal <- match_on(absdist+caliper(absdist,.08),data=meddat)
fm2 <- fullmatch(absdist_cal,data=meddat,min.controls=.5)
summary(fm2, min.controls=0, max.controls=Inf )
table(meddat$nhTrt,fm2)
stratumStructure(fm1)
stratumStructure(fm2)
matched.distances(distance=absdist_cal,matchobj=fm2)
summary(unlist(matched.distances(distance=absdist_cal,matchobj=fm2)))
summary(unlist(matched.distances(distance=absdist,matchobj=fm1)))

meddat %>% group_by(fm1) %>% summarize(mean(nhAboveHS),n_b=n(), ate_b=mean(HomRate08[nhTrt==1])-mean(HomRate08[nhTrt==0]))
```

## Can we improve the design?

Here we (1) require sets to contain neighborhoods that differ by no more than
.08 on proportion educated more than high-school and (2) require than no set
have more than 2 treated neighborhoods.

```{r xbhs3, echo=TRUE}
xbfm <- balanceTest(nhTrt~nhAboveHS+strata(fm1)+strata(fm2),
                  data=meddat)
xbfm$results[,,]
xbfm$overall
```

## Can we improve the design?

Here we (1) require sets to contain neighborhoods that differ by no more than
.08 on proportion educated more than high-school and (2) require than no set
have more than 2 treated neighborhoods.


```{r echo=FALSE, out.width=".9\\textwidth"}
bpfm2 <- ggplot(meddat,aes(x=fm2,y=nhAboveHS)) +
	geom_boxplot() +
	 stat_summary(fun=mean, geom="point", shape=20, size=3, color="red", fill="red")
meddat$nostrata <- rep(1,45)
bporig <- ggplot(meddat,aes(x=nostrata,y=nhAboveHS))+
	 geom_boxplot()+
	 stat_summary(fun=mean, geom="point",
		      shape=20, size=3, color="red", fill="red")

grid.arrange(bpfm2,bporig,ncol=2,layout_matrix=matrix(c(1,1,1,1,2),nrow=1))
```



## Summary of the Day

 - How to justify an adjustment strategy for an observational study? The
   linear model adjustment strategy is difficult to justify. A stratification
   based strategy is easier to justify, inspect, learn from. (We can compare
   our stratification to a block randomized experiment, to a known design, a
   known standard.)

 - How to choose a stratification? We can do it by hand. Or we can delegate to
   a computer (i.e. `optmatch`) --- we can think of it as an optimization
   problem and ask the computer to optimize.

 - Today we created two matched designs aiming to reduce the confounding
   associated with `nhAboveHS`. We judged those designs based on (1) substantive
   information about what kinds of education-based differences seemed small or
   large compared with what we know about the context and (2) statistical
   information about the consistency of our design with a truly randomized
   design.

## References