day8_stratification_balance.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'
 - 'BIB/master.bib'
 - 'BIB/misc.bib'
 - 'BIB/refs.bib'
fontsize: 10pt
geometry: margin=1in
graphics: yes
biblio-style: authoryear-comp
colorlinks: true
biblatexoptions:
  - natbib=true
output:
  beamer_presentation:
    slide_level: 2
    keep_tex: true
    latex_engine: xelatex
    citation_package: biblatex
    template: icpsr.beamer
    incremental: true
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    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 and avoided extrapolation and dependence on functional
     form?**
     a. A simple approach: stratification on one categorical variable
     (using weighted combinations of within-strata estimates for an overall
     estimate). Justify this **design** by comparison of this stratification
     with one or more **standards** of unconfounded designs (like an experiment
     with treatment randomized within strata).
     b. 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: Problems of using linear (or
     other parametric models) for **adjustment** (as well as statistical
     inference) include justifying variables, justifying functional forms (of
     $x_1 \rightarrow Z$ and $x_1 \rightarrow Y$ and $x_1 \rightarrow x_2$, and
     $f(x_1,x_2) \rightarrow Y$ and $f(x_1,x_2) \rightarrow Z$ and $f(x_1,x_2)
     \rightarrow x_3$, diagnosing extrapolation and interpolation, diagnosing
     and confronting overly influential points, and struggling with the
     problems of seeing the causal estimate and test results everytime you try
     a new specification. And, basically, finding a **standard of adjustment**
     for such models where we can answer the question "have we adjusted
     enough"?
  3. Questions arising from the reading or assignments or life?

# Strategies for Causal Inference

## Strategies and Workflow for randomized studies

### Before fielding the study
 - Plan the design for interpretability and power.
 - Pre-register your analysis plan and experimental design.

### After outcome data have been collected
 - Make the case that randomization worked. (often trivial, but in field
   experiments with long chains of responsibility, an important screening step
   even if not dispositive of a "failed experiment").
 - Increase precision with design and test statistics (and even models of
   effects)
 - Make sure that your estimator is estimating the right thing (i.e. is not
   biased or at least is consistent)
 - Make sure that your test has a controlled false positive rate and is as
   powerful as possible (i.e. the coverage rate of a confidence interval is
   nominal, the Type I error rate is controlled).

## Strategies for Adjustment of Observational Studies

 - If you have randomized $Z$ but not $D$, then IV. (Like an observational study within an experiment.)
 - Find a discontinuity/A Natural Experiment (RDD: either natural experiment
   or continuous forcing function)
 - Multiple controls each with different biases (i.e. "Choice as an alternative to control"
   see [@rosenbaum:1999])
 - "Controlling For" directly in linear models, but lots of questions to answer...
 - Difference in Differences (if you have some pre- and post-outcomes)
 - Matched Stratification (approximating a block-randomized experiment)
 - Best matched subset selection (approximating a completely or simply
   randomized experiment)
 - Weighting (the stratification approaches imply weightings, so we could try to calculate good weightings directly)
 - Direct theoretical modeling / Directed Acyclic Graphs (DAGS) to guide other
   data modeling choices.

# Recap

## Regression is not research design {.shrink}

What does this mean?

Research design occurs **before** estimation and testing. You can try out lots
of ideas, simulate, discuss, pre-register your planned analysis and your
design, all before you produce the estimate or test having to do with some
causal effect. This means (1) you are not seeing the estimates and tests when
you are looking for a good design and (2) standards for good design are not the
same as the standards for good estimators and tests.

When you present the result of a regression model after searching for a
specification, readers wonder:

1. is this just the best result out of
thousands? Are most results like this? Or was this cherry picked?
2. how many
$p$-values were looked at before reporting this one with $p < .05$?

\small{Recall
that if we reject $H_0$ when $p < .05$, we are saying that it is ok for 5% of
tests to mislead us --- to have $p < .05$ when there is no effect (when $H_0$
is true). So, a $p < .05$ after one test means that either have a rare testing
error (a false positive result) or that $H_0$ is false.  But an unadjusted $p <  .05$ after 100 tests is virtually
certain to occur (with prob of .99) when $H_0$ is true: $p < .05$ is a signal of a false positive error not of a substantive signal about $H_0$. So we
should not reject the null even if the $p < .05$ in this case. (See
[EGAP Guide on Multiple Comparisons](https://egap.org/resource/10-things-to-know-about-multiple-comparisons/) ).
}

## Regression is not research design

Say we want to describe the relationship between an outcome $y$, a focal explanatory
variable, $z$ and a potential confounder $x$ as a linear and additive function:
$y_i = \beta_0 + \beta_1 z_i + \beta_2 x_i$. What can we say about this?

1. "The **math implies** that the difference in $y$ between any two values of $z$ that differ by 1 must be the same regardless of which two values and regardless of the values that we plug-in for $x$." (Correct)
2. "The **math says** that relationship between differences in $z$ and differences in $y$ ($\frac{dy}{dz}$), is the same for all values of $z$ and $x$. This is a constant relation." (Correct)

## Regression is not research design


Say we fit that mathematical model, $y_i = \beta_0 + \beta_1 z_i + \beta_2 x_i$, to data. What can we say?

```{r, echo=FALSE, out.width=".5\\textwidth"}
load(here::here("day7_dat.rda"))
lm_Y_x1 <- lm(Y ~ x1, data = dat)
lm_Z_x1 <- lm(Z ~ x1, data = dat)
dat$resid_Y_x1 <- resid(lm_Y_x1)
dat$resid_Z_x1 <- resid(lm_Z_x1)
lm2 <- lm(Y~Z+x1,data=dat)
coef(lm2)
```

Here is a plot of Y-without-linear-x1 on X-without-linear-x1 with a slope of `r coef(lm2)[["Z"]]`.

```{r, echo=FALSE, out.width=".5\\textwidth",warning=FALSE}
g1 <- ggplot(dat,aes(x=resid_Z_x1,y=resid_Y_x1,color=x1))+geom_point()+geom_smooth(method="lm",se=FALSE)
print(g1)
```


## Regression is not research design {.shrink}

```{r, echo=FALSE, out.width=".8\\textwidth",warning=FALSE}
g2 <- ggplot(dat,aes(x=x1,y=Y,color=factor(Z)))+geom_point()+
	geom_abline(intercept=coef(lm2)[["(Intercept)"]],slope=coef(lm2)[["x1"]])+
	geom_abline(intercept=(coef(lm2)[["(Intercept)"]]+coef(lm2)[["Z"]]),
		    slope=coef(lm2)[["x1"]],color="turquoise")

##print(g2)
library(cowplot)
plot_grid(g1,g2)
```


1. "Are we comparing $Y$ values with different $Z$ values but the same $x1$ value? Does our data fitting do the same thing as actually holding constant $x1$?" (No.)
2. "Does fitting our mathematical model to data do the same thing as physically holding constant?" (No.)
3. "Do predictions of $Y$ from our model for two values of $Z$ that are 1 apart differ by the same amount regardless of (a) which two values they are and (b) any value that we put in for $x$?" (Yes. This is because we decided to describe all of the relationships with a line.)


## A 3D surface

Notice that the lines have the same slope as you move across the plane.

```{r, fig.keep="last"}
library(rgl)
n <- 20
uniq_x <- seq(0,1,length=n)
uniq_z <- seq(0,1,length=n)
region <- expand.grid(x = uniq_x, z = uniq_z)
y <- matrix((region$x + region$z), n, n)
surface3d(x=uniq_x, y=uniq_z, z=y, back = 'line', front = 'line', col = 'red', lwd = 1.5, alpha = 0.4)
axes3d()
```

## Recap summary:

- Correct interpretation of mathematical linear models may involve the word
  "constant" (after all, lines have constant slope, planes have constant slopes
  (one slope per dimension)).
- Mathematical linear models fit to data are not research designs. We learn
  which linear model fits the data best (where "best" can be defined in least
  squares ways for example), but we cannot learn whether confounding is linear
  from fitting such a model.
- Linear models estimate average treatment effects well (since OLS is a
  mean-difference calculator) and friendly software designers provide tests of
  the weak null of no average effects by default (that are valid given
  assumptions about standard errors and CLTs).
- Linear models are not research designs and should not be used as such.
- Using linear models for adjustment might make sense if you are not worried
  about and/or have investigated: extrapolation, interpolation, functional
  form, influential points, and you have pre-specified your specification to
  avoid multiple testing problems and/or maybe generate a specification curve
  (of the millions of ways to specify the **adjustment part** of the linear
  model).



# How to assess the randomization process in an experiment (to teach us how to assess research designs in observational studies).

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

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

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

- In controlled experiments, random assignment justifies this argument.
- In natural experiments, justified otherwise, this is an article of faith.
- 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).
- **However, the part with the $x$es has testable consequences** if you worried about the success of the randomization --- say, the path between the random numbers on your computer and the application in the field.

## Covariate balance in experiments: What does it look like?

\begin{columns}
\begin{column}{.4\linewidth}
\begin{itemize}
\item \cite{arceneaux:2005}
\item Kansas City, November 2003
\item Completely randomized design: 14 precincts $\rightarrow$ Tx; 14 $\rightarrow $ Control.
\item Substantively large baseline differences (red dots)
\item<2-> Differences not large compared to other possible assignments from same design; compared to other possible experiments with the same design.
\item<2-> $\PP(\chi^{2} > x) = .91$ \citep{hansenbowers2008}. (grey lines)
\end{itemize}
\end{column}
\begin{column}{.6\linewidth}
\only<1>{\igrphx{KC-baseline}}
\only<2>{\igrphx{KC-bal+SDs}}
\end{column}
\end{columns}


## How did we do this?

```{r xb1, echo=TRUE}
acorn <- read.csv("data/acorn03.csv", row.names = 1)
xb1 <- balanceTest(z ~ v_p2003 + v_m2003 + v_g2002 + v_p2002 + v_m2002 + v_s2001 +
  v_g2000 + v_p2000 + v_m2000 + v_s1999 + v_m1999 + v_g1998 +
  v_m1998 + v_s1998 + v_m1997 + v_s1997 + v_g1996 + v_p1996 +
  v_m1996 + v_s1996 + size, p.adjust.method = "none",
data = acorn
)

xb1$results
```

## How did we do this?

```{r xb1overall, echo=TRUE}
xb1$overall
```


```{r, out.width=".7\\textwidth", warning=FALSE}
balplot0 <- plot(xb1,statistic="adj.diff",xlab="Differences in Proportion")
balplot0_dat <- balplot0$data
xb1_df <- as_tibble(xb1$results)
names(xb1_df) <- dimnames(xb1$results)$stat
xb1_df$rowname <- dimnames(xb1$results)$vars

balplot_dat <- left_join(balplot0_dat,xb1_df,by="rowname")

balplot1 <- ggplot(data=balplot_dat,aes(x=adj.diff*100,y=rowname)) +
    geom_point()+
    geom_linerange(aes(xmin=-pooled.sd*100,xmax=pooled.sd*100)) +
    xlim(c(-.1,.1)*100) +
    theme(legend.position = "none")

print(balplot1)
```


## DeMystifying balanceTest

```{r d1, echo=TRUE}
d_stat <- function(zz, mm, ss) {
  ## this is the d_statistic (harmonic mean weighted diff of means statistic)
  ## from Hansen and Bowers 2008 almost directly from balanceTest.Engine
  h.fn <- function(n, m) {
    (m * (n - m)) / n
  }
  myssn <- apply(mm, 2, function(x) {
    sum((zz - unsplit(tapply(zz, ss, mean), ss)) * x)
  })
  hs <- tapply(zz, ss, function(z) {
    h.fn(m = sum(z), n = length(z))
  })
  mywtsum <- sum(hs)
  myadjdiff <- myssn / mywtsum
  return(myadjdiff)
}
```

## DeMystifying balanceTest

Recall our discussion of estimation "holding constant" within strata?

```{r d1v2, echo=TRUE}
## This is another version that might be more clear in regards what is going on.
d_stat_v2 <- function(zz, mm, ss) {
  ## mm is a data.frame
  dat <- cbind(mm, z = zz, s = ss)
  datb <- dat %>%
    group_by(s) %>%
    summarize(across(.cols = all_of(names(mm)), function(x) {
      mean(x[z == 1]) - mean(x[z == 0])
    }),
    nb = n(),
    pib = mean(z),
    nbwt = nb / nrow(dat),
    hbwt0 = pib * (1 - pib) * nbwt
    )
  datb$hbwt <- datb$hbwt0 / sum(datb$hbwt0)
  # datb[,15:27]
  adjmns <- datb %>% summarize(across(.cols = all_of(names(mm)), function(x) {
    sum(x * hbwt)
  }))
  adjmnsmat <- as.matrix(adjmns)
  return(adjmnsmat)
}
```

## DeMystifying balanceTest

```{r nullddistsetup, echo=TRUE}
acorncovs <- c("v_p2003", "v_m2003", "v_g2002", "v_p2002", "v_m2002", "v_s2001", "v_g2000", "v_p2000", "v_m2000", "v_s1999", "v_m1999", "v_g1998", "v_m1998", "v_s1998", "v_m1997", "v_s1997", "v_g1996", "v_p1996", "v_m1996", "v_s1996", "size")

dstats1 <- d_stat(zz = acorn$z, mm = acorn[, acorncovs], ss = rep(1, nrow(acorn)))
dstats2 <- d_stat_v2(zz = acorn$z, mm = acorn[, acorncovs], ss = rep(1, nrow(acorn)))

dstats1[1:5]
dstats2[1:5]
xb1$results[,"adj.diff",]

stopifnot(all.equal(dstats1, dstats2[1, ]))
```

```{r, eval=FALSE,echo=FALSE,results=FALSE}
## Curious whether one function is faster than the other
library(microbenchmark)
microbenchmark(d_stat(zz = acorn$z, mm = acorn[, acorncovs], ss = rep(1, nrow(acorn))),
  d_stat_v2(zz = acorn$z, mm = acorn[, acorncovs], ss = rep(1, nrow(acorn))),
  times = 100
)
```

## The reference distribution of the $d^2$ stat

For all vectors $z \in \Omega$ get `adj.diffs`. This is the distribution of the $d$ statistic for one-by-one balance assessment. Next question is about the distribution of the $d^2$ statistic: does it follow a $\chi^2$ distribution in this case?

```{r nulldist, cache=TRUE}
d_dist <- replicate(10000, d_stat(sample(acorn$z), acorn[, acorncovs], ss = rep(1, nrow(acorn))))
```

Get the randomization-based $p$-values:

```{r echo=TRUE}
xb1ds <- xb1$results[, "adj.diff", ]
xb1ps <- xb1$results[, "p", ]
obs.d <- d_stat(acorn$z, acorn[, acorncovs], rep(1, nrow(acorn)))
dps <- matrix(NA, nrow = length(obs.d), ncol = 1)
for (i in 1:length(obs.d)) {
  dps[i, ] <- 2 * min(mean(d_dist[i, ] >= obs.d[i]), mean(d_dist[i, ] <= obs.d[i]))
}
## You can compare this to the results from balanceTest
round(cbind(randinfps = dps[, 1], xbps = xb1ps, obsdstats = obs.d, xbdstats = xb1ds), 3)
```

## Reference distribution of the $d^2$ stat {.allowframebreaks}

The $d^2$ statistic is a linear function of the probably correlated $d$-statistics: a linear combination of correlated variables is $d^{T} \Sigma_d^{-1} d$ where $d$ is a vector of the stratum adjusted differences in means and $\Sigma_d$ is the variance-covariance matrix of the distribution of the $d$ statistics under the sharp null of no differences. With only one variable, this is basically a standardized $d$ statistic (after taking sqrt, and forced to be positive).

```{r d2stat, echo=TRUE}
d2_stat <- function(dstats, ddist = NULL, theinvcov = NULL) {
  ## d is the vector of d statistics
  ## ddist is the matrix of the null reference distributions of the d statistics
  if (is.null(theinvcov) & !is.null(ddist)) {
    as.numeric(t(dstats) %*% solve(cov(t(ddist))) %*% dstats)
  } else {
    as.numeric(t(dstats) %*% theinvcov %*% dstats)
  }
}
```

## Reference distribution of the $d^2$ stat

The distribution of the $d^2$ statistic arises from the distribution of the d statistics --- for each draw from the set of treatment assignments we can collapse the $d$-statistics into one $d^2$. And so we can calculate the $p$-value for the $d^2$.

```{r d22, echo=TRUE}
## Here we have the inverse of the covariance/variance matrix of the d statistics
invCovDDist <- solve(cov(t(d_dist)))
obs.d2 <- d2_stat(obs.d, d_dist, invCovDDist)

d2_dist <- apply(d_dist, 2, function(thed) {
  d2_stat(thed, theinvcov = invCovDDist)
})
## The chi-squared reference distribution only uses a one-sided p-value going in the positive direction
d2p <- mean(d2_dist >= obs.d2)
cbind(obs.d2, d2p)
xb1$overall
```


```{r, eval=FALSE, echo=FALSE, results=FALSE, out.width=".7\\textwidth"}
plot(density(d2_dist))
rug(d2_dist)
abline(v = obs.d2)
```


## Why differences between balanceTest and d2?

I suspect that $N=28$ is too small. `balanceTest` uses an asymptotic
approximation to the randomization distribution.

```{r echo=FALSE, out.width=".8\\textwidth", cache=FALSE}
par(mfrow = c(1, 2),pty="m",oma=rep(0,4),mgp=c(1.5,.5,0),mar=c(3,3,0,0))
qqplot(rchisq(10000, df = 21), d2_dist)
abline(0, 1)
plot(density(d2_dist),main="",xlim=c(2,60))
rug(d2_dist)
lines(density(rchisq(10000, df = 21)),col="grey")
abline(v=c(obs.d2,xb1$overall$chisquare),col=c("black","grey"))
```

## Summary

 - Randomization balances covariate distributions between treated and control groups.
 - We can use randomization inference to check the randomization procedure (mostly useful if there is a long chain of communication between the random number generator and the field).
 - **Randomization does not imply exact equivalence. Large differences in covariates easily arise in small experiments.**

# Assessing comparisons in observational studies

```{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
```
## 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? {.allowframebreaks}

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))
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:

```{r initialtest, echo=TRUE}
balanceTest(nhTrt ~ HomRate08, data = meddat)
meddat$nhTrtF <- factor(meddat$nhTrt)
test2 <- oneway_test(HomRate08 ~ nhTrtF, data = meddat, distribution = asymptotic())
test3 <- oneway_test(HomRate08 ~ nhTrtF, data = meddat, distribution = approximate(nresample = 1000))
test4 <- wilcox_test(HomRate08 ~ nhTrtF, data = meddat, distribution = approximate(nresample = 1000))
pvalue(test2)
pvalue(test3)
pvalue(test4)
```

## 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 `nhAboveHS` is proportion
with more than a high school education in the neighborhood in 2003 or so and
`nhTrt` is 0=no station built, 1=station built? (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
```

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

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


## 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]
```

## Exactly what does this kind of adjustment do?

So, how would you explain what it means to "control for HS-Education" here?

```{r}
plot(eZX, eYX)
```



## 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=".6\\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(outcomefmla, data = meddat)
```


## 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?
## Putting this together
outcome_analysis_strat <- meddat %>% group_by(nhAboveHS>=.1) %>%
    summarize(nb=n(),nT=sum(nhTrt),nC=nb-nT,pr_trt=mean(nhTrt),bar_y_t=mean(HomRate08[nhTrt==1]),
        bar_y_c=mean(HomRate08[nhTrt==0]),ate_b=bar_y_t - bar_y_c)

outcome_analysis_strat <- outcome_analysis_strat %>% mutate(nbwt=nb/sum(nb),
    prec_wt0 = nbwt * pr_trt * (1-pr_trt))
outcome_analysis_strat$prec_wt <-
    with(outcome_analysis_strat,prec_wt0/sum(prec_wt0))

outcome_analysis_strat

outcome_analysis_strat %>% summarize(ate_block_size= sum(ate_b * nbwt),
    ate_prec_wt = sum(ate_b * prec_wt/sum(prec_wt)))

```

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

## Stratified adjustment V 2.0

One-step stratified estimation.

```{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(HomRate08 ~ nhTrt + I(nhAboveHS >= .1), data = meddat)
coef(lm1d)["nhTrt"]
```

## Stratified adjustment V 2.0 {.allowframebreaks}

One-step stratified testing

```{r echo=TRUE}
xbate1 <- balanceTest(nhTrt ~ HomRate08+ strata(I(nhAboveHS >= .1)), data = meddat)
xbate1$results[, c("adj.diff","p"), ]
xbate1$overall

## Effect of the treatment on the treated weights
outcome_analysis_strat$nTwt <- with(outcome_analysis_strat,nT/sum(nT))
with(outcome_analysis_strat,sum(ate_b*nTwt))

## Approximating the as-if-randomized null distribution with a Normal
## approximation
hstest2 <- independence_test(HomRate08~nhTrt|factor(nhAboveHS >= .1),data=meddat)

## Now using the "as-if-randomized" distribution directly
set.seed(12345)
hstest2_perm <- independence_test(HomRate08~nhTrt|factor(nhAboveHS >= .1),data=meddat,distribution=approximate(nresample=10000))

pvalue(hstest2)
pvalue(hstest2_perm)

## Now trying different test statistics
hstest4 <- independence_test(HomRate08~nhTrt|factor(nhAboveHS >= .1),data=meddat,ytrafo=rank_trafo)
pvalue(hstest4)

hstest5 <- wilcox_test(HomRate08~factor(nhTrt)|factor(nhAboveHS >= .1),data=meddat)
pvalue(hstest5)

```

## Balance assessment after stratification

Did we adjust enough? What would *enough* mean? Use the testing approach but now focus only on the covariate(s) that you are trying to adjust.

```{r xbHS1, echo=TRUE}
xbHS1 <- balanceTest(nhTrt ~ nhAboveHS+strata(I(nhAboveHS >= .1)), data = meddat)
xbHS1$overall
xbHS1$results[1, c("Treatment", "Control", "adj.diff", "std.diff", "z", "p"), ] ## 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

 - Problem of Interepretability: "Controlling for" is  "removing (additive) linear relationships" it is  not "holding constant"
 - Problem of Diagnosis and Assessment: What is the  standard against which we can compare a given linear covariance adjustment specification?
 - Problem of extrapolation and interpolation: Often known as "common support", too.
 - 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).

## The Problem of Using  the Linear Model for  Adjustment

 - Problems of  bias even in randomized experiments:

\begin{equation}
Y_i = \beta_0 + \beta_1 Z_i + e_i
\label{eq:olsbiv}
\end{equation}

This is a common practice because, we know that the formula to estimate
$\beta_1$ in equation \eqref{eq:olsbiv} 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}

\begin{equation}
Y_i = \beta_0 + \beta_1 Z_i + \beta_2 X_i + e_i \label{eq:olscov}
\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 in \eqref{eq:olscov}:

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

## 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...?

**One idea:** 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 from the fullmatch.

```{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

Differences within sets versus raw differences from the pairmatch.

```{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

```{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


```{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
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"]
```

## Summary of the Day

 - We can assess the randomization of a randomized experiment easily using
   covariates ($X$): compare the observed treatment-vs-control differences in
   $X$ with those consistent with differences that would emerge from repeating
   the design. (The differences will not be strictly zero. But shouldn't be
   systematically different.)

 - 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.

## References