day12_sensitivity.Rmd

---
title: |
 | Intro to Sensitivity Analysis: Rosenbaum Style
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(tidyverse)
library(dplyr)
library(ggplot2)
library(coin)
library(RItools)
library(optmatch)
library(estimatr)
library(sensitivitymv)
library(sensitivitymw)
library(sensitivitymult)
library(sensitivityfull)
library(senstrat)
```

## Today

 1. Agenda: Reasoning about how departures from our "as-if randomized" approach
    might change our substantive conclusions: sensitivity analysis, Rosenbaum
    style.
 2. We are now in the "Sensitivity Analysis" part of the course
    (see the reading there).
 3. Questions arising from the reading or assignments or life? (Bipartite matching? Non-bipartite matching? Instrumental variables? Randomized experiments?)

# But first, review

## Statistical Inference for Stratified Designs

Given a research design that (a) can be defended on substantive grounds and (b) compares favorably with a randomized experiment of the same design (i.e. in the case of fullmatching, a block randomized experiment with either 1 treated and 1 or more control units or 1 control unit and 1 or more treated units), how should we do statistical inference? (How should we choose and justify/explain our choices of estimators and tests?)

 - Testing and estimation procedures that have good properties have
   **distributions** that we can describe and work with.
   - Most obviously, $p$-values compare observed test stat. to the
     **distribution** of that test stat. in a hypothesized world. (Recall
     randomization based tests in randomized experiments).
   - However an "unbiased estimator" is an estimator having a **distribution**
     that is centered on the truth (a "precise estimator" has a
     **distribution** that is narrow, a "consistent estimator" has a
     **distribution** that makes all individual estimates closer to the truth
     as information increases)

## Statistical Inference for Stratified Designs

Where can those distributions come from?

  - In a randomized experiment they can come from the **random assignment**
    (Randomization **justifies** our use of certain distributions and not
    others for for test statistics and estimators.)
  - In a random sample from a population the **random sampling** can also
    justify choice of distribution for test statistics and estimators.
  - The Central Limit Theorem (CLT) also gives us reason to believe that
    certain distributions will be Normal.
  - If we knew that the outcome arose from a stochastic/random process (like $Y
    \sim \text{Poisson}(\lambda)$ and $\lambda=f(Z,X,\beta)$ ) then the **CLT**
    *and* **Likelihood** gives us distributions. (An **outcome distribution**
    and functional form plus CLT yields distributions of test statistics and
    estimators).
  - If we knew a **Likelihood** function *and* a **Prior** distribution for the
    parameters of the Likelihood function, then **Bayes Rule** gives us
    distributions.(An **outcome distribution** and functional form plus **prior
    distribution** yields distributions of test statistics and estimators
    (versions of **posterior distributions**).


## The "As-If randomized" approach


Given a research design that (a) can be defended on substantive grounds and (b) compares favorably with a randomized experiment of the same design (i.e. in the case of fullmatching, a block randomized experiment with either 1 treated and 1 or more control units or 1 control unit and 1 or more treated units), how should we do statistical inference? (How should we choose and justify/explain our choices of estimators and tests?)

Why not treat this design **as-if it had been randomized** for the sake of argument? One can say, "If this had been an experiment, the results would be $\hat{\tau}$, $p=.02$."

Advantages:

 - No probability model of outcomes to justify
 - No functional form/structural model of covariates and intervention/treatment to justify
 - No population and sampling needed --- inference can focus on the pool of observations available.
 - No CLT required --- CLT for convenience only
 - We already showed that, for many covariates, the design compares favorably to an experiment.


## The "As-If randomized" approach


Given a research design that (a) can be defended on substantive grounds and (b) compares favorably with a randomized experiment of the same design (i.e. in the case of fullmatching, a block randomized experiment with either 1 treated and 1 or more control units or 1 control unit and 1 or more treated units), how should we do statistical inference? (How should we choose and justify/explain our choices of estimators and tests?)

Why not treat this design **as-if it had been randomized** for the sake of argument? One can say, "If this had been an experiment, the results would be $\hat{\tau}$, $p=.02$."

Disadvantages:

 - **This is not a randomized experiment.** How might our results differ if (1) our design did not adequately capture the intervention/selection process or (2) we used a different basis for statistical inference?

Sensitivity analysis aims to answer such questions --- especially question 1.

```{r loaddat, echo=FALSE}
load(url("http://jakebowers.org/Data/meddat.rda"))
meddat$id <- row.names(meddat)
meddat <- mutate(meddat,
  HomRate03 = (HomCount2003 / Pop2003) * 1000,
  HomRate08 = (HomCount2008 / Pop2008) * 1000,
  HomRate0803 = (HomRate08 - HomRate03)
)# %>% column_to_rownames("id")
options(show.signif.stars = FALSE)
```

## Make a matched design:

I  use a gain score approach using a pairmatch (Notice that the difference in
the gain scores is the difference in the differences. Only here we make no
parallel trends assumption, our assumptions are different from those used by
DiD approaches.)

```{r echo=TRUE}
balfmla <- nhTrt ~ HomRate03 + nhPopD + nhHS + nhAboveHS + nhEmp + nhAgeMid + nhAgeYoung + nhMarDom + nhOwn + nhRent
glm1 <- arm::bayesglm(balfmla, data = meddat)
meddat <- transform(meddat, pairm = pairmatch(glm1, data = meddat))
xb1 <- balanceTest(update(balfmla,.~.+strata(pairm)), data = meddat)
xb1$overall
```

And here is an **outcome analysis** using `HomRate08 - HomRate03` as my outcome:

```{r echo=TRUE}
xbtest3 <- balanceTest(nhTrt ~ HomRate0803+strata(pairm), data = meddat[matched(meddat$pairm), ])
xbtest3$overall
xbtest3$results[, c("adj.diff", "z", "p"), ]
## OR using a different asymptotic approximation:
lm1 <- lm_robust(HomRate0803 ~ nhTrt, fixed_effects = ~pairm, data = meddat, subset = !is.na(pairm))
lm1
```


## What about unobserved confounders?

A high $p$-value from an omnibus balance test gives us some basis to claim
that our comparison contains as much confounding on *observed* covariates
(those assessed by our balance test) as would be seen in a block-randomized
experiment. That is, our treatment-vs-control comparison contains demonstrably
little bias from the variables that we have balanced.

\medskip

But, we haven't said anything about *unobserved* covariates (which a truly
randomized study would balance, but which our study **does not**).

## Sensitivity analysis as a formalized thought experiment {.fragile}

> "In an observational study, a sensitivity analysis replaces qualitative claims about whether unmeasured biases are present with an objective quantitative statement about the magnitude of bias that would need to be present to change the conclusions." (Rosenbaum, sensitivitymv manual)


>  "The sensitivity analysis asks about the magnitude, gamma, of bias in treatment assignment in observational studies that would need to be present to alter the conclusions of a randomization test that assumed matching for observed covariates removes all bias."  (Rosenbaum, sensitivitymv manual)

All non-randomized studies have some bias from unobserved confounders. The question is how big the bias could be without changing our substantive conclusions.


\begin{center}
\begin{tikzcd}[column sep=large,every arrow/.append style=-latex]
u = f(u_1,u_2,\ldots) \arrow[from=1-1,to=2-2, bend right, "\Gamma" near start] \arrow[from=1-1,to=2-3, bend left, "\rho" near start] \\
& Z  \arrow[from=2-2,to=2-3, "\tau"] &  y \\
x_1 \arrow[from=3-1,to=2-2, "\beta_1" ] \arrow[from=3-1,to=2-3,grey] &
x_2 \arrow[from=3-2,to=2-2, "\beta_2" ] \arrow[from=3-2,to=2-3,grey]
\end{tikzcd}
\end{center}




```{r dosens, echo=FALSE,results="hide"}
#' Reshape Optmatch Output for Sensitivity Analysis
#'
#' A function to reformat the output of optmatch::fullmatch for use with sensmv/mw such that the first column is the treated unit and the remaining columns are the control units.  We assume that y,z, and fm have no missing data.
#' @param y is a vector of outcomes
#' @param z is a vector of binary treatment indicators (0=control,1=treated)
#' @param fm is a vector containing the factor indicating set membership
#' @param senfm is TRUE if the output should be formatted for use in [sensitivityfull::senfm]: for 1:K sets, the first column will be treated and for K:1 sets, the first column will be controls.
reshape_sensitivity <- function(y, z, fm, senfm=FALSE) {
    #dat <- data.frame(y = y, z = z, fm = fm)[order(fm, z, decreasing = TRUE), ]
    #dat1 <- dat %>% group_by(fm) %>%    mutate(idb = row_number()) %>%  pivot_wider(names_from=idb,values_from=y)
    # dat2 <- dat1 %>% group_by(fm) %>% summarize()
    numcols <- max(table(fm))
    resplist <- lapply(
        split(data.frame(y,z), fm),
         function(dat) {
           numtrt <- sum(dat$z)
           numctrl <- sum(1-dat$z)
           stopifnot(numtrt==1 | numctrl==1)
           if(numtrt==1){
             res <- with(dat,c(y[z==1],y[z!=1],rep(NA, max(numcols - length(y))), 1))
           }
           if(numctrl==1 & numtrt>1){
             res <- with(dat,c(y[z==0],y[z!=0],rep(NA, max(numcols - length(y))),0))
           }
           return(res)
          #return(c(x, rep(NA, max(numcols - length(x), 0))))
         }
    )
    respmat <- t(simplify2array(resplist))
    if(senfm){
  return(respmat)
    } else {
      ## remove the indicator of whether the singleton is control or treated
      return(respmat[,-ncol(respmat)])
    }
    }
```

## An example of sensitivity analysis with `senmv`.


The workflow: First, reshape the matched design into the appropriate shape (one treated unit in column 1, controls in columns 2+).^[So notice that `senmv` requires 1:K matches although K can vary. The `sensitivityfull::senfm` function allows for unrestricted `fullmatch` output (but still reorganized into a matrix).]

```{r echo=TRUE}
respmat <- with(meddat[matched(meddat$pairm), ], reshape_sensitivity(HomRate0803, nhTrt, pairm))
respmat[1:4, ]
meddat <- transform(meddat, fm = fullmatch(balfmla, data = meddat, min.controls = 1))
respmat2 <- with(meddat[matched(meddat$fm), ], reshape_sensitivity(HomRate0803, nhTrt, fm))
respmat2[12:18, ]
```


## An example of sensitivity analysis: the search for Gamma

The workflow: Second, assess sensitivity at different levels of $\Gamma$. (using `-respmat` here because the differences between treated and controls are negative --- stations are associated with reductions in violence.)

```{r echo=TRUE}
sensG1 <- senmv(-respmat, method = "t", gamma = 1)
sensG2 <- senmv(-respmat, method = "t", gamma = 2)
sensG1$pval
sensG2$pval
```

If we used a paired $t$-test to assess the hypothesis of no effects and:

 - Metrocable stations were really randomized within pair (no unobserved confounding), our $p$-value would be about `r round(sensG1$pval,2)`.

 - An unobserved confounder (or a function of many unobserved confounders) made neighborhoods with Metrocable stations within pair **twice as likely** to receive those stations as the neighborhoods without the stations, then our **maximum $p$-value** would be about `r round(sensG2$pval,2)`.


##  Why $\Gamma$? Can we model unobserved confounding?

In what ways can an unobserved covariate confuse our causal inferences? *We need to have a model to help us reason about this.*  @rosenbaum2002observational starts with a *treatment odds ratio* for two units $i$ and $j$ having the same background values for covariates in a vector $\bx$ (as if they were perfectly matched on $\bx$).

\begin{center}
\begin{align}
\frac{\left(\frac{\pi_i}{1 - \pi_i} \right)}{\left(\frac{\pi_j}{1 - \pi_j} \right)} \ \forall \ i,j \ \text{with } \mathbf{x}_i = \mathbf{x}_j \notag  \implies
& \frac{\pi_i (1 - \pi_j)}{\pi_j (1 - \pi_i)} \ \forall \ i,j \ \text{with } \mathbf{x}_i = \mathbf{x}_j.
\end{align}
\end{center}
 which implies a model that links treatment odds, $\frac{\pi_i}{(1 - \pi_i)}$, to the *observed and unobserved* covariates $(\mathbf{x}_i, u_i)$,

\begin{center}
\begin{equation}
\label{eq: unobserved confounding}
\text{log} \left(\frac{\pi_i}{1 - \pi_i} \right) = \kappa(\mathbf{x}_i) + \gamma u_i,
\end{equation}
\end{center}

where $\kappa(\cdot)$ is an unknown function and $\gamma$ is an unknown parameter.

\note{
\begin{center}
\textbf{Remember}:
\end{center}
A logarithm is simply the power to which a number must be raised in order to get some other number. In this case we're dealing with natural logarithms. Thus, we can read $\text{log} \left(\frac{\pi_i}{1 - \pi_i} \right)$ as asking: $\mathrm{e}$ to the power of what gives us $\left(\frac{\pi_i}{1 - \pi_i} \right)$? And the answer is $\mathrm{e}$ to the power of $\kappa(\mathbf{x}_i) + \gamma u_i$. If $\mathbf{x}_i = \mathbf{x}_j$, then $\text{log} \left(\frac{\pi_i}{1 - \pi_i} \right) = \gamma u_i$, which means that $\mathrm{e}^{\gamma u_i} = \left(\frac{\pi_i}{1 - \pi_i} \right)$.
}

## Why $\Gamma$?

Say, we rescale $u$ to $[0,1]$, then we can write the original ratio of treatment odds using the logistic model and the unobserved covariate $u$:

\begin{center}
\begin{equation}
\frac{\pi_i (1 - \pi_j)}{\pi_j (1 - \pi_i)} = \mathrm{e}^{\gamma(u_i - u_j)} \ \text{if} \ \mathbf{x}_i = \mathbf{x}_j.
\end{equation}
\end{center}

Since the minimum and maximum possible value for $u_i - u_j$ are $-1$ and $1$,
for any fixed $\gamma$ the upper and lower bounds on the treatment odds ratio
are:

\begin{center}
\begin{equation}
\label{eq: treatment odds ratio bounds gamma}
\frac{1}{\mathrm{e}^{\gamma}} \leq \frac{\pi_i (1 - \pi_j)}{\pi_j (1 - \pi_i)} \leq \mathrm{e}^{\gamma}.
\end{equation}
\end{center}

If we use $\Gamma$ for  $\mathrm{e}^{\gamma}$, then we can substitute $\frac{1}{\Gamma}$ for $\mathrm{e}^{-\gamma}$ and $\Gamma$ for $\mathrm{e}^{\gamma}$.

## Why $\Gamma$?

\ldots so we can write the odds of treatment in terms of $\Gamma$ (the effect
of $u$ on the odds of treatment) for any two units $i$ and $j$ with the same
covariates (i.e. in the same matched set):

\begin{center}
\begin{equation}
\frac{1}{\Gamma} \leq \frac{\pi_i (1 - \pi_j)}{\pi_j (1 - \pi_i)} \leq \Gamma \ \forall \ i,j \ \text{with } \mathbf{x}_i = \mathbf{x}_j
\end{equation}
\end{center}

So when $\pi_i = \pi_j$ then $\Gamma=1$: the treatment probabilities are the same for the two units --- just as we would expect in a randomized study.


## An example of sensitivity analysis: the search for Gamma

The workflow: assess sensitivity at different levels of $\Gamma$ (here
using two different test statistics).

```{r echo=TRUE}
somegammas <- seq(1, 5, .1)
sensTresults <- sapply(somegammas, function(g) {
  c(gamma = g, senmv(-respmat, method = "t", gamma = g))
})
sensHresults <- sapply(somegammas, function(g) {
  c(gamma = g, senmv(-respmat, gamma = g))
})
```

## An example of sensitivity analysis: the search for Gamma

The workflow: assess sensitivity at different levels of $\Gamma$ (here
using two different test statistics).

```{r echo=FALSE, out.width=".8\\textwidth"}
par(mar = c(3, 3, 2, 1))
plot(
  x = sensTresults["gamma", ],
  y = sensTresults["pval", ],
  xlab = "Gamma", ylab = "P-Value",
  main = "Sensitivity Analysis", ylim = c(0, .2)
)
points(
  x = sensHresults["gamma", ],
  y = sensHresults["pval", ], pch = 2
)
abline(h = 0.05)
text(sensTresults["gamma", 20], sensTresults["pval", 20], label = "T stat (Mean diff)",pos=2)
text(sensHresults["gamma", 22], sensHresults["pval", 22], label = "Influential point resistent\n mean diff",pos=4)
```

## An example of sensitivity analysis: the search for Gamma

Or you can try to directly find the $\Gamma$ for a given $\alpha$ level test using an optimization routine:


```{r echo=TRUE}
## Recall that senmv produces a p-value
senmv(-respmat, gamma = 1, method = "t")$pval
senmv(-respmat, gamma = 1, method = "h")$pval

findSensG <- function(g, a, method) {
  senmv(-respmat, gamma = g, method = method)$pval - a
}

res1_t <- uniroot(f = findSensG, method = "t", lower = 1, upper = 6, a = .05)
res1_t$root
res1_h <- uniroot(f = findSensG, method = "h", lower = 1, upper = 6, a = .05)
res1_h$root
```


## Some other approaches: senfm

If we have sets with multiple treated units, we can use `senfm`. And here we assess sensitivity of this match to the Gamma found earlier (of `r res1_h$root`)

```{r echo=TRUE}
meddat <- transform(meddat, fm1 = fullmatch(glm1, data = meddat))
respmat3 <- with(meddat[matched(meddat$fm1), ], reshape_sensitivity(HomRate0803, nhTrt, fm1,senfm=TRUE))

## The last column tells us whether this is a 1:K (1) versus K:1 (0) match
respmat4 <- respmat3[,-ncol(respmat3)]
settype <- respmat3[,10]==1

res2_full <- senfm(-respmat4, treated1 = settype, gamma = 1)
res2_full$pval

res2_full_g2 <- senfm(-respmat4, treated1 = settype, gamma = res1_h$root)
res2_full_g2$pval
```

## Some other approaches: senstrat

The `senstrat` package is a bit easier to use for our workflow. It encourages
you to think about different test statistics using transformations of the
outcome.


```{r echo=TRUE}

mscores1 <- mscores(y=meddat$HomRate0803,z=meddat$nhTrt,st=meddat$fm1)
hscores1 <- hodgeslehmann(y=meddat$HomRate0803,z=meddat$nhTrt,st=meddat$fm1)

ss_y_g1 <- senstrat(meddat$HomRate0803,z=meddat$nhTrt,st=meddat$fm1,gamma=1, alternative="less") #, detail=TRUE)
ss_h_g1 <- senstrat(hscores1,z=meddat$nhTrt,st=meddat$fm1,gamma=1, alternative="less")
ss_m_g1 <- senstrat(mscores1,z=meddat$nhTrt,st=meddat$fm1,gamma=1, alternative="less")

ss_y_g2 <- senstrat(meddat$HomRate0803,z=meddat$nhTrt,st=meddat$fm1,gamma=2, alternative="less") #,detail=TRUE)
ss_h_g2 <- senstrat(hscores1,z=meddat$nhTrt,st=meddat$fm1,gamma=2, alternative="less")
ss_m_g2 <- senstrat(mscores1,z=meddat$nhTrt,st=meddat$fm1,gamma=2, alternative="less")

```


## Some other approaches:

See also:

 - https://cran.r-project.org/web/packages/treatSens/index.html (for parametric
models)
 - https://cran.r-project.org/web/packages/sensitivityPStrat/index.html
 - See the piece "Do We Really Know the WTO Cures Cancer?"<http://www.stephenchaudoin.com/CHH_Cancer_bjps.pdf>
 - See the piece "The sensitivity of linear regression coefficients’ confidence limits to the omission of a confounder" <https://projecteuclid.org/euclid.aoas/1280842143>




## Confidence Intervals

Since we have fixed size sets (i.e. all 1:1 or all 1:2...), we can also look at
an example involving point-estimates for bias of at most $\Gamma$ and
confidence intervals assuming an additive effect of treatment. Notice also that
that when $\Gamma$ is greater than 1, we have a range of point estimates
consistent with that $\Gamma$.

```{r cis}
respmatPm <- with(droplevels(meddat[matched(meddat$pairm), ]), reshape_sensitivity(HomRate0803, nhTrt, pairm))
(sensCItwosidedG1 <- senmwCI(-respmatPm, method = "t", one.sided = FALSE))
(sensCIonesidedG1 <- senmwCI(-respmatPm, method = "t", one.sided = TRUE))
(sensCItwosidedG2 <- senmwCI(-respmatPm, method = "t", one.sided = FALSE, gamma = 2))
(sensCIonesidedG2 <- senmwCI(-respmatPm, method = "t", one.sided = TRUE, gamma = 2))
```

\note{
 Notice that the
two-sided intervals have lower bounds that are lower than the one-sided
intervals. }

## Interpreting sensitivity analyses


 As an aid to interpreting sensitivity analyses,
  @rosenbaum2009amplification propose a way decompose $\Gamma$ into two
  pieces: one $\Delta$ gauges the relationship between an unobserved
  confounder and the outcome (it records the maximum effect of the unobserved
  confounder on the odds of a positive response (imagining a binary outcome))
  and the other $\Lambda$ gauges the maximum relationship between the unobserved
  confounder and treatment assignment.

```{r amplify}
lambdas <- seq(round(res1_h$root, 1) + .1, 2 * res1_h$root, length = 100)
ampres1 <- amplify(round(res1_h$root, 1), lambda = lambdas)
ampres2 <- amplify(2, lambda = lambdas)
```

```{r echo=FALSE, out.width=".5\\textwidth"}
par(mar = c(3, 3, 1, 1), mgp = c(1.5, .5, 0))
plot(as.numeric(names(ampres1)), ampres1,
  xlab = "Lambda (maximum selection effect of confounder)",
  ylab = "Delta (maximum outcome effect of confounder)",
  main = paste("Decomposition of Gamma=", round(res1_h$root, 2))
)
## Add a line for another Gamma
## lines(as.numeric(names(ampres2)), ampres2, type = "b")
```

## Interpreting sensitivity analyses

```{r echo=FALSE, out.width=".5\\textwidth"}
par(mar = c(3, 3, 1, 1), mgp = c(1.5, .5, 0))
plot(as.numeric(names(ampres1)), ampres1,
  xlab = "Lambda (maximum selection effect of confounder)",
  ylab = "Delta (maximum outcome effect of confounder)",
  main = paste("Decomposition of Gamma=", round(res1_h$root, 2))
)
## Add a line for another Gamma
## lines(as.numeric(names(ampres2)), ampres2, type = "b")
```


## Sensitivity analysis for the weak null

@cinellihazlett2020 and @hosman2010 expresse sensitivity analysis in terms more familiar to people who are using linear regression for estimation and testing of the weak null. Here is a version using the `sensemakr` package:

```{r echo=TRUE}
library(sensemakr)
res1_lm <- lm(HomRate0803~nhTrt+pairm, data = meddat)
sens1 <-  sensemakr(model = res1_lm, treatment = "nhTrt")

summary(sens1)
## If you have trouble using lm() and want to use lm_robust() or 
## another least squares fitter (say, you have lots of pairs) or want 
## a different standard error
## you can do this:
sens2 <- sensemakr(estimate = coef(lm1)[["nhTrt"]], se = lm1$std.error[["nhTrt"]], treatment = "nhTrt", dof = lm1$df.residual)

summary(sens2)
```

```{r}
plot(sens2,sensitivity.of="t-value")
```


## Summary and Questions:

 - Formal sensitivity analysis allows us to use the as-if randomized approach to statistical inference **provisionally** (versus fearfully): we can directly ask how our distributions would change under different forms of unobserved confounding.
 - The fact that a $p<.01$ would change to $p=.051$ if $\Gamma=3$ (or if $\Lambda$=4.5 and $\Delta$=14.7) does not mean that such an unobserved covariate exists.
 - Since we are reasoning about unobserved covariates, we have a naming problem: We have to ask: What covariate could conceivably play this role?
 - See the Rabb et al (2022) piece for an example of this kind of sensitivity analysis in application.
 - See Hazlett (2020) for an application of the regression based approach: "Angry or Weary?
How Violence Impacts Attitudes toward Peace among Darfurian Refugees".


 
 


## References