In omkarakatta/diftrans: Compute the Differences-in-Transports Estimator

library(knitr)
knitr::opts_chunk$set(
  collapse = FALSE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%"
)

diftrans

Table of Contents

• Introduction
• Installation
• Attribution
• Example

Introduction

The diftrans package applies the novel methods described in Daljord, Pouliot, Hu, and Xiao (2021) to compute the transport costs between two univariate distributions and the differences-in-transports estimator.

Appealing to optimal transport theory, the diftrans package builds off of the transport package to compute the change between the distributions of a univariate variable of interest. Extending this application, the diftrans package allows users to compute the before-and-after estimator (Daljord et al., 2021), which controls for sampling uncertainty by trivializing small changes in the distributions.

The diftrans package also computes the differences-in-transports estimator, which controls for unobservable reasons that the distributions may change by comparing the transport costs to those from another source of measurement.

The only function in the diftrans package is diftrans, which is explained below by way of a toy example.

The documentation file for diftrans::diftrans can be found by running the following in an R console:

?diftrans::diftrans

Installation

Install the diftrans from GitHub with:

# install.packages("devtools")
devtools::install_github("omkarakatta/diftrans")

To install the package with the vignette installed, set the build_vignettes argument to TRUE. Note that the installation will take more time this way.

# install.packages("devtools")
devtools::install_github("omkarakatta/diftrans", build_vignettes = TRUE)

Attribution

To cite the diftrans package, use the BibTeX entry provided by:

citation("diftrans")

Example

The workhorse function in the diftrans package is also called diftrans, which serves two purposes:

• compute the transport cost between two univariate distributions, and
• compute the differences-in-transports estimator (see Daljord et al. (2021)).

Setup

library(dplyr)
library(magrittr)
devtools::load_all()

mykable <- function(dt){
  kable(dt,
        align = 'c')
}

library(dplyr)
library(magrittr)
library(diftrans)

We begin by computing the transport cost between two distributions of some variable x. These distributions should be represented as tibbles with two columns:

• column 1 contains the full support of the distribution, and
• column 2 (labeled "count") contains the mass/counts associated with each value in the support.

Suppose that the shift in the distribution is due to some treatment. We refer to the first and second distributions of some random variable x as the pre-distribution and post-distribution for the treated group, respectively.

Below are the tibbles for the pre- and post-distributions as well as the corresponding plots. Both tibbles contain the full support of our variable of interest, i.e., their support column is the same.

Pre-Distribution for Treated Group

set.seed(1)
support_min <- 1
support_max <- 10
support <- seq(support_min, support_max, 1)
pre_treated <- data.frame(x = sample(support, 100, replace = T)) %>% 
  group_by(x) %>% 
  count() %>% 
  ungroup() %>% 
  rename(count = n)

shift <- 5
for (i in 1:shift){
  pre_treated[pre_treated$x == support_max-i+1, "count"] <- 0
}

mykable(pre_treated)

Post-Distribution for Treated Group

post_treated <- data.frame(x = support,
                           count = lag(pre_treated$count, shift)) %>% 
  tidyr::replace_na(list(count = 0))

mykable(post_treated)

Observe that all the mass from the pre-distribution was shifted by r shift units of the support to form the post-distribution. That is, the treatment resulted in all the mass to change. For instance, the mass that was given given to r pre_treated[1, "x"] in the pre-distribution is now given to r pre_treated[1+shift, "x"].

In this toy example, we should expect that the transport cost is 100% because all the mass was transported due to the treatment.

Compute Transport Cost

We can compute the transport cost as follows:

tc <- diftrans(pre_main = pre_treated, post_main = post_treated,
                  estimator = "tc", var = x,
                  bandwidth = 0)

tc

The transport cost is r scales::percent(tc$main) as expected. The estimator argument is specified to be "tc", which stands for transport cost. (Since we only have two distributions, estimator will take on the value of "tc" by default.)

Since the post-distribution is a r shift-unit shift in the pre-distribution, any bandwidth at least r shift must result in a transport cost of 0. We verify this by computing the transport cost for a sequence of bandwidths from 0 to 10:

tc <- diftrans(pre_main = pre_treated, post_main = post_treated,
                  estimator = "tc", var = x,
                  bandwidth = seq(0, 10))

tc

Setup for Difference-in-Transports

Above, we assumed that the treatment is sole reason why any shifts between the pre- and post-distributions of x might occur. Now suppose there are unobserved trends in our variable that might also explain this shift. To elicit how much of this shift is due to the treatment and how much of this shift is due to unobserved trends, we can compare our distributions above with distributions of the same variable measured in some control group that did not receive the treatment but was also subjected to the same unobserved trends.

The novel differences-in-transports estimator described in Daljord et al. (2021) quantifies the change in distribution due to some treatment relative to the change in distribution of some control group.

control_shift <- 2

pre_control <- data.frame(x = support,
                          count = pre_treated$count)
post_control <- data.frame(x = support,
                           count = lag(pre_treated$count, control_shift)) %>% 
  tidyr::replace_na(list(count = 0))

First, we need to set up the distributions of our variable for our control group, both before and after the treatment was administered to the treatment group. For our purposes, we let the pre-distribution of our control group be the same as that of our treated group. Then, let our post-distribution for our control group be a r control_shift-unit shift of our pre-distribution. We therefore have the following tibbles that represent our control group distributions:

Pre-Distribution for Control Group

mykable(pre_control)

Post-Distribution for Control Group

mykable(post_control)

Compute Differences-in-Transports Estimator ¹

Above, we used estimator = "tc" and specified our pre- and post-distributions for our treated group. Now, we use estimator = "dit" (which stands for differences-in-treatments estimator) while also specifying our pre- and post-distributions for our control group.

dit <- diftrans(pre_main = pre_treated, post_main = post_treated,
                   pre_control = pre_control, post_control = post_control,
                   estimator = "dit", var = x,
                   bandwidth_seq = seq(0, 10, 1),
                   save_dit = TRUE)
dit$out
dit$dit
dit$optimal_bandwidth

Another difference between computing the transport cost and the differences-in-differences estimator is that the differences-in-transports estimator is printed as a message. To save this result in dit, we use save_dit = TRUE.

Thus, the differences-in-transports estimator is r scales::percent(dit$dit) at an optimal bandwidth of r dit$optimal_bandwidth.

Sampling Variability

Often times, the data will be a sample from a population, and sample variability is enough to cause differences between distributions. To account for this discrepancy, we can increase the bandwidth to ignore transfers of mass between nearby values of the support.

d_a <- 1

While Daljord et al. (2021) offer a disciplined way to determine what the appropriate bandwidths are, for simplicity, we will suppose that there will not be any sampling variation with a bandwidth greater than r d_a.

d_a <- 1

dit <- diftrans(pre_main = pre_treated, post_main = post_treated,
                   pre_control = pre_control, post_control = post_control,
                   estimator = "dit", var = x,
                   bandwidth_seq = seq(d_a, 10, 1), # smallest bandwidth will be d_a
                   save_dit = TRUE)
dit$out
dit$dit
dit$optimal_bandwidth

Note that this restriction is not binding, so our result is unaffected. Accounting for sampling variability, our differences-in-transports estimate is still r scales::percent(dit$dit).

Conservative Differences-in-Transports

If we want to be more conservative, we can use conservative = TRUE, which essentially uses twice the bandwidth for computing the treated group's transport costs relative to the bandwidth for computing the control group's transport costs.

dit <- diftrans(pre_main = pre_treated, post_main = post_treated,
                   pre_control = pre_control, post_control = post_control,
                   estimator = "dit", var = x,
                   bandwidth_seq = seq(d_a, 10, 1),
                   save_dit = TRUE,
                   conservative = TRUE)
dit$out
dit$dit
dit$optimal_bandwidth

Finally, we have that the conservative differences-in-transports estimator is r scales::percent(dit$dit) at an optimal bandwidth of r dit$optimal_bandwidth.

Note that estimator = "dit" is not necessary. Since we have two sets of pre- and post-distributions, diftrans is smart enough to return the differences-in-transports estimator by default.

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1 There is more nuance to computing the differences-in-transports estimator than is presented in this expository document. See Daljord et al. (2021) for a more complete treatment on how to compute the differences-in-transports estimator. ↩

omkarakatta/diftrans documentation built on Feb. 24, 2023, 9:06 p.m.