Shifting Borders, Creating Polygons from Lines

One of the most tedious tasks regarding spatial RDDs is the execution of so-called placebo checks. With these, one has to show that the postulated effect disappears when the RD cutoff is shifted around. In other words, the regression coefficient on the treatment indicator has to be statistically insignificant when the RD estimation is carried out on any additional border. Just shifting around the border would be already very cumbersome when it is carried out in a “point and click fashtion” in a GIS API such as QGIS or ArcGIS. Here in R, thanks to the sf package and its simple features, this is not that much of a problem as we saw in the main vignette of SpatialRDD already¹. The more tedious task, however, is that we need to determine which dots on the map are the newly (placebo-)treated ones - after we moved our cutoff. Simple distance calculations to the border do not help because they don’t tell us on which side of the cutoff a point is. Since borders are never straight lines and always have odd shapes, we also cannot just come up with a rule based on x- and y-coordinates. And even if we could it would be very tedious to figure out after every shift which were the exact positions in space below/above the units count as treated. Thus the only generalizeable solution left is to come up with a polygon vector that covers the placebo-treated area. This vector can then be used to do a spatial intersect and assign placebo-treatment and placebo-control for every shifted border. That’s where the SpatialRDD package comes in with a very generic solution that allows the user to carry out a myriad of such placebo checks without having to worry much about details.

library(SpatialRDD); data(cut_off, polygon_full, polygon_treated)
library(tmap)
set.seed(1088) # set a seed to make the results replicable
points_samp.sf <- sf::st_sample(polygon_full, 1000)
points_samp.sf <- sf::st_sf(points_samp.sf) # make it an sf object bc st_sample only created the geometry list-column (sfc)
points_samp.sf$id <- 1:nrow(points_samp.sf) # add a unique ID to each observation

More on placebo bordering

When the border is not approximating a line in space but is curving and bending (i.e. in most cases), “placebo bordering” can be tricky and is not straightforward. The simple subtraction/addition of a specified distance from the distance2cutoff variable is also not a very accurate description of a placebo boundary. On top of that, with such a simple transformation of the distance column we can at best do a placebo check on the “pooled” polynomial specification as the border segments change and thus the assignment of fixed effect categories. A placebo GRD design with an iteration over the boundarypoints is literally impossible in such a case.
For a proper robustness check we thus have to create a new cut-off from which we then can extract the corresponding borderpoints (with discretise_border()) and also assign the border segment categories for fixed effects estimations (with border_segment()).
The shift_border() function can execute three different (affine) transformations at the same time:

"shift" in units of CRS along the x- & y-axis (provided with the option shift = c(dist1, dist2))
"scale" in percent around the centroid, where 0.9 would mean 90%
"rotate" in degrees around the centroid with the standard rotation matrix $$ rotation = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix} $$

Rotate


tm_rotate.sf10 <- shift_border(border = cut_off, operation = "rotate", angle = 10)
tm_rotate.sf25 <- shift_border(border = cut_off, operation = "rotate", angle = 25)
tm_rotate.sf45 <- shift_border(border = cut_off, operation = "rotate", angle = 45)

tm_shape(polygon_full) + tm_polygons() + tm_shape(cut_off) + tm_lines() + 
  tm_shape(tm_rotate.sf10) + tm_lines(col = "red") + 
  tm_shape(tm_rotate.sf25) + tm_lines(col = "red") + 
  tm_shape(tm_rotate.sf45) + tm_lines(col = "red")

Scale


tm_scale.sf.4 <- shift_border(border = cut_off, operation = "scale", scale = .4)
tm_scale.sf.7 <- shift_border(border = cut_off, operation = "scale", scale = .7)
tm_scale.sf1.5 <- shift_border(border = cut_off, operation = "scale", scale = 1.5)

tm_shape(polygon_full) + tm_polygons() + tm_shape(cut_off) + tm_lines() + 
  tm_shape(tm_scale.sf.4) + tm_lines(col = "blue") + 
  tm_shape(tm_scale.sf.7) + tm_lines(col = "red") + 
  tm_shape(tm_scale.sf1.5) + tm_lines(col = "red")

Shift


tm_shift.sf3 <- shift_border(border = cut_off, operation = "shift", shift = c(3000, 0))
tm_shift.sf6 <- shift_border(border = cut_off, operation = "shift", shift = c(6000, 0))
tm_shift.sf_4 <- shift_border(border = cut_off, operation = "shift", shift = c(-4000, 0))


tm_shape(polygon_full) + tm_polygons() + tm_shape(cut_off) + tm_lines() + 
  tm_shape(tm_shift.sf3) + tm_lines(col = "red") + 
  tm_shape(tm_shift.sf6) + tm_lines(col = "red") + 
  tm_shape(tm_shift.sf_4) + tm_lines(col = "blue")  


tm_shift.sf_42 <- shift_border(border = cut_off, operation = "shift", shift = c(-4000, -2000))
tm_shift.sf_44 <- shift_border(border = cut_off, operation = "shift", shift = c(-4000, -4000))

tm_shape(polygon_full) + tm_polygons() + tm_shape(cut_off) + tm_lines() + 
  tm_shape(tm_shift.sf_42) + tm_lines(col = "red") + 
  tm_shape(tm_shift.sf_44) + tm_lines(col = "red") + 
  tm_shape(tm_shift.sf_4) + tm_lines(col = "blue")

From the last shifted line, we can already see that a movement along the x-axis quite often requires also a correction on the y-axis for the cut-off movement to be meaningful. This will be explored in the following section, together with all the other operations.

Full fledged “placebo bordering”

A proper placebo border ideally involves both a shift and a re-scaling for it to be meaningful.


tm_placebo.sf1 <- shift_border(border = cut_off, operation = c("shift", "scale"), shift = c(-5000, -3000), scale = .85)
tm_placebo.sf2 <- shift_border(border = cut_off, operation = c("shift", "scale"), shift = c(4000, 2000), scale = 1.1)
tm_placebo.sf3 <- shift_border(border = cut_off, operation = c("shift", "scale"), shift = c(6000, 3000), scale = 1.2)
tm_shape(polygon_full) + tm_polygons() + tm_shape(cut_off) + tm_lines() + 
  tm_shape(tm_placebo.sf1) + tm_lines(col = "red") +
  tm_shape(tm_placebo.sf2) + tm_lines(col = "red") +
  tm_shape(tm_placebo.sf3) + tm_lines(col = "red")


tm_shift.sf <- shift_border(border = cut_off, operation = c("shift", "rotate", "scale"), 
                              shift = c(-10000, -1000), angle = 0, scale = .9)
tm_shape(cut_off) + tm_lines() + tm_shape(tm_shift.sf) + tm_lines(col = "red")

And the according polygons to assign the treated dummies:

polygon1 <- cutoff2polygon(data = points_samp.sf, cutoff = tm_placebo.sf1, orientation = c("west", "west"), endpoints = c(.8, .2) # corners = 0,
                           #     crs = 32643
                           )
polygon2 <- cutoff2polygon(data = points_samp.sf, cutoff = tm_placebo.sf2, orientation = c("west", "west"), endpoints = c(.8, .2) # corners = 0,
                           #     crs = 32643
                           )
polygon3 <- cutoff2polygon(data = points_samp.sf, cutoff = tm_placebo.sf3, orientation = c("west", "west"), endpoints = c(.8, .2) # corners = 0,
                           #     crs = 32643
                           )

tm_shape(polygon_full) + tm_polygons() + 
  tm_shape(polygon_treated) + tm_polygons(col = "grey") + 
  tm_shape(cut_off) + tm_lines(col = "red") +
  tm_shape(polygon1) + tm_polygons(alpha = .3) +
  tm_shape(polygon2) + tm_polygons(alpha = .3) +
  tm_shape(polygon3) + tm_polygons(alpha = .3)

Even though it is not as straightforward as it sounds because shifting a cutoff by a certain distance up or down and left or right is not quite enough. This is because most of the time borders are not straight lines. Thus we typically also need a shrinkage or enlargement and quite often also a transformation to change the “angle”.↩︎

- More on placebo bordering