# Heterogeneity

author: “Vincent Arel-Bundock”

This short vignette illustrates how to use recursive partitioning to explore treatment effect heterogeneity. This exercise inspired by Scholbeck et al. 2022 and their concept of “cATE”.

As pointed out in other vignettes, most of the quantities estimated by the `marginaleffects`

package are “conditional”, in the sense that they vary based on the values of all the predictors in our model. For instance, consider a Poisson regression that models the number of hourly bike rentals in Washington, DC:

We can use the `comparisons()`

function to estimate how the predicted outcome changes for a 5 celsius increase in temperature:

```
cmp <- comparisons(mod, variables = list(temp = 5))
cmp
```

```
Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
423 55.3 7.65 <0.001 45.5 315 531
384 51.8 7.40 <0.001 42.7 282 485
320 40.1 8.00 <0.001 49.5 242 399
360 43.4 8.29 <0.001 52.9 275 445
370 45.7 8.10 <0.001 50.7 281 460
--- 721 rows omitted. See ?avg_comparisons and ?print.marginaleffects ---
418 48.3 8.66 <0.001 57.6 323 513
426 50.0 8.51 <0.001 55.7 328 524
366 44.0 8.33 <0.001 53.5 280 453
304 40.6 7.50 <0.001 43.9 225 384
343 40.5 8.47 <0.001 55.2 264 422
Term: temp
Type: response
Comparison: +5
Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, count, season, weekday, weather, temp
```

The output printed above includes 727 rows: 1 for each of the rows in the original `bikes`

dataset. Indeed, since the “effect” of a 5 unit increase depends on the values of covariates, different unit of observation will typically be associated with different contrasts.

In such cases, a common strategy is to compute an average difference, as described in the G-Computation vignette:

`avg_comparisons(mod, variables = list(temp = 5))`

```
Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
689 64.1 10.7 <0.001 87.1 564 815
Term: temp
Type: response
Comparison: mean(+5)
Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
```

Alternatively, one may be interested in exploring heterogeneity in effect sizes in different subsets of the data. A convenient way to achieve this is to use the `ctree`

function of the `partykit`

package. This function allows us to use recursive partitioning (conditional inference trees) to find subspaces with reasonably homogenous estimates, and to report useful graphical and textual summaries.

Imagine that we are particularly interested in how the effect of temperature on bike rentals varies based on day of the week and season:

```
tree <- ctree(
estimate ~ weekday + season,
data = cmp,
control = ctree_control(maxdepth = 2)
)
```

Now we can use the `plot()`

function to draw the distributions of estimates for the effect of an increase of 5C on bike rentals, by week day and season:

`plot(tree)`

To obtain conditional average estimates for each subspace, we first use the `predict()`

function in order to place each observation in the dataset in its corresponding “bucket” or “node”. Then, we use the `by`

argument to indicate that `comparisons()`

should compute average estimates for each of the nodes in the tree:

```
dat <- transform(bikes, nodeid = predict(tree, type = "node"))
comparisons(mod,
variables = list(temp = 5),
newdata = dat,
by = "nodeid")
```

```
nodeid Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
3 352 37.2 9.46 <0.001 68.1 279 425
4 433 42.9 10.08 <0.001 76.9 348 517
6 757 70.4 10.74 <0.001 87.0 619 895
7 841 80.9 10.40 <0.001 81.8 683 1000
Term: temp
Type: response
Comparison: mean(+5)
Columns: term, contrast, nodeid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
```

The four `nodeid`

values correspond to the terminal nodes in this tree:

`print(tree)`

```
Model formula:
estimate ~ weekday + season
Fitted party:
[1] root
| [2] season in winter
| | [3] weekday in Monday, Tuesday, Sunday: 351.902 (n = 80, err = 248952.3)
| | [4] weekday in Wednesday, Thursday, Friday, Saturday: 432.617 (n = 101, err = 461053.1)
| [5] season in spring, summer, fall
| | [6] season in spring, fall: 756.522 (n = 362, err = 7548395.3)
| | [7] season in summer: 841.324 (n = 188, err = 2116175.9)
Number of inner nodes: 3
Number of terminal nodes: 4
```