Quantile Treatment Effects in R

What are Quantile Treatment Effects?

The Average Treatment Effect (ATE) or Average Treatment Effect on the Treated (ATT) summarizes the causal effect of a treatment by a single number: the mean difference in outcomes between treated and untreated units. This is often a natural quantity, but it can miss important heterogeneity. If a job training program substantially raises earnings at the bottom of the distribution while having little effect at the top, the ATE obscures this. If a minimum wage increase compresses the wage distribution, an average effect conceals the compression.

The Quantile Treatment Effect at level $\tau$ is $QTE(\tau) = Q_{Y(1)}(\tau) - Q_{Y(0)}(\tau)$ , where $Q_{Y(d)}(\tau)$ is the $\tau$ -th quantile of the potential outcome distribution under treatment status $d$ . Mapping $QTE(\tau)$ across $\tau \in (0,1)$ traces how the treatment shifts the entire outcome distribution. The Quantile Treatment Effect on the Treated ( $QTT$ ) conditions on the treated group.

This vignette demonstrates unc_qte(), the qte package’s estimator for cross-sectional settings (no panel data required). For DiD-based estimators and staggered treatment adoption, see vignette("panel-estimators") and vignette("staggered-adoption").

library(qte)
library(ggplot2)
set.seed(42)
data(lalonde)

Random assignment

Under random assignment, no covariate adjustment is needed: the unconditional quantiles of the treated and control outcome distributions identify the QTE directly. We use the experimental Lalonde data (lalonde.exp), where treatment was randomly assigned.

res_exp <- unc_qte(
  yname  = "re78",
  dname  = "treat",
  data   = lalonde.exp,
  target = "qte",
  probs  = seq(0.1, 0.9, 0.1),
  biters = 50
)
summary(res_exp)
#> 
#> Overall ATE:  
#>       ATE    Std. Error     [ 95%  Conf. Int.]  
#>  1794.343      655.1211   510.3294    3078.357 *
#> 
#> 
#> QTE:
#>  Tau      QTE Std. Error [ 95% Simult.  Conf. Band]  
#>  0.1    0.000     0.0000         0.0000      0.0000  
#>  0.2    0.000   127.5450      -249.9836    249.9836  
#>  0.3  929.884   361.5139       221.3297   1638.4382 *
#>  0.4 1176.944   950.4180      -685.8414   3039.7286  
#>  0.5 1091.468   961.2442      -792.5361   2975.4718  
#>  0.6 1466.690   956.3520      -407.7249   3341.1059  
#>  0.7 1810.696  1047.2267      -241.8308   3863.2223  
#>  0.8 2300.454  1161.5595        23.8390   4577.0686 *
#>  0.9 2856.068  1811.4848      -694.3767   6406.5134  
#> ---
#> Signif. codes: `*' confidence band does not cover 0

autoplot(res_exp, ylab = "QTE (earnings, 1978)")

The confidence band indicates that the QTE is positive across much of the distribution, with the effect somewhat larger in the lower quantiles.

Unconfoundedness

When treatment is not randomly assigned, the unconfoundedness assumption (conditional independence of potential outcomes given covariates) provides identification. We use the observational Lalonde data (lalonde.psid), which combines the treated group from the experiment with a comparison group from the PSID.

unc_qte() supports three estimation methods:

"ipw" — inverse probability weighting (propensity score re-weighting)
"or" — outcome regression (conditional quantile regression)
"aipw" — augmented IPW (doubly robust; recommended)

xf <- ~ age + I(age^2) + education + black + hispanic + married + nodegree

res_psid <- unc_qte(
  yname      = "re78",
  dname      = "treat",
  data       = lalonde.psid,
  xformla    = xf,
  est_method = "aipw",
  target     = "qte",
  probs      = seq(0.1, 0.9, 0.1),
  biters     = 100
)
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
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#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
summary(res_psid)
#> 
#> Overall ATE:  
#>        ATE    Std. Error     [ 95%  Conf. Int.]  
#>  -13194.78      1560.465  -16253.24   -10136.33 *
#> 
#> 
#> QTE:
#>  Tau         QTE Std. Error [ 95% Simult.  Conf. Band]  
#>  0.1     25.8602   580.3426      -1111.590    1163.311  
#>  0.2  -6409.5999  1563.3724      -9473.753   -3345.446 *
#>  0.3  -8519.7248  1940.1842     -12322.416   -4717.034 *
#>  0.4 -11106.7654  2292.5445     -15600.070   -6613.461 *
#>  0.5 -12684.5156  2788.0920     -18149.075   -7219.956 *
#>  0.6 -13532.8367  2706.0240     -18836.546   -8229.127 *
#>  0.7 -14171.4789  2024.7467     -18139.910  -10203.048 *
#>  0.8 -18275.7168  1887.3424     -21974.840  -14576.594 *
#>  0.9 -22572.0260  3364.3645     -29166.059  -15977.993 *
#> ---
#> Signif. codes: `*' confidence band does not cover 0

autoplot(res_psid, ylab = "QTE (earnings, 1978)")

QTT under unconfoundedness

Setting target = "qtt" estimates the Quantile Treatment Effect on the Treated — the distributional effect for the subpopulation that actually received treatment.

res_qtt <- unc_qte(
  yname      = "re78",
  dname      = "treat",
  data       = lalonde.psid,
  xformla    = xf,
  est_method = "aipw",
  target     = "qtt",
  probs      = seq(0.1, 0.9, 0.1),
  biters     = 100
)
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
#> Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm):
#> collapsing to unique 'x' values
summary(res_qtt)
#> 
#> Overall ATT:  
#>        ATT    Std. Error     [ 95%  Conf. Int.]  
#>  -4685.583       891.021  -6431.952   -2939.214 *
#> 
#> 
#> QTT:
#>  Tau         QTT Std. Error [ 95% Simult.  Conf. Band]  
#>  0.1      0.0001   133.4222      -261.5027    261.5028  
#>  0.2  -1002.7420   856.4152     -2681.2850    675.8010  
#>  0.3  -3427.3187  1702.8307     -6764.8055    -89.8318 *
#>  0.4  -5022.1405  1291.7066     -7553.8389  -2490.4421 *
#>  0.5  -4611.6428  1055.6680     -6680.7140  -2542.5715 *
#>  0.6  -5229.1454  1359.9888     -7894.6745  -2563.6164 *
#>  0.7  -5588.8001  1616.7891     -8757.6484  -2419.9517 *
#>  0.8  -7030.0171  1641.2890    -10246.8844  -3813.1498 *
#>  0.9 -10540.3830  2301.9302    -15052.0833  -6028.6828 *
#> ---
#> Signif. codes: `*' confidence band does not cover 0

autoplot(res_qtt, ylab = "QTT (earnings, 1978)")

Reading the output

summary() prints the overall ATT, the QTE/QTT at each quantile, and standard errors with confidence intervals. autoplot() returns a ggplot object so you can add layers:

autoplot(res_qtt) +
  ggplot2::labs(title = "QTT — Lalonde (observational)",
                subtitle = "AIPW with pre-treatment covariates")

By default autoplot() shows a uniform confidence band (cband = TRUE), which provides simultaneous coverage over all quantile levels — a stronger guarantee than pointwise intervals. Pass cband = FALSE for pointwise intervals instead.

Standard errors are computed via the empirical bootstrap. The biters argument controls the number of iterations (default 100). Parallel computation is available via the cl argument.

What are Quantile Treatment Effects?

Random assignment

Unconfoundedness

QTT under unconfoundedness

Reading the output

References