Entropy balancing versus inverse probability tilting

library(WeightIt)
library(tidyverse)

data("lalonde", package = "cobalt")

W_ipt <- weightit(treat ~ age + educ + married +
                  nodegree + re74, data = lalonde,
                method = "ipt", estimand = "ATT")

W_ebal <- weightit(treat ~ age + educ + married +
                  nodegree + re74, data = lalonde,
                method = "ebal", estimand = "ATT")

summary(W_ipt)

                  Summary of weights

- Weight ranges:

          Min                                 Max
treated 1.                 ||               1.   
control 0.017 |---------------------------| 2.263

- Units with the 5 most extreme weights by group:
                                      
             1     2     3     4     5
 treated     1     1     1     1     1
           410   404   224   111    84
 control 1.464 1.485 1.576 1.743 2.263

- Weight statistics:

        Coef of Var   MAD Entropy # Zeros
treated       0.000 0.000   0.000       0
control       0.839 0.707   0.341       0

- Effective Sample Sizes:

           Control Treated
Unweighted  429.       185
Weighted    252.12     185

summary(W_ebal)

                  Summary of weights

- Weight ranges:

         Min                                 Max
treated 1.         ||                      1.   
control 0.04 |---------------------------| 5.247

- Units with the 5 most extreme weights by group:
                                      
             1     2     3     4     5
 treated     1     1     1     1     1
           410   404   224   111    84
 control 3.396 3.443 3.655 4.043 5.247

- Weight statistics:

        Coef of Var   MAD Entropy # Zeros
treated       0.000 0.000   0.000       0
control       0.839 0.707   0.341       0

- Effective Sample Sizes:

           Control Treated
Unweighted  429.       185
Weighted    252.12     185

cobalt::bal.tab(W_ipt)

Balance Measures
               Type Diff.Adj
prop.score Distance   0.0164
age         Contin.  -0.0000
educ        Contin.   0.0000
married      Binary  -0.0000
nodegree     Binary  -0.0000
re74        Contin.  -0.0000

Effective sample sizes
           Control Treated
Unadjusted  429.       185
Adjusted    252.12     185

cobalt::bal.tab(W_ebal)

Balance Measures
            Type Diff.Adj
age      Contin.        0
educ     Contin.        0
married   Binary       -0
nodegree  Binary       -0
re74     Contin.       -0

Effective sample sizes
           Control Treated
Unadjusted  429.       185
Adjusted    252.12     185

weight_comp <- tibble(
  ipt     = W_ipt$weights,
  entropy = W_ebal$weights,
  treat   = lalonde$treat
)

weight_comp |>
  filter(treat == 0) |>
  ggplot() +
  aes(x = entropy, y = ipt) |>
  geom_point() +
  labs(x = "Entropy balancing",
       y = "Inverse probability tilting (logit link)",
       title = "Comparison group ATT weights")

The weights are perfectly correlated:

weight_comp |>
  filter(treat == 0) |>
  cor.test(~ ipt + entropy, data = _)

    Pearson's product-moment correlation

data:  ipt and entropy
t = 980570508, df = 427, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 1 1
sample estimates:
cor 
  1 

Sample sizes:

lalonde |>
  group_by(treat) |>
  tally()

weight_comp |>
  filter(treat == 0) |>
  pivot_longer(
    cols = c(entropy, ipt),
    values_to = "weight",
    names_to = "method"
  ) |>
  group_by(method) |>
  summarise(
    mean = mean(weight),
    sum = sum(weight),
    sd = sd(weight),
    ESS = ESS(weight)
  )

So the entropy balancing weights sum to the comparison group sample size and the IPT weights sum to the intervention group sample size.