# Estimate Heterogeneous Treatment Effects (CATE)

A single average treatment effect can mask important variation and the same intervention may benefit some people substantially and others barely at all. The conditional average treatment effect (CATE) captures this heterogeneity by estimating how the treatment effect changes across the values of individual characteristics.

An interactive causal GAM models both the control surface and the treatment effect surface nonparametrically, allowing the effect of treatment to vary smoothly with confounders rather than assuming a constant additive shift.


# Simulate data

We generate wages where the treatment effect grows with age: older workers benefit more from the training program. The true CATE is `1.0 + 0.08 * age`.


``` python
import numpy as np
import whittaker as wk

# Generate worker covariates
rng = np.random.default_rng(23)
n = 600
age = rng.uniform(25, 55, n)
experience = rng.uniform(0, 20, n)

# Simulate confounded treatment assignment
p_treat = 1 / (1 + np.exp(-(0.05 * age + 0.1 * experience - 3)))
training = rng.binomial(1, p_treat).astype(float)

# Define true CATE and generate wages
cate_true = 1.0 + 0.08 * age
wage = 10 + 0.3 * age + 0.5 * experience + cate_true * training + rng.normal(0, 2, n)
data = {"wage": wage, "training": training, "age": age, "experience": experience}
```


# Fit

Setting `method="interactive"` allows the treatment effect to interact with all confounders. The model estimates separate smooth surfaces for treated and untreated potential outcomes and computes their difference as the CATE.


``` python
model = wk.CausalGAM(
    outcome="wage",
    treatment="training",
    confounders=["age", "experience"],
    method="interactive",
).fit(data, seed=23)
```


# Extract the CATE

`.cate(variable="age")` evaluates the estimated treatment effect on a grid of age values, holding other confounders at representative values. The result shows how the effect changes as age increases.


``` python
cate = model.cate(variable="age")
cate.cate[:5]
```


    array([3.018258  , 3.03995023, 3.06164246, 3.08333471, 3.10502697])


The corresponding age values on the evaluation grid show where each CATE estimate is located.


``` python
cate.x[:5]
```


    array([25.22453211, 25.524614  , 25.82469589, 26.12477779, 26.42485968])


The pointwise confidence band captures uncertainty in the estimated effect at each age value.


``` python
(cate.lower[:5], cate.upper[:5])
```


    (array([1.43870912, 1.48464241, 1.53046117, 1.57615987, 1.6217327 ]),
     array([4.59780688, 4.59525804, 4.59282376, 4.59050954, 4.58832125]))


# Interpret


``` python
model.summary()
```


    'CausalGAM summary\n============================================================\nOutcome:     wage\nTreatment:   training\nConfounders: age, experience\nMethod:      interactive\nN folds:     5\nN obs:       600\n\nTreatment effect:\n  ATE = 4.1136 (SE = 0.0839)\n  95% CI: [3.9492, 4.2779]\n  p-value: 0.0000\n\nCATE model fitted (use .cate() for estimates)'


The CATE curve traces the treatment effect as a function of age. Where the curve slopes upward, older workers gain more from training than younger ones. In this simulation, the effect rises from roughly 3 wage units at age 25 to about 5.4 at age 55, consistent with the true `1.0 + 0.08 * age` relationship. Confidence bands that remain above zero across the full age range confirm that the treatment benefits all subgroups, while the slope of the curve reveals who benefits most.