Activity 12: Compliance status and local average treatment effects

Activity 12: Compliance status and local average treatment effects#

2025-04-03

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns

Part 1: Monotonicity violation#

Here are the four compliance types in a study:

  • Always-taker: takes treatment regardless of the instrument encouragement

  • Never-taker: does not take treatment regardless of the instrument encouragement

  • Complier: takes treatment only when the instrument encourages it

  • Defier: takes treatment only when NOT encouraged by the instrument

Discuss with folks around you which scenario corresponds to a likely violation of the monotonicity assumption (no defiers):

A. Some patients offered a new medication (Z=1) refuse to take it (T=0), while others not offered the medication obtain it through alternative means.

B. Some consumers who see repeated targeted ads for a streaming service (Z=1) become annoyed by the aggressive marketing and decide not to subscribe (T=0), whereas they might have subscribed if they hadn’t been bombarded with ads.

C. Some homeowners given rebates for energy-efficient appliances (Z=1) still choose not to upgrade (T=0), while others upgrade their appliances even without rebates.

D. Some employees who aren’t offered a retirement savings workshop (Z=0) contribute to their 401(k) plan (T=1), while some who are offered the workshop still don’t contribute.

Your response: pollev.com/tliu

Part 2: Calculating compliance types and LATE#

Suppose for our smart thermostat study, we have the following data:

  • \(Z\): household was given a smart thermostat

  • \(T\): used energy-efficient thermostat settings

  • \(Y\): household energy consumption

We collected data on 1000 households:

Group

Z=1

Z=0

Total

T=1

300

100

400

T=0

200

400

600

Total

500

500

1000

 
thermo_df = pd.DataFrame({
    'Z=1': [300, 200],
    'Z=0': [100, 400],
}, index=['T=1', 'T=0'])

thermo_df.head()
Z=1 Z=0
T=1 300 100
T=0 200 400

2.1#

We assume that monotonicity holds, so there are no defiers.

We can calculate the proportions of each compliance type as follows:

  • Always-takers are people who take treatment even without encouragement:

\[P(\text{Always-taker}) = P(T=1|Z=0)\]

  • Never-takers are people who don’t take treatment even with encouragement:

\[P(\text{Never-taker}) = P(T=0|Z=1)\]

  • Defiers: 0, based on monotonicity

Since all of the probabilities need to sum to 1, we can therefore calculate the proportion of compliers as follows:

\[P(\text{Complier}) = 1 - P(\text{Always-taker}) - P(\text{Never-taker})\]

The proportion of compliers is also equivalent to:

\[P(\text{Complier}) = E[T=1|Z=1] - E[T=1|Z=0]\]

Compute, either manually or by using the thermo_df dataframe, the proportions of each compliance type in our data:

  • Always-takers: TODO

  • Never-takers: TODO

  • Compliers: TODO

# TODO your calculations here

2.2#

We saw earlier that the intent-to-treat (ITT) effect is as follows:

\[ \text{ITT} = E[Y|Z=1] - E[Y|Z=0] \]

And we have just defined the local average treatment effect (LATE) as:

\[ \text{LATE} = \frac{E[Y|Z=1] - E[Y|Z=0]}{E[T|Z=1] - E[T|Z=0]} \]

Suppose that the intent-to-treat effect is -10 kilowatt-hours (kWh). What is the local average treatment effect (LATE)?

Your response: pollev.com/tliu

# TODO your calculations here

2.3#

From lecture last time we saw that under linear outcomes, the ATE is identified in the exact same way as the LATE:

\[ \text{ATE} = \frac{E[Y|Z=1] - E[Y|Z=0]}{E[T|Z=1] - E[T|Z=0]} \]

Researchers often choose whether to interpret their instrumental variable results as a LATE or ATE:

  • ATE: \(E[Y(1) - Y(0)]\)

  • LATE: \(E[Y(1) - Y(0) \mid \text{compliers}]\)

Which causal quantity seems more useful in the context of the smart thermostat study? Feel free to discuss with folks around you.

Your response: pollev.com/tliu

Contents