Implementation of the nonparametric bounds for the average treatment effect (ATE) or average causal effect (ACE) of Balke and Pearl (1997). This is a python version of the R-package bpbounds.

To install the bpbounds package from github execute the following commands in your terminal:

pip install git+https://github.com/pgmeiner/bpbounds.git

The formulas for the Balke-Pearl bounds apply in situations were three random variables X, Y, and Z are related via an instrumental setting. That means Z->X->Y and we can have confounding variables C for X and Y. To be precise We assume the following situation:

• Z is a instrumental variable (i.e. Z and C are independent, Z and X are dependent, and Z,Y are independent when conditioned on {X,C})
• X is a cause of Y (i.e. X->Y)
• C is a unobserved confounder (i.e. X<-C->Y)

The Balke-Pearl bounds give lower and upper bounds for the average causal effect between X and Y when X, Y are binary and Z is binary or ternary:

bplb <= (P(Y|do(X=1)) - P(Y|do(X=0))) <= bpub

It has to be checked before if Z is a instrumental variable. However, the package also calculates the so-called instrumental inequality that is a necessary condition that Z is a instrumental variable and can serve as a test to reject inadequate situations (But such a test does not replace non data-driven arguments that Z is a instrumental variable). When we have additionally a monotonic situation we can sometimes get stronger bounds. Also this is an assumption that can be checked via an inequality but since this is also just a necessary condition it is not sufficient for showing that the relation between X and Y is monotonic (see references for more details).

The main function ace_balke_pearl_bounds expects pandas series as parameters that contain data from X, Y, and Z. It calculates all bounds as well as all inequalities and returns a dictionary that can be printed out by using print_results.

import pandas as pd
from bpbounds.balke_pearl_bounds import ace_balke_pearl_bounds, print_results

x = [1, 0, 1, 0, 1]
y = [1, 1, 0, 0, 1]
z = [1, 0, 1, 0, 1]
df = pd.DataFrame({'x': x, 'y': y, 'z': z})
res = ace_balke_pearl_bounds(df['x'], df['y'], df['z'])

print_results(res)

Output:

Instrumental inequality: True
Causal parameter Lower bound Upper bound
            ACE  0.1666667 0.1666667
   P(Y|do(X=0))  0.5000000 0.5000000
   P(Y|do(X=1))  0.6666667 0.6666667
            CRR  1.3333333 1.3333333
Monotonicity inequality: True
Causal parameter Lower bound Upper bound
            ACE  0.1666667 0.1666667
   P(Y|do(X=0))  0.5000000 0.5000000
   P(Y|do(X=1))  0.6666667 0.6666667
            CRR  1.3333333 1.3333333

The results dictionary res contains the following entries:

• nzcats: number of categories in z
• inequality: True if instrumental inequality is fulfilled. If False then we should not trust the lower and upper bounds at all (since we are not in an instrumental setting).
• bplb: lower bound for ACE
• bpup: upper bound for ACE
• p11low: lower bound for P(Y|do(X=1))
• p11upp: upper bound for P(Y|do(X=1))
• p10low: lower bound for P(Y|do(X=0))
• p10upp: upper bound for P(Y|do(X=0))
• crrlb: lower bound for Causal Risk Ratio (CRR) := P(Y=1|do(X=1)) / P(Y=1|do(X=0))
• crrub: upper bound for Causal Risk Ratio
• monoinequality: True if monotonicity inequality is fulfilled. If False we are not in a monotonic situation and the following entries are `0
• monobplb: monotonic lower bound for ACE
• monobpub: monotonic upper bound for ACE
• monop11low: monotonic lower bound for P(Y|do(X=1))
• monop11upp: monotonic upper bound for P(Y|do(X=1))
• monop10low: monotonic lower bound for P(Y|do(X=0))
• monop10upp: monotonic upper bound for P(Y|do(X=0))
• monocrrlb: monotonic lower bound for Causal Risk Ratio
• monocrrub: monotonic upper bound for Causal Risk Ratio

Peter Gmeiner (maintainer, peter.gmeiner@algobalance.com).

Balke A, Pearl J. Bounds on Treatment Effects from studies with imperfect compliance. Journal of the American Statistical Association, 1997, 92, 439, 1171-1176, doi: 10.1080/01621459.1997.10474074.