Double Machine Studying Simplified: Half 1 — Primary Causal Inference Functions | by Jacob Pieniazek | Jul, 2023

DML Functions

Software 1: Converging in direction of Exogeneity/CIA/Ignorability in our Therapy given Non-Experimental/Observational Knowledge

Recall that we mentioned how within the absence of randomized experimental knowledge we should management for all potential confounders to make sure we acquire exogeneity in our therapy of curiosity. In different phrases, once we management for all potential confounders, our therapy is “nearly as good as randomly assigned”. There are two major issues that also persist right here:

1. It’s tough, and unattainable in some instances, to really know the entire confounders and, moreover, to acquire the info for all these confounders. Solutioning this includes sturdy institutional data of the info producing course of, cautious building of the causal mannequin (i.e., constructing a DAG whereas evaluating potential confounders and avoiding colliders), and/or exploiting quasi-experimental designs.

2. If we do care for level 1, we nonetheless have to regulate for the proper purposeful type of confounding, together with interactions and higher-order phrases, when using a parametric mannequin (corresponding to within the regression framework). Merely together with linear phrases in a regression might not sufficiently management for the confounding. That is the place DML steps in; it could actually flexibly partial out the confounding in a extremely non-parametric trend. That is notably useful in saving the info scientist the difficulty of instantly modeling the purposeful types of confounding, and permits extra consideration to be directed in direction of figuring out and measuring the confounders. Let’s see how this works!

Suppose, as a extremely stylized instance, we work for an e-commerce web site and we’re tasked with estimating the ATE of an people time spent on the web site on their buy quantity, or gross sales, prior to now month. Nonetheless, additional assume we solely have observational knowledge to work with, however now we have measured all potential confounders (these variables that affect each time spent on the web site and gross sales). Let the info producing course of be as follows (observe that each one values are chosen arbitrarily):

import numpy as np
import pandas as pd
# Pattern Dimension
N = 100_000
# Noticed Confounders (Age, Variety of Social Media Accounts, & Years Member on Web site)
age = np.random.randint(low=18,excessive=75,measurement=N)
num_social_media_profiles = np.random.alternative([0,1,2,3,4,5,6,7,8,9,10], measurement = N)
yr_membership = np.random.alternative([0,1,2,3,4,5,6,7,8,9,10], measurement = N)
# Extra Covariates (Arbitrary Z)
Z = np.random.regular(loc=50, scale = 25, measurement = N)
# Error Phrases
ε_1 = np.random.regular(loc=20,scale=5,measurement=N)
ε_2 = np.random.regular(loc=40,scale=15,measurement=N)
# Therapy DGP (T) - Hrs spent on web site in previous month
time_on_website = np.most( np.random.regular(loc=10, scale=5, measurement=N) 
- 0.01*age 
- 0.001*age**2 
+ num_social_media_profiles 
- 0.01 * num_social_media_profiles**2
- 0.01*(age * num_social_media_profiles)
+ 0.2 * yr_membership
+ 0.001 * yr_membership**2
- 0.01 * (age * yr_membership)
+ 0.2 * (num_social_media_profiles * yr_membership)
+ 0.01 * (num_social_media_profiles * np.log(age) * age * yr_membership**(1/2))
+ ε_1
,0)
# Consequence DGP (y) - Gross sales in previous month
gross sales = np.most( np.random.regular(loc=25, scale=10, measurement=N)
+  5 * time_on_website # Simulated ATE = $5
- 0.1*age 
- 0.001*age**2 
+ 8 * num_social_media_profiles 
- 0.1 * num_social_media_profiles**2
- 0.01*(age * num_social_media_profiles)
+ 2 * yr_membership
+ 0.1 * yr_membership**2
- 0.01 * (age * yr_membership)
+ 3 * (num_social_media_profiles * yr_membership)
+ 0.1 * (num_social_media_profiles * np.log(age) * age * yr_membership**(1/2))
+ 0.5 * Z
+ ε_2
,0)
collider = np.random.regular(loc=100, scale=50, measurement=N) + 2*gross sales + 7*time_on_website
df = pd.DataFrame(np.array([sales,time_on_website,age,num_social_media_profiles,yr_membership,Z]).T
,columns=["sales","time_on_website","age","num_social_media_profiles","yr_membership","Z"])

By building, our therapy of curiosity (hours spent on the web site prior to now month) and our end result (gross sales prior to now month) have the next confounders: Age, Variety of Social Media Accounts, & Years Member of Web site. Moreover, we are able to see that the constructed floor fact for the ATE is $5 (outlined within the arbitrary and non-linear DGP for gross sales within the code above above). That’s, on common, for each extra hour the person spends on the web site, they spend a further $5. Word, we additionally embrace a collider variable (a variable that’s influenced by each time spent on the web site and gross sales), which will likely be utilized for demonstration beneath on how this biases the ATE.

To show the power of DML to flexibly partial out the extremely non-linear confounding, we’ll run the 4 following fashions:

1. Naïve OLS of gross sales (y) on hours spent on the web site (T)

2. A number of OLS of gross sales (y) on hours spent on the web site (T) and linear phrases of the entire confounders

3. OLS using DML residualization process outlined in eq. (5)

4. OLS using DML residualization process, together with collider variable

The code of that is as follows:

import statsmodels.method.api as smf
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.model_selection import cross_val_predict
# 1 - Naive OLS
naive_regression = smf.ols(method=’gross sales ~ 1 + time_on_website’,knowledge=df).match()
# 2 - A number of OLS
multiple_regression = smf.ols(method=’gross sales ~ 1 + time_on_website + age + num_social_media_profiles + yr_membership’,knowledge=df).match()
# 3 - DML Process
M_sales = GradientBoostingRegressor()
M_time_on_website = GradientBoostingRegressor()
residualized_sales = df["sales"] - cross_val_predict(M_sales, df[["age","num_social_media_profiles","yr_membership"]], df[‘sales’], cv=3)
residualized_time_on_website = df[‘time_on_website’] - cross_val_predict(M_time_on_website, df[["age","num_social_media_profiles","yr_membership"]], df[‘time_on_website’], cv=3)
df[‘residualized_sales’] = residualized_sales
df[‘residualized_time_on_website’] = residualized_time_on_website
DML_model = smf.ols(method=’residualized_sales ~ 1 + residualized_time_on_website’, knowledge = df).match()
# 4 - DML Process w/ Collider
M_sales = GradientBoostingRegressor()
M_time_on_website = GradientBoostingRegressor()
residualized_sales = df["sales"] - cross_val_predict(M_sales, df[["age","num_social_media_profiles","yr_membership","collider"]], df['sales'], cv=3)
residualized_time_on_website = df['time_on_website'] - cross_val_predict(M_time_on_website, df[["age","num_social_media_profiles","yr_membership", "collider"]], df['time_on_website'], cv=3)
df['residualized_sales'] = residualized_sales
df['residualized_time_on_website'] = residualized_time_on_website
DML_model_collider = smf.ols(method='residualized_sales ~ 1 + residualized_time_on_website', knowledge = df).match()

with the corresponding outcomes (see code in appendix for creating this desk):

Recall our simulated supply of fact for the ATE is $5. Discover that the one mannequin that is ready to seize this worth is the DML process! We are able to see that the naïve mannequin has a major constructive bias within the estimate, whereas controlling just for linear phrases of the confounders within the a number of regression barely reduces this bias. Moreover, the DML process w/ a collider demonstrates a detrimental bias; this detrimental affiliation between gross sales and our therapy that arises from controlling for the collider could be loosely demonstrated/noticed by fixing for gross sales in our collider DGP as such:

collider = np.random.regular(loc=100, scale=50, measurement=N) + 2*gross sales + 7*time_on_website
# Word the detrimental relationship between gross sales and time_on_website right here
gross sales = (collider - np.random.regular(loc=100, scale=50, measurement=N) - 7*time_on_website)/2

These outcomes show the unequivocal energy of utilizing versatile, non-parametric ML fashions within the DML process for residualizing out the confounding! Fairly satisfying, no? DML removes the need to appropriately mannequin the purposeful types of the confounding DGP (given the entire confounders are managed for)!

Software 2: Enhancing Precision & Statistical Energy in Experimental Knowledge (Randomized Managed Trial’s (RCTs) or A/B Assessments)

It’s a frequent false impression that if one run’s an experiment with a giant sufficient pattern measurement, one can acquire enough statistical power to precisely measure the therapy of curiosity. Nonetheless, one generally neglected element in figuring out statistical energy in an experiment, and in the end the precision within the ATE estimate, is the variation within the end result you are attempting measure.

For instance, suppose we’re concerned about measuring the influence of a particular commercial on an people buy quantity, and we anticipate the impact to be small, however non-trivial — say an ATE of $5. Nonetheless, suppose the usual deviation in particular person gross sales may be very giant… maybe, within the $100s and even $1000s. On this case, it might be tough to precisely seize the ATE given this excessive variation —that’s, we might acquire very low precision (giant customary errors) in our estimate. Nonetheless, capturing this ATE of $5 could also be economically vital (if we run the experiment on 100,000 households, this may quantity to $500,000). That is the place DML can come to the rescue. Earlier than we show this in motion, let’s first go to the method for the usual error of our ATE estimate from the straightforward regression in equation (1):

Right here we observe that the usual error of our estimate is instantly influenced by the dimensions of our residuals (ε). What does this inform us then? If our therapy is randomized, we are able to embrace covariates in a a number of OLS or DML process, to not acquire exogeneity, however to scale back the variation in our end result. Extra particularly, we are able to embrace variables which are sturdy predictors of our end result to scale back the residuals and, consequently, the usual error of our estimate. Let’s check out this in motion — suppose the next DGP:

import numpy as np
import pandas as pd
N = 100_000
# Covariates (X)
age = np.random.randint(low=18,excessive=75,measurement=N)
num_social_media_profiles = np.random.alternative([0,1,2,3,4,5,6,7,8,9,10], measurement = N)
yr_membership = np.random.alternative([0,1,2,3,4,5,6,7,8,9,10], measurement = N)
Z = np.random.regular(loc=50, scale = 25, measurement = N)
# Error Time period
ε = np.random.regular(loc=40,scale=15,measurement=N)
# Randomized Therapy (T) (50% cut up)
advertisement_exposure = np.random.alternative([0,1],measurement=N,p=[.5,.5])
# Consequence (y)
gross sales = np.most( np.random.regular(loc=500, scale=25, measurement=N)
+  5 * advertisement_exposure # Floor Reality ATE of $5
- 10*age 
- 0.05*age**2 
+ 15 * num_social_media_profiles 
- 0.01 * num_social_media_profiles**2
- 0.5*(age * num_social_media_profiles)
+ 20 * yr_membership
+ 0.5 * yr_membership**2
- 0.8 * (age * yr_membership)
+ 5 * (num_social_media_profiles * yr_membership)
+ 0.8 * (num_social_media_profiles * np.log(age) * age * yr_membership**(1/2))
+ 15 * Z
+ 2 * Z**2
+ ε
,0)
df = pd.DataFrame(np.array([sales,advertisement_exposure,age,num_social_media_profiles,yr_membership, Z]).T,columns=["sales","advertisement_exposure","age","num_social_media_profiles","yr_membership","Z"])

Right here once more, we artificially simulate our floor fact ATE of $5. This time, nonetheless, we generate gross sales such that now we have a really giant variance, thus making it tough to detect the $5 ATE.

To show how the inclusion of covariates which are sturdy predictors of our end result within the DML process enormously enhance the precision of our ATE estimates, we’ll run the next 3 fashions:

1. Naïve OLS of gross sales (y) on randomized publicity to commercial (T)

2. A number of OLS of gross sales (y) on randomized publicity to commercial (T) and linear phrases of the entire gross sales predictors

3. OLS using DML residualization process outlined in eq. (5)

The code is as follows:

import statsmodels.method.api as smf
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.model_selection import cross_val_predict
naive_regression = smf.ols(method='gross sales ~ 1 + advertisement_exposure',knowledge=df).match()
multiple_regression = smf.ols(method='gross sales ~ 1 + advertisement_exposure + age + num_social_media_profiles + yr_membership + Z',knowledge=df).match()
# DML Process
M_sales = GradientBoostingRegressor()
M_advertisement_exposure = GradientBoostingClassifier() # Word binary therapy 
residualized_sales = df["sales"] - cross_val_predict(M_sales, df[["age","num_social_media_profiles","yr_membership","Z"]], df['sales'], cv=3)
residualized_advertisement_exposure = df['advertisement_exposure'] - cross_val_predict(M_advertisement_exposure, df[["age","num_social_media_profiles","yr_membership", "Z"]], df['advertisement_exposure'], cv=3, technique = 'predict_proba')[:,0]
df['residualized_sales'] = residualized_sales
df['residualized_advertisement_exposure'] = residualized_advertisement_exposure
DML_model = smf.ols(method='residualized_sales ~ 1 + residualized_advertisement_exposure', knowledge = df).match()

With the corresponding outcomes of those fashions (see code in appendix for creating this desk):

First, observe that b/c therapy was randomly assigned, there isn’t any true confounding that’s occurring above. The poor estimates of the ATE in (1) and (2) are the direct results of imprecise estimates (see the big customary error’s within the parenthesis). Discover how the usual error will get smaller (precision growing) as we transfer from (1)-(3), with the DML process having probably the most exact estimate. Draw your consideration to the “Residual Std. Error” row outlined within the purple field above. We are able to see how the DML process was in a position to enormously cut back the variation within the ATE mannequin residuals by way of partialling out the variation that was in a position to be learnt (non-parametrically) from the predictors within the ML mannequin of our end result, gross sales. Once more, on this instance, we see DML being the one mannequin to acquire the true ATE!

These outcomes show the good thing about utilizing DML in an experimental setting to extend statistical energy and precision of 1’s ATE estimate. Particularly, this may be utilized in RCT or A/B testing settings the place the variation within the end result may be very giant and/or one is scuffling with reaching exact estimates and one has entry to sturdy predictors of the result of curiosity.

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