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Validating the Causal Impact of the Synthetic Control Method | by Ryan O’Sullivan | Jun, 2024

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Causal AI, exploring the combination of causal reasoning into machine studying

Photograph by Irina Inga on Unsplash

Welcome to my sequence on Causal AI, the place we’ll discover the combination of causal reasoning into machine studying fashions. Anticipate to discover a lot of sensible functions throughout totally different enterprise contexts.

Within the final article we coated measuring the intrinsic causal affect of your advertising campaigns. On this article we’ll transfer onto validating the causal impression of the artificial controls.

In case you missed the final article on intrinsic causal affect, test it out right here:

On this article we’ll deal with understanding the artificial management methodology and exploring how we will validate the estimated causal impression.

The next elements shall be coated:

What’s the artificial management methodology?What problem does it attempt to overcome?How can we validate the estimated causal impression?A Python case research utilizing life like google development knowledge, demonstrating how we will validate the estimated causal impression of the artificial controls.

The total pocket book will be discovered right here:

What’s it?

The artificial management methodology is a causal approach which can be utilized to evaluate the causal impression of an intervention or therapy when a randomised management trial (RCT) or A/B check was not potential. It was initially proposed in 2003 by Abadie and Gardezabal. The next paper features a nice case research that will help you perceive the proposed methodology:

https://net.stanford.edu/~jhain/Paper/JASA2010.pdf

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Let’s cowl a few of the fundamentals ourselves… The artificial management methodology creates a counterfactual model of the therapy unit by making a weighted mixture of management models that didn’t obtain the intervention or therapy.

Handled unit: The unit which receives the intervention.Management models: A set of comparable models which didn’t obtain the intervention.Counterfactual: Created as a weighted mixture of the management models. Intention is to search out weights for every management unit that end in a counterfactual which intently matches the handled unit within the pre-intervention interval.Causal impression: The distinction between the post-intervention therapy unit and counterfactual.

If we wished to actually simplify issues, we might consider it as linear regression the place every management unit is a characteristic and the therapy unit is the goal. The pre-intervention interval is our practice set, and we use the mannequin to attain our post-intervention interval. The distinction between the precise and predicted is the causal impression.

Under are a pair examples to convey it to life once we may think about using it:

When working a TV advertising marketing campaign, we’re unable to randomly assign the viewers into these that may and might’t see the marketing campaign. We might nonetheless, rigorously choose a area to trial the marketing campaign and use the remaining areas as management models. As soon as we’ve got measured the impact the marketing campaign may very well be rolled out to different areas. That is usually referred to as a geo-lift check.Coverage adjustments that are introduced into some areas however not others — For instance an area council could convey a coverage grow to be pressure to cut back unemployment. Different areas the place the coverage wasn’t in place may very well be used as management models.

What problem does it attempt to overcome?

After we mix high-dimensionality (numerous options) with restricted observations, we will get a mannequin which overfits.

Let’s take the geo-lift instance for instance. If we use weekly knowledge from the final 12 months as our pre-intervention interval, this offers us 52 observations. If we then resolve to check our intervention throughout international locations in Europe, that may give us an commentary to characteristic ratio of 1:1!

Earlier we talked about how the artificial management methodology may very well be applied utilizing linear regression. Nevertheless, the commentary to characteristic ratio imply it is extremely possible linear regression will overfit leading to a poor causal impression estimate within the post-intervention interval.

In linear regression the weights (coefficients) for every characteristic (management unit) may very well be destructive or constructive and so they could sum to a quantity higher than 1. Nevertheless, the artificial management methodology learns the weights while making use of the beneath constraints:

Constraining weights to sum to 1Constraining weights to be ≥ 0

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These constraints assist with regularisation and keep away from extrapolation past the vary of the noticed knowledge.

It’s price noting that when it comes to regularisation, Ridge and Lasso regression can obtain this, and in some instances are affordable options. However we’ll check this out within the case research!

How can we validate the estimated causal impression?

An arguably greater problem is the truth that we’re unable to validate the estimated causal impression within the post-intervention interval.

How lengthy ought to my pre-intervention interval be? Are we positive we haven’t overfit our pre-intervention interval? How can we all know whether or not our mannequin generalises effectively within the submit intervention interval? What if I need to check out totally different implementations of artificial management methodology?

We might randomly choose a number of observations from the pre-intervention interval and maintain them again for validation — However we’ve got already highlighted the problem which comes from having restricted observations so we could make issues even worse!

What if we might run some type of pre-intervention simulation? Might that assist us reply a few of the questions highlighted above and achieve confidence in our fashions estimated causal impression? All shall be defined within the case research!

Background

After convincing Finance that model advertising is driving some severe worth, the advertising group method you to ask about geo-lift testing. Somebody from Fb has advised them it’s the following huge factor (though it was the identical one who advised them Prophet was forecasting mannequin) and so they need to know whether or not they might use it to measure their new TV marketing campaign which is developing.

You’re a little involved, because the final time you ran a geo-lift check the advertising analytics group thought it was a good suggestion to mess around with the pre-intervention interval used till that they had a pleasant huge causal impression.

This time spherical, you recommend that they run a “pre-intervention simulation” after which you plan that the pre-intervention interval is agreed earlier than the check begins.

So let’s discover what a “pre-intervention simulation” seems like!

Creating the info

To make this as life like as potential, I extracted some google development knowledge for almost all of nations in Europe. What the search time period was isn’t related, simply faux it’s the gross sales for you firm (and that you simply function throughout Europe).

Nevertheless, in case you are thinking about how I obtained the google development knowledge, try my pocket book:

Under we will see the dataframe. Now we have gross sales for the previous 3 years throughout 50 European international locations. The advertising group plan to run their TV marketing campaign in Nice Britain.

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Now right here comes the intelligent bit. We’ll simulate an intervention within the final 7 weeks of the time sequence.

np.random.seed(1234)

# Create intervention flagmask = (df[‘date’] >= “2024-04-14”) & (df[‘date’] <= “2024-06-02”)df[‘intervention’] = masks.astype(int)

row_count = len(df)

# Create intervention upliftdf[‘uplift_perc’] = np.random.uniform(0.10, 0.20, measurement=row_count)df[‘uplift_abs’] = spherical(df[‘uplift_perc’] * df[‘GB’])df[‘y’] = df[‘GB’]df.loc[df[‘intervention’] == 1, ‘y’] = df[‘GB’] + df[‘uplift_abs’]

Now let’s plot the precise and counterfactual gross sales throughout GB to convey what we’ve got accomplished to life:

def synth_plot(df, counterfactual):

plt.determine(figsize=(14, 8))sns.set_style(“white”)

# Create plotsns.lineplot(knowledge=df, x=’date’, y=’y’, label=’Precise’, shade=’b’, linewidth=2.5)sns.lineplot(knowledge=df, x=’date’, y=counterfactual, label=’Counterfactual’, shade=’r’, linestyle=’–‘, linewidth=2.5)plt.title(‘Artificial Management Technique: Precise vs. Counterfactual’, fontsize=24)plt.xlabel(‘Date’, fontsize=20)plt.ylabel(‘Metric Worth’, fontsize=20)plt.legend(fontsize=16)plt.gca().xaxis.set_major_formatter(plt.matplotlib.dates.DateFormatter(‘%Y-%m-%d’))plt.xticks(rotation=90)plt.grid(True, linestyle=’–‘, alpha=0.5)

# Excessive the intervention pointintervention_date = ‘2024-04-07’plt.axvline(pd.to_datetime(intervention_date), shade=’ok’, linestyle=’–‘, linewidth=1)plt.textual content(pd.to_datetime(intervention_date), plt.ylim()[1]*0.95, ‘Intervention’, shade=’ok’, fontsize=18, ha=’proper’)

plt.tight_layout()plt.present()

synth_plot(df, ‘GB’)

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So now we’ve got simulated an intervention, we will discover how effectively the artificial management methodology will work.

Pre-processing

The entire European international locations aside from GB are set as management models (options). The therapy unit (goal) is the gross sales in GB with the intervention utilized.

# Delete the unique goal column so we do not use it as a characteristic by accidentdel df[‘GB’]

# set characteristic & targetsX = df.columns[1:50]y = ‘y’

Regression

Under I’ve setup a perform which we will re-use with totally different pre-intervention intervals and totally different regression fashions (e.g. Ridge, Lasso):

def train_reg(df, start_index, reg_class):

df_temp = df.iloc[start_index:].copy().reset_index()

X_pre = df_temp[df_temp[‘intervention’] == 0][X]y_pre = df_temp[df_temp[‘intervention’] == 0][y]

X_train, X_test, y_train, y_test = train_test_split(X_pre, y_pre, test_size=0.10, random_state=42)

mannequin = reg_classmodel.match(X_train, y_train)

yhat_train = mannequin.predict(X_train)yhat_test = mannequin.predict(X_test)

mse_train = mean_squared_error(y_train, yhat_train)mse_test = mean_squared_error(y_test, yhat_test)print(f”Imply Squared Error practice: spherical(mse_train, 2)”)print(f”Imply Squared Error check: spherical(mse_test, 2)”)

r2_train = r2_score(y_train, yhat_train)r2_test = r2_score(y_test, yhat_test)print(f”R2 practice: spherical(r2_train, 2)”)print(f”R2 check: spherical(r2_test, 2)”)

df_temp[‘pred’] = mannequin.predict(df_temp.loc[:, X])df_temp[‘delta’] = df_temp[‘y’] – df_temp[‘pred’]

pred_lift = df_temp[df_temp[‘intervention’] == 1][‘delta’].sum()actual_lift = df_temp[df_temp[‘intervention’] == 1][‘uplift_abs’].sum()abs_error_perc = abs(pred_lift – actual_lift) / actual_liftprint(f”Predicted elevate: spherical(pred_lift, 2)”)print(f”Precise elevate: spherical(actual_lift, 2)”)print(f”Absolute error share: spherical(abs_error_perc, 2)”)

return df_temp, abs_error_perc

To start out us off we preserve issues easy and use linear regression to estimate the causal impression, utilizing a small pre-intervention interval:

df_lin_reg_100, pred_lift_lin_reg_100 = train_reg(df, 100, LinearRegression())

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Trying on the outcomes, linear regression doesn’t do nice. However this isn’t stunning given the commentary to characteristic ratio.

synth_plot(df_lin_reg_100, ‘pred’)

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Artificial management methodology

Let’s leap proper in and see the way it compares to the artificial management methodology. Under I’ve setup an identical perform as earlier than, however making use of the artificial management methodology utilizing sciPy:

def synthetic_control(weights, control_units, treated_unit):

artificial = np.dot(control_units.values, weights)

return np.sqrt(np.sum((treated_unit – artificial)**2))

def train_synth(df, start_index):

df_temp = df.iloc[start_index:].copy().reset_index()

X_pre = df_temp[df_temp[‘intervention’] == 0][X]y_pre = df_temp[df_temp[‘intervention’] == 0][y]

X_train, X_test, y_train, y_test = train_test_split(X_pre, y_pre, test_size=0.10, random_state=42)

initial_weights = np.ones(len(X)) / len(X)

constraints = (‘sort’: ‘eq’, ‘enjoyable’: lambda w: np.sum(w) – 1)

bounds = [(0, 1) for _ in range(len(X))]

end result = reduce(synthetic_control, initial_weights, args=(X_train, y_train),methodology=’SLSQP’, bounds=bounds, constraints=constraints,choices=‘disp’: False, ‘maxiter’: 1000, ‘ftol’: 1e-9,)

optimal_weights = end result.x

yhat_train = np.dot(X_train.values, optimal_weights)yhat_test = np.dot(X_test.values, optimal_weights)

mse_train = mean_squared_error(y_train, yhat_train)mse_test = mean_squared_error(y_test, yhat_test)print(f”Imply Squared Error practice: spherical(mse_train, 2)”)print(f”Imply Squared Error check: spherical(mse_test, 2)”)

r2_train = r2_score(y_train, yhat_train)r2_test = r2_score(y_test, yhat_test)print(f”R2 practice: spherical(r2_train, 2)”)print(f”R2 check: spherical(r2_test, 2)”)

df_temp[‘pred’] = np.dot(df_temp.loc[:, X].values, optimal_weights)df_temp[‘delta’] = df_temp[‘y’] – df_temp[‘pred’]

pred_lift = df_temp[df_temp[‘intervention’] == 1][‘delta’].sum()actual_lift = df_temp[df_temp[‘intervention’] == 1][‘uplift_abs’].sum()abs_error_perc = abs(pred_lift – actual_lift) / actual_liftprint(f”Predicted elevate: spherical(pred_lift, 2)”)print(f”Precise elevate: spherical(actual_lift, 2)”)print(f”Absolute error share: spherical(abs_error_perc, 2)”)

return df_temp, abs_error_perc

I preserve the pre-intervention interval the identical to create a good comparability to linear regression:

df_synth_100, pred_lift_synth_100 = train_synth(df, 100)

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Wow! I’ll be the primary to confess I wasn’t anticipating such a major enchancment!

synth_plot(df_synth_100, ‘pred’)

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Comparability of outcomes

Let’s not get too carried away but. Under we run a number of extra experiments exploring mannequin sorts and pre-interventions intervals:

# run regression experimentsdf_lin_reg_00, pred_lift_lin_reg_00 = train_reg(df, 0, LinearRegression())df_lin_reg_100, pred_lift_lin_reg_100 = train_reg(df, 100, LinearRegression())df_ridge_00, pred_lift_ridge_00 = train_reg(df, 0, RidgeCV())df_ridge_100, pred_lift_ridge_100 = train_reg(df, 100, RidgeCV())df_lasso_00, pred_lift_lasso_00 = train_reg(df, 0, LassoCV())df_lasso_100, pred_lift_lasso_100 = train_reg(df, 100, LassoCV())

# run artificial management experimentsdf_synth_00, pred_lift_synth_00 = train_synth(df, 0)df_synth_100, pred_lift_synth_100 = train_synth(df, 100)

experiment_data = “Technique”: [“Linear”, “Linear”, “Ridge”, “Ridge”, “Lasso”, “Lasso”, “Synthetic Control”, “Synthetic Control”],”Knowledge Dimension”: [“Large”, “Small”, “Large”, “Small”, “Large”, “Small”, “Large”, “Small”],”Worth”: [pred_lift_lin_reg_00, pred_lift_lin_reg_100, pred_lift_ridge_00, pred_lift_ridge_100,pred_lift_lasso_00, pred_lift_lasso_100, pred_lift_synth_00, pred_lift_synth_100]

df_experiments = pd.DataFrame(experiment_data)

We’ll use the code beneath to visualise the outcomes:

# Set the stylesns.set_style=”whitegrid”

# Create the bar plotplt.determine(figsize=(10, 6))bar_plot = sns.barplot(x=”Technique”, y=”Worth”, hue=”Knowledge Dimension”, knowledge=df_experiments, palette=”muted”)

# Add labels and titleplt.xlabel(“Technique”)plt.ylabel(“Absolute error share”)plt.title(“Artificial Controls – Comparability of Strategies Throughout Completely different Knowledge Sizes”)plt.legend(title=”Knowledge Dimension”)

# Present the plotplt.present()

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The outcomes for the small dataset are actually attention-grabbing! As anticipated, regularisation helped enhance the causal impression estimates. The artificial management then took it one step additional!

The outcomes of the massive dataset recommend that longer pre-intervention intervals aren’t all the time higher.

Nevertheless, the factor I need you to remove is how worthwhile finishing up a pre-intervention simulation is. There are such a lot of avenues you can discover with your individual dataset!

Right now we explored the artificial management methodology and how one can validate the causal impression. I’ll depart you with a number of remaining ideas:

The simplicity of the artificial management methodology make it some of the broadly used approach from the causal AI toolbox.Sadly it is usually essentially the most broadly abused — Lets run the R CausalImpact package deal, altering the pre-intervention interval till we see an uplift we like. ????That is the place I extremely suggest working pre-intervention simulations to agree check design upfront.Artificial management methodology is a closely researched space. It’s price trying out the proposed adaptions Augmented SC, Sturdy SC and Penalized SC.

Alberto Abadie, Alexis Diamond & Jens Hainmueller (2010) Artificial Management Strategies for Comparative Case Research: Estimating the Impact of California’s Tobacco Management Program, Journal of the American Statistical Affiliation, 105:490, 493–505, DOI: 10.1198/jasa.2009.ap08746



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