DOC Improve documentation regarding some pitfalls in interpretation (#20451)

jygerardy · Jean-Yves Gerardy · Jean-Yves Gerardy · web-flow · commit c892ade83793 · 2023-01-11T14:01:47.000+01:00
Co-authored-by: Jean-Yves Gerardy &lt;jygerardy@DKUMBPjygerardy.lan&gt;
Co-authored-by: Jean-Yves Gerardy &lt;jygerardy@DKU-MBP-jygerardy.local&gt;
Co-authored-by: Thomas J. Fan &lt;thomasjpfan@gmail.com&gt;
Co-authored-by: Olivier Grisel &lt;olivier.grisel@ensta.org&gt;
Co-authored-by: Guillaume Lemaitre &lt;g.lemaitre58@gmail.com&gt;
Co-authored-by: Arturo Amor &lt;86408019+ArturoAmorQ@users.noreply.github.com&gt;
Co-authored-by: Tim Head &lt;betatim@gmail.com&gt;
diff --git a/examples/inspection/plot_causal_interpretation.py b/examples/inspection/plot_causal_interpretation.py
@@ -0,0 +1,185 @@
+"""
+===================================================
+Failure of Machine Learning to infer causal effects
+===================================================
+
+Machine Learning models are great for measuring statistical associations.
+Unfortunately, unless we're willing to make strong assumptions about the data,
+those models are unable to infer causal effects.
+
+To illustrate this, we will simulate a situation in which we try to answer one
+of the most important questions in economics of education: **what is the causal
+effect of earning a college degree on hourly wages?** Although the answer to
+this question is crucial to policy makers, `Omitted-Variable Biases
+<https://en.wikipedia.org/wiki/Omitted-variable_bias>`_ (OVB) prevent us from
+identifying that causal effect.
+"""
+
+# %%
+# The dataset: simulated hourly wages
+# -----------------------------------
+#
+# The data generating process is laid out in the code below. Work experience in
+# years and a measure of ability are drawn from Normal distributions; the
+# hourly wage of one of the parents is drawn from Beta distribution. We then
+# create an indicator of college degree which is positively impacted by ability
+# and parental hourly wage. Finally, we model hourly wages as a linear function
+# of all the previous variables and a random component. Note that all variables
+# have a positive effect on hourly wages.
+import numpy as np
+import pandas as pd
+
+n_samples = 10_000
+rng = np.random.RandomState(32)
+
+experiences = rng.normal(20, 10, size=n_samples).astype(int)
+experiences[experiences < 0] = 0
+abilities = rng.normal(0, 0.15, size=n_samples)
+parent_hourly_wages = 50 * rng.beta(2, 8, size=n_samples)
+parent_hourly_wages[parent_hourly_wages < 0] = 0
+college_degrees = (
+    9 * abilities + 0.02 * parent_hourly_wages + rng.randn(n_samples) > 0.7
+).astype(int)
+
+true_coef = pd.Series(
+    {
+        "college degree": 2.0,
+        "ability": 5.0,
+        "experience": 0.2,
+        "parent hourly wage": 1.0,
+    }
+)
+hourly_wages = (
+    true_coef["experience"] * experiences
+    + true_coef["parent hourly wage"] * parent_hourly_wages
+    + true_coef["college degree"] * college_degrees
+    + true_coef["ability"] * abilities
+    + rng.normal(0, 1, size=n_samples)
+)
+
+hourly_wages[hourly_wages < 0] = 0
+
+# %%
+# Description of the simulated data
+# ---------------------------------
+#
+# The following plot shows the distribution of each variable, and pairwise
+# scatter plots. Key to our OVB story is the positive relationship between
+# ability and college degree.
+import seaborn as sns
+
+df = pd.DataFrame(
+    {
+        "college degree": college_degrees,
+        "ability": abilities,
+        "hourly wage": hourly_wages,
+        "experience": experiences,
+        "parent hourly wage": parent_hourly_wages,
+    }
+)
+
+grid = sns.pairplot(df, diag_kind="kde", corner=True)
+
+# %%
+# In the next section, we train predictive models and we therefore split the
+# target column from over features and we split the data into a training and a
+# testing set.
+from sklearn.model_selection import train_test_split
+
+target_name = "hourly wage"
+X, y = df.drop(columns=target_name), df[target_name]
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)
+
+# %%
+# Income prediction with fully observed variables
+# -----------------------------------------------
+#
+# First, we train a predictive model, a
+# :class:`~sklearn.linear_model.LinearRegression` model. In this experiment,
+# we assume that all variables used by the true generative model are available.
+from sklearn.linear_model import LinearRegression
+from sklearn.metrics import r2_score
+
+features_names = ["experience", "parent hourly wage", "college degree", "ability"]
+
+regressor_with_ability = LinearRegression()
+regressor_with_ability.fit(X_train[features_names], y_train)
+y_pred_with_ability = regressor_with_ability.predict(X_test[features_names])
+R2_with_ability = r2_score(y_test, y_pred_with_ability)
+
+print(f"R2 score with ability: {R2_with_ability:.3f}")
+
+# %%
+# This model predicts well the hourly wages as shown by the high R2 score. We
+# plot the model coefficients to show that we exactly recover the values of
+# the true generative model.
+import matplotlib.pyplot as plt
+
+model_coef = pd.Series(regressor_with_ability.coef_, index=features_names)
+coef = pd.concat(
+    [true_coef[features_names], model_coef],
+    keys=["Coefficients of true generative model", "Model coefficients"],
+    axis=1,
+)
+ax = coef.plot.barh()
+ax.set_xlabel("Coefficient values")
+ax.set_title("Coefficients of the linear regression including the ability features")
+plt.tight_layout()
+plt.show()
+
+# %%
+# Income prediction with partial observations
+# -------------------------------------------
+#
+# In practice, intellectual abilities are not observed or are only estimated
+# from proxies that inadvertently measure education as well (e.g. by IQ tests).
+# But omitting the "ability" feature from a linear model inflates the estimate
+# via a positive OVB.
+features_names = ["experience", "parent hourly wage", "college degree"]
+
+regressor_without_ability = LinearRegression()
+regressor_without_ability.fit(X_train[features_names], y_train)
+y_pred_without_ability = regressor_without_ability.predict(X_test[features_names])
+R2_without_ability = r2_score(y_test, y_pred_without_ability)
+
+print(f"R2 score without ability: {R2_without_ability:.3f}")
+
+# %%
+# The predictive power of our model is similar when we omit the ability feature
+# in terms of R2 score. We now check if the coefficient of the model are
+# different from the true generative model.
+
+model_coef = pd.Series(regressor_without_ability.coef_, index=features_names)
+coef = pd.concat(
+    [true_coef[features_names], model_coef],
+    keys=["Coefficients of true generative model", "Model coefficients"],
+    axis=1,
+)
+ax = coef.plot.barh()
+ax.set_xlabel("Coefficient values")
+_ = ax.set_title("Coefficients of the linear regression excluding the ability feature")
+
+# %%
+# To compensate for the omitted variable, the model inflates the coefficient of
+# the college degree feature. Therefore, interpreting this coefficient value
+# as a causal effect of the true generative model is incorrect.
+#
+# Lessons learned
+# ---------------
+#
+# Machine learning models are not designed for the estimation of causal
+# effects. While we showed this with a linear model, OVB can affect any type of
+# model.
+#
+# Whenever interpreting a coefficient or a change in predictions brought about
+# by a change in one of the features, it is important to keep in mind
+# potentially unobserved variables that could be correlated with both the
+# feature in question and the target variable. Such variables are called
+# `Confounding Variables <https://en.wikipedia.org/wiki/Confounding>`_. In
+# order to still estimate causal effect in the presence of confounding,
+# researchers usually conduct experiments in which the treatment variable (e.g.
+# college degree) is randomized. When an experiment is prohibitively expensive
+# or unethical, researchers can sometimes use other causal inference techniques
+# such as `Instrumental Variables
+# <https://en.wikipedia.org/wiki/Instrumental_variables_estimation>`_ (IV)
+# estimations.
diff --git a/examples/inspection/plot_linear_model_coefficient_interpretation.py b/examples/inspection/plot_linear_model_coefficient_interpretation.py
@@ -704,6 +704,43 @@
 # We observe that the AGE and EXPERIENCE coefficients are varying a lot
 # depending of the fold.
 #
+# Wrong causal interpretation
+# ---------------------------
+#
+# Policy makers might want to know the effect of education on wage to assess
+# whether or not a certain policy designed to entice people to pursue more
+# education would make economic sense. While Machine Learning models are great
+# for measuring statistical associations, they are generally unable to infer
+# causal effects.
+#
+# It might be tempting to look at the coefficient of education on wage from our
+# last model (or any model for that matter) and conclude that it captures the
+# true effect of a change in the standardized education variable on wages.
+#
+# Unfortunately there are likely unobserved confounding variables that either
+# inflate or deflate that coefficient. A confounding variable is a variable that
+# causes both EDUCATION and WAGE. One example of such variable is ability.
+# Presumably, more able people are more likely to pursue education while at the
+# same time being more likely to earn a higher hourly wage at any level of
+# education. In this case, ability induces a positive `Omitted Variable Bias
+# <https://en.wikipedia.org/wiki/Omitted-variable_bias>`_ (OVB) on the EDUCATION
+# coefficient, thereby exaggerating the effect of education on wages.
+#
+# See the :ref:`sphx_glr_auto_examples_inspection_plot_causal_interpretation.py`
+# for a simulated case of ability OVB.
+#
+# Warning: data and model quality
+# -------------------------------
+#
+# Keep in mind that the outcome `y` and features `X` are the product
+# of a data generating process that is hidden from us. Machine
+# learning models are trained to approximate the unobserved
+# mathematical function that links `X` to `y` from sample data. As a
+# result, any interpretation made about a model may not necessarily
+# generalize to the true data generating process. This is especially
+# true when the model is of bad quality or when the sample data is
+# not representative of the population.
+#
 # Lessons learned
 # ---------------
 #
@@ -719,3 +756,7 @@
 #   coefficients could significantly vary from one another.
 # * Inspecting coefficients across the folds of a cross-validation loop
 #   gives an idea of their stability.
+# * Coefficients are unlikely to have any causal meaning. They tend
+#   to be biased by unobserved confounders.
+# * Inspection tools may not necessarily provide insights on the true
+#   data generating process.
