plot_likelihood_ratios.py

"""
=============================================================
Class Likelihood Ratios to measure classification performance
=============================================================

This example demonstrates the :func:`~sklearn.metrics.class_likelihood_ratios`
function, which computes the positive and negative likelihood ratios (`LR+`,
`LR-`) to assess the predictive power of a binary classifier. As we will see,
these metrics are independent of the proportion between classes in the test set,
which makes them very useful when the available data for a study has a different
class proportion than the target application.

A typical use is a case-control study in medicine, which has nearly balanced
classes while the general population has large class imbalance. In such
application, the pre-test probability of an individual having the target
condition can be chosen to be the prevalence, i.e. the proportion of a
particular population found to be affected by a medical condition. The post-test
probabilities represent then the probability that the condition is truly present
given a positive test result.

In this example we first discuss the link between pre-test and post-test odds
given by the :ref:`class_likelihood_ratios`. Then we evaluate their behavior in
some controlled scenarios. In the last section we plot them as a function of the
prevalence of the positive class.

"""

# Authors:  Arturo Amor <david-arturo.amor-quiroz@inria.fr>
#           Olivier Grisel <olivier.grisel@ensta.org>
# %%
# Pre-test vs. post-test analysis
# ===============================
#
# Suppose we have a population of subjects with physiological measurements `X`
# that can hopefully serve as indirect bio-markers of the disease and actual
# disease indicators `y` (ground truth). Most of the people in the population do
# not carry the disease but a minority (in this case around 10%) does:

from sklearn.datasets import make_classification

X, y = make_classification(n_samples=10_000, weights=[0.9, 0.1], random_state=0)
print(f"Percentage of people carrying the disease: {100*y.mean():.2f}%")

# %%
# A machine learning model is built to diagnose if a person with some given
# physiological measurements is likely to carry the disease of interest. To
# evaluate the model, we need to assess its performance on a held-out test set:

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)

# %%
# Then we can fit our diagnosis model and compute the positive likelihood
# ratio to evaluate the usefulness of this classifier as a disease diagnosis
# tool:

from sklearn.metrics import class_likelihood_ratios
from sklearn.linear_model import LogisticRegression

estimator = LogisticRegression().fit(X_train, y_train)
y_pred = estimator.predict(X_test)
pos_LR, neg_LR = class_likelihood_ratios(y_test, y_pred)
print(f"LR+: {pos_LR:.3f}")

# %%
# Since the positive class likelihood ratio is much larger than 1.0, it means
# that the machine learning-based diagnosis tool is useful: the post-test odds
# that the condition is truly present given a positive test result are more than
# 12 times larger than the pre-test odds.
#
# Cross-validation of likelihood ratios
# =====================================
#
# We assess the variability of the measurements for the class likelihood ratios
# in some particular cases.

import pandas as pd


def scoring(estimator, X, y):
    y_pred = estimator.predict(X)
    pos_lr, neg_lr = class_likelihood_ratios(y, y_pred, raise_warning=False)
    return {"positive_likelihood_ratio": pos_lr, "negative_likelihood_ratio": neg_lr}


def extract_score(cv_results):
    lr = pd.DataFrame(
        {
            "positive": cv_results["test_positive_likelihood_ratio"],
            "negative": cv_results["test_negative_likelihood_ratio"],
        }
    )
    return lr.aggregate(["mean", "std"])


# %%
# We first validate the :class:`~sklearn.linear_model.LogisticRegression` model
# with default hyperparameters as used in the previous section.

from sklearn.model_selection import cross_validate

estimator = LogisticRegression()
extract_score(cross_validate(estimator, X, y, scoring=scoring, cv=10))

# %%
# We confirm that the model is useful: the post-test odds are between 12 and 20
# times larger than the pre-test odds.
#
# On the contrary, let's consider a dummy model that will output random
# predictions with similar odds as the average disease prevalence in the
# training set:

from sklearn.dummy import DummyClassifier

estimator = DummyClassifier(strategy="stratified", random_state=1234)
extract_score(cross_validate(estimator, X, y, scoring=scoring, cv=10))

# %%
# Here both class likelihood ratios are compatible with 1.0 which makes this
# classifier useless as a diagnostic tool to improve disease detection.
#
# Another option for the dummy model is to always predict the most frequent
# class, which in this case is "no-disease".

estimator = DummyClassifier(strategy="most_frequent")
extract_score(cross_validate(estimator, X, y, scoring=scoring, cv=10))

# %%
# The absence of positive predictions means there will be no true positives nor
# false positives, leading to an undefined `LR+` that by no means should be
# interpreted as an infinite `LR+` (the classifier perfectly identifying
# positive cases). In such situation the
# :func:`~sklearn.metrics.class_likelihood_ratios` function returns `nan` and
# raises a warning by default. Indeed, the value of `LR-` helps us discard this
# model.
#
# A similar scenario may arise when cross-validating highly imbalanced data with
# few samples: some folds will have no samples with the disease and therefore
# they will output no true positives nor false negatives when used for testing.
# Mathematically this leads to an infinite `LR+`, which should also not be
# interpreted as the model perfectly identifying positive cases. Such event
# leads to a higher variance of the estimated likelihood ratios, but can still
# be interpreted as an increment of the post-test odds of having the condition.

estimator = LogisticRegression()
X, y = make_classification(n_samples=300, weights=[0.9, 0.1], random_state=0)
extract_score(cross_validate(estimator, X, y, scoring=scoring, cv=10))

# %%
# Invariance with respect to prevalence
# =====================================
#
# The likelihood ratios are independent of the disease prevalence and can be
# extrapolated between populations regardless of any possible class imbalance,
# **as long as the same model is applied to all of them**. Notice that in the
# plots below **the decision boundary is constant** (see
# :ref:`sphx_glr_auto_examples_svm_plot_separating_hyperplane_unbalanced.py` for
# a study of the boundary decision for unbalanced classes).
#
# Here we train a :class:`~sklearn.linear_model.LogisticRegression` base model
# on a case-control study with a prevalence of 50%. It is then evaluated over
# populations with varying prevalence. We use the
# :func:`~sklearn.datasets.make_classification` function to ensure the
# data-generating process is always the same as shown in the plots below. The
# label `1` corresponds to the positive class "disease", whereas the label `0`
# stands for "no-disease".

import numpy as np
import matplotlib.pyplot as plt
from sklearn.inspection import DecisionBoundaryDisplay
from collections import defaultdict

populations = defaultdict(list)
common_params = {
    "n_samples": 10_000,
    "n_features": 2,
    "n_informative": 2,
    "n_redundant": 0,
    "random_state": 0,
}
weights = np.linspace(0.1, 0.8, 6)
weights = weights[::-1]

# fit and evaluate base model on balanced classes
X, y = make_classification(**common_params, weights=[0.5, 0.5])
estimator = LogisticRegression().fit(X, y)
lr_base = extract_score(cross_validate(estimator, X, y, scoring=scoring, cv=10))
pos_lr_base, pos_lr_base_std = lr_base["positive"].values
neg_lr_base, neg_lr_base_std = lr_base["negative"].values

# %%
# We will now show the decision boundary for each level of prevalence. Note that
# we only plot a subset of the original data to better assess the linear model
# decision boundary.

fig, axs = plt.subplots(nrows=3, ncols=2, figsize=(15, 12))

for ax, (n, weight) in zip(axs.ravel(), enumerate(weights)):

    X, y = make_classification(
        **common_params,
        weights=[weight, 1 - weight],
    )
    prevalence = y.mean()
    populations["prevalence"].append(prevalence)
    populations["X"].append(X)
    populations["y"].append(y)

    # down-sample for plotting
    rng = np.random.RandomState(1)
    plot_indices = rng.choice(np.arange(X.shape[0]), size=500, replace=True)
    X_plot, y_plot = X[plot_indices], y[plot_indices]

    # plot fixed decision boundary of base model with varying prevalence
    disp = DecisionBoundaryDisplay.from_estimator(
        estimator,
        X_plot,
        response_method="predict",
        alpha=0.5,
        ax=ax,
    )
    scatter = disp.ax_.scatter(X_plot[:, 0], X_plot[:, 1], c=y_plot, edgecolor="k")
    disp.ax_.set_title(f"prevalence = {y_plot.mean():.2f}")
    disp.ax_.legend(*scatter.legend_elements())

# %%
# We define a function for bootstraping.


def scoring_on_bootstrap(estimator, X, y, rng, n_bootstrap=100):
    results_for_prevalence = defaultdict(list)
    for _ in range(n_bootstrap):
        bootstrap_indices = rng.choice(
            np.arange(X.shape[0]), size=X.shape[0], replace=True
        )
        for key, value in scoring(
            estimator, X[bootstrap_indices], y[bootstrap_indices]
        ).items():
            results_for_prevalence[key].append(value)
    return pd.DataFrame(results_for_prevalence)


# %%
# We score the base model for each prevalence using bootstraping.

results = defaultdict(list)
n_bootstrap = 100
rng = np.random.default_rng(seed=0)

for prevalence, X, y in zip(
    populations["prevalence"], populations["X"], populations["y"]
):

    results_for_prevalence = scoring_on_bootstrap(
        estimator, X, y, rng, n_bootstrap=n_bootstrap
    )
    results["prevalence"].append(prevalence)
    results["metrics"].append(
        results_for_prevalence.aggregate(["mean", "std"]).unstack()
    )

results = pd.DataFrame(results["metrics"], index=results["prevalence"])
results.index.name = "prevalence"
results

# %%
# In the plots below we observe that the class likelihood ratios re-computed
# with different prevalences are indeed constant within one standard deviation
# of those computed with on balanced classes.

fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize=(15, 6))
results["positive_likelihood_ratio"]["mean"].plot(
    ax=ax1, color="r", label="extrapolation through populations"
)
ax1.axhline(y=pos_lr_base + pos_lr_base_std, color="r", linestyle="--")
ax1.axhline(
    y=pos_lr_base - pos_lr_base_std,
    color="r",
    linestyle="--",
    label="base model confidence band",
)
ax1.fill_between(
    results.index,
    results["positive_likelihood_ratio"]["mean"]
    - results["positive_likelihood_ratio"]["std"],
    results["positive_likelihood_ratio"]["mean"]
    + results["positive_likelihood_ratio"]["std"],
    color="r",
    alpha=0.3,
)
ax1.set(
    title="Positive likelihood ratio",
    ylabel="LR+",
    ylim=[0, 5],
)
ax1.legend(loc="lower right")

ax2 = results["negative_likelihood_ratio"]["mean"].plot(
    ax=ax2, color="b", label="extrapolation through populations"
)
ax2.axhline(y=neg_lr_base + neg_lr_base_std, color="b", linestyle="--")
ax2.axhline(
    y=neg_lr_base - neg_lr_base_std,
    color="b",
    linestyle="--",
    label="base model confidence band",
)
ax2.fill_between(
    results.index,
    results["negative_likelihood_ratio"]["mean"]
    - results["negative_likelihood_ratio"]["std"],
    results["negative_likelihood_ratio"]["mean"]
    + results["negative_likelihood_ratio"]["std"],
    color="b",
    alpha=0.3,
)
ax2.set(
    title="Negative likelihood ratio",
    ylabel="LR-",
    ylim=[0, 0.5],
)
ax2.legend(loc="lower right")

plt.show()