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# Class Likelihood Ratios to measure classification performance¶

This example demonstrates the `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 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}%")
```

```
Percentage of people carrying the disease: 10.37%
```

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.linear_model import LogisticRegression
from sklearn.metrics import class_likelihood_ratios
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}")
```

```
LR+: 12.617
```

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 `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
`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
SVM: Separating hyperplane for unbalanced classes for
a study of the boundary decision for unbalanced classes).

Here we train a `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
`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”.

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

```
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 bootstrapping.

```
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()
```

**Total running time of the script:** (0 minutes 2.109 seconds)

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