Correlation vs linear regression

Interactively compare and contrast correlation and regression for simple linear models.

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import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
from scipy import stats
from shiny import App, render, ui, reactive

# Define the UI
app_ui = ui.page_sidebar(
    ui.sidebar(
        ui.input_slider("slope", "Slope (β):", -5.0, 5.0, 1.0, step=0.1),
        ui.input_slider("intercept", "Intercept (c):", -10, 10, 0, step=1),
        ui.input_slider("noise", "Noise Level:", 0, 50, 1, step=0.1),
        ui.input_slider("n_points", "Number of Points:", 10, 500, 100, step=10),
        ui.input_checkbox("fix_scale", "Fix Axis Scale", True)
    ),
    ui.row(
        ui.column(
            6,
            ui.output_plot("scatter_plot"),
            ui.output_text("correlation_output"),
            ui.output_text("regression_output")
        ),
        ui.column(
            6,
            ui.output_code("regression_summary")
            )
    )
)

# Define the server logic
def server(input, output, session):
    @reactive.Calc
    def generate_data():
        np.random.seed(42)  # For reproducibility
        x = np.linspace(-10, 10, input.n_points())
        noise = np.random.normal(0, input.noise(), len(x))
        y = input.slope() * x + input.intercept() + noise
        return pd.DataFrame({"x": x, "y": y})

    @output
    @render.plot
    def scatter_plot():
        data = generate_data()
        x, y = data["x"], data["y"]
        max_val = max(abs(x).max(), abs(y).max())
        plt.figure()
        plt.scatter(x, y)
        plt.axhline(0, color='black', linewidth=1)
        plt.axvline(0, color='black', linewidth=1)
        plt.xlabel("X")
        plt.ylabel("Y")
        plt.title("Scatter Plot")
        plt.grid(True)
        if input.fix_scale():
            plt.xlim(-max_val, max_val)
            plt.ylim(-max_val, max_val)
        plt.gca().set_aspect('equal', adjustable='box')
        return plt.gcf()

    @output
    @render.text
    def correlation_output():
        data = generate_data()
        correlation, p_value = stats.pearsonr(data["x"], data["y"])
        return f"Correlation: {correlation:.2f}, p-value: {p_value:.2e}"

    @output
    @render.text
    def regression_output():
        data = generate_data()
        x = sm.add_constant(data["x"])  # Adds a constant term to the predictor
        model = sm.OLS(data["y"], x).fit()
        summary = model.summary2().tables[1]
        slope_p_value = summary.loc["x", "P>|t|"]
        intercept_p_value = summary.loc["const", "P>|t|"]
        return (
            f"Regression Line: y = {model.params['x']:.2f}x + {model.params['const']:.2f}\n"
            f"R-squared: {model.rsquared:.2f}\n"
            f"Slope p-value: {slope_p_value:.2e}\n"
            f"Intercept p-value: {intercept_p_value:.2e}\n"
            f"Standard Error: {model.bse['x']:.2f}"
        )

    @output
    @render.code
    def regression_summary():
        data = generate_data()
        x = sm.add_constant(data["x"])  # Adds a constant term to the predictor
        model = sm.OLS(data["y"], x).fit()
        return model.summary()

app = App(app_ui, server)