半正态离散数据的统计检验选择及多Y数据集相关性分析咨询

Hey there, let's work through this problem for you—since your X follows a half-normal distribution and you've got multiple Y datasets to compare, here's a structured approach to pick the most correlated Y and visualize your data effectively:

Half-normal data (non-negative, right-skewed like the right half of a normal curve) breaks the normality assumption of Pearson correlation, so we’ll focus on non-parametric or adjusted methods:

• Spearman’s Rank Correlation: This is your go-to here. It ignores the actual distribution shape and only looks at the monotonic trend between X and each Yi. It works by ranking the values of X and Yi, then calculating correlation on those ranks. You can compare the absolute values of Spearman’s r across all Yi—larger absolute values mean stronger correlation.
• Kendall’s Tau: Another non-parametric option, great for smaller datasets or when you want to focus on the consistency of paired observations (how often X and Yi move in the same direction). Like Spearman’s, you can rank Yi by their Tau values to find the best fit.
• Transform-then-Pearson (Optional): If you can normalize X with a transformation (square root or log—add a tiny constant like 0.1 if X has zeros), you can use Pearson correlation after verifying the transformed X is roughly normal (via Shapiro-Wilk test). Just be sure to document the transformation step if you go this route.

Since X is continuous (half-normal), bar charts work best if you first bucket X into meaningful groups. Here’s how to make them useful:

• Quantile-based Grouping: Split X into equal-sized groups (e.g., quartiles: Q1 to Q4) instead of arbitrary bins. This ensures each group has the same number of data points, making comparisons fair.
• Grouped Bar Charts for Comparison: Plot the mean/median of each Yi within each X group side-by-side. This lets you visually spot which Yi follows X’s trend most closely.
• Add Context: Label each bar chart with the corresponding correlation coefficient (e.g., Spearman’s r) so viewers can link the visual pattern to statistical significance. For extra depth, pair bars with boxplots to show the spread of Yi values in each X group.

Here’s a snippet to implement the above steps:

import scipy.stats as stats
import matplotlib.pyplot as plt
import pandas as pd

# Load your data (assuming X is the half-normal column, Y1-Yi are the rest)
df = pd.read_csv("your_data.csv")

# Calculate Spearman correlation for each Yi
corr_results = {}
for col in df.columns:
    if col.startswith("Y"):
        corr, p_val = stats.spearmanr(df["X"], df[col])
        corr_results[col] = {"corr": round(corr, 3), "p_value": round(p_val, 3)}

# Sort Ys by correlation strength (absolute value)
sorted_corrs = sorted(corr_results.items(), key=lambda x: abs(x[1]["corr"]), reverse=True)
print(f"Top correlated Y: {sorted_corrs[0][0]} (r = {sorted_corrs[0][1]['corr']})")

# Create quartile groups for X
df["X_quartile"] = pd.qcut(df["X"], q=4, labels=["Q1", "Q2", "Q3", "Q4"])

# Plot top 3 Ys as grouped bar charts
top_ys = [item[0] for item in sorted_corrs[:3]]
grouped_means = df.groupby("X_quartile")[top_ys].mean()

plt.figure(figsize=(10, 6))
grouped_means.plot(kind="bar", color=["#1f77b4", "#ff7f0e", "#2ca02c"])
plt.title("Top 3 Y Datasets by X Quartile")
plt.xlabel("X Quartile Groups")
plt.ylabel("Mean Y Value")
plt.legend(title="Y Datasets")
plt.show()

Note: If your Y variables are categorical instead of continuous, swap correlation tests for point-biserial correlation (for binary Y) or polychoric correlation (for multi-category Y).

内容的提问来源于stack exchange，提问作者Peter Duncan