Can Relative Difference Use Median and IQR Instead of Mean and SD?

✅ Short Answer

No, relative difference (RD) is fundamentally based on means, ratios, or changes in mean values, and cannot be meaningfully used with medians and IQRs. Median and IQR are non-parametric summaries and do not support ratio-based comparisons in the same statistical framework.

🧠 Why It Matters

1. Relative Difference Formula (Mean-based):

The standard formula is:

RD = (Mean_Group1 - Mean_Group2) / Mean_Group2

This formula assumes interval scale data and normal (or approximately symmetric) distribution to be interpretable. It's often expressed as a percentage.

2. Median and IQR Context:

Median and IQR (Interquartile Range) are rank-based statistics.
They are robust to outliers and do not have mathematical properties that support ratio computations.
There’s no meaningful way to calculate (median1 - median2) / median2 as a reliable "relative" metric since the median is not additive or multiplicative across distributions.

🧭 Alternatives for Skewed Data

If your data is non-normal (e.g., cost, LOS, lab values):

➤ Use Ratio of Medians (ROM) cautiously:

ROM = Median_Group1 / Median_Group2

Interpret with caution—this has no formal statistical test unless bootstrapped.

➤ Use Generalized Linear Models (GLM) with log link:

stata glm outcome i.group, family(gaussian) link(log) robust

This estimates mean ratio even for skewed data.

➤ Use quantile regression to model medians:

stata qreg outcome i.group

Estimates difference in medians, not ratio.

🩺 Clinical Tip

For reporting skewed outcomes:

Report medians + IQRs
Test differences using non-parametric methods (e.g., ranksum)
Avoid ratio metrics unless justified with modeling (e.g., GLM with log link)