5. Statistics Which Analyze Relationships

1. CORRELATION

What it is

Correlation tells you how strongly two numerical variables move together.

Examples:

• Height ↑ → Weight ↑
• Socioeconomic class ↑ → Mortality ↓

When you use it

Use correlation when you want to know if two variables have a linear relationship.

(Not to find cause—only association.)

Key term: Correlation coefficient (r)

The coefficient is written as r.

• r = +1 → perfect positive linear relationship
• r = –1 → perfect negative linear relationship
• r = 0 → no linear relationship at all

How to interpret r (book rule-of-thumb)

• 0 to 0.2 → very low, probably meaningless
• 0.2 to 0.4 → low correlation
• 0.4 to 0.6 → reasonable correlation
• 0.6 to 0.8 → high correlation
• 0.8 to 1.0 → very high (check for errors or duplication!)

Applies equally to negative values.

Example from the book

A nurse compared fasting glucose vs HbA1c in 12 diabetics.

Scatter plot showed a straight-line trend → r = 0.88, meaning:

• Very high positive correlation
• As glucose increases, HbA1c increases

Another example:

Activity level vs BMI → r = –0.34

• Low negative correlation
• Higher activity → slightly lower BMI

Spearman vs Pearson

• Pearson’s r: if data are normally distributed
• Spearman’s r (rs): if skewed

(Study used Spearman because prescribing data were skewed.)

Important warning

Correlation ≠ causation.

R² (coefficient of determination)

If r = –0.88 → R² = 0.77

Meaning 77% of variation in HbA1c is explained by glucose variation.

2. REGRESSION

Regression is correlation plus prediction.

What regression does

It draws a best-fit line through the scatter plot to quantify how much one variable changes when the other changes.

Regression line

Written as:

y = a + b x

Where:

• a = regression constant (where line hits vertical axis)
• b = regression coefficient (slope)

b tells you how much y changes when x increases by 1 unit.

Book example

Using the same glucose–HbA1c data:

Regression equation:

HbA1c = 3.2 + (0.45 × glucose)

Meaning:

• Every 1 mmol/L increase in glucose → HbA1c rises by 0.45%
• If glucose = 15 mmol/L → predicted HbA1c = 9.95%

This is exactly the graph shown in the book (Fig. 12).

What R² means in regression

R² = 0.77 → 77% of variation in HbA1c is explained by glucose.

Types of regression

• Linear regression → straight line (book example)
• Logistic regression → outcome has 2 categories (e.g., diseased vs not)
• Poisson regression → rare events / waiting times
• Cox regression → time-to-event (survival) analysis

Warnings

• You can only predict within the range of your data (no extrapolation).
• Regression requires a logical directional relationship (x should come before y).

Summary Table (Based on the Book)