Chapter 7 — Relationships Between Variables

The Question¶

“Does a mother’s age predict the birth weight of her baby?”

All previous chapters studied one variable at a time. Now we ask: do two variables move together? And if so, how strongly?

Scatter Plots¶

The first tool for any relationship: plot one variable against the other.

For NSFG, plot mother’s age (agepreg) on the x-axis vs birth weight (totalwgt_lb) on the y-axis. One dot per pregnancy.

Problem: with 9,000 points, dots overlap completely. You can’t see the pattern.

Solutions:

Jitter — add small random noise to each point so overlapping points spread out
Alpha — make each dot semi-transparent; dense regions appear darker
Bin and mean — divide x into bins, plot mean y per bin (shows trend clearly)
Hexbin — 2D histogram using hexagonal bins

Characterizing Relationships¶

Before computing a correlation coefficient, ask:

Is the relationship linear or curved?
Is it monotone (always increasing or always decreasing)?
Are there subgroups that behave differently?
Are there outliers that dominate the relationship?

A correlation coefficient summarizes point 1 only. Always look at the scatter plot.

Covariance¶

Covariance measures how two variables move together:

\text{Cov}(X, Y) = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})

(1)

Positive covariance: when $X$ is high, $Y$ tends to be high
Negative covariance: when $X$ is high, $Y$ tends to be low
Zero covariance: no linear relationship

Problem: covariance has units (lbs × years for our example). Comparing covariances across different pairs of variables is meaningless.

Pearson’s Correlation¶

Normalize covariance by the product of standard deviations:

r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \sum(y_i - \bar{y})^2}}

(2)

Now $r \in [-1, 1]$ :

$r = +1$ : perfect positive linear relationship
$r = -1$ : perfect negative linear relationship
$r = 0$ : no linear relationship
$|r| > 0.7$ : strong (by convention)

Critical warning: $r = 0$ does not mean “no relationship” — it means no linear relationship. The data could have a strong curved relationship with $r = 0$ .

Nonlinear Relationships¶

For mother’s age vs birth weight, the relationship is nonlinear:

Very young mothers (teens) have lower birth weight babies
Mothers in their 20s and 30s have the heaviest babies
Very old mothers (40+) have slightly lower weight babies

Pearson’s $r$ misses this U-shape. Always plot before computing $r$ .

Spearman’s Rank Correlation¶

Spearman’s $\rho$ is Pearson’s $r$ applied to the ranks of the data, not the values.

Replace each $x_i$ with its rank (1 = smallest, $n$ = largest)
Replace each $y_i$ with its rank
Compute Pearson’s $r$ on the ranks

\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}

(3)

where $d_i$ is the difference in ranks.

Advantages over Pearson’s $r$ :

Works for any monotone relationship (not just linear)
Robust to outliers (one extreme value can’t dominate)
Works on ordinal data

When to use which:

Linear relationship, no outliers → Pearson
Monotone relationship, or outliers present → Spearman

Correlation and Causation¶

The most abused concept in statistics.

Correlation tells you two variables are associated. It says nothing about why.

Ice cream sales and drowning rates are positively correlated. Cause: both increase in summer (confound = season).

In NSFG, older mothers have slightly lighter babies. Possible causes:

Direct: older maternal age causes lower birth weight
Confound: older mothers have more complicated pregnancies
Reverse: nothing — this is an observational study

Establishing causation requires either:

A randomized controlled experiment (randomize the cause)
A natural experiment (quasi-random assignment occurs naturally)
Careful matching/adjustment for confounds

Observational data like NSFG can only establish association.

Exercises¶

Make a scatter plot of mother’s age vs birth weight with alpha=0.1. What pattern do you see?
Compute Pearson’s $r$ between age and birth weight. Is it strong?
Compute Spearman’s $\rho$ for the same pair. How does it compare to Pearson?
Bin mother’s age into 5-year groups and plot mean birth weight per bin. What shape do you see?
Compute $r$ between pregnancy length and birth weight. Which pair is more strongly correlated?

Glossary¶

scatter plot — plot of one variable vs another; one point per observation

jitter — small random noise added to avoid overplotting

covariance — $\text{Cov}(X,Y)$ ; measures direction of linear association; has units

Pearson’s r — normalized covariance; unit-free; measures linear association

Spearman’s ρ — Pearson’s r on ranks; measures monotone association; robust to outliers

monotone relationship — a relationship that is always increasing or always decreasing

confound — a third variable that causes both $X$ and $Y$ , creating spurious correlation

correlation ≠ causation — association between variables does not imply one causes the other