Examining Relationships — Master Notes

Two variables → classify each by role and type before picking any tool.

Roles

Types: Categorical (C) or Quantitative (Q)

The Four Cases (explanatory listed first):

Case	Explanatory	Response	Tool
C→Q	Categorical	Quantitative	Side-by-side boxplots + descriptive stats
C→C	Categorical	Categorical	Two-way table + conditional %
Q→Q	Quantitative	Quantitative	Scatterplot + correlation (r) + regression line
Q→C	Quantitative	Categorical	(not covered in this course)

Principle: First classify the data structure into one of the four cases — only then pick the statistical tool.

Display: side-by-side boxplots
Numbers: descriptive stats (min, Q1, median, Q3, max) computed separately per category
Interpretation = compare the distributions of Y across levels of X (center, spread, shape)
Example takeaway style: "poultry hot dogs have a lower median calorie count than beef/meat" — compare medians, IQRs, ranges.

Display: two-way table (contingency table) — rows/cols = categories, cells = counts, margins = Totals.
Never compare raw counts across groups of different sizes — always convert to conditional percentages.
Conditional % = cell count ÷ total of the row or column the explanatory variable sits in.
- Explanatory in rows → row percentages.
- Explanatory in columns → column percentages.
Compare the % pattern of the response variable across levels of the explanatory variable.
Visual alternative: double bar chart.

Plot: X = explanatory (horizontal), Y = response (vertical). Each point = one individual.
Describe a scatterplot the same way you describe a distribution: overall pattern + deviations.

Overall pattern = Direction + Form + Strength

Direction: positive (both ↑ together), negative (one ↑, other ↓), or none.
Form: linear, curvilinear, clusters, or other. Always state the simplest shape that fits.
Strength: how tightly points hug the form. Hard to judge by eye — the same data can look strong or weak depending on axis scaling, which is exactly why we need a numerical measure (r).

Deviations = Outliers

Points that don't follow the pattern.
Key distinction: an outlier can be extreme in magnitude but still consistent with the direction of the relationship (e.g., the elephant in gestation/longevity data), vs. an outlier that breaks the pattern entirely.

Labeled scatterplots: color/mark subgroups within the data (e.g., hot dog type) to reveal more structure.

Definition: r measures the strength and direction of a LINEAR relationship between two quantitative variables.

Range: −1 ≤ r ≤ 1

Unitless, unaffected by changing units of measurement (feet→meters, etc.) on either variable.
Only captures linear association. A strong curvilinear relationship can have r ≈ 0 (fuel-usage vs. speed example: r = −0.172 despite a near-perfect curve). Conversely, a curved relationship can have r close to 1 (incentive/response-rate example, r = 0.876). → r alone cannot tell you if a relationship is linear — always look at the scatterplot first.
Very sensitive to outliers.
- An outlier that goes against the pattern → weakens (pulls r toward 0).
- An outlier that is consistent with the pattern's direction → strengthens r (pulls it toward ±1).
r and the scatterplot are a package deal — never report r without having looked at the plot.

Used only when the scatterplot shows a linear form.

Equation: Ŷ = a + bX

Least squares criterion: the line that minimizes the sum of squared vertical deviations (residuals) of the points from the line.

Formulas (given from software output):

Interpreting the slope: the average change in Y for a 1-unit increase in X.

Example: slope = −3 → "for every year older a driver is, max legibility distance decreases on average by 3 feet."

Interpreting the intercept: predicted Y when X = 0 (not always meaningful in context).

Prediction: plug an X value into the equation to get a predicted Y (Ŷ).

Extrapolation — know this cold:

Predicting Y for X values outside the range of the observed data = extrapolation.
There's no evidence the linear pattern continues beyond the data → unreliable, avoid it.
Classic exam trap: "predict for age 90" when data only goes up to 82.

Core principle (repeated constantly — will be tested): Association does NOT imply causation.

Lurking variable: a variable not included in the study (not X, not Y) that could substantially change how you interpret the X–Y relationship.

Two ways a lurking variable can operate:

Common cause — the lurking variable affects both X and Y, creating a fake association with no direct causal link (Firefighters vs. Fire Damage → real driver is fire severity).
Confounding — the lurking variable is tangled up with X, so its effect can't be separated from X's effect on Y (US vs. international SAT scores → confounded with which subgroup of each population takes the test).

Simpson's Paradox

When including a lurking variable reverses the direction of an association seen in the aggregated (combined) data.
Classic example: Hospital A has a higher overall death rate than Hospital B, but once you split by severity of illness, Hospital B actually has the higher rate in both severity groups. Explanation: Hospital A treats far more severely-ill patients, skewing the overall (unadjusted) numbers.
Not all lurking variables cause a paradox — sometimes they just deepen understanding without flipping the direction (e.g., SAT vs. ACT by state).

Explanatory = X = predictor. Response = Y = outcome.
C→Q: boxplots. C→C: two-way table + conditional %. Q→Q: scatterplot + r + regression.
Scatterplot description = direction, form, strength + outliers.
r: only linear, only −1 to 1, unitless, outlier-sensitive, useless alone for judging linearity.
Regression line: Ŷ = a + bX, b = r(Sy/Sx), slope = "average change in Y per 1-unit increase in X."
Never trust predictions outside the data range → extrapolation.
Correlation/association ≠ causation. Watch for lurking variables and Simpson's Paradox.