Association Between Variables

Context: FIT1043_MOC, FIT2086_MOC · is there a relationship between two variables? · method depends on the type pair · Pearson via df.corr() FIT2086 framing: Pearson correlation $R(\mathbf{x},\mathbf{y})=\frac{{\sum }_j(x_j-\bar{x})(y_j-\bar{y})}{ns(\mathbf{x})s(\mathbf{y})}\in [-1,1]$ measures linear association only — $R\approx 0$ does not imply independence ($y=x^2$ and points on a circle both give $R\approx 0$ despite deterministic association); and correlation $\ne$ causation.

Quick Revision

• 🎯 Objective: detect a relationship between two variables ➔ pick the tool by the variable-type pair.
• 📦 Core Components: cont–cont → scatter + Pearson $r$ | cat–cat → side-by-side bars | cat–num → side-by-side boxplots.
• ⚡ Key Constraint: Pearson $r$ measures linear association only — $r\approx 0$ can still hide a strong non-linear relationship; and correlation $\ne$ causation.

📝 How It Works

1. Two Continuous Variables

• Pearson correlation ➔ measures linear association; $-1\le r\le 1$ (−1 perfectly negative, +1 perfectly positive, 0 = no linear association).
• Scatter plot ➔ plot $x$ vs $y$ to see the relationship; combine with $r$ for strength + shape.
• Non-linear blind spot ➔ $y=x^2+\text{noise}$ gives $r\approx 0$ though clearly associated; a deterministic curve can give $r=0$.

2. Two Categorical Variables

• Side-by-side bar graphs ➔ compare each category’s distribution; if the bar shapes differ across groups, a possible association (e.g. cancer frequency vs ethnicity — if unchanged, unlikely associated).
• Contingency table ➔ table(a, b) quantifies the same comparison; association shows as a category’s split departing from the overall base rate (mechanics ➔ Categorical Summaries and Cross-Tabulation in R).

3. Categorical + Numeric

• Side-by-side boxplots ➔ split the numeric variable by category and compare boxplots; different distributions ⇒ association (e.g. price varies with number of rooms, but not much between two suburbs).

⚖️ Core Decision Matrix

Variable pair	Visualisation	Quantify
continuous × continuous	scatter plot	Pearson $r$
categorical × categorical	side-by-side bars	compare distributions
categorical × numeric	side-by-side boxplots	compare group boxplots

When It Flips: $r$ near $\pm 1$ tightens the scatter toward a line (e.g. $r\approx 0.9$ vs $0.44$); but $r$ only sees linear trend, so always plot as well as compute.

🔧 Screening many predictors in R (FIT2086 studio — wine.csv)

wine <- read.csv("wine.csv")
wine_corr = c()
for (i in 1:11)                                   # columns 1..11 are the attributes
{
  wine_corr[i] = cor(wine[,i], wine$quality)
  cat("Correlation between", names(wine)[i], "and quality = ", wine_corr[i], "\n")
}
cat("Variable with maximum correlation is:", names(wine)[which.max(abs(wine_corr))], "\n")

Expected output ($n=4898$ wines):

attribute	$r$ with quality	reading
alcohol	$+0.436$	strongest — the one real signal
density	$-0.307$	moderate negative
chlorides	$-0.210$	weak negative
volatile acidity	$-0.195$	weak negative
total sulfur dioxide	$-0.175$	weak negative
fixed acidity · residual sugar · pH · sulphates	$r<0.12$	negligible
citric acid · free sulfur dioxide	$r<0.01$	no linear association

• which.max(abs(...)) not which.max(...) ➔ a strong negative association is just as informative as a positive one; taking the absolute value first is what makes the screen correct.
• Building a vector in a loop ➔ wine_corr = c() creates an empty vector that grows by index assignment; names(wine)[i] recovers the $i$-th column’s name for the printout.
• What this screen is for ➔ ranking candidate predictors before modelling ➔ but a near-zero $r$ only rules out a linear relationship, so a variable dismissed here can still matter in a non-linear model.

⚠️ Common Mistakes

• 💡 Correlation ≠ causation ➔ a strong $r$ never proves one variable causes the other.
• 💡 $r=0$ ≠ no relationship ➔ it means no linear relationship; a scatter may reveal a clear curve (e.g. $y=x^2$).
• 💡 Ranking predictors by signed $r$ ➔ use which.max(abs(r)); the strongest association can be negative, and a signed sort silently buries it.

🧠 Active Recall

Data with $y=x^2+\text{noise}$ has $r\approx 0$ — is $x$ associated with $y$? What does this teach?

• Hint: Linear-only measure.

Answer

• Short answer: Yes — they are clearly associated, but non-linearly, so Pearson $r\approx 0$; always plot the data, don’t rely on $r$ alone.
• Why: Linear scope ➔ Pearson captures only straight-line association; symmetric curves cancel to $r\approx 0$.

Which visualisation detects association for each variable-type pair?

• Hint: Match tool to types.

Answer

• Short answer: cont–cont → scatter plot (+ Pearson $r$); cat–cat → side-by-side bar graphs; cat–num → side-by-side boxplots.
• Why: Distribution comparison ➔ association shows as differing distributions across groups, or a trend in the scatter.