Module 3 ‒ Describing & Presenting Bivariate Data

Module Context & Learning Goals

• Module 3 focuses on Describing and Presenting Bivariate Data (two‐variable situations)
• Central questions introduced with the Florida manatee vs. boating controversy:
  • Do two variables “move together”?
  • If yes, how strong, what shape, and what interpretation?
• Two major statistical tools explored
  • Correlation – quantifies linear association
  • Regression – builds a predictive/causal model (mostly linear in this module)
• Underpinning skills
  • Correct graph choice & construction
  • Recognition of lurking variables, outliers, non-linear patterns, extrapolation risks

Fundamental Concepts & Vocabulary

• Bivariate Data
  • Two measurements taken on the same experimental unit (EU)
  • Data recorded as pairs: ((xi, yi)), assumed independent & unbiased
• Variable Type Combinations
  1. 2 Qualitative → contingency table, side-by-side bars
  2. 1 Qualitative + 1 Quantitative → multiple box-and-whisker plots,
     compare center & spread across categories
  3. 2 Quantitative → scatterplot, correlation, regression (focus of module)
• Key statistical symbols
  • Sample correlation coefficient: $r$ (direction & strength, (-1\le r\le 1))
  • Regression equation: $\hat{y}=b<em>0+b</em>1x$ where
  • $b_0$ = intercept (estimate of (y) when (x=0))
  • $b_1$ = slope (estimated change in (y) for one-unit change in (x))
  • Coefficient of determination: $R^2$ (proportion of (y)-variability explained by (x);
    (0\le R^2\le 1))

Visualization Toolkit

• Contingency Table (hair colour vs. hair type example)
• Side-by-Side Bar Chart – categorical comparison
• Multiple Box-and-Whisker Plots – electric-bill data by U.S. region
  • Reveals medians, IQR, outliers for each qualitative level
• Scatterplot – default for two quantitative variables
  • Horizontal axis = explanatory variable (often denoted (x))
  • Vertical axis = response variable ((y))
  • Assess form (linear? curved?), direction (positive/negative), strength (tightness), outliers
  • Never draw a fitted line on a correlation‐only graphic (save line for regression section)

Case Study: Boating & Manatee Deaths (1991–2000)

• Data pairs ((#\text{boats} \times 10\,000, #\text{deaths})) show striking pattern
• Scatterplot reveals approximate linear trend upward
• Calculated correlation: $r\approx0.92$ (very strong positive)
• Talking points
  • Correlation ≠ causation (boats may hit manatees, but also both variables rise with population/economic growth)
  • Graph conveys pattern clearer than separate bar charts originally shown

Correlation in Depth

• Usage criteria
  • Both variables quantitative
  • Relationship roughly linear
• Interpretation components
  1. Direction
  • Positive: (y) increases with (x) ((r>0))
  • Negative: (y) decreases with (x) ((r<0))
  • None: scatter cloud ((r\approx0)) or flat line
  1. Strength
  • $|r|=1$ perfect linear fit
  • $|r|\approx0.7$ strong (practically meaningful)
  • $|r|<0.3$ weak/negligible (to be tested in Module 13)
• Mathematical formula
  $r=\frac{\sum<em>{i=1}^{n}(x</em>i-\bar{x})(y<em>i-\bar{y})}{\sqrt{\sum</em>{i=1}^{n}(x<em>i-\bar{x})^2}\;\sqrt{\sum</em>{i=1}^{n}(y_i-\bar{y})^2}}$
• Invariance properties
  • Swapping (x,y) leaves (r) unchanged
  • Adding/multiplying one variable by a constant leaves (r) unchanged (linear rescaling)
• Pitfalls & “Correlation Concerns” (A–F mnemonic)
  A. Always Graph First – detect non-linear forms (enzyme vs. pH example, $r=0$ but clear curve)
  B. Beware Outliers – can inflate/deflate (r) dramatically (husband-wife ages: (r=0.44) vs. (0.97) after removing data error)
  C. Correlation ≠ Causation – most studies are surveys; cannot assign cause
  D. Third-Variable (Lurking) Problem – e.g., TV ownership vs. life expectancy mediated by income
  E. Extrapolation Risk – don’t interpret association outside data range (HS GPA vs. TV hours cartoon)
  F. Form Misinterpretation – identical (r) values can mask radically different scatterplot shapes (Anscombe-like slide, all (r=0.7))
• Appropriate wording: tends to, linked, associated, connected, tied to

Regression Essentials

• Aim: predict or explain (y) from (x)
• Best-Fit (least-squares) line minimizes vertical residuals: $e<em>i = y</em>i - \hat{y}_i$
• Simple Linear Formula: $\hat{y}=b<em>0+b</em>1x$
• Example: Onion frying experiment (controlled)
  • $\text{Onion size (cm)} = 24.6 + 2.8\, \text{Salt (tsp)}$
  • $R^2=0.87$ → 87 % of onion-size variability explained by salt concentration
  • Replicates at each salt level confirm experimental design
• Coefficient of Determination ((R^2))
  • Quantifies explanatory power (tightness only, no direction)
  • $R^2 = (\text{Correlation})^2$ for simple linear regression when assumptions met
• Distinction r vs. R²
  • Correlation: magnitude & sign, hypothesis testable (Module 13)
  • Regression: magnitude only, slope sign indicates direction, test slope/intercept separately
• Hypothesis framework (preview)
  • $H<em>0: b</em>1 = 0$ (no linear relation) vs. $H<em>A: b</em>1 \ne 0$
  • Significance of $b_0$ also testable
• Interpretation nuance
  • Causality requires controlled manipulation of (x) (e.g., fertilizer dose experiment)
  • Survey-based regression offers prediction but not guaranteed cause (chlorophyll vs. fluorescence example)
• Presenting Regression Results: include title, properly labelled axes, plotted line only within data range, full equation & $R^2$ value

Regression Concerns (Parallel to Correlation)

• Impact of outliers on slope & $R^2$
• Extrapolation beyond observed (x) unsafe
• Non-linearity → try transformations (e.g., log health spending vs. life expectancy)
• Lurking variables in non-experimental data
• Measurement error in (x): standard linear regression assumes (x) known without error

Real-World & Historical Applications

• Health Spending vs. Life Expectancy
  • Raw relation curved; log transform linearizes; emphasizes need for graphical diagnostics
• Smoking & Lung Cancer (Lawrence Garfinkel’s million-person studies)
  • Early correlations linked smoking to cancer before causal mechanics fully established
• Race & Lung Cancer Rates (Haiman et al., NEJM 2006)
  • Predictive regression by age, cigarettes/day, race – controversial; survey limits causal claims; HPV as potential lurking variable
• Student Projects
  • Tongue surface area vs. licks to Tootsie Pop: $r=-0.40$ (not statistically significant); demonstrates volunteer sampling & result reporting

Ethical & Practical Interpretation Guidelines

• Use association language unless experimental causation established
• Justify any data removal (outliers) transparently in Methods section
• Always complement numerical summaries (r, R²) with visual evidence
• Recognize societal implications of statistical claims (environmental policy, healthcare equity, public health)

Quick Reference: Decision Tree

• Two quantitative variables? → draw scatterplot first
  • Linear? → choose Correlation if question is association; choose Regression if question is prediction/causation and (x) measured reliably
  • Non-linear? → consider transformations or non-linear regression
• Mixed variable types?
  • 1 Qual + 1 Quant → boxplots/means comparison
  • 2 Qual → contingency table/chi-square

Formula Bank & Symbols

• Correlation coefficient:
  $r=\frac{\sum(x-\bar{x})(y-\bar{y})}{\sqrt{\sum(x-\bar{x})^2}\;\sqrt{\sum(y-\bar{y})^2}}$
• Regression line: $\hat{y}=b<em>0+b</em>1x$
• Slope estimator: $b<em>1 = r\left(\frac{s</em>y}{s_x}\right)$
• Intercept estimator: $b<em>0 = \bar{y} - b</em>1\bar{x}$
• Coefficient of determination: $R^2 = 1 - \frac{\text{SS}<em>\text{Residual}}{\text{SS}</em>\text{Total}}$

Key Take-Aways

• Correlation quantifies linear association; independent of units; non-causal
• Regression predicts (y) from (x); provides functional equation & (R^2); causation only under controlled design
• Visualization & critical thinking (outliers, lurking variables, extrapolation) are indispensable
• Always match statistical language (linked, associated, tends to) to design limitations