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Module 7 — Relationships between scale variables

Question: does blood pressure rise with BMI — and by how much, holding age constant?

Open the sample (Ctrl+Shift+O).

Correlation

Analyze → Correlate → Bivariate…, variables Age, BMI, SystolicBP, HeartRate, Run.

The matrix gives each pair's Pearson r and its p. Expect Age–SystolicBP and BMI–SystolicBP clearly positive, and everything involving HeartRate near zero (the planted null). Remember module 4's scatterplots: r is only trustworthy when the cloud is roughly linear — always look first.

Linear regression

Analyze → Regression → Linear…, dependent SystolicBP, predictors Age and BMI, Run.

How to read the three blocks:

  • Model summary — R²: the share of SystolicBP's variance the two predictors explain together.
  • ANOVA — whether the model beats no model at all.
  • Coefficients — the substance. The BMI coefficient is mmHg per BMI unit, holding Age constant — the "holding constant" is what regression adds over correlation. The constant is the fitted value at Age 0, BMI 0 — outside the data, not meaningful on its own.

Write the fitted equation out once by hand: SystolicBP ≈ b₀ + b₁·Age + b₂·BMI. Predict the SBP of a 50-year-old with BMI 30 and sanity-check it against the Explore output from module 3.

Checking the model, not just reading it

A regression will print coefficients for any data you hand it, including data that violate its assumptions. Re-run the model with Diagnostics ticked and you get three plots that check them:

  • Residuals vs Fitted — should be a shapeless cloud. A curve means the relationship is not linear; a funnel means the spread depends on the prediction.
  • Scale-Location — the same homoscedasticity check made easier to read, plotting √|standardized residual| against the fitted value. A rising trend is the funnel again.
  • Normal Q-Q of the standardized residuals — points on the reference line mean the residuals are plausibly normal, which is what the p values assume.

A separate Casewise diagnostics option adds the per-case table — leverage and Cook's distance — which answers a different question: is a handful of cases steering the whole fit?

Reading these before quoting R² is the difference between a model you have run and a model you can defend.

Exercise

  1. Add TotalCholesterol_mmolL as a third predictor. Does R² move much? Is its coefficient significant?
  2. Correlate PhysicalActivity_minWk with TotalCholesterol_mmolL — the codebook's documented null pair. Report it honestly.
  3. Spearman instead of Pearson (the method option): rank-based, robust to outliers and curvature. Do the Age/BMI/SBP conclusions change?
  4. Re-run the regression with Diagnostics on. Is Residuals vs Fitted a shapeless cloud, or does it fan out? Then add Casewise diagnostics — does any single case stand out on Cook's distance?

The syntax trail

correlate variables=[Age, BMI, SystolicBP, HeartRate]
linear_regression dependent=SystolicBP, predictors=[Age, BMI]
linear_regression dependent=SystolicBP, predictors=[Age, BMI], diagnostics=true, casewise=true
correlate variables=[PhysicalActivity_minWk, TotalCholesterol_mmolL]