SO5005 Chap.8 Correlation, Regression and Association
Correlation, Regression and Association
Quantify patterns in units
Correlation summarises direction and strength, while simple regression expresses an expected outcome through an intercept and slope. The slope is the expected outcome change for one predictor unit.
R-squared describes variation captured by the fitted linear model, not the percentage of observations predicted correctly.
Use counts for categorical association
Contingency tables compare observed counts with those expected under independence. Sparse expected counts affect procedure choice. Odds equal probability divided by one minus probability, and an odds ratio compares group odds.
It differs from a risk ratio when outcomes are common.
What this chapter covers
- 01
Scatterplots
- 02
Correlation
- 03
Regression slopes
- 04
R-squared
- 05
Contingency tables
- 06
Odds and odds ratios
Worked example · free
Distinguish odds from risk
- 2Compute exposed odds.
- 2Compute reference odds.
- 2Divide and interpret.
Key terms
- Regression Slope
- The expected change in an outcome for a one-unit increase in a predictor under a fitted model.
- Expected Count
- The cell count predicted under independence from row total times column total divided by grand total.
- Odds
- The probability of an event divided by the probability of its non-event.
Correlation, Regression and Association FAQ
Does correlation establish causation?
No. Correlation records a pattern of co-variation. A causal interpretation requires temporal order, credible control of common causes, appropriate measurement and a design capable of isolating change. A strong coefficient cannot supply those design features.
How should a regression slope be reported?
State the outcome change, predictor increment and units, then limit the interpretation to the fitted range and model. For categorical predictors, name the reference category. Avoid causal verbs unless the design independently supports them.
Why can an odds ratio look larger than a risk ratio?
Odds use events divided by non-events, while risk uses events divided by all people in the group. As an outcome becomes common, these quantities separate. Report the measure actually calculated and avoid translating odds language into risk language.
Assessment move
Redraw one scatterplot and one two-by-two table from raw values. Interpret slope, fitted scatter, expected counts, odds and the reference group aloud. Then write a sentence that describes association without inserting causal language. For scatterplots, label the axes with units and describe form, direction, strength and unusual observations before calculating a coefficient.
Add one extreme point and notice how correlation and slope respond. Draw a curved pattern to see why a near-zero linear correlation can coexist with a strong non-linear relationship. For regression, write the fitted equation and substitute two predictor values within the observed range. The difference between predictions gives the slope a concrete meaning.
Check whether zero is within the observed predictor range before interpreting the intercept. Translate R-squared as the proportion of variation described by the fitted model in this sample, then list the variation it leaves unexplained. For categorical association, reconstruct the row and column totals and calculate one expected count by hand. Identify sparse cells before accepting a chi-square result.
Practise odds from both probability and cell counts, always writing event over non-event. Compute an odds ratio in both directions and observe that reversing the reference takes the reciprocal. Compare it with the risk ratio when the outcome is common so the difference becomes visible rather than memorised.
End each exercise by naming one plausible common cause and one design feature that would strengthen causal interpretation. The final result sentence should use associated with unless the design provides independent grounds for a causal verb. Create a small table with an outcome that is rare and then common, holding the risk ratio roughly similar, and observe how the odds ratio diverges.
This builds intuition for why odds language cannot be replaced casually. In regression practice, change the predictor unit from years to decades and confirm that the slope rescales while fitted values do not. Examine a scatterplot with two clusters and calculate whether a single line hides group structure. For each analysis, state the population and range over which interpretation is intended.
Avoid extrapolating beyond observed values unless a justified model supports it. Finish with one diagnostic graph and one substantive sentence explaining what that graph changes. If it changes nothing, explain why it still confirms an assumption; otherwise revise the model, qualify the range or identify the observation that needs verification. Note the decision in the analysis record. Reproduce it during final checking.
Consistency remains part of reproducibility and transparent interpretation.
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