BB2025 Chap.7 Correlation, Regression and Categorical Data
Correlation, Regression and Categorical Data
Let the outcome scale set the language
Correlation treats two continuous variables symmetrically and summarises their pattern. Regression distinguishes an outcome from explanatory variables and estimates expected change or prediction. Contingency-table methods compare counts and proportions.
The same dataset can support different questions, so state whether the goal is association, prediction, explanation or categorical independence before interpreting coefficients.
Read coefficients conditionally
A linear regression slope gives expected outcome change for one predictor unit while other included predictors are held constant. That condition is mathematical rather than automatically causal.
Predictor selection should follow the conceptual model because unnecessary adjustment can reduce precision or introduce bias. Residual plots test the functional form, spread and influence that the fitted equation assumes.
Preserve categorical denominators
Pearson chi-square addresses independent categorical observations. McNemar's test uses discordant pairs from linked binary measurements.
A p value should be accompanied by cell proportions and an effect measure such as a risk difference, risk ratio or odds ratio where the design supports it. Name the denominator in words so direction and conditioning remain visible.
What this chapter covers
- 01
Pearson and Spearman correlation
- 02
Scatterplots and nonlinear patterns
- 03
Simple and multiple linear regression
- 04
Conditional coefficient interpretation
- 05
Residuals and influential observations
- 06
Collinearity and predictor roles
- 07
Pearson chi-square association
- 08
McNemar paired binary comparison
Worked example · free
Use a regression equation without causal overreach
- 3Substitute the printed predictor values.
- 2Compute the predicted score.
- 3Interpret the slope with its condition and limit.
Key terms
- Pearson Correlation
- A bounded coefficient summarising the direction and strength of linear association between two continuous variables.
- Spearman Correlation
- A rank-based coefficient summarising the direction and strength of a monotonic relationship.
- Regression Slope
- The expected outcome change for one predictor unit under the conditions represented by the fitted model.
- Residual
- The difference between an observed outcome and the value predicted by a fitted model.
- Collinearity
- Overlap among predictors that makes their separate coefficient estimates unstable or imprecise.
- McNemar Test
- A test comparing discordant paired binary outcomes across two linked conditions or times.
Correlation, Regression and Categorical Data FAQ
Does a high correlation mean one variable predicts another well?
Not necessarily. Correlation summarises a sample's linear or monotonic association and does not evaluate calibration, prediction error or performance in new data. Outliers, subgroup mixing and restricted ranges can change it substantially. A prediction model needs explicit direction, validation and error assessment beyond a single pairwise coefficient.
What does holding other predictors constant mean?
It means the coefficient compares fitted outcomes for observations differing by one predictor unit while the model values of the other included predictors are fixed. This conditional comparison may be scientifically useful, but it becomes causal only under design and adjustment assumptions that rule out relevant confounding and inappropriate control variables.
When should I use McNemar instead of chi-square?
Use McNemar when binary results are linked within the same participants or matched pairs across two conditions. It analyses the discordant pairs that changed category. Pearson chi-square independence fits unrelated observations. The tables may look identical, but the reference variability changes because pairing must be preserved.
Exam move
For regression questions, write the equation, units and outcome direction before substitution. Interpret every slope with its predictor unit and conditioning phrase, then return to the design before using causal language. For correlation, draw the possible scatterplot shapes that produce a misleading summary.
For categorical data, label rows and columns with denominators and decide independence or pairing before choosing Pearson chi-square or McNemar. Finish with an effect measure, not a p value alone. Build a residual checklist from four visual patterns: curvature, widening spread, isolated influence and subgroup structure.
For every categorical table, calculate one row proportion and one column proportion and explain why they answer different questions. That exercise makes conditioning visible and reduces odds, risk and prevalence reversals under time pressure. Write one example in which adjustment reduces a coefficient and another in which it increases it.
Explain both through the direction of confounding instead of treating change as proof that one model is correct. Re-express one coefficient after changing the predictor unit, such as years to decades, and verify that fitted values stay unchanged. This separates numerical scaling from substantive effect. Do the same for a reference category and track which comparisons reverse sign.
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