GEA1000 Chap.2 Categorical Data: Proportions and Confounding
Categorical Data: Proportions and Confounding
Denominators carry the question
Marginal and joint proportions use the whole table, while conditional proportions use a restricted row or column. Risk divides outcomes by everyone in a group; odds divide outcomes by non-outcomes in that group. Ratios preserve the comparison order and require the outcome noun in the interpretation.
The words in a question identify the correct total.
Out of everyone points to the grand total. Among exposed points to the exposed row. Among cases points to the outcome column. The numerator can be the same cell while the answers differ because the sample space has changed.
Write the conditioning phrase beside the denominator before entering numbers.
Risk and odds use different comparisons
Risk is outcomes divided by the full group, while odds are outcomes divided by non-outcomes. A risk ratio compares risks and an odds ratio compares odds. Both equal one under no association and generally agree on direction, but they need not agree in size.
When outcomes are common, the odds ratio can sit much farther from one.
Interpretation depends on the outcome label as well as the ratio. A value above one for recovery can be favourable, while a value above one for failure is adverse. Reversing the comparison groups takes the reciprocal.
These checks catch calculations that preserve the digits while losing the question.
Slicing can change the story
Aggregate associations may weaken, disappear or reverse after data are sliced by a third variable.
Simpson's paradox flags a composition problem, while confounding asks whether the third variable provides an alternative explanation for the observed exposure-outcome association.
Always print the overall rates and each stratum rate before naming the pattern. A full reversal differs from a change in magnitude and from opposite directions across strata.
The case counts matter too, because rates from a small stratum are less stable. The table reveals the numerical pattern; the study design and timing of variables determine whether the third variable is a plausible confounder.
Keep association separate from causation
A large risk ratio or odds ratio does not license a causal verb. Return to assignment, selection and alternative explanations.
The categorical measure describes the observed association; the design decides whether that association can support a causal interpretation.
What this chapter covers
- 01
Marginal, joint and conditional proportions
- 02
Two-way table denominators
- 03
Risk and risk ratios
- 04
Odds and odds ratios
- 05
Simpson's paradox
- 06
Confounding and stratified comparisons
Worked example · free
Risk and odds from one table
- 3Calculate both group risks and form the risk ratio.
- 3Calculate both group odds and form the odds ratio.
Key terms
- Marginal Proportion
- A category share calculated using the total number of observations.
- Joint Proportion
- The share of all observations satisfying two categorical conditions together.
- Conditional Proportion
- A category share inside a row or column fixed by a conditioning phrase.
- Risk Ratio
- The risk in one group divided by the risk in a comparison group.
- Odds Ratio
- The odds in one group divided by the odds in another group.
- Simpson's Paradox
- A reversal between an aggregate association and the relevant within-group associations.
Categorical Data: Proportions and Confounding FAQ
How is the correct denominator selected?
Write the conditioning phrase before calculating. Out of everyone signals the grand total; among the exposed signals the exposed row; among cases signals the outcome column. The same cell can produce different proportions because the conditioned totals differ.
Why are a risk ratio and odds ratio different?
Risk compares outcomes with everyone in a group, while odds compare outcomes with non-outcomes. Their ratios usually agree on whether association is above or below one, but odds ratios can sit farther from one when outcomes are common.
Does every confounder create Simpson's paradox?
No. A confounder can distort the size of an association without reversing its direction, and a reversal is a clue rather than a complete causal diagnosis. Inspect stratified results and the data-generating story before deciding what the third variable does.
Exam move
Draw a two-by-two table and practise asking three questions of the same highlighted cell: what share of the whole table is this cell, what share inside its row, and what share inside its column. Then convert the row proportions into risks and the case to non-case comparisons into odds. Always interpret the ratio with the outcome, numerator group and denominator group stated in that order.
For confounding, use an aggregate table and at least two stratified tables. Calculate every proportion before naming Simpson's paradox; the directions may reverse uniformly, differ by stratum or simply shrink. Use the available chapter and tutorial solutions to study which reasoning steps are shown, then invent new counts rather than reusing their questions.
Check every custom table by recomputing row totals and totals across strata. End by asking whether the slicing variable precedes the exposure and can affect the outcome, because arithmetic alone does not prove a causal explanation.
Working through Categorical Data: Proportions and Confounding in GEA1000? Sia is AskSia’s AI Statistics tutor — ask any GEA1000 Categorical Data: Proportions and Confounding question and get a clear, step-by-step explanation grounded in how GEA1000 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
