Week 13 — Categorical outcomes

MATH 21003 · Introduction to Statistical Methods · Fall 2026 · Week 13 (Nov 16–20, 2026)

Why this week matters

For two weeks we have asked the questions of inference about a single number — is this one proportion different from a benchmark? is this one mean what we expected? This week the question becomes a comparison: two groups, a yes/no outcome, and the everyday question that runs through medicine, public health, and the life sciences — does the rate differ between these groups, and by how much? A treated group versus a control group. People who took a drug versus people who did not. Exposed versus unexposed.

The honest answer needs two layers. First, how big is the difference? We have three standard ways to size it up — the risk difference, the relative risk, and the odds ratio — and they are not interchangeable. Second, could the difference be chance? That is the same hypothesis-testing engine from Week 12, now pointed at a two-way table through the chi-square idea. The rhythm: Monday builds the three risk measures from a 2×2 table; Wednesday turns the table into a test — expected counts and the chi-square logic; Friday reads a real clinical categorical-outcome case and writes a careful conclusion. The recurring caution this week: a number that compares two groups can be right and still be misread — a large relative risk on a rare outcome can be a tiny absolute change, and “associated” is still not “caused.”

A bridge from Weeks 10 and 12

You have already done most of the hard thinking. Back in Week 10 you read two-way tables for diagnostic testing, you worked with risk as a conditional probability — the chance of an outcome given a group — and you learned the base-rate caution that a rare condition makes a positive result harder to interpret. Everything this week is built on that table-reading. Relative risk and the odds ratio are just two of those conditional probabilities (or odds) placed side by side as a ratio.

And from Week 12 you already own the inference engine: a test statistic, a p-value, and a significance level, with the p-value read as strength of evidence (the habit from Week 11). This week adds no new logic of testing — it applies the Week 12 engine to a categorical comparison. When you read a chi-square result on Wednesday, you are doing exactly what you did with a \(t\)- or \(z\)-test: ask whether what you see is more than chance would produce.

Monday: rates, risks, and three ways to compare two groups

Start with the object everything rests on — the 2×2 table. Two groups down the side, a yes/no outcome across the top, four counts inside. Label them \(a\), \(b\), \(c\), \(d\):

A two-by-two table with rows exposed and unexposed and columns outcome yes and no, cells labeled a, b, c, d, with row totals a plus b and c plus d. A side panel lists risk in exposed equals a over a plus b, risk in unexposed equals c over c plus d, risk difference equals the difference of the two risks, relative risk equals the ratio of the two risks, and odds ratio equals a over b divided by c over d equals a d over b c, with a note that relative risk compares risks and the odds ratio compares odds. — A course-built schematic of a two-by-two table for two groups and a yes/no outcome. Down the left are two groups, exposed and unexposed; across the top is the outcome, yes and no. The four inside cells are labeled a, b, c, and d: a is exposed with the outcome, b is exposed without, c is unexposed with the outcome, d is unexposed without. The exposed row total is a plus b and the unexposed row total is c plus d. A panel on the right defines four measures from the same four cells: risk in the exposed equals a over a plus b; risk in the unexposed equals c over c plus d; the risk difference equals a over a plus b minus c over c plus d; the relative risk equals the ratio of those two risks; and the odds ratio equals a over b divided by c over d, which simplifies to a times d over b times c. A note reads that relative risk compares risks while the odds ratio compares odds.

The risk of the outcome in a group is just its proportion: for the exposed group, \(a/(a+b)\). From the two risks you can build three comparisons, each answering a slightly different question.

Risk difference — how many more cases per 100? Subtract the two risks. In the opportunity-cost-style clinical world a clean example is the CPR study you met last week: survival was \(11/50 = 0.22\) in the control group and \(14/40 = 0.35\) in the treatment group, a risk difference of

\[0.35 - 0.22 = 0.13,\]

13 more survivors per 100 patients. The risk difference is an absolute measure, on the same scale as the risks themselves, and it travels with a confidence interval just like Week 12: the \(90\%\) interval here is \((-0.027,\ 0.287)\). Because that interval contains \(0\), “no difference” is still plausible — the study cannot rule out chance. Contrast the fish-oil trial, where the heart-attack rate was \(0.0112\) on fish oil versus \(0.0155\) on placebo, a risk difference of \(-0.0043\) with a \(95\%\) interval of \((-0.0071,\ -0.0015)\). That interval lies entirely below \(0\), so fish oil really did lower the rate — by about \(0.15\) to \(0.71\) percentage points. Small, but real.

Relative risk — how many times as likely? Divide the two risks instead of subtracting. A relative risk (RR) of \(2\) means the outcome is twice as likely in one group as the other; an RR of \(0.73\) means the exposed group has \(73\%\) of the risk — a reduction. Relative risk is a ratio, which makes it feel dramatic, and that is exactly its trap: a relative risk can be large while the absolute change is tiny. “Doubles your risk” sounds alarming, but doubling a \(1\)-in-a-million risk is still almost nothing. Always read a relative risk next to the underlying risks.

Odds ratio — how do the odds compare? Odds are not the same as probability: the odds of an outcome are (cases)/(non-cases), so \(a/b\) in the exposed group. The odds ratio (OR) divides the two groups’ odds, and it has a tidy shortcut straight from the table:

\[\text{OR} = \frac{a/b}{c/d} = \frac{ad}{bc}.\]

An odds ratio of \(1\) means no association; above \(1\) the exposure goes with higher odds of the outcome; below \(1\), lower. The odds ratio matters because of how some studies are built. In a case-control study, researchers start by collecting people who already have the outcome (the cases) and people who do not (the controls), then look backward at exposure. That design is the only practical way to study a rare disease — but it means you cannot read the true risk of the disease from the table, because the researchers chose how many cases to include. You can still compute the odds ratio, and that is why case-control studies report it.

A clean clinical example: a case-control study of newborn pulmonary hypertension (PPHN) and a class of antidepressants (SSRIs) sampled \(337\) infants with PPHN and \(836\) without, then checked SSRI exposure. Among the cases, \(14\) were exposed and \(323\) were not; among the controls, \(6\) were exposed and \(830\) were not. The odds ratio is

\[\text{OR} = \frac{14 \times 830}{323 \times 6} = 6.00.\]

The odds of SSRI exposure were six times higher for infants with PPHN than for those without — strong evidence of an association worth investigating. Notice what we did not say: we did not say PPHN is “six times as likely,” because in a case-control design we cannot estimate that risk at all.

(One bridge to keep straight: odds ratios also show up in logistic regression — the Week 9 model for yes/no outcomes — but this week we read them directly from a 2×2 table, no model required.)

When relative risk and the odds ratio disagree

Students often treat RR and OR as the same number. They are not, and the gap between them depends on how common the outcome is.

Two hypothetical two-by-two tables. Left, common outcome about thirty percent: exposed 40 of 100 risk 0.40, unexposed 20 of 100 risk 0.20, relative risk 2.00 but odds ratio 2.67, the odds ratio overstates. Right, rare outcome about three tenths of one percent: exposed 4 of 1000 risk 0.004, unexposed 2 of 1000 risk 0.002, relative risk 2.00 and odds ratio 2.00, the rare-disease approximation. Caption: odds ratio approximates relative risk only when the outcome is rare. — A course-built schematic comparing relative risk and odds ratio on two hypothetical two-by-two tables, both built so the relative risk equals two. In the left table the outcome is common, about thirty percent overall: the exposed group has forty cases of one hundred, a risk of 0.40, and the unexposed group has twenty of one hundred, a risk of 0.20, so the relative risk is 2.00 but the odds ratio is 2.67, meaning the odds ratio overstates the relative risk. In the right table the outcome is rare, about three tenths of a percent overall: the exposed group has four cases of one thousand, a risk of 0.004, and the unexposed group has two of one thousand, a risk of 0.002, so the relative risk is 2.00 and the odds ratio is 2.00 as well, the rare-disease approximation. The takeaway is that the odds ratio approximates the relative risk only when the outcome is rare.

Both tables are rigged to have the same relative risk of \(2.0\). When the outcome is common (about \(30\%\) here), the odds ratio inflates to \(2.67\) — it overstates the relative risk. When the outcome is rare (about \(0.3\%\)), the odds ratio is \(2.00\), essentially equal to the relative risk. This is the rare-disease approximation: an odds ratio is a good stand-in for a relative risk only when the outcome is uncommon. It is why headlines that quote an odds ratio for a common outcome as if it were a relative risk can badly mislead.

Wednesday: two-way tables, expected counts, and the chi-square idea

Sizing up the difference is half the job. The other half is the Week 12 question: could the pattern in this table be chance? For categorical data the tool is the chi-square test of independence, and its logic is simpler than its name.

Begin with a question: what would the table look like if the two variables were unrelated? If a group made no difference, each group should show the same outcome rate as the table overall. Those “if-independent” counts are the expected counts, and they come from a single rule:

\[\text{expected count} = \frac{(\text{row total}) \times (\text{column total})}{\text{table total}}.\]

You do not memorize a derivation; you read the expected count as what we would expect in this cell if the two variables were independent. Then you compare the observed counts to those expected counts. Where the table departs from independence, observed and expected pull apart; where they match, the table looks independent.

Observed versus expected counts for mosquito-net use and malaria. Net row: 22 malaria (expected 26), 78 no malaria (expected 74), total 100. No-net row: 30 malaria (expected 26), 70 no malaria (expected 74), total 100. Column totals 52 and 148, grand total 200. A side panel computes the expected count as row total times column total over n equals 100 times 52 over 200 equals 26, one cell's contribution as 22 minus 26 squared over 26 equals 0.62, the total chi-square as 1.66, with degrees of freedom one, not large, so no clear association; software gives the p-value. — A course-built figure showing observed versus expected counts for a two-by-two table of mosquito-net use and malaria, from an OpenIntro ISLBS exercise. The left side is the table: the net group had 22 children with malaria (expected 26 if net use and malaria were independent) and 78 without (expected 74), totaling 100; the no-net group had 30 with malaria (expected 26) and 70 without (expected 74), totaling 100; column totals are 52 with malaria and 148 without, grand total 200. The net-and-malaria cell is highlighted. The right panel shows the chi-square idea with one cell: the expected count if independent is one hundred times fifty-two over two hundred, which equals 26; that one cell’s contribution is 22 minus 26 squared over 26, equal to 0.62; adding all four cells gives a chi-square of 1.66; with degrees of freedom equal to one, that is not large, so there is no clear association, and software supplies the p-value.

The chi-square statistic adds up, over every cell, how far observed sits from expected — squared, then scaled by the expected count:

\[\chi^2 = \sum \frac{(\text{observed} - \text{expected})^2}{\text{expected}}.\]

You will not crank a full chi-square by hand in this course; the point is the logic the formula encodes. Bigger gaps between observed and expected counts make a bigger \(\chi^2\), and a bigger \(\chi^2\) means a smaller p-value — stronger evidence the variables are related. The p-value itself is a right-tail area, read off a chi-square curve whose shape is set by the degrees of freedom, \(df = (\text{rows} - 1)\times (\text{columns} - 1)\).

Chi-square density curves for degrees of freedom one, two, and four, all right-skewed and positive. On the df-two curve a vertical line marks an observed chi-square of 5.99 and the area to the right is shaded as the p-value equal to 0.05. Text notes that bigger observed-minus-expected gaps give a larger chi-square and a smaller p-value, and that the p-value is the right-tail area. — A course-built figure of the chi-square distribution showing the right-tail p-value idea. Three densities are drawn for degrees of freedom one, two, and four; all are positive and right-skewed. On the degrees-of-freedom-two curve an observed chi-square of 5.99 is marked with a red line, and the area to its right is shaded as the p-value, which equals 0.05. An annotation states that bigger gaps between observed and expected counts give a bigger chi-square and a smaller p, and that the p-value is the right-tail area.

So the whole test reads as one sentence: the chi-square measures how far the observed counts stray from what independence predicts, and the p-value is the chance of straying that far if the variables were truly unrelated.

Two contrasting reads make it concrete. In the mosquito-net table above, \(\chi^2 = 1.66\) on \(1\) degree of freedom — a small statistic, no clear association in these \(200\) children, even though the observed risks (\(0.22\) with a net versus \(0.30\) without) hint at a benefit. The hint is real but the evidence is thin: a point estimate is not proof, and a small study can leave chance on the table. Compare a study of how an iPod-selling script affected whether sellers disclosed a known defect: there the observed and expected counts pulled far apart, giving \(\chi^2 = 40.13\) on \(2\) degrees of freedom — astronomically far into the tail, decisive evidence the script and the disclosure were related.

Comparing two proportions, formally

When the table is exactly 2×2, the chi-square test and the two-proportion \(z\)-test are two routes to the same conclusion, and the \(z\) route fits last week’s machinery exactly. The test statistic keeps the familiar form,

\[z = \frac{(\hat p_1 - \hat p_2) - 0}{\text{SE}},\]

— an estimate, minus the null value of “no difference,” over a standard error — and the p-value is a tail of the normal curve. Take the mammogram study: over \(\approx 90{,}000\) women, breast-cancer deaths were \(500\) of \(44{,}925\) in the screened group and \(505\) of \(44{,}910\) in the control group. The difference in death rates is about \(-0.00012\), giving \(z = -0.17\) and a two-sided p-value of \(0.8650\) — nowhere near significant, so we fail to reject: no detectable difference in breast-cancer deaths. The chi-square test on the same \(2\times2\) table agrees, with \(\chi^2 = 0.02\) on \(1\) degree of freedom (for a \(2\times2\) table the two tests are equivalent, so use whichever the output shows).

The mammogram study also makes a Week-12 point vivid: with a sample this enormous, the test had ample power to detect even a small effect — and it found none. That is the honest opposite of the other classic warning: huge samples can make a trivial difference “significant” — and, just as usefully, they can give us enough precision to rule out any difference large enough to matter in context. A failure to reject is never proof of “no effect”; what a very large, precise study can support is that any remaining difference is small enough to be practically unimportant. Statistical discernibility and practical importance are still two different things.

Friday: reading a clinical categorical-outcome case

Friday pulls the week together on a real case, read end to end. The skill is not computing — it is interpreting: given a 2×2 table or a printed chi-square result, say what the comparison is, choose the right measure, and write a conclusion a careful reader would trust.

Work the mosquito-net case as that exercise. The risk of malaria was \(22/100 = 0.22\) with a net and \(30/100 = 0.30\) without — a risk difference of \(-0.08\) and a relative risk of \(0.22/0.30 \approx 0.73\), meaning netted children had about three-quarters the risk. On its face, a protective effect. But the chi-square is only \(1.66\) on \(1\) degree of freedom — the evidence from these \(200\) children is not strong enough to rule out chance. The responsible conclusion holds both thoughts at once: the observed pattern suggests nets help, but this study is too small to be convincing; a larger trial is needed. That sentence — an effect estimate and an honest read of the evidence behind it — is exactly what Week 13 is training.

A second discipline carries over from all term: a chi-square test, or a risk ratio, shows association, not causation by itself. Whether you can say “caused” still depends on the study design from Weeks 2 and 6 — a randomized experiment earns causal language; an observational comparison does not, no matter how small the p-value.

Common mistakes

Reading an odds ratio as a probability or a relative risk. An OR of \(6\) does not mean “six times as likely.” It compares odds, and it only approximates a relative risk when the outcome is rare.
Quoting a relative risk without the absolute risk. “Doubles the risk” can mean a huge change or a negligible one. Always pair RR with the underlying risks or the risk difference.
Trying to get a risk or a relative risk from a case-control study. The design fixes how many cases there are, so the disease risk is not estimable — report the odds ratio.
Thinking expected counts are something you observed. Expected counts are what independence would predict, not data. The test compares the two.
Treating a big \(\chi^2\) as proof of a strong or important effect. A large \(\chi^2\) (or tiny p) rules out chance; it says nothing about the size of the association, and with huge samples even trivial associations clear the bar.
Saying “associated” and meaning “caused.” Chi-square and risk ratios measure association. Causation still rides on the study design.

What you should be able to do by Friday

By the end of Week 13 you should be able to:

read a 2×2 table and compute a risk, a risk difference, a relative risk, and an odds ratio (\(ad/bc\)), and say in words what each one means;
explain why a case-control study reports an odds ratio rather than a risk, and when an odds ratio approximates a relative risk;
explain expected counts as “what we would expect if the variables were independent,” and read the chi-square logic — bigger observed-versus-expected gaps give a bigger \(\chi^2\) and a smaller p-value;
read a chi-square result (the statistic, the degrees of freedom, the p-value) or a two-proportion test and conclude in context;
keep association separate from causation, and a relative measure separate from an absolute one;
write a careful conclusion that pairs an effect estimate with an honest read of the evidence behind it.

Assignments this week

Monday check. A short in-class concept check on reading a 2×2 table: compute a risk, a risk difference, and a relative risk, and say what each means. Plan for about 3–5 minutes. Sheet: Week 13 Monday exit ticket.
Wednesday check. A short application on a risk/odds table: from a 2×2, find the relative risk and the odds ratio, or read a chi-square result and state the conclusion. Plan for about 8–12 minutes. Sheet: Week 13 Wednesday exit ticket.
🔒 Friday quiz — handled through Blackboard or in class as directed. The quiz prompt is not posted here. Timing and submission details live in Blackboard.
🔒 Project checkpoint — this week falls in the course project window. Your checkpoint deliverable and its due date are posted in Blackboard, not here.
No new homework opens this week. Homework 6 (Weeks 11–12) was the last biweekly set; from here the project carries the synthesis work.

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