This glossary collects the key concepts, people, and ideas you will meet in this lesson. Use it as a reference while you work through the material, or as a review before assessments. Type in the search box to filter entries.

Introduction and Overview

Lesson 5 gave us measures of disease frequency — prevalence, incidence, risk, rate. Lesson 6 took the same probabilistic vocabulary and applied it at the level of a single test. Lesson 7 brings these strands together: it combines the disease-frequency vocabulary with the 2×2 contingency logic to produce measures of association — the quantitative comparison between exposed and unexposed groups that is the central output of analytic epidemiology. The four content sections build up from the three ratio measures (Section 1: risk ratio, rate ratio, odds ratio), through difference measures and exposed-group attributable fractions (Section 2), to population-level attributable measures and how each measure relates to study design (Section 3), and finally to the hypothesis-testing and confidence-interval machinery that turns each of these point estimates into a defensible inference (Section 4).

Why Measure Association?

Measures of association assess the magnitude of the relationship between an exposure (a potential cause) and a disease. Unlike measures of statistical significance, which are heavily dependent on sample size, measures of association indicate the strength of the effect — how much more (or less) likely disease is in exposed compared to non-exposed groups (Tripepi et al., 2007).

Data Layout

Depending on study design, disease frequency can be expressed as incidence risk, incidence rate, prevalence, or odds. For risk data, the standard 2×2 table is:

For rate data, the denominator is person-time at risk rather than the number of individuals:

Three Ratio Measures of Association

Click each card to learn more:

Worked Example: Brazil Water Cistern Study

Worked Example: Migraine Incidence Rates

Relationships Among RR, IR, and OR

In general, IR values are further from the null (1) than RR values, and OR values are even further away (Cornfield, 1951; Knol et al., 2008). This can be visualised on a number line:

Introduction and Overview

Section 1 covered the three ratio measures (RR, IR, OR) — how many times more likely disease is in the exposed group compared to the unexposed. Section 2 turns to the parallel set of difference measures, which answer a different question: not how many times more, but how many extra cases occur because of the exposure. Difference measures lead naturally to attributable fractions in the exposed, which quantify how much disease in the exposed group can be attributed to the exposure itself.

Ratio vs. Difference Measures

The ratio measures from Section 1 (RR, IR, OR) tell us the relative strength of association, but they do not indicate the absolute number of cases attributable to the exposure. Difference (absolute effect) measures address this gap by computing how many additional cases occur because of the exposure; the choice between ratio and difference measures is itself a substantive scientific decision rather than a statistical convenience (Greenland & Pearce, 2015; Tripepi et al., 2007).

Risk Difference (RD) — Attributable Risk

The risk difference (RD), also called attributable risk (Wikipedia; Walter, 1976), is simply the risk in the exposed group minus the risk in the non-exposed group:

Similarly, the incidence rate difference (ID) is the difference between two incidence rates:

Attributable Fraction in the Exposed (AF_e)

The AF_e (also called the attributable fraction among the exposed) expresses the proportion of disease in exposed individuals that is due to the exposure, assuming the relationship is causal (Walter, 1976). It can be viewed as the proportion of disease in the exposed group that would be avoided if the exposure were removed.

AF_e ranges from 0 (where risk is equal, RR = 1) to 1 (where all disease in the exposed group is due to the exposure, RR = ∞). In case-control studies, AF_e can be approximated by substituting OR for RR.

Vaccine Efficacy

Vaccine efficacy is a special form of AF_e, where “not vaccinated” is the exposure (factor positive) and “vaccinated” is the comparison group. If 20% of unvaccinated individuals develop disease versus 5% of vaccinated individuals:

Introduction and Overview

Sections 1 and 2 worked at the level of the exposed group. Section 3 zooms out: even if an exposure powerfully causes disease in the exposed, its public-health importance also depends on how common it is in the population. The population attributable fraction (AF_p) captures this combination, and the section closes by mapping each measure of association onto the study design that produces it — tying back to HSCI 230 Lessons 4–6.

From the Exposed Group to the Entire Population

While RD and AF_e describe the effect of exposure among exposed individuals, public health decisions often require understanding the impact of an exposure on the entire population. Two key population-level measures address this:

Population Attributable Risk (PAR)

The PAR is the increase in overall population risk attributable to the exposure. It reflects both the strength of the association and the frequency of the exposure in the population — an idea originally developed by Levin (1953) for lung cancer and reviewed by Northridge (1995) as the key link between causal inference and public-health action (see also Wikipedia).

Population Attributable Fraction (AF_p)

The AF_p indicates the proportion of disease in the entire population that is attributable to the exposure, and which would be avoided if the exposure were removed (assuming causation and no confounding).

Confounding and AF_p

If confounding is present, adjusted estimates of RR should be used. The AF_p can then be estimated using:

where pd is the proportion of cases exposed to the risk factor, and aRR is the adjusted risk ratio. For multiple exposure categories, a summation formula is used.

Study Design and Measures of Association

Not all measures can be computed from all study designs. The following table summarises which measures are available:

^a Requires independent estimate of p(D+) or p(E+). ^b Estimated using OR. ^c Requires OR and independent estimate of p(E+|D+).

Introduction and Overview

Sections 1–3 produced point estimates of association — single numbers like RR = 2.5 or AF_p = 30%. Those numbers are useless without a quantification of how uncertain they are. Section 4 closes the lesson by introducing the standard error, hypothesis tests, and confidence intervals — the same statistical-inference machinery you previewed in HSCI 230, now applied directly to the measures you just learned to compute.

Standard Error

The standard error (SE) provides a measure of the precision of a point estimate — how much uncertainty exists in the estimate. For difference measures (RD, ID), the variance can be computed directly:

For ratio measures (RR, IR, OR), the variance is computed on the log scale using Taylor series approximations:

Hypothesis Testing

Significance testing is based on specifying a null hypothesis about the population parameter. The null hypothesis typically states there is no association:

• For difference measures (RD, ID): H₀: θ = 0
• For ratio measures (RR, IR, OR): H₀: θ = 1

An alternative hypothesis can be 1-tailed or 2-tailed. In general, 2-tailed hypotheses are preferred because 1-tailed hypotheses are harder to justify.

Test Statistics

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Confidence Intervals

A confidence interval (CI) (Wikipedia) reflects the level of uncertainty in a point estimate. A 95% CI means that if the study were repeated many times under identical conditions, 95% of the computed CIs would contain the true parameter value — it is a property of the procedure, not a probability statement about the parameter (Greenland et al., 2016).

Computing CIs

For difference measures, the CI is computed directly:

For ratio measures, the CI is computed on the log scale and then exponentiated:

The CI is symmetrical about lnθ but not about θ itself — this is why confidence intervals for ratio measures appear asymmetric.

Choosing the Right Statistical Test

The Pearson χ², Fisher’s exact, Wald, and likelihood ratio tests above are the workhorses for 2×2 tables and the regression-based measures of association you will meet in HSCI 410. But epidemiological analyses often require comparing means, proportions, or whole distributions across groups — sometimes paired, sometimes not, sometimes badly skewed. The right test depends on three structural questions about your data:

• Outcome type — continuous (means / medians) or categorical (counts / proportions)?
• Group structure — one group, two groups, or three or more? Independent or paired/matched?
• Distributional assumptions — can you defend approximate normality (or appeal to a large-sample CLT argument), or do you need a non-parametric (rank-based) alternative?

Comparison Table: When Used, How Calculated, How Interpreted

The table below summarises the tests most commonly reported alongside measures of association. The R code that follows runs each on simple built-in datasets so you can copy, paste, and read the output without external data files.

R Tutorial — Running These Tests

R has all of these tests in base (no packages required). The blocks below use the built-in datasets ToothGrowth, sleep, and iris — copy any block straight into your console.

# --- ONE-SAMPLE t-test: is mean tooth length different from 18 mm? ---
t.test(ToothGrowth$len, mu = 18)

# --- TWO-SAMPLE (independent) t-test: OJ vs. VC supplements ---
t.test(len ~ supp, data = ToothGrowth)                 # Welch (default)
t.test(len ~ supp, data = ToothGrowth, var.equal = TRUE)  # Student

# --- PAIRED t-test: sleep gain under two drugs, same 10 subjects ---
t.test(extra ~ group, data = sleep, paired = TRUE)

# --- ONE-WAY ANOVA: tooth length across 3 doses ---
ToothGrowth$dose <- factor(ToothGrowth$dose)
fit <- aov(len ~ dose, data = ToothGrowth)
summary(fit)

# Post-hoc pairwise comparisons (control familywise error)
TukeyHSD(fit)

# --- 2x2 table: exposure x disease ---
exposure <- matrix(c(30, 70, 10, 90), nrow = 2, byrow = TRUE,
                   dimnames = list(exposure = c("E+", "E-"),
                                   disease  = c("D+", "D-")))
exposure

chisq.test(exposure)                       # Pearson chi-square (with Yates' correction)
chisq.test(exposure, correct = FALSE)       # without correction
fisher.test(exposure)                      # exact test (preferred for small cells)

# --- McNemar's test: paired/matched binary data ---
# e.g. agreement between two diagnostic tests on the same 100 patients
paired <- matrix(c(40, 10,
                   25, 25), nrow = 2, byrow = TRUE,
                 dimnames = list(test1 = c("+", "-"),
                                 test2 = c("+", "-")))
mcnemar.test(paired)                       # with continuity correction
mcnemar.test(paired, correct = FALSE)       # without

# --- WILCOXON SIGNED-RANK (one-sample / paired) ---
wilcox.test(ToothGrowth$len, mu = 18)             # one-sample
wilcox.test(extra ~ group, data = sleep, paired = TRUE) # paired

# --- MANN-WHITNEY U / WILCOXON RANK-SUM (two-sample) ---
wilcox.test(len ~ supp, data = ToothGrowth)

# --- KRUSKAL-WALLIS (3+ groups, non-parametric ANOVA) ---
kruskal.test(len ~ dose, data = ToothGrowth)

# Post-hoc pairwise (Bonferroni-adjusted)
pairwise.wilcox.test(ToothGrowth$len, ToothGrowth$dose,
                     p.adjust.method = "bonferroni")

# --- CORRELATION: Pearson (linear) and Spearman (monotonic) ---
cor.test(iris$Sepal.Length, iris$Petal.Length, method = "pearson")
cor.test(iris$Sepal.Length, iris$Petal.Length, method = "spearman")

Bringing It All Together

This lesson translated the disease-frequency measures of Lesson 5 into measures of association between an exposure and an outcome. You worked through the three ratio measures (RR, IR, OR), the difference measures of effect in the exposed (RD, AF_e), the population-level extensions (PAR, AF_p), and the inferential machinery — standard errors, hypothesis tests, confidence intervals — that surrounds every estimate.

Lesson 8 returns to study-design topics first introduced in HSCI 230 and consolidates them; Lessons 9–10 cover hybrid designs (nested case-control, case-cohort, case-crossover) and controlled trials. The measures you computed here will reappear in each of those lessons as the outputs that designs deliver and that systematic reviews eventually pool. For deeper treatment of these measures and their statistical foundations, see Greenland and Pearce (2015) and the historical landmark papers by Cornfield (1951) and Doll and Hill (1950).

Final Assessment

Complete all 15 questions below with 100% accuracy to finish this lesson. You must also complete the reflection above before submitting.