Measures and models
Epidemiology guide, part 1
Every analysis in this guide goes through the same steps. A question (“does smoking cause coronary heart disease, and by how much?”) becomes an estimand: a precise quantity, such as the risk ratio of a 10-year coronary event for smokers against non-smokers with the same age and sex. A design gathers data that can estimate it. A causal diagram says which variables to adjust for. A model turns the data into a number, and the number is interpreted on the scale of the estimand. This page covers the last two steps: the measures epidemiology uses and how the coefficients of its models are read. The next pages cover the designs, the diagrams and mediation (Lash et al. 2021; Hernán and Robins 2020).
Measures of frequency
| measure | definition | example | typical design |
|---|---|---|---|
| Prevalence | proportion of a population that has the condition at a time | 18% of adults have hypertension | cross-sectional |
| Risk (cumulative incidence) | proportion of people free of the outcome at the start who develop it over a period | 8% have a coronary event in 10 years | cohort, trial |
| Rate (incidence density) | new cases per unit of person-time | 2.5 breast cancers per 1000 woman-years | cohort |
| Odds | probability / (1 − probability) | a risk of 0.2 is odds of 0.25 | case-control (only odds ratios) |
| Hazard | instantaneous rate among those still at risk at a given time | cohort with times (Cox) |
Prevalence mixes how often a condition starts and how long it lasts: in a steady population, prevalence ≈ incidence rate × mean duration. A treatment that keeps patients alive longer raises prevalence. A risk always refers to a period (“10-year risk”), and it requires everyone to be followed for that period or a method that handles censoring (Kaplan-Meier). When another event can prevent the outcome (death from other causes before a cancer relapse), the risk of the outcome is a cumulative incidence in the presence of a competing event. One minus Kaplan-Meier, which censors the competing deaths, overstates it.
Measures of association
| measure | definition | estimated by |
|---|---|---|
| Risk difference (RD) | risk1 − risk0 | cohort, trial (2×2 table, binomial model with identity link) |
| Number needed to treat (NNT) | 1 / |RD| | trial |
| Risk ratio (RR) | risk1 / risk0 | cohort, trial, cross-sectional (prevalence ratio) |
| Rate ratio (IRR) | rate1 / rate0 | cohort (Poisson regression with log person-time as offset) |
| Hazard ratio (HR) | hazard1 / hazard0, assumed constant over time | cohort, trial (Cox regression) |
| Odds ratio (OR) | odds1 / odds0 | any design; the only one a case-control study gives |
| Mean difference | mean1 − mean0 | linear regression |
| Ratio of geometric means | exp(mean of log1 − mean of log0) | linear regression of a log outcome |
Difference measures say how many people are affected, and that is what decisions need. Ratio measures say how strong the association is, and they are usually more stable across populations with different baseline risks. Report both when you can: a risk ratio of 2 means 10 extra cases per 1000 people if the baseline risk is 1%, and 200 if it is 20%.
The odds ratio is not a risk ratio
The odds ratio approaches the risk ratio only when the outcome is rare (Figure 2). A case-control study of a rare disease therefore estimates the risk (or rate) ratio. A cross-sectional study of a common condition analysed with logistic regression reports odds ratios that look much stronger than the prevalence ratios.
Conditional and marginal: non-collapsibility
The odds ratio and the hazard ratio have a property that surprises many people. Suppose a treatment multiplies the odds by 3 in every subgroup of patients and the subgroups are perfectly balanced between the arms, so nothing is confounded. The odds ratio of the whole trial is still smaller than 3 (Greenland et al. 1999). Risk differences and risk ratios do not behave this way.
| risk, control | risk, treated | OR | RR | RD | |
|---|---|---|---|---|---|
| low risk | 0.10 | 0.250 | 3.00 | 2.50 | 0.150 |
| high risk | 0.60 | 0.818 | 3.00 | 1.36 | 0.218 |
| whole population (half each) | 0.35 | 0.534 | 2.13 | 1.53 | 0.184 |
Both subgroups have an odds ratio of 3, and the whole population has 2.13. Neither number is wrong: they answer different questions. The conditional odds ratio is the effect for a patient with given characteristics; the marginal one is the effect on the population. In a trial the crude analysis estimates the marginal effect and an analysis adjusted for prognostic variables the conditional one, which is further from 1. studysim generates data with conditional effects, and its answer key shows both for trials (absolute effects in the trial).
| Value | |
|---|---|
| Risk: placebo | 0.1700 |
| Risk: indomethacin | 0.0896 |
| Risk difference, indomethacin vs placebo | -0.0806 |
| Number needed to treat for one event fewer, indomethacin vs placebo | 12.4000 |
| Risk ratio, indomethacin vs placebo | 0.5260 |
| Odds ratio, marginal (population), indomethacin vs placebo | 0.4800 |
| Odds ratio, conditional (true model), indomethacin vs placebo | 0.4700 |
Models and how to read their coefficients
| outcome | model | exp(β) or β is | jamovi |
|---|---|---|---|
| continuous | linear regression | β: difference in the mean per unit of x | Regression ▸ Linear Regression |
| continuous, skewed | linear regression of log(y) | exp(β): ratio of geometric means | Compute log(y), then linear regression |
| binary | logistic regression | odds ratio | Regression ▸ Logistic Regression ▸ 2 Outcomes |
| binary, matched sets | conditional logistic regression | odds ratio within sets | condLogReg |
| count or rate | Poisson (negative binomial) regression, log person-time as offset | rate ratio | GAMLj |
| time to event | Cox regression | hazard ratio | prognosis |
The same rules apply to every model:
- A coefficient is the effect of its variable with the other variables in the model held fixed. It is a conditional effect. Whether it is the effect you want depends on which variables are in the model: the causal diagram decides that, not the model.
- Per unit. A coefficient for age is per year. A hazard ratio of 1.05 per year is 1.0510 = 1.63 per 10 years. Choose units that make sense: per 10 mmHg, per 5 kg/m², per standard deviation.
- Per level. A categorical variable gets one coefficient per level, each against the reference level. Changing the reference changes the numbers but not the model.
- Log scale. Logistic, Poisson and Cox coefficients are logarithms: exp(β) is the ratio, and effects of several variables multiply. A log-transformed exposure gives the effect per doubling (log2) or per tenfold increase (log10).
- Interaction. With a product term X × Z, the coefficient of X is the effect of X when Z is at its reference or zero, and the interaction coefficient is how much that effect changes per unit of Z. If zero is not a meaningful value of Z (a BMI of zero), centre Z first, or the main effects become meaningless. This is the problem studysim’s
center:solves in its own scenarios. - The intercept is the outcome (or its log odds or log rate) when every variable is at zero or its reference: rarely of interest, and meaningless without centring.
- Categorising a continuous variable loses information and leaves residual confounding within the categories. It is useful for showing a shape (dose response), not for adjustment.
- Small numbers. Logistic and Cox models need roughly ten events per parameter; with fewer, coefficients are unstable and biased away from the null (Peduzzi et al. 1996). studysim warns when a study falls short.
What studysim’s answer key reports
| column | estimand |
|---|---|
| Direct (conditional) | the effect in the generating model, every other variable fixed |
| Total | the effect of changing the variable while its consequences change too, given the variables it does not cause (Causal diagrams) |
| Crude | the association without adjustment |
| Adjusted (all) | the conditional association given every measured variable |
| Adjusted (DAG) | the association given the measured variables that the diagram says to adjust for |
| Absolute effects (trials) | risks per arm, RD, NNT, marginal and conditional OR or HR |
Effect modification and interaction
Two words, two questions
- Effect modification: the effect of an exposure X differs between groups of another variable Z. The effect of BMI on breast cancer is protective before menopause and harmful after it. Z need not cause anything: it describes in whom X acts more strongly.
- Interaction: the joint effect of two exposures is not what their separate effects predict. Smoking and asbestos together cause far more lung cancer than the two separately would suggest. Both are causes, and the question is what happens if we act on both.
They are estimated the same way (X within levels of Z, or a product term X × Z), and the words are often used interchangeably. The distinction matters when deciding what to do: a modifier tells you who benefits most from acting on X.
It depends on the scale
Whether an effect “differs” depends on how effects are measured. In this example, exposure X triples the risk both in people without and with a second factor Z:
| risk, X absent | risk, X present | risk ratio of X | risk difference of X | |
|---|---|---|---|---|
| Z absent | 1% | 3% | 3 | +2% |
| Z present | 5% | 15% | 3 | +10% |
On the ratio scale there is no modification: the risk ratio is 3 in both groups. On the difference scale there is a great deal: X adds 2 cases per 100 people without Z and 10 per 100 with Z (Figure 3). Whenever two factors both act, no modification on one scale implies modification on the other.
Logistic and Cox models are multiplicative: their product term tests departure from multiplying the effects. Public health usually cares about the additive scale, because it counts cases: here, removing X prevents five times more cases among people with Z. The relative excess risk due to interaction, RERI = RR11 − RR10 − RR01 + 1, measures additive interaction from ratio measures: here 15 − 3 − 5 + 1 = 8 (0 means no additive interaction).
Reading a model with a product term
In a model logit(risk) = β0 + β1·X + β2·Z + β3·X·Z:
| coefficient | is | holds |
|---|---|---|
| exp(β1) | odds ratio of X | when Z = 0 (its reference level) |
| exp(β2) | odds ratio of Z | when X = 0 |
| exp(β3) | ratio of the odds ratios of X between Z = 1 and Z = 0 | the multiplicative interaction |
| exp(β1 + β3) | odds ratio of X | when Z = 1 |
The “main effects” are therefore not average effects: each is the effect at the reference or zero of the other variable. When the other variable is continuous and zero is meaningless, a main effect is an extrapolation far from the data. Centre the continuous variable first (subtract a typical value), so that the main effect of the modifier holds at that typical value.
Breast cancer risk factors: the effect of BMI is modified by menopause (per 5 kg/m², HR 0.93 before and 1.12 after). The table fits the same interaction model to a large simulated case-control study, with BMI as it is and centred at the population mean.
| BMI as measured | BMI centred at 26.6 | |
|---|---|---|
| BMI per 5 kg/m², before menopause (exp β1) | 0.95 | 0.95 |
| BMI per 5 kg/m², after menopause (exp(β1 + β3)) | 1.15 | 1.15 |
| ratio of the two (exp β3) | 1.21 | 1.21 |
| menopause, post vs pre (exp β2) | 0.37 | 1.02 |
The BMI effects and their ratio do not change with centring. The “effect of menopause” does: with BMI as measured it is the odds ratio at a BMI of 0, which nobody has (Figure 4). Centred, it is the odds ratio at a typical BMI, which is what the question means.
A modifier that reverses the effect
Dexamethasone in hospitalised COVID-19 (RECOVERY): dexamethasone lowers mortality in patients on oxygen or ventilation, but not in those without respiratory support. This is qualitative modification: the effect changes direction. Its overall odds ratio, averaged over the patients of the trial, would hide both the harm and the largest benefit.
In practice
- Decide in advance which modifiers to examine and why (biology, previous evidence). Testing many subgroups guarantees false positives.
- Report the effect within each group, with its confidence interval, and the test of interaction. A significant effect in one subgroup and a non-significant one in another is not evidence that they differ: compare them with the interaction test.
- Interaction tests have little power: detecting an interaction as large as the main effect needs about four times the sample size. A non-significant test does not show that the effect is the same.
- Say which scale you used, and give the additive measures (RD, RERI) when the question is about how many cases could be prevented.
In jamovi: in Regression models, select both variables in Model Builder and add their interaction (centre a continuous modifier first with Compute); Frequencies ▸ Independent Samples with the modifier as Layer gives the odds ratio in each group and a test of homogeneity; the prognosis module gives subgroup hazard ratios for an interaction in a Cox model, and GAMLj simple effects. In studysim: Breast cancer risk factors (BMI × menopause), Dexamethasone in hospitalised COVID-19 (treatment × respiratory support), HPV and cofactors of cervical cancer (IARC multicentric) (cofactors that act only with HPV), and the 2×2 factorial design with an interaction between the two treatments.
References
Greenland, Sander, Judea Pearl, and James M. Robins. 1999. “Confounding and Collapsibility in Causal Inference.” Statistical Science 14 (1). https://doi.org/10.1214/ss/1009211805.
Hernán, Miguel A., and James M. Robins. 2020. Causal Inference: What If. Chapman & Hall/CRC. https://miguelhernan.org/whatifbook.
Lash, Timothy L., Tyler J. VanderWeele, Sebastien Haneuse, and Kenneth J. Rothman. 2021. Modern Epidemiology. 4th ed. Wolters Kluwer.
Peduzzi, Peter, John Concato, Elizabeth Kemper, Theodore R. Holford, and Alvan R. Feinstein. 1996. “A Simulation Study of the Number of Events Per Variable in Logistic Regression Analysis.” Journal of Clinical Epidemiology 49 (12): 1373–79. https://doi.org/10.1016/s0895-4356(96)00236-3.