Notation and glossary
Note
This appendix collects the symbols, estimands, and terminology conventions used throughout the book so that notation stays consistent across modules. Where individual chapters historically used a different word for the same idea, this page states the preferred convention.
Variables and symbols
| Symbol | Meaning |
|---|---|
| \(Y\) | Outcome variable |
| \(A\) | Exposure (the primary variable whose effect we study). We use exposure throughout for observational studies; treatment is used only where it is part of an established term (e.g., “average treatment effect”). |
| \(L\), \(C\), \(X\) | Covariates / potential confounders |
| \(M\) | Mediator |
| \(Y^{a}\) or \(Y(a)\) | Potential (counterfactual) outcome that would be observed if exposure were set to \(A=a\) |
| \(E[\cdot]\) | Expectation. Conditional expectation is written \(E[Y \mid A=a]\); the mean of a potential outcome is \(E[Y(a)]\). |
| \(\hat{\theta}\) | An estimate of a parameter \(\theta\) |
Estimands (what we are trying to estimate)
| Term | Symbol | Definition |
|---|---|---|
| Average Treatment Effect | ATE | \(E[Y(1) - Y(0)]\) — the average effect in the whole target population, on the difference (risk-difference) scale |
| Average Treatment effect on the Treated | ATT | \(E[Y(1) - Y(0) \mid A=1]\) — average effect among the exposed |
| Individual Treatment Effect | ITE | \(Y_i(1) - Y_i(0)\) for a single unit \(i\) (generally not identifiable) |
| Total Effect | TE | The full effect of \(A\) on \(Y\), i.e. direct + indirect (mediated) effect |
| Natural Direct Effect | NDE | Effect of \(A\) on \(Y\) not through \(M\), with \(M\) left at its natural value |
| Natural Indirect Effect | NIE | Effect of \(A\) on \(Y\) operating through \(M\) |
| Controlled Direct Effect | CDE | Effect of \(A\) on \(Y\) when \(M\) is fixed to a specific value (e.g., \(M=0\)) |
Tip
Effect terminology. Use ITE for an individual-level effect, TE for the total effect, and CDE vs NDE/NIE to distinguish fixing the mediator from leaving it at its natural value. Earlier drafts occasionally wrote “TE” for an individual effect or “TCE” for the total effect — prefer ITE and TE respectively.
Measures of association/effect
- RD (risk difference), RR (risk ratio), OR (odds ratio), HR (hazard ratio).
- Marginal vs conditional. A marginal effect is averaged over the covariate distribution; a conditional effect holds covariates fixed. For non-linear models (e.g., logistic), the OR is non-collapsible: a conditional OR and a marginal OR generally differ even with no confounding. State which one a quantity represents.
- Collapsibility. RD and RR are collapsible; OR and HR are not.
Weights
Different “weights” appear in different modules; they are not interchangeable:
| Weight | Where used | Purpose |
|---|---|---|
| Survey (sampling) weight | Complex Survey Data (D) | Inverse probability of selection into the sample; makes estimates representative of the target population |
| IPTW (inverse-probability-of-treatment weight) | Propensity Score (S), Causal ML (C) | Inverse probability of the observed exposure; creates a pseudo-population in which exposure is independent of measured confounders |
| Matching weight | Propensity Score (S) | Weight induced by a matching scheme (e.g., 1:k matching, matching with replacement) |
| MI / analysis weight | Missing Data (M) | Combining/aggregating across multiply imputed datasets |
When survey weights and IPTW (or matching weights) both apply, they are multiplied to form a combined weight.
Key causal assumptions
- SUTVA (stable unit treatment value assumption): no interference between units and one version of each exposure level.
- Exchangeability / no unmeasured confounding: \(Y(a) \perp A \mid L\).
- Positivity: \(0 < P(A=a \mid L) < 1\) for all covariate strata.
- Consistency: the observed outcome under the observed exposure equals the corresponding potential outcome.
Missing-data mechanisms
- MCAR (missing completely at random): missingness unrelated to observed or unobserved data.
- MAR (missing at random): missingness depends only on observed data.
- MNAR (missing not at random): missingness depends on unobserved values.
Other recurring terms
- MSM (marginal structural model): a model for the marginal mean of the potential outcomes, \(E[Y(a)]\), typically estimated by an IPTW-weighted GLM/GEE. The weighting (not the GEE machinery alone) is what handles time-varying treatment–confounder feedback.
- SMD (standardized mean difference): a scale-free balance measure. This book uses SMD < 0.2 as the working balance threshold consistently across tutorials and exercise solutions.
- R Markdown: written as two words (“R Markdown”) throughout; file extension
.Rmd. This book is itself built with Quarto (.qmd), a successor to R Markdown / bookdown.