Appendix A — Notation Glossary
This appendix is the reference index for the book’s notation. It has two parts. The first repeats every symbol introduced in Notation, grouped by the same topics, and adds for each symbol a one-line gloss together with the chapter that introduces it, naming the defining environment by its slug where one exists. The second is an alphabetical glossary of every formally defined term in the book; each entry gives a one-sentence statement of the term and a pointer to the chapter and the definition slug that fixes it. The conventions themselves, that bold lowercase letters denote vectors and bold uppercase letters denote matrices, that random variables are capitalized when the distinction from a realized value matters, that an unsubscripted probability refers to the population under discussion, and that \(\square\) closes a proof while \(\blacktriangleleft\) closes a definition, example, or remark, are stated in Notation and are not repeated here. Nothing in this appendix is proved; it is a single place to look up a symbol or a definition and to follow the slug to the statement that establishes it.
Throughout, “Ch.” abbreviates chapter, and a slug such as @def-effect-modifier names the cross-referenceable definition environment in the chapter where the term is established; that slug is the exact cross-reference an author would use to cite the result.
A.1 How to use this appendix
There are two ways in. If you have seen a symbol and want to know what it is, use the symbol tables of Section A.6, grouped by role from linear algebra through population adjustment. If you have heard a term and want its meaning, use the alphabetical glossary of Section A.8; the clusters in Section A.7 gather related terms so a whole family can be read at once. In either place the final column names the chapter and, where a formal definition fixes the item, the slug of that definition. That slug is the cross-reference key that takes you to the formal definition in the chapter that establishes it. The chapter legend in Table A.1 turns each “Ch.” number into its subject and part, so that “Ch. 14” reads as “Bucher indirect comparison,” not a bare number.
Four short aids come before the tables: a compact key to the conventions (Section A.2), a worked notation key that maps the symbols onto a single anchored comparison (Section A.3), a table of symbols that carry more than one meaning (Section A.4), and a plain-language note on effect scales (Section A.5).
| Ch. | Subject | Part |
|---|---|---|
| 1 to 3 | Linear algebra: spaces, orthogonality, spectral theory | I, Foundations |
| 4 to 5 | Probability; statistical inference | I, Foundations |
| 6 to 8 | Linear regression; generalized linear models; hierarchical and Bayesian models | I, Foundations |
| 9 to 12 | Potential outcomes; adjustment methods; effect modification and collapsibility; transportability | II, Causal foundations |
| 13 to 16 | Pairwise meta-analysis; Bucher; network meta-analysis; limits of standard NMA | III, Evidence synthesis |
| 17 to 19 | Population adjustment theory; MAIC; STC | IV, Population adjustment |
| 20 to 23 | ML-NMR: aggregation, discrete approximation, integration, estimands | V, Multilevel network meta-regression |
| 24 to 26 | Bayesian computation and diagnostics; general likelihoods; ML-UMR | V, Multilevel network meta-regression |
| 27 to 30 | Distance matching; quantitative bias analysis; simulation; the Coq formalization | VI, Frontier and formalization |
A.2 A compact key to the conventions
The full conventions live in Notation; the few a reader most often needs are repeated here, since a reader who reaches for a glossary is usually already unsure. Bold lowercase letters (\(\mathbf{x}\), \(\boldsymbol{\beta}\)) are vectors; bold uppercase letters (\(\mathbf{X}\), \(\boldsymbol{\Sigma}\)) are matrices; plain letters are scalars. A capital letter is a random variable and the matching lowercase letter is a value it takes: \(X\) is the random covariate, \(x\) a realized covariate value, \(\mathbf{x}\) the covariate vector for one patient, and \(\mathcal{X}\) the set of all possible covariate values. A probability \(P(\cdot)\) with no subscript refers to the population under discussion. The index letters are fixed: \(i\) is a patient, \(j\) a study, \(k\) an arm, and \(a,b,c\) name treatments, so \(\eta_{ijk}\) reads “patient \(i\), study \(j\), arm \(k\).” The symbol \(\square\) closes a proof and \(\blacktriangleleft\) closes a definition, example, or remark.
A.3 A worked notation key: one anchored comparison
To pin the symbols to a familiar picture, read them off a single indirect comparison. Treatments \(A\) and \(B\) were never studied head to head. An \(AC\) trial compares \(A\) against a common comparator \(C\); a separate \(BC\) trial compares \(B\) against the same \(C\). A health technology assessment wants the \(A\) versus \(B\) effect in a target population \(\mathcal{P}^{*}\). The symbols then read as follows.
| Symbol | In this anchored comparison |
|---|---|
| \(j\) | the study: here \(j=AC\) or \(j=BC\) |
| \(i\) | a patient inside one of the two trials |
| \(\mathbf{x}_i\) | patient \(i\)’s baseline covariate vector (age, disease severity, and so on) |
| \(\mathcal{P}^{*}\) | the target population in which the answer is wanted |
| \(\hat d_{AC},\ \hat d_{BC}\) | the within-trial estimates, each a \(\hat d_{ab(j)}\), of \(A\) versus \(C\) and of \(B\) versus \(C\) |
| \(d_{AB(\mathcal{P}^{*})}\) | the wanted marginal \(A\) versus \(B\) contrast in \(\mathcal{P}^{*}\), on a stated scale |
| the Bucher step | \(\hat d_{AB}=\hat d_{AC}-\hat d_{BC}\), valid when relative effects are constant across the populations (@def-anchored-comparison) |
Had the two trials shared no comparator, the comparison would be unanchored and would rest on the stronger prognostic assumptions of Chapters 17 and 26; the same \(i\), \(j\), \(\mathbf{x}_i\), and \(\mathcal{P}^{*}\) still apply.
A.4 Overloaded symbols at a glance
A few glyphs carry more than one meaning. Context always fixes the intended one, but a reader arriving in the middle of a chapter may want the map. Each row lists a reused symbol and its separate roles.
| Symbol | One meaning | Another meaning |
|---|---|---|
| \(\perp\) | “is independent of,” as in \(X\perp Y\mid Z\) (Ch. 4) | “is orthogonal to,” inner product zero (Ch. 2) |
| \(\Pi\) | prior probability measure (Ch. 8) | the coupling set \(\Pi(\mathcal{P},\mathcal{Q})\) of a Wasserstein distance (Ch. 27) |
| \(S\) | study or population membership indicator (Ch. 12) | a generic set, as in \(\mathrm{span}(S)\) (Ch. 1) |
| \(\mathrm{ESS}\) | effective sample size of a weighted sample (Ch. 18) | of an MCMC chain, written \(N_{\mathrm{eff}}\) to keep them apart (Ch. 24); and \(\mathrm{ESS}_{\mathrm{reg}}\), the explained regression sum of squares (Ch. 6) |
| \(\tau\) | average treatment effect (Ch. 9) | restricted mean survival time horizon (Ch. 25); and \(\tau^2\), the heterogeneity variance (Chs. 8, 13) |
| \(\mu\) | a mean, and \(\mu_t(x)\) a mean outcome (Chs. 4, 9) | \(\mu_j\), a study baseline intercept (Ch. 15) |
| \(\sigma\) | \(\sigma^2\), an error or sampling variance (Ch. 6) | \(\sigma_i(A)\), the \(i\)-th singular value (Ch. 3) |
| \(\boldsymbol{\beta}\) | regression coefficients (Ch. 6) | matching-weight parameters (Ch. 18); prognostic effects \(\boldsymbol{\beta}_1\) and interactions \(\boldsymbol{\beta}_{2,k}\) (Ch. 17) |
| \(\alpha\) | matching-weight normalizing constant in \(w_i=\exp(\alpha+\boldsymbol{\beta}^\top\mathbf{x}_i)\) (Ch. 18) | a significance level in hypothesis testing (Ch. 29) |
| \(d\) versus \(\delta\) | \(d_{ab(\mathcal{P})}\), a population-level relative effect (Ch. 14) | \(\delta_{j,\,t_1 t_k}\), the within-study contrast that estimates it (Ch. 15) |
| \(D\) | deviance, and \(\bar D\) the posterior mean deviance (Chs. 7, 15) | \(D_N^{*}(P)\), the star discrepancy (Ch. 22) |
A.5 A note on effect scales
Many entries below turn on two distinctions that run through the whole book, so they are worth stating plainly once. The first is marginal versus conditional. A conditional effect \(\tau^g_{ab}(x)\) compares treatments within a fixed covariate value \(x\); a marginal effect compares treatments after averaging outcomes over a whole population. On a non-collapsible scale, such as the odds ratio, these two differ even when nothing is confounded, which is exactly why the book keeps them apart. The second is outcome scale versus link scale. The outcome scale is the natural scale of the response, a probability, a rate, or a survival time; the link scale, also called the linear-predictor scale, is the scale on which the model is linear, reached by applying the link \(g\) (for a probability, the log-odds). Relative effects such as \(d_{ab(\mathcal{P})}\) live on a stated scale, and odds ratios, risk ratios, risk differences, and hazard ratios are not interchangeable: an assumption true on one scale can fail on another. The book names the scale whenever it matters; the defined terms @def-scale-alignment, @def-collapsibility, and @def-marginal-effect make the point precisely.
A.6 Symbols by topic
Each table below corresponds to one group of Notation. The “Introduced” column records the chapter of first use and, where a formal definition fixes the symbol, the slug of that definition. Many symbols recur across the book; the chapter named is the one that introduces and is responsible for the symbol, not the only one that uses it. One table has no counterpart in Notation: the Bayesian computation table Table A.8 collects symbols from Chapters 8, 15, and 24 that the Notation chapter does not yet list. The groups run from general mathematics toward the notation specific to indirect comparison; the turning point is Table A.10, where the symbols begin to describe trials, populations, and contrasts rather than generic mathematical objects.
A.6.1 Linear algebra and matrices
| Symbol | Meaning | Introduced |
|---|---|---|
| \(\mathbb{R},\ \mathbb{R}^n,\ \mathbb{R}^{m\times n}\) | reals, real \(n\)-vectors, real \(m\times n\) matrices | Ch. 1 |
| \(V,\ W\) | vector spaces over \(\mathbb{R}\) (sets where adding and scaling stay inside the set, like the space of covariate vectors) | Ch. 1, @def-vector-space |
| \(\dim V,\ \mathrm{span}(S)\) | dimension of a space; linear span of a set (everything you can build from \(S\) by adding and scaling) | Ch. 1, @def-dimension, @def-span |
| \(\mathbf{e}_i\) | \(i\)-th standard basis vector (all zeros but a \(1\) in slot \(i\); for example \(\mathbf{e}_1=(1,0,\dots,0)^\top\)) | Ch. 1, @def-basis |
| \(\mathbf{1}_n,\ \mathbf{0}_n\) | all-ones and all-zeros \(n\)-vectors | Ch. 1 |
| \(I_n,\ J_n=\mathbf{1}_n\mathbf{1}_n^\top\) | identity matrix; all-ones matrix | Ch. 1 |
| \(A^\top,\ A^{-1},\ A^{+}\) | transpose; inverse; Moore-Penrose pseudoinverse (a stand-in inverse that works even when \(A\) is not square or not invertible) | Ch. 1; pseudoinverse Ch. 3, @def-pseudoinverse |
| \(\mathrm{rank}(A),\ \det(A),\ \mathrm{tr}(A)\) | rank (number of independent columns); determinant; trace (sum of the diagonal) | rank Ch. 1 (@def-rank); \(\det,\ \mathrm{tr}\) Ch. 3 |
| \(\mathcal{C}(A),\ \mathcal{N}(A)\) | column space (all vectors \(A\) can output); null space (all vectors \(A\) sends to \(\mathbf{0}\)) | Ch. 1, @def-column-space, @def-kernel-image |
| \(\langle\mathbf{x},\mathbf{y}\rangle=\mathbf{x}^\top\mathbf{y}\) | Euclidean inner product (zero means the two vectors sit at right angles) | Ch. 2, @def-inner-product |
| \(\lVert\mathbf{x}\rVert_2=\sqrt{\mathbf{x}^\top\mathbf{x}},\ \lVert\mathbf{x}\rVert_A=\sqrt{\mathbf{x}^\top A\,\mathbf{x}}\) | Euclidean norm (ordinary length); quadratic-form norm | Ch. 2, @def-norm |
| \(P_{\mathcal{M}},\ P^{\perp}_{\mathcal{M}}=I-P_{\mathcal{M}}\) | orthogonal projection onto \(\mathcal{M}\) (the nearest point in the subspace) and onto its complement (everything left over) | Ch. 2, @def-projection-matrix |
| \(A\succeq 0,\ A\succ 0\) | positive semidefinite; positive definite (the quadratic form \(\mathbf{x}^\top A\,\mathbf{x}\) is always \(\ge 0\), or strictly \(>0\) for \(\mathbf{x}\ne\mathbf{0}\)) | Ch. 3, @def-definiteness |
| \(\lambda_i(A),\ \sigma_i(A)\) | \(i\)-th eigenvalue (stretch factor along a special direction of \(A\)); \(i\)-th singular value (stretch factor of the map \(\mathbf{x}\mapsto A\,\mathbf{x}\)) | Ch. 3, @def-eigenvalue |
| \(A=Q\Lambda Q^\top,\ A=U\Sigma V^\top\) | spectral decomposition; singular value decomposition (factorizations that expose those stretch factors) | Ch. 3 |
A.6.2 Regression and least squares
| Symbol | Meaning | Introduced |
|---|---|---|
| \(\mathbf{X}\in\mathbb{R}^{n\times p}\) | design (model) matrix; rows are observations, columns are regressors | Ch. 6, @def-linear-model |
| \(\mathbf{y},\ \hat{\mathbf{y}}\) | response vector; fitted values (\(\hat{\mathbf{y}}=H\mathbf{y}\)) | Ch. 6 |
| \(\boldsymbol{\beta},\ \hat{\boldsymbol{\beta}}\) | regression coefficients and their estimator (the slopes the fit chooses) | Ch. 6 |
| \(\boldsymbol{\varepsilon},\ \mathbf{r}=\mathbf{y}-\hat{\mathbf{y}}\) | error vector (unobserved noise); residual vector (its observed proxy) | Ch. 6 |
| \(H=\mathbf{X}(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top\) | hat matrix; it puts the “hat” on \(\mathbf{y}\), projecting it onto \(\mathcal{C}(\mathbf{X})\) | Ch. 6, @def-hat-matrix |
| \(M=I-H\) | residual maker (\(M\mathbf{y}=\mathbf{r}\), the part \(\mathbf{X}\) cannot explain) | Ch. 6, @def-hat-matrix |
| \(\mathrm{RSS},\ \mathrm{TSS},\ \mathrm{ESS}_{\mathrm{reg}}\) | residual, total, explained sums of squares (the subscript keeps \(\mathrm{ESS}_{\mathrm{reg}}\) apart from the weighting \(\mathrm{ESS}\)) | Ch. 6, @def-sums-of-squares |
| \(\sigma^2,\ s^2\) | error variance and its unbiased estimator | Ch. 6 |
| \(R^2\) | coefficient of determination (fraction of response variation the fit explains) | Ch. 6, @def-r-squared |
A.6.3 Probability and random variables
| Symbol | Meaning | Introduced |
|---|---|---|
| \((\Omega,\mathcal{F},P)\) | probability space: sample space, event \(\sigma\)-algebra (the events probabilities attach to), probability measure | Ch. 4, @def-probability-space |
| \(\mathbb{E}[X],\ \mathbb{E}[X\mid Z]\) | expectation; conditional expectation (best mean-square prediction of \(X\) from \(Z\)) | Ch. 4, @def-conditional-expectation |
| \(\mathrm{Var}(X),\ \mathrm{Cov}(X,Y),\ \boldsymbol{\Sigma}\) | variance; covariance; covariance matrix | Ch. 4, @def-variance-covariance |
| \(X\perp Y\mid Z\) | conditional independence: given \(Z\), knowing \(X\) tells you nothing about \(Y\) (here \(\perp\) reads “is independent of,” not the “perpendicular” of Ch. 2) | Ch. 4, @def-conditional-independence |
| \(\mathcal{N}(\mu,\sigma^2),\ \mathcal{N}_p(\boldsymbol{\mu},\boldsymbol{\Sigma})\) | univariate and \(p\)-variate normal | Ch. 4, @def-normal, @def-mvn |
| \(\mathrm{Bin}(n,p),\ \mathrm{Poi}(\lambda),\ \mathrm{Ber}(p)\) | binomial, Poisson, Bernoulli | Ch. 4 |
| \(\mathrm{PoBin}(\mathbf{p})\) | Poisson binomial (sum of independent coin flips with different success probabilities) | Ch. 4, @def-poisson-binomial |
| \(F_X,\ F_X^{-1}\) | distribution function and its generalized inverse (the quantile function; “generalized” so it still works for discrete laws) | Ch. 4 |
| \(\Phi,\ \phi\) | standard normal distribution function and density | Ch. 4, @def-normal |
| \(M_X(\mathbf{t})=\mathbb{E}[e^{\mathbf{t}^\top X}]\) | moment generating function (when finite near \(\mathbf{0}\) it determines the whole distribution) | Ch. 4, @def-mgf |
| \(\lVert\mu-\nu\rVert_{\mathrm{TV}}\) | total variation distance (largest probability gap two laws assign any event) | Ch. 21, @def-total-variation |
| \(\xrightarrow{p},\ \xrightarrow{d},\ \xrightarrow{\text{a.s.}}\) | convergence in probability (settles near a fixed value as \(n\) grows), in distribution (its histogram settles to a fixed shape), almost surely (settles for all but a negligible set of outcomes) | Ch. 4 |
| \(O_p,\ o_p,\ O(\cdot),\ o(\cdot)\) | stochastic and deterministic order symbols (\(O(1/n)\) means “shrinks about like \(1/n\)”; little-\(o\) means “shrinks strictly faster than its argument”) | Ch. 5 |
A.6.4 Statistical inference
| Symbol | Meaning | Introduced |
|---|---|---|
| \(L(\boldsymbol{\theta}),\ \ell(\boldsymbol{\theta})=\log L(\boldsymbol{\theta})\) | likelihood and log-likelihood (the data’s probability read as a function of the parameter) | Ch. 5, @def-likelihood |
| \(\mathbf{s}(\boldsymbol{\theta})=\nabla\ell(\boldsymbol{\theta})\) | score function (gradient of the log-likelihood; it vanishes at the MLE) | Ch. 5, @def-score |
| \(\mathcal{I}(\boldsymbol{\theta})\) | Fisher information matrix (curvature of the log-likelihood; larger means sharper estimates) | Ch. 5, @def-fisher-information |
| \(\hat{\boldsymbol{\theta}}_{\mathrm{MLE}}\) | maximum likelihood estimator | Ch. 5, @def-mle |
| \(\mathbb{V}\) | sandwich (robust) variance estimator, \(\mathbb{V}=A^{-1}BA^{-\top}\); stays valid under model misspecification | Ch. 5, @def-sandwich-variance |
A.6.5 Bayesian computation and model comparison
These symbols run through the Bayesian chapters (8, 15, 24) but have no home in Notation, so they are collected here: a reader who meets \(\hat R\) or \(\pi(\boldsymbol{\xi}\mid\mathbf{y})\) in a chapter will look for them in this appendix. The concepts behind them are glossed in the alphabetical table of Section A.8.
| Symbol | Meaning | Introduced |
|---|---|---|
| \(\pi(\boldsymbol{\xi}),\ \pi(\boldsymbol{\xi}\mid\mathbf{y})\) | prior and posterior density of the parameter vector (belief before, then after, seeing the data \(\mathbf{y}\)) | Ch. 8 |
| \(p(\mathbf{y}\mid\boldsymbol{\xi})\) | likelihood (probability of the data given the parameters) | Ch. 8 |
| \(\Pi\) | prior probability measure, used when the prior has no density (a different \(\Pi\) than the coupling set of Ch. 27) | Ch. 8 |
| \(\tau^2\) | between-study heterogeneity variance (spread of true study effects around their mean) | Chs. 8, 13, @def-random-effects-ma |
| \(\hat R\) | Gelman-Rubin potential scale reduction statistic; a value near \(1\) signals the chains have converged | Ch. 24, @def-comp-rhat-pieces |
| \(N_{\mathrm{eff}}\) | effective sample size of an autocorrelated MCMC chain (how many independent draws it is worth); reported in bulk and tail forms, and distinct from the weighting \(\mathrm{ESS}\) | Ch. 24, @def-mcmc-ess |
| \(\bar D,\ D(\bar{\boldsymbol{\theta}}),\ p_D\) | posterior mean deviance, deviance at the posterior mean, and effective number of parameters \(p_D=\bar D-D(\bar{\boldsymbol{\theta}})\) | Ch. 15, @def-dic |
| \(\mathrm{lppd},\ \widehat{\mathrm{elpd}}\) | log pointwise predictive density and the estimated expected log pointwise predictive density (out-of-sample fit) | Ch. 24, @def-comp-criteria |
| \(\mathrm{WAIC},\ \mathrm{LOO}\) | widely applicable information criterion and leave-one-out cross-validation scores | Ch. 24, @def-comp-criteria |
| PSIS, \(\widehat k\) | Pareto-smoothed importance sampling and its per-observation tail-index diagnostic (a large \(\widehat k\), above about \(0.7\), flags an unreliable, influential point) | Ch. 24, @def-psis-loo |
A.6.6 Causal inference
| Symbol | Meaning | Introduced |
|---|---|---|
| \(Y(0),\ Y(1)\) | potential outcomes under control and active treatment (what a patient’s outcome would be under each option) | Ch. 9, @def-potential-outcome |
| \(T\in\{0,1\}\) | treatment indicator | Ch. 9 |
| \(Y=TY(1)+(1-T)Y(0)\) | observed outcome (you see only the potential outcome for the treatment actually given; consistency under SUTVA) | Ch. 9, @def-sutva |
| \(X\in\mathcal{X}\) | baseline covariates (\(X\) the random covariate, \(x\) a value, \(\mathbf{x}\) a patient’s vector, \(\mathcal{X}\) the space of values) | Ch. 9 |
| \(e(x)=P(T=1\mid X=x)\) | propensity score, the coarsest balancing score; for example \(e(x)=0.3\) means 30 percent of patients with covariates \(x\) were treated | Ch. 10, @def-propensity-score |
| \(\mu_t(x)=\mathbb{E}[Y(t)\mid X=x]\) | conditional mean outcome under treatment \(t\) (the outcome model at covariate value \(x\)) | Ch. 9 |
| \(\tau(x)=\mu_1(x)-\mu_0(x)\) | conditional average treatment effect (the effect for patients with covariates \(x\); average it over \(X\) to get \(\tau\)) | Ch. 9, @def-cate |
| \(\tau=\mathbb{E}[\tau(X)]\) | average treatment effect (the conditional effect averaged over the covariate distribution) | Ch. 9, @def-ate |
| \(S\) | study or population membership indicator (a different \(S\) than the generic set of Ch. 1) | Ch. 12, @def-transport-invariance |
A.6.7 Studies, treatments, and populations
This is where the appendix turns from general mathematics to the notation specific to indirect comparison: the symbols now describe trials, the patients in them, and treatment contrasts. Read them against the worked key of Section A.3, where \(j\), \(i\), \(\mathbf{x}_i\), \(\mathcal{P}^{*}\), and \(d_{ab(\mathcal{P})}\) are placed in a concrete \(AC\)-versus-\(BC\) comparison.
| Symbol | Meaning | Introduced |
|---|---|---|
| \(j\) | study index, for example \(j=AB,\ BC\) (which trial) | Ch. 13 |
| \(k\) | treatment index, or arm (which arm within a trial) | Ch. 15 |
| \(i\) | individual index within a study or arm (which patient) | Ch. 6 |
| \(\mathcal{P},\ \mathcal{P}^{*}\) | populations as distributions over \(\mathcal{X}\), density \(f_{\mathcal{P}}\); \(\mathcal{P}^{*}\) is the target population the comparison is transported to | Ch. 11; target Ch. 12 |
| \(f_j(x),\ F_j(x)\) | covariate density and distribution in study \(j\) (who is in trial \(j\)) | Ch. 14 |
| \(\bar{\mathbf{x}}_j,\ \boldsymbol{\Sigma}_j\) | covariate mean and covariance in study \(j\) (the published baseline table, as means and spread) | Ch. 17 |
| \(d_{ab(\mathcal{P})}\) | marginal relative effect of \(b\) versus \(a\) in population \(\mathcal{P}\) on a stated scale (the target treatment contrast in a named population) | Ch. 14, @def-constancy-relative-effects |
| \(\hat d_{ab(j)}\) | estimator of \(d_{ab}\) computed from study \(j\)’s data | Ch. 13 |
| \(\delta_{j,\,t_1 t_k}\) | study-\(j\) contrast of treatment \(t_k\) versus \(t_1\); \(\delta\) is the within-one-study contrast, whereas \(d_{ab(\mathcal{P})}\) is the population-level estimand it targets | Ch. 15 |
A.6.8 Outcome models and likelihoods
| Symbol | Meaning | Introduced |
|---|---|---|
| \(g,\ g^{-1}\) | link function and its inverse (maps between the outcome scale and the linear-predictor scale) | Ch. 7, @def-link-function |
| \(\eta_{ijk}\) | linear predictor for individual \(i\), study \(j\), arm \(k\) (the additive score \(\mu_j+\mathbf{x}^\top\boldsymbol{\beta}_1+\mathbf{x}^\top\boldsymbol{\beta}_{2,k}+\gamma_k\) before the link is inverted) | Ch. 20, @def-individual-model |
| \(\theta_{ijk}=g^{-1}(\eta_{ijk})\) | conditional mean on the outcome scale (the linear predictor pushed back to a probability, rate, or mean) | Ch. 20, @def-individual-model |
| \(\mu_j\) | study-\(j\) baseline (intercept), the reference-arm outcome level in trial \(j\) | Ch. 15, @def-nma-arm-model |
| \(\boldsymbol{\beta}_1\) | main (prognostic) covariate effects (how covariates shift outcome regardless of treatment) | Ch. 17 |
| \(\boldsymbol{\beta}_{2,k}\) | treatment-by-covariate interactions for arm \(k\) (effect modification: how covariates change the treatment effect) | Ch. 17, @def-sema |
| \(\gamma_k\) | treatment main effect of arm \(k\) relative to the reference | Ch. 17 |
| \(L_i(\boldsymbol{\xi};y_i),\ \bar L_j(\boldsymbol{\xi};\bar y_j)\) | individual and aggregate likelihood contributions (one IPD patient versus one integrated AgD arm) | Ch. 20, @def-aggregate-model |
| \(\boldsymbol{\xi}\) | full parameter vector; in ML-NMR it bundles \(\mu_j,\ \boldsymbol{\beta}_1,\ \boldsymbol{\beta}_{2,k},\ \gamma_k\) | Ch. 20 |
A.6.9 Population adjustment: MAIC, STC, ML-NMR, ML-UMR
| Symbol | Meaning | Introduced |
|---|---|---|
| \(w_i,\ w(x)\) | matching weight for individual \(i\), or weight function (tilts the IPD sample to match the comparator’s covariate profile) | Ch. 18, @def-moment-balance |
| \(\mathrm{ESS}=\left(\sum_i w_i\right)^2/\sum_i w_i^2\) | effective sample size of a weighted sample; a small \(\mathrm{ESS}\) warns of unstable weights (not the MCMC \(N_{\mathrm{eff}}\) of Ch. 24) | Ch. 18, @def-maic-ess |
| \(\alpha,\ \boldsymbol{\beta}\) | matching-weight parameters, \(w_i=\exp(\alpha+\boldsymbol{\beta}^\top\mathbf{x}_i)\); the exponential form is forced by the balance constraints (entropy balancing) | Ch. 18, @def-entropy-balancing |
| \(\hat\mu_t^{\mathrm{STC}}\) | simulated treatment comparison predicted mean under \(t\) (the outcome model evaluated on the target population) | Ch. 19, @def-stc-marginal |
| \(C(u_1,\dots,u_d)\) | copula (a function that glues marginal distributions into a joint law) | Ch. 22, @def-copula |
| \(C_\Omega(\mathbf{u})=\Phi_\Omega\big(\Phi^{-1}(u_1),\dots,\Phi^{-1}(u_d)\big)\) | Gaussian copula with correlation matrix \(\Omega\); the tractable default for correlating covariates in ML-NMR integration | Ch. 22, @def-gaussian-copula |
| \(V_{\mathrm{HK}}(f)\) | Hardy-Krause variation of \(f\) (how much \(f\) wiggles; it bounds quasi-Monte Carlo error) | Ch. 22, @def-hardy-krause-variation |
| \(D_N^{*}(P)\) | star discrepancy of a point set \(P\) (how far \(P\) is from uniform; the other half of the QMC error bound) | Ch. 22, @def-star-discrepancy |
| SEMA, SPFA | shared effect modifier and shared prognostic factor assumptions | Ch. 17 (@def-sema); Ch. 26 (@def-spfa) |
A.6.10 Abbreviations
| Short form | Expansion | Developed in |
|---|---|---|
| ITC | indirect treatment comparison | Ch. 14 |
| PAIC | population-adjusted indirect comparison | Ch. 17 |
| (N)MA | (network) meta-analysis | Ch. 13, 15 |
| MAIC | matching-adjusted indirect comparison | Ch. 18 |
| STC | simulated treatment comparison | Ch. 19 |
| ML-NMR | multilevel network meta-regression | Ch. 20 |
| ML-UMR | multilevel unanchored meta-regression | Ch. 26 |
| IPD, AgD | individual patient data, aggregate data | Ch. 17, @def-pat-data-structures |
| EM, PV | effect modifier, prognostic variable | Ch. 11 |
| ATE, CATE | average and conditional average treatment effect | Ch. 9 |
| IPW, AIPW | (augmented) inverse probability weighting | Ch. 10 |
| OR, RR, HR, RD | odds ratio, risk ratio, hazard ratio, risk difference | Ch. 7, 11; HR Ch. 25 |
| RMST | restricted mean survival time | Ch. 25, @def-rmst |
| QMC | quasi-Monte Carlo | Ch. 22 |
| HTA | health technology assessment | Ch. 16 |
A.7 Finding related terms: clusters by role
The alphabetical table that follows is exhaustive but scatters related ideas: everything about STC or about collapsibility is spread across the alphabet. This index groups the terms a reader most often wants together, so a whole family can be found at once, and it flags which clusters are core to indirect comparison and which are specialized later tools. Each term is named as it appears in the alphabetical table, so it can be located there directly.
| Cluster (and reading priority) | Terms, as they appear in the alphabetical table |
|---|---|
| Mathematical background, Part I | Real vector space; Subspace; Basis; Dimension; Linear map; Rank and nullity; Four fundamental subspaces; Orthogonality; Orthogonal projection matrix; Eigenvalue and eigenvector; Definiteness; Quadratic form; Moore-Penrose pseudoinverse |
| Probability and inference, Part I | Probability space; Random variable and its law; Conditional expectation; Conditional independence; Normal distribution; Multivariate normal; Likelihood and log-likelihood; Score function; Fisher information; Maximum likelihood estimator; Consistency; Sandwich (robust) variance estimator |
| Bayesian and computation (Chs. 8, 15, 24) | Bayesian hierarchical model; Exchangeability; Fixed effects and random effects; Hamiltonian Monte Carlo; Detailed balance (reversibility); Transition kernel and stationary distribution; Credible set; Posterior predictive distribution; Deviance information criterion; DIC, WAIC, and LOO-CV; Pareto-smoothed importance sampling LOO; Effective sample size of an MCMC chain; Within-chain, between-chain, and pooled variance estimators |
| Causal foundations (core ITC) | Potential outcomes; Individual treatment effect; Average treatment effect; Conditional average treatment effect; Ignorability; Positivity (overlap); Propensity score; Balancing score; Stable unit treatment value assumption (SUTVA) |
| Effect modification and scale (core ITC) | Effect modifier; Prognostic variable; Statistical interaction; Conditional effect; Marginal effect; Collapsibility and direct collapsibility; Collapsibility and non-collapsibility of an effect measure; Shared effect modifier assumption (SEMA); Shared prognostic factor assumption (SPFA) |
| Evidence synthesis (core ITC) | Fixed-effect meta-analysis model; Random-effects meta-analysis model; Cochran’s \(Q\) statistic; \(I^2\) and \(H^2\) heterogeneity statistics; Anchored indirect comparison; Unanchored (naive single-arm) comparison; Transitivity (anchored triangle); Constancy of relative effects; Arm-based network model; Node-splitting |
| Transportability and population adjustment (core ITC) | Generalizability; Transportability; Structural invariance of the conditional model; Scale alignment; Individual patient data and aggregate data; Anchored and unanchored setups; Conditional constancy of relative effects (anchored); Conditional constancy of absolute effects (unanchored); Balance and the adjustment set |
| MAIC, STC, ML-NMR, ML-UMR | Method-of-moments covariate balance; Entropy-balancing problem; Effective sample size; STC outcome model; STC estimands; Conditional (plug-in) STC; Marginal STC; Bayesian g-computation for STC; Individual-level model (ML-NMR); Aggregate-level model (ML-NMR); Target-population marginal and conditional contrasts; Unanchored disconnected two-study network |
| Integration and numerics | Copula; Gaussian copula; Hardy-Krause variation; Star discrepancy; Digital \((t,m,s)\)-nets and the Sobol’ sequence; Doubling diagnostic for integration error |
| Survival and general likelihoods | Survival function; Hazard function and cumulative hazard; Restricted mean survival time; Right-censored individual likelihood; Proportional-odds model |
| Distance matching | Euclidean and standardized Euclidean distance; Mahalanobis distance; Distance variants; Comparator selection (aggregate-level) |
| Bias analysis, simulation, formalization | Quantitative bias analysis; E-value (indirect-comparison); Probabilistic bias analysis; Tipping point; ADEMP framework; Coverage; Monte Carlo bias estimator, relative bias, and Monte Carlo standard error; Proof-status tiers; Active classical axioms |
A.8 Alphabetical glossary of defined terms
Every formally stated definition in the book appears below, sorted alphabetically by its leading term (ignoring case, punctuation, and parentheticals). The “Defined in” column gives the chapter and the slug of the definition; that slug is the cross-reference that links to the statement in the rendered book. Terms whose meaning is scale-dependent, notably effect modification and collapsibility, are stated for a fixed link \(g\), as in the chapter that defines them.
| Term | One-sentence meaning | Defined in |
|---|---|---|
| Active classical axioms | The classical logical axioms, such as functional extensionality and the law of the excluded middle, that the Coq development assumes. | Ch. 30, @def-active-axioms |
| ADEMP framework | A protocol that specifies a simulation study by its Aims, Data-generating mechanisms, Estimands, Methods, and Performance measures. | Ch. 29, @def-ademp |
| Aggregate-level model (ML-NMR) | The individual-level model integrated over a study’s covariate distribution so that aggregate study arms can enter the likelihood. | Ch. 20, @def-aggregate-model |
| Anchored and unanchored setups | The two evidence configurations in which the index and comparator studies either share a common treatment arm (anchored) or do not (unanchored). | Ch. 17, @def-pat-setups |
| Anchored indirect comparison | The Bucher estimator \(\hat d_{AB}=\hat d_{AC}-\hat d_{BC}\) that differences two within-study contrasts to a shared comparator. | Ch. 14, @def-anchored-comparison |
| Arm-based network model | A network meta-analysis written at the level of arm means, each one a study baseline plus a treatment effect on the link scale. | Ch. 15, @def-nma-arm-model |
| Average treatment effect | The population-mean potential-outcome contrast \(\tau=\mathbb{E}[Y(1)-Y(0)]\). | Ch. 9, @def-ate |
| Average treatment effect on the treated | The mean potential-outcome contrast among the units that actually received treatment. | Ch. 9, @def-att |
| Balance and the adjustment set | The set of covariates whose cross-study imbalance biases an unadjusted comparison and must therefore be adjusted for. | Ch. 17, @def-pat-imbalance |
| Balancing score | A covariate summary \(b(X)\) such that treatment assignment and covariates are independent given \(b(X)\). | Ch. 10, @def-balancing-score |
| Baseline-shift parameterization | A network parameterization that writes each arm mean as a study-specific baseline plus a treatment contrast. | Ch. 15, @def-baseline-shift-param |
| Basis | A linearly independent set that spans a vector space. | Ch. 1, @def-basis |
| Bayesian g-computation for STC | Posterior predictive averaging of modeled outcomes over the target covariate distribution to obtain a marginal estimate. | Ch. 19, @def-stc-bayes |
| Bayesian hierarchical model | A model whose unit-level parameters are themselves drawn from shared population-level priors. | Ch. 8, @def-hierarchical-model |
| Bernoulli distribution | The law of a single binary trial with success probability \(p\). | Ch. 4, @def-bernoulli |
| Binomial distribution | The law of the number of successes in \(n\) independent identical Bernoulli trials. | Ch. 4, @def-binomial |
| Canonical link | The link that sets the linear predictor equal to the exponential family’s natural parameter (the \(\theta\) in the exponential-family density; for logistic regression it is the log-odds). | Ch. 7, @def-canonical-link |
| Centered and non-centered parameterization | Two equivalent ways to write a hierarchical model that induce different posterior geometries for sampling. | Ch. 24, @def-centered-noncentered |
| Characteristic polynomial | The polynomial \(\det(A-\lambda I)\) whose roots are the eigenvalues of \(A\). | Ch. 3, @def-characteristic-polynomial |
| Cochran’s \(Q\) statistic | The inverse-variance-weighted sum of squared deviations of study estimates from their pooled value, the basis of heterogeneity tests. | Ch. 13, @def-cochran-q-stat |
| Coefficient of determination | \(R^2\), the fraction of total response variation explained by the regression. | Ch. 6, @def-r-squared |
| Collapsibility and direct collapsibility | Whether the marginal effect equals the common conditional effect (collapsible) or always equals a covariate-weighted average of conditional effects (directly collapsible). | Ch. 11, @def-collapsibility |
| Collapsibility and non-collapsibility of an effect measure | Whether a measure’s marginal value coincides with its conditional value when that conditional value is constant across strata. | Ch. 7, @def-non-collapsibility |
| Comparator selection (aggregate-level) | Choosing, among candidate aggregate studies, the one whose covariate profile is closest to the target population. | Ch. 27, @def-comparator-selection |
| Conditional average treatment effect | The covariate-conditional contrast \(\tau(x)=\mu_1(x)-\mu_0(x)\). | Ch. 9, @def-cate |
| Conditional constancy of absolute effects (unanchored) | The condition that each arm’s conditional mean outcome is the same function of covariates across the populations being linked. | Ch. 17, @def-conditional-constancy-absolute |
| Conditional constancy of relative effects (anchored) | The condition that the conditional treatment effect \(\tau^g_{ab}(x)\) is the same function of covariates across populations. | Ch. 17, @def-conditional-constancy-relative |
| Conditional effect | The stratum-specific link-scale contrast \(\tau^g_{ab}(x)=g(\mu_b(x))-g(\mu_a(x))\). | Ch. 11, @def-conditional-effect |
| Conditional expectation | The mean-square-optimal predictor \(\mathbb{E}[X\mid Z]\) of \(X\) given \(Z\). | Ch. 4, @def-conditional-expectation |
| Conditional independence | \(X\perp Y\mid Z\): the conditional joint law of \(X\) and \(Y\) given \(Z\) factorizes. | Ch. 4, @def-conditional-independence |
| Conditional (plug-in) STC | The relative effect produced by evaluating the fitted outcome model at the comparator covariate mean. | Ch. 19, @def-stc-conditional |
| Consistency | The property that an estimator converges in probability to its target as the sample size grows. | Ch. 5, @def-consistent-estimator |
| Constancy of relative effects | The assumption that the marginal relative effect \(d_{ab}\) takes the same value in every study population. | Ch. 14, @def-constancy-relative-effects |
| Copula | A distribution on the unit cube with uniform margins that couples given marginals into a joint law (Sklar’s theorem). | Ch. 22, @def-copula |
| Coverage | With the empirical and model-based standard errors, the probability that a stated confidence interval contains the estimand. | Ch. 29, @def-coverage |
| Credible set | A region carrying a stated posterior probability, in equal-tailed or highest-posterior-density form. | Ch. 8, @def-credible-interval |
| Definiteness | The sign pattern of a symmetric matrix’s quadratic form: positive or negative (semi)definite, or indefinite. | Ch. 3, @def-definiteness |
| Detailed balance (reversibility) | The pairwise probability-flux symmetry that guarantees a target stationary distribution for a Markov chain. | Ch. 8, @def-detailed-balance |
| Deviance and residual deviance | Twice the log-likelihood gap between the saturated model and the fitted model. | Ch. 7, @def-deviance |
| Deviance information criterion | A Bayesian model-comparison score combining posterior fit with the effective number of parameters. | Ch. 15, @def-dic |
| DIC, WAIC, and LOO-CV | The three predictive criteria used to compare fitted hierarchical models. | Ch. 24, @def-comp-criteria |
| Diagonalizable and similar matrices | A matrix similar to a diagonal one, and the relation \(B=S^{-1}AS\) between matrices. | Ch. 3, @def-diagonalizable |
| Digital \((t,m,s)\)-nets and the Sobol’ sequence | Low-discrepancy point sets with guaranteed equidistribution across dyadic boxes. | Ch. 22, @def-sobol-net |
| Dimension | The common cardinality shared by every basis of a vector space. | Ch. 1, @def-dimension |
| Distance variants | The Manhattan, Minkowski, mean standardized-mean-difference, and Gower covariate distances. | Ch. 27, @def-distance-variants |
| Doubling diagnostic for integration error | A check that bounds numerical integration error by comparing estimates computed at \(N\) and at \(2N\) nodes. | Ch. 24, @def-integration-error-monitoring |
| Ecological (aggregation) bias | The gap between the plug-in arm mean \(g^{-1}(\eta(\bar x))\) and the covariate-averaged mean \(\int g^{-1}(\eta(x))\,f(x)\,dx\). | Ch. 16, @def-ecological-bias |
| Effect modifier | A covariate on which the conditional treatment effect depends, a property defined relative to a chosen scale. | Ch. 11, @def-effect-modifier |
| Effective sample size | \(\mathrm{ESS}=\left(\sum_i w_i\right)^2/\sum_i w_i^2\), the equivalent unweighted size of a weighted sample. | Ch. 18, @def-maic-ess |
| Effective sample size of an MCMC chain | The autocorrelation-deflated equivalent number of independent posterior draws. | Ch. 24, @def-mcmc-ess |
| Efficiency and asymptotic efficiency | The property of attaining the Cramér-Rao lower bound exactly or in the large-sample limit. | Ch. 5, @def-efficient-estimator |
| Eigenvalue and eigenvector | A scalar \(\lambda\) and nonzero vector \(v\) with \(Av=\lambda v\). | Ch. 3, @def-eigenvalue |
| Entropy-balancing problem | The convex program minimizing weight divergence from uniformity subject to covariate-balance constraints, whose solution is an exponential tilt. | Ch. 18, @def-entropy-balancing |
| Estimator, bias, variance, mean squared error | A statistic used to target a parameter, together with its bias, variance, and mean squared error. | Ch. 5, @def-estimator |
| Euclidean and standardized Euclidean distance | The ordinary and variance-scaled \(\ell_2\) distances between covariate-mean vectors. | Ch. 27, @def-euclidean-distance |
| E-value (indirect-comparison) | The smallest unreported-covariate association that could reduce an unanchored estimate to the null. | Ch. 28, @def-itc-evalue |
| Exchangeability | The property that a sequence’s joint law is invariant under permutation of its indices. | Ch. 8, @def-exchangeability |
| Exponential dispersion family | Densities of the form \(\exp\{(y\theta-b(\theta))/a(\phi)+c(y,\phi)\}\), the common template for the normal, binomial, and Poisson laws; \(\theta\) is the natural parameter and \(\phi\) the dispersion that scales the variance. | Ch. 7, @def-exponential-family |
| Fisher information | The variance of the score, equivalently minus the expected Hessian of the log-likelihood. | Ch. 5, @def-fisher-information |
| Fixed-effect meta-analysis model | A pooling model in which all studies estimate one common effect, combined by inverse-variance weighting. | Ch. 13, @def-fixed-effect-ma |
| Fixed effects and random effects | Parameters treated as fixed unknowns versus parameters modeled as draws from a population distribution. | Ch. 8, @def-random-effects |
| Four fundamental subspaces | The column space, null space, row space, and left null space of a matrix. | Ch. 1, @def-column-space |
| Gaussian copula | The copula \(C_\Omega(\mathbf{u})=\Phi_\Omega\big(\Phi^{-1}(u_1),\dots,\Phi^{-1}(u_d)\big)\) induced by a correlated normal vector. | Ch. 22, @def-gaussian-copula |
| Generalizability | A within-population property: a sample’s estimand equals the estimand of its own source population. | Ch. 12, @def-generalizability |
| Generalized linear model | A regression with an exponential-family response and a link-transformed linear mean. | Ch. 7, @def-glm |
| Hamiltonian Monte Carlo | An MCMC method that proposes distant, high-acceptance moves by simulating gradient-driven Hamiltonian dynamics. | Ch. 8, @def-hmc |
| Hardy-Krause variation | A multidimensional measure of a function’s variation that bounds quasi-Monte Carlo error. | Ch. 22, @def-hardy-krause-variation |
| Hat matrix and residual maker | The projection \(H\) onto \(\mathcal{C}(\mathbf{X})\) and its complement \(M=I-H\). | Ch. 6, @def-hat-matrix |
| Hazard function and cumulative hazard | The instantaneous event rate \(h(t)\) and its integral \(H(t)\). | Ch. 25, @def-hazard |
| \(I^2\) and \(H^2\) heterogeneity statistics | The share of total variability attributable to between-study heterogeneity, and its ratio form. | Ch. 13, @def-i-squared |
| Ignorability | Conditional independence of treatment and potential outcomes given covariates, in strong, weak, or mean form. | Ch. 9, @def-ignorability |
| Independence | The property that a joint law factorizes into the product of its marginals. | Ch. 4, @def-independence |
| Individual-level model (ML-NMR) | The link-scale model \(g(\theta_{ijk})=\mu_j+\mathbf{x}^\top\boldsymbol{\beta}_1+\mathbf{x}^\top\boldsymbol{\beta}_{2,k}+\gamma_k\) for each individual. | Ch. 20, @def-individual-model |
| Individual patient data and aggregate data | Per-patient covariates and outcomes (IPD) versus published summary statistics (AgD). | Ch. 17, @def-pat-data-structures |
| Individual treatment effect | The unit-level, generally unobservable contrast \(Y(1)-Y(0)\). | Ch. 9, @def-individual-effect |
| Inner product space | A vector space carrying a symmetric, bilinear, positive-definite form. | Ch. 2, @def-inner-product |
| Invertible matrix | A square matrix possessing a two-sided inverse, equivalently one of full rank. | Ch. 1, @def-invertible |
| Kernel and image | The set of inputs a linear map sends to zero and the set of outputs it attains. | Ch. 1, @def-kernel-image |
| Likelihood and log-likelihood | The data density read as a function of the parameter, and its logarithm. | Ch. 5, @def-likelihood |
| Linear independence | The property that the only vanishing linear combination of a set of vectors is the trivial one. | Ch. 1, @def-linear-independence |
| Linear map | A function between vector spaces that preserves addition and scalar multiplication. | Ch. 1, @def-linear-map |
| Linear model | The model \(\mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\boldsymbol{\varepsilon}\) with mean-zero, constant-variance, uncorrelated errors. | Ch. 6, @def-linear-model |
| Link function | A monotone smooth map \(g\) with \(g(\mu)=\eta\) relating the mean to the linear predictor. | Ch. 7, @def-link-function |
| M-estimator and estimating equation | A parameter value solving an estimating equation \(\sum_i\psi(X_i;\theta)=\mathbf{0}\). | Ch. 5, @def-m-estimator |
| Mahalanobis distance | The covariance-whitened distance \(\sqrt{(x-y)^\top\Sigma^{-1}(x-y)}\). | Ch. 27, @def-mahalanobis-distance |
| Marginal and average-conditional relative effects | The two population-level contrasts that diverge whenever the effect measure is non-collapsible. | Ch. 16, @def-conditional-marginal-contrast |
| Marginal effect | The contrast of link-transformed population-mean outcomes, \(g(\mathbb{E}[\mu_b(X)])-g(\mathbb{E}[\mu_a(X)])\). | Ch. 11, @def-marginal-effect |
| Marginal likelihood and Bayes factor | The prior-averaged data density, and the ratio of two models’ marginal likelihoods. | Ch. 8, @def-marginal-likelihood |
| Marginal link-scale position | The link-transformed population-mean outcome of a single arm. | Ch. 14, @def-marginal-link-position |
| Marginal STC | The estimator that averages predicted individual outcomes over the comparator population before contrasting on the link scale. | Ch. 19, @def-stc-marginal |
| Maximum likelihood estimator | The parameter value that maximizes the likelihood. | Ch. 5, @def-mle |
| Method-of-moments covariate balance | Choosing weights so that the weighted individual covariate moments match the comparator’s published moments. | Ch. 18, @def-moment-balance |
| Moment generating function | \(M_X(\mathbf{t})=\mathbb{E}[e^{\mathbf{t}^\top X}]\), which fixes the law when finite near the origin. | Ch. 4, @def-mgf |
| Monte Carlo bias estimator, relative bias, and Monte Carlo standard error | The empirical bias of an estimator over simulation repetitions, and the Monte Carlo uncertainty in that estimate. | Ch. 29, @def-bias-mcse |
| Moore-Penrose pseudoinverse | The unique matrix \(A^{+}\) satisfying the four Penrose conditions, giving the minimum-norm least-squares solution. | Ch. 3, @def-pseudoinverse |
| Multivariate normal | The \(p\)-variate Gaussian \(\mathcal{N}_p(\boldsymbol{\mu},\boldsymbol{\Sigma})\) fixed by its mean vector and covariance matrix. | Ch. 4, @def-mvn |
| Node-splitting | An inconsistency check that compares direct and indirect evidence on a single network contrast. | Ch. 15, @def-node-splitting |
| Norm and induced norm | A nonnegative, homogeneous, triangle-inequality length function, here induced by an inner product. | Ch. 2, @def-norm |
| Normal distribution | The Gaussian law \(\mathcal{N}(\mu,\sigma^2)\), with distribution function \(\Phi\) and density \(\phi\). | Ch. 4, @def-normal |
| Orthogonal complement | The subspace of all vectors orthogonal to every vector of a given subspace. | Ch. 2, @def-orthogonal-complement |
| Orthogonal projection matrix | A symmetric matrix with \(P=P^\top=P^2\) (idempotent: applying it twice does no more than applying it once) sending each vector to its nearest point in a subspace. | Ch. 2, @def-projection-matrix |
| Orthogonality | The relation between two vectors whose inner product is zero. | Ch. 2, @def-orthogonality |
| Parametric statistical model | A family of distributions indexed by a finite-dimensional parameter \(\theta\in\Theta\). | Ch. 5, @def-statistical-model |
| Pareto-smoothed importance sampling LOO | A stabilized leave-one-out cross-validation estimate carrying a tail-index diagnostic \(\hat k\). | Ch. 24, @def-psis-loo |
| Performance measures as functionals | Simulation performance written as functionals of the sampling distribution, distinguishing the data and repetition sample sizes. | Ch. 29, @def-sim-performance-functional |
| Poisson binomial distribution | The law of a sum of independent, non-identical Bernoulli trials with probability vector \(\mathbf{p}\). | Ch. 4, @def-poisson-binomial |
| Poisson distribution | The count law with mean equal to variance \(\lambda\), the rare-event limit of the binomial. | Ch. 4, @def-poisson |
| Positivity (overlap) | The requirement that each treatment has positive probability in every covariate stratum, \(0<e(x)<1\). | Ch. 9, @def-positivity |
| Posterior predictive distribution | The law of new data averaged over the parameter posterior. | Ch. 8, @def-posterior-predictive |
| Potential outcomes | The outcomes \(Y(0)\) and \(Y(1)\) a unit would exhibit under each treatment level. | Ch. 9, @def-potential-outcome |
| Probabilistic bias analysis | Assigning bias parameters prior distributions and propagating them into a bias-adjusted interval. | Ch. 28, @def-probabilistic-bias-analysis |
| Probability generating function | \(\mathbb{E}[s^N]\) for a nonnegative-integer variable, which encodes its mass function. | Ch. 21, @def-pgf |
| Probability space | The triple \((\Omega,\mathcal{F},P)\) of sample space, event \(\sigma\)-algebra (the collection of events that can be assigned a probability), and probability measure. | Ch. 4, @def-probability-space |
| Prognostic variable | A covariate on which the conditional mean outcome depends, whether or not it modifies the treatment effect. | Ch. 11, @def-prognostic-variable |
| Proof-status tiers | The classification of the book’s results by their degree of machine verification in the Coq development. | Ch. 30, @def-coq-proof-status |
| Propensity score | The conditional treatment probability \(e(x)=P(T=1\mid X=x)\), the coarsest balancing score. | Ch. 10, @def-propensity-score |
| Proportional-odds model | An ordinal regression in which treatment shifts every cumulative logit by a common amount. | Ch. 25, @def-proportional-odds |
| Quadratic form | The map \(x\mapsto x^\top A x\) associated with a symmetric matrix \(A\). | Ch. 3, @def-quadratic-form |
| Quantitative bias analysis | The practice of quantifying how strong an unmeasured bias would have to be to overturn a study’s conclusion. | Ch. 28, @def-quantitative-bias-analysis |
| Random-effects meta-analysis model | A model with exchangeable study effects drawn from a distribution of heterogeneity variance \(\tau^2\). | Ch. 13, @def-random-effects-ma |
| Random variable and its law | A function on the sample space, measurable so that probabilities can attach to its values, together with the distribution it induces. | Ch. 4, @def-random-variable |
| Rank and nullity | The dimension of the image and the dimension of the kernel of a map or matrix. | Ch. 1, @def-rank |
| Rayleigh quotient | The ratio \(x^\top A x/x^\top x\), whose extrema over \(x\neq\mathbf{0}\) are the extreme eigenvalues of symmetric \(A\). | Ch. 3, @def-rayleigh-quotient |
| Real vector space | A set closed under addition and scalar multiplication satisfying the eight vector-space axioms over \(\mathbb{R}\). | Ch. 1, @def-vector-space |
| Reference-treatment parameterization | Expressing every treatment effect relative to one designated network reference treatment. | Ch. 15, @def-reference-treatment-param |
| Regularity conditions | Smoothness and identifiability assumptions under which the score and information behave classically. | Ch. 5, @def-regular-model |
| Residual deviance | The deviance of a fitted network model relative to the saturated model, used as a goodness-of-fit summary. | Ch. 15, @def-residual-deviance |
| Residual, total, and explained sums of squares | The orthogonal decomposition \(\mathrm{TSS}=\mathrm{RSS}+\mathrm{ESS}_{\mathrm{reg}}\) of response variation. | Ch. 6, @def-sums-of-squares |
| Restricted maximum likelihood | Variance-component estimation from the likelihood of error contrasts, removing the downward bias of full maximum likelihood. | Ch. 8, @def-reml |
| Restricted mean survival time | The area under the survival curve to a horizon \(\tau\), \(\int_0^\tau S(t)\,dt\). | Ch. 25, @def-rmst |
| Right-censored individual likelihood | The contribution \(f(t)^\delta S(t)^{1-\delta}\) combining observed events and right-censored observations. | Ch. 25, @def-censored-likelihood |
| Sandwich (robust) variance estimator | The \(A^{-1}BA^{-\top}\) variance for M-estimators, valid under model misspecification. | Ch. 5, @def-sandwich-variance |
| Saturated model | The GLM with one parameter per observation, reproducing the data exactly. | Ch. 7, @def-saturated-model |
| Scale alignment | Choosing the effect scale on which a transportability or invariance assumption actually holds. | Ch. 12, @def-scale-alignment |
| Score function | The gradient of the log-likelihood with respect to the parameter. | Ch. 5, @def-score |
| Shared effect modifier assumption (SEMA) | The assumption that the compared active treatments carry identical effect-modifier interactions, \(\boldsymbol{\beta}_{2,A}=\boldsymbol{\beta}_{2,B}\). | Ch. 17, @def-sema |
| Shared prognostic factor assumption (SPFA) | The assumption that covariate-outcome slopes coincide across treatments, which forces the absence of effect modification. | Ch. 26, @def-spfa |
| Span | The set of all finite linear combinations of a collection of vectors. | Ch. 1, @def-span |
| Stable unit treatment value assumption (SUTVA) | No interference between units and a single version of each treatment, giving consistency \(Y=Y(T)\). | Ch. 9, @def-sutva |
| Star discrepancy | The worst-case gap between the empirical and uniform measures over anchored boxes, the quasi-Monte Carlo accuracy criterion. | Ch. 22, @def-star-discrepancy |
| Statistical interaction | A nonzero product term in the linear predictor, equivalently a covariate-dependent treatment coefficient. | Ch. 11, @def-statistical-interaction |
| STC estimands | The marginal and the conditional-at-the-mean targets a simulated treatment comparison may estimate. | Ch. 19, @def-stc-estimands |
| STC outcome model | The individual patient outcome regression, carrying treatment-covariate interactions, used for cross-study prediction. | Ch. 19, @def-stc-outcome-model |
| Structural invariance of the conditional model | The condition that the conditional mean potential outcome \(\mu_t(x)\) does not depend on population membership. | Ch. 12, @def-transport-invariance |
| Subspace | A nonempty subset of a vector space that is closed under addition and scalar multiplication. | Ch. 1, @def-subspace |
| Survival function | \(S(t)=P(T>t)\), the probability of surviving beyond time \(t\). | Ch. 25, @def-survival-function |
| Target-population marginal and conditional contrasts | The ML-NMR estimands obtained by integrating the fitted model over a chosen target population. | Ch. 23, @def-mlnmr-target-contrast |
| Tipping point | The magnitude of bias at which an adjusted estimate crosses a decision threshold such as the null. | Ch. 28, @def-tipping-point |
| Total-variation distance | The largest probability discrepancy two laws can assign any event, \(\sup_A\lvert\mu(A)-\nu(A)\rvert\). | Ch. 21, @def-total-variation |
| Transition kernel and stationary distribution | The one-step transition law of a Markov chain and the measure it leaves invariant. | Ch. 8, @def-markov-kernel |
| Transitivity (anchored triangle) | The assumption that each of the three triangle relative effects takes a common value across the three trial populations. | Ch. 14, @def-transitivity |
| Transportability | A between-population property: a target estimand is identified from source outcomes together with target covariates. | Ch. 12, @def-transportability |
| Unanchored (naive single-arm) comparison | A comparison of active arms across studies that discards the shared comparator. | Ch. 14, @def-unanchored-comparison |
| Unanchored disconnected two-study network | Two studies without a common comparator, linkable only under prognostic-factor assumptions. | Ch. 26, @def-unanchored-network |
| Variance, covariance, and the covariance matrix | The second central moments of a single variable, of a pair, and of a random vector. | Ch. 4, @def-variance-covariance |
| Wald, score, and likelihood-ratio statistics | The three asymptotically equivalent classical statistics for testing a parametric hypothesis. | Ch. 5, @def-test-statistics |
| Wald test, Type I error rate, and power | The Wald rejection rule together with its rejection probabilities under the null and under an alternative. | Ch. 29, @def-power-type1 |
| Within-chain, between-chain, and pooled variance estimators | The variance components from which the \(\hat R\) convergence diagnostic is built. | Ch. 24, @def-comp-rhat-pieces |
A.9 Notes and sources
The source for this appendix is the book itself: the symbol tables reproduce Notation, which is the single source of truth for the symbols and conventions, extended by one table (Bayesian computation and model comparison, Table A.8) that gathers symbols from Chapters 8, 15, and 24 not yet listed there; the term glossary is assembled from the definition environments of Chapters 1 through 30. Where a symbol carries a formal definition, the slug in the “Introduced” column is that definition; where a term is glossed, the slug in the “Defined in” column links to its statement. The companion catalogues that the notation extends are the proof catalogue and the ITC_Coq development described in Chapter 30 and in Appendix G. Related reference appendices collect the objects that recur across chapters: the probability families of Appendix C, the matrix calculus identities of Appendix B, and the special functions of Appendix D.