Notation and Conventions
This chapter is the single source of truth for the symbols used throughout the book. Symbols listed here are used in later chapters without reintroduction. The notation extends the system of the companion proof catalogue so that the two can be read together.
Conventions
- Bold lowercase letters such as \(\mathbf{x}\) and \(\boldsymbol{\beta}\) denote vectors. Bold uppercase letters such as \(\mathbf{X}\) and \(\boldsymbol{\Sigma}\) denote matrices. Plain letters denote scalars or abstract elements.
- Random variables are capitalized when the distinction from a realized value matters, as in \(X\) versus \(x\). Vectors of random variables follow the same rule.
- Integrals \(\int f(x)\,dx\) are taken over the natural domain, with respect to Lebesgue measure unless stated otherwise. For discrete covariates, sums replace integrals throughout.
- A probability statement without an explicit subscript refers to the population under discussion in the current section. When several populations are in play they are named explicitly.
- The word “effect” without qualification means the relative effect on the linear-predictor scale, that is, the contrast in the linear predictor \(\eta\). The marginal versus conditional distinction and the identity-scale versus link-scale distinction are made explicit wherever they matter.
- The symbol \(\square\) marks the end of a proof. The symbol \(\blacktriangleleft\) marks the end of a definition, example, or remark when needed for clarity.
- A theorem with a machine-checked counterpart carries a tag of the form
[machine-checked: theories/File.v, Theorem name], referring to the companion Coq development.
Linear algebra and matrices
| Symbol | Meaning |
|---|---|
| \(\mathbb{R}, \mathbb{R}^n, \mathbb{R}^{m\times n}\) | reals, real \(n\)-vectors, real \(m \times n\) matrices |
| \(V, W\) | vector spaces over \(\mathbb{R}\) |
| \(\dim V\), \(\mathrm{span}(S)\) | dimension of \(V\), linear span of a set \(S\) |
| \(\mathbf{e}_i\) | \(i\)-th standard basis vector |
| \(\mathbf{1}_n\), \(\mathbf{0}_n\) | \(n\)-vectors of ones and of zeros |
| \(I_n\), \(J_n = \mathbf{1}_n\mathbf{1}_n^\top\) | \(n\times n\) identity, \(n\times n\) all-ones matrix |
| \(A^\top\), \(A^{-1}\), \(A^{+}\) | transpose, inverse, Moore–Penrose pseudoinverse |
| \(\mathrm{rank}(A)\), \(\det(A)\), \(\mathrm{tr}(A)\) | rank, determinant, trace |
| \(\mathcal{C}(A)\), \(\mathcal{N}(A)\) | column space and null space of \(A\) |
| \(\langle \mathbf{x},\mathbf{y}\rangle = \mathbf{x}^\top\mathbf{y}\) | Euclidean inner product |
| \(\|\mathbf{x}\|_2 = \sqrt{\mathbf{x}^\top\mathbf{x}}\) | Euclidean norm; \(\|\mathbf{x}\|_A=\sqrt{\mathbf{x}^\top A\,\mathbf{x}}\) a quadratic-form norm |
| \(P_{\mathcal{M}}\), \(P^{\perp}_{\mathcal{M}}=I-P_{\mathcal{M}}\) | orthogonal projection onto subspace \(\mathcal{M}\) and its complement |
| \(A \succeq 0\), \(A \succ 0\) | positive semidefinite, positive definite |
| \(\lambda_i(A)\), \(\sigma_i(A)\) | \(i\)-th eigenvalue, \(i\)-th singular value of \(A\) |
| \(A = Q\Lambda Q^\top\), \(A=U\Sigma V^\top\) | spectral decomposition, singular value decomposition |
| \(\mathcal{M}^{\perp}=\{\mathbf{z}:\mathbf{z}^\top\mathbf{m}=0\ \forall\,\mathbf{m}\in\mathcal{M}\}\) | orthogonal complement of a subspace \(\mathcal{M}\) |
| \(\langle\mathbf{x},\mathbf{y}\rangle_A=\mathbf{x}^\top A\,\mathbf{y}\) | weighted (\(A\)-)inner product, \(A \succ 0\); induces the norm \(\|\mathbf{x}\|_A\) |
| \(A \prec 0\), \(A \preceq 0\) | negative definite, negative semidefinite |
| \(A^{1/2}\), \(A^{-1/2}\) | symmetric positive semidefinite square root and its inverse |
| \(\chi_A(t)=\det(tI-A)\) | characteristic polynomial of \(A\) |
| \(\mathrm{spec}(A)\), \(E_\lambda=\mathcal{N}(A-\lambda I)\) | spectrum (set of eigenvalues) and the \(\lambda\)-eigenspace |
| \(\langle A,B\rangle_F=\mathrm{tr}(A^\top B)\), \(\|A\|_2\), \(\|A\|_F\) | Frobenius inner product; operator (spectral) and Frobenius norms |
Regression and least squares
| Symbol | Meaning |
|---|---|
| \(\mathbf{X} \in \mathbb{R}^{n\times p}\) | design (model) matrix; rows are observations, columns are regressors |
| \(\mathbf{y}\), \(\hat{\mathbf{y}}\) | response vector, fitted values |
| \(\boldsymbol{\beta}\), \(\hat{\boldsymbol{\beta}}\) | regression coefficient vector and its estimator |
| \(\boldsymbol{\varepsilon}\), \(\mathbf{r}=\mathbf{y}-\hat{\mathbf{y}}\) | error vector, residual vector |
| \(H = \mathbf{X}(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top\) | hat matrix (orthogonal projection onto \(\mathcal{C}(\mathbf{X})\)) |
| \(M = I - H\) | residual maker (projection onto the orthogonal complement) |
| \(\mathrm{RSS}\), \(\mathrm{TSS}\), \(\mathrm{ESS}_{\mathrm{reg}}\) | residual, total, and explained sums of squares |
| \(\sigma^2\), \(s^2\) | error variance and its unbiased estimator |
| \(R^2\) | coefficient of determination |
The abbreviation \(\mathrm{ESS}_{\mathrm{reg}}\) for the explained regression sum of squares is written with a subscript to distinguish it from the effective sample size \(\mathrm{ESS}\) of weighting methods, defined below.
Probability and random variables
| Symbol | Meaning |
|---|---|
| \((\Omega,\mathcal{F},P)\) | probability space: sample space, \(\sigma\)-algebra, probability measure |
| \(\mathbb{E}[X]\), \(\mathbb{E}[X\mid Z]\) | expectation, conditional expectation |
| \(\mathrm{Var}(X)\), \(\mathrm{Cov}(X,Y)\), \(\boldsymbol{\Sigma}\) | variance, covariance, covariance matrix |
| \(X \perp Y \mid Z\) | conditional independence of \(X\) and \(Y\) given \(Z\) |
| \(\mathcal{N}(\mu,\sigma^2)\), \(\mathcal{N}_p(\boldsymbol{\mu},\boldsymbol{\Sigma})\) | univariate and \(p\)-variate normal |
| \(\mathrm{Bin}(n,p)\), \(\mathrm{Poi}(\lambda)\), \(\mathrm{Ber}(p)\) | binomial, Poisson, Bernoulli |
| \(\mathrm{PoBin}(\mathbf{p})\) | Poisson binomial with success-probability vector \(\mathbf{p}\) |
| \(F_X\), \(F_X^{-1}\) | cumulative distribution function and its generalized inverse (quantile) |
| \(\Phi\), \(\phi\) | standard normal distribution function and density |
| \(M_X(\mathbf{t})=\mathbb{E}[e^{\mathbf{t}^\top X}]\) | moment generating function |
| \(\|\mu-\nu\|_{\mathrm{TV}}\) | total variation distance between measures \(\mu\) and \(\nu\) |
| \(\xrightarrow{p}\), \(\xrightarrow{d}\), \(\xrightarrow{\text{a.s.}}\) | convergence in probability, in distribution, almost surely |
| \(O_p\), \(o_p\), \(O(\cdot)\), \(o(\cdot)\) | stochastic and deterministic order symbols |
Statistical inference
| Symbol | Meaning |
|---|---|
| \(L(\boldsymbol{\theta})\), \(\ell(\boldsymbol{\theta})=\log L(\boldsymbol{\theta})\) | likelihood and log-likelihood |
| \(\mathbf{s}(\boldsymbol{\theta})=\nabla\ell(\boldsymbol{\theta})\) | score function |
| \(\mathcal{I}(\boldsymbol{\theta})\) | Fisher information matrix |
| \(\hat{\boldsymbol{\theta}}_{\mathrm{MLE}}\) | maximum likelihood estimator |
| \(\mathbb{V}\) | sandwich or robust variance estimator (M-estimation) |
Bayesian computation and model comparison
| Symbol | Meaning |
|---|---|
| \(\pi(\boldsymbol{\xi})\), \(\pi(\boldsymbol{\xi}\mid\mathbf{y})\) | prior and posterior density of the parameter vector |
| \(p(\mathbf{y}\mid\boldsymbol{\xi})\) | likelihood of the data given the parameters |
| \(\Pi\) | prior probability measure (used when the prior has no density) |
| \(\tau^2\) | between-study heterogeneity variance |
| \(\hat R\) | Gelman-Rubin potential scale reduction statistic (near \(1\) at convergence) |
| \(N_{\mathrm{eff}}\) | effective sample size of an autocorrelated MCMC chain, distinct from the weighting \(\mathrm{ESS}\) |
| \(\bar D\), \(D(\bar{\boldsymbol{\theta}})\), \(p_D\) | posterior mean deviance, deviance at the posterior mean, and effective number of parameters \(p_D=\bar D-D(\bar{\boldsymbol{\theta}})\) |
| \(\mathrm{lppd}\), \(\widehat{\mathrm{elpd}}\) | log pointwise predictive density and estimated expected log pointwise predictive density |
| \(\mathrm{WAIC}\), \(\mathrm{LOO}\) | widely applicable information criterion and leave-one-out cross-validation scores |
| \(\mathrm{PSIS}\), \(\widehat k\) | Pareto-smoothed importance sampling and its per-observation tail-index diagnostic |
Causal inference
| Symbol | Meaning |
|---|---|
| \(Y(0), Y(1)\) | potential outcomes under control and active treatment |
| \(T \in \{0,1\}\) | treatment indicator |
| \(Y = TY(1)+(1-T)Y(0)\) | observed outcome (consistency under SUTVA) |
| \(X \in \mathcal{X}\) | baseline covariates |
| \(e(x)=P(T=1\mid X=x)\) | propensity score |
| \(\mu_t(x)=\mathbb{E}[Y(t)\mid X=x]\) | conditional mean outcome under treatment \(t\) |
| \(\tau(x)=\mu_1(x)-\mu_0(x)\) | conditional average treatment effect |
| \(\tau=\mathbb{E}[\tau(X)]\) | average treatment effect |
| \(S\) | study or population membership indicator (used in transportability) |
Studies, treatments, and populations
| Symbol | Meaning |
|---|---|
| \(j\) | study index, for example \(j=AB,\,BC\) |
| \(k\) | treatment index, or arm |
| \(i\) | individual index within a study or arm |
| \(\mathcal{P}, \mathcal{P}^{*}\) | populations; a population is a distribution over \(\mathcal{X}\), with density \(f_{\mathcal{P}}\) |
| \(f_j(x)\), \(F_j(x)\) | covariate density and distribution function in study \(j\) |
| \(\bar{\mathbf{x}}_j=\int \mathbf{x}\,f_j(x)\,dx\), \(\boldsymbol{\Sigma}_j\) | covariate mean and covariance in study \(j\) |
| \(d_{ab(\mathcal{P})}\) | marginal relative effect of treatment \(b\) versus \(a\) in population \(\mathcal{P}\) on a stated scale |
| \(\hat d_{ab(j)}\) | estimator of \(d_{ab}\) from study \(j\) |
| \(\delta_{j,\,t_1 t_k}\) | study-\(j\) contrast of treatment \(t_k\) versus \(t_1\) |
Outcome models and likelihoods
| Symbol | Meaning |
|---|---|
| \(g\), \(g^{-1}\) | link function and its inverse |
| \(\eta_{ijk}\) | linear predictor for individual \(i\) in study \(j\) on arm \(k\) |
| \(\theta_{ijk}=g^{-1}(\eta_{ijk})\) | conditional mean on the outcome scale |
| \(\mu_j\) | study-\(j\) baseline (intercept) |
| \(\boldsymbol{\beta}_1\) | main (prognostic) covariate effects |
| \(\boldsymbol{\beta}_{2,k}\) | treatment-by-covariate interactions for arm \(k\) (effect modification) |
| \(\gamma_k\) | treatment main effect relative to the reference treatment |
| \(L_i(\boldsymbol{\xi};y_i)\), \(\bar L_j(\boldsymbol{\xi};\bar y_j)\) | individual and aggregate-level likelihood contributions |
| \(\boldsymbol{\xi}\) | full parameter vector |
Population adjustment: MAIC, STC, ML-NMR, ML-UMR
| Symbol | Meaning |
|---|---|
| \(w_i\), \(w(x)\) | matching weight for individual \(i\), or weight function at \(x\) |
| \(\mathrm{ESS}=\left(\sum_i w_i\right)^2 / \sum_i w_i^2\) | effective sample size of a weighted sample |
| \(\alpha, \boldsymbol{\beta}\) | matching-weight parameters in \(w_i=\exp(\alpha+\boldsymbol{\beta}^\top \mathbf{x}_i)\) |
| \(\hat\mu_t^{\mathrm{STC}}\) | simulated treatment comparison predicted mean under treatment \(t\) |
| \(C(u_1,\dots,u_d)\) | copula |
| \(C_\Omega(\mathbf{u})=\Phi_\Omega\!\big(\Phi^{-1}(u_1),\dots,\Phi^{-1}(u_d)\big)\) | Gaussian copula with correlation matrix \(\Omega\) |
| \(V_{\mathrm{HK}}(f)\) | Hardy–Krause variation of \(f\) |
| \(D_N^{*}(P)\) | star discrepancy of a point set \(P\) |
| SEMA, SPFA | shared effect modifier assumption, shared prognostic factor assumption |
Abbreviations
| Short form | Expansion |
|---|---|
| ITC | indirect treatment comparison |
| PAIC | population-adjusted indirect comparison |
| (N)MA | (network) meta-analysis |
| MAIC | matching-adjusted indirect comparison |
| STC | simulated treatment comparison |
| ML-NMR | multilevel network meta-regression |
| ML-UMR | multilevel unanchored meta-regression |
| IPD, AgD | individual patient data, aggregate data |
| EM, PV | effect modifier, prognostic variable |
| ATE, CATE | average treatment effect, conditional average treatment effect |
| IPW, AIPW | inverse probability weighting, augmented inverse probability weighting |
| OR, RR, HR, RD | odds ratio, risk ratio, hazard ratio, risk difference |
| RMST | restricted mean survival time |
| QMC | quasi-Monte Carlo |
| HTA | health technology assessment |