Notation Glossary

Sample, covariates, and the design matrix

Symbol	Meaning
\(n\)	Total sample size (number of subjects in a completed/completing design).
\(p\)	Number of covariate columns.
\(X\)	The covariate design matrix, \(n \times p\) (or \(n \times q\) after intercept/expansion), as consumed by a fast_* C++ backend. Whether \(X\) includes an intercept column is backend-specific and is stated per function — some (e.g. fast_coxph_regression_cpp) never take one because the model has none; others expect the caller to add it.
\(x_i\)	Row \(i\) of \(X\): subject \(i\)’s covariate vector (as a column vector in formulas, \(x_i^\top\beta\)).
\(\eta_i\)	The linear predictor for subject \(i\), \(\eta_i = x_i^\top\beta\) (or with a treatment/offset term folded in, per model).

Treatment assignment

Symbol	Meaning
\(W\) / w	The treatment assignment vector, length \(n\). Public convention throughout the package: \(\{0,1\}\) encoding, \(1\) = treated, \(0\) = control (see ?Design, “Details”). Every Design public method that returns or accepts w (get_w(), draw_ws_according_to_design(), …) uses this encoding.
\(w_i\)	Subject \(i\)’s treatment indicator, \(w_i \in \{0,1\}\).
Signed recoding	A handful of variance estimators (InferenceIncidCMH, InferenceIncidExtendedRobins) recode internally to a signed \(\{-1,+1\}\) contrast where their formulas require it. This recoding is local to those classes and does not change the public \(\{0,1\}\) convention.
SimulationFramework custom hooks	custom_apply_treatment_and_noise/make_estimand_fn receive w in \(\{-1,+1\}\) format for historical reasons specific to that framework’s internal DGP machinery; convert with (w+1)/2 to get the \(\{0,1\}\) convention used everywhere else. This is a documented, isolated exception, not evidence the package-wide convention is inconsistent.

Response, censoring, and the y/y_L/y_R schema

Symbol	Meaning
\(Y\) / y	The (exact, uncensored) response vector. Scale/type depends on response_type: continuous, incidence (binary), count (non-negative integer), proportion (in \((0,1)\) or \([0,1]\)), ordinal (categorical, coded \(1,\dots,K\)), or survival (a time).
y_L, y_R	For interval-censored survival responses, the lower/upper bounds of the interval a subject’s true event time is known to fall in. Supplied XOR with y on Design$add_one_subject_response(): a subject has either an exact y or a censored (y_L, y_R) pair, never both, never just one bound. Right-censored: y_L = last known survival time, y_R = Inf. Left-censored: y_L = 0.
dead	The event/censoring indicator used by the survival-specific C++/Python backends and simulation DGP: 1 = event observed (maps to an exact y), 0 = right-censored (maps to y_L = y, y_R = Inf). This is the DGP-facing two-column contract that Design’s three-column y/y_L/y_R storage schema is bridged to/from (see dead_to_response_bounds() in other_helpers.R).
\(\delta_i\)	An event/censoring indicator in survival model notation (Cox partial likelihood, martingale residuals) — the modeling-formula analogue of dead. Not to be confused with the null-hypothesis \(\delta\) below; which meaning applies is always clear from context (a per-subject subscript \(\delta_i\) is always the censoring indicator, an unsubscripted \(\delta\) in a hypothesis/CI context is always the null value).

Coefficients and treatment effects

Symbol	Meaning
\(\beta\)	A regression coefficient vector, same order as the columns of the design matrix it multiplies.
\(\hat\beta\)	The fitted (maximum-likelihood, or otherwise estimated) coefficient vector.
\(\beta_T\)	The treatment-effect coefficient specifically — the entry of \(\beta\) multiplying the treatment column. Its position is passed explicitly to C++/Python backends as j_treat/j_T (1-based in R/Rcpp, 0-based in the Python bindings — always stated per function).
ssq_b_j / ssq_b_T	The estimated variance of \(\hat\beta_j\) (generically, index j) or specifically of \(\hat\beta_T\); se_beta_hat/s_beta_hat_T is its square root, the standard error.
\(\alpha_k\)	Ordinal-model context only: the \(k\)-th cumulative-category threshold/intercept in a proportional-odds, adjacent-category, continuation-ratio, or stereotype-logit model, \(k = 1,\dots,K-1\), with \(\alpha_1 < \cdots < \alpha_{K-1}\) enforced (directly, or via a log-difference reparameterization during optimization). Not the same symbol as the significance level below — always disambiguated by whether the surrounding text is about an ordinal model’s thresholds or about a confidence level.
\(\alpha\) (no subscript)	Inference context: the significance level; a computed interval has nominal coverage \(1-\alpha\), and \(H_0\) is rejected at level \(\alpha\) when a two-sided p-value is \(< \alpha\).
\(\delta\)	The null value a hypothesis test or confidence-interval inversion is built around: \(H_0: \theta = \delta\) for the estimand \(\theta\) (a risk difference, mean difference, quantile shift, etc.), with \(\delta = 0\) the default “no effect” null used by compute_asymp_two_sided_pval(delta = 0)-style methods package-wide. Confidence intervals are obtained by inverting the test over a grid/bisection of candidate \(\delta\) values (see mn_ci_cpp, newcombe_independent_ci_cpp, and the various compute_rand_confidence_interval() implementations).
\(\tau\)	The target quantile in a quantile-regression class (InferenceContinQuantileRegr, InferencePropQuantileRegr, and their KK variants), \(0 < \tau < 1\); \(\tau = 0.5\) (the default) is the median.
\(\phi\)	The precision parameter of a beta-distributed response (fast_beta_regression/InferencePropBetaRegr; larger \(\phi\) = less dispersion around the mean \(\mu\)), optimized on the log scale (log_phi) for positivity.
\(\theta\)	The dispersion/shape parameter of a negative-binomial response (fast_neg_bin/InferenceCountNegBin; NB2 parameterization, \(\mathrm{Var}(Y) = \mu + \mu^2/\theta\), smaller \(\theta\) = more overdispersion relative to Poisson), optimized on the log scale (log_theta).
\(\sigma\) / log_sigma	The standard deviation of a Gaussian random effect (frailty, random intercept) in a GLMM/mixed-model backend (fast_ordinal_glmm, fast_poisson_glmm, fast_gaussian_lmm, fast_weibull_frailty, …), always optimized on the log scale and typically clamped to \([-\texttt{max\_abs\_log\_sigma}, \texttt{max\_abs\_log\_sigma}]\) during optimization to keep Gauss-Hermite quadrature well-behaved.

Design structure: blocks, matched pairs, clusters, and the reservoir

Symbol	Meaning
m	A per-subject integer vector of block (or matched-pair) identifiers: subjects sharing the same value of m are in the same block/pair. Populated either directly (a supplied m argument) or computed internally (e.g. DesignFixedBinaryMatch’s non-bipartite matching); the only sanctioned way to assign it outside design_*.R is set_m().
Block	A group of subjects (via strata_cols/m) that randomization or resampling respects as a unit — e.g. DesignFixedBlocking randomizes within blocks; the Bayesian bootstrap can draw one Dirichlet weight per block rather than per subject when the design’s exchangeable resampling unit is a block.
Matched pair	The special case of a block of size exactly 2, with within-pair treatment randomization (DesignFixedBinaryMatch) or on-the-fly sequential matching (the KK family: DesignSeqOneByOneKK14/KK21/KK21stepwise). “Matched-pair” inference classes (InferenceIncidKKCondLogitOneLik and friends) condition out the pair’s nuisance intercept via a conditional-logit-style likelihood.
Reservoir (subjects)	In a sequential KK-matched design, subjects who have not yet been paired at the time an allocation decision is needed — they are assigned via the design’s fallback (unmatched) randomization rule rather than within-pair randomization, and are the “concordant”/marginal-model component in combined pair-plus-reservoir inference kernels (fast_cpoisson_combined, fast_clogit_plus_glmm), contrasted with the “discordant”/matched-pair component.
Cluster	A group of subjects sharing a cluster_col value, randomized/resampled as a unit at the whole-cluster level (DesignFixedCluster, DesignFixedBlockedCluster — the latter combines cluster-level and block-level structure, strata-then-cluster).
group_id	The generic per-row grouping identifier passed to GLMM/frailty C++ backends (fast_ordinal_glmm, fast_poisson_glmm, fast_weibull_frailty, …) identifying which rows share a random effect — the member-level analogue of a block/pair/cluster ID for backends that don’t themselves know whether the grouping is a matched pair or something else.

Resampling and randomization

Symbol	Meaning
B_boot / num_boot	Number of nonparametric or Bayesian bootstrap resamples.
r_rand / r	Number of randomization-distribution draws (permutations of w consistent with the design) used for a randomization p-value or as bisection steps of a randomization-based confidence interval.
Bootstrap weights	Nonparametric bootstrap: implicit multinomial resample counts (one full resample = drawing \(n\) subjects/blocks with replacement). Bayesian bootstrap: explicit Dirichlet\((1,\dots,1)\) weights, one per exchangeable resampling unit (subject or block — whichever the design supports; see subject_or_block_weights on compute_estimate_with_bootstrap_weights() methods package-wide), summing to \(n\) (or the unit count) in expectation, used to reweight every subject’s/block’s contribution to the estimating equations rather than physically resampling rows.
Randomization permutation	One re-draw of w from the design’s own randomization distribution (respecting blocks/pairs/clusters exactly as the original design would have), used to build a reference (null or shifted-null) distribution for a treatment-effect statistic under \(H_0\) — the basis of every compute_rand_two_sided_pval()/compute_rand_confidence_interval() implementation.
seed	An integer RNG seed. Supplying seed makes a design’s/simulation’s draws reproducible; every current concrete Design class is seed-reproducible (see vignette("reproducibility") for the mechanism, including DesignFixedGreedyDOptimal’s local-generator approach and DesignFixedGreedy’s parallel-safe per-thread seeding).

Treatment-effect scales

The estimand returned by compute_estimate() (and the scale a confidence interval/p-value is computed on) is family-specific; individual class documentation states which of these applies, but as a map:

Response family	Typical effect scale
Continuous	Mean difference, or a quantile shift (tau-quantile regression)
Incidence (binary)	Risk difference, log risk ratio, or log odds ratio, depending on the link (identity/log/logit)
Count	Log rate ratio (Poisson/NegBin log link)
Proportion	Difference in standardized (G-computation) mean, or a logit-scale quantile shift
Survival	Log hazard ratio (Cox-family), log-time ratio (Weibull AFT), RMST difference, or a KM-median difference
Ordinal	A cumulative-log-odds shift (proportional-odds \(\beta\)), a mean-rank/ridit statistic, or a standardized mean-category-score difference (G-computation)

Transformed outcomes

Several inference classes work on a transformed scale internally while reporting the estimand on that same transformed scale (never silently back-transforming to the raw scale unless the class’s own documentation says so):

• Logit transform (logit()/inv_logit() in other_helpers.R): used to map a proportion/incidence response to the real line before applying continuous-response machinery (e.g. InferencePropQuantileRegr fits quantile regression on logit(y), so its estimand is a log-odds-ratio shift, not a raw-scale quantile shift). Both directions clamp near the \(\{0,1\}\) boundary to avoid non-finite values.
• Log transform: count and rate models (Poisson, NegBin, log-binomial) use a log link, so coefficients are log rate/risk ratios unless exponentiated in the returned summary.
• Quantile transform: InferenceContinQuantileRegr/ InferencePropQuantileRegr report a shift in the response’s tau-quantile (on the logit scale for the proportion variant), not a mean difference.

Where this glossary does not apply

This page documents the package-wide conventions. A small number of low-level numerical kernels (fast_pchisq_upper, fast_qnorm, and other fast_* math utilities in bindings_fast_math.cpp/fast_math_utils.cpp) use single-letter mathematical arguments (x, p, df, a, b) with their ordinary textbook meanings, unrelated to the design/inference notation above; those are documented locally on each function instead.