Appendices

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Appendix A: Math-to-code crosswalk

Each displayed equation maps to the package code that implements it. The locations refer to the development version of mlumr; search for the quoted expression if the file has moved.

Table 9: Equation-to-code crosswalk.

Equation	What it says	Code location
Equation 13	IPD GLM, $g(\mathbb{E}[y]) = \mu_{\mathrm{idx}} + x^\top\beta$	`inst/stan/mlumr_binary_spfa.stan` (transformed parameters, `eta_ipd`)
Equation 21, Equation 25	binomial AgD, $\bar p_{\mathrm{cmp}} = \mathrm{mean}(g^{-1})$	`inst/stan/include/binary_functions.stan`: `integrated_binomial_lpmf()`
Equation 26	normal AgD likelihood	`mlumr_normal_spfa.stan`: `y_agd[k] ~ normal(theta_agd_bar, se_agd[k])`
Equation 27	Poisson AgD log-sum-exp	`mlumr_poisson_spfa.stan`: `log_lambda_agd_bar`
Equation 29, Table 6	survival likelihood by status	`inst/stan/include/survival_functions.stan` (`surv_ll_status`)
Equation 33	integrated pseudo-IPD likelihood	`mlumr_survival_spfa.stan`: `agd_ll[j] = log_sum_exp(ll) - log(n_int)`
Equation 31, Equation 32	M-spline hazard, simplex	`mlumr_survival_mspline_spfa.stan`: `scoef = softmax(append_row(0, lscoef))`
Equation 22, Equation 23	Sobol, copula, quantile transform	`R/integration.R`: `.generate_copula_uniforms()` (Sobol points, Cholesky factor of $\Omega$), `.transform_integration_points()`
Equation 24	Spearman to copula correlation	`R/utils.R`: `cor_adjust_spearman()`
Equation 38	covariate centering	`R/mlumr.R`: `.mlumr_center_covariates()`
Equation 39	thin QR reparameterization	`R/mlumr.R`: `.mlumr_qr_design()`; Stan `allbeta = R_inv * beta_tilde`
Equation 36	LOR (always logit), RD, RR	`mlumr_binary_spfa.stan` (generated quantities, `lor_*` from the log event and non-event probabilities)
Equation 37	survivor-weighted marginal hazard	`survival_functions.stan`: `log_mean_haz()`; `survival_mspline_functions.stan`: `mspline_log_mean_haz()`
RMST (Section 14.4)	trapezoidal rule on $\bar S(t)$	`survival_functions.stan`: `rmst_param()`; `survival_mspline_functions.stan`: `mspline_rmst()`
Equation 42	DIC, $p_D = \tfrac12\mathrm{Var}(D)$	`R/diagnostics.R`: `pD <- 0.5 * var(D)`
Equation 41, Equation 40	naive contrasts, delta method	`R/naive.R`
(STC standardization)	Comparator-population G-computation	`R/stc.R`

Appendix B: Notation glossary

Table 10: Notation used in this document.

Symbol	Meaning
$y_i$, $x_i$	outcome and covariate vector of IPD patient $i$
$p$	number of covariates
$\mu_{\mathrm{idx}}, \mu_{\mathrm{cmp}}$	index and comparator treatment intercepts
$\beta$	shared prognostic coefficient vector (SPFA)
$\beta_{\mathrm{idx}}, \beta_{\mathrm{cmp}}$	treatment-specific coefficients (relaxed)
$\Delta\beta$	effect modification, $\beta_{\mathrm{idx}} - \beta_{\mathrm{cmp}}$
$\eta_i$	linear predictor $\mu + x_i^\top\beta$
$g$, $g^{-1}$	link and inverse-link function
$f_{\mathrm{cmp}}(x)$	comparator covariate density
$\tilde x_m$, $M$	integration point $m$, number of points (`n_int`)
$\bar\theta_{\mathrm{cmp}}$	model-implied comparator mean outcome (Equation 21)
$\bar p_{\mathrm{cmp}}$	average comparator event probability (Equation 25)
$\Omega$, $\rho_S$	copula correlation matrix, Spearman correlation
$S(t), h(t), H(t)$	survival, hazard, cumulative hazard
$h_0(t)$, $M_j(t)$, $I_j(t)$	baseline hazard, M-spline and I-spline basis
$s$, $\sigma_{\text{smooth}}$	spline simplex coefficients, smoothing scale
$\sigma$	residual SD (normal family)
$D$, $p_D$	deviance, effective number of parameters
$Q$, $R$	thin QR factors of the design matrix

Appendix C: Survival distribution reference

All nine parametric survival distributions, with the linear predictor $\eta = \mu + x^\top\beta$. PH distributions act on the hazard; AFT distributions rescale time. Auxiliary shape and scale parameters carry half-normal priors by default (Section 12).

Table 11: The nine parametric survival distributions. $k$ is a shape, $\gamma$ the Gompertz rate, $\sigma$ a log-scale SD, $\kappa$ a gamma shape, $\gamma_{\text{reg}}$ the regularized lower incomplete gamma function.

Distribution	Type	Survival $S(t)$	Hazard $h(t)$
Exponential	PH	$\exp(-t\,e^{\eta})$	$e^{\eta}$
Weibull	PH	$\exp(-t^{k} e^{\eta})$	$k\,t^{k-1} e^{\eta}$
Gompertz	PH	$\exp\!\big(-\tfrac{e^{\eta}}{\gamma}(e^{\gamma t}-1)\big)$	$e^{\eta} e^{\gamma t}$
Exponential	AFT	$\exp(-t\,e^{-\eta})$	$e^{-\eta}$
Weibull	AFT	$\exp\!\big(-(t\,e^{-\eta})^{k}\big)$	$k\,e^{-\eta}(t e^{-\eta})^{k-1}$
Log-normal	AFT	$1 - \Phi\!\big(\tfrac{\log t - \eta}{\sigma}\big)$	$f(t)/S(t)$
Log-logistic	AFT	$\big(1 + (t e^{-\eta})^{k}\big)^{-1}$	$\tfrac{(k/ e^{\eta})(t e^{-\eta})^{k-1}}{1 + (t e^{-\eta})^{k}}$
Gamma	AFT	$1 - \gamma_{\text{reg}}(\kappa, t e^{-\eta})$	$f(t)/S(t)$
Generalized gamma	AFT	(three-parameter; see flexsurv)	$f(t)/S(t)$

Appendix D: Reproducibility

Every code chunk in this document runs on base R plus three packages that are already mlumr dependencies: ggplot2, randtoolbox, and splines2 (the M-spline figure also uses patchwork if available). No chunk fits a Stan model or calls mlumr itself, so the document renders in seconds. Every chunk that uses randomness sets set.seed(2026) for exact reproducibility.

To render:

quarto render mathematical-foundations.qmd

sessionInfo()

R version 4.6.0 (2026-04-24)
Platform: aarch64-apple-darwin23
Running under: macOS 27.0

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.6/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.6/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1

locale:
[1] en_CA.UTF-8/en_CA.UTF-8/en_CA.UTF-8/C/en_CA.UTF-8/en_CA.UTF-8

time zone: America/Toronto
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] splines2_0.5.4    randtoolbox_2.0.5 rngWELL_0.10-10   ggplot2_4.0.3    

loaded via a namespace (and not attached):
 [1] gtable_0.3.6       jsonlite_2.0.0     dplyr_1.2.1        compiler_4.6.0    
 [5] tidyselect_1.2.1   Rcpp_1.1.2         scales_1.4.0       yaml_2.3.12       
 [9] fastmap_1.2.0      R6_2.6.1           labeling_0.4.3     generics_0.1.4    
[13] patchwork_1.3.2    isoband_0.3.0      knitr_1.51         tibble_3.3.1      
[17] pillar_1.11.1      RColorBrewer_1.1-3 rlang_1.3.0        xfun_0.60         
[21] S7_0.2.2           otel_0.2.0         cli_3.6.6          withr_3.0.3       
[25] magrittr_2.0.5     digest_0.6.39      grid_4.6.0         lifecycle_1.0.5   
[29] vctrs_0.7.3        evaluate_1.0.5     glue_1.8.1         farver_2.1.2      
[33] rmarkdown_2.32     tools_4.6.0        pkgconfig_2.0.3    htmltools_0.5.9

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Contents | Previous: Part V: Computation, benchmarks, diagnostics