macros.qmd

\providecommand{\cbl}[1]{\mathopen{}\left\{#1\right.}
\providecommand{\cb}[1]{\mathopen{}\left\{#1\right\}\mathclose{}}
\providecommand{\cbmat}[1]{\mathopen{}\left\{\begin{matrix}#1\end{matrix}\right\}\mathclose{}}
\providecommand{\paren}[1]{\mathopen{}\left(#1\right)\mathclose{}}
\providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}
\providecommand{\sparen}[1]{(#1)}
\providecommand{\sb}[1]{\mathopen{}\left[#1\right]\mathclose{}}
\def\distop{\operatorname}
\providecommand{\sbmat}[1]{\begin{bmatrix}#1\end{bmatrix}}
\providecommand{\Prf}[1]{\Pr\paren{#1}}
\def\am{\arg \max}
\def\argmax{\arg \max}
\def\amin{\arg \min}
\def\argmin{\arg \min}
\def\p{\distop{p}}
\def\f{\distop{f}}
\def\F{\distop{F}}
\def\h{\distop{h}}
\def\H{\distop{H}}
\def\pr{\p}
\def\dens{\p}
\def\pdens{\p}
\def\fdens{\f}
\def\mass{\P}
\def\pmass{\P}
\def\fmass{\F}
\def\S{\distop{S}}
\providecommand{\Sf}[1]{\S\paren{#1}}
\def\surv{\distop{S}}
\providecommand{\survf}[1]{\surv\paren{#1}}
\def\survratio{\ratio_{\surv}}
\def\rsurv{\survratio}
\def\survr{\survratio}
\providecommand{\survratiof}[1]{\survratio\paren{#1}}
\def\rsurvf{\survratiof}
\def\survrf{\survratiof}
\def\riskset{\mathcal{R}}
\providecommand{\risksetf}[1]{\riskset\paren{#1}}
\def\Q{\distop{Q}}
\def\quantile{\distop{Q}}
\providecommand{\Qf}[1]{\Q\paren{#1}}
\providecommand{\quantilef}[1]{\quantile\paren{#1}}
\def\hQ{\mathop{\hat{\Q}}\nolimits}
\def\hquantile{\mathop{\hat{\quantile}}\nolimits}
\providecommand{\hQf}[1]{\hQ\paren{#1}}
\providecommand{\hquantilef}[1]{\hquantile\paren{#1}}
\def\ph{\mathop{\hat{\p}}\nolimits}
\def\P{\distop{P}}
\def\hp{\mathop{\hat{\p}}\nolimits}
\def\ga{\alpha}
\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\providecommand{\floor}[1]{\mathopen{}\left\lfloor#1\right\rfloor\mathclose{}}
\providecommand{\ceiling}[1]{\mathopen{}\left\lceil#1\right\rceil\mathclose{}}
\providecommand{\ceil}[1]{\mathopen{}\left\lceil#1\right\rceil\mathclose{}}
\def\Ber{\operatorname{Ber}}
\def\Bernoulli{\operatorname{Bernoulli}}
\def\Pois{\operatorname{Pois}}
\def\Poisson{\operatorname{Poisson}}
\def\Gaus{\operatorname{Gaussian}}
\def\Normal{\operatorname{N}}
\providecommand{\ndist}[1]{\Normal\paren{#1}}
\def\NB{\operatorname{NegBin}}
\def\NegBin{\operatorname{NegBin}}
\def\vbeta{\vec \beta}
\def\vb{\vec \b}
\def\vzero{\vec{0}}
\def\vone{\vec{1}}
\def\gb{\beta}
\def\gg{\gamma}
\def\gd{\delta}
\def\eps{\varepsilon}
\def\veps{\vec{\varepsilon}}
\providecommand{\epsp}[1]{\eps\paren{#1}}
\providecommand{\epsf}[1]{\eps\paren{#1}}
\def\om{\omega}
\def\m{\mu}
\def\mean{\mu}
\def\pmean{\mu}
\def\popmean{\mu}
\def\vm{\vec \m}
\def\s{\sigma}
\def\l{\lambda}
\def\gl{\lambda}
\def\lam{\lambda}
\def\k{\kappa}
\def\cs{\kappa}
\def\condsurv{\kappa}
\def\gs{\sigma}
\def\gm{\mu}
\def\M{M}
\def\gM{M}
\def\Mu{M}
\def\cd{\cdot}
\def\cds{\cdots}
\def\lds{\ldots}
\def\eqdef{\stackrel{\text{def}}{=}}
\def\defeq{\stackrel{\text{def}}{=}}
\def\hb{\hat \beta}
\def\hvb{\hat{\vb}}
\def\hR{\hat{R}}
\def\hk{\hat \kappa}
\def\hkappa{\hat \kappa}
\def\hl{\hat \lambda}
\def\hy{\hat y}
\def\yh{\hat y}
\def\V{{\operatorname{Var}}}
\def\hs{\hat \sigma}
\def\hsig{\hat \sigma}
\def\hS{\hat \Sigma}
\def\hSig{\hat \Sigma}
\def\hSigma{\hat \Sigma}
\def\hsurv{\mathop{\hat{\surv}}\nolimits}
\def\hSurv{\mathop{\hat{\surv}}\nolimits}
\providecommand{\hSurvf}[1]{\hSurv\paren{#1}}
\def\emp{\text{emp}}
\def\km{\text{KM}}
\def\naa{\text{NA}}
\def\hsemp{\hsurv_{\emp}}
\def\hskm{\hsurv_{\km}}
\def\hsnaa{\hsurv_{\naa}}
\providecommand{\hsempf}[1]{\hsemp\paren{#1}}
\providecommand{\hskmf}[1]{\hskm\paren{#1}}
\providecommand{\hsnaaf}[1]{\hsnaa\paren{#1}}
\def\dist{\ \sim \ }
\def\ddist{\ \dot{\sim} \ }
\def\dsim{\ \dot{\sim} \ }
\def\za{z_{1 - \frac{\alpha}{2}}}
\def\ciradsym{\xi}
\def\cirad{\za \cdot \hse{\hb}}
\providecommand{\ciradf}[1]{\za \cdot \hse{#1}}
\def\ci{\hb {\color{red}\pm} \cirad}
\def\th{\theta}
\def\vth{\vec{\theta}}
\def\Th{\Theta}
\def\xbar{\bar{x}}
\def\hth{\hat\theta}
\def\hthml{\hth_{\text{ML}}}
\def\ba{\begin{aligned}}
\def\ea{\end{aligned}}
\def\ind{\perp\!\!\!\perp}
\def\indpt{\ind}
\def\indold{⫫}
\def\all{\forall}
\def\iid{\operatorname{iid}}
\def\ciid{\operatorname{ciid}}
\def\simind{\ \sim_{\ind}\ }
\def\siid{\ \sim_{\iid}\ }
\def\sciid{\ \sim_{\ciid}\ }
\def\simiid{\siid}
\def\distiid{\siid}
\def\tf{\therefore}
\def\Lik{\mathcal{L}}
\def\llik{\ell}
\def\loglik{\ell}
\def\logl{\ell}
\providecommand{\llikf}[1]{\llik \paren{#1}}
\providecommand{\Likf}[1]{\Lik\paren{#1}}
\def\loglikf{\llikf}
\def\loglf{\llikf}
\def\likf{\llikf}
\def\deviation{e}
\def\devn{\deviation}
\providecommand{\deviationv}{\vec{\deviation}}
\def\devnv{\deviationv}
\def\Deviation{E}
\def\Devn{\Deviation}
\def\residual{r}
\def\resid{\residual}
\def\Residual{R}
\def\Resid{\Residual}
\def\stdresidual{r^\prime}
\def\stdresid{\stdresidual}
\def\StdResidual{R^\prime}
\def\StdResid{\StdResidual}
\def\modresidual{r^*}
\def\modresid{\modresidual}
\def\ModResidual{R^*}
\def\ModResid{\ModResidual}
\def\score{\ell'}
\def\scorev{\vec{\ell'}}
\def\scorei{\vec{\ell_i'}}
\def\scorevi{\vec{\ell_i'}}
\providecommand{\scoref}[1]{\score \paren{#1}}
\def\hess{\llik''}
\def\hessm{\llik''}
\def\hessian{\hess}
\def\hessij{\llik_{ij}''}
\def\hessijalt{\sb{\hess}_{ij}}
\providecommand{\hessf}[1]{\hess \paren{#1}}
\providecommand{\hessianf}[1]{\hess \paren{#1}}
\providecommand{\starf}[1]{#1^*}
\def\lik{\ell}
\providecommand{\est}[1]{\hat{#1}}
\providecommand{\Est}[1]{\widehat{#1}}
\providecommand{\esttmp}[1]{{\Est{#1}}^*}
\def\esttmpl{\esttmp{\lambda}}
\def\cR{\mathcal{R}}
\def\range{\mathcal{R}}
\def\Range{\mathcal{R}}
\providecommand{\rangef}[1]{\cR(#1)}
\providecommand{\seq}[2]{#1{:}#2}
\def\~{\approx}
\def\dapp{\dot\approx}
\providecommand{\deriv}[1]{\frac{\partial}{\partial #1}}
\providecommand{\dderiv}[1]{\frac{\partial^2}{\partial #1^2}}
\providecommand{\derivf}[2]{\frac{\partial #1}{\partial #2}}
\providecommand{\derivff}[2]{\frac{\partial #2}{\partial #1}}
\providecommand{\derivfnd}[2]{\derivf{#1}{#2}}
\providecommand{\derivfdn}[2]{\derivff{#1}{#2}}
\providecommand{\dderivf}[2]{\frac{\partial^2 #1}{\partial #2^2}}
\providecommand{\xderiv}[2]{\frac{\partial^2}{\partial #1 \partial #2}}
\providecommand{\xderivf}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}}
\providecommand{\xderivff}[3]{\frac{\partial^2 #3}{\partial #1 \partial #2}}
\providecommand{\dif}[2]{\frac{d #1}{d #2}}
\providecommand{\red}[1]{{\color{red}{#1}}}
\providecommand{\green}[1]{{\color{green}{#1}}}
\providecommand{\blue}[1]{{\color{blue}{#1}}}
\providecommand{\cyan}[1]{{\color{cyan}{#1}}}
\providecommand{\magenta}[1]{{\color{magenta}{#1}}}
\providecommand{\yellow}[1]{{\color{yellow}{#1}}}
\providecommand{\black}[1]{{\color{black}{#1}}}
\providecommand{\gray}[1]{{\color{gray}{#1}}}
\providecommand{\white}[1]{{\color{white}{#1}}}
\providecommand{\darkgray}[1]{{\color{darkgray}{#1}}}
\providecommand{\lightgray}[1]{{\color{lightgray}{#1}}}
\providecommand{\brown}[1]{{\color{brown}{#1}}}
\providecommand{\lime}[1]{{\color{lime}{#1}}}
\providecommand{\olive}[1]{{\color{olive}{#1}}}
\providecommand{\orange}[1]{{\color{orange}{#1}}}
\providecommand{\pink}[1]{{\color{pink}{#1}}}
\providecommand{\purple}[1]{{\color{purple}{#1}}}
\providecommand{\teal}[1]{{\color{teal}{#1}}}
\providecommand{\violet}[1]{{\color{violet}{#1}}}
\providecommand{\hE}[1]{\mathop{\hat{\operatorname{E}}}\nolimits\sb{#1}}
\providecommand{\hExp}[1]{\mathop{\hat{\operatorname{E}}}\nolimits\sb{#1}}
\providecommand{\hExpf}[1]{\mathop{\hat{\operatorname{E}}}\nolimits\sb{#1}}
\def\hmu{\hat{\mu}}
\providecommand{\hmuf}[1]{\hmu\paren{#1}}
\def\Expp{\distop{E}}
\def\Ep{\distop{E}}
\def\expit{\operatorname{expit}}
\providecommand{\expitf}[1]{\expit\cb{#1}}
\providecommand{\dexpitf}[1]{\expit'\cb{#1}}
\def\logit{\operatorname{logit}}
\providecommand{\logitf}[1]{\logit\cb{#1}}
\providecommand{\E}[1]{\distop{E}\sb{#1}}
\providecommand{\Ef}[1]{\distop{E}\sb{#1}}
\providecommand{\Exp}[1]{\distop{E}\sb{#1}}
\providecommand{\Expf}[1]{\distop{E}\sb{#1}}
\providecommand{\Expfc}[2]{\distop{E}_{#1}\sb{#2}}
\providecommand{\evalAt}[2]{\left.{#1}\right|_{#2}}
\def\Varr{\operatorname{Var}}
\def\Vart{\operatorname{Var}}
\def\hvar{\mathop{\widehat{\Vart}}\nolimits}
\def\hvart{\mathop{\widehat{\Vart}}\nolimits}
\def\Covt{\operatorname{Cov}}
\def\hcovt{\mathop{\widehat{\Covt}}\nolimits}
\def\hCovt{\mathop{\widehat{\Covt}}\nolimits}
\providecommand{\var}[1]{\operatorname{Var}\paren{#1}}
\providecommand{\varf}[1]{\operatorname{Var}\paren{#1}}
\providecommand{\Var}[1]{\operatorname{Var}\paren{#1}}
\providecommand{\hVar}[1]{\mathop{\widehat{\operatorname{Var}}}\nolimits\paren{#1}}
\providecommand{\Varf}[1]{\operatorname{Var}\paren{#1}}
\providecommand{\hvarf}[1]{\hvart\paren{#1}}
\providecommand{\Cov}[1]{\Covt \paren{#1}}
\providecommand{\Covs}[2]{\Covt \paren{#1, #2}}
\providecommand{\Covf}[1]{\Covt \paren{#1}}
\providecommand{\covh}[1]{\hcovt\paren{#1}}
\providecommand{\covhf}[1]{\hcovt\paren{#1}}
\providecommand{\hcovf}[1]{\hcovt\paren{#1}}
\def\Cort{\operatorname{Cor}}
\def\hcort{\mathop{\widehat{\Cort}}\nolimits}
\providecommand{\Cor}[1]{\Cort \paren{#1}}
\providecommand{\Corf}[1]{\Cort \paren{#1}}
\providecommand{\Cors}[2]{\Cort \paren{#1, #2}}
\providecommand{\hCor}[1]{\hcort\paren{#1}}
\providecommand{\hCorf}[1]{\hcort\paren{#1}}
\providecommand{\hCors}[2]{\hcort\paren{#1, #2}}
\def\med{\operatorname{med}}
\providecommand{\medf}[1]{\med\paren{#1}}
\def\IQR{\operatorname{IQR}}
\providecommand{\IQRf}[1]{\IQR\paren{#1}}
\def\varht{\mathop{\widehat{\operatorname{Var}}}\nolimits}
\providecommand{\varh}[1]{\varht\paren{#1}}
\providecommand{\varhf}[1]{\varht\paren{#1}}
\providecommand{\vc}[1]{\boldsymbol{#1}}
\providecommand{\sd}[1]{\operatorname{sd}\paren{#1}}
\providecommand{\SD}[1]{\operatorname{SD}\paren{#1}}
\providecommand{\hSD}[1]{\mathop{\widehat{\operatorname{SD}}}\nolimits\paren{#1}}
\providecommand{\hSDf}[1]{\mathop{\widehat{\operatorname{SD}}}\nolimits\paren{#1}}
\providecommand{\se}[1]{\operatorname{se}\paren{#1}}
\def\hset{\mathop{\widehat{\operatorname{se}}}\nolimits}
\def\hSE{\mathop{\widehat{\operatorname{SE}}}\nolimits}
\def\hSEt{\mathop{\widehat{\operatorname{SE}}}\nolimits}
\providecommand{\sef}[1]{\operatorname{se}\paren{#1}}
\providecommand{\hse}[1]{\hset\paren{#1}}
\providecommand{\hsef}[1]{\hset\paren{#1}}
\providecommand{\SE}[1]{\operatorname{SE}\paren{#1}}
\providecommand{\SEf}[1]{\operatorname{SE}\paren{#1}}
\providecommand{\SDf}[1]{\operatorname{SD}\paren{#1}}
\providecommand{\HSE}[1]{\hSE\paren{#1}}
\providecommand{\hSEf}[1]{\hSE\paren{#1}}
\def\logt{\operatorname{log}}
\providecommand{\logf}[1]{\logt\cb{#1}}
\def\dlog{\logt'}
\providecommand{\dlogf}[1]{\dlog \cb{#1}}
\renewcommand{\exp}[1]{\operatorname{exp}\cb{#1}}
\def\expt{\operatorname{exp}}
\providecommand{\expf}[1]{\operatorname{exp}\cb{#1}}
\providecommand{\expfn}[1]{\expf{#1}}
\def\dexp{\operatorname{exp}'}
\providecommand{\dexpf}[1]{\dexp \cb{#1}}
\providecommand{\e}[1]{\text{e}^{#1}}
\providecommand{\ef}[1]{\text{e}^{#1}}
\providecommand{\inv}[1]{\mathopen{}\left(#1\right)^{-1}\mathclose{}}
\providecommand{\invf}[1]{\mathopen{}\left(#1\right)^{-1}\mathclose{}}
\providecommand{\invp}[1]{\mathopen{}\left(#1\right)^{-1}\mathclose{}}
\providecommand{\invcb}[1]{\mathopen{}\left\{#1\right\}^{-1}\mathclose{}}
\providecommand{\invsb}[1]{\mathopen{}\left[#1\right]^{-1}\mathclose{}}
\def\oinf{I}
\def\oinfm{\mat{I}}
\def\Nat{\mathbb{N}}
\providecommand{\oinff}[1]{\oinf\paren{#1}}
\def\einf{\mathcal{I}}
\def\EInf{\einf}
\def\FInf{\einf}
\def\einfm{\mathcal{I}}
\providecommand{\einff}[1]{\einf\paren{#1}}
\def\heinf{\hat{\einf}}
\providecommand{\heinff}[1]{\heinf \paren{#1}}
\providecommand{\1}[1]{\text{1}_{#1}}
\providecommand{\indic}[1]{\mathbb{1}_{#1}}
\providecommand{\indicp}[1]{\mathbb{1}\paren{#1}}
\providecommand{\indiccb}[1]{\mathbb{1}\cb{#1}}
\providecommand{\indicf}[1]{\mathbb{1}_{#1}}
\providecommand{\set}[1]{\cb{#1}}
\providecommand{\pf}[1]{\p \paren{#1}}
\providecommand{\Pf}[1]{\P \paren{#1}}
\providecommand{\Bias}[1]{\operatorname{Bias}\paren{#1}}
\providecommand{\bias}[1]{\operatorname{Bias}\paren{#1}}
\def\ss{\sigma^2}
\providecommand{\ssqf}[1]{\sigma^2\paren{#1}}
\providecommand{\mselr}[1]{\operatorname{MSE}\paren{#1}}
\providecommand{\maelr}[1]{\operatorname{MAE}\paren{#1}}
\providecommand{\abs}[1]{\mathopen{}\left|#1\right|\mathclose{}}
\providecommand{\norm}[1]{\mathopen{}\left\lVert#1\right\rVert\mathclose{}}
\providecommand{\normsq}[1]{\mathopen{}\left\lVert#1\right\rVert^2\mathclose{}}
\providecommand{\sqf}[1]{\mathopen{}\left(#1\right)^2\mathclose{}}
\providecommand{\sq}{^2}
\def\err{\eps}
\providecommand{\erf}[1]{\err\paren{#1}}
\renewcommand{\vec}[1]{\tilde{#1}}
\providecommand{\vecf}[1]{\tilde{#1}}
\renewcommand{\v}[1]{\vecf{#1}}
\providecommand{\matr}[1]{\mathbf{#1}}
\def\mX{\matr{X}}
\def\mx{\matr{x}}
\def\va{\vecf{a}}
\def\vx{\vecf{x}}
\def\vxs{{\vec{x}^*}}
\def\diff{\Delta}
\def\diffvx{\diff\vx}
\def\vxi{\vecf{x}_i}
\def\vX{\vecf{X}}
\def\vy{\vecf{y}}
\def\vY{\vecf{Y}}
\def\vz{\vecf{z}}
\def\vZ{\vecf{Z}}
\def\vD{\vecf{D}}
\def\vd{\vecf{d}}
\def\vT{\vecf{T}}
\def\vt{\vecf{t}}
\def\vpi{\vecf{\pi}}
\providecommand{\mat}[1]{\mathbf{#1}}
\providecommand{\dsn}[1]{#1_1, \ldots, #1_n}
\def\X1n{\dsn{X}}
\def\Xin{\dsn{X}}
\def\x1n{\dsn{x}}
\def\y1n{\dsn{y}}
\def\'{^{\top}}
\providecommand{\pt}[1]{\paren{#1}\'}
\providecommand{\tp}[1]{{#1}\'}
\def\dpr{\cdot}
\def\Xx1n{X_1=x_1, \ldots, X_n = x_n}
\providecommand{\dprod}[2]{#1 \cdot #2}
\def\dprodf{\dprod}
\def\dpf{\dprodf}
\providecommand{\tprod}[2]{\tp{#1} #2}
\def\tprodf{\tprod}
\providecommand{\iprod}[2]{\left\langle #1, #2 \right\rangle}
\def\iprodf{\iprod}
\providecommand{\oprod}[2]{#1 \tp{#2}}
\providecommand{\stprod}[1]{\tprod{#1}{#1}}
\providecommand{\siprod}[1]{\iprod{#1}{#1}}
\providecommand{\soprod}[1]{#1 \tp{#1}}
\providecommand{\sdprod}[1]{\dprod{#1}{#1}}
\def\sdprodf{\sdprod}
\providecommand{\dsvn}[2]{#1_1=#2_1, \ldots, #1_n = #2_n}
\providecommand{\sumn}[1]{\sum_{#1=1}^n}
\def\sumin{\sum_{i=1}^n}
\def\sumiN{\sum_{i=1}^N}
\def\sumip{\sum_{i=1}^p}
\def\sumjn{\sum_{j=1}^n}
\def\sumjp{\sum_{j=1}^p}
\def\sumi1n{\sum_{i=1}^n}
\def\prodin{\prod_{i=1}^n}
\def\prodi1n{\prod_{i=1}^n}
\providecommand{\lp}[1]{\dprodf{#1}{\vb}}
\def\lpdp{\lp{\vx}}
\def\lincomp{\eta}
\def\odds{\omega}
\def\OR{\operatorname{OR}}
\def\ratio{\theta}
\providecommand{\ratiof}[1]{\ratio\paren{#1}}
\def\ror{\ratio_{\odds}}
\providecommand{\orf}[1]{\ror\paren{#1}}
\def\logodds{\lincomp}
\providecommand{\logoddsf}[1]{\logodds \paren{#1}}
\def\difflogodds{\diff \logodds}
\def\hdifflogodds{\widehat{\difflogodds}}
\def\diffeta{\diff \eta}
\def\oddst{\operatorname{odds}}
\def\probst{\operatorname{probs}}
\def\probt{\operatorname{prob}}
\def\riskt{\operatorname{risk}}
\def\logoddst{\text{log-odds}}
\def\prob{\pi}
\def\probs{\pi}
\def\risk{\pi}
\def\prev{\pi}
\providecommand{\potout}[1]{Y_{#1}}
\providecommand{\potprob}[1]{\prob_{#1}}
\providecommand{\potmean}[1]{\mean_{#1}}
\providecommand{\obsprob}[1]{\prob\paren{#1}}
\providecommand{\obsmean}[1]{\mean\paren{#1}}
\def\py{\potout}
\def\ppi{\potprob}
\def\pmu{\potmean}
\def\opi{\obsprob}
\def\omu{\obsmean}
\def\probit{\operatorname{probit}}
\providecommand{\oddsf}[1]{\oddst\cb{#1}}
\providecommand{\doddsf}[1]{{\oddst}'\cb{#1}}
\def\oddsinv{\oddst^{-1}}
\def\invodds{\operatorname{invodds}}
\providecommand{\oddsinvf}[1]{\oddsinv\cb{#1}}
\providecommand{\invoddsf}[1]{\invodds\cb{#1}}
\providecommand{\doddsinvf}[1]{{\oddsinv}'\cb{#1}}
\providecommand{\dinvoddsf}[1]{{\invodds}'\cb{#1}}
\def\rate{{\lambda}}
\def\lograte{\lincomp}
\def\haz{{\lambda}}
\def\hazratio{\ratio_{\haz}}
\def\hr{\hazratio}
\def\rhaz{\hazratio}
\def\hazr{\hazratio}
\providecommand{\hrf}[1]{\hazratio\paren{#1}}
\def\rhazf{\hrf}
\def\hazrf{\hrf}
\def\rateratio{\ratio_{\rate}}
\def\rater{\rateratio}
\def\rrate{\rateratio}
\providecommand{\rateratiof}[1]{\rateratio\paren{#1}}
\def\raterf{\rateratiof}
\def\rratef{\rateratiof}
\def\riskratio{\ratio_{\risk}}
\def\rrisk{\riskratio}
\def\riskr{\riskratio}
\providecommand{\riskratiof}[1]{\riskratio\paren{#1}}
\def\rriskf{\riskratiof}
\def\riskrf{\riskratiof}
\def\prevratio{\ratio_{\prev}}
\def\prr{\prevratio}
\providecommand{\prevratiof}[1]{\prevratio\paren{#1}}
\def\prrf{\prevratiof}
\def\loghaz{\lincomp}
\def\diffloghaz{\diff \loghaz}
\def\cuhaz{{\Lambda}}
\def\cumhaz{\cuhaz}
\def\cuhazratio{\ratio_{\cuhaz}}
\def\rcuhaz{\cuhazratio}
\def\cuhazr{\cuhazratio}
\providecommand{\cuhazratiof}[1]{\cuhazratio\paren{#1}}
\def\rcuhazf{\cuhazratiof}
\def\cuhazrf{\cuhazratiof}
\def\factor{\theta}
\providecommand{\factorf}[1]{\factor\paren{#1}}
\def\oddsfactor{\factor_{\odds}}
\providecommand{\oddsfactorf}[1]{\oddsfactor\paren{#1}}
\def\fodds{\oddsfactor}
\def\foddsf{\oddsfactorf}
\def\hazfactor{\factor_{\haz}}
\providecommand{\hazfactorf}[1]{\hazfactor\paren{#1}}
\def\fhaz{\hazfactor}
\def\fhazf{\hazfactorf}
\def\ratefactor{\factor_{\rate}}
\providecommand{\ratefactorf}[1]{\ratefactor\paren{#1}}
\def\frate{\ratefactor}
\def\fratef{\ratefactorf}
\def\riskfactor{\factor_{\risk}}
\providecommand{\riskfactorf}[1]{\riskfactor\paren{#1}}
\def\frisk{\riskfactor}
\def\friskf{\riskfactorf}
\def\prevfactor{\factor_{\prev}}
\providecommand{\prevfactorf}[1]{\prevfactor\paren{#1}}
\def\fprev{\prevfactor}
\def\fprevf{\prevfactorf}
\def\cuhazfactor{\factor_{\cuhaz}}
\providecommand{\cuhazfactorf}[1]{\cuhazfactor\paren{#1}}
\def\fcuhaz{\cuhazfactor}
\def\fcuhazf{\cuhazfactorf}
\def\survfactor{\factor_{\surv}}
\providecommand{\survfactorf}[1]{\survfactor\paren{#1}}
\def\fsurv{\survfactor}
\def\fsurvf{\survfactorf}
\def\ff{\factorf}
\def\oddsff{\oddsfactorf}
\def\hazff{\hazfactorf}
\def\rateff{\ratefactorf}
\def\riskff{\riskfactorf}
\def\prevff{\prevfactorf}
\def\cuhazff{\cuhazfactorf}
\def\survff{\survfactorf}
\def\Surv{\surv}
\def\incidence{\bar{\haz}}
\def\phaz{\Expf{\haz}}
\def\idmat{\mathbb{I}}
\def\and{\cap}
\def\reglincomb{\vx \cdot \vb}
\def\regbetasum{\beta_1 x_1+ \dots + \beta_p x_p}
\def\pdf{\distop{f}}
\providecommand{\pmf}[1]{\distop{P}\paren{#1}}
\def\cdf{\distop{F}}
\providecommand{\cdff}[1]{\cdf\paren{#1}}
\def\hcdf{\mathop{\hat{\cdf}}\nolimits}
\providecommand{\hcdff}[1]{\hcdf\paren{#1}}
\def\defLik{\Lik(\theta) \eqdef \p(\vX = \vx | \Theta = \theta)}
\def\defLogLik{\lik \eqdef \logf{\Lik(\vx|\th)}}
\def\defScore{\score \eqdef \deriv{\th} \lik(\vx|\th)}
\def\defHess{\hess \eqdef \deriv{\vth}\deriv{\vth\'} \llik(\vx | \vth)}
\def\thmHessij{\hessij =
\deriv{\th_i}\deriv{\th_j} \llik(\vX = \vx | \vth)}
\def\defOInf{\oinf \eqdef -\hess(\vx|\vth)}
\def\defEInf{\einf \eqdef \E{\oinf(\vx|\theta)}}
\def\thmADistMLE{\hth_{ML}\ \dot \sim\ \ndist{\th,\sb{\einf(\vth)}^{-1}}}
\def\eqGLMRatio{\ror(\vx,\vxs) = \exp{(\diffvx) \cdot \vb}}
\def\defHR{\hr(t| \vx : \vxs) \eqdef \frac{\haz(t|\vx)}{\haz(t|\vxs)}}
\def\kerngaus{\frac{-1}{2}\frac{(x-\mu)^2}{\ss}}
\def\likgaus{\paren{2\pi\ss}^{-1/2}\expf{\kerngaus}}
\def\llgaus{\frac{-1}{2}\paren{\logf{2\pi\ss} + \frac{\devn^2}{\ss}}}
\def\scoreexpf{\frac{\devn}{\Varf{X}}}
\def\llexpmean{-\logf{\pmean} - \frac{x}{\pmean}}
\def\signt{\operatorname{sign}}
\providecommand{\signf}[1]{\signt\cb{#1}}
\def\reals{\mathbb{R}}
\def\NA{\text{NA}}
\def\resid{r}
\def\stdresid{r'}
\def\modresid{r^*}
\def\ub{\underbrace}
\providecommand{\ubf}[2]{\ub{#1}_{#2}}
\providecommand{\dims}[2]{\ub{#1}_{#2}}
\providecommand{\residv}{\vec{r}}
\def\hprob{\hat{\prob}}
\def\hprobs{\hprob}
\def\hrisk{\hprob}
\def\hprev{\hprob}
\def\hodds{\hat{\odds}}
\def\hrate{\hat{\rate}}
\def\hhaz{\hrate}
\def\hcuhaz{\hat{\cuhaz}}
\def\hcumhaz{\hcuhaz}
\def\hratio{\hat{\ratio}}
\providecommand{\hratiof}[1]{\hratio\paren{#1}}
\def\hror{\hat{\ror}}
\providecommand{\horf}[1]{\hror\paren{#1}}
\def\hhazratio{\hat{\hazratio}}
\def\hhr{\hhazratio}
\def\hrhaz{\hhazratio}
\def\hhazr{\hhazratio}
\providecommand{\hhrf}[1]{\hhazratio\paren{#1}}
\def\hrhazf{\hhrf}
\def\hhazrf{\hhrf}
\def\hrateratio{\hat{\rateratio}}
\def\hrater{\hrateratio}
\def\hrrate{\hrateratio}
\providecommand{\hrateratiof}[1]{\hrateratio\paren{#1}}
\def\hraterf{\hrateratiof}
\def\hrratef{\hrateratiof}
\def\hriskratio{\hat{\riskratio}}
\def\hrrisk{\hriskratio}
\def\hriskr{\hriskratio}
\providecommand{\hriskratiof}[1]{\hriskratio\paren{#1}}
\def\hrriskf{\hriskratiof}
\def\hriskrf{\hriskratiof}
\def\hprevratio{\hat{\prevratio}}
\def\hprr{\hprevratio}
\providecommand{\hprevratiof}[1]{\hprevratio\paren{#1}}
\def\hprrf{\hprevratiof}
\def\hsurvratio{\hat{\survratio}}
\def\hrsurv{\hsurvratio}
\def\hsurvr{\hsurvratio}
\providecommand{\hsurvratiof}[1]{\hsurvratio\paren{#1}}
\def\hrsurvf{\hsurvratiof}
\def\hsurvrf{\hsurvratiof}
\def\hcuhazratio{\hat{\cuhazratio}}
\def\hrcuhaz{\hcuhazratio}
\def\hcuhazr{\hcuhazratio}
\providecommand{\hcuhazratiof}[1]{\hcuhazratio\paren{#1}}
\def\hrcuhazf{\hcuhazratiof}
\def\hcuhazrf{\hcuhazratiof}
\def\hfactor{\hat{\factor}}
\providecommand{\hfactorf}[1]{\hfactor\paren{#1}}
\def\hff{\hfactorf}
\def\hoddsfactor{\hat{\oddsfactor}}
\providecommand{\hoddsfactorf}[1]{\hoddsfactor\paren{#1}}
\def\hfodds{\hoddsfactor}
\def\hfoddsf{\hoddsfactorf}
\def\hoddsff{\hoddsfactorf}
\def\hhazfactor{\hat{\hazfactor}}
\providecommand{\hhazfactorf}[1]{\hhazfactor\paren{#1}}
\def\hfhaz{\hhazfactor}
\def\hfhazf{\hhazfactorf}
\def\hhazff{\hhazfactorf}
\def\hratefactor{\hat{\ratefactor}}
\providecommand{\hratefactorf}[1]{\hratefactor\paren{#1}}
\def\hfrate{\hratefactor}
\def\hfratef{\hratefactorf}
\def\hrateff{\hratefactorf}
\def\hriskfactor{\hat{\riskfactor}}
\providecommand{\hriskfactorf}[1]{\hriskfactor\paren{#1}}
\def\hfrisk{\hriskfactor}
\def\hfriskf{\hriskfactorf}
\def\hriskff{\hriskfactorf}
\def\hprevfactor{\hat{\prevfactor}}
\providecommand{\hprevfactorf}[1]{\hprevfactor\paren{#1}}
\def\hfprev{\hprevfactor}
\def\hfprevf{\hprevfactorf}
\def\hprevff{\hprevfactorf}
\def\hcuhazfactor{\hat{\cuhazfactor}}
\providecommand{\hcuhazfactorf}[1]{\hcuhazfactor\paren{#1}}
\def\hfcuhaz{\hcuhazfactor}
\def\hfcuhazf{\hcuhazfactorf}
\def\hcuhazff{\hcuhazfactorf}
\def\hsurvfactor{\hat{\survfactor}}
\providecommand{\hsurvfactorf}[1]{\hsurvfactor\paren{#1}}
\def\hfsurv{\hsurvfactor}
\def\hfsurvf{\hsurvfactorf}
\def\hsurvff{\hsurvfactorf}
\def\eQ{\mathop{\est{\Q}}\nolimits}
\def\equantile{\mathop{\est{\quantile}}\nolimits}
\providecommand{\eQf}[1]{\eQ\paren{#1}}
\providecommand{\equantilef}[1]{\equantile\paren{#1}}
\def\ep{\mathop{\est{\p}}\nolimits}
\def\eb{\est{\beta}}
\def\evb{\est{\vb}}
\def\eR{\est{R}}
\def\ek{\est{\kappa}}
\def\ekappa{\est{\kappa}}
\def\el{\est{\lambda}}
\def\ey{\est{y}}
\def\es{\est{\sigma}}
\def\esig{\est{\sigma}}
\def\eS{\est{\Sigma}}
\def\eSig{\est{\Sigma}}
\def\eSigma{\est{\Sigma}}
\def\esurv{\mathop{\est{\surv}}\nolimits}
\def\eSurv{\mathop{\est{\surv}}\nolimits}
\providecommand{\eSurvf}[1]{\eSurv\paren{#1}}
\def\esemp{\esurv_{\emp}}
\def\eskm{\esurv_{\km}}
\def\esnaa{\esurv_{\naa}}
\providecommand{\esempf}[1]{\esemp\paren{#1}}
\providecommand{\eskmf}[1]{\eskm\paren{#1}}
\providecommand{\esnaaf}[1]{\esnaa\paren{#1}}
\def\eth{\est{\theta}}
\def\ethml{\eth_{\text{ML}}}
\providecommand{\eE}[1]{\mathop{\est{\operatorname{E}}}\nolimits\sb{#1}}
\providecommand{\eExp}[1]{\mathop{\est{\operatorname{E}}}\nolimits\sb{#1}}
\providecommand{\eExpf}[1]{\mathop{\est{\operatorname{E}}}\nolimits\sb{#1}}
\def\emu{\est{\mu}}
\providecommand{\emuf}[1]{\emu\paren{#1}}
\def\evar{\mathop{\Est{\Vart}}\nolimits}
\def\evart{\mathop{\Est{\Vart}}\nolimits}
\def\ecovt{\mathop{\Est{\Covt}}\nolimits}
\def\eCovt{\mathop{\Est{\Covt}}\nolimits}
\providecommand{\eVar}[1]{\mathop{\Est{\operatorname{Var}}}\nolimits\paren{#1}}
\providecommand{\evarf}[1]{\evart\paren{#1}}
\providecommand{\ecovf}[1]{\ecovt\paren{#1}}
\def\ecort{\mathop{\Est{\Cort}}\nolimits}
\providecommand{\eCor}[1]{\ecort\paren{#1}}
\providecommand{\eCorf}[1]{\ecort\paren{#1}}
\providecommand{\eCors}[2]{\ecort\paren{#1, #2}}
\providecommand{\eSD}[1]{\mathop{\Est{\operatorname{SD}}}\nolimits\paren{#1}}
\providecommand{\eSDf}[1]{\mathop{\Est{\operatorname{SD}}}\nolimits\paren{#1}}
\def\eset{\mathop{\Est{\operatorname{se}}}\nolimits}
\def\eSE{\mathop{\Est{\operatorname{SE}}}\nolimits}
\def\eSEt{\mathop{\Est{\operatorname{SE}}}\nolimits}
\providecommand{\ese}[1]{\eset\paren{#1}}
\providecommand{\esef}[1]{\eset\paren{#1}}
\providecommand{\eSEf}[1]{\eSE\paren{#1}}
\def\eeinf{\est{\einf}}
\providecommand{\eeinff}[1]{\eeinf \paren{#1}}
\def\edifflogodds{\Est{\difflogodds}}
\def\ecdf{\mathop{\est{\cdf}}\nolimits}
\providecommand{\ecdff}[1]{\ecdf\paren{#1}}
\def\eprob{\est{\prob}}
\def\eprobs{\eprob}
\def\erisk{\eprob}
\def\eprev{\eprob}
\def\eodds{\est{\odds}}
\def\erate{\est{\rate}}
\def\ehaz{\erate}
\def\ecuhaz{\est{\cuhaz}}
\def\ecumhaz{\ecuhaz}
\def\eratio{\est{\ratio}}
\providecommand{\eratiof}[1]{\eratio\paren{#1}}
\def\eror{\est{\ror}}
\providecommand{\eorf}[1]{\eror\paren{#1}}
\def\ehazratio{\est{\hazratio}}
\def\ehr{\ehazratio}
\def\erhaz{\ehazratio}
\def\ehazr{\ehazratio}
\providecommand{\ehrf}[1]{\ehazratio\paren{#1}}
\def\erhazf{\ehrf}
\def\ehazrf{\ehrf}
\def\erateratio{\est{\rateratio}}
\def\erater{\erateratio}
\def\errate{\erateratio}
\providecommand{\erateratiof}[1]{\erateratio\paren{#1}}
\def\eraterf{\erateratiof}
\def\erratef{\erateratiof}
\def\eriskratio{\est{\riskratio}}
\def\errisk{\eriskratio}
\def\eriskr{\eriskratio}
\providecommand{\eriskratiof}[1]{\eriskratio\paren{#1}}
\def\erriskf{\eriskratiof}
\def\eriskrf{\eriskratiof}
\def\eprevratio{\est{\prevratio}}
\def\eprr{\eprevratio}
\providecommand{\eprevratiof}[1]{\eprevratio\paren{#1}}
\def\eprrf{\eprevratiof}
\def\esurvratio{\est{\survratio}}
\def\ersurv{\esurvratio}
\def\esurvr{\esurvratio}
\providecommand{\esurvratiof}[1]{\esurvratio\paren{#1}}
\def\ersurvf{\esurvratiof}
\def\esurvrf{\esurvratiof}
\def\ecuhazratio{\est{\cuhazratio}}
\def\ercuhaz{\ecuhazratio}
\def\ecuhazr{\ecuhazratio}
\providecommand{\ecuhazratiof}[1]{\ecuhazratio\paren{#1}}
\def\ercuhazf{\ecuhazratiof}
\def\ecuhazrf{\ecuhazratiof}
\def\efactor{\est{\factor}}
\providecommand{\efactorf}[1]{\efactor\paren{#1}}
\def\eff{\efactorf}
\def\eoddsfactor{\est{\oddsfactor}}
\providecommand{\eoddsfactorf}[1]{\eoddsfactor\paren{#1}}
\def\efodds{\eoddsfactor}
\def\efoddsf{\eoddsfactorf}
\def\eoddsff{\eoddsfactorf}
\def\ehazfactor{\est{\hazfactor}}
\providecommand{\ehazfactorf}[1]{\ehazfactor\paren{#1}}
\def\efhaz{\ehazfactor}
\def\efhazf{\ehazfactorf}
\def\ehazff{\ehazfactorf}
\def\eratefactor{\est{\ratefactor}}
\providecommand{\eratefactorf}[1]{\eratefactor\paren{#1}}
\def\efrate{\eratefactor}
\def\efratef{\eratefactorf}
\def\erateff{\eratefactorf}
\def\eriskfactor{\est{\riskfactor}}
\providecommand{\eriskfactorf}[1]{\eriskfactor\paren{#1}}
\def\efrisk{\eriskfactor}
\def\efriskf{\eriskfactorf}
\def\eriskff{\eriskfactorf}
\def\eprevfactor{\est{\prevfactor}}
\providecommand{\eprevfactorf}[1]{\eprevfactor\paren{#1}}
\def\efprev{\eprevfactor}
\def\efprevf{\eprevfactorf}
\def\eprevff{\eprevfactorf}
\def\ecuhazfactor{\est{\cuhazfactor}}
\providecommand{\ecuhazfactorf}[1]{\ecuhazfactor\paren{#1}}
\def\efcuhaz{\ecuhazfactor}
\def\efcuhazf{\ecuhazfactorf}
\def\ecuhazff{\ecuhazfactorf}
\def\esurvfactor{\est{\survfactor}}
\providecommand{\esurvfactorf}[1]{\esurvfactor\paren{#1}}
\def\efsurv{\esurvfactor}
\def\efsurvf{\esurvfactorf}
\def\esurvff{\esurvfactorf}