equations.html
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2023-06-03 Sat 16:55 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Equations for the rstpm2 package</title>
<meta name="generator" content="Org mode" />
<meta name="author" content="Mark Clements" />
<style type="text/css">
<!--/*--><![CDATA[/*><!--*/
.title { text-align: center;
margin-bottom: .2em; }
.subtitle { text-align: center;
font-size: medium;
font-weight: bold;
margin-top:0; }
.todo { font-family: monospace; color: red; }
.done { font-family: monospace; color: green; }
.priority { font-family: monospace; color: orange; }
.tag { background-color: #eee; font-family: monospace;
padding: 2px; font-size: 80%; font-weight: normal; }
.timestamp { color: #bebebe; }
.timestamp-kwd { color: #5f9ea0; }
.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
.org-center { margin-left: auto; margin-right: auto; text-align: center; }
.underline { text-decoration: underline; }
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
p.verse { margin-left: 3%; }
pre {
border: 1px solid #ccc;
box-shadow: 3px 3px 3px #eee;
padding: 8pt;
font-family: monospace;
overflow: auto;
margin: 1.2em;
}
pre.src {
position: relative;
overflow: visible;
padding-top: 1.2em;
}
pre.src:before {
display: none;
position: absolute;
background-color: white;
top: -10px;
right: 10px;
padding: 3px;
border: 1px solid black;
}
pre.src:hover:before { display: inline;}
/* Languages per Org manual */
pre.src-asymptote:before { content: 'Asymptote'; }
pre.src-awk:before { content: 'Awk'; }
pre.src-C:before { content: 'C'; }
/* pre.src-C++ doesn't work in CSS */
pre.src-clojure:before { content: 'Clojure'; }
pre.src-css:before { content: 'CSS'; }
pre.src-D:before { content: 'D'; }
pre.src-ditaa:before { content: 'ditaa'; }
pre.src-dot:before { content: 'Graphviz'; }
pre.src-calc:before { content: 'Emacs Calc'; }
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
pre.src-fortran:before { content: 'Fortran'; }
pre.src-gnuplot:before { content: 'gnuplot'; }
pre.src-haskell:before { content: 'Haskell'; }
pre.src-hledger:before { content: 'hledger'; }
pre.src-java:before { content: 'Java'; }
pre.src-js:before { content: 'Javascript'; }
pre.src-latex:before { content: 'LaTeX'; }
pre.src-ledger:before { content: 'Ledger'; }
pre.src-lisp:before { content: 'Lisp'; }
pre.src-lilypond:before { content: 'Lilypond'; }
pre.src-lua:before { content: 'Lua'; }
pre.src-matlab:before { content: 'MATLAB'; }
pre.src-mscgen:before { content: 'Mscgen'; }
pre.src-ocaml:before { content: 'Objective Caml'; }
pre.src-octave:before { content: 'Octave'; }
pre.src-org:before { content: 'Org mode'; }
pre.src-oz:before { content: 'OZ'; }
pre.src-plantuml:before { content: 'Plantuml'; }
pre.src-processing:before { content: 'Processing.js'; }
pre.src-python:before { content: 'Python'; }
pre.src-R:before { content: 'R'; }
pre.src-ruby:before { content: 'Ruby'; }
pre.src-sass:before { content: 'Sass'; }
pre.src-scheme:before { content: 'Scheme'; }
pre.src-screen:before { content: 'Gnu Screen'; }
pre.src-sed:before { content: 'Sed'; }
pre.src-sh:before { content: 'shell'; }
pre.src-sql:before { content: 'SQL'; }
pre.src-sqlite:before { content: 'SQLite'; }
/* additional languages in org.el's org-babel-load-languages alist */
pre.src-forth:before { content: 'Forth'; }
pre.src-io:before { content: 'IO'; }
pre.src-J:before { content: 'J'; }
pre.src-makefile:before { content: 'Makefile'; }
pre.src-maxima:before { content: 'Maxima'; }
pre.src-perl:before { content: 'Perl'; }
pre.src-picolisp:before { content: 'Pico Lisp'; }
pre.src-scala:before { content: 'Scala'; }
pre.src-shell:before { content: 'Shell Script'; }
pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
/* additional language identifiers per "defun org-babel-execute"
in ob-*.el */
pre.src-cpp:before { content: 'C++'; }
pre.src-abc:before { content: 'ABC'; }
pre.src-coq:before { content: 'Coq'; }
pre.src-groovy:before { content: 'Groovy'; }
/* additional language identifiers from org-babel-shell-names in
ob-shell.el: ob-shell is the only babel language using a lambda to put
the execution function name together. */
pre.src-bash:before { content: 'bash'; }
pre.src-csh:before { content: 'csh'; }
pre.src-ash:before { content: 'ash'; }
pre.src-dash:before { content: 'dash'; }
pre.src-ksh:before { content: 'ksh'; }
pre.src-mksh:before { content: 'mksh'; }
pre.src-posh:before { content: 'posh'; }
/* Additional Emacs modes also supported by the LaTeX listings package */
pre.src-ada:before { content: 'Ada'; }
pre.src-asm:before { content: 'Assembler'; }
pre.src-caml:before { content: 'Caml'; }
pre.src-delphi:before { content: 'Delphi'; }
pre.src-html:before { content: 'HTML'; }
pre.src-idl:before { content: 'IDL'; }
pre.src-mercury:before { content: 'Mercury'; }
pre.src-metapost:before { content: 'MetaPost'; }
pre.src-modula-2:before { content: 'Modula-2'; }
pre.src-pascal:before { content: 'Pascal'; }
pre.src-ps:before { content: 'PostScript'; }
pre.src-prolog:before { content: 'Prolog'; }
pre.src-simula:before { content: 'Simula'; }
pre.src-tcl:before { content: 'tcl'; }
pre.src-tex:before { content: 'TeX'; }
pre.src-plain-tex:before { content: 'Plain TeX'; }
pre.src-verilog:before { content: 'Verilog'; }
pre.src-vhdl:before { content: 'VHDL'; }
pre.src-xml:before { content: 'XML'; }
pre.src-nxml:before { content: 'XML'; }
/* add a generic configuration mode; LaTeX export needs an additional
(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
pre.src-conf:before { content: 'Configuration File'; }
table { border-collapse:collapse; }
caption.t-above { caption-side: top; }
caption.t-bottom { caption-side: bottom; }
td, th { vertical-align:top; }
th.org-right { text-align: center; }
th.org-left { text-align: center; }
th.org-center { text-align: center; }
td.org-right { text-align: right; }
td.org-left { text-align: left; }
td.org-center { text-align: center; }
dt { font-weight: bold; }
.footpara { display: inline; }
.footdef { margin-bottom: 1em; }
.figure { padding: 1em; }
.figure p { text-align: center; }
.equation-container {
display: table;
text-align: center;
width: 100%;
}
.equation {
vertical-align: middle;
}
.equation-label {
display: table-cell;
text-align: right;
vertical-align: middle;
}
.inlinetask {
padding: 10px;
border: 2px solid gray;
margin: 10px;
background: #ffffcc;
}
#org-div-home-and-up
{ text-align: right; font-size: 70%; white-space: nowrap; }
textarea { overflow-x: auto; }
.linenr { font-size: smaller }
.code-highlighted { background-color: #ffff00; }
.org-info-js_info-navigation { border-style: none; }
#org-info-js_console-label
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
.org-info-js_search-highlight
{ background-color: #ffff00; color: #000000; font-weight: bold; }
.org-svg { width: 90%; }
/*]]>*/-->
</style>
<script type="text/javascript">
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
Copyright (C) 2012-2020 Free Software Foundation, Inc.
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
<!--/*--><![CDATA[/*><!--*/
function CodeHighlightOn(elem, id)
{
var target = document.getElementById(id);
if(null != target) {
elem.cacheClassElem = elem.className;
elem.cacheClassTarget = target.className;
target.className = "code-highlighted";
elem.className = "code-highlighted";
}
}
function CodeHighlightOff(elem, id)
{
var target = document.getElementById(id);
if(elem.cacheClassElem)
elem.className = elem.cacheClassElem;
if(elem.cacheClassTarget)
target.className = elem.cacheClassTarget;
}
/*]]>*///-->
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
displayAlign: "center",
displayIndent: "0em",
"HTML-CSS": { scale: 100,
linebreaks: { automatic: "false" },
webFont: "TeX"
},
SVG: {scale: 100,
linebreaks: { automatic: "false" },
font: "TeX"},
NativeMML: {scale: 100},
TeX: { equationNumbers: {autoNumber: "AMS"},
MultLineWidth: "85%",
TagSide: "right",
TagIndent: ".8em"
}
});
</script>
<script type="text/javascript"
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"></script>
</head>
<body>
<div id="content">
<h1 class="title">Equations for the rstpm2 package</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgfa17b91">1. Generalised survival model</a></li>
<li><a href="#org92fd60c">2. Mixture cure models</a></li>
<li><a href="#orgab1b650">3. Integral equation for AFT models</a></li>
<li><a href="#org9f78459">4. Integral equation for AFT models with functional constraints on gamma</a></li>
<li><a href="#orgf26dba5">5. Non-integral equation for AFT models</a></li>
<li><a href="#org3f1b7de">6. Non-integral equation for AFT models without time-varying effect</a></li>
<li><a href="#org806037c">7. Integral equations for AFT models with \(H = B(\ldots)\)</a></li>
</ul>
</div>
</div>
<div id="outline-container-orgfa17b91" class="outline-2">
<h2 id="orgfa17b91"><span class="section-number-2">1</span> Generalised survival model</h2>
<div class="outline-text-2" id="text-1">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">></span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> G(eta(t,z,theta))<span style="color: #a020f0;">*</span>S0(t);
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,theta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,theta));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),theta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),theta));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"ll:"</span>);
ll <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">delta</span><span style="color: #a020f0;">*</span><span style="color: #483d8b;">log</span>(h) <span style="color: #a020f0;">-</span> H <span style="color: #a020f0;">-</span> <span style="color: #483d8b;">diff</span>(eta(t,z,<span style="color: #483d8b;">beta</span>),t)<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span>;
<span style="color: #483d8b;">tex</span>(ll);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, beta)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">beta</span>));
</pre>
</div>
<p>
H:
\[-\log G\left(\eta\left(t , z , \vartheta\right)\right)-\log S_{0}
\left(t\right)\]
h:
\[-{{\eta_{\left(1 , 0 , 0\right)}(t,z,\vartheta)\,G_{\left(1\right)}
(\eta\left(t , z , \vartheta\right))}\over{G\left(\eta\left(t , z ,
\vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(t)}\over{S_{0}
\left(t\right)}}\]
diff(H,theta):
\[-{{\eta_{\left(0 , 0 , 1\right)}(t,z,\vartheta)\,G_{\left(1\right)}
(\eta\left(t , z , \vartheta\right))}\over{G\left(\eta\left(t , z ,
\vartheta\right)\right)}}\]
diff(log(h),theta):
\[{{-{{\eta_{\left(0 , 0 , 1\right)}(t,z,\vartheta)\,\eta_{\left(1 ,
0 , 0\right)}(t,z,\vartheta)\,G_{\left(2\right)}(\eta\left(t , z ,
\vartheta\right))}\over{G\left(\eta\left(t , z , \vartheta\right)
\right)}}+{{\eta_{\left(0 , 0 , 1\right)}(t,z,\vartheta)\,\eta_{
\left(1 , 0 , 0\right)}(t,z,\vartheta)\,G_{\left(1\right)}(\eta
\left(t , z , \vartheta\right))^2}\over{G\left(\eta\left(t , z ,
\vartheta\right)\right)^2}}-{{\eta_{\left(1 , 0 , 1\right)}(t,z,
\vartheta)\,G_{\left(1\right)}(\eta\left(t , z , \vartheta\right))
}\over{G\left(\eta\left(t , z , \vartheta\right)\right)}}}\over{-{{
\eta_{\left(1 , 0 , 0\right)}(t,z,\vartheta)\,G_{\left(1\right)}(
\eta\left(t , z , \vartheta\right))}\over{G\left(\eta\left(t , z ,
\vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(t)}\over{S_{0}
\left(t\right)}}}}\]
ll:
\[\delta\,\log \left(-{{\eta_{\left(1 , 0 , 0\right)}(t,z,\vartheta)
\,G_{\left(1\right)}(\eta\left(t , z , \vartheta\right))}\over{G
\left(\eta\left(t , z , \vartheta\right)\right)}}-{{S_{0}_{\left(1
\right)}(t)}\over{S_{0}\left(t\right)}}\right)+\log G\left(\eta
\left(t , z , \vartheta\right)\right)+\log S_{0}\left(t\right)-\eta
_{\left(1 , 0 , 0\right)}(t,z,\beta)^2\]
diff(ll, beta)
\[-2\,\eta_{\left(1 , 0 , 0\right)}(t,z,\beta)\,\eta_{\left(1 , 0 , 1
\right)}(t,z,\beta)\]
</p>
</div>
</div>
<div id="outline-container-org92fd60c" class="outline-2">
<h2 id="org92fd60c"><span class="section-number-2">2</span> Mixture cure models</h2>
<div class="outline-text-2" id="text-2">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
S <span style="color: #a020f0;">:</span> S0(t)<span style="color: #a020f0;">*</span>(pi(theta)<span style="color: #a020f0;">+</span>(<span style="color: #008b8b;">1</span><span style="color: #a020f0;">-</span>pi(theta))<span style="color: #a020f0;">*</span><span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>Hu(t,theta)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">subst</span>(hu(t,theta), <span style="color: #483d8b;">diff</span>(Hu(t,theta),t), h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,theta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,theta));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),theta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),theta));
<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),theta) <span style="color: #a020f0;">-</span> <span style="color: #483d8b;">diff</span>(H,theta);
</pre>
</div>
<p>
H:
\[-\log \left(\pi\left(\vartheta\right)+e^ {- {\it Hu}\left(t ,
\vartheta\right) }\,\left(1-\pi\left(\vartheta\right)\right)\right)-
\log S_{0}\left(t\right)\]
h:
\[{{e^ {- {\it Hu}\left(t , \vartheta\right) }\,{\it hu}\left(t ,
\vartheta\right)\,\left(1-\pi\left(\vartheta\right)\right)}\over{\pi
\left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,
\left(1-\pi\left(\vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(
t)}\over{S_{0}\left(t\right)}}\]
diff(H,theta):
\[-{{-e^ {- {\it Hu}\left(t , \vartheta\right) }\,\pi_{\left(1\right)
}(\vartheta)+\pi_{\left(1\right)}(\vartheta)-e^ {- {\it Hu}\left(t
, \vartheta\right) }\,{\it Hu}_{\left(0 , 1\right)}(t,\vartheta)\,
\left(1-\pi\left(\vartheta\right)\right)}\over{\pi\left(\vartheta
\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi
\left(\vartheta\right)\right)}}\]
diff(log(h),theta):
\[{{-{{e^ {- {\it Hu}\left(t , \vartheta\right) }\,{\it hu}\left(t ,
\vartheta\right)\,\left(1-\pi\left(\vartheta\right)\right)\,\left(-e
^ {- {\it Hu}\left(t , \vartheta\right) }\,\pi_{\left(1\right)}(
\vartheta)+\pi_{\left(1\right)}(\vartheta)-e^ {- {\it Hu}\left(t ,
\vartheta\right) }\,{\it Hu}_{\left(0 , 1\right)}(t,\vartheta)\,
\left(1-\pi\left(\vartheta\right)\right)\right)}\over{\left(\pi
\left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,
\left(1-\pi\left(\vartheta\right)\right)\right)^2}}-{{e^ {- {\it Hu}
\left(t , \vartheta\right) }\,{\it hu}\left(t , \vartheta\right)\,
\pi_{\left(1\right)}(\vartheta)}\over{\pi\left(\vartheta\right)+e
^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi\left(
\vartheta\right)\right)}}+{{e^ {- {\it Hu}\left(t , \vartheta\right)
}\,{\it hu}_{\left(0 , 1\right)}(t,\vartheta)\,\left(1-\pi\left(
\vartheta\right)\right)}\over{\pi\left(\vartheta\right)+e^ {-
{\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi\left(\vartheta
\right)\right)}}-{{e^ {- {\it Hu}\left(t , \vartheta\right) }\,
{\it Hu}_{\left(0 , 1\right)}(t,\vartheta)\,{\it hu}\left(t ,
\vartheta\right)\,\left(1-\pi\left(\vartheta\right)\right)}\over{\pi
\left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,
\left(1-\pi\left(\vartheta\right)\right)}}}\over{{{e^ {- {\it Hu}
\left(t , \vartheta\right) }\,{\it hu}\left(t , \vartheta\right)\,
\left(1-\pi\left(\vartheta\right)\right)}\over{\pi\left(\vartheta
\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi
\left(\vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(t)}\over{
S_{0}\left(t\right)}}}}\]
</p>
</div>
</div>
<div id="outline-container-orgab1b650" class="outline-2">
<h2 id="orgab1b650"><span class="section-number-2">3</span> Integral equation for AFT models</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">></span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span><span style="color: #483d8b;">exp</span>(B(<span style="color: #483d8b;">log</span>(<span style="color: #483d8b;">integrate</span>(<span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>x(v)<span style="color: #a020f0;">*</span><span style="color: #483d8b;">beta</span>),v,<span style="color: #008b8b;">0</span>,t)),<span style="color: #483d8b;">gamma</span>)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"log(h):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">log</span>(h));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">gamma</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">gamma</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"ll:"</span>);
ll <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">delta</span><span style="color: #a020f0;">*</span><span style="color: #483d8b;">log</span>(h) <span style="color: #a020f0;">-</span> H <span style="color: #a020f0;">-</span> (<span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(<span style="color: #483d8b;">integrate</span>(<span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>x(v)<span style="color: #a020f0;">*</span><span style="color: #483d8b;">beta</span>),v,<span style="color: #008b8b;">0</span>,t)), <span style="color: #483d8b;">diff</span>(B(u,<span style="color: #483d8b;">gamma</span>),u)))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span>;
<span style="color: #483d8b;">tex</span>(ll);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, beta)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, gamma)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">gamma</span>));
</pre>
</div>
<p>
H:
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} ,
\gamma\right)}\]
h:
\[{{e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}
, \gamma\right)-\beta\,x\left(t\right)}\,B_{\left(1 , 0\right)}(
\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}}}\]
log(h):
\[\log B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v
\right) }\;dv},\gamma)+B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(
v\right) }\;dv} , \gamma\right)-\log \int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv}-\beta\,x\left(t\right)\]
diff(H,beta):
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
\right)\;dv}\right)\,e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv} , \gamma\right)}\,B_{\left(1 , 0\right)}(\log
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}}}\]
diff(H,gamma):
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} ,
\gamma\right)}\,B_{\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta
\,x\left(v\right) }\;dv},\gamma)\]
diff(log(h),beta):
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
\right)\;dv}\right)\,B_{\left(2 , 0\right)}(\log \int_{0}^{t}{e^ {-
\beta\,x\left(v\right) }\;dv},\gamma)}\over{\left(\int_{0}^{t}{e
^ {- \beta\,x\left(v\right) }\;dv}\right)\,B_{\left(1 , 0\right)}(
\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}-{{
\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v\right)
\;dv}\right)\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,
x\left(v\right) }\;dv},\gamma)}\over{\int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv}}}+{{\int_{0}^{t}{e^ {- \beta\,x\left(v\right)
}\,x\left(v\right)\;dv}}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v
\right) }\;dv}}}-x\left(t\right)\]
diff(log(h),gamma):
\[{{B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v
\right) }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(\log \int_{0}^{t
}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}+B_{\left(0 , 1
\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
\gamma)\]
ll:
\[\delta\,\left(\log B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {-
\beta\,x\left(v\right) }\;dv},\gamma)+B\left(\log \int_{0}^{t}{e
^ {- \beta\,x\left(v\right) }\;dv} , \gamma\right)-\log \int_{0}^{t
}{e^ {- \beta\,x\left(v\right) }\;dv}-\beta\,x\left(t\right)\right)-
B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v
\right) }\;dv},\gamma)^2-e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv} , \gamma\right)}\]
diff(ll, beta)
\[\delta\,\left(-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }
\,x\left(v\right)\;dv}\right)\,B_{\left(2 , 0\right)}(\log \int_{0
}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{\left(\int_{
0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}\right)\,B_{\left(1 , 0
\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
\gamma)}}-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x
\left(v\right)\;dv}\right)\,B_{\left(1 , 0\right)}(\log \int_{0}^{t
}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{\int_{0}^{t}{e
^ {- \beta\,x\left(v\right) }\;dv}}}+{{\int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\,x\left(v\right)\;dv}}\over{\int_{0}^{t}{e^ {-
\beta\,x\left(v\right) }\;dv}}}-x\left(t\right)\right)+{{2\,\left(
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v\right)\;dv}
\right)\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv},\gamma)\,B_{\left(2 , 0\right)}(\log \int_{0
}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{\int_{0}^{t
}{e^ {- \beta\,x\left(v\right) }\;dv}}}+{{\left(\int_{0}^{t}{e^ {-
\beta\,x\left(v\right) }\,x\left(v\right)\;dv}\right)\,e^{B\left(
\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , \gamma
\right)}\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv},\gamma)}\over{\int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv}}}\]
diff(ll, gamma)
\[\delta\,\left({{B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {-
\beta\,x\left(v\right) }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(
\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}+B_{
\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }
\;dv},\gamma)\right)-2\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e
^ {- \beta\,x\left(v\right) }\;dv},\gamma)\,B_{\left(1 , 1\right)}(
\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)-e^{B
\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , \gamma
\right)}\,B_{\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv},\gamma)\]
</p>
</div>
</div>
<div id="outline-container-org9f78459" class="outline-2">
<h2 id="org9f78459"><span class="section-number-2">4</span> Integral equation for AFT models with functional constraints on gamma</h2>
<div class="outline-text-2" id="text-4">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">></span><span style="color: #008b8b;">0</span>)$
<span style="color: #483d8b;">gamma</span> <span style="color: #a020f0;">:</span> gamma0 <span style="color: #a020f0;">+</span> <span style="color: #483d8b;">exp</span>(alpha);
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span><span style="color: #483d8b;">exp</span>(B(<span style="color: #483d8b;">log</span>(<span style="color: #483d8b;">integrate</span>(<span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>x(v)<span style="color: #a020f0;">*</span><span style="color: #483d8b;">beta</span>),v,<span style="color: #008b8b;">0</span>,t)),<span style="color: #483d8b;">gamma</span>)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,alpha):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,alpha));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma0):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,gamma0));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),alpha):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),alpha));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma0):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),gamma0));
</pre>
</div>
<p>
H:
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} ,
\gamma_{0}+e^{\alpha}\right)}\]
h:
\[{{e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}
, \gamma_{0}+e^{\alpha}\right)-\beta\,x\left(t\right)}\,B_{\left(1
, 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
\gamma_{0}+e^{\alpha})}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v
\right) }\;dv}}}\]
diff(H,beta):
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
\right)\;dv}\right)\,e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv} , \gamma_{0}+e^{\alpha}\right)}\,B_{\left(1 ,
0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
\gamma_{0}+e^{\alpha})}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v
\right) }\;dv}}}\]
diff(H,alpha):
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} ,
\gamma_{0}+e^{\alpha}\right)+\alpha}\,B_{\left(0 , 1\right)}(\log
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{
\alpha})\]
diff(H,gamma0):
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} ,
\gamma_{0}+e^{\alpha}\right)}\,B_{\left(0 , 1\right)}(\log \int_{0
}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{\alpha})\]
diff(log(h),beta):
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
\right)\;dv}\right)\,B_{\left(2 , 0\right)}(\log \int_{0}^{t}{e^ {-
\beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{\alpha})}\over{\left(
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}\right)\,B_{\left(1
, 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
\gamma_{0}+e^{\alpha})}}-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v
\right) }\,x\left(v\right)\;dv}\right)\,B_{\left(1 , 0\right)}(\log
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{
\alpha})}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}}}+{{
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v\right)\;dv}
}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}}}-x\left(t
\right)\]
diff(log(h),alpha):
\[{{e^{\alpha}\,B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta
\,x\left(v\right) }\;dv},\gamma_{0}+e^{\alpha})}\over{B_{\left(1 , 0
\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
\gamma_{0}+e^{\alpha})}}+e^{\alpha}\,B_{\left(0 , 1\right)}(\log
\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{
\alpha})\]
diff(log(h),gamma0):
\[{{B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v
\right) }\;dv},\gamma_{0}+e^{\alpha})}\over{B_{\left(1 , 0\right)}(
\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{
\alpha})}}+B_{\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv},\gamma_{0}+e^{\alpha})\]
</p>
</div>
</div>
<div id="outline-container-orgf26dba5" class="outline-2">
<h2 id="orgf26dba5"><span class="section-number-2">5</span> Non-integral equation for AFT models</h2>
<div class="outline-text-2" id="text-5">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">></span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span><span style="color: #483d8b;">exp</span>(B(<span style="color: #483d8b;">log</span>(t<span style="color: #a020f0;">*</span><span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>eta(X,<span style="color: #483d8b;">log</span>(t),<span style="color: #483d8b;">beta</span>))),<span style="color: #483d8b;">gamma</span>)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"log(h):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">log</span>(h));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">gamma</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">gamma</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"ll:"</span>);
ll <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">delta</span><span style="color: #a020f0;">*</span><span style="color: #483d8b;">log</span>(h) <span style="color: #a020f0;">-</span> H <span style="color: #a020f0;">-</span> (<span style="color: #008b8b;">1</span><span style="color: #a020f0;">-</span><span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(t), <span style="color: #483d8b;">diff</span>(eta(X,u,<span style="color: #483d8b;">beta</span>),u)))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span> <span style="color: #a020f0;">-</span>
<span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(t)<span style="color: #a020f0;">-</span>eta(X,<span style="color: #483d8b;">log</span>(t),<span style="color: #483d8b;">beta</span>), <span style="color: #483d8b;">diff</span>(B(u,<span style="color: #483d8b;">gamma</span>),u))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span>;
<span style="color: #483d8b;">tex</span>(ll);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, beta)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, gamma)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">gamma</span>));
</pre>
</div>
<p>
H:
\[e^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma
\right)}\]
h:
\[\left({{1}\over{t}}-{{\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)
}\over{t}}\right)\,e^{B\left(\log t-\eta\left(X , \log t , \beta
\right) , \gamma\right)}\,B_{\left(1 , 0\right)}(\log t-\eta\left(X
, \log t , \beta\right),\gamma)\]
log(h):
\[\log B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta
\right),\gamma)+B\left(\log t-\eta\left(X , \log t , \beta\right) ,
\gamma\right)+\log \left({{1}\over{t}}-{{\eta_{\left(0 , 1 , 0
\right)}(X,\log t,\beta)}\over{t}}\right)\]
diff(H,beta):
\[-\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,e^{B\left(\log t-
\eta\left(X , \log t , \beta\right) , \gamma\right)}\,B_{\left(1 , 0
\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)\]
diff(H,gamma):
\[e^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma
\right)}\,B_{\left(0 , 1\right)}(\log t-\eta\left(X , \log t , \beta
\right),\gamma)\]
diff(log(h),beta):
\[-{{\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,B_{\left(2 , 0
\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)}\over{B
_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
\gamma)}}-\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,B_{\left(1
, 0\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)-{{
\eta_{\left(0 , 1 , 1\right)}(X,\log t,\beta)}\over{\left({{1}\over{
t}}-{{\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)}\over{t}}\right)
\,t}}\]
diff(log(h),gamma):
\[{{B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t , \beta\right)
,\gamma)}\over{B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t ,
\beta\right),\gamma)}}+B_{\left(0 , 1\right)}(\log t-\eta\left(X ,
\log t , \beta\right),\gamma)\]
ll:
\[\delta\,\left(\log B_{\left(1 , 0\right)}(\log t-\eta\left(X ,
\log t , \beta\right),\gamma)+B\left(\log t-\eta\left(X , \log t ,
\beta\right) , \gamma\right)+\log \left({{1}\over{t}}-{{\eta_{\left(
0 , 1 , 0\right)}(X,\log t,\beta)}\over{t}}\right)\right)-B_{\left(1
, 0\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)^2-e
^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma\right)}
-\left(1-\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)\right)^2\]
diff(ll, beta)
\[\delta\,\left(-{{\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,B_{
\left(2 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
\gamma)}\over{B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t ,
\beta\right),\gamma)}}-\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)
\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
\gamma)-{{\eta_{\left(0 , 1 , 1\right)}(X,\log t,\beta)}\over{\left(
{{1}\over{t}}-{{\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)}\over{
t}}\right)\,t}}\right)+2\,\eta_{\left(0 , 0 , 1\right)}(X,\log t,
\beta)\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta
\right),\gamma)\,B_{\left(2 , 0\right)}(\log t-\eta\left(X , \log t
, \beta\right),\gamma)+\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta
)\,e^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma
\right)}\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta
\right),\gamma)+2\,\left(1-\eta_{\left(0 , 1 , 0\right)}(X,\log t,
\beta)\right)\,\eta_{\left(0 , 1 , 1\right)}(X,\log t,\beta)\]
diff(ll, gamma)
\[\delta\,\left({{B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t
, \beta\right),\gamma)}\over{B_{\left(1 , 0\right)}(\log t-\eta
\left(X , \log t , \beta\right),\gamma)}}+B_{\left(0 , 1\right)}(
\log t-\eta\left(X , \log t , \beta\right),\gamma)\right)-2\,B_{
\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
\gamma)\,B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t , \beta
\right),\gamma)-e^{B\left(\log t-\eta\left(X , \log t , \beta\right)
, \gamma\right)}\,B_{\left(0 , 1\right)}(\log t-\eta\left(X , \log
t , \beta\right),\gamma)\]
</p>
</div>
</div>
<div id="outline-container-org3f1b7de" class="outline-2">
<h2 id="org3f1b7de"><span class="section-number-2">6</span> Non-integral equation for AFT models without time-varying effect</h2>
<div class="outline-text-2" id="text-6">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">></span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span><span style="color: #483d8b;">exp</span>(B(<span style="color: #483d8b;">log</span>(t<span style="color: #a020f0;">*</span><span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>eta(X,<span style="color: #483d8b;">beta</span>))),<span style="color: #483d8b;">gamma</span>)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"log(h):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">log</span>(h));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">gamma</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">gamma</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"ll:"</span>);
ll <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">delta</span><span style="color: #a020f0;">*</span><span style="color: #483d8b;">log</span>(h) <span style="color: #a020f0;">-</span> H <span style="color: #a020f0;">-</span> (<span style="color: #008b8b;">1</span><span style="color: #a020f0;">-</span><span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(t), <span style="color: #483d8b;">diff</span>(eta(X,u,<span style="color: #483d8b;">beta</span>),u)))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span> <span style="color: #a020f0;">-</span>
<span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(t)<span style="color: #a020f0;">-</span>eta(X,<span style="color: #483d8b;">log</span>(t),<span style="color: #483d8b;">beta</span>), <span style="color: #483d8b;">diff</span>(B(u,<span style="color: #483d8b;">gamma</span>),u))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span>;
<span style="color: #483d8b;">tex</span>(ll);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, beta)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, gamma)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">gamma</span>));
</pre>
</div>
<p>
H:
\[e^{B\left(\log t-\eta\left(X , \beta\right) , \gamma\right)}\]
h:
\[{{e^{B\left(\log t-\eta\left(X , \beta\right) , \gamma\right)}\,B_{
\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),\gamma)}\over{
t}}\]
log(h):
\[\log B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),
\gamma)+B\left(\log t-\eta\left(X , \beta\right) , \gamma\right)-
\log t\]
diff(H,beta):
\[-\eta_{\left(0 , 1\right)}(X,\beta)\,e^{B\left(\log t-\eta\left(X
, \beta\right) , \gamma\right)}\,B_{\left(1 , 0\right)}(\log t-\eta
\left(X , \beta\right),\gamma)\]
diff(H,gamma):
\[e^{B\left(\log t-\eta\left(X , \beta\right) , \gamma\right)}\,B_{
\left(0 , 1\right)}(\log t-\eta\left(X , \beta\right),\gamma)\]
diff(log(h),beta):
\[-{{\eta_{\left(0 , 1\right)}(X,\beta)\,B_{\left(2 , 0\right)}(\log
t-\eta\left(X , \beta\right),\gamma)}\over{B_{\left(1 , 0\right)}(
\log t-\eta\left(X , \beta\right),\gamma)}}-\eta_{\left(0 , 1\right)
}(X,\beta)\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right)
,\gamma)\]
diff(log(h),gamma):
\[{{B_{\left(1 , 1\right)}(\log t-\eta\left(X , \beta\right),\gamma)
}\over{B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),
\gamma)}}+B_{\left(0 , 1\right)}(\log t-\eta\left(X , \beta\right),
\gamma)\]
ll:
\[\delta\,\left(\log B_{\left(1 , 0\right)}(\log t-\eta\left(X ,
\beta\right),\gamma)+B\left(\log t-\eta\left(X , \beta\right) ,
\gamma\right)-\log t\right)-B_{\left(1 , 0\right)}(\log t-\eta\left(
X , \log t , \beta\right),\gamma)^2-e^{B\left(\log t-\eta\left(X ,
\beta\right) , \gamma\right)}-\left(1-\eta_{\left(0 , 1 , 0\right)}(
X,\log t,\beta)\right)^2\]
diff(ll, beta)
\[2\,\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,B_{\left(1 , 0
\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)\,B_{
\left(2 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
\gamma)+\delta\,\left(-{{\eta_{\left(0 , 1\right)}(X,\beta)\,B_{
\left(2 , 0\right)}(\log t-\eta\left(X , \beta\right),\gamma)}\over{
B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),\gamma)}}-
\eta_{\left(0 , 1\right)}(X,\beta)\,B_{\left(1 , 0\right)}(\log t-
\eta\left(X , \beta\right),\gamma)\right)+\eta_{\left(0 , 1\right)}(
X,\beta)\,e^{B\left(\log t-\eta\left(X , \beta\right) , \gamma
\right)}\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),
\gamma)+2\,\left(1-\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)
\right)\,\eta_{\left(0 , 1 , 1\right)}(X,\log t,\beta)\]
diff(ll, gamma)
\[-2\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta
\right),\gamma)\,B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t
, \beta\right),\gamma)+\delta\,\left({{B_{\left(1 , 1\right)}(\log
t-\eta\left(X , \beta\right),\gamma)}\over{B_{\left(1 , 0\right)}(
\log t-\eta\left(X , \beta\right),\gamma)}}+B_{\left(0 , 1\right)}(
\log t-\eta\left(X , \beta\right),\gamma)\right)-e^{B\left(\log t-
\eta\left(X , \beta\right) , \gamma\right)}\,B_{\left(0 , 1\right)}(
\log t-\eta\left(X , \beta\right),\gamma)\]
</p>
</div>
</div>
<div id="outline-container-org806037c" class="outline-2">
<h2 id="org806037c"><span class="section-number-2">7</span> Integral equations for AFT models with \(H = B(\ldots)\)</h2>
<div class="outline-text-2" id="text-7">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">></span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>B(<span style="color: #483d8b;">integrate</span>(<span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>x(v)<span style="color: #a020f0;">*</span><span style="color: #483d8b;">beta</span>),v,<span style="color: #008b8b;">0</span>,t),<span style="color: #483d8b;">gamma</span>));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">gamma</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">gamma</span>));
</pre>
</div>
<p>
H:
\[B\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , \gamma
\right)\]
h:
\[e^ {- \beta\,x\left(t\right) }\,B_{\left(1 , 0\right)}(\int_{0}^{t
}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)\]
diff(H,beta):
\[-\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v\right)
\;dv}\right)\,B_{\left(1 , 0\right)}(\int_{0}^{t}{e^ {- \beta\,x
\left(v\right) }\;dv},\gamma)\]
diff(H,gamma):
\[B_{\left(0 , 1\right)}(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }
\;dv},\gamma)\]
diff(log(h),beta):
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
\right)\;dv}\right)\,B_{\left(2 , 0\right)}(\int_{0}^{t}{e^ {- \beta
\,x\left(v\right) }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(\int_{
0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}-x\left(t\right)\]
diff(log(h),gamma):
\[{{B_{\left(1 , 1\right)}(\int_{0}^{t}{e^ {- \beta\,x\left(v\right)
}\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(\int_{0}^{t}{e^ {-
\beta\,x\left(v\right) }\;dv},\gamma)}}\]
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Mark Clements</p>
<p class="date">Created: 2023-06-03 Sat 16:55</p>
</div>
</body>
</html>
