Equations for the rstpm2 package

Table of Contents

1 Generalised survival model

logexpand:all;
derivsubst:true;
load(pdiff)$
assume(t>0)$
S : G(eta(t,z,theta))*S0(t);
H : -log(S);
h : diff(H,t);
print("H:");
tex(H);
print("h:");
tex(h);

print("diff(H,theta):");
tex(diff(H,theta));

print("diff(log(h),theta):");
tex(diff(log(h),theta));

print("ll:");
ll : delta*log(h) - H -  diff(eta(t,z,beta),t)^2;
tex(ll);
print("diff(ll, beta)");
tex(diff(ll, beta));

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) \[-2\,\eta_{\left(1 , 0 , 0\right)}(t,z,\beta)\,\eta_{\left(1 , 0 , 1 \right)}(t,z,\beta)\]

2 Mixture cure models

logexpand:all;
derivsubst:true;
load(pdiff)$
S : S0(t)*(pi(theta)+(1-pi(theta))*exp(-Hu(t,theta)));
H : -log(S);
h : diff(H,t);
h : subst(hu(t,theta), diff(Hu(t,theta),t), h);

print("H:");
tex(H);
print("h:");
tex(h);

print("diff(H,theta):");
tex(diff(H,theta));

print("diff(log(h),theta):");
tex(diff(log(h),theta));
diff(log(h),theta) - diff(H,theta);

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): \[{{-{{e^ {- {\it Hu}\left(t , \vartheta\right) }\,{\it hu}\left(t , \vartheta\right)\,\left(1-\pi\left(\vartheta\right)\right)\,\left(-e ^ {- {\it Hu}\left(t , \vartheta\right) }\,\pi_{\left(1\right)}( \vartheta)+\pi_{\left(1\right)}(\vartheta)-e^ {- {\it Hu}\left(t , \vartheta\right) }\,{\it Hu}_{\left(0 , 1\right)}(t,\vartheta)\, \left(1-\pi\left(\vartheta\right)\right)\right)}\over{\left(\pi \left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\, \left(1-\pi\left(\vartheta\right)\right)\right)^2}}-{{e^ {- {\it Hu} \left(t , \vartheta\right) }\,{\it hu}\left(t , \vartheta\right)\, \pi_{\left(1\right)}(\vartheta)}\over{\pi\left(\vartheta\right)+e ^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi\left( \vartheta\right)\right)}}+{{e^ {- {\it Hu}\left(t , \vartheta\right) }\,{\it hu}_{\left(0 , 1\right)}(t,\vartheta)\,\left(1-\pi\left( \vartheta\right)\right)}\over{\pi\left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi\left(\vartheta \right)\right)}}-{{e^ {- {\it Hu}\left(t , \vartheta\right) }\, {\it Hu}_{\left(0 , 1\right)}(t,\vartheta)\,{\it hu}\left(t , \vartheta\right)\,\left(1-\pi\left(\vartheta\right)\right)}\over{\pi \left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\, \left(1-\pi\left(\vartheta\right)\right)}}}\over{{{e^ {- {\it Hu} \left(t , \vartheta\right) }\,{\it hu}\left(t , \vartheta\right)\, \left(1-\pi\left(\vartheta\right)\right)}\over{\pi\left(\vartheta \right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi \left(\vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(t)}\over{ S_{0}\left(t\right)}}}}\]

3 Integral equation for AFT models

logexpand:all;
derivsubst:true;
load(pdiff)$
assume(t>0)$
S : exp(-exp(B(log(integrate(exp(-x(v)*beta),v,0,t)),gamma)));
H : -log(S);
h : diff(H,t);
print("H:");
tex(H);
print("h:");
tex(h);
print("log(h):");
tex(log(h));

print("diff(H,beta):");
tex(diff(H,beta));
print("diff(H,gamma):");
tex(diff(H,gamma));

print("diff(log(h),beta):");
tex(diff(log(h),beta));
print("diff(log(h),gamma):");
tex(diff(log(h),gamma));

print("ll:");
ll : delta*log(h) - H - (subst(u=log(integrate(exp(-x(v)*beta),v,0,t)), diff(B(u,gamma),u)))^2;
tex(ll);
print("diff(ll, beta)");
tex(diff(ll, beta));
print("diff(ll, gamma)");
tex(diff(ll, gamma));

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) \[\delta\,\left({{B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}( \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}+B_{ \left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) } \;dv},\gamma)\right)-2\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e ^ {- \beta\,x\left(v\right) }\;dv},\gamma)\,B_{\left(1 , 1\right)}( \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)-e^{B \left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , \gamma \right)}\,B_{\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x \left(v\right) }\;dv},\gamma)\]

4 Integral equation for AFT models with functional constraints on gamma

logexpand:all;
derivsubst:true;
load(pdiff)$
assume(t>0)$
gamma : gamma0 + exp(alpha);
S : exp(-exp(B(log(integrate(exp(-x(v)*beta),v,0,t)),gamma)));
H : -log(S);
h : diff(H,t);
print("H:");
tex(H);
print("h:");
tex(h);

print("diff(H,beta):");
tex(diff(H,beta));
print("diff(H,alpha):");
tex(diff(H,alpha));
print("diff(H,gamma0):");
tex(diff(H,gamma0));

print("diff(log(h),beta):");
tex(diff(log(h),beta));
print("diff(log(h),alpha):");
tex(diff(log(h),alpha));
print("diff(log(h),gamma0):");
tex(diff(log(h),gamma0));

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): \[{{B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v \right) }\;dv},\gamma_{0}+e^{\alpha})}\over{B_{\left(1 , 0\right)}( \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{ \alpha})}}+B_{\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x \left(v\right) }\;dv},\gamma_{0}+e^{\alpha})\]

5 Non-integral equation for AFT models

logexpand:all;
derivsubst:true;
load(pdiff)$
assume(t>0)$
S : exp(-exp(B(log(t*exp(-eta(X,log(t),beta))),gamma)));
H : -log(S);
h : diff(H,t);
print("H:");
tex(H);
print("h:");
tex(h);
print("log(h):");
tex(log(h));

print("diff(H,beta):");
tex(diff(H,beta));
print("diff(H,gamma):");
tex(diff(H,gamma));

print("diff(log(h),beta):");
tex(diff(log(h),beta));
print("diff(log(h),gamma):");
tex(diff(log(h),gamma));

print("ll:");
ll : delta*log(h) - H - (1-subst(u=log(t), diff(eta(X,u,beta),u)))^2 -
subst(u=log(t)-eta(X,log(t),beta), diff(B(u,gamma),u))^2;
tex(ll);
print("diff(ll, beta)");
tex(diff(ll, beta));
print("diff(ll, gamma)");
tex(diff(ll, gamma));

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) \[\delta\,\left({{B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)}\over{B_{\left(1 , 0\right)}(\log t-\eta \left(X , \log t , \beta\right),\gamma)}}+B_{\left(0 , 1\right)}( \log t-\eta\left(X , \log t , \beta\right),\gamma)\right)-2\,B_{ \left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right), \gamma)\,B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t , \beta \right),\gamma)-e^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma\right)}\,B_{\left(0 , 1\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)\]

6 Non-integral equation for AFT models without time-varying effect

logexpand:all;
derivsubst:true;
load(pdiff)$
assume(t>0)$
S : exp(-exp(B(log(t*exp(-eta(X,beta))),gamma)));
H : -log(S);
h : diff(H,t);
print("H:");
tex(H);
print("h:");
tex(h);
print("log(h):");
tex(log(h));

print("diff(H,beta):");
tex(diff(H,beta));
print("diff(H,gamma):");
tex(diff(H,gamma));

print("diff(log(h),beta):");
tex(diff(log(h),beta));
print("diff(log(h),gamma):");
tex(diff(log(h),gamma));

print("ll:");
ll : delta*log(h) - H - (1-subst(u=log(t), diff(eta(X,u,beta),u)))^2 -
subst(u=log(t)-eta(X,log(t),beta), diff(B(u,gamma),u))^2;
tex(ll);
print("diff(ll, beta)");
tex(diff(ll, beta));
print("diff(ll, gamma)");
tex(diff(ll, gamma));

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) \[-2\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta \right),\gamma)\,B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)+\delta\,\left({{B_{\left(1 , 1\right)}(\log t-\eta\left(X , \beta\right),\gamma)}\over{B_{\left(1 , 0\right)}( \log t-\eta\left(X , \beta\right),\gamma)}}+B_{\left(0 , 1\right)}( \log t-\eta\left(X , \beta\right),\gamma)\right)-e^{B\left(\log t- \eta\left(X , \beta\right) , \gamma\right)}\,B_{\left(0 , 1\right)}( \log t-\eta\left(X , \beta\right),\gamma)\]

7 Integral equations for AFT models with \(H = B(\ldots)\)

logexpand:all;
derivsubst:true;
load(pdiff)$
assume(t>0)$
S : exp(-B(integrate(exp(-x(v)*beta),v,0,t),gamma));
H : -log(S);
h : diff(H,t);
print("H:");
tex(H);
print("h:");
tex(h);

print("diff(H,beta):");
tex(diff(H,beta));
print("diff(H,gamma):");
tex(diff(H,gamma));

print("diff(log(h),beta):");
tex(diff(log(h),beta));
print("diff(log(h),gamma):");
tex(diff(log(h),gamma));

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): \[{{B_{\left(1 , 1\right)}(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}\]

Author: Mark Clements

Created: 2023-06-03 Sat 16:55