#+title: Equations for the rstpm2 package
#+author: Mark Clements
#+latex_header: \usepackage[landscape,margin=1cm]{geometry}

* Generalised survival model

#+begin_src maxima :results raw :exports both
logexpand:all;
derivsubst:true;
load(pdiff)$
assume(t>0)$
S : S0(t)*G(eta(t,z,theta));
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));
#+end_src


* Mixture cure models

#+begin_src maxima :results raw :exports both
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);
#+end_src

* Integral equation for AFT models

#+begin_src maxima :results raw :exports both
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));
#+end_src

* Integral equation for AFT models with functional constraints on gamma

#+begin_src maxima :results raw :exports both
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));
#+end_src


* Non-integral equation for AFT models

#+begin_src maxima :results raw :exports both
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));

#+end_src


# * Constraints

# 1. Baseline log cumulative hazard function:

# Delta Q gamma >= 0 <=> Q[i+1,] gamma >= Q[i,] gamma 

# And/or: hazard is positive <=> B'(log(tstar),gamma)>0

# What if we use B-splines with a constraint? This would lead to 
# gamma[i+1] >= gamma[i]

# 2. Cumulative covariate effects on the acceleration factor:
# If g(X,t,beta) = X beta + \sum_j X_j(t)*B_j(log(t))^T gamma_j

# This constraint is not needed for the integral model. Should we focus on that model?


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

#+begin_src maxima :results raw :exports both
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));

#+end_src


# * Constraints

# 1. Baseline log cumulative hazard function:

# Delta Q gamma >= 0 <=> Q[i+1,] gamma >= Q[i,] gamma 

# And/or: hazard is positive <=> B'(log(tstar),gamma)>0

# What if we use B-splines with a constraint? This would lead to 
# gamma[i+1] >= gamma[i]

# 2. Cumulative covariate effects on the acceleration factor:
# If g(X,t,beta) = X beta + \sum_j X_j(t)*B_j(log(t))^T gamma_j

# This constraint is not needed for the integral model. Should we focus on that model?


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

#+begin_src maxima :results raw :exports both
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));
#+end_src


# /* non-integral equation for AFT model (non mixture cure)
# - cumulative effect modelled directly (does this make any sense?)
# */
# assume(t>0)$
# S : exp(-exp(B(exp(g(X,log(t),beta)),gamma)));

# /* non-integral equation for AFT model (non mixture cure)
#   H = B(...)
# */
# assume(t>0)$
# S : exp(-B(t*exp(-g(X,t,beta)),gamma));