rgs: What the Package Does (One Line, Title Case)

This function calculates the asymptotic covariance of n1**(-1/2)U1(t) and n2*(-1/2)U2(t) in two weighted logrank tests Z1 and Z2, where Zj = nj*(-1/2)*Uj(t) / sqrt(Vj(t)/n).

wlr.Ucov(
  DCO = c(24, 32),
  r = 1,
  h0 = function(t) {     log(2)/12 },
  S0 = function(t) {     exp(-log(2)/12 * t) },
  h1 = function(t) {     log(2)/12 * 0.7 },
  S1 = function(t) {     exp(-log(2)/12 * 0.7 * t) },
  cuts = c(6),
  rho = c(0, 0),
  gamma = c(0, 0),
  tau = c(NULL, NULL),
  s.tau = c(0, 0),
  f.ws = list(f.ws1 = NULL, f.ws2 = NULL),
  Lambda = function(t) {     (t/18) * as.numeric(t <= 18) + as.numeric(t > 18) },
  G0 = function(t) {     0 },
  G1 = function(t) {     0 },
  Hypo = "H0"
)

`DCO`	Analysis time, calculated from first subject in.
`r`	Randomization ratio of experimental arm : control arm as r:1. When r = 1, it is equal allocation. Default r = 1.
`h0`	Hazard function of control arm. h0(t) = log(2)/m0 means T~exponential distribution with median m0.
`S0`	Survival function of control arm. In general, S0(t) = exp(- integral of h0(u) for u from 0 to t). but providing S0(t) can improves computational efficiency and usually the survival function is known in study design. The density function f0(t) = h0(t) * S0(t).
`h1`	Hazard function of experimental arm. h1(t) = log(2)/m1 means T~exponential distribution with median m0.
`S1`	Survival function of experimental arm. In general, S1(t) = exp(- integral of h1(u) for u from 0 to t). but providing S1(t) can improves computational efficiency and usually the survival function is known in study design. The density function f1(t) = h1(t) * S1(t).
`cuts`	A vector of cut points to define piecewise distributions. If cuts is not specified or incorrectly specified, it might occasionally have numerical integration issue.
`rho`	Parameter for Fleming-Harrington (rho, gamma) weighted log-rank test.
`gamma`	Parameter for Fleming-Harrington (rho, gamma) weighted log-rank test. For log-rank test, set rho = gamma = 0.
`tau`	Cut point for stabilized FH test, sFH(rho, gamma, tau); with weight function defined as w(t) = s_tilda^rho*(1-s_tilda)^gamma, where s_tilda = max(s(t), s.tau) or max(s(t), s(tau)) if s.tau = NULL tau = Inf reduces to regular Fleming-Harrington test(rho, gamma)
`s.tau`	Survival rate cut S(tau) at t = tau1; default 0.5, ie. cut at median. s.tau = 0 reduces to regular Fleming-Harrington test(rho, gamma)
`f.ws`	Self-defined weight function of survival rate, eg, f.ws = function(s)1/max(s, 0.25) When f.ws is specified, sFH parameter will be ignored.
`Lambda`	Cumulative distribution function of enrollment.
`G0`	Cumulative distribution function of drop-off for control arm, eg, G.ltfu=function(t)1-exp(-0.03/12*t) is the distribution function for 3 percent drop-off in 12 months of followup.
`G1`	Cumulative distribution function of drop-off for experimental arm.
`Hypo`	Hypothesis: "H0", "H1", "H1.LA". Under H0, the pooled hazard h.bar(t) = f.tilda(t)/S.tilda(t).

#1:1 randomization, control~exp(median=12);
#experiment~delayed effect 6 mo and HR 0.6 after delay
#Fleming-Harrington (1,1) test; Enrollment: 18 mo with weight 1.5
#drop-off rate: 3% in 1-year of follow-up in control arm; no drop-off in exp arm
HR = 0.6; delay = 6; lam0 = log(2) / 12; 
h0 = function(t){lam0}; S0 = function(t){exp(-lam0 * t)}
h1.D6 = function(t){lam0*as.numeric(t < delay)+HR*lam0*as.numeric(t >= delay)}
c = exp(-delay*lam0*(1-HR)); 
S1.D6 = function(t){exp(-lam0*t)*as.numeric(t<delay) + c*exp(-HR*lam0*t)*as.numeric(t>=delay)}
Lambda = function(t){(t/18)^1.5*as.numeric(t <= 18) + as.numeric(t > 18)}
drop0 = 0.03/12; drop1 = 0

wlr.Ucov(DCO = c(12, 24), r = 1, h0 = h0, S0 = S0,h1 = h1.D6,S1 = S1.D6, 
cuts=c(6), rho = c(0,0), gamma = c(0,0), tau = c(NULL,NULL), s.tau = c(0,0), f.ws = list(f.ws1=NULL, f.ws2=NULL),
Lambda = Lambda, G0 = function(t){1-exp(-drop0 * t)}, G1 = function(t){0},
Hypo = "H1")

wlr.Ucov(DCO = c(24, 24), r = 1, h0 = h0, S0 = S0,h1 = h1.D6,S1 = S1.D6, 
cuts=c(6), rho = c(0,0), gamma = c(0,1), tau = c(NULL,NULL), s.tau = c(0,0), f.ws = list(f.ws1=NULL, f.ws2=NULL),
Lambda = Lambda, G0 = function(t){1-exp(-drop0 * t)}, G1 = function(t){0},
Hypo = "H0")