The 2-D survival function is expressed as
\(S(t1,t2)=C(S1(t1),S2(t2))\), where \(S1(t1)\), \(S2(t2)\)
are marginal survival functions and \(C(u1,u2)\) is a 2-D copula.
The marginal survival functions are estimated via the marginal
hazards as in sshzd
, and the copula is estimated
nonparametrically by calling sscopu2
.
When symmetry=TRUE
, a common marginal survial function
S1(t)=S2(t) is estimated, and a symmetric copula is estimated such
that \(C(u1,u2)=C(u2,u1)\).
Covariates can be incorporated in the marginal hazard models as in
sshzd
, including parametric terms via partial
and frailty terms via random
. Arguments formula1
and
formula2
are typically model formulas of the same form as the
argument formula
in sshzd
, but when
partial
or random
are needed, formula1
and
formula2
should be lists with model formulas as the first
elements and partial
/random
as named elements; when
necessary, variable configurations (that are done via argument
type
in sshzd
) should also be entered as named
elements of lists formula1
/formula2
.
When symmetry=TRUE
, parallel model formulas must be
consistent of each other, such as
formula1=list(Surv(t1,d1)~t1*u1,partial=~z1,random=~1|id1) |
formula2=list(Surv(t2,d2)~t2*u2,partial=~z2,random=~1|id2) |
where pairs t1
-t2
, d2
-d2
respectively
are different elements in data
, pairs u1
-u2
,
z1
-z2
respectively may or may not be different
elements in data
, and factors id1
and id2
are typically the same but at least should have the same levels.