The specification of competing risks survival data that we presented in Section 2.1 and 2.2 is based on the joint modeling of the failure time and the failure cause (or failure type). The traditional route is based on assigning a set of latent lifetime variables corresponding to the multiple failure causes. Although this approach is intuitive at first sight, it is either accompanied by the additional assumption of independence between these latent lifetimes, which may not be applicable to real applications; or without the independence assumption, there is a certain issue of model identifiability presented in detail in Chapter 7 of Crowder (2001).
2.3.1
Introduction of Latent Lifetimes
In the traditional approach for describing competing risks survival data, it is assumed that there is a potential failure time associated with each of the p risks to which an individual is exposed. Let Tj denotes the time to failure from cause j ( j = 1, 2, · · · , p). Then the
smallest Tj( j= 1, 2, · · · , p) determines the time T to overall failure, and we use its index C
to denote the cause of failure, i.e., T = min {T1, T2, · · · , Tp}= TC. Once the individual has
failed, the remaining lifetimes corresponding to the individual causes are lost to observation (Crowder, 2001).
The joint distribution of the random vector T = (T1, T2, · · · , Tp) is used to statistically
describe the competing risks data. The joint distribution function is defined as G(t) = P(T ≤ t) = P(T1 ≤ t1, T2 ≤ t2, · · · , Tp ≤ tp); and similarly, the joint survivor function
is G(t) = P(T > t). If the Tj ( j = 1, 2, · · · , p) are jointly continuous, the joint density is
∂p
Independenceof the Tjs is defined by G(t)= p Y j=1 Gj(tj) ; (2.3) or equivalently, G(t)= p Y j=1 Gj(tj) , (2.4)
where Gj(tj)= P(Tj ≤ tj) is the marginal distribution function of Tjand Gj(tj)= P(Tj > tj)
is the marginal survivor function of Tj, where Tj corresponds to the time to failure due to
cause j specifically.
It will be assumed that Tj( j= 1, 2, · · · , p) are continuous and that ties cannot happen,
that is, P(Tj = Tj0)= 0 for all j , j0, otherwise C is not easily defined (refer to Section 7.2
of Crowder (2001)).
2.3.2
Marginal Distribution of the Latent Failure Times and the Sub-
distribution
In the older terminology, the sub-survivor function S ( j, t), defined in Section 2.1, was called the crude survivor function; and the marginal distribution function Gj(t) was called the
net survivor function. These two ways of modeling competing risks survival data are not irrelevant. If the joint survivor function G(t) of T has known form, then the overall survivor function S (t) can be obtained as G(t1p), where 1p= (1, 1, · · · , 1) is of length p. Moreover, the following theorem shows the relation between the sub-density function and the joint survivor function of the latent failure times.
the joint suvrivor function of the latent failure times as
f( j, t) = ∂G(t) ∂tj
|t1
p . (2.5)
The proof of the theorem can be found on Page 38 of Crowder (2001) and is omitted here.
It follows from the theorem that the sub-hazard function can also be calculated directly from the joint survivor function of the latent time variables G(t) as
h( j, t)= f( j, t) S(t) = −∂log G(t) ∂tj |t1 p .
2.3.3
The Identifiability Crisis of Latent Failure Times Approach
Describing competing risks from the point of view of latent lifetimes seems very natural, and statistically, one can specify a parametric model for the joint multivariate distribution of the latent time variables G(t), fit it to the data, and do the inference, etc. However, one problem of the approach of latent time variables is that the Gj(t) ( j = 1, 2, · · · , p) do not
describe events that physically occur - “they only describe failures from isolated causes in situations where all the other risks have been removed somehow” (Crowder, 2001). It is the S ( j, t), the sub-distributions, not the Gj(t), the marginal distribution of latent variables,
that are truly observed in the real situation. Moreover, it often happens that the whole mechanism is changed after the removal of the other causes, and therefore it is not valid to assume that when Tjis observed without other competing risks, its distribution is the same
as its marginal distribution derived from the joint distribution (Crowder, 2001). Even more so, without the assumption of independence between the latent times, which is often the reality, the modeling of latent times can have a serious identifiability problem brought out by Cox (1959). Cox (1959) studied various parametric models with two causes, and Tsiatis (1975) extended to the general case of p risks. They showed that, “given any joint survivor
function with arbitrary dependence between the component variates (i.e., the latent times), there exists a different joint survivor function in which the variates are independent and which reproduces the sub-densities f ( j, t) precisely” (Crowder, 2001). This implies that from the same set of observations on (C, T ) alone, we can have two different models that fit the data equally well; that is, model identifiability problem arises with the latent time variables approach. The following theorem theoretically describes this problem. The proof follows that in Chapter 7 of Crowder (2001) with additional details and correction of a mistake, and can be found in the appendix.
THEOREM 2.4 (Tsiatis (1975)). Suppose that the set of S ( j, t) is given for some model with dependent risks. Then there exists a unique proxy model with independent risks yield- ing identical S( j, t). It is defined by G(t)= Qpj=1G∗j(tj), where G
∗ j(tj) = exp n −Rt 0 h( j, s)ds o
and the sub-hazard functiion h( j, s) derives from the given S ( j, t).
The theorem establishes only that to each dependent-risks model there corresponds a unique independent-risks proxy model with the same sub-survivor functions. Moreover, it has been shown (Crowder, 1991) that each independent-risks model actually has a whole class of satellite dependent-risks models and that this class can be further partitioned into sets with the same marginal functions. Therefore, “it is not possible to obtain from the observations of (C, T ) unique information about the distributions of cause-specific failure times or on the dependence structure between them” (Crowder, 2001). An exception is the case of regression model where there are explanatory variables in the model, identification is possible within a certain framework (Heckman and Honor, 1989), though Kalbfleisch and Prentice (2002, page 261) pointed out that it is still more straightforward to consider only specifying the cause-specific hazards since in fact only the cause-specific hazards (i.e. sub-hazards) enter the likelihood function and they are all that is needed to specify the joint distribution of the failure time and the cause (T, C).
For more discussions and some examples about the problem of non-identifiability, the reader can refer to Chapter 7 of Crowder (2001).