Matemáticas

¹ School of Statistics, Universidad Nacional de Colombia, Sede Medellín,
² Experimental Statistics, Louisiana State University, * Correspondencia: Sergio Yáñez, syanez@unal.edu.co.

Recibido: 23 de junio de 2014. Aceptado: 28 de agosto de 2014

Abstract

Lifetime and survival data are usually the lives of subjects, units, or systems that has been exposed to multiple risks or modes of failure. In the analysis of the data, however, is common to ignore the modes of failure because they are unknown, they are not recorded, or because the complexity of including them in the modeling. It is of interest knowing when the conclusions might be robust to ignoring the failure modes in the analysis. In particular, it would be useful to characterize situations where it could be safely said that the modes of failure effect on the analysis would be negligible or that failing to include such information could completely invalidate the conclusions drawn from the study. As a first step in identifying when the failure modes have little or large influence on the competing risks model, this article studies two different competing risks models: (a) a model with independent risks; (b) a model derived from a multivariate Weibull with dependence.

Key words: competing risks, probability plots, log-location-scale family, Weibull distribution, multivariate Weibull with dependence, lognormal distribution.

Resumen

Los datos de tiempos de falla y sobrevivencia se refieren generalmente a la vida de individuos, unidades o sistemas que han sido expuestos a múltiples riesgos o modos de falla. Sin embargo, en el análisis de los datos es común ignorar los modos de falla porque son desconocidos, no se registran o por la complejidad de la modelación al incluirlos. Es importante, por lo tanto, determinar la robustez del análisis ignorando los modos de falla. En particular, sería útil caracterizar situaciones donde pudiera decirse que el efecto de los modos de falla en el análisis es despreciable o donde no incluir tal información pudiera invalidar completamente las inferencias del estudio. Como un primer paso para identificar cuando los modos de falla tienen poca o mucha influencia en el modelo de riesgos en competencia, este artículo estudia dos modelos de riesgos en competencia: (a) un modelo con riesgos independientes; (b) un modelo derivado de una distribución Weibull multivariada con dependencia.

Palabras clave: riesgos en competencia, gráficos de probabilidad, familia de log-localización y escala, distribución Weibull, distribución Weibull multivariada con dependencia, distribución lognormal.

1. Introduction

In the analysis of lifetimes one is usually dealing with time to an event of interest like failure of a component or system, dead of an individual, or end to a subscription service. In most cases, there are several competing reasons (risks or modes of failure) that cause the event of interest. Say, for example, that we were interested in the first time that a bicycle fails. In this situation, the competing risks for failure include: a flat tire, a broken chain, or the rupture of a brake cable.

The interest in competing risks is old but the formal development and application of the methodology to problems in engineering, survival analysis, and other applied areas is relatively new. Crowder (2001, 2012) and Pintilie (2006) provide historical accounts, relevant references, and methodology.

It is interesting that, for many years, competing risks were not in the front end of reliability and survival analysis. Beyond the statistical challenges of handling properly multiple risks in an analysis, there are, however, some compelling reason for which the competing risks might have been ignored in the analysis of the data: (a) sometimes the competing risks are hidden or unknown to the observer; (b) the competing risks may be well known but it is difficult or expensive to identify the risk that caused the event; (c) in other cases the cost and time spent on determining the failure cause ends being a waste of resources because ignoring the competing risks in the analysis is as good as the analysis that takes them in consideration. This is the case for example, with the analysis of the Shock Absorber data in Meeker and Escobar (1998) where a simple Weibull analysis, that ignores the failure modes, is basically indistinguishable of aWeibull analysis that takes in consideration the two observed failure modes.

It would be useful to have a clear understanding on the situations (e.g., model, data type, risk type) where the use of the competing risks information makes a difference in the analysis. Meeker, Escobar, and Hong (2009, page 157), in the context of a complex model and analysis, came to the conclusion that in a competing risks model with two risks that are log-location-scale distributed with similar shape parameter, the distribution of the competing risks can be approximated by the same log-location-scale distribution with a shape parameter that is in between those for the marginal distributions and that the adequacy of the approximation does not depend strongly on amount of association between the two risks. One, however, would need a formal study of the problem to provide a more definitive answer.

The purpose of this article is to study two simple competing risks models with Weibull risks to assess the effect of the risks on the time to the event of interest and the inter-relationships between the two models. This might be useful in setting a simulation study that could provide broader guidelines on the situations where ignoring the competing risks information in the analysis could lead to erroneous conclusions.

The rest of the paper is organized as follows. Section 2 is an introduction to competing risks models (CRMs) and some features of their probability plots. Section 3 discusses the competing risks model with Weibull independent risks and its characteristics. Section 4 presents a competing risks model derived from a bivariate Weibull distribution with dependent risks. Section 5 describes generalization of the results to CRMs with k > 2 risks and anticipate some of the difficulties in generalizing the results to log-location-scale families in general. Section 6 summarizes the results in the paper and outlines future related work. Section 7 contains appendixes with technical details for some of the results given in the paper.

2. Competing Risks Models and Their Probability Plots

Lifetimes of units or individuals subject to two risks (or modes of failure), k = 2, can be modeled as a seriessystem. Each risk is like a component in a series system with two components. The unit has a potential lifetime associated with each risk. The observed lifetime for the unit is the minimum of these individual potential lifetimes T_i, i = 1, 2. Informally, we refer to T_i as the Risk i and to the model as the competing risks model (CRM).

For a CRM with two risks, a quantity of interest is T_m = min (T₁, T₂) which has the cumulative distribution function (cdf)

where Pr(T₁ > t, T₂ > t) is computed with respect to the joint distribution of (T₁, T₂).

When the risks are independent

where F_i(t) is the cdf for T_i, i = 1, 2.

In the case of k risks, F_m(t) = 1 - Pr(T₁ > t, . . . , T_k > t) for dependent risks and F_m(t) = 1 - P^k_{i =1} Pr(T_i > t) = 1 - P^k_{i =1} [1 - F_i(t)] for independent risks, where F_i(t) is the cdf for the i risk. See Meeker and Escobar (1998, chapter 15) for additional information.

2.1. Probability Plots and CRMs

In theoretical and applied work with lifetime data, probability plots have shown to be useful for multiple purposes, see Meeker and Escobar (1998, chapter 6) for detailed explanation on their construction, interpretation, and use. In this paper we will use probability plots to display the distribution function of the competing risks model along with the distribution of the individual risks. As illustrated later, with proper choice of the plot scales, the distributions of the individuals risks, F_i(t), show up as straight lines in the probability plot, but the CRM cdf, F_m(t), is usually a non-linear curve in the plot.

Graph of a cdf on a log-location-scale plot

Consider a log-location-scale probability plot defined by the scales [ln t, F^-1(p)] where F(z) is a standardized continuous cdf that does not depend on unknown parameters, 0 < p < 1, t > 0, and ln t denotes the natural logarithm of t. It is known, see for example Meeker and Escobar (1998, chapter 6), that in these scales any cdf of the form F[(ln(t) - m)/s] (where s > 0) plots as a straight line with slope 1/s.

In a log-location scale probability plot, if a cdf F(t) is a fairly linear curve, then there is strong information that, with properly chosen (m, s), the cdf F(t) is well approximated by the distribution function F[(ln(t) - m)/s]. We will use this feature of probability plots later in studying the properties of some CRMs.

The following result considers a probability plot defined by the scales [ln t, F^-1(p)] and the plot {ln t, F^-1 [F(t)]} of a cdf F(t), t > 0 on it.

Result 1. For an absolutely continuous cdf F(t) = Pr(T ≤ t), t > 0 the slope of the curve {x = ln t, y = F^-1[F(t)]} in the probability plot [ln t, F^-1(p)] is given by

where f (t) and f(z) are the probability density functions (pdfs) corresponding to F(t) and F(z), respectively.

The proof of (1) follows after differentiation of y with respect to ln t.

Because t, f (t), and the function f(z) are all nonnegative, the slope in (1) is non-negative. This is the expected behavior because the cdf F(t) is a non-decreasing function of ln(t).

2.2. The competing risks model for independent positive continuous variables

For the CRM with two independent risks, k = 2,

where F_i(t) and f_i(t) are the cdf and pdf of T_i and S_i = 1 - F_i(t) for i = 1, 2. Then, using (1) with F(t) = F_m(t),

The competing risks model for two independent log-location-scale continuous variables

Now suppose that the individual risks are independent and log-location-scale distributed. That is, T₁ and T₂ are independent, and Ti ∼ F(z_i) , i = 1, 2, where

-∞ < m_i < ∞, s_i > 0, and F(z) is a differentiable cdf that does not depend on unknown parameters. In this case,

where f_i(z) is the pdf corresponding to the cdf Fi(z), i = 1, 2.

Thus for a competing risks model with two independent log-location-scale risks, the slope of the curve {x = ln t, y = F^-1[Fm(t)]} in the probability plot [ln t, F^-1(p)] is given by

In this case, the cdfs of the individual risks T_i appear as straight lines in the probability plot. But the line corresponding to the CRM cdf is not necessarily linear. For an example, see Figure 1 which is described in the following section.

3. Competing RisksModelWith Independent Weibull Risks

A Weibull cdf F(t) is often written as

where h > 0 is a scale parameter and b > 0 is a shape parameter. F_sev(z) = 1 - exp[-exp(z)] is the standard smallest extreme value distribution, m = ln(h), and s= 1/b. When T has a Weibull distribution, we indicate it by T ∼ WEI(h, b).

Although (5) and (6) are equivalent specifications of the Weibull cdf, (5) is commonly used in engineering and (6) is convenient on theoretical developments, for numerical stability in computations, and for notational standardization with other log-location scale distributions. For additional details see Meeker and Escobar (1998) and Lawless (2003).

Then the Weibull competing risks model with two independent risks has the cdf

Figure 1 shows p_miw(t) in aWeibull probability plot with scales [ln t, F^-1_sev(p)]. The plot also shows, as straight lines, the cdfs for the two Weibull independent risks defining the CRM model. In this case, because the plot of p_miw(t) is non-linear, we know that the distribution of the CRM is not a Weibull distribution. For this and other figures in the paper, the model parameters values were chosen for illustration purposes.

Important properties of the CRM with independent Weibull risks

The following results characterize the CRM with independent Weibull risks.

Result 2. Consider the graph {x = ln t, y = F^-1_sev[p_miw(t)]} of the cdf p_miw, for the CRM with independent Weibull risks, on aWeibull probability plot. The following results hold.

(a)

The slope of p_miw(t) in the probability plot is the convex combination


To prove this result, note that the density and the quantile functions for F_sev(z) are



Thus fsev[F^-1_sev(p)] = (1 - p)[-ln(1 - p)] and we get

f_sev[F^-1_sev(p_miw(t))] = [1- F_sev(z₁)][1 - F_sev(z₂)][exp(z₁) + exp(z₂)].

From (3) with F(z) = F_sev(z) and f(z) = f_sev(z), one obtains (7).

A more detailed proof of (7) can be obtained as the special case with q = 1 in Appendix B.1.

(b)

When b₁ = b₂ = b, p_miw(t) is the cdf of a WEI(h, b) where


To prove this result, use (7) with s₁ = s₂ = 1/b to obtain



Thus p_miw(t) is linear in the Weibull probability plot which implies that it is a Weibull cdf. The particular form of the shape parameter is obtained from the anti-log of m_t in (11) with s₁ = s₂ = 1/b.

The fact that, in this case, p_miw(t) is a Weibull cdf is in agreement with the following known result: the minimum of k independent Weibulls, that have a common shape parameter b but possibly differing scale parameters, is Weibull distributed with shape parameter equal to b. For details, see, for example, Meeker and Escobar (1998, page 372) and the related Result 3 below.

(c)

The slope of the curve {x = ln t, y = F^-1_sev[p_miw(t)]} is bounded by (b₁, b₂) as follows. Using (7)


Or equivalently,



This is consistent with the observation made in Meeker et al. (2009, page 157) where they found through simulation and sensitivity work that in modeling competing risks data with a single Weibull distribution (i.e., ignoring the failure mode), the estimated Weibull shape parameter tend to fall between the two Weibull shape parameters used to simulate the data. This result suggest that the closeness between the shape parameters of the risks' cdfs might be an important factor in determining if a single Weibull model could fit well a Weibull CRM with independent risks.

(d)

In the limit


These limiting behaviors follow directly from (7) and the relationships b_i = 1/s_i, for i = 1, 2.

In Figure 1, as ln(t) → -∞, the distribution p_miw(t) approaches the individual risk F₁(t) and also the derivative of p_miw(t) with respect to ln(t) converges toward b₁. Similarly, when ln(t) → ∞, p_miw(t) approaches F₂(t) and the derivative of p_miw(t) with respect to ln(t) converges toward b₂.

(e) The curve {x = ln t, y = F^-1_sev[p_miw(t)]} is concave when b₁ ≠ b₂.

To verify this result, obtaining the derivative of (7) with respect to ln t to obtain



which is positive when s₁ ≠ s₂. This implies that the curve (ln t, y) is concave. Because the curvature of the [ln t, y] is proportional to (10), see for example Courant and John (1965, page 357), then is also proportional to (1/s₁ - 1/s₂)² = (b₁ - b₂)². Note that derivative (10) is a special case of Appendix B.2 when q = 1.

Large values of (10) are associate with large curvatures of the curve (ln t, y) and we can use curvature as another criteria to help in the decision when is that a simple Weibull distribution fits well the CRM model with independent Weibull risks. In particular, a single Weibull model would fit well aWeibull CRM cdf with independent risks if the two Weibull shape parameters in the CRM model are not far apart. There are situations, however, where the curvature is large just in an interval with negligible probability content with respect to the CRM cdf, in those case the simple Weibull distribution will fit well the CRM model in an interval of high probability with respect to the CRM cdf, regardless of sizes of the scale parameters. See the discussion at the end or Result 3 for a case where the curvature is large just in an interval with negligible probability content with respect to the CRM cdf.

All the outcomes in Result 2 can be extended readily, with trivial minor changes, to a CRM with k > 2 independent Weibull risks. See Jiang and Murthy (2003) for results related to items (a) and (d) in Result 2.

CRM with independent Weibull risks and similar shape parameters

An important problem is the modeling of competing risks data when the risks are Weibull distributed with similar shape parameter. The following result considers that setting in the special case when the risks are independent.

Result 3. Consider the cdf p_miw(t) of the CRM T = min(T₁, T₂) with independent risks. Suppose T_i ∼ WEI(h_i, b_i), i = 1, 2. Then when b_i → b > 0, i = 1, 2, p_miw(t) converges to a WEI(h, b) cdf where h = (h₁h₂)/(h^b₁ + h^b₂ )^1/b.

To prove this result, it can be shown that (use Appendix C with q = 1),

where

The proposed results follows after letting b_i → b, i = 1, 2 (which is equivalent to s_i → s, i = 1, 2 where s = 1/b). Then

Where

Equation (12) corresponds to a WEI(h, b), where b = 1/s, h = exp(m), with

Note that (11) is well approximated by a Weibull distribution when m_t is fairly constant. That is, when m_t does not depend strongly on t. Values of s₁ and s₂ closed to each other might have that effect on m_t. Observe, however, that m_t could also be fairly linear in an interval of high probability content (with respect to the CRM cdf), if exp(-m₁/s₁) dominates t^1/s2-1/s1 exp (-m₂/s₂) in that interval. Or equivalently, when F₁(t) > F₂(t) in a interval with high p_miw(t) probability content. A similarly situation arises when the second risk is the dominant one, that is when F₂(t) > F₁(t) in a interval with high p_miw(t) probability content. Thus to anticipate the linearity of the CRM cdf in a probability plot, two factors emerge as important to be considered: (a) the closeness of the shape parameters and (b) the crossing point of the cdfs F₁(t) and F₂(t) associated with the two risks in the model. These two factors may or may not interact depending of the CRM model parameters. Extreme cases are when the crossing point is too far to the right or to the left in the plot in whose case the CRM cdf can be approximated well by a single Weibull distribution.

Next section studies a CRM derived from a bivariate Weibull distribution with dependence between the risks. We will see that the properties for the CRM model with independent Weibull risks extend to this new CRM.

4. Competing Risks Model Derived From a Bivariate Weibull Distribution

In practical application the competing risks might not be independent and the dependence can greatly complicate the statistical modeling and the data analysis, see a detailed account in Crowder (2001, chapter 7).

Here we consider a simple model that allows for positive dependence. In the analysis of lifetime data, models that allow for risks with positive dependence are reasonable because in those cases the risks are jointly leading toward shorter lives, but see Crowder (1989) for a somehow dissenting opinion.

The setting here is that the risks (T₁, T₂) are jointly distributed with a joint survival function S(t₁, t₂) = Pr(T₁ > t₁, T₂ > t₂) given by

where t_i ≥ 0, b_i > 0, h_i > 0, i = 1, 2, and q > 1. The h_is are scale parameters and the b_is are shape parameters. q characterizes the association between the two variables. In particular, the Kendall's coefficient of association between T₁ and T₂ is equal to 1 - 1/q.

Johnson and Kotz (1972, pages 268-269) have an early description of this distribution. Lee (1979, pages 268-269) discusses this distribution in a larger context of multivariate distributions with Weibull properties. Kotz, Balakrishnan, and Johnson (2000, page 408) call it a Weibull form B and show how to derive it through power trans- formations of the variables from a bivariate exponential distribution. It can be shown that (13) corresponds to a Gumbel-Hougaard survival copula evaluated at two Weibull survival marginals, see Nelsen (2006, page 96). This bivariate Weibull has has been used in dependent failure-times analysis, among others, by Hougaard (1986), Hougaard (1989), and Lu and Bhattacharyya (1990).

Using the log-location scales w_i = [ln(t_i) - m_i]/s_i, with m_i = ln(h_i), s_i = 1/b_i, the survival distributions in (13) becomes

The joint distribution function of (T₁, T₂) is

where F₁ (t₁) and F₂ (t₂) are the Weibull marginal distributions of T₁ and T₂ given by

where the z_i are as in (2).

The distribution F(t₁ , t₂ ) is absolutely continuous and its density is (see Appendix D)

where for i = 1, 2

Figure 2 shows contours of the bivariate Weibull for a specific and convenient choice of the parameters. The southwest to northeast orientation of the plot is an effect of the positive dependence (q = 1.5) between the two Weibull risks. With independence, that is q = 1, and everything else being the same, the density contours will be fairly symmetric about a vertical line drawn at time t₁ = 106.

The CRM derived from the Bivariate Weibull

As before, using T_m = min(T₁, T₂), it follows that the survival function s_mdw (t) of T_m is

where z_i = [ln(t) - m_i]/s_i, m_i = ln(h_i), and s_i = 1/b_i. The related distribution function p_mdw for the CRM T_m

is

With q = 1, the pdf p_mdw(t) is equal to the cdf p_miw(t) for the CRM with independent risks T₁ and T₂ .

Result 4. Consider the cdf, p_mdw(t), given in (16). Then (see Appendix A)

1. For each t, p_mdw(t) is a monotone decreasing function of q.

2. Assuming, without loss of generality, that b₁ < b₂ ,


where t_c is the time at which the marginal distributions F₁(t) and F₂(t) cross. That is, the time at which z₁ = z₂ , or equivalently

3. When b₁ = b₂,


for all t > 0.

Result 4 implies that the cdf p_mdw(t) of T_m , for the bivariate case with dependence, is bounded from above by the cdf p_miw of T_m in the model with independent risks and below by the marginals associated with the Weibull bivariate risks. For example, in Figure 3, the dashed line (the one closest to the northwest corner) corresponds to the CRM with q = 1. The solid line is the CRM with q = ∞. This cdf agrees with the distribution of Risk 1 for times before the time at which the two marginals cross each other and with the distribution of Risk 2 for times after that crossing point of the marginals.

Important properties of the CRM with dependent Weibull risks

Result 5. Consider the Weibull probability plot {x = ln t, y = Φ_sev^-1 [ p_mdw (t)]} of the cdf p_mdw (t) in (16) for the CRM derived from the bivariate Weibull distribution in (14). The following results hold.

(a)

The slope of p_mdw(t) in the probability plot is the convex combination


where z_i = [ln(t) - m_i]/s_i .

To verify this result, use (15) to obtain the y-ordinate

y = ln[-ln(s_mdw(t))]
=1/q ln [exp(qz₁) + exp(qz₂)].

Then taking the derivative of y with respect to ln t yields (17), see Appendix B.1.

The similarity between (7) and (17) is more than a simple coincidence. We proceed to show that the CRM p_mdw(t) is an extension of the CRM p_miw(t) with characteristics very similar to the p_miw(t) model. The similarities between the two models was unknown to us at the outset of this study.

(b)

When b₁ = b₂ = b, p_mdw(t) is the cdf of a WEI(h, b) where


With b₁ = b₂ = b, then s₁ = s₂ = 1/b and it follows from (17) that



which implies that p_mdw(t) is a WEI(h, b). The specific value of h is derived in Appendix C.

It is interesting that, even in the presence of dependence, the equality of the shape parameters implies a Weibull distribution for the CRM cdf, p_mdw(t).

See the related Result 6 below which shows that when the shape parameters of the risks are closed to each other, the CRM has a distribution that might be approximated by a simple Weibull distribution.

(c)

The slope of the curve [ln t, y] is bounded by (b₁, b₂) as follows


To prove this, suppose that max_i {1/s_i} = 1/s₂ then using (17) it can be verified that



As in the case of independent risks it is useful to know that the CRM cdf has a derivative with respect to ln t that is bounded by the shape parameters of the risks defining the model.

(d)

In the limit


To prove this, note that if s₁ = s₂ the result holds in view of item (b) above. Then assume that s₁ ≠ s₂. If max_i {1/s_i} = 1/s_k = b_k then



and



Using (20) and (21), when taking the limits of dy/d ln t, yield the results in (19) for the limiting behavior of the derivative function.

(e)

The curve [ln t, y] is concave when b₁ ≠ b₂.

Taking the derivative of (17) with respect to ln t (see Appendix B.2)



Because this second derivative is positive, the concavity of the curve follows. The concavity, however, is also proportional to q and large values of q could imply large concavity and curvature of the curve even in the presence of shape parameters that are closed to each other. This is different to what happens in the independent risks case where similar shape parameters imply small concavity.

CRM with dependent Weibull risks and similar shape parameters

Result 6. Consider the cdf p_mdw(t) of the CRM given in (16) and derived from the bivariate model with dependence in (14). When b_i → b > 0, i = 1, 2, p_mdw(t) converges in distribution to a WEI(h, b) where

See Appendix C for justification of the result.

Note that the limiting Weibull distribution WEI(h, b) was already obtained in (18) as a special case of equal shape parameters. The special case with independent risks given in Result 3 is obtained with the value q = 1.

5. Other Competing Risks Models

As noted earlier, the findings in Result 2 extend directly to a CRM with k > 2 independent Weibull risks and cdf p_miw(t) = 1 - P^k_i=1 [1 - Fsev(z_i)], where z_i = [ln(t) - m_i]/s_i , i = 1, . . . , k.

The bivariate Weibull model described in (13) has been extended to the case of k > 2 dependent risks. In this case, the survival function is S(t₁, . . . , t_k) = exp { - [S^k_{i =1} exp(qw_i)]^1/q}, where w_i = [ln(t_i) - m_i ]/si. Lee and Wen (2010) derive the density and the general moments of the distribution for any integer k ≥ 2. They also apply the model to a data set with three variables. All the properties in Result 5 extend readily to the CRM cdf, p_mdw(t) = 1 - exp {- [S^k_i =1 exp(qz_i)]^1/q }, z_i = [ln(t) - m_i ]/s_i, derived from this k > 2 dimensional Weibull distribution with dependent risks.

There is a large number of other Weibull multivariate models. Murthy, Xie, and Jiang (2004) provide a comprehensive collection of univariate and multivariate Weibull models and Crowder (1989) proposes an interesting multivariate Weibull model. There is not, however, assurance that the properties discussed here are satisfied by those models. For other models of interest, their properties need to be studied separately.

A natural inquire is if the results presented here extend to the CRM model derived from log-location-families. We have not studied this in detail, but the items in Result 2 do not extend in their full generality to a CRM derived from a log-location-scale family. This was surprising and unexpected to us. For example, suppose T_i ∼ LOGNOR(m_i , s_i), i = 1, 2 with T₁ and T₂ independent. As before T_m = min(T₁, T₂). Thus, F_m(t) = 1 - F_nor(-z₁) F_nor(-z₂) and the slope of the curve { x = ln t, y = F^-1 _nor[F_m(t)] } , see (4), in a lognormal probability plots is

where f_nor(z) and F_nor(z) are the standardized normal pdf and cdf, respectively. Figure 4 shows the slope of the curve when m₁ = 5, m₂ = 6, s₁ = 3, and s₂ = 2. Clearly,

for t > 236. In contrast to the Weibull case, the slope of the F_m(t) in the lognormal probability plot is not bounded by the s_is as in (9).

6. Conclusions and Future Work

The characteristics of the Weibull CRM with independent risks and the CRM derived from the bivariate Weibull with dependence are very similar. For a CRM with two risks, based on the information drawn from this work, we can say that important factors in deciding when the inference is robust to ignoring the failure mode include: (a) the relative sizes of the Weibull scale parameters; (b) the crossing point of the marginal distributions; and (c) the size of the dependence among the factors.

Similar conclusions apply to the models with k > 2 risks considered here. A single Weibull model might describe the CRMwell if: (a) the ratio between the largest and the smallest shape parameters is small and the dependence among the risks is not large; or (b) there is a dominant risk.

It is plausible that these continue to be important factors when studying the CRM for other distributions, but additional work is needed to corroborate these preliminary findings.

There is need to study other distributions beside the Weibull and the lognormal and to consider other situations with dependent risks. The brief look into the lognormal case indicates that caution should be exercised when trying to generalize the conclusions to other distributions because the results do not seem to be completely generalizable to other competing risks situations of interest like CRMs derived from general log-location-scale families.

Eventually, the model robustness problem will have to be addressed using meaningful and well formulated simulation studies with complete, dependent, and censored data. Toward that end, the results of studies like the one pursued here would be useful because a good understanding of the important factors in a model facilitate the test planning and the interpretation of a simulation study.

Acknowledgments

This work was supported by Colciencias, Colombia. This is part of the research project "Characterization of Dependence in Competing Risks Models in Industrial Reliability Data," code 110152128913. This project is directed by Sergio Yáñez, Luis Escobar, and Nelfi González.

7. Appendixes

A. Monotonicity of the CRM cdf p_mdw(t) as a function of q

From (15), s_mdw(t) = exp (-R^1/q), where R = exp(qz₁) +exp(qz₂), and z_i = (ln(t) - m_i)/s_i, i = 1, 2.

Consider

Note that ln(R) = ln[exp(qz₁) + exp(qz₂)] > ln[exp(qz_i)] = qz_i , for i = 1, 2. Thus for fix t, [j ln(R^1/q)/jq] < 0. This implies that R^1/q is monotone decreasing on q. Consequently, s_mdw(t) is increasing on q and p_mdw(t) = 1- s_mdw(t) is decreasing on q.

B. Important Properties of the CRM

This appendix provides details of the properties of the CRM derived from the bivariate Weibull defined in (13). The correspondent properties for the CRM with independent Weibull risks are obtained by setting the dependence parameter to q = 1.

B.1. Slope of the curve {ln t, F^-1_sev[p_mdw(t)]} in a Weibull probability plot

From the definition of fsev(z) and F^-1_sev(p) in (8) and using (15) and (16), we get that for 0 < p < 1,

Now we compute t d_mdw(t), where d_mdw(t) is the density corresponding to the cdf p_mdw(t) given in (16).

Direct differentiation of p_mdw(t) yields

Taking the ratio between (23) and (22) and using (1), we get

which is the result suggested in (17). Note that with q = 1, one obtains the independent risks case suggested in (7).

B.2. Concavity of the curve {ln t, F^-1_sev[p_mdw(t)]} in a Weibull probability plot

To show that the second derivative of y with respect to ln t is positive, we write

Then

For the CRM with independent Weibull risks, use q = 1. In this case the second partial derivative is still positive and takes the value suggested in (10).

C. Cdf for CRM with Weibull Dependent Risks and Similar or Equal Shape Parameters

where

When b_i → b > 0, i = 1, 2, it follows that s_i → s = 1/b > 0, i = 1, 2, and

Thus p_mdw(t) is the cdf of a WEI(h, b), where

Note that: (a) when b₁ = b₂ = b, p_mdw(t) is the cdf of a WEI(h, b) with h given by (24); (b) when b₁ = b₂ = b and q = 1, p_mdw(t) = p_miw(t) and they are the cdf of a WEI(h, b) with h given by (24) evaluated at q = 1. This is the CRM cdf with independent Weibull risks considered in Result 3.

D. Bivariate Density

The density is given by the mixed second partial derivative of S(t₁, t₂) with respect to t₁ and t₂. For simplicity write S = S(t₁, t₂) and define M = -ln(S) = [exp(qw₁) + exp(qw₂)]^1/q . Using w_i = [ln(t_i) - m_i]/s_i and taking the derivative of ln(S) with respect to t₂,

Equivalently,

Taking the derivative with respect to t₁

Direct computations yield

Substituting (26) into (25)

where for i = 1, 2

See Lu and Bhattacharyya (1990, page 554) for an equivalent expression for the density.

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