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\begin{document}
\begin{slide}
\thispagestyle{empty}

\sloppy



\begin{center}
Competing Risks - A Review
\vsp
Isha Dewan\\
Indian Statistical Institute\\
New Delhi
\end{center}
\end{slide}
\begin{slide}
The competing risks situation was first considered in the eighteenth century when small pox vaccination
 was being discovered and popularised. 
 
 D. Bernoulli (1761) posed the question : How much would the mortality be reduced or expected life be increased
  if the risk of death due to small pox is totally eliminated, the other risks persisting as before?

  The demographers , over the years, have taken up this question enthusastically.
  
   A strong motivation has come  from the insurance business in which different premium rates 
   have to be established for groups of persons exposed to different sorts of risks.
   
    Of course now it is relevant in several disciplines - medicine, reliability engineering , economics ,
     manufacturing and even the game of cricket (Crowder (2001)).
\end{slide}
\begin{slide}

Let $X_1,X_2, \ldots, X_k$ denote the latent failure times of individuals subject to $k$ risks,
 where $X_i$ represents the age at death if cause $i$ were the only cause of failure. 
 
 What is actually observed is the time to failure $T$, where $T= \min  (X_1, X_2, \ldots, X_k)$, 
 and the cause of failure $\delta$ where $\delta=j$ if $X_j = \min (X_1, X_2, \ldots, X_k)$.
 

The joint probability distribution of $(T, \delta)$ is specified by the set of $k$ sub-distribution functions 

$F(i,t) = P[T \le t, \delta = i],$

or equivalently by the subsurvival functions

$
S(i,t) = P[T > t, \delta = i], \; i=1,2, \ldots,k.$

Let $H(t)$ and $ S(t)$ denote , respectively the distribution function and the survivor function of $T$.
 \end{slide}
\begin{slide}

Let $f(i,t)$ denote the sub-density function corresponding to $ith$ risk.

The density function of $T$ is $h(t) = \sum_{i=1}^k f(i,t)$

 $p_i = F(i,\ity)$ is the probability of failure due to the $ith$ risk. 
 
 The cause specific hazard rate be given by $\lambda(i,t) = \frac{f(i,t)}{S(t)}$.

  Note that if $X_1, X_2, \ldots, X_k$ are independent and identically distributed random variables with
   a common distribution function $F(x)$, then it is uniquely determined by the distribution function
     of the minimum, that is,
$$ H(x) = P[T \le x] = 1 - [ 1-F(x)]^k .$$

When $X's$ are not identically distributed , let $F_i(x)$ denote the distribution of $X_i$.
\end{slide}
\begin{slide}
If $X's$ are independent, but not identically distributed , Berman (1963) showed that the joint distribution 
of $(T, \delta)$ uniquely determines 
$F_i$, the distribution function of $X_i$.
$$F_i(x) = 1- \exp [ 1 - \int_{-\ity}^x [ 1- \sum_{j=1}^k F(j,t)]^{-1} dF(i,t)], $$


 Keyfitz et al (1972) have strongly argued  that
 risks to human life can be grouped into four non-overlapping sets, which are independent of each other. 

 In this, and other such situations, one can use methods based on the apriori independence assumption 
 while dealing with pooled risks.

If the latent failure times are not independent, then the underlying distributions are not identifiable 
on the basis of $(T, \delta)\;$ ( see    Tsiatis (1975), Crowder (1991, 1993)).
 \end{slide}
\begin{slide}
{\bf Theorem (Tsiatis (1975))} Suppose that the set of sub-survival functions $S(j,t)$ is given for some model with dependent risks. Then there exists a unique proxy model with independent risks yielding identical $S(j,t)$. It is defined by 

$ \bar{G}^*(t) = \prod_{j=1}^k \bar{G}_j(t_j), $  where 

$ \bar{G}_j(t) = exp(-\int_0^t \lambda(j,s)ds) $.

The independence or otherwise of the latent lifetimes   
 $(X_1, X_2, \ldots, X_k)$ cannot be statistically tested from any data
  collected on $(T, \delta)$. 
  
  Also, the marginal distribution functions $F_i(x)$ may not represent the probability 
   distribution of lifetimes in any practical situation. 
   
   Elimination of $jth$ risk may change the environment in such a way that $F_i(x)$ does not 
   represent the lifetime of $X_i$ in the changed scenario. 
    \end{slide}
\begin{slide}

   In view of the above considerations, unless one can assume independence, it is necessary to suggest 
   appropriate models, develop methodology and carry out the data analysis in terms of the observable 
   random variables $(T, \delta)$ alone.
 
 For recent work on these lines see 
 Deshpande (1990), Aras and Deshpande (1992),Aly, Kochar and McKeague (1994),  Sun and Tiwari (1995),
  Deshpande and Dewan ( 2000) , Crowder (2001),
  Kalbfleisch and Prentice (2002), Dewan and Kulathinal (2004), 
  and  Dewan, Deshpande and Kulathinal (2004).



{\bf Locally Most Powerful Rank Tests}

Suppose $k=2$, that is , a unit is exposed to two risks of failure denoted by  $1$ and $0$.
 
When $n$ units are put to trial , the data consists of   $(T_i, \delta^*_i), i=1, \ldots,n$ where $\delta^*= 2-\delta $. 

Suppose we wish to test the hypothesis

  $H_0 : F(1,t) = F(2,t), \mbox {for all} \;\;t$. 

 \end{slide}
\begin{slide}

Under the null hypothesis the two risks are equally effective at all ages in the prevailing environment. 
However, the alternative hypothesis is that the two risks are not equally effective atleast at some ages.

 The likelihood function is given by (see , Aras and Deshpande (1992)) 
\begin{equation}
L( \underline{t}, \underline{\delta^*}) = \prod_{i=1}^n [f(1,t_i)]^{\delta^*_i} 
 [f(2,t_i)]^{ 1- \delta^*_i}
\end{equation}
where $\underline{t} = (T_1, \ldots, T_n), \underline{\delta^*} = (\delta^*_1, \ldots, \delta^*_n)$. 

If $F(i,t)$ depends upon the parameter $\theta$ , then inference about it can be based on 
the above likelihood function. 
When  $T$ and $\delta^*$ are independent then Deshpande (1990) proposed the 
model $F(1,t) = \theta H(t) , \;\; F(2,t) =  (1-\theta) H(t).$
 Here $
 \theta = P[\delta^* =1]$.

 Then the likelihood reduces to

$L (\theta,\underline{T}, \underline{\delta^*} ) = 
 \theta^{\sum_{i=1}^n \delta^* _i} (1-  \theta)^{\sum_{i=1}^n (1-\delta^* _i)}  \prod_{i=1}^n h(t_i).$

Testing of the
 hypothesis $F(1,t)= F(2,t)$ reduces to testing that  $\theta=1/2$. 
  \end{slide}
\begin{slide}

 The obvious statistic is the sign statistic 

$U_1 = \frac{1}{n} \sum_{i=1}^n \delta^*_i.$

 $n U_1$ has $B(n, \theta)$ distribution and there exist optimal estimation and
 testing procedures based on it. 
 
 However, if $F(1,t)$ and $F(2,t)$ depend on a parameter    $\theta$ in a more complicated manner then
  one needs to look at locally most powerful rank tests.

 Let $f(1,t) = f(t, \theta), f(2,t) = h(t) -  f(t, \theta)$ where $h(t)$ and 
$f(t, \theta)$ are known density functions and incidence density 
such that $f(t, \theta_0)= \frac{1}{2} h(t).$


\noindent Let $ T_{(1)} \le T_{(2)} \le \ldots \le T_{(n)}$ denote the ordered failure times. Let
\begin{eqnarray}
W_i &=& \left \{ \begin{array}{ll}
  1 & \mbox {if}\; T_{(i)}\; \mbox{corresponds to first risk,} \\
0 & \mbox{otherwise}. \end{array} \right .\nonumber
\end{eqnarray}

Let $R_j$ be the rank of $T_j$ among $T_1, \ldots, T_n$ . 
 \end{slide}
\begin{slide}

Let
$ \underline{R}^{\prime} = (R_1, R_2, \ldots, R_n), \;\; \underline{W}^{\prime} = (W_1, W_2, \ldots, W_n)$
denote the vector of ranks and indicator functions corresponding to ordered minima. 

The likelihood of $(\underline{R}, \underline{W})$ is given by
\bea
&&P(\theta, \underline{R}, \underline{W})\nonumber\\
& =& \int  _{0 <t_1 < \ldots < t_n < \infty} \prod_{i=1}^n [ f(t_i, \theta)  ]^{w_i}[ h(t_i) - f(t_i, \theta)  ]^{1- w_i} dt_i \nonumber
\eea



\begin{theorem} If $f^{\prime} (t, \theta)$ is the derivative of $f(t, \theta)$ with respect to $\theta$, then 
the locally most powerful rank test for 
$H_0: \theta= \theta_0$ against $H_1: \theta >  \theta_0$ is given by: reject $H_0$ for
 large values of

 $L_c = \sum_{i=1}^n w_i a_i$, where

$
a_i =  \int \ldots \int _{0 <t_1 < \ldots < t_n < \ity} \frac{f^{\prime} (t_i, \theta_0)}{ f(t_i, \theta_0)} \prod_{i=1}^n [ f(t_i, \theta_0) dt_i ].
$
\end{theorem}

\noindent(i) If Deshpande's model holds with $\theta_0 =1/2$, then sign test is the LMPR test.

 \end{slide}
\begin{slide}

\noindent(ii) If $f(1,t) = \frac{1}{2} g(t,0)$ and $f(2,t) = \frac{1}{2} g(t,\theta), \linebreak \theta > 0$
 and $g(t, \theta)$ is the logistic density function
 $ g(t, \theta) = \frac {e^{(x- \theta)}}{ [ 1 + e^{(x- \theta)}]^2}$, then the LMPR test is based on the statistic 

$W^+ = \sum_{i=1}^n W_i R_i$, 

which is the analogue 
of the Wilcoxon signed rank statistic for competing risks data.

\noindent(iii) In case of Lehmann type alternative \linebreak  defined by $F(1,t) =[ \frac{H(t)}{2}]^ {\theta}, \;\;F(2,t) = H(t) - [ \frac{H(t)}{2}]^ {\theta}, $ the  LMPR test is based on scores \linebreak  $a_i = E(E_{(j)})$ where $E_{(j)}$ is the $jth$ order statistic from a random sample of size $n$ from the standard exponential distribution.

But  for more complicated families of distributions, e.g., Gumbel (1960), the scores are complicated and need to be solved using numerical integration (see Aras and Deshpande (1992)).

 \end{slide}
\begin{slide}

 {\bf Tests for bivariate symmetry}

Suppose that the latent failure times $X$ and $Y$ are dependent with their joint 
distribution  given by $F(x,y)$.

 On the basis of $n$ independent pairs $(T_i, \delta^*_i)$ we want to test whether the  forces of two risks are equivalent against the alternative that the force of one risk is greater than that of the other.

 That is we test the null hypothesis of bivariate symmetry

$H_0 : F(x,y) = F(y,x) \;\mbox{ for every} \; (x,y).$

 {\bf Theorem 3.1:} Under the null hypothesis of bivariate symmetry we have

(i)$ F(1,t) = F(2,t) \;\; \mbox{for all} \;\; t,$

(ii)$ S(1,t) = S(2,t) \;\; \mbox{for all} \;\; t,$

(iii) $ \lambda(1,t) = \lambda(2,t) \;\; \mbox{for all} \;\; t,$
 \end{slide}
\begin{slide}

(iv) $ P [ \delta^*=1] = P[\delta^* =0]$

(v) $T \;\; \mbox{and} \;\; \delta^* $ are independent.

Consider the following alternatives

$H_1 :  \lambda(1,t) < \lambda(2,t);\;\; H_2:  F(1,t) < F(2,t;)$

$ \;\;H_3:S(1,t) > S(2,t) .$

All these alternatives say that risk II is more potent  than risk I at all ages $t$ in some stochastic sense.

Sen (1979) considered fixed sample and sequential tests for the null hypothesis of bivariate symmetry of the joint distribution of $(X,Y)$.

 The alternatives are expressed in terms of \linebreak $\pi_1(t) = Pr [ \delta^* =1| T=t]$, the conditional probability that the failure is due to first risk , given that failure occurs at time $t$.
  \end{slide}
\begin{slide}

He  derived optimal score statistics for such parametric situations. But the statistics cannot be used without the knowledge of the joint distribution $F(x,y)$.  

We look at various distribution-free test procedures for testing $H_0$ against above alternatives. 

For testing $H_0$ against $H_1$ consider

$\psi(t) = F(1,t) - F(2,t) = \int_0^t S(t) [ \lambda(1,u) - \lambda(2,u)] du. $

$H_1$ holds iff  the above function is non-increasing in $t$.

Consider the following measure of deviation between $H_0$ and
$H_1$,

$\Delta  =  \displaystyle {\int _{0 < x < y < \infty}}[\psi(x) - \psi(y)]\, dF(x)\, dF(y) ,$

and its empirical estimator $\Delta_n$ as the test statistic where $\Delta_n$ is given by

$\Delta_n  =  \int _{0 < x < y < \infty}[\psi _n(x) - \psi
_n(y)]\,  dF_n(x)\, dF_n(y),$
 \end{slide}
\begin{slide}

where $
F_{1n}(t)= \frac {1}{n} \sum ^n_{j=1} I \{ \delta_j = 1 , T_j \le t \},\;\;$

$F_n(t) = \frac {1}{n} \sum ^n_{j=1} I \{ T_j \le t \},
\;$ 

and
$ \; \psi _n(t)  =  2F_{1n}(t) - F_n(t)$

are the empirical estimators of $F_1, F$ and $\psi$, respectively. 
Then 

$\Delta _n =  \frac{1}{n^3} [\frac{n(n^2 - 1)}{6} - 2 \sum ^n_{i=1} (i-1)(n -
i + 1 ) W_i].$

Under $H_0 , \;  \Delta = 0$  but under the alternative 
$\Delta > 0$. 

Large values of the statistic are significant.

Rejecting $H_0$ for large values of $\Delta _n$ is equivalent to
rejecting it for small values of the statistic 

$U_2 = \sum ^{n-1}_{i=1} i(n-i) W_{i+1}.$


The exact null distribution of $U_2$ can be obtained
following the approach of Hettmansperger (1984 pp 35). 
 \end{slide}
\begin{slide}

For $n = 5 (1) 20$, the
1\% and 5\% critical values of $U_2$ are given in second and fourth columns of Table 2.1 and corresponding  exact significance levels in third and fifth columns respectively.
\vskip 1em
\centerline {TABLE 2.1}
\vskip 1em
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
n & $\alpha \approx $ 0.01 &  $\alpha \approx $ 0.01 &$\alpha \approx $ 0.05 &$\alpha \approx $ 0.05\\ \hline
5 & -&- & 0& 0.062500 \\
6 & - &-& 0& 0.031250 \\
7 & 0 & 0.015625 & 6 & 0.046875 \\
8 & 7 & 0.015625 & 14 & 0.046875 \\
9 & 8 & 0.011719 & 22 & 0.054688 \\
10 & 18 & 0.011719 & 34 & 0.054688 \\
11 & 28 & 0.013672 & 50 & 0.050781 \\
12 & 38 & 0.010254 & 67 & 0.051758 \\
13 & 54 & 0.010498 & 90 & 0.051514 \\
14 & 74 & 0.010010 & 118 & 0.052734 \\
15 & 98 & 0.010132 & 148 & 0.050293 \\
16 & 126 & 0.010010 & 186 & 0.051300 \\
17 & 158 & 0.010010 & 228 & 0.050308 \\
18 & 198 & 0.010269 & 278 & 0.050613 \\
19 & 242 & 0.010311 & 332 & 0.049911 \\
20 & 290 & 0.010050 & 396 & 0.050467 \\ \hline
\end{tabular}
\end{center}
\vskip 1em
Under $H_0, \;
E[U_2] = \frac {n(n^2-1)}{12}
, \;\;Var[U_2] = \frac {n(n^4-1)}{120}.
$
 \end{slide}
\begin{slide}

By the Central Limit Theorem it can be shown that, under $H_0$,
$
n^{\frac{1}{2}} \{ \frac{U_2}{n^3} - \frac{1}{12} \}\stackrel {{\cal L}} {\rightarrow}
 N(0, \frac{1}{120})$. 

After change of order of  integration $\Delta$ can also be expressed as 

$\Delta = \int^\infty_0 S^2(x) F(x) [\lambda_2(x) - \lambda_1(x)]\, dx.$

It is seen from this representation that the test based on 
$U_2$ is equivalent to the test V of Yip and Lam (1992) proposed for
the case of independent risks without censoring. They have not discussed its small sample 
exact null distribution.

Deshpande (1990)  proposed two tests for testing $H_0$ versus $H_2$
 on heuristic grounds. 
The first test is the Wilcoxon signed rank type statistic 

$W^+ = \sum_{i=1}^n(1- \delta^*_i) R_i.$


It is argued that $W^+$ will be large when the alternative $H_2$ is true, there being a greater incidence of the second risk upto any fixed time $t$.
 \end{slide}
\begin{slide}

This consideration leads to another test based on the U-statistic
\bea
U_3 &=& \frac{1}{{ n \choose 2}} \sum_{1 \le i < j \le n} \phi_3 (T_i, \delta^* _i, T_j, \delta^*_j) \nonumber \\
&=& \frac{1}{{ n \choose 2}}\sum_{i=1}^n (n- R_i + 1) \delta^*_i,\nonumber 
\eea
where $\phi_3$ is given by
\begin{eqnarray}
\phi_3(T_i, \delta^*_i, T_j, \delta^*_j) &=& \left \{ \begin{array}{ll}
1 & \mbox{if} \; \delta^*_j =0, T_i > T_j ,  \\
& \mbox{or } \; \delta^*_i =0, T_i <  T_j ,  \\
0 & \mbox{otherwise}.\end{array} \right. \nonumber 
\end{eqnarray}
 $E(U_3)= 1/2$ under $H_0$ and strictly larger than $1/2$ under $H_2$.

$U_3$ is the same as the statistic proposed earlier to test for $H_0$ against the alternative $H_{A1}$ of stochastic dominance of distribution functions of independent latent failure times  (see, Deshpande and Dewan (2005)).  It is also consistent for testing bivariate symmetry against dominance of incidence functions. 
 \end{slide}
\begin{slide}

For testing $H_0$ against $H_2$, one can  consider the measure of deviation

 $F(2,t) - F(1,t)$,

 which is non-negative under $H_2$. Then

$\int_0^{\infty}[ F(2,t) - F(1,t)] dH(t)$

$ = P[\delta^*_1 =0, T_1 \le T_2] -\frac{1}{2}, $

and its   U-statistic estimator  is the statistic $U_3$ discussed above.


Similarly for testing $H_0$ against $H_3$ consider the measure of deviation $S(1,t) - S(2,t)$, which is non-negative under $H_3$. Then

$\int_0^{\infty}[ S(1,t) - S(2,t)] dH(t) = P[\delta^*_1 =1, T_1 > T_2] -\frac{1}{2}.$
 \end{slide}
\begin{slide}

Consider the kernel 
\begin{eqnarray}
\phi_4 (T_i, \delta_i, T_j, \delta_j) &=& \left \{ \begin{array}{ll}
1 & \mbox{if} \; \delta^*_i =1, T_i > T_j ,  \\
& \mbox{or } \; \delta^*_j =1, T_i <  T_j  , \\
0 ,& \mbox{otherwise},\end{array} \right. \nonumber
\end{eqnarray}
and  the corresponding U-statistic is given by
\bea
U_4 &=& \frac{1}{{ n \choose 2}} \sum_{1 \le i < j \le n} \phi_4 (T_i, \delta_i, T_j, \delta_j)\nonumber\\ 
&=& \frac{1}{{ n \choose 2}}\sum_{i=1}^n ( R_i - 1) \delta^*_i.\nonumber
\eea
This statistic was earlier proposed by Bagai et al (1989) to test for equality of failure rates of independent latent competing risks.

 Aly et al (1994) proposed Kolmogrov-Smirnov type tests for testing the equality of two competing risks against the alternatives $H_1$ and $H_2$. Here we discuss their approach.
Consider 
\bea
\psi^*(t) &=&  F(2,t)- F(1,t) \nonumber \\
&=& \int_0^t  S(u) ( \lambda (2,u) -\lambda(1,u) ) du.\nonumber 
\eea

 Under $H_0 , \; \psi^*(t)   = 0$. $H_1$ holds iff $\psi^*(t)$ is nondecreasing in $t$ . 
 \end{slide}
\begin{slide}

 Let $\psi_n^*(t)$ be its empirical estimator. Consider

$D_{1n} = \sup_{0 \le s < t < \infty} \{ \psi^*_n(t) - \psi^*_n(s)\}.$

Large values of the statistic are significant.


If  one is interested in a general two sided alternative $ F(1,t) \ne F(2,t) $ for some $t$ or equivalently $ \lambda_1(t) \ne \lambda_2(t) $ for some $t$ , then one can use the Kolmogrov-Smirnov type statistic 

$D_n = \sup_{t \ge 0} |\psi^*_n(t)|.$

Under $H_0, \sqrt{n} D_n $ converges in distribution to $\sup_{0 \le x \le 1} |W(x)|$.  This test is consistent against arbitrary departures from $H_0$.

For testing $H_0$ against $H_2$, Aly , Kochar and McKeague (1994) proposed the statistic 

$D_{2n} = \sup_{0 \le t \infty} \psi_n(t).$

Large values of $D_{2n}$ are significant for testing $H_0$ against $H_2$.

For exact null and asymptotic distributions of these statistics see Aly et al (1994).
 \end{slide}
\begin{slide}

{\bf Censored data}

Most of
the above tests can be generalized to the case when the data is right censored. 

Let $C$ be the censoring random variable independent of the latent
failure times $X$ and $Y$.

Denote the survival function of $C$ by
$ {S}_c$ and assume that $ {S}_C(t) > 0$  for all $t.$

Now the available information consists of $\linebreak (\tilde {T}_i, \tilde {\delta}_i),
\, i=1,2,\cdots ,n$ where 

$\tilde {T} = \min (T ,C)$ and $\tilde {\delta} = \delta^*
I(T\le C).$ 

Aly et al (1994) generalised the function $\psi^*$ so as to capture departures of $H_0$ from $H_2$ in case of censored data.
Let

$\phi(t) = \int^t_0 S(u-) ({S}_C(u-))^{\frac{1}{2}}(\lambda(2,u)
- \lambda(1,u)) du,$

which is the $\psi^*$ function when there is no censoring.
 \end{slide}
\begin{slide}

The integrand $S_C(u-)^{1/2}$ is the function required to compensate for censoring in order that the $D$ statistics remain \linebreak asymptotically distribution-free. 

Under $H_0, \phi(t) = 0$.

 $H_1$ holds iff $\phi(t)$ is increasing in $t$.

An obvious choice of $\phi _n$ is 

$\phi _n(t) = \int ^t_0 \hat {S}(u-) (\hat { {S}}_C(u-))^{1/2}
\,d(\hat \Lambda _1 - \hat \Lambda _2)(u),$

where $\hat {S}$ and $\hat { {S}}_C$ are the product limit
estimators of $\bar {S}$ and $ {S}_C$, and $\hat \Lambda_j$ is the
Aalen estimator of the cumulative CSHR function

 $\Lambda _j(t) = \int
^t_0 \lambda_j(u)\, du.$

A suitable modification of $U_2$  to censored data is  given by 
the statistic 

$K_n = \int _{0<x<y<\infty } (\phi_ n(x) - \phi_ n(y)) \, d \hat {S}(x) \,d \hat {S}(y),$

where $\phi_n$ and $\hat{S}$ are as defined above.
$K_n$ can be expressed as 
 \end{slide}
\begin{slide}

$K_n = \sum _{i|\tilde {\delta}_i \ne 0} \phi _n(i) (2\hat { S}(\tilde {T}_{(i-1)
}) - 1) (\hat {S}(\tilde {T}_{(i-1)}) \frac {1}{n-i+1}.$
 

\noindent Large values of $K_n$ are significant for testing $H_0$ against $H_1$.

Aly et al (1994) showed that, under $H_0$,

$
n^{1/2} \phi _n  \stackrel {{\cal L}} {\rightarrow} W(S(.))$
where W(.) is the standard Brownian Motion.

Using the continuous mapping theorem, we have 
$n^{1/2} K_n   \stackrel {{\cal L}}{\rightarrow}  N (0 , \frac{1}{30}) .
$
Sun and Tiwari (1995) modified  the statistic $U_3$  so that it can be used for censored data. Consider 
\bea
V &=& \int_0^{\infty}(F(2,t) - F(1,t)) dH(t) \nonumber \\
&=& \int_0^{\infty} [S(t)]^2 d (\Lambda_2(t) - \Lambda_1(t)). \nonumber
\eea
 A natural estimator of $V$ is given by $V_n$ where
$$ V_n =  \int_0^{\infty} [\hat{S}(t-)]^2 d (\hat{\Lambda}_2(t) - \hat{\Lambda}_1(t)),$$
where $\hat{S}(t)$  and $\hat{\Lambda_j}$ are as defined above.

In the absence of censoring $V_n$ reduces to the statistic $U_3$.
 \end{slide}
\begin{slide}

 Sun and Tiwari (1995) discussed the asymptotic distribution of $V_n.$

Aly et al (1994) extended $D_{1n}$ and $D_{2n}$ to included censored data.

{\bf SIMULATION RESULTS}

Given below are the results of  a simulation study done 
for power comparisons of various tests for the uncensored case listed above.

Random samples were generated from absolutely continuous bivariate
exponential (ACBVE) due to Block and Basu (1974) with density
\begin{eqnarray}
f(x,y) & = & \left \{ \begin{array}{lll}
\frac {\lambda \lambda _1 (\lambda _2 + \lambda _0)}{\lambda _1 +
\lambda _2} e^{-\lambda _1x - (\lambda _2 + \lambda _0)y} & \mbox
{if} & x < y , \\
\frac {\lambda \lambda _2 (\lambda _1 + \lambda _0)}{\lambda _1 +
\lambda _2} e^{-\lambda _2y - (\lambda _1 + \lambda _0)x} & \mbox
{if} & x > y ,  \end{array} \right .\nonumber
\end{eqnarray}
where $(\lambda _0 , \lambda _1 , \lambda _2)$ are the parameters and
 $\lambda = \lambda _0 + \lambda _1 + \lambda _2$. 

The CSHR's are
$ \lambda_j(t) = \frac {\lambda _j \lambda}{\lambda _1 + \lambda _2} ,
\quad j = 1,2.$

Under $H_1, \; \lambda _1 < \lambda _2$.

$X$ and $Y$ are independent if and only if $\lambda _0 = 0$.

 We set
$\lambda _1 = 1$ and consider $\lambda _2 = 1.0,1.4 , 1.8 , 2.2 $ indicating
larger and larger departures from $H_0$. 
 \end{slide}
\begin{slide}

The case $\lambda_2=1.0$ corresponds to the null hypothesis. $ n=100$ and there are 10000 replications.
\vskip 1em
\centerline {TABLE 3.1 - UNCENSORED CASE }
\begin{center}
\begin{tabular}
{|c|c|c|c|c|}\hline
 & \multicolumn {4}{c|} {$\lambda_2$}\\\hline
Test & 1.0 & 1.4 & 1.8 & 2.2 \\\hline
$D_1$ & 3.76 & 41.98 & 82.53 & 96.83 \\
$D_2$ & 4.85 & 47.71 & 86.98 & 98.14 \\
$U_2$ & 5.09 & 44.60 & 83.96 & 96.92 \\
$U_3$ & 4.79 & 43.06 & 80.54 & 95.42 \\
$U_4$ & 4.99 & 42.67 & 80.81 & 95.77 \\
$Sign$ & 4.39 & 49.50 & 88.29 & 98.66 \\
\hline
\end{tabular}
\end{center}


Next we look at the censored case. The censoring distribution
was exponential with parameters $1$ and $3$, respectively. We use
asymptotic critical levels of $5$ percent. Results are based on
$5,000$ replications.

 \end{slide}
\begin{slide}

\centerline {TABLE 3.2}

Observed levels and powers of  $K_n$ at an asymptotic level
of 5 percent. The underlying distribution of $( X,Y )$ is ACBVE with $\lambda _1 = 1$.

\vskip 1em
\noindent (a) CENSORED (EXP(1))
\vskip 1em
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
 &\multicolumn {2}{c|}{n=50} & \multicolumn {2}{c|}{n=100} \\ \hline
$\lambda _2$ & $\lambda _0=0$ &  $\lambda _0=1$ & $\lambda _0=0$ &
 $\lambda _0=1$ \\ \hline
1.0 & .0218 & .0312 & .0360 & .0376 \\
1.5 & .1864 & .2192 & .3732 & .4302 \\
2.0 & .4482 & .5080 & .7862 & .8342 \\
2.5 & .6928 & .7414 & .9546 & .9704 \\ \hline
\end{tabular}
\end{center}

\vskip 1em
\noindent (b) CENSORED (EXP(3))
\vskip 1em
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
 &\multicolumn {2}{c|}{n=50} & \multicolumn {2}{c|}{n=100} \\ \hline
$\lambda _2$ & $\lambda _0=0$ &  $\lambda _0=1$ & $\lambda _0=0$ &
 $\lambda _0=1$ \\ \hline
1.0 & .0048 & .0124 & .0084 & .0172 \\
1.5 & .0342 & .0762 & .1012 & .1834 \\
2.0 & .1202 & .1986 & .3216 & .4860 \\
2.5 & .2496 & .3774 & .5862 & .7344 \\ \hline
\end{tabular}
\end{center}
 \end{slide}
\begin{slide}

From the table it is clear that the asymptotic critical levels give
conservative tests for the censored case, with the effect increasing
as the censoring becomes more severe. 

There is slight effect on the
levels or the power due to lack of independence of $X$ and $Y$ in the presence of censoring. 

The
results are comparable with the results for test proposed by Aly et al (1994) for
the lightly censored case.

{\bf Remarks}

(i) The various tests are consistent against their intended alternatives.

(ii We can also use these  tests for the hypothesis $\lambda_1(t) = \lambda_2(t)$ against the alternative that cause-specific hazards are ordered.

(iii) The tests are distribution-free under $H_0$. The null distribution of the tests $U_3$ and $U_4$ is same as in the case of independent latent failures (see, Deshpande and Dewan (2003)).
 \end{slide}
\begin{slide}

(iv) It is important to note that  $T$ and $\delta$ continue to be independent under the null hypothesis of bivariate symmetry . 

 (v) The statistic $U_2$ puts more weight on the middle observations and is less sensitive to the observations in the beginning and the end of the experiment. 

On the other hand, $U_3$ puts more weight to later observations .

$U_4$ puts higher weight to observations in the beginning. 


(vi) Deshpande and Dewan (2000) proposed tests for testing bivariate symmetry aginst  dispersive asymmetry. Here the alternatives can be expressed in terms of ordering of sub-survival functions and ordering of sub-distributions of the maximum of observations and $\delta$ . 

The statistic is a linear combination of two statistics, the first one is a U-statistic based on minimum and $\delta$
and the other one is a U-statistic based on maximum and $\delta$. 

The  one  based on minimum and $\delta$ is the statistic $U_4$

(vii) The statistics $U_2, U_3, U_4$ are all linear combinations of  
the sign statistic and the Wilcoxon-signed rank type statistic.

(viii) Tests proposed by Aly et al (1994) can be extended to the case of multiple risks in which any two of the cause-specific risks are to be compared. The statistic can be modified to test dominance of one risk over the other in a specified interval.



 It is observed that many of the statistics 
which have been originally proposed in the context of independent risks 
continue to be useful for testing similar hypotheses regarding dependent 
risks. This raises the interesting possibility that in general such rank 
based procedures work whether the risks are independent or not.
 \end{slide}
\begin{slide}

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\end{slide}

\end{document}