Chapter 4 Testing quasi-independence
4.4 Modification for Right Censoring
4.4.3 Asymptotic Analysis under Censoring
≤
•
•
⎭⎬
⎫
⎩⎨
⎧ −
−
=
y
x R x y
dy x N y dx dy N
dx N y x v
L ( , )
) , ( ) , ) (
, ( ) ,
ˆ( ρ 11 1 1
ρ , (4.16)
where vˆ(x,y) (1/n) I(X x,Z y)/SˆC(y)
j
j
∑
j ≤ >= and SˆC(y) is the Lynden-Bell’s
estimator for Pr(C>y)=SC(y) based on data {(Xi,Zi,1−δi) )}(i=1,...,n . Note that the weight vˆ(x,y)ρ mimics π(x,y)ρ by applying the idea of inverse probability of censoring weighting .
For the weight choice W(x,y)=R(x,y)/n, the expression in the discordance form becomes
∑
<⎟⎠
⎜ ⎞
⎝
⎛Δ −
=
j i
ij ij n
R I B
L n
2 } 1
2 {
/ , (4.17)
which is equivalent to the modified statistics proposed by Tsai (1990).
4.4.2 The Conditional Score Test under Censoring
Under model assumption (i) and (ii), the conditional score function has the same form as (4.9), where the cell and marginal counts are redefined for the censored case. For a semi-survival AC model in (4.11), the model assumption (i) holds for
) ( / ) ( )
(η ηφ η φ η
θα =− α′′ α′ , and the nuisance parameter becomes η(x,y)=c*ν(x,y), where )
* Pr(
Z X
c = ≤ and
) ( / )
| ,
Pr(
) ,
(x y X x Z y X Z S y
v = ≤ > ≤ C .
The nuisance parameters can be estimated by
∏ ∑
< ⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧ =
−
=
Xj
X
j j j
k k j
X X R
X X I X
X R c n
) 1
: (
) 1 ( ) 1 (
*
) , (
) 1 (
) ,
ˆ ( (4.18)
where j
j X
X(1) =min , and vˆ(x,y−)=R(x,y)/SˆC(y).
4.4.3 Asymptotic Analysis under Censoring
Now we discuss the asymptotic normality of the classes
∫∫
As we could see in the proofs of Theorem 4/1 and 4.2, the formulas under censoring are very complicated. Hence here we describe a brief sketch of proving the asymptotic normality under
H0. For the empirical process
it can be shown in Appendix A (part III) that
) each defined in Appendix A (part V). The asymptotic normality follows from the functional delta method that is applied based on the fact that both Lw and L*w are Hadamard
differentiable function of Hˆ and that the standardized process n1/2(Hˆ −H) converges weakly to some Gaussian process.
Similar to the uncensored case, the consistency of jackknife estimator can be proven by checking the continuous Gateaux differentiability of the functional expression. The proof follows the same lines as that for Theorem 4.3 and is omitted.
4.5. Numerical Analysis
The analysis has several objectives. First we want to choose a better variance estimator via simulations. Then we will study the size and power of the proposed tests. In particular, we want to confirm whether our conjecture that the score statistics leads a more powerful test when the dependence structure under the alternative hypothesis is specified. The rejection rule is determined based on the normal approximation using the Jackknife variance estimator in the standardization.
4.5.1 Comparison of two Variance Estimators
We generated truncated data (X,Y) which follow exponential distributions with hazards λX =1 and λY =1. Total 500 replications of samples n =50,100and 200 were examined for comparing the analytic and Jackknife estimators for variance estimation. The true variances were approximated by the sample variance of 30,000 separate Monte Carlo replications. To obtain the size of the tests, we compute the empirical proportion of rejection based on the standard normal approximation.
Table 4.1: Comparison of Two Variance Estimators
Recall that based on the asymptotic mean-zero linear expression of the test statistic, we can derive an analytic estimator for the variance using the ideas of method of moment and the
Average of V /ˆ n Size ρ n Var(n−1/2Lρ)
Analytic Jackknife Analytic Jackknife
0 50 0.759 0.613 0.840 0.088 0.060
0 100 0.843 0.744 0.913 0.076 0.062
0 200 0.906 0.829 0.946 0.070 0.058
1 50 0.0469 0.0512 0.0523 0.048 0.046
1 100 0.0466 0.0476 0.0481 0.044 0.040
1 200 0.0458 0.0460 0.0463 0.060 0.060
plug-in approach. However based in Table 4.1, this complicated formula slightly underestimates the true variance and hence inflate the type I error rates for small sample sizes.
It improves as the number of sample size increase. The jackknife method has much smaller bias and the empirical sizes are satisfactory in all the sample sizes considered here.
4.5.2 Size of the Proposed Tests
The main purpose here is to examine the size of the proposed tests, namely Lρ=0, Lρ=1
and Linvlog, under the null hypothesis of quasi-independence. The nominal level is set to be 05
.
=0
α . Note that the variance of each test statistic was estimated using the Jackknife method. We consider three sample sizes with n=50, 100 and 200. For each sample size, we evaluate four configurations of (λX,λY,λC) . Specifically we set (λX,λY,λC)=(1,1,0), (1,0.5,0), (0.5,1,0), (1.5,1,0.5), which yields c=Pr(X ≤Y) = 0.5, 0.667, 0.333 and
) Pr(
* X Z
c = ≤ = 0.5 respectively. The rejection rule is determined by whether the standardized statistic falls outside the 95% confidence interval based on the standard normal distribution.
Table 4.2 presents summary of the results including the means of the Jackknife variance estimator (Ave(Vˆ/n)), the true variance (the number in the parenthesis) and the size of the test. The Jackknife variance estimates slightly overestimate the true variance. Note that this kind of positive bias may be common for using the jackknife method, which has been explained by Theorem 4.1 of Efron (1982). The rejection rates of the three tests are close to the nominal 5% level.
Table 4.2. Empirical Size of the Proposed Tests (based on 500 runs) at nominal level α =0.05 under different truncation proportions
=0
Lρ Lρ=1 Linvlog
n c/c*
) ˆ/ ( AveV n
(True)
Size Ave(Vˆ/n) (True)
Size Ave(Vˆ/n) (True)
Size
50 c=0.5 0.864
(0.752) 0.070 0.0517
(0.0499) 0.074 0.195
(0.179) 0.066 50 c=0.67 0.853
(0.749) 0.050 0.0646
(0.0606) 0.054 0.314
(0.281) 0.034 50 c=0.33 0.830
(0.733) 0.058 0.0433
(0.0396) 0.056 0.134
(0.120) 0.036 100 c=0.5 0.894
(0.843) 0.062 0.0483
(0.0466) 0.048 0.178
(0.174) 0.040 100 c=0.67 0.912
(0.851) 0.042 0.0609
(0.0597) 0.044 0.286
(0.276) 0.036 100 c=0.33 0.915
(0.822) 0.056 0.0388
(0.0374) 0.056 0.1225
(0.115) 0.044 100 c* =0.5 0.614
(0.549) 0.044 0.0666
(0.0625) 0.050 0.185
(0.1709) 0.044 200 c=0.5 0.949
(0.906) 0.054 0.0459
(0.0455) 0.058 0.175
(0.172) 0.048 200 c=0.67 0.963
(0.899) 0.042 0.0592
(0.0580) 0.062 0.280
(0.269) 0.048 200 c=0.33 0.946
(0.892) 0.048 0.0374
(0.0368) 0.044 0.119
(0.114) 0.044 200 c* =0.5 0.647
(0.592) 0.052 0.0634
(0.0610) 0.060 0.177
(0.168) 0.054
4.5.3 Empirical Power of the Tests
To examine the power of the proposed weighted Log-rank statistics, we generate (X,Y) from three semi-survival AC models, namely the Clayton, Frank and Gumbel models. Then we apply the conditional score tests, Lρ=0, Lρ=1 and Linvlog to the above all the three settings respectively. All the marginal distributions are exponentially distributed. Marginal hazards are fixed to be (λX,λY,λC) =(1,1,0) and (1.5,1,0.5) which yield c=0.5 and
5 . 0
*=
c respectively. Tables 4.3 and 4,4 show the empirical powers of the three tests based on 500 replications. Also two sample sizes n=100 and 200 are evaluated. The power functions are also depicted in Figures 4.2 and 4.3.
The tests based on Lρ=0 and Lρ=1 are uniformly more powerful under correct specification of Clayton and Frank model respectively. It indicates that the weight choice based on the score test yields high efficiency when the model assumption (I) is correctly specified. The large discrepancy between the powers of Lρ=0 and Lρ=1 can be explained by the obvious difference in the suggested weight functions for the Clayton and Frank models.
Note that, under the Frank model, the performance in presence of censoring is deceptively better since we changed the parameter values for the marginal distributions are different.
Table 5.5 shows the empirical powers under the semi-survival Gumbel model which only permits negative association. Five five selected levels of association are examined. In contrast to the Clayton and Frank models, the discrepancy amongr the power curves becomes less clear for n=100. Nevertheless Linvlog still performs slightly better than the other two tests for n=200. To explain why the power improvement is less obvious for the Gumbel case, we suspect that the problem is caused by the estimation of the nuisance parameter
) , ( ) ,
(x y cπ x y
η = which is used in w(η(x,y))=1/log(cπ(x,y)). The extra variation due to cˆ and πˆ(x,y−) may bring extra variation especially for n=100 which offset the correct
choice of the weighting form. In other simulations not provided here, we have seen that the test based on Linvlog clearly dominant the other two tests for both sample size n=100 and 200 when the true weight function 1/log(cπ(x,y)) is used.
Table 4.3: Empirical Power of L*ρ=0, Lρ=1 and Linvlog at level α =0.05 for the Clayton model (based on 500 runs)
* denotes the score test when the alternative is correctly specified.
n=100 n=200
5 .
=0
c ,cen%=0 c* =0.5,cen%=33 c=0.5,cen%=0 c* =0.5,cen%=33 Tau
*
=0
Lρ Lρ=1 Linvlog L*ρ=0 Lρ=1 Linvlog L*ρ=0 Lρ=1 Linvlog L*ρ=0 Lρ=1 Linvlog
-0.25 0.976 0.944 0.962 0.930 0.844 0.858 1.000 0.996 0.998 0.998 0.990 0.994 -0.20 0.910 0.782 0.844 0.810 0.676 0.706 0.996 0.988 0.994 0.966 0.916 0.940 -0.15 0.710 0.560 0.630 0.562 0.450 0.472 0.942 0.876 0.916 0.838 0.736 0.790 -0.10 0.414 0.276 0.344 0.312 0.210 0.224 0.676 0.504 0.602 0.566 0.430 0.470 -0.05 0.176 0.116 0.118 0.140 0.098 0.096 0.284 0.184 0.212 0.218 0.154 0.172 0.05 0.176 0.146 0.136 0.128 0.108 0.098 0.290 0.226 0.252 0.192 0.160 0.168 0.10 0.484 0.348 0.398 0.344 0.260 0.266 0.860 0.654 0.804 0.656 0.520 0.600 0.15 0.904 0.696 0.822 0.702 0.516 0.564 0.998 0.968 0.998 0.980 0.872 0.946 0.20 0.998 0.910 0.976 0.938 0.768 0.850 1.000 0.996 1.000 1.000 0.972 0.998 0.25 1.000 0.982 1.000 0.944 0.944 0.984 1.000 1.000 1.000 1.000 0.994 0.996
-0.2 -0.1 0.0 0.1 0.2
0.00.20.40.60.81.0
n=100
tau
power
rho=0 rho=1 invlog
-0.2 -0.1 0.0 0.1 0.2
0.00.20.40.60.81.0
n=200
tau
power
rho=0 rho=1 invlog
Figure 4.2: Power function under Clayton alternative
Table 4.4: Empirical Power of Lρ=0, L*ρ=1 and Linvlog at level α =0.05 for the Frank model (based on 500 runs)
* denotes the score test when the alternative is correctly specified.
-0.4 -0.2 0.0 0.2 0.4
0.00.20.40.60.81.0
n=100
tau
power
rho=0 rho=1 invlog
n=100 n=200
5 .
=0
c ,cen%=0 c* =0.5,cen%=33 c=0.5,cen%=0 c* =0.5,cen%=33 Tau
=0
Lρ L*ρ=1 Linvlog Lρ=0 L*ρ=1 Linvlog Lρ=0 L*ρ=1 Linvlog Lρ=0 L*ρ=1 Linvlog
-0.5 0.912 0.990 0.980 0.944 0.976 0.972 0.998 1.000 1.000 0.998 1.000 1.000 -0.4 0.670 0.808 0.788 0.700 0.818 0.790 0.904 0.988 0.982 0.942 0.982 0.982 -0.3 0.390 0.466 0.462 0.458 0.468 0.460 0.608 0.770 0.754 0.680 0.806 0.798 -0.2 0.184 0.192 0.180 0.204 0.212 0.204 0.296 0.376 0.336 0.328 0.416 0.396 -0.1 0.074 0.086 0.072 0.090 0.090 0.076 0.094 0.118 0.082 0.096 0.136 0.122 0.2 0.092 0.098 0.084 0.082 0.092 0.068 0.080 0.090 0.082 0.080 0.104 0.088 0.10 0.090 0.124 0.084 0.140 0.164 0.138 0.156 0.198 0.156 0.206 0.292 0.246 0.3 0.172 0.196 0.166 0.204 0.288 0.206 0.256 0.330 0.272 0.396 0.536 0.464 0.4 0.184 0.244 0.180 0.338 0.428 0.348 0.336 0.418 0.354 0.614 0.750 0.668 0.5 0.216 0.274 0.206 0.494 0.576 0.492 0.416 0.528 0.454 0.774 0.900 0.800
-0.4 -0.2 0.0 0.2 0.4
0.00.20.40.60.81.0
n=200
tau
power
rho=0 rho=1 invlog
Figure 4.3: Power function under Frank alternative
Table 4.5 Empirical powers of Lρ=0, Lρ=1 and Linvlog at level α =0.05 for the Gumbel model ( based on 500 runs).
* denotes the conditional score test under the alternative.
n=100 n=200
5 .
=0
c ,cen%=0 c* =0.5,cen%=33 c=0.5,cen%=0 c* =0.5,cen%=33 Tau
=0
Lρ Lρ=1 L*nvlog Lρ=0 L*ρ=1 Linvlog Lρ=0 L*ρ=1 L*nvlog Lρ=0 L*nvlog L*nvlog -0.5 0.930 0.928 0.934 0.908 0.900 0.914 0.998 0.996 1.000 0.994 0.988 0.998 -0.4 0.692 0.698 0.708 0.656 0.668 0.644 0.932 0.940 0.946 0.894 0.936 0.940 -0.3 0.352 0.336 0.334 0.346 0.320 0.318 0.630 0.658 0.671 0.612 0.612 0.620 -0.2 0.182 0.160 0.162 0.144 0.158 0.150 0.290 0.282 0.306 0.296 0.300 0.302 -0.1 0.066 0.060 0.060 0.098 0.078 0.076 0.100 0.084 0.096 0.102 0.080 0.088
-0.5 -0.4 -0.3 -0.2 -0.1 0.0
0.00.20.40.60.81.0
n=100
tau
power
rho=0 rho=1 invlog
-0.5 -0.4 -0.3 -0.2 -0.1 0.0
0.00.20.40.60.81.0
n=200
tau
power
rho=0 rho=1 invlog
Figure 4.4: Power function under Gumbel alternative
4.6. Data Analysis
We applied the proposed tests to the contaminated blood transfusion AIDS dataset provided in Lagakos et al. (1988). The variables included the infection times T measured from April 1, 1978, and the induction period X measured from their infection times. The sample contained 258 adults and 37 children. Only those who developed AIDS within the 8 years study period can be included in the sample, and thus X + T ≤8 is the truncation criteria. We set the new variable Y = 8−T so that the pair (X,Y) is observed subject to
Y
X ≤ . Lagakos et al. (1988) applied the product-limit estimator for the survival function of X under the quasi-independence of (X,Y) for adults and children groups separately. Now we examine the validity of this assumption.
Applying the proposed Log-rank tests for the adult group, we found that the Z-values of the test statistics Lρ=0, Lρ=1 and Linvlog, standardized by the jackknife estimators, were -5.012, -2.918 and -3.795 respectively. The negative sign of the Z-values indicates the positive association for (X,Y). The corresponding two-sided p-values of the three test statistics were 5.4×10−7, 3.5×10−3and 1.5×10−4 respectively. All p-values in the adult group showed significant deviation from quasi-independence, but the test based on Lρ=0 produced the smallest p-value.
For the children group, the Z-values of the test statistics Lρ=0, Lρ=1 and Linvlog, after standardized by the jackknife estimator, were -1.838, -1.379 and -1.373 respectively. The positive association on (X,Y) can be found in the children group as well. The p-values for the two sided alternative were 0.0661, 0.1679 and 0.1697 respectively. The smallest p-value was also achieved by the Lρ=0 statistics, showing 10% significance level. In this case, the other statistics Lρ=1 and Linvlog could not reveal significant departure form quasi-independence.
In both groups, the significance level from Lρ=1 is the highest and that from Lρ=1 , which is equivalent to Tsai’s test statistics, was the lowest. One possible explanation of this result is that the data is better approximated by the Clayton semi-survival model than the Frank model. As we have seen in the simulation studies, the statistics Lρ=0 has the highest
efficiency while the statistics Lρ=1 is the worst under the Clayton model. This data analysis also indicates that choosing an appropriate weight function is essential for power improvement especially when the sample size is small.
4.7. Conclusion
In the second project, we have proposed a general class of tests in the form of the weighted Log-rank statistics for testing quasi-independence for truncation data. Tsai’s test (1990) turns out to be a special case of our proposal.
We also utilize the distributional property of the 2× tables in constructing the 2 proposed score test. Our results show that the score test belongs to the proposed class of weighted Log-rank tests with an appropriate choice of the weight function. Our simulations confirm that the score test yields a more powerful testing procedure if the pattern of dependence under the alternative hypothesis is correctly specified. It is important to note that optimal properties of the score test cannot be derived by applying the results for parametric models or the efficiency theory under a semi-parametric framework (Van Der Vaart, 1998, Chapter 25). The difficulty comes from the fact that each term in the product of the likelihood function (4.9) is neither the conditional likelihood nor partial likelihood since the probabilities are calculated conditional on an un-nested sequence of conditioning events. Further theoretical investigation on the likelihood formulation would be helpful.
For establishing the asymptotic normality, we have applied the functional delta method
which can handle more general situations than the U-statistics or rank statistics approaches.
Furthermore, the expression of the proposed statistics in the statistically differentiable functional allows us to verify the consistency of the jackknife estimator. These theoretical justifications allow us to safely use a computationally simpler way for finding the cut-off values.
Another important and practical problem is how to choose the best weight in real data analysis where the association pattern on (X,Y) is unknown in a nonparametric setting.
Now we discuss the possible approaches based on the literature of survival analysis. A common, but somewhat ad-hoc way of choosing weight function is to rely on the researchers’
own experience, or their knowledge on the association structure. Another more elaborate approach is to use a combination of several weighed Log-rank statistics (Tarone, 1981;
Chapter 7 of Fleming & Harrington, 1991 and Kosorok & Lin, 1999). Such an approach is considered to be a robust test (Kosorok & Lin, 1999) in that one may avoid using the worst weight choice in data analysis. To implement this methodology, the joint distribution for several weighted Log-rank statistics must be derived in some sense, and it would be our future problem for investigation.
Appendices : Project 2
Appendix 4.A. Asymptotic Analysis
Let }D{[0,∞)2 be the collection of all right-continuous functions with left-side limit defined on [0,∞)2 , whose norm is defined by f(x,y)∞ =supx,y | f(x,y)| for continuous. The empirical process on the plane is defined as:
∑
=The functional delta method is applied based on the weak convergence result of )) structure given by
)
Part I: Proof of Theorem 4.1
After some algebraic manipulations involving (6), we obtain
∫∫
≤the above expression can be written as
By direct calculations, we can show the Hadamard differentiability of Φ . The differential (⋅) map of Φ at (⋅) π∈ D{[0,∞)2} with direction h∈ D{[0,∞)2} is:
By applying the functional delta method (Van Der Vaart, 1998, p. 297), we obtain the following asymptotic linear expression
),
)
are iid random variables with mean-zero. From the central limit theorem, n−1/2Lw converges weakly to a mean-zero normal distribution with the variance σ2 = E[U(Xj,Yj)2].
Part II: Analytic Variance Estimator for the G Class ρ
The statistics in the Gρ class are special cases of Lw. For this class, it is relatively easier to obtain an analytic formula for estimating σ based on asymptotic linear 2 expressions. Specifically, the derivative map is given by
∫∫∫∫
The asymptotic expression
∑
= empirical estimator:
.
Part III: Proof of Theorem 2
*
Lw involves the estimator of the truncation probability c. From the result of He and Yang (1998), cˆ has an algebraically equivalent expression
∫
∞= 0 ˆ ( ) ˆ ( )
ˆ S u dF u
c Y X .
The product limit estimators (Lynden-Bell, 1971; Wang, Jewell & Tsai , 1986) for (X,Y) are defined as:
∏
< ⎭⎬⎫ Hadamard differentiable maps:∫
0∞ ˆ ( ) ˆ ( )It is well-known for right-censored data that the product limit estimator is Hadamard differentiable function of the empirical process. For truncation data, we apply the arguments of example 20.15 of Van Der Vaart (1998) to show the Hadamard differentiability of maps from }D{[0,∞)2 to D{[0,∞)}:
To prove the former statement, we decompose the map into three differentiable maps
,
where the Hadamard differentiability of the second map follows from Lemma 20.10 of Van Der Vaart (1998) and the last map follows from the Hadamard differentiability of product integral (Andersen et al., 1993, proposition II.8.7). The Hadamard differentiability of the map
)
π can be established by the same arguments. The Hadamard differentiability of the second map in (A.1) can be found in Lemma 20.10 of Van Der Vaart (1998).Using chain rules (Van Der Vaart, 1998, theorem 20.9), the map g is shown to be Hadamard differentiable. Let gπ′ )(h ∈R be the differential map of g at π∈ D{[0,∞)2} with expansion
|).
.
By applying the functional delta method, we obtain the following asymptotic linear expression:
),
where the sequences,
)
are mean-zero i.i.d. random variables. From the central limit theorem, n−1/2L*w converges
weakly to a mean-zero normal distribution with the variance σ2*.
Part IV: Consistency of the Jackknife Estimator
Now we show the consistency of the jackknife estimator for L . We have shown that w statistics of the form L have asymptotic normal distributions with finite variances. w According to the Theorem 3.1 of Shao (1993), we need to show the continuous Gateaux differentiability of Φ(π) at π∈ D{[0,∞)2}. Note that the Hadamard differentiability is stronger than the Gateaux differentiability, and hence the Gateaux derivative map is already available from Section A1. We only need to show the continuous requirement of the derivative map. For sequence πk∈ D{[0,∞)2} satisfying πk−π ∞ →0 and tk →0, we
need to show proving the continuous Gateaux differentiability is essentially the same manner as the example 2.6 in Shao (1993). The continuous differentiability of w(⋅) and the assumption
→0
−π ∞
πk ensure the following expansion
),
uniformly in (u,v). Hence a straightforward but tedious calculation shows that )
)
To show the consistency of the jackknife estimator for L , we only need to check whether *w the continuity of the Gateaux differential map of Ψ(π) which is available in Section A.3.
We can obtain the continuity requirement after tedious algebraic operations similar to the above arguments in A4.
Part V: Asymptotic Analysis in Presence of Censoring
Based on the product integral form of the Lynden-Bell’s estimator SˆC(y), we obtain the expression
{ }
From equation (?), (A-1) and (A-2), we obtain the following functional expression:).
Here, the last equation follows from the property
⎩⎨
Based on the similar arguments with Section A3, we can express the estimator c as a ˆ* function of Hˆ such that cˆ*=g*(Hˆ). The similar algebraic operation can be applied to obtain the functional expression for L . *w
Appendix 4.B: Proof of Equivalence Formula For right censored data, we show the identity:
∫∫ ∑
.
Similar algebraic manipulation shows that
∑ ∑
Combining these formulae, we obtain
∑ ∑
~ . 1
~ 2 } {
~ 1
~ 2 } {
1 : 2
1
∑
∑ ∑
<
= <
−
= Δ
−
= Δ
−
j
i ij
ij ij ij n
i jX X ij
ij ij ij
W R B I
W R B I I
I
i j
The last equation follows from the permutation symmetry of each term with respect to arguments (i,j).
Chapter 5 Future Work
In Chapter 3, we consider semi-parametric inference for semi-survival AC models and propose a likelihood-based approach for estimating the association parameter. A nice equivalent condition for different types of estimating functions is established. Similar idea is used again to construct a score test. Despite that we have seen efficiency gain or power improvement by choosing an appropriate weight function, optimality results are still not available. As mentioned earlier, each term in the product of the likelihood function is neither the conditional likelihood nor the partial likelihood since the probabilities are calculated conditional on an un-nested sequence of conditioning events. Further investigation is needed to elucidate the proposed likelihood, and it is hoped that we develop more understanding for the theoretical properties of the proposed methods.
For establishing the asymptotic normality, the functional delta method is applied for two problems. For the Log-rank statistics in Chapter 4, its expression has been shown to be a statistically differentiable functional that allows us to verify the consistency of the jackknife estimator. This theoretical justification allows us to safely use a computationally simple way for determining the decision rule of the testing procedure. Theoretical property of the jackknife estimator is only proven for the simple case of the Log-rank statistics with no censoring. For other complicated cases, the jackknife method is still a useful tool even though it may lack theoretical justification. Nevertheless finding a tractable and theoretically valid way of constructing confidence intervals or bands still deserves further investigation.
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