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STS353 H. Zhao et al.
For the analysis, define to be 0 if subject was given the new treatment

and 1 otherwise. Note that here we only have cluster-specific covariates.
Table 2 contains the results obtained by the application of the proposed
estimation procedure and includes the estimated treatment effect on the
time to the clearance of the worms, the estimated standard error (SE), and
the − values for testing the covariate effects equal to zero. They suggest
that there seems no significant difference between the two treatment groups.
Williamson et al. (2008) and Zhang and Sun (2013) gave similar conclusions.
Note that here we used different values but the results seem to be robust.
On the other hand, one may be careful about the conclusions due to the
small number of subjects.

4. Discussion and Conclusion
A main feature of the models considered is their generality and flexibility
as they allow one to describe covariate effects in various forms. For inference
about regression parameters, a WCR-based estimating equation approach
was presented, and although the method may be computationally intensive,
it is highly intuitive and can be easily implemented. Also similar to the partial
likelihood approach, the proposed method has the advantage that it does
not require the estimation of the nonparametric function involved.
In the above, the focus has been on regression parameters, but
sometimes one may be interested in making inference about the unknown
function () too. One such situation is when the survival prediction is of
0
interest. On the other hand, the derivation of the limiting distribution of
̂
̂ (; ) is quite challenging even if under right censoring mechanism. A

̂
main reason for this is that the estimator (), the NPMLE of (), used

above has a non-normal limiting distribution only with a convergence rate of
̂
1 ⁄ 3. Thus it is reasonable to postulate that the estimator (̂ (; ) −

̂

) also has a very complicated asymptotic distribution with a

convergence rate of 1 ⁄ 3.

References
1. Chen, L., Sun, J., Xiong, C. (2016). A Multiple imputation approach to
the analysisof clustered interval-censored failure time data with the
additive hazards model. Computational Statistics and Data Analysis,
103, 242-249.
2. Chen, L., Feng, Y., Sun, J. (2017). Regression analysis of clustered failure
time datawith informative cluster size under the additive
transformation models. Lifetime Data Analysis, 23, 651-670.
3. Cong, X., Yin, G., Shen, Y. (2007). Marginal analysis of correlated failure
time datawith informative cluster sizes. Biometrics 63, 663–672.

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