Regression analysis of competing risks data with general missing pattern in failure types

TL;DR: In this paper, the cause-specific hazard rates under the general missing pattern were estimated using some semi-parametric models, and the regression coefficients and the baseline hazards were investigated.

About: This article is published in Statistical Methodology.The article was published on 2016-03-01 and is currently open access. It has received 2 citations till now. The article focuses on the topics: Missing data & Nelson–Aalen estimator.

Summary

1 Introduction

• Competing risks models are usually employed to analyse such type of data.
• Due to inadequacy on the diagnostic mechanism one often uncertain about the true failure type or it is reluctant to reveal the value of J for certain individuals.
• In carcinogenicity studies, death of individuals can be classified into death s with tumour or deaths due to other reasons.

2 Data and Models

• Note that these hazard rates in (3) can also be viewed as another set of cause-specific hazard rates with the different g’s in G representing the different failure types.
• This may be a strong assumption; but the authors relax this to some extent in the second model.
• The baseline cause-specific hazard rates, λ0j(t)’s, are now assumed to be proportional to each other under this model with eγj ’s being the proportionality constants.
• Both the models can be independently tested from standard competing risks data without any missingness.
• Under the general missing pattern as discussed, none of these can be tested.

3 Estimation under Model 1

• Also, these cause-specific hazard rates for the ‘modified’ competing risks problem are of the same semi-parametric form as those for the original cause-specific hazard rates in (4).
• Hence, the following partial likelihood is the most appropriate to estimate the regression parameters β, in the absence of any knowledge on the baseline cause-specific hazards λ∗0g(t), based on ‘modified’ competing risks data which is available without any missing failure type (Kalbfleisch and Prentice, 1980, Sec. 7.2.3).

3.1 Estimation of regression parameters

• Then, at each of these g-type failure times, say t(gi), the authors consider the conditional probability that the individual (gi) with covariate value z(gi) fails at time t(gi), given the history up to time t(gi)− and that one failure with missing pattern g occurs at time t(gi).
• Clearly, this partial likelihood (7) can accommodate tied failure times with different missing pattern, but an approximation may be needed to deal with tied failure times with the same missing pattern.
• Note that the standard asymptotic likelihood techniques can be applied to this partial likelihood (7) and to the estimate β̂ to make inference on β.

3.2 Estimation of baseline cumulative cause-specific hazards

• The baseline cumulative cause-specific hazards Λ∗0g(t) = ∫ t 0 λ∗0g(u)du for the ‘modified’ competing risks problem can also be estimated as follows.
• Under the same set of regularity conditions, as required for the asymptotic normality of β̂, the process( Λ̂∗0g(t)−.
• There have been some works concerning estimation of these masking probabilities usually requiring either additional modeling assumptions or secondary data.
• In practice, one can use “pooling-the-adjacent-violators” algorithm to achieve monotonicity.
• If some of the {Ng(t)}’s are not observed to have any jump during the study, the corresponding Λ̂∗0g(t)’s, and the associated entries of Σ̂(t), turn out to be zero; the corresponding rows of P are also estimated, as given in Dewanji and Sengupta (2003), to be zero.

4 Estimation under Model 2

• These have the similar semi-parametric form as those for the original cause-specific hazard rates in (5), except that the parametric component fg(z, t, θ ∼ ), for different g’s, are not of the simple exponential form.
• Nevertheless, from Kalbfleisch and Prentice (1980, Sec. 7.2.3), the partial likelihood in (13) is the most appropriate to estimate the vector of regression parameters θ ∼ , in the absence of any knowledge on the baseline cause-specific hazards λ0(t), based on the ‘modified’ competing risks data which is available without any missingness.

4.1 Estimation of regression parameters

• Then, at each of these failure times, say t(i), the authors consider the conditional probability that the individual (i) with covariate value z(i) fails at time t(i) with missing pattern g(i), given the history up to time t(i)− and that one failure occurs at time t(i).
• See Dewanji (1992) for a special case of this partial likelihood.
• An approximation may be needed to deal with tied failure times.
• Note that the model (12) cannot be written as the underlying multiplicative hazard competing risks model of Andersen et al. (1993, Ch. VII.2) for the asymptotic results therein to be readily available, as for the model in (6).
• The proofs of the asymptotic results follow the similar steps, as those in Andersen and Gill (1982) and Andersen et al. (1993, Ch VII.2), with little modification, as worked out by Prentice and Self (1983) in the context of ordinary survival data with general relative risk form.

4.2 Estimation of baseline cumulative cause-specific hazards

• A note on a test for competing risks with missing failure type.
• Nonparametric prevalence and mortality estimators for animal experiments with incomplete cause of death data.

Frequently asked questions

Q1. What have the authors contributed in "Regression analysis of competing risks data with general missing pattern in failure types" ?

In this work, the authors deal with the regression problem, in which the cause-specific hazard rates may depend on some covariates, and consider estimation of the regression coefficients and the cause-specific baseline hazards under the general missing pattern using some semi-parametric models. The authors consider two different proportional hazards type semi-parametric models for their analysis. The authors also consider an example from an animal experiment to illustrate their methodology.