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A quantile regression estimator for interval-censored data

  • Paolo Frumento EMAIL logo
Veröffentlicht/Copyright: 3. Juni 2022
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The International Journal of Biostatistics
Aus der Zeitschrift The International Journal of Biostatistics Band 19 Heft 1

Abstract

We describe an estimating equation that can be used to fit quantile regression models to interval-censored data. The proposed estimator presents important advantages over the existing methods, and can be applied when the data are a mixture of interval-censored, left-censored, and right-censored observations. We describe estimation and inference, report simulation results, and apply the proposed method to analyze the Signal Tandmobiel® data. The necessary R code has been incorporated in the existing R package c t q r .

Keywords: interval-censored quantile regression; R package ctqr; signal Tandmobiel®data; two-step estimation
Corresponding author: Paolo Frumento, University of Pisa, Pisa, Italy, E-mail:

Acknowledgments

I would like to thank Professor Emmanuel Lesaffre (Biostatistical Centre Katholieke Universiteit Leuven) for his permission to use the Signal Tandmobiel® data, and for his kind answers to my requests of clarification about the variables in the dataset.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: The first-step estimator

In this appendix, we briefly describe the implementation of the first-step estimator F ̂ in the ctqr R package. The choice of using a flexible parametric model, instead of a fully nonparametric estimator, is motivated by convenience as suggested in Section 4. The decision of using a piecewise-constant hazards model, however, does not have any particular theoretical explanation, and is only dictated by simplicity and flexibility.

The proposed first-step estimator is implemented in the pchreg function from the pch R package [62], which is a dependency of ctqr. The calculation of F ̂ proceeds as follows. The range of the response is divided into k sub-intervals with breaks {t 0, t 1, …, t k }. The user can either supply the number of intervals, or the exact position of the breaks. By default, k = max(5, min(10, ⌈n 1/(5q)⌉)), where n 1 is the sample size excluding left- and right-censored observations, q is the dimension of x , and ⌈a⌉ denotes the ceiling operator. The default breaks are placed at the empirical quantiles of the response, calculated excluding left- and right-censored observations and replacing interval-censored observations by their middle point. Within each interval, the hazard function is held constant over time but is allowed to depend on covariates, i.e., proportionality of hazards is assumed to hold locally. The log-hazard in the jth interval is parametrized as log  λ j = x T θ j where θ j is a vector of parameters, j = 1, …, k. The total number of parameters is k × q. To fit the model, pchreg applies standard Newton-Raphson algorithm to maximize the log-likelihood function.

Appendix B: Sensitivity to first-step misspecification

The flexibility of the first-step model depends on the number of intervals, k. If k = 1, the model reduces to an Exponential distribution with parameter log(λ) = x T θ . As k increases, the model becomes more flexible and behaves like a nonparametric estimator. There is no obvious way to select an optimal value of k. A possible option is to fit a variety of first-step models, and choose the “best” one based on some information criterion, such as AIC or BIC. In the simulations presented in Section 5 we used the default value, that corresponds to k = 10. An important question is how the final estimators of β p behave if the first-step model is misspecified. To perform a robustness check, we repeated the simulation by selecting values of k that are too small to provide a good approximation of the true distribution function. Tables 6 and 7 report the average estimates and the empirical standard errors of β p ̂ obtained with k = 2, 6, and 10. Results show that misspecification of the first-step model does not impact the final estimators of β p in a significant way. The observed robustness can be explained by the fact that, in practice, F ̂ only contributes to the estimating equation when T L i ≤ x i T β p ̂ ≤ T R i , i.e., it is only relevant when the predicted quantile regression function lies in the interval ( T L i , T R i ) . For all other observations (those with x i T β p ̂ > T R i or x i T β p ̂ < T L i ), the indicator I T ≤ x i T β p ̂ is known and the value of F ̂ is irrelevant.

Table 6:

Sensitivity to the first-step model (simulation 1).

n = 250 n = 500
k = 10 k = 6 k = 2 k = 10 k = 6 k = 2
p True est. s.e. est. s.e. est. s.e. est. s.e. est. s.e. est. s.e.
Intercept 0.10 0.11 0.11 0.16 0.10 0.16 0.10 0.14 0.10 0.10 0.11 0.10 0.11 0.09
0.25 0.29 0.29 0.27 0.30 0.26 0.28 0.25 0.29 0.18 0.30 0.18 0.29 0.16
0.75 1.39 1.39 0.55 1.42 0.53 1.42 0.53 1.41 0.39 1.40 0.39 1.39 0.39
0.90 2.30 2.34 0.69 2.39 0.68 2.37 0.67 2.38 0.49 2.34 0.50 2.32 0.50
x 1 0.10 0.30 0.30 0.29 0.31 0.27 0.31 0.26 0.30 0.20 0.29 0.19 0.29 0.17
0.25 0.75 0.73 0.46 0.74 0.45 0.76 0.45 0.74 0.33 0.74 0.32 0.75 0.32
0.75 2.25 2.26 0.87 2.23 0.87 2.23 0.86 2.25 0.62 2.26 0.62 2.26 0.61
0.90 2.70 2.70 1.06 2.67 1.03 2.68 1.03 2.65 0.74 2.70 0.74 2.71 0.74
x 2 0.10 0.03 0.06 0.11 0.06 0.11 0.07 0.11 0.05 0.07 0.05 0.07 0.06 0.07
0.25 0.19 0.22 0.20 0.22 0.21 0.23 0.20 0.20 0.14 0.20 0.14 0.21 0.14
0.75 1.69 1.67 0.44 1.67 0.43 1.67 0.43 1.66 0.31 1.67 0.31 1.67 0.31
0.90 2.43 2.34 0.47 2.35 0.46 2.35 0.46 2.36 0.33 2.38 0.33 2.38 0.33
x 3 0.10 0.95 0.95 0.24 0.96 0.22 0.88 0.21 0.95 0.16 0.96 0.15 0.89 0.14
0.25 1.50 1.49 0.31 1.48 0.30 1.50 0.30 1.50 0.22 1.49 0.22 1.51 0.22
0.75 2.60 2.58 0.50 2.56 0.49 2.55 0.49 2.57 0.36 2.58 0.36 2.57 0.36
0.90 2.85 2.84 0.59 2.79 0.61 2.80 0.61 2.82 0.43 2.83 0.44 2.85 0.43

  1. Simulation 1: average estimates of β p and empirical standard errors for different values of k, the number of intervals used to fit the first-step piecewise-constant hazards model. The value k = 10 is the same used in Section 5. If k is small, the first-step model can be severely misspecified.

Table 7:

Sensitivity to the first-step model (simulation 2).

n = 250 n = 500
k = 10 k = 6 k = 2 k = 10 k = 6 k = 2
p True est. s.e. est. s.e. est. s.e. est. s.e. est. s.e. est. s.e.
Intercept 0.10 0.11 0.13 0.18 0.12 0.17 0.10 0.13 0.11 0.12 0.11 0.12 0.10 0.09
0.25 0.29 0.33 0.29 0.32 0.27 0.29 0.24 0.30 0.20 0.30 0.18 0.27 0.17
0.75 1.39 1.42 0.56 1.48 0.57 1.48 0.56 1.42 0.39 1.45 0.40 1.43 0.40
0.90 2.30 2.39 0.74 2.44 0.72 2.41 0.70 2.39 0.49 2.39 0.49 2.35 0.48
x 1 0.10 0.30 0.27 0.31 0.28 0.29 0.30 0.25 0.28 0.22 0.29 0.21 0.29 0.18
0.25 0.75 0.70 0.49 0.71 0.46 0.75 0.45 0.73 0.36 0.73 0.33 0.76 0.33
0.75 2.25 2.23 0.85 2.15 0.87 2.15 0.85 2.24 0.62 2.20 0.61 2.19 0.60
0.90 2.70 2.65 1.06 2.58 1.06 2.60 1.07 2.64 0.74 2.63 0.74 2.66 0.74
x 2 0.10 0.03 0.06 0.11 0.06 0.11 0.08 0.10 0.05 0.08 0.06 0.07 0.06 0.08
0.25 0.19 0.20 0.20 0.22 0.20 0.25 0.20 0.20 0.14 0.22 0.14 0.22 0.14
0.75 1.69 1.66 0.44 1.61 0.45 1.61 0.44 1.66 0.30 1.65 0.30 1.65 0.30
0.90 2.43 2.33 0.46 2.31 0.46 2.32 0.46 2.35 0.33 2.37 0.33 2.37 0.33
x 3 0.10 0.95 0.95 0.28 0.98 0.27 0.82 0.22 0.95 0.20 0.99 0.18 0.89 0.18
0.25 1.50 1.48 0.34 1.49 0.33 1.55 0.35 1.49 0.26 1.48 0.24 1.52 0.21
0.75 2.60 2.59 0.53 2.60 0.53 2.56 0.51 2.57 0.38 2.55 0.37 2.53 0.36
0.90 2.85 2.82 0.64 2.80 0.63 2.85 0.62 2.80 0.44 2.77 0.44 2.78 0.44

  1. Simulation 2: average estimates of β p and empirical standard errors for different values of k.

Appendix C: Simulations with left- and right-censored data

In this appendix we present additional simulations in which some of the data are either left- or right-censored. The data-generating process is the same as in Section 5: the quantile function is Q ( p ∣ x ) = − log ( 1 − p ) + ( 3 p ) x 1 + ( 3 p 2 ) x 2 + ( 3 p ) x 3 , with x 1 ∼ U(0, 1), x 2 ∼ Exp(1), and x 3 ∼ Ber(0.5), and examinations times t j are simulated by cumulating random draws from an Exp(λ) distribution. However, there is no examination at time t 0 = 0, which generates left censoring whenever T < t 1, and there is a maximum follow-up time M = 10, which generates right censoring when T > max j {t j : t j < 10}. As in Section 5, we consider two different values of λ: in scenario 1, where λ = 2, the average proportion of left-censored data is about 0.10, and the average proportion of right-censored data is around 0.06; in scenario 2, where λ = 1, the average proportion of left-censored data is about 0.18, and the average proportion of right-censored data is around 0.07.

The presence of left and right censoring prevents estimating some of the extreme quantiles. For example, if more than 50% of the data are right-censored, the sample median is not identifiable. In our simulation, we estimated the quartiles, that correspond to p = {0.25, 0.50, 0.75}. Results are summarized in Tables 8 and 9, where we report the average estimates and the standard errors of our method and compare them with ynh’s estimator. Note that zfd’s estimator is not applicable in this context, as some of the intervals (T L , T R ) have infinite width.

Table 8:

Results of simulation 1 with left- and right-censored data.

n = 250 n = 500
c t q r ynh c t q r ynh
p True est. s.e. est. s.e. est. s.e. est. s.e.
Intercept 0.25 0.29 0.29 0.25 0.29 0.29 0.30 0.18 0.30 0.20
0.50 0.69 0.73 0.40 0.72 0.42 0.72 0.29 0.72 0.31
0.75 1.39 1.27 0.64 1.39 0.56 1.33 0.47 1.41 0.41
x 1 0.25 0.75 0.75 0.45 0.75 0.51 0.74 0.32 0.75 0.35
0.50 1.50 1.47 0.67 1.48 0.70 1.47 0.48 1.48 0.52
0.75 2.25 2.28 0.87 2.27 0.89 2.23 0.60 2.22 0.63
x 2 0.25 0.19 0.22 0.20 0.22 0.21 0.19 0.14 0.19 0.15
0.50 0.75 0.76 0.35 0.76 0.36 0.74 0.24 0.74 0.26
0.75 1.69 1.92 0.82 1.68 0.46 1.82 0.52 1.67 0.33
x 3 0.25 1.50 1.48 0.30 1.48 0.32 1.49 0.24 1.50 0.25
0.50 2.12 2.09 0.43 2.09 0.45 2.11 0.30 2.11 0.32
0.75 2.60 2.62 0.53 2.59 0.53 2.61 0.38 2.59 0.39

  1. Average estimates of β p and empirical standard errors (simulation 1). In the table, c t q r refers to our proposal, and ynh refers to Yang, Narisetty, and He’s estimator computed using the DArq R package. In this simulation, the average proportion of left-censored data is about 0.10, and the average proportion of right-censored data is around 0.06.

Table 9:

Results of simulation 2 with left- and right-censored data.

n = 250 n = 500
c t q r ynh c t q r ynh
p True est. s.e. est. s.e. est. s.e. est. s.e.
Intercept 0.25 0.29 0.31 0.28 0.30 0.33 0.29 0.20 0.30 0.23
0.50 0.69 0.73 0.44 0.72 0.47 0.72 0.31 0.70 0.33
0.75 1.39 1.35 0.70 1.43 0.59 1.33 0.49 1.37 0.44
x 1 0.25 0.75 0.71 0.48 0.74 0.56 0.73 0.34 0.73 0.40
0.50 1.50 1.45 0.73 1.48 0.79 1.48 0.50 1.49 0.54
0.75 2.25 2.15 0.92 2.19 0.93 2.24 0.65 2.27 0.69
x 2 0.25 0.19 0.22 0.20 0.22 0.23 0.21 0.14 0.20 0.16
0.50 0.75 0.76 0.35 0.76 0.37 0.74 0.26 0.75 0.27
0.75 1.69 1.88 0.85 1.66 0.46 1.82 0.62 1.68 0.34
x 3 0.25 1.50 1.50 0.34 1.49 0.36 1.50 0.25 1.50 0.26
0.50 2.12 2.11 0.44 2.11 0.46 2.13 0.32 2.13 0.35
0.75 2.60 2.60 0.53 2.59 0.54 2.61 0.38 2.60 0.40

  1. Average estimates of β p and empirical standard errors (simulation 2). In this simulation, the average proportion of left-censored data is about 0.18, and the average proportion of right-censored data is around 0.07.

Based on our simulation, c t q r estimators are generally reliable, although some bias is found for β 2 at quantile 0.75. This may be due to misspecification of the first-step estimator F ̂ , that contributes to the estimating equation for all observations that are right-censored before the quantile T L i ≤ x i T β p ≤ ∞ or left-censored after the quantile − ∞ ≤ x i T β p ≤ T R i . In our simulation, the default definition of F ̂ might fail to capture some important features of the conditional distribution. Possible improvements can be achieved by increasing the number k of intervals, choosing a better positioning of the knots, or incorporating nonlinear/interaction terms in the first-step model. ynh’s estimator also performed well, and was usually less efficient than c t q r at quantiles 0.25 and 0.50, and more efficient at the third quartile.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/ijb-2021-0063).

Received: 2021-07-12
Revised: 2022-05-06
Accepted: 2022-05-16
Published Online: 2022-06-03

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Two-sample t α -test for testing hypotheses in small-sample experiments
  4. Estimating risk and rate ratio in rare events meta-analysis with the Mantel–Haenszel estimator and assessing heterogeneity
  5. Estimating population-averaged hazard ratios in the presence of unmeasured confounding
  6. Commentary
  7. Comments on ‘A weighting analogue to pair matching in propensity score analysis’ by L. Li and T. Greene
  8. Research Articles
  9. Variable selection for bivariate interval-censored failure time data under linear transformation models
  10. A quantile regression estimator for interval-censored data
  11. Modeling sign concordance of quantile regression residuals with multiple outcomes
  12. Robust statistical boosting with quantile-based adaptive loss functions
  13. A varying-coefficient partially linear transformation model for length-biased data with an application to HIV vaccine studies
  14. Application of the patient-reported outcomes continual reassessment method to a phase I study of radiotherapy in endometrial cancer
  15. Borrowing historical information for non-inferiority trials on Covid-19 vaccines
  16. Multivariate small area modelling of undernutrition prevalence among under-five children in Bangladesh
  17. The optimal dynamic treatment rule superlearner: considerations, performance, and application to criminal justice interventions
  18. Estimators for the value of the optimal dynamic treatment rule with application to criminal justice interventions
  19. Efficient estimation of pathwise differentiable target parameters with the undersmoothed highly adaptive lasso
https://doi.org/10.1515/ijb-2021-0063
Schlagwörter für diesen Artikel
interval-censored quantile regression; R package ctqr; signal Tandmobiel®data; two-step estimation

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Two-sample t α -test for testing hypotheses in small-sample experiments
  4. Estimating risk and rate ratio in rare events meta-analysis with the Mantel–Haenszel estimator and assessing heterogeneity
  5. Estimating population-averaged hazard ratios in the presence of unmeasured confounding
  6. Commentary
  7. Comments on ‘A weighting analogue to pair matching in propensity score analysis’ by L. Li and T. Greene
  8. Research Articles
  9. Variable selection for bivariate interval-censored failure time data under linear transformation models
  10. A quantile regression estimator for interval-censored data
  11. Modeling sign concordance of quantile regression residuals with multiple outcomes
  12. Robust statistical boosting with quantile-based adaptive loss functions
  13. A varying-coefficient partially linear transformation model for length-biased data with an application to HIV vaccine studies
  14. Application of the patient-reported outcomes continual reassessment method to a phase I study of radiotherapy in endometrial cancer
  15. Borrowing historical information for non-inferiority trials on Covid-19 vaccines
  16. Multivariate small area modelling of undernutrition prevalence among under-five children in Bangladesh
  17. The optimal dynamic treatment rule superlearner: considerations, performance, and application to criminal justice interventions
  18. Estimators for the value of the optimal dynamic treatment rule with application to criminal justice interventions
  19. Efficient estimation of pathwise differentiable target parameters with the undersmoothed highly adaptive lasso
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