Tests for comparing estimated survival functions

Chauvel, Cécile; O’Quigley, John

doi:10.1093/biomet/asu015

Cited by 16 publications

(34 citation statements)

References 16 publications

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“…No other tools such as smoothing, the projection on a basis of functions, or kernel estimation are needed (Hastie and Tibshirani, ; Cai and Sun, ; Scheike and Martinussen, ). For instance, as seen in Chauvel and O'Quigley (), a constant effect until time

τ

followed by a null effect is easily detectable, especially with moderate and larger sample sizes. Assume that the time‐dependent regression parameter can be expressed as

β false(t false) = false(β_{1} false(t false), \dots, β_{p} false(t false) false)

, where

β_{j} false(t false) = β_{0, j} B_{j} false(t false)

(

j = 1, \dots, p

) with

{boldβ}_{0} = false(β_{0, 1}, \dots, β_{0, p} false) \in ℝ^{p}

an unknown regression parameter and

B = false(B_{1}, \dots, B_{p} false)

a known

ℝ^{p}

‐valued function of the time.…”

Section: Interplay Of Fit and Predictionmentioning

confidence: 90%

“…We first need the number of time points,

k_{n}

at which the covariate distribution is not degenerate and this is

\begin{matrix} right k_{n} & center = & left # {i : i = 1, \dots, n, δ_{i} = 1, x^{T} V_{boldβ (X_{i})} (Z | X_{i}) x > 0, \\ right \forall x & center \in & left {double-struckℝ}^{p} ∖ {0}}, \end{matrix}

where

# A

denotes the cardinality of the set A . Chauvel and O'Quigley () defined the transformed times

φ_{n} false(X_{i} false)

where

φ_{n} false(X_{i} false) = \frac{\overset{N}{true} false(X_{i} false)}{k_{n}} (1 + (1 - δ_{i}) \frac{# ({}, j : j = 1, \dots, n, X_{j} < X_{i}, \overset{N}{true} false(X_{j} false) = \overset{N}{true} false(X_{i} false))}{# ({}, j : j = 1, \dots, n, \overset{N}{true} false(X_{j} false) = \overset{N}{true} false(X_{i} false))})

The inverse transformation of

φ_{n}

false[0, 1 false]

is defined by

φ_{n}^{- 1}

such that

φ_{n}^{- 1} false(t false) = inf \{X_{i},\}

…”

Section: Model‐based Empirical Processesmentioning

confidence: 99%

“…Corollary 1 suggests that a good approximation for this process is a Brownian motion with a linear drift. An indication of the plausibility of this is shown in Figure 1 in Chauvel and O'Quigley () where

β false(t false)

is set to

0.5

. Departures from the proportional hazards assumption can be of various forms.…”

Section: Model‐based Empirical Processesmentioning

confidence: 99%

“…Corollary 1 implies that the shape of the drift of the process

U_{n}^{*} false(0, \cdot false)

will reflect the shape of the cumulated regression coefficient. A further illustration from Chauvel and O'Quigley () represents a simulated process under the non‐proportional hazards model, with

β false(t false)

piecewise constant. Before

t = 0.5

, there is a linear trend corresponding to

β = 1

and for

t > 0.5

β

equals zero and the process

U_{n}^{*} false(0, t false)

is constant in expectation over time.…”

Section: Model‐based Empirical Processesmentioning

confidence: 99%

“…The goodness‐of‐fit is evaluated with a version of the score process introduced by (O'Quigley, , Chap. 8), O'Quigley () and Chauvel and O'Quigley (). We study this process more closely here, in particular in the multivariate setting.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Survival Model Construction Guided by Fit and Predictive Strength

Chauvel

O’Quigley

2016

Biometrics

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We describe a unified framework within which we can build survival models. The motivation for this work comes from a study on the prediction of relapse among breast cancer patients treated at the Curie Institute in Paris, France. Our focus is on how to best code, or characterize, the effects of the variables, either alone or in combination with others. We consider simple graphical techniques that not only provide an immediate indication as to the goodness of fit but, in cases of departure from model assumptions, point in the direction of a more involved alternative model. These techniques help support our intuition. This intuition is backed up by formal theorems that underlie the process of building richer models from simpler ones. Goodness-of-fit techniques are used alongside measures of predictive strength and, again, formal theorems show that these measures can be used to help identify models closest to the unknown non-proportional hazards mechanism that we can suppose generates the observations. We consider many examples and show how these tools can be of help in guiding the practical problem of efficient model construction for survival data.

show abstract

τ

followed by a null effect is easily detectable, especially with moderate and larger sample sizes. Assume that the time‐dependent regression parameter can be expressed as

β false(t false) = false(β_{1} false(t false), \dots, β_{p} false(t false) false)

, where

β_{j} false(t false) = β_{0, j} B_{j} false(t false)

(

j = 1, \dots, p

) with

{boldβ}_{0} = false(β_{0, 1}, \dots, β_{0, p} false) \in ℝ^{p}

an unknown regression parameter and

B = false(B_{1}, \dots, B_{p} false)

a known

ℝ^{p}

‐valued function of the time.…”

Section: Interplay Of Fit and Predictionmentioning

confidence: 90%

“…We first need the number of time points,

k_{n}

at which the covariate distribution is not degenerate and this is

\begin{matrix} right k_{n} & center = & left # {i : i = 1, \dots, n, δ_{i} = 1, x^{T} V_{boldβ (X_{i})} (Z | X_{i}) x > 0, \\ right \forall x & center \in & left {double-struckℝ}^{p} ∖ {0}}, \end{matrix}

where

# A

denotes the cardinality of the set A . Chauvel and O'Quigley () defined the transformed times

φ_{n} false(X_{i} false)

where

φ_{n} false(X_{i} false) = \frac{\overset{N}{true} false(X_{i} false)}{k_{n}} (1 + (1 - δ_{i}) \frac{# ({}, j : j = 1, \dots, n, X_{j} < X_{i}, \overset{N}{true} false(X_{j} false) = \overset{N}{true} false(X_{i} false))}{# ({}, j : j = 1, \dots, n, \overset{N}{true} false(X_{j} false) = \overset{N}{true} false(X_{i} false))})

The inverse transformation of

φ_{n}

false[0, 1 false]

is defined by

φ_{n}^{- 1}

such that

φ_{n}^{- 1} false(t false) = inf \{X_{i},\}

…”

Section: Model‐based Empirical Processesmentioning

confidence: 99%

β false(t false)

is set to

0.5

. Departures from the proportional hazards assumption can be of various forms.…”

Section: Model‐based Empirical Processesmentioning

confidence: 99%

“…Corollary 1 implies that the shape of the drift of the process

U_{n}^{*} false(0, \cdot false)

will reflect the shape of the cumulated regression coefficient. A further illustration from Chauvel and O'Quigley () represents a simulated process under the non‐proportional hazards model, with

β false(t false)

piecewise constant. Before

t = 0.5

, there is a linear trend corresponding to

β = 1

and for

t > 0.5

β

equals zero and the process

U_{n}^{*} false(0, t false)

is constant in expectation over time.…”

Section: Model‐based Empirical Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Survival Model Construction Guided by Fit and Predictive Strength

Chauvel

O’Quigley

2016

Biometrics

View full text Add to dashboard Cite

show abstract

A flexible and coherent test/estimation procedure based on restricted mean survival times for censored time‐to‐event data in randomized clinical trials

Horiguchi

Cronin

Takeuchi

et al. 2018

Statistics in Medicine

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In randomized clinical trials where time-to-event is the primary outcome, almost routinely, the logrank test is prespecified as the primary test and the hazard ratio is used to quantify treatment effect. If the ratio of 2 hazard functions is not constant, the logrank test is not optimal and the interpretation of hazard ratio is not obvious. When such a nonproportional hazards case is expected at the design stage, the conventional practice is to prespecify another member of weighted logrank tests, eg, Peto-Prentice-Wilcoxon test. Alternatively, one may specify a robust test as the primary test, which can capture various patterns of difference between 2 event time distributions. However, most of those tests do not have companion procedures to quantify the treatment difference, and investigators have fallen back on reporting treatment effect estimates not associated with the primary test. Such incoherence in the "test/estimation" procedure may potentially mislead clinicians/patients who have to balance risk-benefit for treatment decision. To address this, we propose a flexible and coherent test/estimation procedure based on restricted mean survival time, where the truncation time τ is selected data dependently. The proposed procedure is composed of a prespecified test and an estimation of corresponding robust and interpretable quantitative treatment effect. The utility of the new procedure is demonstrated by numerical studies based on 2 randomized cancer clinical trials; the test is dramatically more powerful than the logrank, Wilcoxon tests, and the restricted mean survival time-based test with a fixed τ, for the patterns of difference seen in these cancer clinical trials.

show abstract

Improving testing and description of treatment effect in clinical trials with survival outcomes

Yang

2018

Statistics in Medicine

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Cox model inference and the log-rank test have been the cornerstones for design and analysis of clinical trials with survival outcomes. In this article, we summarize some recently developed methods for analyzing survival data when the hazards may possibly be nonproportional and also propose some new estimators for summary measures of the treatment effect. These methods utilize the short-term and long-term hazard ratio model proposed in Yang and Prentice (2005), which contains the Cox model and also accommodates various nonproportional hazards scenarios. Without the proportional hazards assumption, these methods often improve the log-rank test and inference procedures based on the Cox model, as well as nonparametric procedures currently available in the literature. The proposed methods have sound theoretical justifications and can be computed quickly. R codes for implementing them are available. Detailed illustrations with 3 clinical trials are provided.

show abstract

Tests for comparing estimated survival functions

Cited by 16 publications

References 16 publications

Survival Model Construction Guided by Fit and Predictive Strength

Survival Model Construction Guided by Fit and Predictive Strength

A flexible and coherent test/estimation procedure based on restricted mean survival times for censored time‐to‐event data in randomized clinical trials

Improving testing and description of treatment effect in clinical trials with survival outcomes

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