On Confidence Intervals for the Hazard Ratio in Randomized Clinical Trials

Lin, Dan; Dai, Luyan; Cheng, Gang; Sailer, Martin Oliver

doi:10.1111/biom.12528

Cited by 17 publications

(24 citation statements)

References 13 publications

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“…For all the methods except one, there was no material difference in coverage (or mean width) between score and corresponding Wald intervals. The exception was the Lin et al method for which the score intervals were about 10% narrower than the Wald intervals (results not shown but available upon request).…”

Section: Resultsmentioning

confidence: 95%

“…For method m ,

L_{i}^{[m]} ((), β)

S_{i}^{[m]} ((), β)

, and

I_{i}^{[m]} ((), β)

serve as inputs for Wald and score tests, with

{\overset{true^}{β}}^{[m]}

denoting the estimate of β . A recently proposed method by Lin et al, labeled L, is also studied; it can be viewed as a hybrid of the B and C methods. All the methods are identical when there are no ties in the event times.…”

Section: Methodsmentioning

confidence: 99%

“…Using our notation, the statistic proposed by Lin et al can be written as

U^{[normalL]} ((), β) = \frac{{([], \sum_{i = 1}^{k} ((), d_{iA} - \frac{d_{i} n_{iA} e^{β}}{n_{iA} e^{β} + n_{iB}}))}^{2}}{\sum_{i = 1}^{k} \frac{d_{i} n_{iA} n_{iB} e^{β}}{{((), n_{iA} e^{β} + n_{iB})}^{2}} \frac{(n_{italiciA} + n_{italiciB} - d_{i})}{(n_{italiciA} + n_{italiciB} - 1)}} .

…”

Section: Methodsmentioning

confidence: 99%

“…Equations and reveal that the statistic in is constructed by using the same numerator but a modified denominator in the Breslow score statistic; the modification inserts the multiplier ( n iA + n iB − d i )/( n iA + n iB − 1) in . It follows that

{\overset{true^}{β}}^{[normalL]} = {\overset{true^}{β}}^{[normalB]}

and that U [L] ( β ) ≥ U [B] ( β ), so the test proposed by Lin et al is at least as powerful as the Breslow score test. However, since

{\overset{true^}{β}}^{[normalL]} ((), = {\overset{true^}{β}}^{[normalB]})

is generally biased, the test‐based confidence interval for β based on will sometimes suffer from inadequate coverage; we confirm this using simulations reported in the next section.…”

Section: Methodsmentioning

confidence: 99%

“…However, in practice, many tied event times are often encountered, as evidenced by the examples in Table 1. The four most widely discussed tie-handling approximations for the proportional hazards model were developed by Cox [C], 10 Kalbfleisch and Prentice [KP], 19 20 labeled L, is also studied; it can be viewed as a hybrid of the B and C methods. All the methods are identical when there are no ties in the event times.…”

Section: Wald and Score Tests Based On Competing Methodsmentioning

confidence: 99%

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Hazard ratio estimation and inference in clinical trials with many tied event times

Mehrotra

Zhang

2018

Statistics in Medicine

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The medical literature contains numerous examples of randomized clinical trials with time-to-event endpoints in which large numbers of events accrued over relatively short follow-up periods, resulting in many tied event times. A generally common feature across such examples was that the logrank test was used for hypothesis testing and the Cox proportional hazards model was used for hazard ratio estimation. We caution that this common practice is particularly risky in the setting of many tied event times for two reasons. First, the estimator of the hazard ratio can be severely biased if the Breslow tie-handling approximation for the Cox model (the default in SAS and Stata software) is used. Second, the 95% confidence interval for the hazard ratio can include one even when the corresponding logrank test p-value is less than 0.05. To help establish a better practice, with applicability for both superiority and noninferiority trials, we use theory and simulations to contrast Wald and score tests based on well-known tie-handling approximations for the Cox model. Our recommendation is to report the Wald test p-value and corresponding confidence interval based on the Efron approximation. The recommended test is essentially as powerful as the logrank test, the accompanying point and interval estimates of the hazard ratio have excellent statistical properties even in settings with many tied event times, inferential alignment between the p-value and confidence interval is guaranteed, and implementation is straightforward using commonly used software.

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Section: Resultsmentioning

confidence: 95%

“…For method m ,

L_{i}^{[m]} ((), β)

S_{i}^{[m]} ((), β)

, and

I_{i}^{[m]} ((), β)

serve as inputs for Wald and score tests, with

{\overset{true^}{β}}^{[m]}

Section: Methodsmentioning

confidence: 99%

“…Using our notation, the statistic proposed by Lin et al can be written as

U^{[normalL]} ((), β) = \frac{{([], \sum_{i = 1}^{k} ((), d_{iA} - \frac{d_{i} n_{iA} e^{β}}{n_{iA} e^{β} + n_{iB}}))}^{2}}{\sum_{i = 1}^{k} \frac{d_{i} n_{iA} n_{iB} e^{β}}{{((), n_{iA} e^{β} + n_{iB})}^{2}} \frac{(n_{italiciA} + n_{italiciB} - d_{i})}{(n_{italiciA} + n_{italiciB} - 1)}} .

…”

Section: Methodsmentioning

confidence: 99%

{\overset{true^}{β}}^{[normalL]} = {\overset{true^}{β}}^{[normalB]}

and that U [L] ( β ) ≥ U [B] ( β ), so the test proposed by Lin et al is at least as powerful as the Breslow score test. However, since

{\overset{true^}{β}}^{[normalL]} ((), = {\overset{true^}{β}}^{[normalB]})

is generally biased, the test‐based confidence interval for β based on will sometimes suffer from inadequate coverage; we confirm this using simulations reported in the next section.…”

Section: Methodsmentioning

confidence: 99%

Section: Wald and Score Tests Based On Competing Methodsmentioning

confidence: 99%

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Hazard ratio estimation and inference in clinical trials with many tied event times

Mehrotra

Zhang

2018

Statistics in Medicine

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Association of Titin-Truncating Genetic Variants With Life-threatening Cardiac Arrhythmias in Patients With Dilated Cardiomyopathy and Implanted Defibrillators

et al. 2019

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Hazard ratio inference in stratified clinical trials with time‐to‐event endpoints and limited sample size

Mehrotra

Shaw

2019

Pharmaceutical Statistics

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The stratified Cox model is commonly used for stratified clinical trials with time-to-event endpoints. The estimated log hazard ratio is approximately a weighted average of corresponding stratum-specific Cox model estimates using inverse-variance weights; the latter are optimal only under the (often implausible) assumption of a constant hazard ratio across strata. Focusing on trials with limited sample sizes (50-200 subjects per treatment), we propose an alternative approach in which stratum-specific estimates are obtained using a refined generalized logrank (RGLR) approach and then combined using either sample size or minimum risk weights for overall inference. Our proposal extends the work of Mehrotra et al, to incorporate the RGLR statistic, which outperforms the Cox model in the setting of proportional hazards and small samples. This work also entails development of a remarkably accurate plug-in formula for the variance of RGLR-based estimated log hazard ratios. We demonstrate using simulations that our proposed two-step RGLR analysis delivers notably better results through smaller estimation bias and mean squared error and larger power than the stratified Cox model analysis when there is a treatment-by-stratum interaction, with similar performance when there is no interaction. Additionally, our method controls the type I error rate while the stratified Cox model does not in small samples. We illustrate our method using data from a clinical trial comparing two treatments for colon cancer.KEYWORDS minimum risk weights, refined generalized logrank statistic, stratified cox model, treatment-by-stratum interaction, weighted average 366

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On Confidence Intervals for the Hazard Ratio in Randomized Clinical Trials

Cited by 17 publications

References 13 publications

Hazard ratio estimation and inference in clinical trials with many tied event times

Hazard ratio estimation and inference in clinical trials with many tied event times

Association of Titin-Truncating Genetic Variants With Life-threatening Cardiac Arrhythmias in Patients With Dilated Cardiomyopathy and Implanted Defibrillators

Hazard ratio inference in stratified clinical trials with time‐to‐event endpoints and limited sample size

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