Median event time test

Let Y₁, …, Y_n be random variables from an exponential distribution with mean μ. Then, the median ϕ is equivalent to μ log 2. Motivated by single arm Phase II studies whose primary endpoint is time-to-event, we formulate a hypothesis test based on median event time with H₀: ϕ = ϕ₀ versus H_a: ϕ = ϕ₁ for some values of ϕ₀ and ϕ₁. In practice for intervention study, we use median time of standard drug or therapy and the expected median time from the intervention (for improvement) to specify the values of ϕ₀ and ϕ₁, respectively. Let $$ {\widehat{\varphi }}_n\left(\mathbf{y}\right)$$ be a sample median event time obtained from a sample of size n. Then, for any λ ≥ 0, the rejection region for the test is $$ \{\mathbf{y}:{\widehat{\varphi }}_n\left(\mathbf{y}\right)>\lambda \}$$. For any α and β in unit interval (0, 1), this test is the level α test with power 1 − β such that $$ \text{Pr}\{{\widehat{\varphi }}_n\left(\mathbf{y}\right)>\lambda |H_0\}=\alpha $$ and $$ \text{Pr}\{{\widehat{\varphi }}_n\left(\mathbf{y}\right)\le \lambda |H_{\text{a}}\}=\beta $$. Since the first conditional probability indicates the probability of rejecting the null hypothesis when the null hypothesis is true, the value of α indicates the type I error rate. Also, the second conditional probability indicates the probability of failure of rejecting null hypothesis when the null hypothesis is not true, and the value of β indicates the type II error rate. The distribution function of the sample median event time is derived in the following theoretical result.

Theorem 1 Let Y₁, …, Y_n be random variables from an exponential distribution with median ϕ. Then, the probability density function (pdf) of sample median of Y₁, …, Y_n is either

The proof is in Appendix A. Theorem 1 provides a cumulative distribution function (cdf) of median event time given by $$ G_n(\lambda )={\int }_0^{\lambda }{g}_n\left(m\right)\text{d}m$$, where g_n denotes the pdf of the sample median, for λ ≥ 0. Since the cdf of sample median event time has no closed form, a numerical search over grid is required to identify sample size n and threshold λ such that the empirical errors $$ \widehat{\alpha }(n,\lambda )$$ and $$ \widehat{\beta }(n,\lambda )$$ are close to the nominal target values of type I and II error rate (i.e., α and β), respectively. The empirical errors are calculated as

Let’s consider a clinical trial with hypothesis testing of median progression free survivals (PFS) ϕ₀ = 10 months versus ϕ₁ = 17 months. Suppose that maximum number of patients for this study is 100. We consider a grid search over the integer n between 1 and 100 and a real number λ between 10 and 17 with the increment 0.1. For the target error rates of α = 0.05 and β = 0.2, our numerical study results in n = 42 and λ = 14.1 months minimizing deviation of the empirical errors from the target rates, i.e., $$ {\{\widehat{\alpha }(n,\lambda )-\alpha \}}^2+{\{\widehat{\beta }(n,\lambda )-\beta \}}^2$$. Fig 1 shows results for numerical search of n and λ, and Table 1 summarizes the decision rule for testing median survival ϕ₀ versus ϕ₁. In Table 1, scenarios with different value of ϕ₀ and ϕ₁ are considered. The choice of ϕ₀ and ϕ₁ is determined intentionally. To see the performance of method, we fixed ϕ₀ or ϕ₁ to vary ϕ₁ or ϕ₀, respectively.

Fig 1

Plot of $$ \widehat{\alpha }(n,\lambda )$$ and $$ \widehat{\beta }(n,\lambda )$$ for median event time test with ϕ₀ = 10 and ϕ₁ = 17 months: Left panel shows the result when n = 42 and right panel shows the results when λ = 14.1 months.

Table 1

Numerical search is done over the integer n between 1 and 100 and a real number λ between ϕ₀ and ϕ₁ with the increment 0.1.

Decision rules for hypothesis testing of median PFS ϕ₀ months versus ϕ₁ months at target error rates of α = 0.05 and β = 0.2.

ϕ0	ϕ1	n	λ	$$	$$
10	17	42	14.1	0.049	0.2019
8	17	21	12.9	0.049	0.1974
8	14	38	11.5	0.048	0.2016
3	7	16	5.2	0.042	0.2007
3	6	24	4.7	0.047	0.2027
3	5	48	4.2	0.042	0.2030

We have proposed median event time test for exponential survivals. The proposed method is extended for other parametric survival distributions. We provide the distribution function of the sample median event time for uniform and weibull survivals in Appendix B, which determines the decision rule.

This proposed median event time test is attractive with three reasons. First, this is newly proposed for an exact median event time test using the observed median event time. Second, author provides a software to calculate sample size and identify the threshold for the test, which is open for public use (https://yeonhee.shinyapps.io/METTshinyapp/). It does not require complicated statistical analysis for decision but implements easily. The simple and straightforwardly interpretable rule can save lots of time for drug development. Third, it has potential to extend for many applications. For example, the decision rule can be used to monitor futility in group sequential designs (for Phase II or III studies), and the threshold λ provides a good candidate (or range) for the Bayesian monitoring rule. Specifically, it provides foundation for optimal two-stage design based on median event time test which is described later.

Before we move to next section, we provide a remark on the median event time test. To develop the exact median event time test, we considered hypothesized values for median of survivals Y_i = min(S_i, U_i), where S_i denote the time-to-event and U_i denote the administrative censoring time for the ith patient. Our setting in this section uses Y_i and does not require information for accrual rate and follow-up time. However, when we consider a hypothesis testing with median of survivals S_i, we should care the censoring information. Assume that the time to arrival of the patient and survival time (i.e., S_i) follow some exponential distributions. Then, U_i, which is the time to arrival of the last patient plus follow-up time minus time to arrival of the ith patient, follows another exponential distribution. We notice that minimum of two exponential random variables (i.e., Y_i) follow some exponential distribution. Therefore, this setting for hypothesis testing with Y_i looks reasonable with the exponential survival assumption for S_i and the right censoring.

Optimal two-stage design based on median event time test

The proposed median event time is straightforward for clinicians to interpret and justify the sample size for the exact level of errors. It is critical, especially in rare disease trials, to obtain promising evidence with the target error rates and minimize the expected sample size. Moreover, from an ethical and practical viewpoint, it is desirable to stop the trial early if the therapeutic intervention is not effective. This motivates to propose a single arm Phase II trial with an interim planned when the total accrual reached n₁ patients. The final analysis will be performed after the follow-up of all planned number of n patients. A two-stage design using median event time test is proposed as follows. In the first stage (i.e., at interim), we determine go/no-go of the trial based on the observed median event time for those n₁ patients. When the observed median event time is less than or equal to the threshold λ, the trial is stopped for futility. Otherwise, the trial continues to enroll (n − n₁) patients in the second stage. At final analysis, we argue that the drug is sufficiently promising to evaluate against the standard therapy if the observed median event time based on all trial data is larger than the threshold.

The expected sample size for the two-stage design above is EN = n₁ + (1 − PET)(n − n₁), where PET denotes the probability of early termination after the first stage. EN depends on n, n₁ and λ, and the optimal two-stage design using median event time test is proposed in that EN is minimized under the null hypothesis. In the following, we describe how to specify n, n₁ and λ for the optimal two-stage design. Let M be a maximum sample size for the study, e.g., this can be specified by clinicians or by sample size calculator for one-arm nonparametric statistics (https://stattools.crab.org/Calculators/oneNonParametricSurvival.htm). Let ϵ₁ and ϵ₂ denote the acceptable difference between the estimated type I and II error rates from the trial design and the target error rates. For each prespecified values of ϕ₀, ϕ₁, α and β,

Step 1For each n₁ between 1 and M − 1, search n and λ minimizing $$ {\{\widehat{\alpha }(n_1,n,\lambda )-\alpha \}}^2+{\{\widehat{\beta }(n_1,n,\lambda )-\beta \}}^2$$, where



over n₁ < n ≤ M and ϕ₀ ≤ λ ≤ ϕ₁.

Step 2Choose an optimal pair of (n, n₁, λ) minimizing EN among the pairs satisfying $$ |\widehat{\alpha }(n_1,n,\lambda )-\alpha |<{ϵ}_1$$ and $$ |\widehat{\beta }(n_1,n,\lambda )-\beta |<{ϵ}_2$$.

As seen in median event time test, we don’t have closed forms for the marginal or joint distributions of sample median event times to calculate empirical probabilities in (1) and (2). The numerical search is used in the first step. The criteria given in the second step targets at minimizing the expected sample size, EN. This optimality criteria can be modified according to the study objective, e.g., the design minimizes expected total study length. Moreover, our design is flexible to use different thresholds, λ₁ for the interim analysis and λ₂ for the final analysis. Investigators fix a threshold λ₁ to stop early for futility and search threshold λ₂ satisfying the target error rates (or they can search both thresholds):

Although the proposed method does not assume specific survival distribution, the probability calculation requires to specify the survival distribution. We can borrow information from the previous research to specify the survival distribution (e.g., exponential, uniform, or weibull distribution). As an example to illustrate the optimal two-stage design based on median event time test, we assume that the survivals follow from an exponential distribution with median ϕ_⋅ and will be right-censored for subjects who have not yet met the criteria at the date of the last valid disease assessment. Setting with median survivals ϕ₀ = 10 and ϕ₁ = 17 for the standard therapy and a new therapy, respectively, and target error rates α = 0.05 and β = 0.2, we consider the maximum allowable sample size M = 50 (which is obtained from a sample size calculator for one-arm nonparametric statistics). Patients arrived according to a Poisson process with the accrual rate of 1.04 patients per month. We continue follow-up for 24 months after the last patient was enrolled. Empirical estimate of type I and II errors were obtained based on 1000 simulation trials. Then, our method with ϵ₁ = 0.02 and ϵ₂ = 0.025 attains the optimal results at n₁ = 32, n = 42 and λ = 15.2. It implies that our two-stage design accrue n₁ = 32 patient for the first stage. At interim, if the observed median event time based on these n₁ patients is less than or equal to λ = 15.2 months, the study will be early stopped for futility. Otherwise, additional n − n₁ = 10 patients will be accrued in stage 2, resulting in a total sample size of n = 42. At the end of trial, if the observed median event time based on all n = 42 patients is larger than λ = 15.2 months, we reject the null hypothesis and claim that the treatment is sufficiently promising. Different null and alternative median event times can be considered, and the results of the optimal two-stage design are summarized in Table 2.

Table 2

Note M denotes the maximum sample size for the study calculated from one-arm nonparametric statistics. Notations such as EN₀ and PET₀ are used to denote EN and PET, respectively, under the null hypothesis.

Summary results of optimal two-stage design for hypothesis testing of median survivals ϕ₀ versus ϕ₁ when survivals are assumed to follow an exponential distribution.

			n is searched
ϕ0	ϕ1	M	n1	n	λ	$$	$$	EN0	PET0
10	17	50	32	42	15.2	0.06	0.222	32.6	0.94
8	17	26	22	26	12.1	0.065	0.212	22.26	0.94
8	14	45	29	37	12.6	0.067	0.199	29.54	0.93
3	7	21	12	21	5.1	0.063	0.204	12.57	0.94
3	6	31	13	30	5	0.062	0.225	14.05	0.94
3	5	54	17	44	4.9	0.067	0.221	18.81	0.93
			n = M is prespecified
ϕ0	ϕ1	M	n1		λ	$$	$$	EN0	PET0
10	17	50	34		15.3	0.062	0.21	34.99	0.94
8	17	26	23		12.4	0.066	0.207	23.30	0.93
8	14	45	28		12.4	0.062	0.224	29.05	0.94
3	7	21	12		5.1	0.062	0.198	12.56	0.94
3	6	31	13		5	0.061	0.223	14.10	0.94
3	5	54	17		4.9	0.063	0.225	19.33	0.94

The proposed optimal two-stage design finds, in the first stage, the total sample size n and threshold λ for the given interim size n₁ and target error rates (i.e., α and β). In other words, both n and λ are searched. In case where study has a certain planned total sample size, the proposed design is tailored to identify the threshold λ in the first stage for the given interim size n₁, total sample size, and target error rates. In other words, only λ is searched, and n is prespecified. As an example, the value of M obtained from one-arm nonparametric statistics in Table 2 is used for the prespecified total sample size n in scenarios. The results are also summarized in Table 2.

Replacing M = 100 in Table 1 with the value obtained from one-arm nonparametric statistics, we obtained the same results. Tables 1 and 2 show that decision rule of two-stage design yields smaller expected sample size under the null than the one-stage design except the case with ϕ₀ = 8 and ϕ₁ = 17 requires 2 or 3 more patients to be enrolled. When ϕ₀ = 3 and ϕ₁ = 5, the total sample size n for two-stage design is smaller than one-stage design, and the expected sample size under the null is much smaller. More patients can avoid from treating the ineffective drug under the two-stage design.

Application: Trial illustration

We provide an application of the proposed methods with a trial NCT00780494. This is a Phase II, single arm, single-institution study of bevacizumab in combination with carboplatin and capecitabine for patients with unresectable or metastatisc gastroesophageal junction or gastric cancers. The study was started on February 2009 and completed on December 2017 to accrue enrollment of 35 participants. It enrolled two patients per month and follow up at time of study completion for 12 months. The study objective is to investigate the efficacy of the addition of bevacizumab to standard chemotherapy based on progression-free survival (PFS), which is defined as the duration of time from the start of treatment to time of disease progression or death. The median PFS is 5 months with the standard treatment therapy and the study team hypothesized the addition of bevacizumab to standard chemotherapy improve PFS by 90%, i.e., the median PFS is 9.5 months (https://clinicaltrials.gov/ProvidedDocs/94/NCT00780494/Prot_SAP_000.pdf).

We applied the proposed methods to redesign the trial based on median PFS. Target error rates are α = 0.05 and β = 0.2, and the maximum sample size is 35 obtained from one-arm nonparametric test. First, we considered a case where investigators want a trial without interim monitoring to collect the data with all 35 patients. The exact median event time test was applied, and provided the decision rule with threshold 7.5 months for a sample of size 29. This yielded 80.48% power. Second, we supposed that investigators want an interim for futility monitoring. Assuming the exponential survivals, we applied optimal two-stage design based on median event time test. We set ϵ₁ = ϵ₂ = 0.005 to get closer empirical error rates to the target rates, and it determined n₁ = 33, n = 34 and λ = 8 months with $$ \widehat{\alpha }=0.047$$ and $$ \widehat{\beta }=0.196$$. The rule implies that an additional patient will be enrolled after the first stage, and it is inappropriate from a practical point of view. Rather than considering two-stage design with a threshold, two-stage design with different thresholds λ₁ and λ₂ would be suggested to obtain the reasonable decision rule. Table 5 provides the summary results of the optimal two-stage design, which search n₁, n, and λ₂ for a fixed threshold λ₁ for the first stage. When survivals are assumed to follow the exponential distribution and threshold for the first stage is 5.5 months, the optimal two-stage design based on median event time test determined n₁ = 17, n = 32, λ₂ = 8.5 months. Specifically, the interim analysis will be performed based on the first 17 enrolled patients by comparing the observed median PFS with 5.5 months. If the observed median PFS is smaller than or equal to 5.5 months, the study is stopped early for futility. Otherwise, additional 15 patients are enrolled to have the total sample size of 32. At the final analysis, the observed median PFS with 32 patients is compared with the threshold 8.5 months, and we claim the addition of bevacizumab to the standard therapy improves PFS if the observed median PFS is larger than the threshold. This trial decision rule yielded power of 80.2%. When the true median PFS is 5 months, the expected sample size for this trial (i.e., EN) is 17.77 and the probability of early stopping (i.e., PET) is 94%. The required sample size and decision rule are changed for the different choice of λ₁.

Table 5

Summary results of optimal two-stage design with a fixed threshold λ₁ when a maximum sample size M is 35.

	n1	n	λ1	λ2	$$	$$	EN0	PET0
Exponential	17	32	5.5	8.5	0.051	0.198	17.77	0.949
	20	29	6	8.7	0.050	0.201	20.45	0.950
Uniform	16	24	5.5	5.8	0.054	0.202	16.43	0.946
	18	31	6	5.6	0.051	0.200	18.66	0.949
Weibull	15	31	5.5	5.9	0.052	0.202	15.83	0.948
	16	34	6	5.7	0.054	0.204	16.97	0.946

In this trial illustration, PFS was assumed to follow exponential distribution, and the decision rule was identified for the trial under the assumption. We further investigated with the nonexponential assumption. As seen in Table 5, decision rules, especially interim monitoring rules, for exponential versus nonexponential survivals are different, which implies that survival distribution assumption matters to design the clinical trials. In practice, earlier phase trials or experts’ knowledge would be able to provide information of survival distribution for Phase II study. We can borrow the information to identify the appropriate decision rule for the study.

A. Proof of theoretical result in methods

Proof of Theorem 1 Let F(y) and f(y) be the cdf and pdf, respectively, of the random variable whose median is ϕ (i.e., mean μ = ϕ/log 2). Then, we have f(y) = log 2 exp(−y log 2/ϕ)/ϕ and F(y) = 1 − exp(−y log 2/ϕ). Suppose that we have a sample of size n = 2k + 1 for some k = 0, 1, 2, …. Then, by Cramér [26], the pdf of the sample median is

We now derive the pdf of the sample median for a sample of size n = 2k for some k = 1, 2, …. When n is an even number, the sample median is $$ \widehat{m}=(Y_{\left(k\right)}+Y_{(k+1)})/2$$, where Y_(k) denotes the kth order statistic of the sample. Note that the joint pdf of Y_(k) and Y_(k+1) is

B. Median event time test for other parametric survival distributions

We provide two theoretical results stating the distribution function of the sample median event time for uniform and weibull distributions.

Theorem 2

Let Y₁, …, Y_n be random variables from a uniform distribution with lower bound parameter a and median ϕ (i.e., upper bound parameter is 2ϕ − a). Then,


for n = 2k + 1 and

where c = n!/{(k − 1)!(n − k − 1)!}, for n = 2k.

Let Y₁, …, Y_n be random variables from a uniform distribution with upper bound parameter b and median ϕ (i.e., lower bound parameter is 2ϕ − b). Then,


for n = 2k + 1 and

where c = n!/{(k − 1)!(n − k − 1)!}, for n = 2k.

Proof. We first consider a uniform random variable with lower bound parameter a and median ϕ. Then, we have pdf f(y) = 1/{2(ϕ − a)} for a ≤ y ≤ 2ϕ − a and cdf

By Cramér [26], if sample size is n = 2k + 1 for some k = 0, 1, 2, …, the pdf of the sample median is

When the sample size is n = 2k for some k = 1, 2, …, we use the joint pdf of Y_(k) and Y_(k+1), where Y_(k) denotes the kth order statistic of the sample, given by

Similarly, for a uniform random variable with upper bound parameter b and median ϕ, we have pdf f(y) = 1/{2(b − ϕ)} for 2ϕ − b ≤ y ≤ b and cdf

It is easy to verify that

Theorem 3 Let Y₁, …, Y_n be random variables from a weibull distribution with shape parameter τ and median ϕ (i.e., scale parameter is ϕ/(log 2)^1/τ). Then,

Proof. Let Y be a weibull random variable with scale parameter τ and median ϕ. Then, we have pdf f(y) = τ(log 2)^1/τ{(log 2)^1/τ y/ϕ}^τ−1 exp[−{(log 2)^1/τ y/ϕ}^τ]/ϕ and cdf F(y) = 1 − exp[−{(log 2)^1/τ y/ϕ}^τ]. By Cramér [26], if sample size is n = 2k + 1 for some k = 0, 1, 2, …, the pdf of the sample median is

When the sample size is n = 2k for some k = 1, 2, …, by proof of Theorem 1, the pdf of the sample median is