Integrating subgroups with mixed-type endpoints in early phase oncology trials

Abstract

Testing anti-cancer agents with multiple disease subtypes is challenging and it becomes more complicated when the subgroups have different types of endpoints (such as binary endpoints of tumor response and progression-free survival endpoints). When this occurs, one common approach in oncology is to conduct a series of small screening trials in specific patient subgroups, and these trials are typically run in parallel, independent of each other. However, this approach does not consider the possibility that some of the patient subpopulations respond similarly to therapy. In this article, we developed a simple approach to jointly model subgroups with mixed-type endpoints, which allows borrowing strength across subgroups for efficient estimation of treatment effects.

1. Introduction

Early phase ontology clinical trials typically are single-arm studies conducted to decide whether an experimental treatment is sufficiently promising, compared to the standard of care treatment, to warrant further investigation. In these trials, tumor response is measured over time (e.g. at two cycles, six months, one year and two years), and at each time point, it is conventionally categorized as complete response, partial response, stable, or progressive disease. To evaluate treatment efficacy, a commonly used endpoint is tumor response rate (i.e. the percentage of patients having a complete or partial response). Another potential endpoint is progression-free survival (PFS).

Because of the heterogeneity of human tumors, cancer patient populations are usually composed of multiple subgroups with different histologic types and primary sites. An example of this is a trial to evaluate the activity of dasatinib in combination with everolimus for children with gliomas harboring PDGFR or FGFR alterations. This trial has four subgroups including high-grade glioma (HGG) and diffuse intrinsic pontine glioma (DIPG) for newly diagnosed and recurrent/refractory patients. Moreover, subgroups have different types of endpoints of interest. Newly diagnosed patients generally live much longer than recurrent patients; thus, PFS is a commonly used endpoint to evaluate the treatment efficacy,^1–3 whereas the response rate at two cycles is a more established endpoint for recurrent patients.^4,5 One common approach in oncology is to conduct a series of small screening trials in specific patient subgroups; these trials are typically run in parallel, independent of each other. However, performing separate analyses is not feasible for this gliomas trial due to the relative rarity of the molecular and clinical subgroups. The expected enrollment is 20 subjects per subgroup, and the numbers are even smaller at the interim analysis.

As glioma subpopulations in this study are expected to respond similarly to the new therapy, an approach is needed that can build connections across subgroups to allow treatment effects in some groups to provide information about treatment effects in other groups. The information sharing between subgroups will enable an efficient estimation of the treatment effect in each group, especially when the sample size is small. Toward this end, we propose a statistical approach that models different types of endpoints simultaneously.

2. Methods

2.1. A unified framework for jointly modelling mixed-type endpoints

In order to model the binary tumor response and the PFS endpoint in a unified framework, we convert the PFS to a binary endpoint; that is, we estimate the progression rate at a landmark time point rather than the survival function. Then, we link these two binary endpoints by using a “failure” indicator. For the binary response endpoint, “failure” is defined as a subject not responding to treatment, whereas for a PFS endpoint, “failure” means that a subject has progressed before or at a pre-specified landmark time point. An indicator as to whether subject i is a failure (y_i= 1) or not (y_i =0) is sufficient to characterize the binary data, but not for the PFS data. For example, the endpoint is PFS at 12 months. If all subjects complete the 12-month follow-up, failure can be determined based on whether the subject had progressed during the 12 months’ follow-up window. However, if the subject is free-of-progression before reaching the 12-month landmark time point, the data are partially missing because it is unclear whether the subject would experience progression if he/she were followed to 12 months. The investigators could consider closing the trial for accrual until all patients complete the 12-month follow-up in that subgroup, but this will increase the trial duration and potentially turn away patients who may benefit from the therapy. To avoid the trial suspension, we use the actual amount of follow-up, t_i, collected on subject i, as well as the failure indicator for the PFS endpoint. The proposed model is described by

$$\text{log}\left(\frac{w\left(t_i\right)p_{g\left(i\right)}}{1-w\left(t_i\right)p_{g\left(i\right)}}\right)={\alpha }_{g\left(i\right)}+{\beta }_{g\left(i\right)}$$ (1)

where g(i) is the subgroup indicator for subject i and it takes a value 1, 2,…, or K if there are a total of K subgroups, p_g(i) is the failure rate of the experimental treatment in subgroup g(i). The amount of follow-up for subject i is entered in model (1) through a weight function, w(t_i). When all weights are equal to one, the model in equation (1) is reduced to a standard logistic regression model. The left-hand side of the equation is log-odds of failure, the intercept α_g(i) is the log-odds of failure in the standard of care treatment, that is, α_g(i) = log (p_0g(i)/(1 – p_0g(i))), where p_0g(i) is historical non-response rate in subgroup g(i), which is calculated by subtracting the historical response rate, θ_0g(i), from one. This historical response rate can be obtained from the literature, then converted to the log-odds, α_g(i), and finally entered into model (1) as a fixed parameter. The unknown parameter, β_g(i), is the log odds ratio (logOR) of the experimental treatment over the historical control in subgroup g(i), which measures the treatment effect in subgroup i.

2.2. Specification of the weight function

Theoretically, the weight, w(t_i), should reflect the expected distribution of progression times, which is often unknown and difficult to estimate when sample size is small. A simple solution, which has been shown to work well in such settings, is to assume progression occurs uniformly over the interval [0, τ_g(i)],^6,7 where τ_g(i) is the landmark time in subgroup g(i). Thus, the weight for subject i is defined as w(t_i) = t_i/τ_g(i). This weight function is between zero and one, and it reflects the amount of information available from a patient. For subjects with a PFS endpoint, if they have reached the landmark time point or had experienced progression, we should claim full information from this subject by letting w(t_i) = 1. For subjects with a binary endpoint, the weights are always one. The use of a weight function in the logistic regression model allows modelling both types of endpoints in a unified framework. Generally, the proposed model in equation (1) is a weighted logistic regression model, and the model parameters are estimated through a weighted binomial likelihood

$$L_n(\beta )=\prod _{i=1}^n{\left\{p_{g(i)}\right\}}^{y_i}{\left\{1-w\left(t_i\right)p_{g(i)}\right\}}^{1-y_i}$$

When y_i = 1, the weight is one because the subject who experienced progression before the landmark time point has complete information.

2.3. Borrowing strength across subgroups

If treatment effects in subgroups are independent and a strong biologic rationale exists for expecting similar treatment effects in subgroups (i.e. using statistical terminology, treatment effects are exchangeable⁸), we can allow sharing information across subgroups by assuming that β_g(i)’s are drawn from a common normal distribution with mean μ and variance σ². A commonly used prior distribution for μ is normal with mean 0 and variance 100. This vague prior allows the common logOR to be estimated mainly from the observed data. The conventional prior distributions for the variance parameter, σ², include inverse-gamma and uniform distributions. We consider the uniform prior, denoted by U (0, A), for the standard deviation, σ, as suggested by Gelman.⁹ Different values of the hyper-parameter, A, in the uniform distribution allow different degrees of information borrowing. The strongest borrowing occurs when we assume that β₁ = … = β_k = β, which is equivalent to a single pooled analysis where all data are pooled to estimate the common treatment effect, β. A hierarchical model is a compromised approach between the single pooled analysis and the separate analysis, by acknowledging the population heterogeneity while allowing “borrowing” information across the subgroups.^10–13 When there is strong rationale for expecting the same treatment effect across the subgroups, the single pooled analysis may be better than the hierarchical model.¹⁴ To decide which model to use for the real trial data, Bayesian model selection criteria, such as the Deviance Information Criterion,¹⁵ can be used.

2.4. Parameter estimation

Parameter estimations were implemented in JAGS,¹⁶ a statistical software package that uses Markov Chain Monte Carlo to generate samples from the relevant posterior distributions. JAGS codes to implement the proposed weighted logistic regression model can be found in Appendix 1. The convergence of the Markov chain was assessed visually and by the Gelman-Rubin Diagnostic criterion.¹⁷

2.5. Decision making in a two-stage design

Decisions at the interim and at the end of the study for subgroup i are made using the posterior distribution of θ_g(i), which is the success rate calculated by 1 − p_g(i). At the first stage (i.e. the interim), the trial is stopped for futility in subgroup i if Pr(θ_g(i) > θ_0g(i) + δ |data) < p_L. Thus, the trial is stopped early if the posterior probability that θ_g(i) exceeds a clinical meaningful threshold (δ) over the historical control θ_0g(i) by less than a pre-specified cut-off, p_L. At the end of the study, if Pr(θ_g(i) > θ_0g(i)data) >p_U, we declare that the experimental treatment is promising subgroup i and worth of further study; otherwise, we declare the experimental treatment is uninteresting in subgroup i. Both p_L and p_U can be calibrated by simulations to obtain good trial properties such as probability of early termination, desired type I error and power.¹⁸ It is worth noting that decisions are made for each subgroup separately at both stages. For example, the trial might be stopped in one subgroup for futility at stage one, while continued in another subgroup. The two-stage design protects the trial from sharing information between subgroups at the second stage, when treatment effects are very different (e.g. no treatment effect in one subgroup and a strong treatment effect in another subgroup), as it discontinues the trial in subgroups with no clear evidence of treatment efficacy at the first stage.

In certain circumstances, if we want to make decision for the overall treatment effect, parameter μ can be used (remember β₁, …, β_k ~ N(μ, σ²)). If treatment effects in subgroups are expected to be the same, the single pooled model can be used and decisions are based on the common parameter β.

2.6. Description of the gliomas trial

The goal of the trial is to evaluate the activity of dasatinib in combination with everolimus as an adjuvant therapy following surgery and radiation therapy for children and young adults with gliomas harboring PDGFR or FGFR alterations. We considered three subgroups for this trial: (1) newly diagnosed HGG, (2) newly diagnosed DIPG, (3) recurrent/refractory glioma (HGG and DIPG were pooled together in this subgroup due to very similar historical response rates). Newly diagnosed patients generally live much longer than recurrent patients; thus, the PFS is a commonly used endpoint to evaluate the treatment, whereas the response rate is a more established endpoint for recurrent patients.

For recurrent patients, previous studies have shown that the response rate at two cycles of standard-care-treatment is approximately 10%,^4,5 and the combination therapy is expected to increase the rate to 25% (i.e. δ = 15%), we call this subgroup stratum A. Based on previous clinical trials,^1–3 with the standard care, the estimated median PFS is 12 and 8 months for children with newly diagnosed HGG and DIPG, respectively. If we define the landmark time point at 12 and 8 months for HGG and DIPG, respectively, the uninteresting PFS rate in the historical control is 50% in both subgroups. Based on shared tumor biology, patient population and historical response rates, we believe that the treatment effects are very similar in these two subpopulations and expect the new combination therapy will increase the PFS rate to 75% (i.e. δ = 25%) in both subgroups. Thus, we combined these two subgroups into one subgroup and call it stratum B.

3. Simulation studies

We evaluated our proposed method using Monte Carlo simulations. For each simulated phase II trial, we fixed the sample size to be 20 in each stratum and simulated patient arrival time via a Poisson distribution such that each potential patient arrived an average of every 2.4 months in both strata, and then randomly allocate patients to HGG or DIPG group in stratum A. Patient PFS times were generated from exponential distributions. The interim analysis was done at the time when the 11th subject was entered into the study such that the interim decision was made based on the first 10 subjects’ data in each stratum. The final analysis was conducted after all the enrolled patients had complete data.

We considered four scenarios based on the hypothesized difference of historical control vs new therapy: (1) 0.1 vs. 0.3 in stratum A and 0.5 vs. 0.75 in stratum B, (2) 0.1 vs. 0.3 in stratum A and 0.5 vs. 0.7 in stratum B, (3) 0.1 vs 0.25 in stratum A and 0.5 vs. 0.7 in stratum B and (4) 0.1 vs. 0.25 in stratum A and 0.5 vs. 0.75 in stratum B. In all the studied cases, the difference, δ, is in the range of 15%−25%, which is commonly seen in early phase oncology trials. The treatment effects (measured on the logOR scale) are similar between two strata, and the differences in the treatment effect are 0.25, 0.5, 0.25 and 0, respectively, in the above four scenarios.

In our Joint Bayesian hierarchical model (JBHM), information are shared between two strata by assuming β₁, β₂ ~ N(μ, σ²). We used a normal prior, N(0,100), for μ and a uniform distribution U(0, A) for σ, where A is the maximum value; specifically, we considered A=0.1, 1, 10 and 100 for different degrees of information borrowing (a larger value of A indicates a lower degrees of borrowing). We also applied Simon’s two-stage design separately for each stratum. Simon’s design¹⁹ is very popular in single-armed ontology trials with a binary endpoint. Under the error rate constraints, it selects sample sizes and critical values by minimizing the expected sample size (called Simon’s Optimal Design), or the total sample size (called the Simon’s Minimax design). To use these designs, we need to pause the trial until each subject with the PFS endpoint completes the study period to know whether it is a success or failure. To make the comparison easier, we first fixed p_L = 0.1 and p_U = 0.9 in our model and then used the type I error and power obtained to calculate the required sample sizes in the Simon’s Minimax and Optimal designs.

3.1. Results

All simulation results are summarized based on 2000 simulated trials. Trial properties were evaluated with type I error, power, maximum sample size (n), expected sample size (En) and probability of early stopping under the null hypothesis of no treatment effect. Table 1 shows trial properties under four uniform distributions. This table indicates that U(0,1) had the best trial properties, and it induced the most efficient information borrowing between groups, which is weaker than the U(0,0.1) prior and stronger than the U(0,10) and U(0,100) priors. In designing real trials, we can treat parameter A in the uniform prior as a tuning parameter, and adjust it to obtain desirable trial properties. We then selected results under the prior σ ~ U(0,1) and σ·~ U(0,10) and compared them to the Simon’s designs (see Tables 2 and 3). Simulation results show that JBHM is more efficient than Simon’s designs under both priors, as evidenced by the dramatically reduced sample sizes given the same type I error and power as in the Simon’s designs.

Table 1.

Trial properties under different uniform prior distributions for σ using our Joint Bayesian hierarchical model.

Scenario	Stratum	Hypothesis	Design	Type I	Power	En	PET
I	A	0.1 vs. 0.3	U (0,0.1)	0.05	0.75	11	0.88
			U (0,1)	0.04	0.83	12	0.85
			U (0,10)	0.05	0.78	13	0.75
			U (0,100)	0.04	0.77	13	0.75
	B	0.5 vs. 0.75	U (0,0.1)	0.07	0.85	12	0.79
			U (0,1)	0.08	0.90	13	0.73
			U (0,10)	0.10	0.86	14	0.62
			U (0,100)	0.10	0.86	14	0.60
II	A	0.1 vs. 0.3	U (0,0.1)	0.04	0.69	11	0.88
			U (0,1)	0.04	0.79	11	0.86
			U (0,10)	0.05	0.77	13	0.75
			U (0,100)	0.04	0.77	13	0.75
	B	0.5 vs. 0.7	U (0,0.1)	0.09	0.82	13	0.66
			U (0,1)	0.10	0.82	14	0.58
			U (0,10)	0.11	0.74	15	0.50
			U (0,100)	0.12	0.74	15	0.49
III	A	0.1 vs. 0.25	U (0,0.1)	0.07	0.70	12	0.78
			U (0,1)	0.05	0.73	13	0.73
			U (0,10)	0.05	0.66	13	0.70
			U (0,100)	0.05	0.63	13	0.72
	B	0.5 vs. 0.7	U (0,0.1)	0.09	0.77	13	0.66
			U (0,1)	0.10	0.80	14	0.58
			U (0,10)	0.11	0.73	15	0.50
			U (0,100)	0.11	0.73	15	0.49
IV	A	0.1 vs. 0.25	U (0,0.1)	0.07	0.78	12	0.80
			U (0,1)	0.05	0.78	13	0.73
			U (0,10)	0.05	0.69	13	0.71
			U (0,100)	0.05	0.65	13	0.72
	B	0.5 vs. 0.75	U (0,0.1)	0.07	0.78	12	0.80
			U (0,1)	0.07	0.87	13	0.73
			U (0,10)	0.10	0.86	14	0.63
			U (0,100)	0.10	0.85	14	0.60

En: expected sample size; PET: probability of early stopping.

Table 2.

Comparing trial properties of our JBHM with Simon’s designs. JBHM had the U(0, 1) prior for σ.

Scenario	Stratum	Hypothesis	Design	Type I	Power	n	En	PET
I	A	0.1 vs. 0.3	Optimal	0.04	0.83	33	18	0.7
			Minimax			26	20	0.55
			JBHM			20	12	0.85
	B	0.5 vs. 0.75	Optimal	0.08	0.9	31	19	0.61
			Minimax			26	22	0.81
			JBHM			20	13	0.73
II	A	0.1 vs. 0.3	Optimal	0.04	0.79	27	16	0.7
			Minimax			28	18	0.62
			JBHM			20	11	0.86
	B	0.5 vs. 0.7	Optimal	0.1	0.82	32	21	0.6
			Minimax			30	22	0.5
			JBHM			20	14	0.58
III	A	0.1 vs. 0.25	Optimal	0.05	0.73	36	20	0.79
			Minimax			33	23	0.55
			JBHM			20	13	0.73
	B	0.5 vs. 0.7	Optimal	0.1	0.8	32	20	0.61
			Minimax			28	22	0.5
			JBHM			20	14	0.58
IV	A	0.1 vs. 0.25	Optimal	0.05	0.78	44	23	0.76
			Minimax			34	29	0.54
			JBHM			20	13	0.73
	B	0.5 vs. 0.75	Optimal	0.07	0.87	29	18	0.71
			Minimax			26	21	0.87
			JBHM			20	13	0.73

En: expected sample size; PET: probability of early stopping; JBHM: Joint Bayesian hierarchical model.

Table 3.

Comparing trial properties of our JBHM with Simon’s designs. JBHM had the U(0, 10) prior for σ.

Scenario	Stratum	Hypothesis	Design	Type I	Power	n	En	PET
I	A	0.1 vs. 0.3	Optimal	0.05	0.78	28	15	0.73
			Minimax			21	15	0.66
			JBHM			20	13	0.75
	B	0.5 vs. 0.75	Optimal	0.1	0.86	23	15	0.62
			Minimax			21	16	0.5
			JBHM			20	14	0.62
II	A	0.1 vs. 0.3	Optimal	0.05	0.77	30	14	0.77
			Minimax			21	15	0.66
			JBHM			20	13	0.75
	B	0.5 vs. 0.7	Optimal	0.11	0.74	29	16	0.64
			Minimax			23	16	0.5
			JBHM			20	16	0.35
III	A	0.1 vs. 0.25	Optimal	0.05	0.66	28	16	0.7
			Minimax			26	22	0.48
			JBHM			20	13	0.70
	B	0.5 vs. 0.7	Optimal	0.11	0.73	29	16	0.64
			Minimax			22	18	0.83
			JBHM			20	15	0.50
IV	A	0.1 vs. 0.25	Optimal	0.05	0.69	37	17	0.74
			Minimax			27	21	0.51
			JBHM			20	13	0.71
	B	0.5 vs. 0.75	Optimal	0.1	0.86	23	15	0.62
			Minimax			21	16	0.5
			JBHM			20	14	0.63

En: expected sample size; PET: probability of early stopping; JBHM: Joint Bayesian hierarchical model.

Additionally, we tried a smaller variance in the normal prior for μ, i.e. N(0,10), while fixing the U(0,1) prior for σ. We found that the results are the same as using the original N(0,100) prior, which suggests N(0,10) is an uninformative prior relative to the likelihood. In designing real trials, we can consider a N(0,10) or N(0,100) prior for parameter μ.

In the last scenario where the ORs are the same in both strata, we also tried a single pooled analysis by assuming a common treatment effect (i.e. β) in the model. In this simulation study, parameters p_L and p_U were adjusted to obtain type I error of 5% and power of 80%. We found that, with the same type I error and power, Simon’s Minimax design requests a maximum of 63 subjects in both strata with an expected sample size of 47, and Simon’s Optimal design requests a maximum of 68 subjects in both strata with an expected sample size of 40, whereas our method only needs a maximum of 32 subjects with an expected sample size of 24. Using the single pooled analysis, the sample size was reduced to approximately half as we gained the most efficiency by pooling the data if the treatment effects were truly the same.

It is also important to note that the proposed method allows a continuous patient enrollment at the interim, whereas the Simon’s design has to pause at the interim analysis until all subjects without progression reached the landmark time point (i.e. eight months in DIPG or 12 months in HGG group).

3.2. Sensitivity analysis

3.2.1. Weight function

We had chosen a uniform weight function because we assumed that the event occurs uniformly within the follow-up window. To assess the robustness of our JBHM, we generated failure event times from a Weibull distribution with a shape parameter smaller or larger than one, which resulted in the event occurring more often at early or late times, respectively. When the shape parameter is equal to one, the Weibull distribution reduces to the exponential distribution we used for other simulations. Simulation results in Table 4 show that the uniform weight function is robust to different event time distributions, as evidenced by similar type I error and power for the three event time distributions. Furthermore, if there is no treatment effect, the trial had a higher probability of stopping at the first stage when the shape parameter is smaller, as we have more information to make the correct early decision when more events occurred at the first stage.

Table 4.

Trial properties using our JBHM with progression times generated from a Weibull distribution with a shape parameter (S) equal to 0.5, 1 or 2.

Scenario	Stratum	Hypothesis	S	Type I	Power	En	PET
I	A	0.1 vs. 0.3	0.5	0.04	0.82	11	0.87
			1	0.04	0.83	12	0.85
			2	0.04	0.84	12	0.83
	B	0.5 vs. 0.75	0.5	0.08	0.89	12	0.78
			1	0.08	0.90	13	0.73
			2	0.09	0.91	13	0.67
II	A	0.1 vs. 0.3	0.5	0.04	0.78	11	0.87
			1	0.04	0.79	11	0.86
			2	0.04	0.81	12	0.82
	B	0.5 vs. 0.7	0.5	0.10	0.81	13	0.65
			1	0.10	0.82	14	0.58
			2	0.11	0.83	15	0.52
III	A	0.1 vs. 0.25	0.5	0.05	0.73	12	0.75
			1	0.05	0.73	13	0.73
			2	0.05	0.74	13	0.69
	B	0.5 vs. 0.7	0.5	0.09	0.78	13	0.65
			1	0.10	0.80	14	0.58
			2	0.10	0.80	15	0.51
IV	A	0.1 vs. 0.25	0.5	0.04	0.77	12	0.76
			1	0.05	0.78	13	0.73
			2	0.05	0.79	13	0.69
	B	0.5 vs. 0.75	0.5	0.07	0.85	12	0.78
			1	0.07	0.87	13	0.73
			2	0.08	0.88	13	0.67

En: expected sample size; PET: probability of early stopping.

3.2.2. Different treatment effects

The hierarchal model works well when treatment effects are similar between groups. We conducted a sensitivity analysis to evaluate the performance of the JBHM when the treatment effects are very different. Specifically, we generated data assuming no treatment effect in stratum A (i.e. the true success rate is equal to the null value of 0.1) and a strong treatment effect in stratum B (i.e. the true success rate is equal to value in the alternative hypothesis). In this simulation study, we tried both U(0,1) and U(0,10) prior for σ. Simulation results in Table 5 show that JBHM with the weakly informative prior, U(0,10), was able to make the correct decision in most of the simulated trials; it correctly rejected the null hypothesis in stratum B in 72–83% of the simulated trials; it correctly stopped the trial at the first stage in stratum A in 55–74% of the trials and incorrectly rejected the null hypothesis in stratum A in only 8–10% of the simulated trials. Therefore, the weakly informative prior combined with the two-stage design guarded against violation of the exchangeability assumption in a hierarchical model.

Table 5.

Trial properties of using our JBHM with U(0, 1) and U(0, 10) prior for σ, when there is no treatment effect in stratum A and a strong treatment effect in stratum B. The percentage of times that the trial rejects the null hypothesis (Reject) and the probability of early stopping (PET) are summarized.

Scenario	Hypothesis	True success rate	U (0,1)		U (0, 10)
			Reject	PET	Reject	PET
I	0.1 vs. 0.3	0.1	0.17	0.61	0.09	0.74
	0.5 vs. 0.75	0.75	0.75	0.21	0.83	0.12
II	0.1 vs. 0.3	0.1	0.14	0.66	0.08	0.75
	0.5 vs. 0.7	0.7	0.68	0.19	0.72	0.11
III	0.1 vs. 0.25	0.1	0.16	0.48	0.09	0.59
	0.5 vs. 0.7	0.7	0.66	0.18	0.72	0.11
IV	0.1 vs. 0.25	0.1	0.21	0.41	0.10	0.55
	0.5 vs. 0.75	0.75	0.73	0.21	0.82	0.12

En: expected sample size; PET: probability of early stopping.

4. Discussion

One of the major challenges in the development of anti-cancer agents is the heterogeneity of patient populations. With expanding number of clinical and molecular subgroups, it has become more difficult to meet accrual goals with traditional clinical trial designs.²⁰ Therefore, it is critical to use as much information as possible for assessing activity of new anti-cancer agents by borrowing strength across subgroups. In this paper, we developed a simple approach to jointly model subgroups with mixed-type endpoints, which allows integrating data across subgroups for more efficient estimation of treatment effects. If treatment effects are expected to be similar, our joint modelling approach is more powerful than performing separate analysis in each subgroup, as evidenced by the significantly reduced sample sizes given the same type I error and power in the simulation studies. The degree of information borrowing across subgroups can be adjusted by specifying different prior distributions. When there is strong rationale for expecting a uniform treatment effect across the subgroups, a single pooled analysis can be performed using the proposed method by assuming a common logOR parameter in the model.

Theoretically, the use of a hierarchical Bayesian model would gain more efficiency when the number of strata/subgroups is larger.¹⁴ However, we have demonstrated the advantage of the hierarchical Bayesian model when there are only two strata (a PFS endpoint for one stratum and a binary response endpoint for another stratum) under the assumption that the two stratums have similar treatment effects. We are in the process of prospectively evaluating the proposed method in multiple ongoing early phase molecularly adapted pediatric glioma trials conducted at the University of Michigan.

The proposed joint modelling method is very simple to implement, which requires only a few lines of codes in JAGS software (see Appendix 1), and the complete R code for running the simulations is freely available in the Github repository at https://github.com/lilizhaoUM/JBHM. The proposed method can be easily extended to analyze randomized trials. In such trials, the only change is that intercepts are also random parameters, and they will be estimated from the data. A prior distribution will be assigned to this parameter in each subgroup. The treatment effects are still represented by the logOR parameter.

In this study, we considered a uniform weight function. It may appear to be an oversimplified choice but has been shown to be adequate in our simulation studies and has demonstrated its success in small phase I oncology trials.^6,7 When the sample size is larger (e.g. > 40), a more complicated weight function, such as the adaptive weight, or obtaining weights from a parametric survival distribution or the Kaplan-Meier curve, can be considered.

The proposed hierarchical model relies on the assumption that treatment effects are similar across subgroups. However, a two-stage design with the use of a weakly informative prior seems to work very well in the simulated studies, where there is a strong treatment effect in one subgroup, while no treatment effect in another subgroup. Alternatively, the method proposed by Wathen et al.²¹ can be considered. It directly models the treatment-subgroup interactions to account for the different treatment effects between subgroups. However, it requires a reasonably large sample size. Our method offers an attractive and relatively simple strategy for small oncology trials with mixed-type endpoints, such as the glioma trial, where it is difficult to meet accrual goals with traditional clinical trial designs.

Appendix 1

# JAGS code when logOR are assumed to be SIMILAR in subgroups (i.e. a hierarchical model)

# N: total number subjects in the study

# K: total number of strata

# y[i]: an indicator for failure, which is defined as a non-response or progression depending on the subgroup

# g[i]: subgroup indicator for subject i, which takes a value 1, …, or K.

# w[i]: weight for subject i.

model {

for(i in 1:N){

y[i] - dbin(p[i]*w[i], 1)

logit(p[i]) < - alpha[g[i]] + beta[g[i]]

}

for(k in 1:K){

beta[k] — dnorm(mu.beta,tau)

}

mu.beta —dnorm(0,0.01)

# uniform prior for standard deviation

tau < − pow(sigma, −2)

sigma ~ dunif(0, A)

#calculate success rate in the experimental treatment

for(k in 1:K){

theta[k]= 1/(1 +exp(alpha[k]+beta[k]))

}

}

# JAGS code when logOR are assumed to be the SAME in subgroups (i.e. a single pooled analysis) model {

for(i in 1:N){

y[i] - dbin(p[i]*w[i], 1)

logit(p[i]) < - alpha[g[i]] + beta

}

beta ~ dnorm(0,0.01)

}