Isotonic designs for phase I trials in partially ordered groups

Review of the original Yuan–Chappell method

In Yuan and Chappell, ¹ there are J dose levels under consideration, and the goal of the study is to find the “maximum tolerated dose” (MTD) in each of G risk groups. By definition, the MTD in group g is the highest dose under consideration that can be administered to patients in that group with an “acceptable level” of toxicity. “Acceptable” toxicity means limiting the proportion of patients in that group who experience a sufficiently severe, protocol-specified adverse event, usually called a “dose-limiting toxicity” (DLT). With cytotoxic agents, it is generally assumed that regardless of the group, the greater the dose administered, the greater the probability that a patient will experience a DLT.

The isotonic regression continual reassessment method proposed in Yuan and Chappell ¹ is built on the continual reassessment method. ¹⁷ Originally proposed for a single risk group, the continual reassessment method assigns doses adaptively, with the dose for the current patient determined by the observed toxicities and dose allocations from the prior patients on the trial. Specifically, at any point in the trial, we have observed n j patients on dose level j, j = 1 , … , J and among these patients, we have observed Y j patients experiencing a DLT. The number of patients with a DLT on dose level j, Y j , is distributed as independent B i n ( n j , π j ) , where π j is the probability that a patient will experience a DLT, given dose level j. The continual reassessment method uses a working model (1) for the π j

where 0 < ψ 1 < ψ 2 < ⋯ < ψ J < 1 are pre-specified constants, often referred to as the “skeleton” values and a is a scalar parameter. The parametrization exp ( a ) ensures that the probability of toxicity increases in dose for all − ∞ < a < ∞ . Advice on choosing skeleton values is given in Lee and Cheung. ¹⁸

With these specifications, at any point in the trial, the likelihood from the previously observed DLTs and dose allocations is

Current estimates of the DLT probabilities are π ^ j = ψ j exp ( a ^ ) , where ( a ^ ) is found by maximizing equation (2) as a function of a. Based on the current estimates, the next patient is assigned to the dose level j * with the smallest value of Δ ( π ^ j , θ ) , where Δ ( u , v ) is a pre-specified measure of distance, usually quadratic distance, between u and v. The parameter θ is a target toxicity probability specified prior to the trial. The process of updating a and allocating patients to doses continues until a pre-specified number of patients have been observed. At the end of the study, the MTD is taken to be the dose that the next patient would have received had the trial not ended.

Yuan and Chappell ¹ generalize the continual reassessment method to G completely ordered groups, where by convention, group 1 is the lowest risk group and patient in group G have the highest risk of a DLT. At any point in the trial, within group g, we have observed n g j patients allocated to dose level j and of these, Y g j patients have experienced a DLT. Assuming skeleton values 0 < ψ 1 < ψ 2 < ⋯ < ψ J < 1 , the working model for the probability π g j that a patient in risk group g, given dose level j, experiences a DLT is given by

for j = 1 , … , J and g = 1 , … , G . The likelihood as a function of ( a 1 , a 2 , … , a G ) is

Estimates ( a ^ 1 , a ^ 2 , … , a ^ G ) can be found by maximizing equation (4), and from these, we can obtain the current estimates of π g j

The skeleton values and the working model guarantee that within groups, the estimated probabilities of toxicity increase with dose. Across groups, however, there may be “reversals,” meaning that at a given dose, the probability of a DLT is estimated to be greater in a lower risk group than in a higher risk group. To eliminate these reversals, Yuan and Chappell ¹ create estimates π ^ g j * by applying the two-way isotonic regression procedure of Robertson et al. ² to the model-based estimates, π ^ g j , from equation (5). The two-way isotonic regression estimation is an iterative process of adjusting the model-based estimates within rows and within columns to guarantee that the estimated probabilities of a DLT have no reversals, that is, the estimates are non-decreasing within rows and within columns. As in the original continual reassessment method, if the next patient is from group g, that patient is assigned the dose level with the “isotonized” estimate closest to the target toxicity probability, θ , using a quadratic distance measure. The process of estimation and allocation continues until a pre-specified number of patients have been observed in each group.

A modification

As in the original Yuan and Chappell ¹ method, we obtain the estimate a ^ = ( a ^ 1 , a ^ 2 , … , a ^ G ) by maximizing equation (4), and we let v j ( a ^ ) denote the j t h diagonal element of the observed information matrix based on the log of the likelihood (4) evaluated at a ^ . Instead of estimating the group-dose DLT probabilities from the model and using two-way isotonic regression, we apply the estimation procedure of Hwang and Peddada ¹⁶ to a ^ = ( a ^ 1 , a ^ 2 , … , a ^ G ) using weights 1 v j ( a ^ ) to obtain estimates subjected to the following restriction

We denote the resulting estimates by a ^ H P = ( a ^ 1 H P , a ^ 2 H P … , a ^ G H P ) , which satisfy

After “isotonizing” the estimates, we compute estimates of the probabilities of a DLT for dose d in group g

Within groups, the estimated probabilities of a DLT increase with dose and for a fixed dose, the probabilities increase across groups. While Hwang and Peddada ¹⁶ provide a general estimation procedure for parameters subjected to partial orders, in the case of a complete order, the estimation procedure is identical to using isotonic regression on − a ^ = ( − a ^ 1 , − a ^ 2 , … , − a ^ G ) using weights 1 v j ( a ^ ) , as in Robertson et al. ² We would argue that this is a simpler method than the original to avoid reversals in the DLT probability estimates. In general, the number of groups is small, and the modification requires isotonic regression of only a small number of parameters. In order to make sure we have finite estimates a ^ g , we use a pseudo-data prior as in Whitehead and Brunier ¹⁹ or Legedza and Ibrahim. ²⁰ with total weight equal to one observation per group.

Comparing the original and the modification using a family of dose–toxicity curves

We denote the true MTD in group g by M T D g . With G groups and J dose levels under consideration, complete ordering of the groups implies that

We generated two scenarios for each of the possible configurations of the MTDs consistent with equation (9). Table 1 lists the number of such configurations for three or four groups and four or six dose levels.

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Table 1.

Possible MTD configurations.

Groups	Dose levels	Number
3	4	20
3	5	35
3	6	56
4	4	35
4	5	70
4	6	126

Letting γ = ( γ 1 , γ 2 , γ 3 ) denote one of these configurations; we generated group-dose toxicity probabilities π g j , g = 1 , … , G and j = 1 , … , J starting from group 1. We define π g , 0 = 0 and π 0 , j = 0 . In group g, for j = γ g , we set π g , j = θ . To generate probabilities for j ≠ γ g , we define “Lower” to equal max { π g − 1 , j , π g , j − 1 } . For j < γ g , we define “Upper” to equal the target, θ and for j > γ g , we define Upper to equal 1. With these definitions, we sequentially generated the true group-dose toxicity probabilities, π g , j as

where β ( 1 , 4 ) are independently generated β -distributed random variables with mean 0.2 and variance 0.0267. This method of generating the group-dose toxicity probabilities guarantees that the resulting probabilities will follow the complete order.

For each scenario, we simulated 1000 trials with either a total of 36 or 54 patients, and either equal or unequal group probabilities. For three groups, the group probabilities were equal to 0.45, 0.40, and 0.15 to match the approximate group proportions in Innocenti et al.[8] For four groups, the group probabilities were chosen to be equal to 0.4, 0.3, 0.2, and 0.1. For each combination of sample size and group probabilities, we computed for both the original and the modified methods, the “percent correct selection,” defined as the proportion of times that the method correctly identifies the MTD in each group. We also computed the “accuracy index” of Cheung ²¹ within groups. Figure 1 displays the distribution of differences in the percent correct selection, modified minus original, and Figure 2 displays the distribution of differences in the accuracy index. The figures show a small difference between the original and the modified methods with, in general, the percent correct selection and the accuracy index with the original method greater than for the modification.

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Figures 1.

Comparison of percent correct selection for original and modified methods.

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Figures 2.

Comparison of accuracy indices for original and modified methods.