Multivariate Markov models for the conditional probability of toxicity in phase II trials

Abstract

In addition to getting a preliminary assessment of efficacy, phase II trials can also help to determine dose(s) that have an acceptable toxicity profile over repeated cycles as well as identify subgroups with particularly poor toxicity profiles. Correct modeling of the dose-toxicity relationship in patients receiving multiple cycles of the same dose in oncology trials is crucial. A major challenge lies in taking advantage of the conditional nature of data collection, that is each cycle is observed conditional on having no previous toxicities on earlier cycles. We develop a novel and parsimonious model for the probability of toxicity during a k^th cycle of therapy, conditional on not seeing toxicity in any of the k − 1 previous cycles using a Markov model, hereafter we refer to these probabilities as conditional probabilities of toxicity. Our model allows the conditional probability of toxicity to depend on randomized dose group, cumulative dose from prior cycles, a measure of how consistently a patient responds to the same dose exposure and individual risk factors influencing the ability to tolerate the treatment regimen. Simulations studying finite sample properties of the model are given. Finally, the approach is demonstrated in a phase II trial studying two dose levels of ifosfamide plus doxorubicin and granulocyte colony-stimulating factor in soft tissue sarcoma patients over four cycles. The Markov model provides correct estimates of the probabilities of toxicity in finite sample simulations. It also correctly models the data from the phase II clinical trial, and identifies particularly high cumulative toxicity in females.

1. Background and significance

The study of dose toxicity relationships in oncology occurs in phase I and phase II clinical trials and variations of these. We focus on the setting where patients are randomized to two, or more, dose groups administered over several cycles. An example of this study design can be found in Worden et al. (2005); Chugh et al. (2007), which investigated two randomized doses of ifosamide plus doxorubicin and granulocyte colony-stimulating factor, hereafter called ifosamide for brevity. Depending on whether they had metastases, patients received either six or four cycles of either 6 g/m² or 12 g/m² of ifosamide given over a 4 day period at the beginning of a 21 day cycle. As is often the case, patients who experienced treatment toxicity were not continued in subsequent cycles. Adverse events are classified by National Cancer Institute (NCI) Common Toxicity Criteria and grade 3 or higher toxic responses are considered to be a dose-limiting toxicity (DLT) (National Cancer Institute, 2003). Proportions of continuing patients in each cycle who had a hemoglobin DLT are shown in Fig. 1. Two important features of the data are (1) the conditional nature of toxicity proportions observed at each cycle that are based on previously toxicity-free patients and (2) the trend toward higher conditional toxicity rates as dose accumulates over cycles in the high dose group. Possible analytical tools for this data are generalized estimating equations (GEE) or generalized linear-mixed models (GLMM) that account for correlation within a patient treated over multiple cycles. These models allow covariates for cycle, dose, cumulative dose, and other mitigating factors in modeling the probability of toxicity. An example of analyzing such data can also be found in Legedza and Ibrahim (2000) that applies to phase I clinical trials and as a special case of Doussau et al. (2013) that uses ordinal outcomes also in a phase I setting. All these methods do not explicitly model the tendency to discontinue cycles for patients who have demonstrated previous DLT, although the resulting estimated toxicity rates may be conditional in nature. In fact, conditional probabilities of toxicity are precisely what are needed in order to advise patients during subsequent therapy. Developing models that explicitly capture these conditional probabilities are key.

Figure 1.

Observed proportion of low HGB in continuing subjects with one standard error intervals.

A large number of parameters may be necessary to capture all features of the dose-toxicity relationship, and with the typically small sample size available for modeling purposes, a Bayesian approach is an attractive alternative. Our model is an extension to transitional models with first-order Markov chains (Agresti, 2002), considering only the previous cycle. In this manuscript, we use a Markov model to explicitly model conditional probabilities of toxicity in a cycle given that dosed patients did not have a DLT in the past. The proposed model allows for a cumulative effect of dose on toxicity after the first cycle and allows covariates to influence the toxicity profile over cycles. A parameter is included to reflect an individual’s tendency to respond consistently with past dose experience. In Section 2, we formulate a model useful in this phase II clinical trial setting, describe the Bayesian estimation method and provide intuition on model behavior. We then study finite sample operating characteristics through simulation in Section 3. In Section 4, we apply the methods to the ifosfamide study described earlier and follow with a discussion in Section 5.

2. Methodology

In Section 2.1, we define the data structure and the proposed dose-toxicity model. Calculations for the expected total dose over K potential treatment cycles as well as the expected number of completed cycles is presented in Section 2.2. Technical details of dose modeling via skeleton probabilities of toxicity during the first cycle of treatment are covered in Section 2.3. Selection of priors and the formulation of the posterior distribution is presented in Section 2.4 followed by a Section 2.5 reviewing model selection strategies.

2.1. Proposed Markov model

We assume patients i = 1 … N are randomized to one of G doses S_g, g = 1, 2, …, G. The most common trials have G = 2 doses, and we present our method in this case for simplicity. However, the method extends easily to the more general case. In Section 2.3, we will show that it is convenient to rescale doses S_g to d_g; a strategy useful for incorporating beliefs about toxicity on the first cycle, similar to what was proposed in O’Quigley et al. (1990); Lee and Cheung (2009). For convenience, d_g terms will hereafter be referred to as the dose assigned per cycle to group g, even though numerically d_g must be transformed back to the S_g scale to reflect actual doses.

Each patient i completes K_i ≤ 6 cycles, where K_i may be less than six if a patient experiences a DLT or if the patient drops out for other reasons. On each cycle k = 1, …, K_i, patient i randomized to group g receives dose d_i,k = d_g. Since dose is constant across cycles, we will typically use the notation d_i,1 for the dose given to patient i at each cycle. A patient’s cumulative dose prior to cycle k is D_i,k = (k − 1) × d_i,1, with D_i,1 = 0. We also use the convention that d_i,k−1 = 0 for cycle k = 1.

A Bernoulli random variable, Y_i,k, denotes the occurrence of a DLT for patient i on cycle k. In general, Y_i,k = 0 for k = 1, …, K_i − 1, indicating no DLT on these cycles. If patient i completes K_i = K cycles, then Y_i,K may be either zero or one, depending on manifestation of a DLT in the final cycle. Additional patient covariates (Z) are available for modeling the dose-toxicity relationship. So the observed data becomes $\left(Y_{i,1},\dots ,Y_{i,K_i},Z_i,d_{i,1}\right)$.

We define p_i,k = P(Y_i,k = 1|Y_i,k−1 = 0, …, Y_i,1 = 0, Z_i, d_i,1) as the conditional probability of a DLT for patient i on cycle k given that patient i has experienced no previous DLTs. Model 1 for p_i,k is

$$\log \left(1-p_{i,k}\right)=-g_1(\alpha ,Z)\left(d_{i,1}-g_2(\rho ,Z)d_{i,k-1}\right)-g_3(\beta ,Z)D_{i,k}$$ (1)

where α, ρ, and β parameterize the relationship between dose, Z and p_i,k. In the simplest case, g₁(·), g₂(·), and g₃(·) are identity functions not involving Z, so that Model 1 reduces to Model 2 below:

$$\log \left(1-p_{i,k}\right)=-\alpha \left(d_{i,1}-\rho d_{i,k-1}\right)-\beta D_{i,k}$$ (2)

$$\text{or}\text{equivalently},p_{i,k}=1-\exp \left[-\alpha \left(d_{i,1}-\rho d_{i,k-1}\right)-\beta D_{i,k}\right].$$ (3)

Additional examples of g₁(·), g₂(·), and g₃(·) that involve covariates are given throughout the manuscript; in particular in Sections, 2.3, 2.4.2, and 4. Intuition behind the parameters is easiest to follow for the special case in Model 2, where α, ρ, and β are 1-dimensional parameters. The probability of a DLT on cycle 1 is determined solely by the parameter α. The parameter, 0 ≤ ρ ≤ 1, allows for a reduced probability of toxicity related to dose d_i,1 in a subsequent cycle if patient i has previously tolerated this dose; this term captures dependency in short-term toxicity outcomes between cycles. The effect of cumulative dose from previous cycles is captured by β. We note in Model 2 that D_i,k is the sum of previous rescaled doses, it would be possible to modify this such that D_i,k is the sum of the real doses S_g, or the sum of h(S_g) for some monotone function h.

Figure 2 shows the dose-toxicity relationship across cycles and p_i,k for two dose levels with β = (0, 0.2) and ρ = (0, 0.75, 1) with K = 4 cycles. The parameter α is set to one to reflect probability skeleton dose rescaling as described in Section 2.3. For instance, in the top left panel where α is the only parameter driving the dose-toxicity relationship, a patient has an independent DLT response at each dose administration so that p_i,k is constant across cycles. In the top right panel, the parameter ρ equals 1 indicating that a patient tolerating dose d_i,1 on cycle 1, will not experience future toxicity at this dose level. In the lower three panels, the influence of increasing β alters the dose-toxicity relationship based on the effect of cumulative dose.

Figure 2.

Conditional P(Toxicity) on cycle k, p_i,k based on Eq. (2), for two dose levels with β = (0, 0.2) and ρ = (0, 0.75, 1). The parameter α is set to one to reflect probability skeleton dose rescaling as described in Section 2.3. Solid triangles and circles correspond to the high and low dose groups, respectively.

2.2. Expected total dose and completed cycles

A higher dose for each cycle might not be attractive if fewer cycles can be completed at that dose level based on DLTs. Investigators should gain a clear understanding of the expected number of completed cycles for a dose level as well as the expected total dose over the entire trial based on Model 1 or the special case without covariates, Model 2.

An individual i’s number of total cycles is 1 with probability p_i,1, is K_i with probability

$$\left(1-p_{i,1}\right)\left(1-p_{i,2}\right)\cdots \left(1-p_{i,K_i-1}\right)p_{i,K_i}=p_{i,K_i}\prod _{j=1}^{K_i-1}\left(1-p_{i,j}\right)$$ (4)

for K_i = 2, … K − 1 and is K with probability

$$P\left(Y_{i,1}=0,\dots ,Y_{i,K-1}=0\right)=\prod _{j=1}^{K-1}\left(1-p_{i,j}\right).$$ (5)

Hence for person i, the expected number of completed cycles is

$$=p_{i,1}+\sum _{k=2}^{K-1}k{p}_{i,k}\prod _{j=1}^{k-1}\left(1-p_{i,j}\right)+K\prod _{j=1}^{K-1}\left(1-p_{i,j}\right)$$ (6)

and the expected total dose is

$$=p_{i,1}{d}_{i,1}+\sum _{k=2}^{K-1}k{d}_{i,1}{p}_{i,k}\prod _{j=1}^{k-1}\left(1-p_{i,j}\right)+K{d}_{i,1}\prod _{j=1}^{K-1}\left(1-p_{i,j}\right).$$ (7)

2.3. Model calibration using skeleton probabilities

For ease of interpretation of the parameter α, it is convenient to rescale the doses. As explained below they are rescaled based on initial guesses (skeleton probabilities) of the toxicity rate for the first cycle. For simplicity, we first describe use of skeleton probabilities for the simple case with no covariates, as in Model 2, and later suggest modifications for the more complex settings. Our strategy of defining skeleton probabilities and corresponding (transformed) dose values consistent with Model 2 is similar to that described by O’Quigley et al. (1990); Lee and Cheung (2009) in the context of the continual reassessment method in phase I studies as well as other contexts Lee et al. (2011); Cheung and Elkind (2010).

In our phase II setting, patients are given S₁ = 6 g/m² of ifosamide or S₂ = 12 g/m² of ifosamide over K = 4 cycles of treatment. Suppose q_g is an initial guess (skeleton probability) of a DLT in cycle one for dose S_g. Instead of using dose S_g in Model 2, we use dose values d_g, g = 1, 2, satisfying ln(1 − q_g) = −d_g. For example, if we choose q₁ = 0.10, the resulting d₁ that stands in for 6 g/m² of ifosamide in Model 2 is d₁ = −ln(1 − 0.10) ≈ 0.11. In defining d_g, we have conveniently normalized α to 1.0 if the skeleton probability for toxicity at cycle 1 is correct, making it easier to see deviations from the skeleton in the posterior distribution of α. Skeleton probabilities may be elicited from clinicians, previous animal studies or earlier phase clinical trials.

In Model 1, g₁ (α, Z), will be a known function of covariates, Z, and skeleton probabilities for toxicity will need to be defined for reference values of Z in this relationship. For example, in Section 4, we consider the effect of gender (M, F) on α via g₁ (α, Z) = α₁ + α₂I(F). In this case, we use the male gender as the reference group, normalizing α₁ to one, as before, in obtaining d₁ and d₂ for this group. So the posterior distribution for α₁ deviating from one gives a sense of how on target the skeleton probabilities for the men were on the two doses. The posterior distribution of α₂ gives a sense of the effect of female gender on the toxicity rates for cycle 1.

2.4. Prior selection and posterior distribution

Based on the study design, patients contribute to the likelihood until they experience a DLT or the final K^th cycle is completed. That is, a person with toxicity on cycle K_i gives data $(Y_{i,1}=0,Y_{i,K_i-1}=0,\dots ,Y_{i,K_i}=1,Z_i,d_{i,1})$ and a contribution to the likelihood as in Eq. (4). And a person completing K cycles without toxicity gives data (Y_i,1 = 0, …, Y_i,K−1 = 0, Y_i,K = 0, Z_i, d_i,1) with likelihood contribution as in Eq. (5). In general, subject i on cycle k contributes $L_{i,k}\left(Y_{i,k}\mid \alpha ,\beta ,\rho \right)={\left(p_{i,k}\right)}^{Y_{i,k}}{\left(1-p_{i,k}\right)}^{1-Y_{i,k}}$ to the likelihood, with p_i,k parameterized as in Eq. (1) and interpreted as the the probability of toxicity on cycle k conditional on having no prior DLTs in previous cycles. The resulting likelihood is,

$$\begin{gathered} L(Y\mid \alpha ,\beta ,\rho )=\prod _{i=1}^N\prod _{k=1}^{K_i}{L}_{i,k}\left(Y_{i,k}\mid \alpha ,\beta ,\rho \right)= \\ =\prod _{i=1}^N\prod _{k=1}^{K_i}{\left(1-exp\left[-g_1(\alpha ,Z)\left\{d_{i,1}-g_2(\rho ,Z)d_{i,k-1}\right\}-g_3(\beta ,Z)D_{i,k}\right]\right)}^{Y_{i,k}}\times \\ \times {\left(\exp \left[-g_1(\alpha ,\mathbf{Z})\left\{d_{i,1}-g_2(\rho ,\mathbf{Z})d_{i,k-1}\right\}-g_3(\beta ,Z)D_{i,k}\right]\right)}^{1-Y_{i,k}}. \end{gathered}$$

Our goal lies in estimating the posterior distribution of α, β, and ρ and hence of p_i,k. Prior distributions on these parameters should reflect any auxiliary knowledge, with a large prior variance when this knowledge is limited. We first consider priors in the case with no covariates, so that g₁ (α, Z)=α, g₂ (ρ, Z)=ρ, and g₃ (β, Z) = β. Modifications of this approach for more complex settings will be discussed after this more simple case is described. The priors are programmed using just another Gibbs sampler (JAGS) rjags (Plummer, 2011) package through R (R Development Core Team, 2011). Examples of the code to fit these models is provided in the Supplementary Information.

2.4.1. Special case with no covariates

Recall that with no covariates, α relates to the probability of toxicity on cycle one, with α = 0 implying no toxicity and increasing values of α giving larger toxicity probabilities. The skeleton described in Section 2.3 is calibrated to give α =1 when correctly specified. Hence, α captures both deviation from prior beliefs via the skeleton and the effect of current dose. For many phase II trials the low dose is associated with very little or no toxicity on the first cycle, so that a prior with a prespecified point mass at α = 0 and ranging across nonnegative values is desirable. In the Supplementary Information, we describe our recommended prior on α, with cumulative distribution function (cdf) F_α(α), mean μ_α = 1 and variance ${\sigma }_{\alpha }^2$, as a mixture distribution of a lognormal density and a q_α × 100% point mass at zero. We denote the probability density function (pdf) of the lognormal component of the mixture distribution as g_α (x), with mean μ_g and variance ${\sigma }_g^2$. The cdf of α then becomes $F_{\alpha }(x)=q_{\alpha }+\left(1-q_{\alpha }\right){\int }_0^x{g}_{\alpha }(u)du$. The μ_g parameter of the lognormal distribution shifts as a function of the point mass percentage to maintain a mean one prior; that is, μ_g = (1 − q_α)⁻¹ yields μ_α = 1. The lognormal variance parameter, ${\sigma }_g^2$ depends on the desired variance for the prior mixture, ${\sigma }_{\alpha }^2$, as well as the point mass probability, q_α, that is, ${\sigma }_g^2={\left(1-q_{\alpha }\right)}^{-1}{\sigma }_{\alpha }^2-{\mu }_g^2{q}_{\alpha }$. For convenience, an example of JAGS code for generating this prior is included on lines 12 through 18 of the example in Supplementary Information.

The ρ parameter (0 ≤ ρ ≤ 1) captures dependency in toxicity outcomes within a patient, with values near zero indicating that the current toxicity outcome is virtually unaffected by previous tolerance of dose and values near one indicating an almost certain chance of tolerating previously administered doses. In supplementary material, we develop this prior as a mixture distribution of prespecified q_ρ × 100% point masses at 0 and 1 and a trapezoidal density with height b at zero and slope m = 2(1 − b) comprising the remainder of the distribution over (0, 1). The cdf of ρ is $F_{\rho }(x)=q_{\rho }+(1-2q_{\rho }){{\int }_0^x[b+2(1-b)u]du+q}_{\rho }I(x=1)$. The b parameter governs whether the trapezoidal shape favors low or high values of ρ, with b = 1 reducing to a uniform shape and b = 0 or b = 2 reducing to triangular shapes with positive and negative slopes. The Supplementary Information has an example code in JAGS for this prior on lines 19 through 30.

The remaining prior that we define is for the β parameter, which captures the effect of cumulative dose on the conditional probability of toxicity. When β = 0, the probability of toxicity does not change based on cumulative dose. When β > 0, toxicities are more likely to occur as dose accumulates. The model also allows the possibility of developing an increased tolerance for dose with repeated exposure, that is, β may be negative subject to the constraint that toxicity probabilities remain in the [0, 1] range. A lower bound for β is obtained by noting that d_i,k = d_i,1 for k ≥ 1, in equation (3) giving 0 < α (d_i,1 − ρd_i,1) + β(k − 1)d_i,1 < ∞, ∀k. It is convenient to define the prior in terms of a shared boundary at all cycles, β > −α(1 − ρ)/(K − 1). In particular, for the ifosamide study we investigate later, K = 4 cycles and the lower boundary on the prior for β becomes −α(1 − ρ)/3, depending on α and ρ.

In constructing a prior for β conditional on α and ρ, with cdf F_β (β|α, ρ), we use a Normal(μ, σ²) distribution truncated on the left by −α(1 − ρ)/3. To program this truncated Normal prior with mean μ_β and standard deviation σ_β in JAGS, we need to input mean μ and standard deviation σ of an untruncated Normal distribution giving mean μ_β and variance σ_β upon left truncation at −α(1 − ρ)/3. This can be achieved in the following way using JAGS. Expressions for deriving μ_β and σ_β based on known parameters (μ, σ²) of the untruncated Normal distribution exist (Johnson and Kotz, 1970; Greene, 2003). The variance, σ, of the untruncated distribution is involved in calculating both μ_β and σ_β for the truncated distribution so that arbitrary specification of these prior parameters does not always give a viable untruncated distribution to work with. The R function, findbetaroots, in Supplementary Information solves for parameters (μ, σ²) required by the JAGS program based on desired prior parameters for β, (μ_β, σ_β) or indicates that no possible solution exists for that combination of values. For convenience, Tables 2 and 3 in the Supplementary Information lists helpful examples of prior parameters for β and the corresponding parameters of the untruncated distribution that are used in JAGS. An example of JAGS code for this prior is given in the Supplementary Information on lines 31 through 37 of the example code. We generally recommend large values of ${\sigma }_{\alpha }^2$ and ${\sigma }_{\beta }^2$ to be used in practice; for instance a coefficient of variance (CV) of two or more is a reasonable choice. Our feeling is that a conservative prior mean for β should reflect high cumulative toxicity that the posterior distribution may reduce based on the observed data. Additional supplementary figure shows various cumulative toxicity profiles associated with a range of β values. Our experience indicates that a prior mean of μ_β = 2 and variance of σ_β = 4 works well in practice.

Table 2.

Estimates of parameters based on Model 2 from 500 simulated datasets containing N = 100 patients receiving one of two dose groups d₁ and d₂ over four cycles. Results presented for α = 1 and various combinations of β = (0, 0.2) and ρ = (0.25, 0.75).

Prior mean (SD)	α 1.04 (3.055)	β 2 (2.000)	ρ 0.6227 (0.286)
Case 1 : α = 1, β = 0, ρ = 0.25
Estimated value	0.965	0.133	0.437
Mean bias	−0.035	0.133	0.187
Mean SD	0.326	0.167	0.265
Empirical SD	0.323	0.145	0.136
CI coverage rate	93.0	89.2	99.6
Case 2 : α = 1, β = 0, ρ = 0.75
Estimated value	0.889	0.002	0.602
Mean bias	−0.111	0.002	−0.148
Mean SD	0.315	0.103	0.256
Empirical SD	0.323	0.073	0.137
CI coverage rate	89.8	99.8	99.8
Case 3 : α = 1, β = 0.2, ρ = 0.25
Estimated value	1.002	0.345	0.465
Mean bias	0.002	0.145	0.215
Mean SD	0.338	0.198	0.276
Empirical SD	0.320	0.161	0.129
CI coverage rate	95.2	93.2	99.8
Case 4 : α = 1, β = 0.2, ρ = 0.75
Estimated value	0.891	0.176	0.563
Mean bias	−0.109	−0.024	−0.187
Mean SD	0.315	0.144	0.272
Empirical SD	0.333	0.119	0.133
CI coverage rate	89.8	98.0	99.0

Table 3.

Estimates of conditional probability of no toxicity based on Model 2 from 500 simulated datasets containing N = 100 patients receiving one of two dose groups d₁ and d₂ over four cycles. Results presented for α = 1, β = 0.2, and ρ = 0.75 for different misspecified probability skeletons.

Cycle	1	2	3	4
Scenario 1: d1 = 0.078, d2 = 0.162
d1
Mean estimated ${\widehat{p}}_{ik}$	0.046	0.027	0.036	0.044
Mean bias ${\widehat{p}}_{ik}$	−0.004	0.004	0.003	0.001
Mean SD ${\widehat{p}}_{ik}$	0.015	0.0084	0.009	0.014
Empirical SD ${\widehat{p}}_{ik}$	0.016	0.006	0.009	0.013
Coverage rate of true pik	93.2	97.6	94.8	96.2
Average patients	47.5	46.432	44.8	42.9
d2
Mean estimated ${\widehat{p}}_{ik}$	0.092	0.056	0.073	0.089
Mean bias ${\widehat{p}}_{ik}$	−0.008	0.009	0.007	0.004
Mean SD ${\widehat{p}}_{ik}$	0.030	0.017	0.018	0.027
Empirical SD ${\widehat{p}}_{ik}$	0.031	0.013	0.018	0.026
Coverage rate of true pik	94.4	97.6	94.6	96.6
Average patients	45.0	42.9	40.1	36.7
Scenario 2: d1 = 0.062, d2 = 0.162
d1
Mean estimated ${\widehat{p}}_{ik}$	0.039	0.023	0.031	0.038
Mean bias ${\widehat{p}}_{ik}$	−0.011	4e-04	−0.002	−0.005
Mean SD ${\widehat{p}}_{ik}$	0.013	0.007	0.008	0.011
Empirical SD ${\widehat{p}}_{ik}$	0.013	0.005	0.008	0.012
Coverage rate of true pik	84.2	97.6	93.4	92.8
Average patients	47.5	46.4	44.8	42.9
d2
Mean estimated ${\widehat{p}}_{ik}$	0.098	0.059	0.078	0.096
Mean bias ${\widehat{p}}_{ik}$	−0.002	0.013	0.012	0.010
Mean SD ${\widehat{p}}_{ik}$	0.032	0.018	0.019	0.029
Empirical SD ${\widehat{p}}_{ik}$	0.033	0.014	0.019	0.028
Coverage rate of true pik	93.0	96.0	92.2	95.0
Average patients	45.0	42.9	40.1	36.7

A few more definitions using Stieltjes notation help in characterizing the posterior distribution. Let h_α(α) = q_αI(α = 0) + {1 − q_α}I(α > 0)g(α) and h_ρ(ρ) = q_ρI(ρ = 0) + {1 − 2q_ρ}I(0 < ρ < 1){b + 2(1 − b)ρ} + q_ρI(ρ = 1) capture either a probability mass or a density function as appropriate. In addition define dF_α(α) = h_α(α)dα and dF_ρ(ρ) = q_ρI(ρ = 0) + {1 − 2q_ρ}I(0 < ρ < 1){b + 2(1 − b)ρ}dρ + q_ρI (ρ = 1). Also let f_β(β|α, ρ) be the prior density function of β. The posterior distribution for α, β and ρ given the observed data Y is then,

$$f(\alpha ,\beta ,\rho \mid Y)=\frac{{}_{i=1}^N{}_{k=1}^{K_i}{L}_{i,k}\left(Y_{i,k}\mid \alpha ,\beta ,\rho \right)f_{\beta }(\beta \mid \alpha ,\rho )h_{\alpha }(\alpha )h_{\rho }(\rho )}{\begin{array}{lllll}1& \infty & \infty & N& K_i\\ 0& 0& \frac{-\alpha (1-\rho )}{K-1}& i=1& k=1\end{array}{L}_{i,k}\left(Y_{i,k}\mid \alpha ,\beta ,\rho \right)f_{\beta }(\beta \mid \alpha ,\rho )d\beta ]d{F}_{\alpha }(\alpha )d{F}_{\rho }(\rho )}.$$ (8)

The posterior distribution of α, β, ρ from Model 2 can be estimated via Markov Chain Monte Carlo (MCMC) methods (Robert and Casella, 1999) using JAGS (Plummer, 2011) called in R (R Development Core Team, 2011). JAGS includes several algorithms for sampling from the posterior distributions produced from the MCMC iterations, for instance the standard Gibbs sampler is available for this purpose. A brief review of the MCMC and Gibbs sampler methods is located in the Supplementary Information. Parallel chains starting from different initial values for each parameter (α, β, ρ) are followed through to convergence after an appropriate burn-in period. After convergence the posterior distributions of the parameters are available as well as functions of these parameters, such as the desired conditional toxicity profiles. The mean of the posterior distributions are used as estimates of the quantities of interest.

2.4.2. Covariate specific priors

Priors from Section 2.4.1 can be easily extended to allow dependence of α, ρ, β on Z in Model 1. As an instructive example, we again consider the case where Z includes gender (M, F) and dose group (d₁, d₂) so that g₁(α, Z) = α₁I (M, d₁) + α₂I (M, d₂) + α₃I (F, d₁) + α₄I (F, d₂) for a total of four required priors. Each of these four priors can be built just as in Section 2.4.1 if there is no prior information suggesting deviations from Model 2, that is the model that parameterizes a single α to account for dose-toxicity on cycle 1. We include an additional subscript to distinguish between priors for α. That is, with a $q_{{\alpha }_j}$ point mass at zero, the prior for α_j, j = 1, …, 4 has CDF $F_{{\alpha }_j}(x)=q_{{\alpha }_j}+{\left\{1-q_{{\alpha }_j}\right\}}_0^x{g}_{{\alpha }_j}(u)du$, where density function $g_{{\alpha }_j}(u)$ is a lognormal $\left\{{\left(1-q_{{\alpha }_j}\right)}^{-1},{\sigma }_{{\alpha }_j}^2={\left(1-q_{{\alpha }_j}\right)}^{-1}{\sigma }_{{\alpha }_j}^2-{\mu }_{g_j}^2{q}_{{\alpha }_j}\right\}$. When prior information on gender related toxicity is available, prior means of α₃ and α₄ may be chosen to reflect this additional knowledge. The priors for α₁ and α₂ are generally left with a mean of one since the skeleton discussed in Section 2.3 calibrated these values to one based on initial assumptions about dose-toxicity on cycle 1. In the case where we desire a prior with mean, ${\mu }_{{\alpha }_j}$, and variance, ${\sigma }_{{\alpha }_j}^2$, we would define density function $g_{{\alpha }_j}(x)$ as lognormal $\left\{{\mu }_{{\alpha }_j}{\left(1-q_{{\alpha }_j}\right)}^{-1},\left(1-q_{{\alpha }_j}\right)\left({\mu }_{g_j}^2{q}_{{\alpha }_j}+{\sigma }_{g_j}^2\right)\right\}$.

Prior parameterization of ρ is technically straightforward using trapezoidal shapes described in Section 2.4.1. However for limited sample sizes it makes sense to assume a common prior for ρ. Priors for α and ρ affect prior definition of β. Recall that when parameterizing the prior for β in Section 2.4.1, the range of the prior was [−α(1 − ρ)/(K − 1), ∞), with negative β indicating an increased dose tolerance upon repeated exposure and positive β indicating increased toxicity with accumulating dose. Continuing our instructive example, suppose g₃(β, Z) = β₁I (M, d₁) + β₂I (M, d₂) + β₃I (F, d₁) + β₄I (F, d₂). Priors for each of the four β^′_js can be constructed as in Section 2.4.1, provided that the lower bound of each β_j is maintained to be consistent with values of α_j and ρ for those with the same covariates, Z. In particular, in the case with gender and dose influencing all parameters except ρ, the range of β_j is restricted on the left by −α_j(1 − ρ)/3, j = 1, …, 4.

Depending upon the number of priors set up on α, ρ, β the likelihood and the posterior distribution will change accordingly in Eq. (8).

For instance, assuming four subgroups for α_j and β_j, j = 1, …, 4 that correspond to levels of gender and dose and assuming a common ρ across all Z, the posterior distribution, f(α₁,…,α₄,β₁,…,β₄, ρ|Y), becomes

$$\frac{{{\prod }_{j=1}^4{\mathrm{ }}_{i=1}^{N_j}}_{k=1}^{K_i}{L}_{i,k}\left(Y_{i,k}\mid {\alpha }_j,{\beta }_j,\rho \right)f_{{\beta }_j}\left({\beta }_j\mid {\alpha }_j,\rho \right)h_{{\alpha }_j}\left({\alpha }_j\right)h_{\rho }(\rho )}{\begin{array}{llllll}1\hfill & 4\hfill & \infty \hfill & \infty \hfill & N_j\hfill & K_i\hfill \\ 0\hfill & j=1\hfill & 0\hfill & \frac{-{\alpha }_j(1-\rho )}{3}\hfill & i=1\hfill & k=1\hfill \end{array}{L}_{i,k}\left(Y_{i,k}\mid {\alpha }_j,{\beta }_j,\rho \right)f_{{\beta }_j}\left({\beta }_j\mid {\alpha }_j,\rho \right)d{\beta }_j\}d{F}_{{\alpha }_j}\left({\alpha }_j\right)]d{F}_{\rho }(\rho )},$$ (9)

where ${\sum }_{j=1}^4{N}_j=N$ is the sum of patients in the four categories of gender and dose.

2.5. Model selection

There is no restriction requiring the same covariates be included in parameterizations of g₁(α, Z), g₂(ρ, Z), and g₃(β, Z). We recommend two common model selection criteria: (1) a plot of observed and predicted values of toxicity, along with 95% credible bands for the true probabilities of toxicity over the cycles. (2) The deviance information criteria (DIC) (Spiegelhalter et al., 2002) is calculated by adding the effective number of parameters (p_D) to the expected deviance, where the expected deviance is the deviance at the posterior mean parameter values and the effective number of parameters are estimated using the approach suggested by Plummer (2002, 2008). Smaller DIC values indicate the preferred model.

2.6. Treatment comparisons based on posterior distributions

Clinicians may be interested in various posterior distributions relating to overall toxicity and expected dose received by treatment group, and the Bayesian paradigm is very flexible in producing this information. For instance, clinicians may wish to consider the posterior probability of no toxicity on all the K cycles for a particular treatment group that can be estimated using the following expression,

$$P\left(Y_{i,1}=0,\dots ,Y_{i,K-1}=0,Y_{i,K}=0\right)=\prod _{j=1}^K\left(1-p_{i,j}\right)$$

Posterior distributions of differences between overall probability of toxicity on different dose levels can also be easily estimated and displayed.

3. Operating characteristics

To study operating characteristics of Model 2, we consider a trial with two dose groups (S₁, S₂) receiving a maximum of K = 4 cycles with 50 patients per group for a total of N = 100. Following Section 2.3, skeleton probabilities are set at (0.05, 0.10), that is, 5% and 10% of patients are expected to have a DLT on the first cycle in the low and high dose groups, respectively. Then the transformed doses, d_g, g = 1, 2, used to stand in for S_g, g = 1, 2, in Model 2 become d₁ = −ln(1 − 0.05) and d₂ = −ln(1 − 0.10). Model 2 defines conditional probabilities of toxicity during the trial with α = 1 and varying values of β = {0, 0.2} and ρ = {0.25, 0.75}, that is four different simulated cases. Simulated toxicity outcomes across cycles are based on Bernoulli(p_i,k) random variables until a DLT is observed or the 4^th cycle is completed. Five hundred independent datasets (simulation replications) were created to assess coverage rates of credible intervals, bias, and standard error.

Priors are set on α, β, ρ as in Section 2.4.1 for Model 2. In particular, the prior on α is a mixture with point mass q_α = 4% on zero and a lognormal density component with parameters μ_g = 1.04 and ${\sigma }_g^2=9.33$ giving prior mean and variance for α, μ_α = 1 and ${\sigma }_{\alpha }^2=9$. The prior for ρ is a mixture with point masses q_ρ = 2% at zero and one and an intercept b = 0.20 trapezoidal density, resulting in prior mean and variance μ_ρ = 0.6280 and ${\sigma }_{\rho }^2=0.0736$. This prior indicates a moderate to high correlation in toxicity responses within patient. The prior set on β is a Normal distribution truncated at −1/3 having mean, μ_β = 2 and variance, ${\sigma }_{\beta }^2=4$, providing a coefficient of variance of two. Since truncation of this distribution of β is conditional on prior values of α and ρ, current values from the MCMC simulation are used for the truncation point at each sampling. Based on Model 2, toxicity probabilities are also sampled and monitored for convergence; an adaptive phase of 1000 samples is used to choose the best sampling algorithm and an additional 10,000 samples are discarded as part of the burn-in period.

The conditional probability of toxicity for each of the four simulated cases are presented in Table 1 and displayed in Fig. 3. Each of the four columns of Table 1 indicate treatment cycles during the study. Rows are separated according to different parameter selections (cases) and dose group within case, d₁ or d₂. Reported values are (1) the estimated conditional probability of toxicity for an arbitrary patient i at cycle k, ${\widehat{p}}_{i,k}$, (2) the mean bias, ${\widehat{p}}_{i,k}-p_{i,k}$, across 500 replications (3) the mean standard deviation (SD) of ${\widehat{p}}_{i,k}$ across 500 replications, (4) the empirical SD of the 500 ${\widehat{p}}_{i,k}$ estimates, (5) the credible interval coverage rate of p_i,k across the 500 replications and (6) the mean number of patients who enter the following cycle toxicity free. Within a particular case (1–4), the same posterior values of α, β, and ρ are used to calculate finite sample characteristics of the eight cells of dose and cycle combinations.

Table 1.

Estimates of conditional probability of toxicity based on Model 2 from 500 simulated datasets containing N = 100 patients receiving one of two dose groups d₁ and d₂ over four cycles. Results presented for α = 1 and various combinations of β = (0, 0.2) and ρ = (0.25, 0.75).

Cycle	1	2	3	4
Case 1: α = 1 β = 0 ρ = 0.25
d1
Mean estimated ${\widehat{p}}_{ik}$	0.0480	0.0335	0.0400	0.0464
Mean bias ${\widehat{p}}_{ik}$	−0.0020	−0.0043	0.0023	0.0087
Mean SD ${\widehat{p}}_{ik}$	0.0158	0.0093	0.0097	0.0153
Empirical SD ${\widehat{p}}_{ik}$	0.0157	0.0077	0.0092	0.0145
Coverage rate of pik	93.0	94.6	95.2	92.8
Average patients	47.5	45.8	44.0	42.4
d2
Mean estimated ${\widehat{p}}_{ik}$	0.0956	0.0674	0.0803	0.0926
Mean bias ${\widehat{p}}_{ik}$	−0.0044	−0.0086	0.0043	0.0166
Mean SD ${\widehat{p}}_{ik}$	0.0305	0.0183	0.0191	0.0297
Empirical SD ${\widehat{p}}_{ik}$	0.0305	0.0152	0.0181	0.0282
Coverage rate of pik	93.0	94.6	95.2	92.8
Average patients	45.1	41.7	38.5	35.6
Case 2: α = 1 β = 0 ρ = 0.75
d1
Mean estimated ${\widehat{p}}_{ik}$	0.0443	0.0161	0.0162	0.0163
Mean bias ${\widehat{p}}_{ik}$	−0.0057	0.0034	0.0035	0.0036
Mean SD ${\widehat{p}}_{ik}$	0.0153	0.0071	0.0059	0.0085
Empirical SD ${\widehat{p}}_{ik}$	0.0157	0.0051	0.0051	0.0073
Coverage rate of pik	89.8	99.2	94.6	96.2
Average patients	47.5	46.9	46.8	45.8
d2
Mean estimated ${\widehat{p}}_{ik}$	0.0883	0.0327	0.0330	0.0331
Mean bias ${\widehat{p}}_{ik}$	−0.0117	0.0067	0.0070	0.0071
Mean SD ${\widehat{p}}_{ik}$	0.0297	0.0142	0.0119	0.0170
Empirical SD ${\widehat{p}}_{ik}$	0.0305	0.0102	0.0103	0.0147
Coverage rate of pik	89.8	99.2	94.6	96.2
Average patients	44.9	43.9	42.7	41.6
Case 3: α = 1 β = 0.2 ρ = 0.25
d1
Mean estimated ${\widehat{p}}_{ik}$	0.0498	0.0435	0.0603	0.0765
Mean bias ${\widehat{p}}_{ik}$	−2e−04	−0.0040	0.0030	0.0096
Mean SD ${\widehat{p}}_{ik}$	0.0163	0.0103	0.0123	0.0191
Empirical SD ${\widehat{p}}_{ik}$	0.0155	0.0086	0.0122	0.0183
Coverage rate of pik	95.2	95.0	95.0	92.6
Average patients	47.4	45.1	42.4	39.5
d2
Mean estimated ${\widehat{p}}_{ik}$	0.0991	0.0872	0.1195	0.1501
Mean bias ${\widehat{p}}_{ik}$	−9e–04	−0.0081	0.0054	0.0176
Mean SD ${\widehat{p}}_{ik}$	0.0315	0.02	0.0235	0.0359
Empirical SD ${\widehat{p}}_{ik}$	0.0301	0.0169	0.0234	0.0344
Coverage rate of pik	95.2	95.0	95.0	92.6
Average patients	44.9	40.7	36.1	31.3
Case 4: α = 1 β = 0.2 ρ = 0.75
d1
Mean Estimated ${\widehat{p}}_{ik}$	0.0444	0.0272	0.0359	0.0444
Mean Bias ${\widehat{p}}_{ik}$	−0.0056	0.0044	0.0031	0.0018
Mean SD ${\widehat{p}}_{ik}$	0.0153	0.0084	0.0091	0.0139
Empirical SD ${\widehat{p}}_{ik}$	0.0162	0.0064	0.0088	0.0134
Coverage rate of pik	89.8	98.0	95.2	96.4
Average patients	47.5	46.5	44.9	43.1
d2
Mean Estimated ${\widehat{p}}_{ik}$	0.0885	0.0549	0.0721	0.0887
Mean Bias ${\widehat{p}}_{ik}$	−0.0115	0.0086	0.0060	0.0031
Mean SD ${\widehat{p}}_{ik}$	0.0297	0.0166	0.0180	0.0271
Empirical SD ${\widehat{p}}_{ik}$	0.0315	0.0128	0.0173	0.0261
Coverage rate of pik	89.8	98.0	95.2	96.4
Average patients	45.1	42.9	40.1	36.8

Notes

^a1SD refers to the standard deviation of the estimates.

Figure 3.

Simulation study results. Plot of conditional P(Toxicity), based on Eq. (2) for two dose levels with α = 1 β = (0, 0.2) and ρ = (0.25, 0.75). Cases 1 through 4 in the Table (1). The panels in the first and second column correspond to β = 0 and β = 0.2 respectively while the top row corresponds to ρ = 0.25 and the bottom row to ρ = 0.75. The solid circles and triangles are the true values of the conditional P(toxicity) at each of the cycles on the lower and higher dose group respectively. The corresponding hollow circles and triangles are the average of the estimates of the values and the average of the limits of the 95% credible bands.

Table 1 indicates that estimates of the conditional probability of toxicity have very low bias and that mean and empirical standard errors are comparable. This low bias is evident in Fig. 3 where solid shapes (true conditional probabilities) and hollow shapes (estimated conditional probabilities) are very close to one another. In cases 1 and 3, where ρ takes on the lower value of 0.25, there is a higher decrease in the average number of patients making it through successive cycles as additional patients exhibit toxicity patterns that remove them from the study. Cases 2 and 4, with ρ = 0.75, have a higher tendency to avoid toxicity once they have tolerated their first cycle. In cases 3 and 4, where β = 0.2, there is a tendency for slightly more patients to discontinue due to accumulated toxicity, particularly in cycles 3 and 4. Case 4 shows the most impact of patients dropping out due to accumulating toxicity in cycles 3 and 4, since in this case the high value of ρ = 0.75 usually causes those with single dose susceptibility to be eliminated in cycle one rather than later cycles.

The major focus of estimation in these small studies is typically on estimated probabilities of toxicity, which seem to have little bias regardless of the model’s ability to clearly identify individual parameter estimates. However, we summarize results from simulation studies of parameter estimates $\widehat{\alpha }$, $\widehat{\beta }$, and $\widehat{\rho }$ in Table 2. Each row corresponds to one of the four simulated cases and presents (1) the mean of the estimated values, (2) the mean bias of the estimates from the true value of the parameter, (3) the mean SD of the parameter estimates, (4) the empirical SD from the 500 simulated datasets, and (5) the coverage rate for the true parameter value in the credible intervals.

Higher values of ρ make it easier to isolate information on β since after tolerating cycle 1, toxicity observed after the first cycle is more likely to be caused by accumulating toxicity captured by β. This is reflected in Table 2 cases 2 and 4 where ρ = 0.75 and bias for β is at its lowest. Lower values of ρ allow the current dose on every cycle to play a higher role in the manifestation of the toxicity, so that information on α is better identified. Table 2 cases 1 and 3, where ρ = 0.25, provide the lowest bias in estimation of α by correctly attributing toxicity to the effect of the current dose. The ability to estimate ρ, when the prior is not compatible with the true model, is a challenge in these small studies. In additional simulations, not shown, larger sample sizes do improve estimation of ρ, as well as the other parameters. As in all early phase studies, model assumptions are relied upon in making inferences. In the following section, we perform additional sensitivity analyses for misspecification of the skeleton probabilities on conditional toxicity probability estimation.

3.1. Sensitivity to choice of skeleton probabilities

Given fixed values of α = 1, β = 0.2, and ρ = 0.75, as in case 4 above, data were simulated as before. Recall that the probabilities of toxicity during the first cycle on doses 1 and 2 are 0.05 and 0.10 and that these probabilities were assumed in creating the skeleton used for analysis in the previous section. As opposed to the previous section, this section misspecifies the probability skeleton when performing the analysis. In scenario 1, a skeleton that is 1.5 times higher than that used to generate the data is used in the analysis. That is, the probabilities of toxicity on cycle one are assumed to be 0.075 and 0.150 for doses 1 and 2, respectively. This user misspecified skeleton results in different values of d₁ = −ln(1 − 0.075) = 0.0780 and d₂ = −ln(1 − 0.150) = 0.1625, respectively, used in the analysis.

In scenario 2, the probabilities of toxicity on cycle 1 were assumed to be 0.06 and 0.15 for the two dose levels, that is 20% and 50% overestimates of toxicity on cycle 1 for dose levels 1 and 2, respectively. This user misspecified skeleton results in values of d₁ = −ln(1 − 0.06) = 0.0619 and d₂ = −ln(1 − 0.150) = 0.1625, respectively, being used in the analysis. Otherwise, the assumed priors and estimation procedure remained unchanged from the previous section.

Table 3 provides results on estimation of the conditional probabilities of toxicity across 500 replications in the simulation study. The columns correspond to the four cycles. Within different skeleton misspecifications (scenarios 1 and 2) the rows are grouped by the two dose levels, d₁ or d₂. Reported values are (1) the estimated conditional probability of toxicity for an arbitrary patient i at cycle k, ${\widehat{p}}_{i,k}$, (2) the mean bias, ${\widehat{p}}_{i,k}-p_{i,k}$, across 500 replications (3) the mean SD of ${\widehat{p}}_{i,k}$ across 500 replications, (4) the empirical SD of the 500 ${\widehat{p}}_{i,k}$ estimates, (5) the credible interval coverage rate of p_i,k across the 500 replications, and (6) the mean number of patients who enter the following cycle toxicity free. Within a particular scenario (1 or 2), the same posterior values of α, β, and ρ are used to calculate finite sample characteristics of the eight cells of dose and cycle combinations.

Results are comparable to those from case 4 in Table 1, where the correct skeleton was used in the analysis. The only exception is seen in scenario 2, cycle 1, dose 1, where bias is higher and coverage is lower than desired. Since the empirical and mean standard deviations (SD) are very close in this case, the coverage is likely being effected by the bias for this term.

4. Application to the ifosamide study

The original ifosamide study was a phase II randomized clinical trial comparing the toxicity and efficacy of doxorubicin with high-dose ifosamide or standard-dose ifosamide in patients with soft-tissue sarcoma (Worden et al., 2005; Chugh et al., 2007). The treatment was given for four consecutive days at the beginning of each 21 day cycle. The original study considered six cycles for metastatic disease and four cycles for localized disease but for convenience we consider just the first four cycles for each group. We evaluate 77 patients with data on toxicity, where 39 of these were randomized to the standard 6 g/m² ifosamide dose group and 38 of these were randomized to receive 12 g/m² of ifosamide. We use a patient’s minimum hemoglobin (HGB) value during a cycle to define a DLT in this example, so that a patient is removed from the study if their HGB value drops below 8 mg. This criteria is defined as a grade 3/4 toxicity by NCI. Table 4 and Fig. 1 present the empirical data from the study with conditional probabilities at each of the four cycles for the two dose groups.

Table 4.

Grade 3/4 dose-limiting toxicities (DLTs) observed when the hemoglobin levels dropped below 8 mg on the two dose groups in patients completing the previous cycle without a DLT over the four cycles of treatment.

	Cycle 1	Cycle 2	Cycle 3	Cycle 4
dose 6	2/39	2/35	2/29	3/24
dose 12	3/38	4/31	9/24	6/15

Based on DIC criteria, Model 2 was improved by allowing differential cumulative effects of dose by dose group resulting in Model 10 of the form:

$$\ln \left(1-p_{i,k}\right)=-\alpha \left(d_{i,1}-\rho d_{i,k-1}\right)-{\beta }_1{D}_{i,k}I\left(d_1\right)-{\beta }_2{D}_{i,k}I\left(d_2\right).$$ (10)

Resulting empirical and model-based conditional probabilities of toxicity by dose group on each of four cycles are shown in Fig. 4A, along with estimated 95% credible intervals for the conditional probabilities of toxicity as given by Model 10. Priors for α and ρ used to perform the analysis are identical to those used in Section 3. Priors for β₁ and β₂ are identical to the prior used for β in Section 3. Parameter estimates shown in Table 5 indicate a particularly high cumulative effect of the dose on patients in the high group.

Figure 4.

Application of the Markov model to the ifosamide dataset

Table 5.

Parameter estimates of the parameters with SD obtained using Model 10.

	Prior mean (SD)	Estimate	SD
α	1.04 (3.05)	0.76	0.33
β1	2.0 (2.0)	0.81	0.37
β2	2.0 (2.0)	1.66	0.44
ρ	0.62 (0.28)	0.54	0.30
DIC	–	167.90	–

Following Section 2.2 Eq. (6), the estimated expected number of completed cycles for the low and high dose groups, respectively, are 3.69 (SD = 0.08) and 3.18 (SD = 0.13) (empirical values 3.68 and 3.23 observed on average). Thus the expected total doses are 22.119 (SD = 0.481) g/m² and 38.259 (SD = 1.563) g/m² for the low and high dose groups, respectively, (22.06 g/m² and 38.70 g/m² observed empirically on average). A patient in the high-dose group tends to receive more total ifosamide than a patient in the low-dose group before a DLT, but is less likely to successfully complete all four cycles without a DLT 28.7% and 70.4%, respectively.

Upon further study, DLTs in the high-dose group are especially high in women, but not necessarily men, as seen in Fig. 4B and Table 6. DIC criteria suggested further improvement in the model with inclusion of gender terms for the parameter α and gender by dose group terms for the cumulative toxicity. To account for the steep quadratic trend in toxicities for the females on the high-dose group a quadratic (squared) term was also included in the model and results in Model 11 as follows,

$$\begin{gathered} \log \left(1-p_{i,k}\right)=-\left\{{\alpha }_1+{\alpha }_{2}I(F)\right\}\left(d_{i,1}-\rho d_{i,k-1}\right)-{\beta }_1{D}_{i,k}I\left(M,d_1\right)- \\ -{\beta }_2{D}_{i,k}I\left(F,d_1\right)-{\beta }_3{D}_{i,k}I\left(M,d_2\right)-{\beta }_4{D}_{i,k}^{2}I\left(F,d_2\right). \end{gathered}$$ (11)

Table 6.

Dose-limiting toxicities (DLTs), grouped by gender, in patients completing the previous cycle without a DLT when the hemoglobin levels dropped below 8 mg over four cycles of the treatment.

	Cycle 1	Cycle 2	Cycle 3	Cycle 4
	Males
dose 6	1/20	1/17	0/14	1/13
dose 12	0/18	2/15	3/12	1/9
	Females
dose 6	1/19	1/18	2/15	2/11
dose 12	3/20	2/16	6/12	5/6

Figures 4C and D display the empirical and model-based conditional probabilities of toxicity, along with estimated 95% credible intervals for the conditional probabilities of toxicity as given by Model 11, by gender on the low and high dose groups, respectively. Priors were identical to those used in Section 3. That is, priors on α₁ and α₂ were identical to the prior on α in Section 3, the prior on ρ was left unchanged, and priors on β₁, β₂, β₃, and β₄ were identical to the prior used for β in Section 3. Parameter estimates of the model shown in Table 7 indicate that the toxicity due to cumulative effect of the dose is high in females in comparison to males on both the dose groups.

Table 7.

Parameter estimates of the parameters with SD obtained using Model 11.

	Prior mean (SD)	Estimate	SD
α1	1.04 (3.05)	0.49	0.33
α2	1.04 (3.05)	0.60	0.53
β1	2.0 (2.0)	0.58	0.42
β2	2.0 (2.0)	1.33	0.65
β3	2.0 (2.0)	0.97	0.43
β4	2.0 (2.0)	15.62	4.70
ρ	0.62 (0.28)	0.55	0.30
DIC	–	160.49	–

The total expected completed cycles for the low-dose female group is 3.54 (SD = 0.12, 3.62 observed) with a total expected dose of 21.22 g/m² (SD = 0.74, 21.71 g/m² observed). Women on the high dose are expected to complete 2.91 cycles (SD = 0.17, 2.97 observed) with a total expected dose of 34.87 g/m² (SD = 2.05, 35.59 g/m² observed). Men are expected to complete roughly the same number of cycles in the low and high dose groups, 3.78 (SD = 0.09, 3.58 observed) and 3.44 (SD = 0.16, 3.36 observed) cycles, respectively, for total expected doses of 22.66 g/m² (SD = 0.56, 21.49 g/m² observed) and 41.29 g/m² (SD = 1.90, 40.28 g/m² observed) in the corresponding dose groups.

A generalized linear-mixed model (GLMM) was fit to the data as an alternative to compare benefits from using the Markov model 2 to standard methods. A simple logistic model with dose group and cycle as main effects and a random effect to account for the repeated measures provided model fit as shown in Fig. 5A and Table 8. The 95% confidence intervals seem a little wide as compared to Fig. 4C and D. When gender was included as a covariate the model fit from this model is depicted in Fig. 5B and C for males and females in the high and low dose groups and in Table 9. In both instances the estimate of the variance of the random effect was negligible because of the limited amount of values of Y = 1 in the data, with at most one per person.

Figure 5.

Application of the Markov model to the ifosamide dataset with GLLM.

Table 8.

Probability estimates of the model with random effect, fixed effects of dose, and cycle.

Dose	Cycle	Estimate (SD)	Lower CI	Upper CI
6	1	0.027 (0.014)	0.0004	0.0546
6	2	0.049 (0.019 )	0.0127	0.0866
6	3	0.088 (0.028)	0.0318	0.1443
6	4	0.151 (0.052)	0.0471	0.2559
12	1	0.101 (0.035)	0.0313	0.1701
12	2	0.172 (0.038)	0.0957	0.2473
12	3	0.277 (0.053)	0.1716	0.3819
12	4	0.414 (0.091)	0.2328	0.5959

Table 9.

Probability estimates of model with random effect, fixed effects of dose, gender, and cycle.

Dose	Cycle	Gender	Estimate (SD)	Lower CI	Upper CI
6	1	0	0.038(0.020)	0	0.077
6	2	0	0.073(0.028)	0.017	0.129
6	3	0	0.136(0.044)	0.047	0.224
6	4	0	0.237(0.081)	0.077	0.398
6	1	1	0.011(0.007)	−0.003	0.026
6	2	1	0.022(0.012)	−0.001	0.046
6	3	1	0.044(0.020)	0.004	0.084
6	4	1	0.083(0.038)	0.006	0.159
12	1	0	0.147(0.051)	0.045	0.249
12	2	0	0.255(0.059)	0.138	0.371
12	3	0	0.404(0.078)	0.247	0.56
12	4	0	0.573(0.109)	0.356	0.79
12	1	1	0.048(0.024)	0.001	0.095
12	2	1	0.090(0.034)	0.023	0.157
12	3	1	0.164(0.053)	0.059	0.269
12	4	1	0.280(0.092)	0.097	0.464

The Chi-squared value from the Markov model is 1.23 as compared to 2.48 from the GLMM without gender as a covariate. When gender is used as a covariate the Chi-squared values from the Markov model and GLMM are 6.47 and 8.28, respectively. Since a smaller Chi-squared value indicates a better fit between the observed and predicted values the Markov model provides a better fit to the data. In comparing the results from both the Markov model and the GLMM a better fit is provided by the Markov model and the unsatisfactory zero variance component of the random effects makes the Markov model preferred.

5. Discussion

We have presented a novel conditional probability model for the dose toxicity relationship in data arising from a Phase II study setting having patients with multiple cycles of the same dose over their treatment course. The conditional nature of the model takes into account that patients having a DLT on a particular cycle will not continue to further cycles. The use of ρ allows dependence of toxicity in a current cycle to depend on tolerance on previous cycles. The α and β parameters capture the effect of current and cumulative dose effects at each cycle. This three parameter model may be all that is estimable in small studies, but the model offers flexibility to include additional parameters to account for covariate-dependent effects on toxicity.

Priors and skeletons described in this work offer a wide variety of prior beliefs to be included in the analyses. Our investigation of misspecified skeleton probabilities showed very little effect on model performance.

One limitation of the model is that there must be sufficient cycles in the study to allow plausible estimation of ρ and β parameters that are based on data beyond cycle one; two cycles would not be sufficient to disentangle the effects of cumulative dose from tolerance to previous dose. As most studies of this type have between three and six cycles of therapy, this limitation should not impede use of the model in practice.

Extensions to this work could include interim looks at the posterior distribution of toxicity differences as described in Section 2.6. Our method does not preclude one from calculating the posterior distribution at frequent intervals during the trial, but to date we have not formally investigated this.

In the phase II setting planning to administer the same dose in all cycles is the norm, but in practice dose reductions do occur, based on patient responses or other factors. In phase I trials planned intrapatient dose escalation or deescalation could be considered, but in practice is not. Most phase I trials consider only the dose and the toxicity from the first cycle in the analysis, and data from subsequent cycles is not explicitly used. A different approach to the design of phase I trials is to allow planned intrapatients dose changes, with the analysis based on data from all cycles. We are currently investigating this, using an adaptation of the model presented in this paper.