This manuscript was accepted for publication in the Journal of Statistical Software.

1 Background

Confirmatory clinical trials are conducted in a strictly regulated environment. A key quality criterion put forward by the relevant agencies (US Food and Drug Administration and others (2019), Committee for Medicinal Products for Human Use and others (2007)) for a study that is supposed to provide evidence for the regulatory acceptance of a new drug or treatment is strict type one error rate control. This requirement was often seen as conflicting with the perceived need to make trials more flexible by, e.g., early stopping for futility, group-sequential enrollment, or even adaptive sample size recalculation. An excellent historical review of the development of the field of adaptive clinical trial designs and the struggles along the way is given in Bauer et al. (2015).

In this manuscript, the focus lies exclusively on adaptive two-stage designs with one unblinded interim analysis. Both early stopping for futility and efficacy are allowed and the final sample size as well as the critical value to reject the null hypothesis is chosen in a data-driven way. There is a plethora of methods for modifying the design of an ongoing trial based on interim results without compromising type one error rate control (Bauer et al. 2015) but the criteria for deciding which adaptation should be performed during an interim analysis and when to perform the interim analysis are still widely based on heuristics. Bauer et al. mention this issue of guiding adaptive decisions at interim in a principled (i.e., ‘optimal’) way by stating that ‘[t]he question might arise if potential decisions made at interim stages might not be better placed to the upfront planning stage.’ Following Mehta and Pocock (2011), Jennison and Turnbull (2015) developed a principle approach to optimal interim sample size modifications, i.e., to conduct the interim decision (conditional on interim results) such that it optimizes an unconditional performance score. Their approach, however, was still restricted unnecessarily. Recently, Pilz et al. (2019) extended the work to a fully general variational problem where the optimization problem for any given performance score (optionally under further constraints) is solved over both the sample size adaptation function and the critical value function and the time point of the interim decision simultaneously. This approach is an application of ideas which have been put forward in single-arm trials with binary endpoint for several years (Englert and Kieser (2013), Kunzmann and Kieser (2016), Kunzmann and Kieser (2020)) to a setting with continuous test statistics. Clearly, by relaxing the problem to continuous sample sizes and test statistics, the theory becomes much more tractable, and important connections between conditional and unconditional optimality can be discussed much easier (Pilz et al. 2019).

A key insight from this recent development is the fact that the true challenge in designing an adaptive trial is less the technical methodology for controlling the type one error rate but rather the choice of the optimality criterion. This issue is much less pressing in single-stage designs since most sensible criteria will be equivalent to minimizing the overall sample size. Thus, in this case, a ‘design’ is often completely specified by given power and type one error rate constraints. For the more complex adaptive designs, however, there are much more sensible criteria (minimize maximal sample size, expected sample size, expected costs, etc.) and the balance between conditional and unconditional properties must be explicitly specified (cf. Section 6). This added complexity might be seen as daunting by practitioners, but it is also a chance for tailoring adaptive designs more specifically to a particular situation. The R-package (R Core Team 2019) adoptr aims at providing a simple yet customizable interface to specifying a broad class of objective functions and constraints for single- or two-arm, one- or two-stage designs with approximately normally distributed endpoints. The goal of adoptr is to enable relatively easy experimentation with different notions of optimality to shift the focus from how to optimize to what to optimize.

In the following, we first give a definition of the problem setting addressed in adoptr and the technicalities of translating the underlying variational problem to a simple multivariate optimization problem before motivating the need for an R-package. We then present the core functionality of adoptr before addressing the issue of facilitating validation of open-source software in a regulated environment and discussing potential future work on adoptr.

2 Setting

We consider the problem of a two-stage, two-arm design to establish superiority of treatment over placebo with respect to the mean difference. Assume that to that end data \(Y_j^{g, i}\) is observed for the \(j\)-th individual of the trial in stage \(i\in\{1,2\}\) under treatment (\(g=T\)) or placebo (\(g=C\)). Let \(n_i\) be the per-group sample size in stage \(i\) and consider the stage-wise test statistics \[ X_i := \frac{\sum_{j=1}^{n_i}Y_j^{T,i} - \sum_{j=1}^{n_i}Y_j^{C,i}}{\sigma\,\sqrt{2\,n_i}} \] for \(i=1,2\). Under the assumption that the \(Y_j^{g, i}\stackrel{iid}{\sim} F_g\) with \(\boldsymbol{E}[F_T] - \boldsymbol{E}[F_C] = \theta\) and common variance \(\sigma^2\), by the central limit theorem, the asymptotic distribution of \(X_1\) is \(\mathcal{N}(\sqrt{n_1/2}\, \theta, 1)\). Formally, the null hypothesis for the superiority test is thus \(\mathcal{H}_0:\theta\leq 0\). Based on the interim outcome \(X_1\), a decision can be made whether to either stop the trial early for futility if \(X_1<c_1^f\), to stop the trial early for efficacy (early rejection of the null hypothesis) if \(X_1>c_1^e\), or to enter stage two if \(X_1\in[c_1^f, c_1^e]\). Conditional on proceeding to a second stage, it holds that \(X_2\,|\,X_1\in[c_1^f, c_1^e]\sim\mathcal{N}(\sqrt{n_2/2}\, \theta, 1)\). In the second stage, the null hypothesis is rejected if and only if \(X_2 > c_2(X_1)\) for a stage-two critical value \(c_2:x_1\mapsto c_2(x_1)\). To test \(\mathcal{H}_0\) at a significance level of \(\alpha\), the stage-one critical values \(c_1^f\) and \(c_1^e\) as well as \(c_2(\cdot)\) must be chosen in a way that protects the overall maximal type one error rate \(\alpha\). Note that it is convenient to define \(c_2(x_1) = \infty\) if \(x_1<c_1^f\) and \(c_2(x_1) = -\infty\) if \(x_1>c_1^e\) since the power curve of the design is then given by \(\theta\mapsto\boldsymbol{Pr}_\theta\big[X_2>c_2(X_1)\big]\). This results in a classical group-sequential design and several methods were proposed in the literature for choosing the early-stopping boundaries \(c_1^f\) and \(c_1^e\) (O’Brien and Fleming (1979), Pocock (1977)) and for defining the stage-two rejection boundary function \(c_2(\cdot)\) (Bauer and Köhne (1994), Hedges and Olkin (1985)). Often, the inverse-normal combination test (Lehmacher and Wassmer 1999) is applied and \(c_2(\cdot)\) is defined as a linear function of the stage-one test-statistic \[ c_2(x_1) = \frac{c - w_1 x_1}{w_2}‚ \] for a critical value \(c\) and predefined weights \(w_1\) and \(w_2\). Most commonly, the stage-wise test statistics are weighted in terms of their respective sample sizes, i.e., \(w_1 = \sqrt{n_1 / (n_1+n_2)}\) and \(w_2 = \sqrt{n_2 / (n_1+n_2)}\). This choice of the weights is optimal in the sense that it minimizes the variance of the final test statistic if the assumed sample sizes are indeed realized (Zaykin 2011). Note, however, that such prespecified weights become inefficient if the sample size deviates strongly from the anticipated value (cf. Wassmer and Brannath (2016), chapter 6.2.5). A natural extension of this group-sequential framework is to allow the second stage sample size to also depend on the observed interim outcome, i.e., to consider a function \(n_2:x_1\mapsto n_2(x_1)\) instead of a fixed value \(n_2\). Such ‘adaptive’ two-stage designs are thus completely characterized by a five-tuple \(\mathcal{D}:=\big(n_1, c_1^f, c_1^e, n_2(\cdot), c_2(\cdot)\big)\).

While the required sample size and the critical value for a single-stage design are uniquely defined by given type one error rate and power constraints, it is much less clear how the design parameters of a two-stage design should be selected. This is especially true since both \(n_2\) and \(c_2\) are functions and thus the parameter space is in fact infinite-dimensional. In order to compare different choices of the design parameters, appropriate scoring criteria are essential. A widely applied criterion is the expected sample size under the alternative hypothesis (see, e.g., Jennison and Turnbull (2015)). However, there is a variety of further scoring criteria that could be incorporated or even combined in order to rate a two-stage design. For instance, conditional power is defined as the probability to reject the null hypothesis under the alternative given the interim result \(X_1=x_1\): \[ \operatorname{CP}_\theta(x_1) := \boldsymbol{Pr}_\theta \big[X_2 > c_2(X_1) \, \big| \, X_1=x_1\big]. \] Hence, conditional power is a conditional score given as the power conditioned on the first-stage outcome \(X_1=x_1\). Vice versa, power can be seen as an unconditional score that is obtained by integrating conditional power over all possible stage-one outcomes, i.e., \[ \operatorname{Power_\theta} = \boldsymbol{E}_\theta\big[\operatorname{CP}(X_1)\big]. \] Intuitively, it makes sense to require a minimal conditional power upon continuation to the second stage since one might otherwise continue a trial with little prospect of still rejecting the null hypothesis. We demonstrate the consequences of this heuristic in Section 6.4. Once the scoring criterion is selected, the design parameters may be chosen in order to optimize this objective. The first ones to address this problem were Jennison and Turnbull (2015) who minimized the expected sample size \[ \text{ESS}_{\theta}(\mathcal{D}) := \boldsymbol{E}_{\theta} \big[n(X_1) \big] := \boldsymbol{E}_{\theta} \big[n_1 + n_2(X_1) \big] \] of a two-stage design for given \(n_1,c_1^f, c_1^e\) with respect to \(n_2(\cdot)\) for given power and type one error rate constraints. The function \(c_2(\cdot)\), however, was not optimized. Instead, Jennison and Turnbull (2015) used a combination test approach to derive \(c_2\) given \(n_2(\cdot)\) and \(n_1\) (cf. Equation (??)). In Pilz et al. (2019), the authors demonstrated that this restriction is not necessary and that the variational problem of deriving both functions \(n_2(\cdot)\) and \(c_2(\cdot)\) given \(n_1,c_1^f, c_1^e\) to minimize expected sample size can be solved by analyzing the corresponding Euler-Lagrange equation. Nesting this step in a standard optimization over the stage-one parameters allows identifying an optimal set of all design parameters without imposing parametric assumptions on \(c_2(\cdot)\). As a result, a fully optimal design \(\mathcal{D}^*:=\big(n_1^*, c_1^{f,*}, c_1^{e, *}, n_2^*(\cdot), c_2^*(\cdot)\big)\) for the following general optimization problem was derived \[\begin{align} & \text{minimize} && \operatorname{ESS}_{\theta_1}(\mathcal{D}) &&&& \\ & \text{subject to:} && \boldsymbol{Pr}_{\theta_0}\big[X_2>c_2(X_1)\big] &&\leq \alpha, &&&& \\ &&& \boldsymbol{Pr}_{\theta_1}\big[X_2>c_2(X_1)\big] &&\geq 1-\beta &&&& \end{align}\] where \(\theta_0=0\).

3 Direct variational perspective

In adoptr, a simpler solution strategy than solving the Euler-Lagrange equation locally is applied to the same problem class. We propose to embed the entire problem in a finite-dimensional parameter space and solve the corresponding problem over both stage-one and stage-two design parameters simultaneously using standard numerical libraries. I.e., we adopt a direct approach to solving the variational problem. This is done by defining a discrete set of pivot points \(\widetilde{x}_1^{(i)}\in(c_1^f, c_1^e), i=1,\ldots,k\), and interpolating \(c_2\) and \(n_2\) between these pivots. We use cubic Hermite splines (Fritsch and Carlson 1980) which are sufficiently flexible, even for a moderate number of pivots, to approximate any realistic stage-two sample size and critical value functions. Since the optimal functions are generally very smooth (Pilz et al. 2019) they are well suited to spline interpolation. Within the adoptr validation report (cf. Section 7) we investigate empirically the shape of the approximated functions and that increasing the number of pivots above a value of 5 to 7 does not improve the optimization results. The latter implies that a relatively small number of pivot points appears to be sufficient to obtain valid spline approximations of the optimal functions. Note that the pivots are only needed in the continuation region since both functions are (piecewise) constant within the early stopping regions. In adoptr, the pivots are defined as nodes of a Gaussian quadrature rule of degree \(k\). This choice allows fast and precise numerical integration of any conditional score over the continuation region, e.g., \[\begin{align} \text{ESS}_{\theta}(\mathcal{D}) = \int n(x_1) f_{\theta} (x_1) \operatorname{d} x_1 \approx n_1 + \sum_{i=1}^{k} \omega_i\, n_2\big(\widetilde{x}_1^{(i)}\big) f_{\theta} \big(\widetilde{x}_1^{(i)}\big), \end{align}\] where \(f_\theta\) is the probability density function of \(X_1\,|\,\theta\) and \(\omega_i\) are the corresponding weights of the integration rule. The weights only depend on \(k\) and the nodes just need to be scaled to the integration interval. Consequentially, this objective function is smooth in the optimization parameters and the resulting optimization problem is of dimension \(2k+3\), where the tuning parameters are \(\big(n_1, c_1^f, c_1^e, n_2\big(\widetilde{x}_1^{(1)}\big), \dots, n_2\big(\widetilde{x}_1^{(k)}\big), c_2\big(\widetilde{x}_1^{(1)}\big), \dots, c_2\big(\widetilde{x}_1^{(k)}\big)\big)\). Standard numerical solvers may then be employed to minimize it. Since adoptr enables generic objectives (cf. Section 6.3), it uses the gradient-free optimizer COBYLA (Powell 1994) internally via the R-package nloptr (Johnson (2018), Ypma, Borchers, and Eddelbuettel (2018)).

Most commonly used unconditional performance scores \(S(\mathcal{D})\) can be seen as expected values over conditional scores \(S(\mathcal{D}|X_1)\) by \(S(\mathcal{D}) = \boldsymbol{E}\big[S(\mathcal{D}|X_1)\big]\) in a similar way as power and expected sample size. Any such ‘integral score’ can be computed quickly and reliably in adoptr via the choice of pivots outlined above. The correctness of numerically integrated scores is checked in the adoptr validation report by comparing the numerical integrals to simulated results.

Note that we tacitly relaxed all sample sizes to be real numbers in the above argument while they are in fact restricted to positive integers. Integer-valued \(n_1\) and \(n_2\) would, however, lead to an NP-hard mixed-integer problem. In our experiments, we found that merely rounding both \(n_1\) and \(n_2\) after the optimization works fine. The extensive validation suite (cf. Section 7) evaluates by numerical integration and simulation whether the included constraints are fulfilled for optimal designs with rounded sample sizes. Up to now, neither the constraints were violated nor an efficiency loss with respect to the underlying objective function was observed. In theory, one could re-adjust the decision boundaries for these rounded sample sizes, but we failed to see any practical benefit from this, even for small trials where the rounding error is largest (data not shown).

4 The need for an R package

Bauer et al. (2015) state that adequate statistical software for adaptive designs ‘is increasingly needed to evaluate the adaptations and to find reasonable strategies’.

Commercial software such as JMP (SAS Institute Inc., n.d.a) or Minitab (Minitab, Inc. 2020) allow planning and analyzing a wide range of experimental setups. Amongst others, they provide tools for randomization, stratification, block-building, or D-optimal designs. These general purpose statistical software packages do not, however, allow planning of more specialized multi-stage designs encountered in clinical trials. For group-sequential designs, some planning capabilities are available in the SAS procedure seqdesign (SAS Institute Inc., n.d.b), PASS (NCSS 2019), or ADDPLAN (ICON plc 2020). East (Cytel 2020) also supports design, simulation and analysis of experiments with interim analyses. The East ADAPT and the East SURVADAPT modules support sample size recalculation. Furthermore, there are various open-source R-packages for the analysis of multi-stage designs. The package adaptTest (Vandemeulebroecke 2009) implements combination tests for adaptive two-stage designs. AGSDest (Hack, Brannath, and Brueckner 2019) allows estimation and computation of confidence intervals in adaptive group-sequential designs. More detailed overviews on software for adaptive clinical trial designs can be found in Bauer et al. (2015), chapter 6, or in Tymofyeyev (2014). The choice of software for optimally designing two- or multi-stage designs, however, is much more limited. Current R-packages concerned with optimal clinical trial designs are OptGS (Wason and Burkardt (2015), Wason (2015)) and rpact (Wassmer and Pahlke 2019). These are, however, exclusively focused on group-sequential designs and lack the ability to specify custom objective functions and constraints.

The lack of flexibility in formulating the objective function and constraints might lead to off-the-shelf solutions not entirely reflecting the needs of a particular trial consequentially resulting in inefficient designs. The R-package adoptr aims at providing a simple and interactive yet flexible interface for addressing a range of optimization problems encountered with two-stage one- or two-arm clinical trials. In particular, adoptr allows to model a priori uncertainty over \(\theta\) via prior distributions and thus supports optimization under uncertainty (cf. Section 6.2). adoptr also supports the combination of conditional (on \(X_1\)) and unconditional scores and constraints to address concerns such as type-one-error-rate control (unconditional score) and, e.g., a minimal conditional power (conditional score) simultaneously (cf. Section 6.3). To facilitate the adoption of these advanced trial designs in the clinical trials community, adoptr also features an extensive test and validation suite (cf. Section 7).

In the following, we outline the key design principles for adoptr.

  1. Interactivity: A major advantage of the R-programming language is its powerful metaprogramming capabilities and flexible class system. With a combination of non-standard evaluation and S4 classes, we hope to achieve a structured and modular way of expressing optimization problems in clinical trials that integrates nicely with an interactive workflow. We feel that a step-wise problem formulation via the creation of modular intermediate objects, which can be explored and modified separately, encourages exploration of different options.
  2. Reliability: A crux in open-source software development for clinical trials is achieving demonstrable validation. Potential users need to be convinced of the software quality and need to be able to comply with their respective validation requirements which often require the ability to produce a validation report. This burden typically results in innovative software not being used at all - simply because the validation effort cannot be stemmed. We address this issue with an extensive unit test suite and a companion validation report (cf. Section 7).
  3. Extensibility: We do not want to impose a particular choice of scores or constraints or promote a particular notion of optimality for clinical trial designs. In cases where the composition of existing scores is not sufficient, the object-oriented approach of adoptr facilitates the definition of custom scores and constraints that seamlessly integrate with the remainder of the package.

5 Adoptr’s structure

The package adoptr is based on R’s S4 class system. This allows to use multiple dispatch on the classes of multiple arguments to a method. In this section, the central components of adoptr are described briefly. Figure 5.1 gives a structural overview of the main classes in adoptr.

Overview of the most important classes and methods (in italic) in the R-package adoptr. A subclassing relationship is indicated by a connecting line to the  corresponding super class above it. The most important methods for each class are listed under the respective class name in italic font.

Figure 5.1: Overview of the most important classes and methods (in italic) in the R-package adoptr. A subclassing relationship is indicated by a connecting line to the corresponding super class above it. The most important methods for each class are listed under the respective class name in italic font.

To compute optimal designs, an object of class UnconditionalScore must be defined as objective criterion. adoptr distinguishes between ConditionalScores and UnconditionalScores (cf. Section 2). All Scores can be evaluated using the method evaluate. For unconditional scores, this method only requires a Score object and a TwoStageDesign object, for conditional scores (like conditional power), it also requires the interim outcome \(x_1\). Note that any ConditionalScore \(S(\mathcal{D}|X_1=x_1)\) can be converted to an UnconditionalScore \(S(\mathcal{D}) = \boldsymbol{E}\big[S(\mathcal{D}|X_1)\big]\) using the method expected. The two most widely used conditional scores are pre-implemented as ConditionalPower and ConditionalSampleSize. Their unconditional counterparts are Power and ExpectedSampleSize. Further predefined unconditional scores are MaximumSampleSize, evaluating the maximum sample size, N1, measuring the first-stage sample size, and AverageN2, evaluating the average of the stage-two sample size (improper prior). These scores may be used for regularization if variable stage-two sample sizes or a high stage-one sample size are to be penalized. Users are free to define their own Scores (cf. the vignette ‘Defining New Scores’ (Kunzmann and Pilz 2020)). Moreover, different Scores can be composed to a single one by the function composite (cf. Section 6.3). Both conditional and unconditional scores can also be used to define constraints - the most common case being constraints for power and maximal type one error rate. The function minimize takes an unconditional score as objective and a set of constraints and optimizes the design parameters.

In adoptr, different kinds of designs are implemented. The most frequently applied case is a TwoStageDesign, i.e., a design with one interim analysis and a sample size function that varies with the interim test statistic. Another option is the subclass GroupSequentialDesign which restricts the sample size function on the continuation region to a single number, i.e., \({n_2(x_1) = n_2\ \forall x_1\in[\,c_1^f, c_1^e\,]}\). Additionally, adoptr supports the computation of optimal OneStageDesigns, i.e., designs without an interim analysis. Technically, one-stage designs are implemented as subclasses of TwoStageDesign since they can be viewed as the limiting case for \(n_2\equiv0\) and \(c_1^f=c_1^e\). Hence, all methods that are implemented for TwoStageDesigns also work for GroupSequentialDesigns and OneStageDesigns. Users can chose to keep some elements of a design fixed during optimization using the methods make_fixed (cf. Section 6.5).

The joint data distribution in adoptr consists of two elements. The distribution of the test statistic is specified by an object of class DataDistribution. Currently, the three options Normal, Binomial, and Student are implemented. The logical variable two_armed allows the differentiation between one- and two-armed trials. Furthermore, adoptr supports prior distributions on the effect size. These can be PointMassPriors (cf. Section 6.1) as well as ContinuousPriors (cf. Section 6.2).

In the following section, more hands on examples demonstrate the capabilities of adoptr and its syntax.

6 Examples

6.1 Standard case

Consider the case of a randomized controlled clinical trial where efficacy is to be demonstrated in terms of superiority of the treatment over placebo with respect to the population mean difference \(\theta\) of an outcome. Let the null hypothesis be \(\mathcal{H}_0:\theta\leq 0\). Assume that the maximal type one error rate is to be controlled at a one-sided level \(\alpha=2.5\%\) and a minimal power of \(90\%\) at a point alternative of \(\theta_1=0.3\) is deemed necessary. For simplicity’s sake, we assume \(\sigma^2=1\) without loss of generality. The required sample size for a one-stage design with analysis by the one-sided two-sample \(t\)-test would then be roughly 235 per group.

Using adoptr, the two-stage design minimizing the expected sample size under the alternative hypothesis can be derived for the very same situation. First, the data distribution is specified to be normal. The two_armed parameter allows to switch between single-armed and two-armed trials.

R> datadist <- Normal(two_armed = TRUE)

In this example, we use simple point priors for both the null and alternative hypotheses. The hypotheses and the corresponding scores (power values) can be specified as:

R> null        <- PointMassPrior(theta = .0, mass = 1.0)
R> alternative <- PointMassPrior(theta = .3, mass = 1.0)
R> power       <- Power(dist = datadist, prior = alternative)
R> toer        <- Power(dist = datadist, prior = null)
R> mss         <- MaximumSampleSize()

A Power score requires the data distribution and the prior to be specified. For this example, we choose PointMassPriors with the entire probability mass of \(1\) on a single point, the null hypothesis \(\theta = 0\) to compute the type one error rate, and the alternative hypothesis \(\theta = 0.3\) to compute the power. The objective function is the expected sample size under the alternative.

R> ess <- ExpectedSampleSize(dist = datadist, prior = alternative)

Since adoptr internally relies on the COBYLA implementation of nloptr, an initial design is required. A heuristic initial choice is provided by the function get_initial_design. It is based on a fixed design that fulfills constraints on type one error rate and power. The type of the design (two-stage, group-sequential, or one-stage) and the data distribution have to be defined. For the Gaussian quadrature used during optimization, one also has to specify the order of the integration rule, i.e., the number of pivot points between early stopping for futility and early stopping for efficacy. In practice, order 7 turned out to be sufficiently flexible to obtain valid results (data not shown).

R> initial_design <- get_initial_design(theta = 0.3, alpha = 0.025,
+                                       beta = 0.1, type = "two-stage",
+                                       dist = datadist, order = 7)

It is easy to check that the initial design does not fulfill any of the constraints (minimal power of 90% and maximal type one error rate of 2.5%) with equality by evaluating the respective scores:

R> evaluate(toer, initial_design)
## [1] 0.0246875
R> evaluate(power, initial_design)
## [1] 0.8130973

Alternatively, one might also evaluate a constraint object directly via

R> evaluate(toer  <= .025, initial_design)
## [1] -0.0003125
R> evaluate(power >= .9, initial_design)
## [1] 0.08690272

All constraint objects are normalized to the form \(h(\mathcal{D}) \leq 0\) (unconditional) or \(h(\mathcal{D}, x_1) \leq 0\) (conditional on \(X_1=x_1\)). Calling evaluate on a constraint object then simply returns the left-hand side of the inequality. The actual optimization is started by invoking minimize

R> opt1 <- minimize(ess, subject_to(power >= 0.9, toer  <= 0.025),
+                   initial_design)

The modular structure of the problem specification is intended to facilitate the inspection or modification of individual components. The call to minimize() is designed to be as close as possible to the mathematical formulation of the optimization problem and returns both the optimized design (opt1$design) as well as the full nloptr return value with details on the optimization procedure (opt1$nloptr_return).

A summary method for objects of the class TwoStageDesign is available to quickly evaluate a set of ConditionalScores such as conditional power as well as UnconditionalScores such as power and expected sample size.

R> cp <- ConditionalPower(dist = datadist, prior = alternative)
R> summary(opt1$design, "Power" = power, "ESS" = ess, "CP" = cp)
## TwoStageDesign: n1 = 120 
          futility |                  continue                 | efficacy
      x1:     0.28 |  0.33  0.54  0.87  1.27  1.68  2.01  2.22 |  2.27
  c2(x1):     +Inf | +2.70 +2.53 +2.23 +1.82 +1.31 +0.74 +0.19 |  -Inf
  n2(x1):        0 |   229   214   188   154   116    79    51 |     0
  CP(x1):     0.00 |  0.69  0.72  0.75  0.79  0.83  0.87  0.91 |  1.00
   Power:      0.899
     ESS:    176.126

adoptr also implements a default plot method for the overall sample size and the stage-two critical value as functions of the first-stage test statistic \(x_1\). The plot method also accepts additional ConditionalScores such as conditional power. Calling the plot method produces several plots with the interim test statistic \(x_1\) on the \(x\)-axis and the respective function on the \(y\)-axis.

R> plot(opt1$design, `Conditional power` = cp)
Optimal sample size, critical value, and conditional power plotted against the interim test statistic (built-in plot method).

Figure 6.1: Optimal sample size, critical value, and conditional power plotted against the interim test statistic (built-in plot method).

Note the slightly bent shape of the \(c_2(\cdot)\) function (cf. Figure 6.1, second plot). For two-stage designs based on the inverse-normal combination function, \(c_2(\cdot)\) would be linear by definition (cf. Equation (??)). Since the optimal shape of \(c_2(\cdot)\) is not linear (but almost), inverse-normal combination methods are slightly less efficient (cf. Pilz et al. (2019) for a more detailed discussion of this issue).

6.2 Optimization under uncertainty

adoptr is not limited to point priors but also supports arbitrary continuous prior distributions. Consider the same situation as before but now assume that the prior over the effect size is given by a much more realistic truncated normal distribution with mean \(0.3\) and standard deviation \(0.1\), i.e., \(\theta\sim\mathcal{N}_{[-1, 1]}(0.3, 0.1^2)\). The order of integration is set to 25 to obtain precise results.

R> prior <- ContinuousPrior(
+    pdf     = function(theta) dnorm(theta, mean = .3, sd = .1),
+    support = c(-1, 1),
+    order   = 25)

The objective function is the expected sample size under the prior

R> ess <- ExpectedSampleSize(dist = datadist, prior = prior)

and we replace power with expected power \[ \boldsymbol{E} \Big[ \boldsymbol{Pr}_\theta\big[X_2>c_2(X_1)\big] \, \Big| \, \theta \geq 0.1 \Big] \] which is the expected power given a relevant effect (here we define the minimal relevant effect as \(0.1\)). This score can be defined in adoptr by first conditioning the prior.

R> epower <- Power(dist = datadist, prior = condition(prior, c(.1, 1)))

The optimal design under the point prior only achieves an expected power of

R> evaluate(epower, opt1$design)
## [1] 0.8143933

The optimal design under the truncated normal prior fulfilling the expected power constraint is then given by

R> opt2 <- minimize(ess, subject_to(epower >= 0.9, toer <= 0.025),
+                   initial_design,
+                   opts = list(algorithm = "NLOPT_LN_COBYLA",
+                               xtol_rel = 1e-5, maxeval = 20000))

Note that the increased complexity of the problem requires a larger maximal number of iterations for the underlying optimization procedure. adoptr exposes the nloptr options via the argument opts. In cases where the maximal number of iterations is exhausted, a warning is thrown.

The expected sample size under the prior of the obtained optimal design equals 236.2. This shows that an increased uncertainty on \(\theta\) requires larger sample sizes to fulfill the expected power constraint since the expected sample size under the continuous prior considered in this section of the optimal design derived under a point alternative (see Section 6.1) is only 176.4.

6.3 Utility maximization and composite scores

adoptr also supports composite scores. This can be used to derive utility maximizing designs by defining an objective function combining both expected power and expected sample size instead of imposing a hard constraint on expected power. For example, in the above situation one could be interested in a utility-maximizing design. Here, we consider the utility function \[ u(\mathcal{D}) := 200000\, \boldsymbol{E} \Big[ \boldsymbol{Pr}_\theta\big[X_2>c_2(X_1)\big] \, \Big| \, \theta \geq 0.1 \Big] - \boldsymbol{E}\Big[n(X_1)^2\Big], \] thus allowing a direct trade-off between power and sample size. Here, the expected sample size is chosen because the practitioner might prefer flatter sample size curves. This can be achieved with expected squared sample size by penalizing large sample sizes stronger than low sample sizes. Furthermore, there is no longer a strict expected power constraint but the expected power becomes part of the utility function which allows a direct trade-off between the two quantities. This can be interpreted as a pricing mechanism (cf. Kunzmann and Kieser (2020)): Every additional percent point of expected power has a (positive) value of \(\$ 2'000\) while an increase of \(\boldsymbol{E}\big[n(X_1)^2\big]\) by \(1\) incurs costs of \(\$ 1\). The goal is then to compute the design which is maximizing the overall utility defined by the utility function \(u(\mathcal{D})\) (or equivalently minimize costs).

A composite score can be defined via any valid numerical R expression of score objects. We start by defining a score for the expected quadratic sample size

R> `n(X_1)`      <- ConditionalSampleSize()
R> `E[n(X_1)^2]` <- expected(composite({`n(X_1)`^2}),
+                            data_distribution = datadist,
+                            prior = prior)

before minimizing the corresponding negative utility without a hard expected power constraint.

R> opt3 <- minimize(composite({`E[n(X_1)^2]` - 200000*epower}),
+                   subject_to(toer <= 0.025), initial_design)

The expected power of the design is

R> evaluate(epower, opt3$design)
## [1] 0.7965527

The three optimal designs which have been computed so far are depicted in a joint plot (cf. Figure 6.2). The design using the continuous prior requires higher sample sizes due to the higher uncertainty about \(\theta\). The utility maximization approach results in similar shapes of \(n(\cdot)\) and \(c_2(\cdot)\) as the constraint optimization. However, the sample sizes are lower due to the design’s lower power which is only possible by allowing a trade-off between expected power and expected sample size. In particular, the maximal sample size of the utility-based design equals 256 and is distinctly smaller than in the case of a hard power constraint under a point prior (maximal sample size: 352) or a continuous prior (maximal sample size: 527).

Comparison of optimal designs under a point prior, a continuous prior, and a utility maximization approach.

Figure 6.2: Comparison of optimal designs under a point prior, a continuous prior, and a utility maximization approach.

6.4 Conditional power constraint

adoptr also allows the incorporation of hard constraints on conditional scores such as conditional power. Conditional power constraints are intuitively sensible to make sure that a trial which continues to the second stage maintains a high chance of rejecting the null hypothesis at the end. For this example, we return to the case of a point prior on the effect size.

R> prior <- PointMassPrior(theta = .3, mass = 1.0)
R> ess   <- ExpectedSampleSize(dist = datadist, prior = prior)
R> cp    <- ConditionalPower(dist = datadist, prior = prior)
R> power <- expected(cp, data_distribution = datadist, prior = prior)

Here, power is derived as expected score of the corresponding conditional power. A conditional power constraint is added in exactly the same way as unconditional constraints.

R> opt4 <- minimize(ess, subject_to(toer <= 0.025, power >= 0.9, cp >= 0.8),
+                   initial_design)

Comparing the optimal design that has been computed here with the same constraints but without a conditional power constraint (cf. beginning of this chapter), the optimal design with the additional constraint requires larger sample sizes in regions where the conditional power would usually be below the given threshold (cf. Figure 6.3, first and third plot). Overall, the additional constraint reduces the feasible solution space and consequently increases the expected sample size (176.6 with conditional power constraint vs. 176.1 without). This example demonstrates, that any additional binding conditional constraints do come at costs for global optimality. Whether or not the loss in unconditional performance is outweighed by more appealing conditional properties must be decided on a case by case basis.

Optimal designs with and without conditional power constraint.

Figure 6.3: Optimal designs with and without conditional power constraint.

6.5 Keeping design parameters fixed

In clinical practice, non-statistical considerations may impose direct constraints on design parameters. For instance, a sponsor might be subject to logistical constraints that render it necessary to design a trial with a specific stage one sample size. Returning to the standard case discussed in Section 6.1, assume that a stage-one per-group sample size of exactly 80 individuals per group is required (\(n_1 = 80\)) instead of the optimal value of \(n_1^* = 120\). Furthermore, assume that the sponsor wants to stop early for futility if and only if there is a negative effect size at the interim analysis, i.e., \(c_1^f = 0\). adoptr supports such considerations by allowing to fix specific values of a design:

R> initial_design@n1  <- 80
R> initial_design@c1f <- 0
R> initial_design     <- make_fixed(initial_design, n1, c1f)

Any ‘fixed’ parameter will be kept constant during optimization. Note that it is also possible to ‘un-fix’ parameters again using the make_tunable function.

R> opt5 <- minimize(ess, subject_to(toer <= 0.025, power >= 0.9),
+                   initial_design)

Figure 6.4 visually compares the original design with the new, more restricted design. The designs are qualitatively similar, but fixing \(n_1\) and \(c_1^f\) does come at the price of slightly increased expected sample size (187.7 compared to 176.1 in the less restricted case).