Savings In Sample Size Research Articles

A multi-arm multi-stage platform design that allows preplanned addition of arms while still controlling the family-wise error.

There is growing interest in platform trials that allow for adding of new treatment arms as the trial progresses as well as being able to stop treatments part way through the trial for either lack of benefit/futility or for superiority. In some situations, platform trials need to guarantee that error rates are controlled. This paper presents a multi-stage design, that allows additional arms to be added in a platform trial in a preplanned fashion, while still controlling the family-wise error rate, under the assumption of known number and timing of treatments to be added, and no time trends. A method is given to compute the sample size required to achieve a desired level of power and we show how the distribution of the sample size and the expected sample size can be found. We focus on power under the least favorable configuration which is the power of finding the treatment with a clinically relevant effect out of a set of treatments while the rest have an uninteresting treatment effect. A motivating trial is presented which focuses on two settings, with the first being a set number of stages per active treatment arm and the second being a set total number of stages, with treatments that are added later getting fewer stages. Compared to Bonferroni, the savings in the total maximum sample size are modest in a trial with three arms, <1% of the total sample size. However, the savings are more substantial in trials with more arms.

Explicit lot inspection by attributes using minimal prior information

An explicit optimal attribute sampling plan for lot acceptance with maximum allowable expected risks is presented without specifying the full prior model of the defective rate. The suggested methodology to find an approximate solution of the underlying integer nonlinear programming problem is simple, fast and sufficiently accurate in most practical cases. The smallest number of units to be inspected per lot in order to reach the required protections for customers and manufacturers, as well as the maximum tolerable number of defectives in the selected sample, is derived in closed-form using Taylor series expansions of the operating characteristic function. The proposed approach allows the practitioners to determine a nearly optimal inspection scheme using only prior estimations of the means and variances of the defective rates for the acceptable and rejectable lots. Furthermore, it provides easy ways to adequately combine multiple expert opinions and to update the current optimal test plan when new information becomes available. The incorporation of prior knowledge yields considerable savings in sample size, as well as improved evaluations of the current expected producer and consumer risks. Some applications in industrial reliability and quality control are included for illustrative and comparative purposes.

Tandem-width sequential confidence intervals for a Bernoulli proportion

We propose a two-stage sequential method for obtaining tandem-width confidence intervals for a Bernoulli proportion p. The term “tandem-width” refers to the fact that the half-width of the confidence interval is not fixed beforehand; it is instead required to satisfy two different half-width upper bounds, h0 and h1, depending on the (unknown) values of p. To tackle this problem, we first propose a simple but useful sequential method for obtaining fixed-width confidence intervals for p, whose stopping rule is based on the minimax estimator of p. We observe Bernoulli(p) trials sequentially, and for some fixed half-width h = h0 or h1, we develop a stopping time T such that the resulting confidence interval for p, [], covers the parameter with confidence at least where is the maximum likelihood estimator of p at time T. Furthermore, we derive theoretical properties of our proposed fixed-width and tandem-width methods and compare their performances with existing alternative sequential schemes. The proposed minimax-based fixed-width method performs similarly to alternative fixed-width methods, while being easier to implement in practice. In addition, the proposed tandem-width method produces effective savings in sample size compared to the fixed-width counterpart and provides excellent results for scientists to use when no prior knowledge of p is available.

Limb-Leaf designs for adaptive exploration of the dose-response curve

We propose a two-stage strategy, called the Limb-Leaf method, to explore the dose-response curve using dose promotion and addition in the context of adaptive seamless Phase II/III trials. Strong control of the overall type 1 familywise error rate of the proposed method is enforced by the closed testing principle. The design constants are determined to minimize the risk-adjusted expected total sample size while maintaining a target power. In the case of a nonmonotonic dose response curve where more doses are required to adequately explore the curve, substantial savings in sample size are achieved compared with a traditional strategy which offers only selection and promotion from among initial first stage doses.

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OBJECTIVES/SPECIFIC AIMS: The objective of this research is to determine under what conditions endpoints based on estimated glomerular filtration rate (eGFR) slope or on relatively small declines in eGFR provide valid and useful surrogate endpoints for pivotal clinical trials in chronic kidney disease (CKD) patients. METHODS/STUDY POPULATION: We consider 2 classes of surrogate endpoints. The first class includes endpoints defined by the average rate of change in eGFR during defined portions of the follow-up period of the trial, following initiation of the randomized treatment interventions. The second class includes composite endpoints defined by the time from randomization until the occurrence of a designated decline in eGFR or kidney failure. The true clinical endpoint is considered to be the time from randomization until kidney failure, irrespective of the trajectory in eGFR measurements prior to kidney failure. We apply statistical simulation to determine conditions under which alternative endpoints within the 2 classes are (1) valid surrogate endpoints, in the sense of preserving a low probability of rejecting the null hypothesis of no treatment effect on the surrogate endpoint when there is no treatment effect on the clinical endpoints and are also (2) useful surrogate endpoints, in the sense of providing increased statistical power that allows significant reductions in sample size and/or duration of follow-up. Input parameters for the simulations include (a) characteristics of the joint distribution of the longitudinal eGFR measurements and the time to occurrence of renal failure, (b) characteristics of the short-term and long-term effects of the treatment, and (c) design parameters, including the duration of accrual and follow-up and the spacing of eGFR measurements during the follow-up period. We use joint analyses of 19 treatment comparisons across 13 previous clinical trials of CKD patients to guide the selection of input parameters for the simulations. We apply longitudinal mixed effects models for analysis of endpoints based on eGFR slope, and Cox regression for analyses of the composite time-to-event endpoints. RESULTS/ANTICIPATED RESULTS: We have previously shown that surrogate endpoints defined by eGFR declines of 30% or 40% can provide valid and useful alternative endpoints in CKD clinical trials for interventions that do not produce short-term effects on eGFR which differ from the longer-term effects of the interventions. Other factors influencing the validity and utility of these endpoints include the average baseline eGFR, the mean rate of change in eGFR, and the extent to which the size of the treatment effect depends on the patient’s underling rate of eGFR decline. We will extend these results by presenting preliminary results describing conditions under which outcomes based on eGFR slope provide valid and useful alternatives to the clinical endpoint of time until occurrence of kidney failure. DISCUSSION/SIGNIFICANCE OF IMPACT: The statistical simulation strategy described in this research can be used during the design of clinical trials of chronic kidney disease to assist in the selection of endpoints that maximize savings in sample size and duration of follow-up while retaining a low risk of producing a false positive conclusion in the absence of a true effect of the treatment on the time until kidney failure.

Explicit quasi-optimal inspection schemes from nonconformity count data with controlled producer and consumer risks

Assuming that maximum allowable risks are specified for both producer and consumer, an integer nonlinear programming problem is formulated in order to find the optimal inspection scheme for lot acceptance purpose based on defect count data and limited prior information. The number of nonconformities per unit is assumed to follow a Poisson distribution. Closed-form quasi-optimal inspection schemes for screening submitted lots of manufactured material are determined using normal transformations of chi-square distributions and second-order Taylor series expansions of the operating characteristic function. The full prior model on the process average is not needed. Instead only prior estimations of the means and variances for the quality levels of the acceptable and rejectable lots are necessary to derive explicit and accurate approximations of the smallest number of units to be tested per lot and the maximum tolerable number of nonconformities in the selected sample. The suggested methodology requires little prior information to judge the quality of the submitted material and greatly simplifies and quickens the determination of optimal acceptance sampling plans with controlled average risks. In general, the inclusion of prior knowledge yields substantial savings in sample size, as well as improved evaluations of the actual average producer and consumer risks. Moreover, our approach is quite robust against slight changes in the available prior knowledge and also provides a suitable way to properly combine multiple expert judgments. The developed procedure is applied to the manufacturing of paper and glass to show the accuracy of the proposed quasi-optimal plans and the corresponding average risks.

A curve-free Bayesian decision-theoretic design for two-agent Phase I trials

ABSTRACTAlthough Bayesian statistical methods are gaining attention in the medical community, as they provide a natural framework for incorporating prior information, the complexity of these methods limited their adoptions in clinical trials. This article proposes a Bayesian design for two-agent Phase I trials that is relatively easy for clinicians to understand and implement, yet performs comparably to more complex designs, so that it is more likely to be adopted in actual trials. In order to reduce model complexity and computational burden, we choose a working model with conjugate priors so that the posterior distributions have analytical expressions. Furthermore, we provide a simple strategy to facilitate the specification of priors based on the toxicity information accrued from single-agent Phase I trials. The proposed method should be useful in terms of the ease of implementation and the savings in sample size without sacrificing performance. Moreover, the conservativeness of the dose-finding algorithm renders it a relatively safe method.

Hypothesis testing for Markov chain Monte Carlo

Testing between hypotheses, when independent sampling is possible, is a well developed subject. In this paper, we propose hypothesis tests that are applicable when the samples are obtained using Markov chain Monte Carlo. These tests are useful when one is interested in deciding whether the expected value of a certain quantity is above or below a given threshold. We show non-asymptotic error bounds and bounds on the expected number of samples for three types of tests, a fixed sample size test, a sequential test with indifference region, and a sequential test without indifference region. Our tests can lead to significant savings in sample size. We illustrate our results on an example of Bayesian parameter inference involving an ODE model of a biochemical pathway.

Optimal schemes for resubmitted lot acceptance using previous defect count data

Optimal defects-per-unit inspection schemes for screening batches of manufactured material are obtained by minimizing the expected sampling effort. Nonaccepted lots may be resubmitted for resampling inspection, whereas the Poisson model is used to describe the random behavior of the number of nonconformities per sampled unit. A coefficient is presented to assess the similarity degree between the available previous information and the current inspection, and truncated gamma distributions are adopted to quantify the natural prior uncertainty about the defect rate using past count data and expert opinions. A step-by-step computational procedure is proposed to solve the underlying integer nonlinear programming problem in order to find the best resubmitted lot sampling plan with controlled expected producer and consumer risks based on previous objective and subjective knowledge. In many practical cases, the inclusion of lot resubmissions and past information into the inspection process provides substantial savings in sample size, as well as more reliable evaluations of the existing producer and consumer risks. The proposed approach allows the practitioners to consider a restricted interval for the defect rate, which is reasonable in practice and unfeasible under the frequentist perspective. Moreover, a mechanism is suggested to update the prior distribution based on past performance of the inspection plan. For illustrative purposes, the methodology developed is applied to the manufacturing of glass.

Optimum attributes component test plans for k-out-of-n:F Weibull systems using prior information

Assuming that the proportion defective in a production process p is variable and the component lifetimes are Weibull distributed, integer nonlinear programming problems are formulated and solved in order to determine the optimum component inspection scheme by attributes for k-out-of-n:F system reliability demonstration using available prior knowledge. A limited beta prior model is adopted to reflect the random fluctuations on p. The required quantity of components to test and the permissible number of component failures up to a specified censoring time are found by solving a minimisation problem with nonlinear constraints which are related to the tolerable average producer and consumer risks. First-order Taylor polynomials of the operating characteristic function are used to derive a quite accurate approximate solution. Lower and upper bounds are also deduced. Optimal solutions are usually robust to small variations in the Weibull parameters and prior information. Existing technical knowledge and experience are shown to be of great practical value in designing optimal sampling plans for lot acceptance purposes. The proposed approach generalises the classical viewpoint to those cases in which appreciable prior information on the fraction of defective systems exists, and also allows the analyst to determine the acceptability of a k-out-of-n:F system before assembly and to limit the range of p. Moreover, the practitioners may attain substantial savings in sample size and improved assessments of the true producer and consumer risks. A 4-out-of-5:F system of independent water pumps for cooling a reactor is considered to illustrate the suggested component test plans.

Optimal Defects-Per-Unit Acceptance Sampling Plans Using Truncated Prior Distributions

Optimal sampling inspection plans for defects per unit with fixed acceptance numbers and limiting quality levels are developed to provide the appropriate protection to customers when the number of nonconformities per sampled item follows a Poisson distribution. The best inspection scheme assures the customer, who has to judge the quality of the submitted material, that a supplier's lot is released only when there is conclusive evidence that it is satisfactory. The underlying integer nonlinear programming problem is formulated and solved in the frequentist setting, and a practically exact approximation to the minimum sample size is presented. Because there is often no reason to assume that the process average is constant, the classical perspective is then extended to those situations in which there is substantial prior information on the supplier's process. A family of generalized truncated gamma models and several restricted maximum entropy distributions satisfying typical constraints are adopted to describe the stochastic fluctuations in the process average. Optimal defects-per-unit acceptance plans are determined by solving the corresponding constrained minimization problems. Lower and upper bounds on the required sample size are deduced in closed-forms. A general procedure based on Taylor series expansions of the operating characteristic function around the mean quality level of the rejectable lots is proposed to derive an explicit, accurate, easily computable approximation to the smallest sample size that provides the required average customer protection. This procedure greatly simplifies the determination of optimal plans from defect or failure count data and prior knowledge, and also requires little prior information, namely the prior mean and variance of the quality level of the rejectable lots, which could be estimated from past data and expert opinions. The suggested methodology is applied to the manufacturing of paper and glass for illustrative purposes. Our approach allows the practitioners to incorporate into the quality analysis a reduced parameter space for the process average. Furthermore, the proposed sampling plans are reasonably insensitive to small disturbances in the prior knowledge on the process average, and the effective use of the available information on the supplier's process provides a more realistic assessment of the actual customer protection, as well as considerable savings in testing time and sample size.

Bayesian Sample Size Determination for a Clinical Trial with Correlated Continuous and Binary Outcomes

In clinical trials, multiple outcomes are often collected in order to simultaneously assess effectiveness and safety. We develop a Bayesian procedure for determining the required sample size in a regression model where a continuous efficacy variable and a binary safety variable are observed. The sample size determination procedure is simulation based. The model accounts for correlation between the two variables. Through examples we demonstrate that savings in total sample size are possible when the correlation between these two variables is sufficiently high.

A general approach for sample size calculation for the three‐arm ‘gold standard’ non‐inferiority design

In the three-arm 'gold standard' non-inferiority design, an experimental treatment, an active reference, and a placebo are compared. This design is becoming increasingly popular, and it is, whenever feasible, recommended for use by regulatory guidelines. We provide a general method to calculate the required sample size for clinical trials performed in this design. As special cases, the situations of continuous, binary, and Poisson distributed outcomes are explored. Taking into account the correlation structure of the involved test statistics, the proposed approach leads to considerable savings in sample size as compared with application of ad hoc methods for all three scale levels. Furthermore, optimal sample size allocation ratios are determined that result in markedly smaller total sample sizes as compared with equal assignment. As optimal allocation makes the active treatment groups larger than the placebo group, implementation of the proposed approach is also desirable from an ethical viewpoint.

The power of FDG-PET to detect treatment effects is increased by glucose correction using a Michaelis constant.

BackgroundWe recently showed improved between-subject variability in our [18F]fluorodeoxyglucose positron emission tomography (FDG-PET) experiments using a Michaelis-Menten transport model to calculate the metabolic tumor glucose uptake rate extrapolated to the hypothetical condition of glucose saturation: , where Ki is the image-derived FDG uptake rate constant, KM is the half-saturation Michaelis constant, and [glc] is the blood glucose concentration. Compared to measurements of Ki alone, or calculations of the scan-time metabolic glucose uptake rate (MRgluc = Ki * [glc]) or the glucose-normalized uptake rate (MRgluc = Ki*[glc]/(100 mg/dL), we suggested that could offer increased statistical power in treatment studies; here, we confirm this in theory and practice.MethodsWe compared Ki, MRgluc (both with and without glucose normalization), and as FDG-PET measures of treatment-induced changes in tumor glucose uptake independent of any systemic changes in blood glucose caused either by natural variation or by side effects of drug action. Data from three xenograft models with independent evidence of altered tumor cell glucose uptake were studied and generalized with statistical simulations and mathematical derivations. To obtain representative simulation parameters, we studied the distributions of Ki from FDG-PET scans and blood [glucose] values in 66 cohorts of mice (665 individual mice). Treatment effects were simulated by varying and back-calculating the mean Ki under the Michaelis-Menten model with KM = 130 mg/dL. This was repeated to represent cases of low, average, and high variability in Ki (at a given glucose level) observed among the 66 PET cohorts.ResultsThere was excellent agreement between derivations, simulations, and experiments. Even modestly different (20%) blood glucose levels caused Ki and especially MRgluc to become unreliable through false positive results while remained unbiased. The greatest benefit occurred when Ki measurements (at a given glucose level) had low variability. Even when the power benefit was negligible, the use of carried no statistical penalty. Congruent with theory and simulations, showed in our experiments an average 21% statistical power improvement with respect to MRgluc and 10% with respect to Ki (approximately 20% savings in sample size). The results were robust in the face of imprecise blood glucose measurements and KM values.ConclusionsWhen evaluating the direct effects of treatment on tumor tissue with FDG-PET, employing a Michaelis-Menten glucose correction factor gives the most statistically powerful results. The well-known alternative ‘correction’, multiplying Ki by blood glucose (or normalized blood glucose), appears to be counter-productive in this setting and should be avoided.

Optimal Reliability Demonstration Test Plans for $k$-out-of-$n$ Systems of Gamma Distributed Components

Constrained optimization problems are solved to determine the best reliability sampling plans for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$k$</tex> </formula> -out-of- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex> </formula> :F systems of gamma distributed components. As usual, the constraints are related to the quality levels required by the producer and the consumer, whereas the decision criterion is based on the uniformly most powerful reliability test. Excellent approximate solutions are derived in closed-forms assuming a frequentist perspective. This classical methodology is generalized to those situations in which the prior information on the fraction effective, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$r$</tex></formula> , is appreciable. Because prior knowledge on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$r$</tex> </formula> often suggests a restricted parameter interval, a limited beta distribution is used to describe the probabilistic behavior of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$r$</tex></formula> . Optimal sample sizes and acceptance factors are obtained by solving mixed integer nonlinear programming problems. Lower and upper bounds on the smallest sample size, as well as a quite accurate approximation, are also provided. An efficient step-by-step procedure, which only needs a few iterations in most practical cases, is proposed for solving the optimization problem. The incorporation of prior knowledge into the reliability analysis provides better assessment of the true producer's and consumer's risks, and considerable savings in sample size and test duration. The optimal solutions are usually quite robust to small variations in the gamma shape parameter and the prior information. An example concerning a system composed of five water pumps for cooling a reactor is included for illustration. The proposed component test sampling plans can be recommended for use in practical applications, particularly when it is economically infeasible, or even impossible, to test a whole system.

Optimal acceptance sampling plans for log-location–scale lifetime models using average risks

Assuming that the probability of obtaining a defective unit in a production process, p , is not constant, a versatile methodology is presented for determining optimal failure-censored reliability sampling plans for log-location–scale lifetime models. The optimization procedure to decide the acceptability of a product is usually sufficiently accurate for the most widely used parametric lifetime models, such as the Weibull and lognormal distributions, and fairly robust to small deviations in the prior knowledge. Moreover, lower and upper bounds on the optimal sample size, and the corresponding acceptance constants, are derived in closed-forms. The proposed approach extends the traditional sampling plans to those cases in which appreciable prior information on p exists, and also allows the analyst the flexibility to delimitate the range of p and to incorporate into the reliability analysis prior impartiality between the producer and the consumer. In addition, the practitioners may achieve substantial savings in sample size, better information on the production process and better assessment of the true producer and consumer risks. An example related to the acceptability of a certain kind of gyroscope is included for illustrative purposes. Various practical prior distributions are considered to describe the random fluctuations in the proportion defective.

Problems with Using the Normal Distribution – and Ways to Improve Quality and Efficiency of Data Analysis

BackgroundThe Gaussian or normal distribution is the most established model to characterize quantitative variation of original data. Accordingly, data are summarized using the arithmetic mean and the standard deviation, by ± SD, or with the standard error of the mean, ± SEM. This, together with corresponding bars in graphical displays has become the standard to characterize variation.Methodology/Principal FindingsHere we question the adequacy of this characterization, and of the model. The published literature provides numerous examples for which such descriptions appear inappropriate because, based on the “95% range check”, their distributions are obviously skewed. In these cases, the symmetric characterization is a poor description and may trigger wrong conclusions. To solve the problem, it is enlightening to regard causes of variation. Multiplicative causes are by far more important than additive ones, in general, and benefit from a multiplicative (or log-) normal approach. Fortunately, quite similar to the normal, the log-normal distribution can now be handled easily and characterized at the level of the original data with the help of both, a new sign, x/, times-divide, and notation. Analogous to ± SD, it connects the multiplicative (or geometric) mean * and the multiplicative standard deviation s* in the form * x/s*, that is advantageous and recommended.Conclusions/SignificanceThe corresponding shift from the symmetric to the asymmetric view will substantially increase both, recognition of data distributions, and interpretation quality. It will allow for savings in sample size that can be considerable. Moreover, this is in line with ethical responsibility. Adequate models will improve concepts and theories, and provide deeper insight into science and life.

Statistical Efficiency in Multiple-to-One Comparison Trials with Optimal Allocation Ratio

This paper discusses multiple-to-one comparison trials testing a multivalent vaccine product against multiple comparators with respect to immunologic responses. An optimal subject allocation ratio between the multivalent vaccine group and any of the comparators is introduced. This optimal ratio leads to higher statistical efficiency in terms of savings in sample size compared to equal allocation or any other allocation ratios. Several trial design scenarios are considered including assumptions of equal standardized effect size, unequal variance, and unequal standardized effect sizes of the multivalent vaccines versus the multiple comparators.

Multistage Tests of Multiple Hypotheses

Conventional multiple hypothesis tests use step-up, step-down, or closed testing methods to control the overall error rates. We will discuss marrying these methods with adaptive multistage sampling rules and stopping rules to perform efficient multiple hypothesis testing in sequential experimental designs. The result is a multistage step-down procedure that adaptively tests multiple hypotheses while preserving the family-wise error rate and extends Holm's (1979) step-down procedure to the sequential setting, yielding substantial savings in sample size with small loss in power.

A novel sequential sampling technique for the surveillance of transmitted HIV drug resistance by cross-sectional survey for use in low resource settings

This article describes the development of a novel sequential sampling method for the surveillance of transmitted HIV drug resistance by cross-sectional survey. Two commonly used sequential sampling methods are described and their applicability to the problem of classifying the prevalence of transmitted HIV drug resistance investigated. Both methods are rejected due to insufficient savings in sample size and operational complexity. A novel method is proposed and this is tested using computer-based simulation. This method provides useful sample size savings and operational simplicity and could provide the basis for a rapid and reliable survey method for classifying the prevalence of transmitted HIV drug resistance in circumstances where monitoring HIV drug resistance is an important issue, but resources do not allow full-scale surveillance to be established. The method is currently being used in several such settings.