Optimal crossover designs in the presence of carryover effects

Under either the random patient-effect model with sequence effects or the fixed patient-effect model, the usual two-period, two-treatment crossover design, AB,BA, cannot be used to estimate the contrast between direct treatment effects when unequal carryover effects are present. If baseline observations are available, the design AB,BA can validly be used to estimate a treatment contrast. However, the design AB,BA,AA,BB with baseline observations is more efficient. In fact, we show that this design is optimal whether or not baseline observations are available. For experiments with more than two periods, universally optimal designs are found for both models, with and without carryover effects. It is shown that uncertainty about the presence of carryover effects is of little or no consequence, and the addition of baseline observations is of little or no added value for designs with three or more periods; however, if the experiment is limited to only two periods the investigator pays a heavy penalty.     

Confidence Interval for Rate Ratio in a 2 × 2 Table with Structural Zero: An Application in Assessing False-Negative Rate Ratio When Combining Two Diagnostic Tests

Summary . In this article, we consider problems with correlated data that can be summarized in a 2 × 2 table with structural zero in one of the off‐diagonal cells. Data of this kind sometimes appear in infectious disease studies and two‐step procedure studies. Lui (1998, Biometrics 54, 706–711) considered confidence interval estimation of rate ratio based on Fieller‐type, Wald‐type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false‐negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false‐negative rate ratio. Score test–based confidence interval construction for false‐negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test–based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre‐assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

Exact confidence bounds following adaptive group sequential tests

Summary .  We provide a method for obtaining confidence intervals, point estimates, and p-values for the primary effect size parameter at the end of a two-arm group sequential clinical trial in which adaptive changes have been implemented along the way. The method is based on applying the adaptive hypothesis testing procedure of Müller and Schäfer (2001, Biometrics 57, 886–891) to a sequence of dual tests derived from the stage-wise adjusted confidence interval of Tsiatis, Rosner, and Mehta (1984, Biometrics 40, 797–803). In the nonadaptive setting this confidence interval is known to provide exact coverage. In the adaptive setting exact coverage is guaranteed provided the adaptation takes place at the penultimate stage. In general, however, all that can be claimed theoretically is that the coverage is guaranteed to be conservative. Nevertheless, extensive simulation experiments, supported by an empirical characterization of the conditional error function, demonstrate convincingly that for all practical purposes the coverage is exact and the point estimate is median unbiased. No procedure has previously been available for producing confidence intervals and point estimates with these desirable properties in an adaptive group sequential setting. The methodology is illustrated by an application to a clinical trial of deep brain stimulation for Parkinson's disease.     

Induced Resistance by β‐Aminobutyric Acid in Artichoke against White Mould Caused by Sclerotinia sclerotiorum

β‐aminobutyric acid (BABA) was assessed for the ability to protect two artichoke cultivars, C3 and Exploter, against white mould caused by Sclerotinia sclerotiorum, which represents a major problem in the cultivation of this crop in many growing areas of Central Italy. Changes in the activity and isoenzymatic profiles of the pathogenesis‐related (PR) proteins β‐1,3‐glucanase, chitinase and peroxidase in plantlets upon BABA treatment and following inoculation of the pathogen in plantlets and leaves detached from adult plants were also investigated as molecular markers of induced resistance and priming. BABA treatments by soil drenching induced a high level of resistance against S. sclerotiorum in artichoke plantlets of both cultivars C3 and Exploter with a similar level of protection and determined a consistent increase in peroxidase activity paralleled with the differential induction of alkaline isoenzyme with a pI 8.6. A consistent change was found in Exploter in the peroxidase activity following BABA treatments and pathogen inoculation and was paralleled with the expression of an anionic band in plantlets and both anionic and cationic bands in leaves. Our results showed a correlation between BABA‐induced resistance (BABA‐IR) and a augmented capacity to express basal defence responses, more pronounced in cultivar C3 and associated β‐1,3‐glucanase accumulation in both plantlets and leaves inoculated with the pathogen, whereas chitinase resulted affected only at plantlet stage. The present results represent the first one showing the effect of BABA in inducing resistance in artichoke and associated accumulation of selected PRs. If confirmed in field tests, the use of BABA at early plant stages may represent a promising approach to the control soilborne pathogens, such as the early infection of S. sclerotiorum.     

Carryover and the two-period crossover clinical trial

The use of the two-period crossover trial for comparing two noncurative treatments in patients with a chronic disease is appealing since it eliminates the between-subject variability. However, the possibility of the existence of carryover effect leads many authors to advise that a parallel design be used whenever carryover is suspected. We examine this advice and quantify the degree of carryover required to make the parallel design preferable in terms of the power of the test of treatment effect and precision of the estimate of treatment difference. We conclude that in many situations this amount of carryover is substantial and unlikely to exist.     

Improved Repeated Confidence Bounds in Trials with a Maximal Goal

Repeated confidence intervals can be computed at every interim analysis of a flexible group sequential design without the need to stop the trial with a pre‐planned stopping rule. Often, however, there is a maximal goal such that the trial is surely stopped if this goal is reached. This induces a maximal stopping rule, and repeated confidence intervals are strictly conservative, when adhering to it. A modification is proposed which uniformly improves the one sided repeated confidence interval in such a situation. It preserves the monitoring character, and leads to uniformly smaller intervals, when reaching the maximal goal at an interim analysis. The modification is worked out for two stage designs and is indicated for multi‐stage trials. The extent of the improvement is quantified for two simple scenarios.     

A Bayesian analysis of the two-period crossover design for clinical trials

Statisticians have been critical of the use of the two-period crossover designs for clinical trials because the estimate of the treatment difference is biased when the carryover effects of the two treatments are not equal. In the standard approach, if the null hypothesis of equal carryover effects is not rejected, data from both periods are used to estimate and test for treatment differences; if the null hypothesis is rejected, data from the first period alone are used. A Bayesian analysis based on the Bayes factor against unequal carryover effects is given. Although this Bayesian approach avoids the "all-or-nothing" decision inherent in the standard approach, it recognizes that with small trials it is difficult to provide unequivocal evidence that the carryover effects of the two treatments are equal, and thus that the interpretation of the difference between treatment effects is highly dependent on a subjective assessment of the reality or not of equal carryover effects.     

Confidence interval for rate ratio in a 2 x 2 table with structural zero: an application in assessing false-negative rate ratio when combining two diagnostic tests

In this article, we consider problems with correlated data that can be summarized in a 2 x 2 table with structural zero in one of the off-diagonal cells. Data of this kind sometimes appear in infectious disease studies and two-step procedure studies. Lui (1998, Biometrics54, 706-711) considered confidence interval estimation of rate ratio based on Fieller-type, Wald-type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false-negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false-negative rate ratio. Score test-based confidence interval construction for false-negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test-based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre-assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

Using the maximum test statistic in the two-period crossover clinical trial

In a two-period crossover trial where residual carryover is suspected, it is often advised that first-period data only be used in an analysis appropriate for a parallel design. However, it has been shown (Willan and Pater, 1986, Biometrics 42, 593-599) that the crossover analysis is more powerful than the parallel analysis if the residual carryover, expressed as a proportion of treatment effect, is less than 2- square root of 2(1 - rho), where rho is the intrasubject correlation coefficient. Choosing between the analyses based on the empirical evaluation of this condition is equivalent to choosing the analysis with the larger corresponding test statistic. Approximate nominal significance levels are presented that maintain the desired level when basing the analysis on the maximum test statistic. Furthermore, the power and precision of the analysis based on the maximum test statistic are compared to the crossover and parallel analyses.     

Testing and estimation of proportion (or risk) ratio under the matched‐pair design with multiple binary endpoints

The proportion ratio (PR) of responses between an experimental treatment and a control treatment is one of the most commonly used indices to measure the relative treatment effect in a randomized clinical trial. We develop asymptotic and permutation‐based procedures for testing equality of treatment effects as well as derive confidence intervals of PRs for multivariate binary matched‐pair data under a mixed‐effects exponential risk model. To evaluate and compare the performance of these test procedures and interval estimators, we employ Monte Carlo simulation. When the number of matched pairs is large, we find that all test procedures presented here can perform well with respect to Type I error. When the number of matched pairs is small, the permutation‐based test procedures developed in this paper is of use. Furthermore, using test procedures (or interval estimators) based on a weighted linear average estimator of treatment effects can improve power (or gain precision) when the treatment effects on all response variables of interest are known to fall in the same direction. Finally, we apply the data taken from a crossover clinical trial that monitored several adverse events of an antidepressive drug to illustrate the practical use of test procedures and interval estimators considered here.     

Stepwise Procedures under Unknown Variances for Toxicological Evaluation

Toxicological study is of practical importance in modern drug development. Proper statistical methodologies for toxicological evaluation of new developed drugs are undoubtedly necessary. In toxicological studies, it is practically desirable for a method to not declare the safety of a developed drug at a higher dosage prior to the declaration of the safety at lower dosages. Hsu and Berger 's stepwise confidence interval method was recently proposed for this purpose. Unfortunately, their procedure necessitates the homogeneity of variances among dosages, which is seldom satisfied in practice. In this article, via the application of the Stein 's two‐stage sampling method, we propose a stepwise confidence interval procedure for the same task without the homoscedasticity restriction. In addition, our procedure is shown to control its family‐wise type I error rate at the pre‐chosen nominal level. A simulation study will be conducted to compare our method, Hsu and Berger 's stepwise confidence interval method, and a single stage stepwise testing procedure based on Welch 's approximation. Our procedure is empirically shown to outperform Hsu and Berger 's procedure under heteroscedasticity and perform similarly with Welch 's procedure. An example will be used to illustrate our method.     

A note on the conditional approach to interval estimation in the calibration problem.

In the calibration problem, the need to construct a confidence interval to estimate the unknown chi 0 arises when the null hypothesis of zero slope is rejected. Otherwise, the resulting confidence interval will be infinite to reflect the fact that the slope of the regression line may be zero. Under the condition of rejecting the hypothesis of zero slope, we study the properties of the conditional coverage rate of the calibration confidence interval. The conditional coverage rate (P1) is a function of the slope, distance between chi 0 and the mean of the trailing sample means, the sum of squares of chi, and n. When the true slope is close to 0 and chi 0 is away from means, P1 can go down to 0. On the other hand, as the power of testing zero slope reaches 1, with or without chi 0 close to means, P1 will tend to the desired nominal coverage rate. In summary, one should choose a reasonably small alpha in testing zero slope to avoid constructing a confidence interval for chi 0 when the true slope is 0. In addition, it is desirable to have high power in testing zero slope so that the resulting confidence interval will maintain the desired coverage rate when using the conditional approach in the calibration problem.     

Accumulation patterns of endogenous β‐aminobutyric acid during plant development and defence in Arabidopsis thaliana

• We recently discovered that β‐aminobutyric acid (BABA), a molecule known for its ability to prime defences in plants, is a natural plant metabolite. However, the role played by endogenous BABA in plants is currently unknown. In this study we investigated the systemic accumulation of BABA during pathogen infection, levels of BABA during plant growth and development and analysed mutants possibly involved in BABA transport or regulation.
• BABA was quantified by LC‐MS using an improved method adapted from a previously published protocol. Systemic accumulation of BABA was determined by analysing non‐infected leaves and roots after localised infections with Plectosphaerella cucumerina or Pseudomonas syringae pv. tomato (Pst) DC3000 avrRpt2. The levels of BABA were also quantified in different plant tissues and organs during normal plant growth, and in leaves during senescence. Mutants affecting amino acid transport (aap6, aap3, prot1 and gat1), γ‐aminobutyric acid levels (pop2) and senescence/defence (cpr5‐2) were analysed.
• BABA was found to accumulate only locally after bacterial or fungal infection, with no detectable increase in non‐infected systemic plant parts. In leaves, BABA content increased during natural and induced senescence. Reproductive organs had the highest levels of BABA, and the mutant cpr5‐2 produced constitutively high levels of BABA.
• Synthetic BABA is highly mobile in the receiving plant, whereas endogenous BABA appears to be produced and accumulated locally in a tissue‐specific way. We discuss a possible role for BABA in age‐related resistance and propose a comprehensive model for endogenous and synthetic BABA.

     

Priming of the Arabidopsis pattern‐triggered immunity response upon infection by necrotrophic Pectobacterium carotovorum bacteria

Boosted responsiveness of plant cells to stress at the onset of pathogen‐ or chemically induced resistance is called priming. The chemical β‐aminobutyric acid (BABA) enhances Arabidopsis thaliana resistance to hemibiotrophic bacteria through the priming of the salicylic acid (SA) defence response. Whether BABA increases Arabidopsis resistance to the necrotrophic bacterium Pectobacterium carotovorum ssp. carotovorum (Pcc) is not clear. In this work, we show that treatment with BABA protects Arabidopsis against the soft‐rot pathogen Pcc. BABA did not prime the expression of the jasmonate/ethylene‐responsive gene PLANT DEFENSIN 1.2 (PDF1.2), the up‐regulation of which is usually associated with resistance to necrotrophic pathogens. Expression of the SA marker gene PATHOGENESIS RELATED 1 (PR1) on Pcc infection was primed by BABA treatment, but SA‐defective mutants demonstrated a wild‐type level of BABA‐induced resistance against Pcc. BABA primed the expression of the pattern‐triggered immunity (PTI)‐responsive genes FLG22‐INDUCED RECEPTOR‐LIKE KINASE 1 (FRK1), ARABIDOPSIS NON‐RACE SPECIFIC DISEASE RESISTANCE GENE (NDR1)/HAIRPIN‐INDUCED GENE (HIN1)‐LIKE 10 (NHL10) and CYTOCHROME P450, FAMILY 81 (CYP81F2) after inoculation with Pcc or after treatment with purified bacterial microbe‐associated molecular patterns, such as flg22 or elf26. PTI‐mediated callose deposition was also potentiated in BABA‐treated Arabidopsis, and BABA boosted Arabidopsis stomatal immunity to Pcc. BABA treatment primed the PTI response in the SA‐defective mutants SA induction deficient 2‐1 (sid2‐1) and phytoalexin deficient 4‐1 (pad4‐1). In addition, BABA priming was associated with open chromatin configurations in the promoter region of PTI marker genes. Our data indicate that BABA primes the PTI response upon necrotrophic bacterial infection and suggest a role for the PTI response in BABA‐induced resistance.     

Interaction Biases in Two-Period Crossover Studies: A Modified Analysis to Test with More Sensitivity

The two-period crossover trial is considered the most powerful means of determining the efficacy of new drugs. However, this study design is frequently invalidated by treatment-by-period interaction. If, for example, the effect of the first treatment period carries on into the next one, then it influences the response to the latter period (carryover effect). A second problem is, that the standard approach (Hills-Armitage analysis) for interaction bias has a low statistical sensitivity. The author recently described an alternative method entitled the clinical approach because it looks at the clinical performance of the separate treatment groups and not, as the standard approach, at the means of the groups. It may be hypothesized that this alternative approach is statistically more sensitive than the standard in situations where there is interaction in just one of the treatment groups. The present study uses two examples and a mathematical model. It shows that in case of single-group interaction the clinical approach can, indeed, detect carryover effect at a 30% lower level than the standard. On the other hand, however, the standard approach does so even at a 40% lower level than the clinical in case of two-group interaction. I conclude that one approach supplements the other and that they be used in future studies simultaneously.     

Design and Analysis of Crossover Trials for Absorbing Binary Endpoints

Summary The crossover is a popular and efficient trial design used in the context of patient heterogeneity to assess the effect of treatments that act relatively quickly and whose benefit disappears with discontinuation. Each patient can serve as her own control as within‐individual treatment and placebo responses are compared. Conventional wisdom is that these designs are not appropriate for absorbing binary endpoints, such as death or HIV infection. We explore the use of crossover designs in the context of these absorbing binary endpoints and show that they can be more efficient than the standard parallel group design when there is heterogeneity in individuals' risks. We also introduce a new two‐period design where first period “survivors” are rerandomized for the second period. This design combines the crossover design with the parallel design and achieves some of the efficiency advantages of the crossover design while ensuring that the second period groups are comparable by randomization. We discuss the validity of the new designs and evaluate both a mixture model and a modified Mantel–Haenszel test for inference. The mixture model assumes no carryover or period effects while the Mantel–Haenszel approach conditions out period effects. Simulations are used to compare the different designs and an example is provided to explore practical issues in implementation.     

The effect of β-aminobutyric acid on the growth of herbivorous insects feeding on Brassicaceae

dl ‐β‐Aminobutyric acid (BABA) is a nonprotein amino acid that can enhance defences in a variety of plants against a wide range of pathogens. BABA can also reduce infestation by phytopathogenic nematodes and has recently been shown to suppress the growth of aphids feeding on legumes. This investigation examined the effect of applying BABA as a root drench to a range of Brassicaceae, including Arabidopsis thaliana, on the performance of two species of aphid (Myzus persicae and Brevicoryne brassicae) and the larvae of two species of Lepidoptera (Trichoplusia ni and Plutella xylostella). Application of BABA reduced the performance of all four insect species, and inhibition of insects occurred on all the plants tested. The results illustrate that BABA‐induced resistance (BABA‐IR) can affect generalist and specialist insect herbivores and inhibit insects feeding with mandibulate as well as sap‐feeding mouthparts. The BABA‐induced suppression of B. brassicae and P. xylostella feeding on A. thaliana provides a means to further examine the mechanisms of BABA‐IR to insects using this model plant.     

Evaluation of aerial sampling methods for detecting waterbird colonies

ABSTRACT Estimating the number of waterbird colonies in a given area can have important conservation implications, including assessment of the regional or global importance of an area and the impacts of conservation efforts (e.g., habitat restoration) and human disturbance (e.g., oil spills). Our objective was to examine differences in estimates of the number of waterbird colonies determined using strip‐transect (ST) surveys, distance sampling, and adaptive cluster sampling (ACS), and to compare these estimates to the minimum number of known colonies (MNKC) obtained using point‐to‐point surveys. We conducted aerial surveys in May 2004 and May and June 2005 at two sites in southern Louisiana: the Atchafalaya Basin (AB), a large forested wetland, and the Barataria‐Terrebonne estuary (BTE), a large coastal marsh with isolated clumps of woody vegetation suitable for nesting. In AB, we detected nine and eight colonies using the ACS and ST/distance sampling methods, respectively. Neither ACS estimator of number of colonies (Horvitz–Thompson and Hansen–Hurwitz) was within the 95% confidence interval of the estimate determined from ST; ST estimated—two to three times more colonies than either ACS estimator. The MNKC for the AB was 33, well within the 95% confidence interval of ? by ST sampling. For the BTE, ACS estimators (?_HT= 20.49, CI = 9.3–31.7; ?_HH= 14.15, CI = 2.3–26.0) were similar to the MNKC (20), whereas the ST (?= 87.94, CI = 82.9–92.9) and distance sampling (?= 60, CI = 31–113) methods produced much larger estimates. Our results suggest that the ACS method performed better when waterbird colonies were spatially clumped (BTE) and the ST method performed better in areas where colonies were more uniformly distributed (AB). Depending on management objectives, a complete, systematic survey of a study area may be required if the potential for missing large colonies is unacceptable. If surveying an area with no previous information about colony location or dispersion, we recommend a coarse‐scale analysis of the availability and contiguity of habitat likely to contain waterbird colonies; this analysis will help determine the most appropriate survey method.     

Constructing a confidence interval for the fraction who benefit from treatment,using randomized trial data

The fraction who benefit from treatment is the proportion of patients whose potential outcome under treatment is better than that under control. Inference on this parameter is challenging since it is only partially identifiable, even in our context of a randomized trial. We propose a new method for constructing a confidence interval for the fraction, when the outcome is ordinal or binary. Our confidence interval procedure is pointwise consistent. It does not require any assumptions about the joint distribution of the potential outcomes, although it has the flexibility to incorporate various user‐defined assumptions. Our method is based on a stochastic optimization technique involving a second‐order, asymptotic approximation that, to the best of our knowledge, has not been applied to biomedical studies. This approximation leads to statistics that are solutions to quadratic programs, which can be computed efficiently using optimization tools. In simulation, our method attains the nominal coverage probability or higher, and can have narrower average width than competitor methods. We apply it to a trial of a new intervention for stroke.     

Blinded and unblinded sample size reestimation in crossover trials balanced for period

The determination of the sample size required by a crossover trial typically depends on the specification of one or more variance components. Uncertainty about the value of these parameters at the design stage means that there is often a risk a trial may be under‐ or overpowered. For many study designs, this problem has been addressed by considering adaptive design methodology that allows for the re‐estimation of the required sample size during a trial. Here, we propose and compare several approaches for this in multitreatment crossover trials. Specifically, regulators favor reestimation procedures to maintain the blinding of the treatment allocations. We therefore develop blinded estimators for the within and between person variances, following simple or block randomization. We demonstrate that, provided an equal number of patients are allocated to sequences that are balanced for period, the proposed estimators following block randomization are unbiased. We further provide a formula for the bias of the estimators following simple randomization. The performance of these procedures, along with that of an unblinded approach, is then examined utilizing three motivating examples, including one based on a recently completed four‐treatment four‐period crossover trial. Simulation results show that the performance of the proposed blinded procedures is in many cases similar to that of the unblinded approach, and thus they are an attractive alternative.