Interim analysis incorporating short‐ and long‐term binary endpoints

Designs incorporating more than one endpoint have become popular in drug development. One of such designs allows for incorporation of short‐term information in an interim analysis if the long‐term primary endpoint has not been yet observed for some of the patients. At first we consider a two‐stage design with binary endpoints allowing for futility stopping only based on conditional power under both fixed and observed effects. Design characteristics of three estimators: using primary long‐term endpoint only, short‐term endpoint only, and combining data from both are compared. For each approach, equivalent cut‐off point values for fixed and observed effect conditional power calculations can be derived resulting in the same overall power. While in trials stopping for futility the type I error rate cannot get inflated (it usually decreases), there is loss of power. In this study, we consider different scenarios, including different thresholds for conditional power, different amount of information available at the interim, different correlations and probabilities of success. We further extend the methods to adaptive designs with unblinded sample size reassessments based on conditional power with inverse normal method as the combination function. Two different futility stopping rules are considered: one based on the conditional power, and one from P‐values based on Z‐statistics of the estimators. Average sample size, probability to stop for futility and overall power of the trial are compared and the influence of the choice of weights is investigated.     

An iterative method to protect the type I error rate in bioequivalence studies under two‐stage adaptive 2×2 crossover designs

Bioequivalence studies are the pivotal clinical trials submitted to regulatory agencies to support the marketing applications of generic drug products. Average bioequivalence (ABE) is used to determine whether the mean values for the pharmacokinetic measures determined after administration of the test and reference products are comparable. Two‐stage 2×2 crossover adaptive designs (TSDs) are becoming increasingly popular because they allow making assumptions on the clinically meaningful treatment effect and a reliable guess for the unknown within‐subject variability. At an interim look, if ABE is not declared with an initial sample size, they allow to increase it depending on the estimated variability and to enroll additional subjects at a second stage, or to stop for futility in case of poor likelihood of bioequivalence. This is crucial because both parameters must clearly be prespecified in protocols, and the strategy agreed with regulatory agencies in advance with emphasis on controlling the overall type I error. We present an iterative method to adjust the significance levels at each stage which preserves the overall type I error for a wide set of scenarios which should include the true unknown variability value. Simulations showed adjusted significance levels higher than 0.0300 in most cases with type I error always below 5%, and with a power of at least 80%. TSDs work particularly well for coefficients of variation below 0.3 which are especially useful due to the balance between the power and the percentage of studies proceeding to stage 2. Our approach might support discussions with regulatory agencies.     

Trimmed Weighted Simes' Test for Two One‐Sided Hypotheses With Arbitrarily Correlated Test Statistics

The two‐sided Simes test is known to control the type I error rate with bivariate normal test statistics. For one‐sided hypotheses, control of the type I error rate requires that the correlation between the bivariate normal test statistics is non‐negative. In this article, we introduce a trimmed version of the one‐sided weighted Simes test for two hypotheses which rejects if (i) the one‐sided weighted Simes test rejects and (ii) both p‐values are below one minus the respective weighted Bonferroni adjusted level. We show that the trimmed version controls the type I error rate at nominal significance level α if (i) the common distribution of test statistics is point symmetric and (ii) the two‐sided weighted Simes test at level 2α controls the level. These assumptions apply, for instance, to bivariate normal test statistics with arbitrary correlation. In a simulation study, we compare the power of the trimmed weighted Simes test with the power of the weighted Bonferroni test and the untrimmed weighted Simes test. An additional result of this article ensures type I error rate control of the usual weighted Simes test under a weak version of the positive regression dependence condition for the case of two hypotheses. This condition is shown to apply to the two‐sided p‐values of one‐ or two‐sample t‐tests for bivariate normal endpoints with arbitrary correlation and to the corresponding one‐sided p‐values if the correlation is non‐negative. The Simes test for such types of bivariate t‐tests has not been considered before. According to our main result, the trimmed version of the weighted Simes test then also applies to the one‐sided bivariate t‐test with arbitrary correlation.     

Adaptive patient enrichment designs in therapeutic trials

The utility of clinical trial designs with adaptive patient enrichment is investigated in an adequate and well‐controlled trial setting. The overall treatment effect is the weighted average of the treatment effects in the mutually exclusive subsets of the originally intended entire study population. The adaptive enrichment approaches permit assessment of treatment effect that may be applicable to specific nested patient (sub)sets due to heterogeneous patient characteristics and/or differential response to treatment, e.g. a responsive patient subset versus a lack of beneficial patient subset, in all patient (sub)sets studied. The adaptive enrichment approaches considered include three adaptive design scenarios: (i) total sample size fixed and with futility stopping, (ii) sample size adaptation and futility stopping, and (iii) sample size adaptation without futility stopping. We show that regardless of whether the treatment effect eventually assessed is applicable to the originally studied patient population or only to the nested patient subsets; it is possible to devise an adaptive enrichment approach that statistically outperforms one‐size‐fits‐all fixed design approach and the fixed design with a pre‐specified multiple test procedure. We emphasize the need of additional studies to replicate the finding of a treatment effect in an enriched patient subset. The replication studies are likely to need fewer number of patients because of an identified treatment effect size that is larger than the diluted overall effect size. The adaptive designs, when applicable, are along the line of efficiency consideration in a drug development program.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

Rotation gene set testing for longitudinal expression data

Gene set analysis methods are popular tools for identifying differentially expressed gene sets in microarray data. Most existing methods use a permutation test to assess significance for each gene set. The permutation test's assumption of exchangeable samples is often not satisfied for time‐series data and complex experimental designs, and in addition it requires a certain number of samples to compute p‐values accurately. The method presented here uses a rotation test rather than a permutation test to assess significance. The rotation test can compute accurate p‐values also for very small sample sizes. The method can handle complex designs and is particularly suited for longitudinal microarray data where the samples may have complex correlation structures. Dependencies between genes, modeled with the use of gene networks, are incorporated in the estimation of correlations between samples. In addition, the method can test for both gene sets that are differentially expressed and gene sets that show strong time trends. We show on simulated longitudinal data that the ability to identify important gene sets may be improved by taking the correlation structure between samples into account. Applied to real data, the method identifies both gene sets with constant expression and gene sets with strong time trends.     

Adaptive Two Stage Designs and the Conditional Error Function

Proschan and Hunsberger (1995) suggest the use of a conditional error function to construct a two stage test that meets the α level and allows a very flexible reassessment of the sample size after the interim analysis. In this note we show that several adaptive designs can be formulated in terms of such an error function. The conditional power function defined similarly provides a simple method for sample size reassessment in adaptive two stage designs.     

Improved power of familywise error rate procedures for discrete data under dependency

In many applications where it is necessary to test multiple hypotheses simultaneously, the data encountered are discrete. In such cases, it is important for multiplicity adjustment to take into account the discreteness of the distributions of the p‐values, to assure that the procedure is not overly conservative. In this paper, we review some known multiple testing procedures for discrete data that control the familywise error rate, the probability of making any false rejection. Taking advantage of the fact that the exact permutation or exact pairwise permutation distributions of the p‐values can often be determined when the sample size is small, we investigate procedures that incorporate the dependence structure through the exact permutation distribution and propose two new procedures that incorporate the exact pairwise permutation distributions. A step‐up procedure is also proposed that accounts for the discreteness of the data. The performance of the proposed procedures is investigated through simulation studies and two applications. The results show that by incorporating both discreteness and dependency of p‐value distributions, gains in power can be achieved.     

On the False Discovery Rate and Expected Type I Errors

The paper is concerned with expected type I errors of some stepwise multiple test procedures based on independent p‐values controlling the so‐called false discovery rate (FDR). We derive an asymptotic result for the supremum of the expected type I error rate(EER) when the number of hypotheses tends to infinity. Among others, it will be shown that when the original Benjamini‐Hochberg step‐up procedure controls the FDR at level α, its EER may approach a value being slightly larger than α/4 when the number of hypotheses increases. Moreover, we derive some least favourable parameter configuration results, some bounds for the FDR and the EER as well as easily computable formulae for the familywise error rate (FWER) of two FDR‐controlling procedures. Finally, we discuss some undesirable properties of the FDR concept, especially the problem of cheating.     

Adaptive dose-finding: Proof of concept with type I error control

We consider an adaptive dose‐finding study with two stages. The doses for the second stage will be chosen based on the first stage results. Instead of considering pairwise comparisons with placebo, we apply one test to show an upward trend across doses. This is a possibility according to the ICH‐guideline for dose‐finding studies (ICH‐E4). In this article, we are interested in trend tests based on a single contrast or on the maximum of multiple contrasts. We are interested in flexibly choosing the Stage 2 doses including the possibility to add doses. If certain requirements for the interim decision rules are fulfilled, the final trend test that ignores the adaptive nature of the trial (naïve test) can control the type I error. However, for the more common case that these requirements are not fulfilled, we need to take the adaptivity into account and discuss a method for type I error control. We apply the general conditional error approach to adaptive dose‐finding and discuss special issues appearing in this application. We call the test based on this approach Adaptive Multiple Contrast Test. For an example, we illustrate the theory discussed before and compare the performance of several tests for the adaptive design in a simulation study.     

Tests of Independence and Zero Correlations Among P Random Variables

Suppose it is desired to determine whether there is an association between any pair of p random variables. A common approach is to test H₀ : R = I, where R is the usual population correlation matrix. A closely related approach is to test H₀ : R_pb = I, where R_pb is the matrix of percentage bend correlations. In so far as type I errors are a concern, at a minimum any test of H₀ should have a type I error probability close to the nominal level when all pairs of random variables are independent. Currently, the Gupta-Rathie method is relatively successful at controlling the probability of a type I error when testing H₀: R = I, but as illustrated in this paper, it can fail when sampling from nonnormal distributions. The main goal in this paper is to describe a new test of H₀: R_pb = I that continues to give reasonable control over the probability of a type I error in the situations where the Gupta-Rathie method fails. Even under normality, the new method has advantages when the sample size is small relative to p. Moreover, when there is dependence, but all correlations are equal to zero, the new method continues to give good control over the probability of a type I error while the Gupta-Rathie method does not. The paper also reports simulation results on a bootstrap confidence interval for the percentage bend correlation.     

IMMUNOHISTOCHEMICAL LOCALIZATION OF TYPE I AND II COLLAGENS IN THE INVOLUTING CHICK NOTOCHORDS IN VIVO AND IN VITRO

Embryonic chick notochords were studied during their metabolically active and involuting periods for the expression of collagen type I and II. The staining was carried out on notochords in vivo at stage 20 and stage 35 and on mesenchyme-contaminated and mesenchyme-free notochords at stage 20, which were cultured in vitro for 6 days. The results show that type II collagen is demonstrable in the notochords, at all the examined stages, both in vivo and in vitro. However, the expression of type I collagen was stage-dependent in vivo and in vitro. At stage 20, the perinotochordal sheath is positively immunostained for collagen type I, but the notochord itself is negative. At stage 35, the perinotochordal sheath as well as the notochord are positively immunostained for collagen type I. The mesenchyme-contaminated and the mesenchyme-free notochords and their sheaths are also positively immunostained for the type I collagen after6 days in vitro. The current results, at late developmental stages, indicate that the involuting notochords express collagen type I, which seems not to be altered by changing the micro-environment in vivo.     

VNN3 is a potential novel biomarker for predicting prognosis in clear cell renal cell carcinoma

Although pathological observations provide approximate prognoses, it is difficult to achieve prognosis in patients with existing prognostic factors. Therefore, it is very important to find appropriate biomarkers to achieve accurate cancer prognosis. Renal cell carcinoma (RCC) has several subtypes, the discrimination of which is crucial for proper treatment. Here, we present a novel biomarker, VNN3, which is used to prognose clear cell renal cell carcinoma (ccRCC), the most common and aggressive subtype of kidney cancer. Patient information analyzed in our study was extracted from The Cancer Genome Atlas (TCGA) and International Cancer Genome Consortium (ICGC) cohorts. VNN3 expression was considerably higher in stages III and IV than in stages I and II. Moreover, Kaplan–Meier curves associated high VNN3 expression with poor prognoses (TCGA, p?p?=?.00076), confirming that ccRCC prognosis can be predicted via VNN3 expression patterns. Consistent with all patient results, the prognosis of patients with higher VNN3 expression was worse in both low stage (I and II) and high stage (III and IV) (TCGA, p < 0.0001 in stage I and II; ICGC, p = 0.028 in stage I and II; TCGA, p = 0.005 in stage III and IV). Area under the curve and receiver operating characteristic curves supported our results that highlighted VNN3 expression as a suitable ccRCC biomarker. Multivariate analysis also verified the prognostic performance of VNN3 expression (TCGA, p?p?=?.017). Altogether, we suggest that VNN3 is applicable as a new biomarker to establish prognosis in patients with ccRCC.     

TYPE I ERROR RATES FOR TESTING GENETIC DRIFT WITH PHENOTYPIC COVARIANCE MATRICES: A SIMULATION STUDY

Studies of evolutionary divergence using quantitative genetic methods are centered on the additive genetic variance–covariance matrix ( G ) of correlated traits. However, estimating G properly requires large samples and complicated experimental designs. Multivariate tests for neutral evolution commonly replace average G by the pooled phenotypic within‐group variance–covariance matrix ( W ) for evolutionary inferences, but this approach has been criticized due to the lack of exact proportionality between genetic and phenotypic matrices. In this study, we examined the consequence, in terms of type I error rates, of replacing average G by W in a test of neutral evolution that measures the regression slope between among‐population variances and within‐population eigenvalues (the Ackermann and Cheverud [AC] test) using a simulation approach to generate random observations under genetic drift. Our results indicate that the type I error rates for the genetic drift test are acceptable when using W instead of average G when the matrix correlation between the ancestral G and P is higher than 0.6, the average character heritability is above 0.7, and the matrices share principal components. For less‐similar G and P matrices, the type I error rates would still be acceptable if the ratio between the number of generations since divergence and the effective population size (t/N_e) is smaller than 0.01 (large populations that diverged recently). When G is not known in real data, a simulation approach to estimate expected slopes for the AC test under genetic drift is discussed.     

An adaptive two-stage design with treatment selection using the conditional error function approach

As an approach to combining the phase II dose finding trial and phase III pivotal trials, we propose a two-stage adaptive design that selects the best among several treatments in the first stage and tests significance of the selected treatment in the second stage. The approach controls the type I error defined as the probability of selecting a treatment and claiming its significance when the selected treatment is indifferent from placebo, as considered in Bischoff and Miller (2005). Our approach uses the conditional error function and allows determining the conditional type I error function for the second stage based on information observed at the first stage in a similar way to that for an ordinary adaptive design without treatment selection. We examine properties such as expected sample size and stage-2 power of this design with a given type I error and a maximum stage-2 sample size under different hypothesis configurations. We also propose a method to find the optimal conditional error function of a simple parametric form to improve the performance of the design and have derived optimal designs under some hypothesis configurations. Application of this approach is illustrated by a hypothetical example.     

Application of adaptive designs--a review

A literature search has been performed to review applications of the adaptive design methodology based on the combination test or conditional error function approach. Some features of the 60 papers identified are summarized, e.g., the specific methodology used, calendar year, country, impact factor of the journal, number of planned and performed stages respectively, stopping for futility boundaries, type of adaptations and others. A selection of the ten recent publications in journals with the highest impact factors is discussed in more detail. Most applications up to now aim at sample size reassessment, the majority of papers is coming from Germany. Although we found that renowned journals allow for sufficient space to present the new statistical methodology in all its necessary details, the general impression is that the presentation of the adaptive designs methodology in applied papers has to be improved. Education and development of standards could help to achieve this.     

DELAYED‐RESPONSE PHYLOGENETIC CORRELATION: AN OPTIMIZATION‐BASED METHOD TO TEST COVARIATION OF CONTINUOUS CHARACTERS

A new phylogenetic comparative method is proposed, based on mapping two continuous characters on a tree to generate data pairs for regression or correlation analysis, which resolves problems of multiple character reconstructions, phylogenetic dependence, and asynchronous responses (evolutionary lags). Data pairs are formed in two ways (tree‐down and tree‐up) by matching corresponding changes, Δx and Δy. Delayed responses (Δy occurring later in the tree than Δx) are penalized by weighting pairs using nodal or branch‐length distance between Δx and Δy; immediate (same‐node) responses are given maximum weight. All combinations of character reconstructions (or a random sample thereof) are used to find the observed range of the weighted coefficient of correlation r (or weighted slope b). This range is used as test statistic, and the null distribution is generated by randomly reallocating changes (Δx and Δy) in the topology. Unlike randomization of terminal values, this procedure complies with Generalized Monte Carlo requirements while saving considerable computation time. Phylogenetic dependence is avoided by randomization without data transformations, yielding acceptable type‐I error rates and statistical power. We show that ignoring delayed responses can lead to falsely nonsignificant results. Issues that arise from considering delayed responses based on optimization are discussed.     

Adaptive enrichment designs with a continuous biomarker

A popular design for clinical trials assessing targeted therapies is the two-stage adaptive enrichment design with recruitment in stage 2 limited to a biomarker-defined subgroup chosen based on data from stage 1. The data-dependent selection leads to statistical challenges if data from both stages are used to draw inference on treatment effects in the selected subgroup. If subgroups considered are nested, as when defined by a continuous biomarker, treatment effect estimates in different subgroups follow the same distribution as estimates in a group-sequential trial. This result is used to obtain tests controlling the familywise type I error rate (FWER) for six simple subgroup selection rules, one of which also controls the FWER for any selection rule. Two approaches are proposed: one based on multivariate normal distributions suitable if the number of possible subgroups, k, is small, and one based on Brownian motion approximations suitable for large k. The methods, applicable in the wide range of settings with asymptotically normal test statistics, are illustrated using survival data from a breast cancer trial.     

Oral health in institutionalised elderly people in Oslo, Norway and its relationship with dependence and cognitive impairment

doi: 10.1111/j.1741‐2358.2011.00490.x
Oral health in institutionalised elderly people in Oslo, Norway and its relationship with dependence and cognitive impairment Objective: Investigating oral health’s relationship with dependency and cognitive state. Background: Oral hygiene is poor in the institutionalised elderly. There are problems regarding the oral care of residents having poor mobility or cognitive impairment. Material and methods: Cross‐sectional study involving 135 participants (mean age 85.7, SD 8.8 years) in two categories: nurses doing tooth cleaning and residents doing tooth cleaning. Those cleaned by nurses were categorised as co‐operative or unco‐operative. The oral hygiene status, presence of caries, retained roots and denture‐related stomatitis were recorded. Results: Of the participants, 70% had only natural teeth. The prevalence of caries was 28%. A significant correlation showed that having more teeth gave a poorer Simplified Oral Hygiene Index (OHI‐S) (p = 0.018). The number of retained roots increased with the severity of cognitive impairment (p < 0.05). Significant differences were found between nurses or residents doing the tooth cleaning on the OHI‐S (p = 0.05) and percentage of dental plaque (p = 0.003). Unco‐operative residents had poorer oral hygiene (p = 0.028), more caries (p = 0.008) and were more often moderate–severe cognitive impaired (p = 0.016). Conclusions: A high percentage of participants had unacceptable oral hygiene. Residents whose teeth were cleaned by the nurses had poorer oral hygiene. Unco‐operative residents had the worst oral hygiene and more caries.     

Chemical Composition and Antimicrobial Activity of Achillea tenuifolia Lam. Essential Oil at Different Phenological Stages from Khoy

Achillea species and in particular Achillea tenuifolia Lam . is generally used as a food flavor and traditional remedies, especially in the initial developmental stage for medical conditions in the Mediterranean part of Iran. In this report, we extracted the essential oil from the aerial parts of A. tenuifolia (collected from Khoy), at various developmental stages (i. e., vegetative, flowering and fruiting), characterized them and studied their antibacterial activities. Of 46, 51 and 38 components found in the vegetative, flowering, and fruiting stages, respectively, 35 were present in all three stages, including oxygenated terpenes such as carvacrol (30.85–34.11), germacrene C (16.21–17.87), spathulenol (7.26–8.96), β‐sesquiphellandrene (4.11–4.25), τ‐muurolol (2.27–3.25) and α‐cadinol (2.01–3.29). We witnessed that the composition of the essential oils varies with phenological stages and geographic regions. The essential oil demonstrated substantial antibacterial properties against both Gram‐positive and Gram‐negative bacteria, indicated by disk method, Minimum Inhibitory Concentration (MIC) and Minimum Bactericidal Concentration (MBC) assays. Except Pseudomonas aeruginosa, the essential oils of various phenological stages showed higher antibacterial activity against tested bacteria, with Bacillus anthracis as the most sensitive strain. Moreover, although antibacterial characteristics of the essential oil from the vegetative and flowering stages were similar (p=0.91), they were significantly different from those of fruiting stage (p<0.005 in both MIC and MBC tests). This emphasizes the importance of the developmental stage of the plant in the biological properties of its essential oil and justifies the widespread application of this plant in the vegetative stage.