Nonparametric Methods in Crossover Trials

The crossover design is often used in biomedical trials since it eliminates between subject variability. This paper is concerned with the statistical analysis of data arising from such trials when assumptions like normality do not necessarily apply. Nonparametric analysis of the two-period, two-treatment design was first described by Koch in a paper 1972. The purpose of this paper is to study nonparametric methods in crossover designs with three or more treatments and an equal number of periods. The proposed test for direct treatment effects is based on within subject comparisons after removing a possible period effect. With only two treatments this test reduces to the twosided Wilcoxon signed rank test. By simulation experiments the validity of the significance level of the test when using the asymptotic distribution of the test statistic are manifested and the power against different alternatives illustrated. A test for first order carryover effects can be constructed by a straightforward generalization of the test proposed by Koch in 1972. However, since this test is based on between subject comparisons its power will be low. Our recommendation is to consider the crossover design rather than the parallel group design if the carryover effects are assumed to be neglible or positive and smaller then the direct treatment effects.     

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.     

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.     

A Note on the Analysis of the Two-Period Crossover Design When the Period-Treatment Interaction is Significant

In the analysis of a two-period crossover study Grizzle (1965) suggests that, if a preliminary test for a period by treatment interaction (residual effect, carryover effect) is significant at the 10 % level, the direct effects of the treatments should be compared by performing a t-test on the data from the first period only. In this note it is shown that under Grizzle's model the comparison of the direct treatment effects is equivalent to a Behrens-Fisher problem. Depending on one's viewpoint–Bayesian, fiducial, or sampling theory–different solutions are possible. The solutions are illustrated using three well-known data sets.     

Testing the Hardy-Weinberg equilibrium in the HLA system

A new method of testing the Hardy-Weinberg equilibrium in the human leukocyte antigen (HLA) system is proposed and applied to real data. The derivation is based on the maximum likelihood method and closely related to standard regression theory. The test statistic has a closed representation of residual sum of squares by a projection mapping of data onto the estimated regression plane. Under the Hardy-Weinberg law the noniterative estimates for the gene frequencies are suggested by the use of the projection mapping. The test statistic and gene frequency estimates are shown to be asymptotically equivalent to the maximum likelihood method and to be more efficient than the other suggested test statistic when there are more than two identified alleles.     

Estimating individualized treatment regimes from crossover designs

The field of precision medicine aims to tailor treatment based on patient-specific factors in a reproducible way. To this end, estimating an optimal individualized treatment regime (ITR) that recommends treatment decisions based on patient characteristics to maximize the mean of a prespecified outcome is of particular interest. Several methods have been proposed for estimating an optimal ITR from clinical trial data in the parallel group setting where each subject is randomized to a single intervention. However, little work has been done in the area of estimating the optimal ITR from crossover study designs. Such designs naturally lend themselves to precision medicine since they allow for observing the response to multiple treatments for each patient. In this paper, we introduce a method for estimating the optimal ITR using data from a 2 × 2 crossover study with or without carryover effects. The proposed method is similar to policy search methods such as outcome weighted learning; however, we take advantage of the crossover design by using the difference in responses under each treatment as the observed reward. We establish Fisher and global consistency, present numerical experiments, and analyze data from a feeding trial to demonstrate the improved performance of the proposed method compared to standard methods for a parallel study design.     

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.     

Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression

In geo-statistics, the Durbin-Watson test is frequently employed to detect the presence of residual serial correlation from least squares regression analyses. However, the Durbin-Watson statistic is only suitable for ordered time or spatial series. If the variables comprise cross-sectional data coming from spatial random sampling, the test will be ineffectual because the value of Durbin-Watson’s statistic depends on the sequence of data points. This paper develops two new statistics for testing serial correlation of residuals from least squares regression based on spatial samples. By analogy with the new form of Moran’s index, an autocorrelation coefficient is defined with a standardized residual vector and a normalized spatial weight matrix. Then by analogy with the Durbin-Watson statistic, two types of new serial correlation indices are constructed. As a case study, the two newly presented statistics are applied to a spatial sample of 29 China’s regions. These results show that the new spatial autocorrelation models can be used to test the serial correlation of residuals from regression analysis. In practice, the new statistics can make up for the deficiencies of the Durbin-Watson test.     

Critical Values of ‘Maximum Studentized Residual’ Statistics in Multiple Linear Regression

In multiple linear regression, test for the discordancy of a single outlier in the response variable is usually based on the ‘maximum studentized residual’ statistic. Exact critical values for the test statistic t are not available. Upper bounds for the critical values have been found by SRIKANTAN (1961), PRESCOTT (1975) and LUND (1975). In this note we show that all these upper bounds are algebraically equivalent.     

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.     

The performance of a two‐stage analysis of ABAB/BABA crossover trials

Freeman has considered the following two‐stage procedure for finding a confidence interval for the treatment difference theta, using data from an AB/BA crossover trial. In the first stage, a preliminary test of the null hypothesis that the differential carryover is zero is carried out. If this hypothesis is accepted then the confidence interval for theta is constructed assuming that the differential carryover is zero. If, on the other hand, this hypothesis is rejected then this confidence interval is constructed using only data from the first period. Freeman has shown that this confidence interval has minimum coverage probability far below nominal. He therefore concludes that this confidence interval should not be used. In the present paper, we analyze the performance of a similar two‐stage procedure for an ABAB/BABA crossover trial. This trial differs in very significant ways from an AB/BA crossover trial, including the fact that for an ABAB/BABA crossover trial there is an unbiased estimator of the differential carryover that is unaffected by between‐subject variation. Despite these great differences, we arrive at the same conclusion as Freeman. Namely, that the confidence interval resulting from the two‐stage procedure should not be used.     

Phylogenetic analysis of genes of the influenza virus. Relationship between adaptability and neutrality

Detailed phylogenetic analysis of the gene family of hemagglutinin H3 of influenza A-type virus was fulfilled, taking into account the domain structure of protein and positions of antigen determinants. The densities of distribution of fixed synonimic replacements between domains HA1 and HA2 were shown to be actually equal (rho (HA1) = rho (HA2], and those of nonsynonimic ones to be unequal: their ratios were rho (HA1): rho (HA2) = 2.8 for nonepidemic branches, and rho (HA1): rho (HA2) = 7.7 for epidemic ones. For the positions of antigen determinants (agd) these densities differ still stronger from HA2: rho (agd): rho (HA2) = 10 for nonepidemic branches, and rho (agd): rho (HA2) = 36 for epidemic ones. In total, the rate of fixation of nonsynonimic replacements per position of antigen determinants for epidemic branches is 32 times higher than for nonepidemic ones. The absolute value of this estimation is K(ns)d = (9.1 +/- 0.7).10(-3) of nonsynonimic replacements per nonsynonimic position per year and seems to be twice as much as the maximum rate of neutral fixations Ks = (4.28 +/- 0.68).10(-3). Therefore, the epidemic reproduction of influenza virus is highly adaptive, exactly being focused on positions of antigen determinants. The evolution of influenza virus is stochastic process, both with the neutral and adaptive fixations.     

Modified test statistics by inter-voxel variance shrinkage with an application to f MRI

Functional magnetic resonance imaging (f MRI) is a noninvasive technique which is commonly used to quantify changes in blood oxygenation and flow coupled to neuronal activation. One of the primary goals of f MRI studies is to identify localized brain regions where neuronal activation levels vary between groups. Single voxel t-tests have been commonly used to determine whether activation related to the protocol differs across groups. Due to the generally limited number of subjects within each study, accurate estimation of variance at each voxel is difficult. Thus, combining information across voxels is desirable in order to improve efficiency. Here, we construct a hierarchical model and apply an empirical Bayesian framework for the analysis of group f MRI data, employing techniques used in high-throughput genomic studies. The key idea is to shrink residual variances by combining information across voxels and subsequently to construct an improved test statistic. This hierarchical model results in a shrinkage of voxel-wise residual sample variances toward a common value. The shrunken estimator for voxel-specific variance components on the group analyses outperforms the classical residual error estimator in terms of mean-squared error. Moreover, the shrunken test statistic decreases false-positive rates when testing differences in brain contrast maps across a wide range of simulation studies. This methodology was also applied to experimental data regarding a cognitive activation task.     

Unbalanced regression analysis with residuals having a covariance structure of intra-class form.

Let Yi be an ni X 1 vector of observations, Xi an ni X p matrix of known values, and beta an unknown p X 1 with the structure Yi = Xi beta + epsilon i, where the covariance matrix of epsilon i is of intra-class form, that is Cov (epsilon i) = sigma2[(1 - rho) Ii + rho e i e i'] where Ii is the ni X ni identity matrix and e i is the ni X 1 vector each element of which is unity. This article develops the maximum likelihood estimators of beta, sigma2, and rho when one observes N pairs (Xi, Yi). This situation arises typically in biological problems where one samples clusters of related organisms. The estimation procedure is illustrated in a commonly occurring genetics situation.     

A nonparametric approach to the analysis of longitudinal data via a set of level crossing problems with application to the analysis of microarray time course experiments

Here we develop a completely nonparametric method for comparing two groups on a set of longitudinal measurements. No assumptions are made about the form of the mean response function, the covariance structure or the distributional form of disturbances around the mean response function. The solution proposed here is based on the realization that every longitudinal data set can also be thought of as a collection of survival data sets where the events of interest are level crossings. The method for testing for differences in the longitudinal measurements then is as follows: for an arbitrarily large set of levels, for each subject determine the first time the subject has an upcrossing and a downcrossing for each level. For each level one then computes the log rank statistic and uses the maximum in absolute value of all these statistics as the test statistic. By permuting group labels we obtain a permutation test of the hypothesis that the joint distribution of the measurements over time does not depend on group membership. Simulations are performed to investigate the power and it is applied to the area that motivated the method-the analysis of microarrays. In this area small sample sizes, few time points and far too many genes to consider genuine gene level longitudinal modeling have created a need for a simple, model free test to screen for interesting features in the data.     

Detecting genomic clustering of risk variants from sequence data: cases versus controls

As the ability to measure dense genetic markers approaches the limit of the DNA sequence itself, taking advantage of possible clustering of genetic variants in, and around, a gene would benefit genetic association analyses, and likely provide biological insights. The greatest benefit might be realized when multiple rare variants cluster in a functional region. Several statistical tests have been developed, one of which is based on the popular Kulldorff scan statistic for spatial clustering of disease. We extended another popular spatial clustering method—Tango’s statistic—to genomic sequence data. An advantage of Tango’s method is that it is rapid to compute, and when single test statistic is computed, its distribution is well approximated by a scaled χ ² distribution, making computation of p values very rapid. We compared the Type-I error rates and power of several clustering statistics, as well as the omnibus sequence kernel association test. Although our version of Tango’s statistic, which we call “Kernel Distance” statistic, took approximately half the time to compute than the Kulldorff scan statistic, it had slightly less power than the scan statistic. Our results showed that the Ionita-Laza version of Kulldorff’s scan statistic had the greatest power over a range of clustering scenarios.     

Group sequential distribution-free methods for the analysis of multivariate observations.

Many studies involve the collection of multivariate observations, such as repeated measures, on two groups of subjects who are recruited over time, i.e., with staggered entry of subjects. Various marginal distribution-free multivariate methods have been proposed for the analyses of such multivariate observations where some measures may be missing at random. Using the multivariate U statistic of Wei and Johnson (1985, Biometrika 72, 359-364), we describe the group sequential analysis of such a study where the multivariate observations are observed sequentially--both within and among subjects. We describe a multivariate generalization of the Hodges and Lehmann (1963, Annals of Mathematical Statistics 34, 598-611) estimator of a location shift that can be obtained via the multivariate U statistic with the Mann-Whitney-Wilcoxon kernel. We then describe large-sample group sequential interval estimators and tests based on an aggregate estimate of the location shift combined over all of the repeated measures. We also describe how the same steps could be employed to perform a group sequential analysis based on any one of the variety of marginal multivariate methods that have been proposed. These methods are applied to a real-life example.     

Oxygen consumption and expression of the adenine nucleotide translocator in cells lacking mitochondrial DNA

It has been shown previously that human rho degrees cells, deprived of mitochondrial DNA and consequently of functional oxidative phosphorylation, maintain a mitochondrial membrane potential, which is necessary for their growth. The goal of our study was to determine the precise origin of this membrane potential in three rho degrees cell lines originating from the human HepG2, 143B, and HeLa S3 cell lines. Residual cyanide-sensitive oxygen consumption suggests the persistence of residual mitochondrial respiratory chain activity, about 8% of that of the corresponding parental cells. The fluorescence emitted by the three rho degrees cell lines in the presence of a mitochondrial specific fluorochrome was partially reduced by a protonophore, suggesting the existence of a proton gradient. The mitochondrial membrane potential is maintained both by a residual proton gradient (up to 45 to 50% of the potential) and by other ion movements such as the glycolytic ATP(4-) to mitochondrial ADP(3-) exchange. The ANT2 gene, encoding isoform 2 of the adenine nucleotide translocator, is overexpressed in rho degrees HepG2 and 143B cells strongly dependent on glycolytic ATP synthesis, as compared to the corresponding parental cells, which present a more oxidative metabolism. In rho degrees HeLa S3 cells, originating from the HeLa S3 cell line, which already displays a glycolytic energy status, ANT2 gene expression was not higher as in parental cells. Mitochondrial oxygen consumption and ANT2 gene overexpression vary in opposite ways and this suggests that these two parameters have complementary roles in the maintenance of the mitochondrial membrane potential in rho degrees cells.     

Hierarchical selection theory and sex ratios I. General solutions for structured populations

Models of sex-ratio evolution in structured populations are derived with G.R. Price's covariance form for the hierarchical analysis of natural selection (1970, Nature 227, 520-521). Previous work on competition among related males for mates (local mate competition), competition among related females for a limiting resource (local resource competition), inbreeding, group selection, and asymmetry of genetic inheritance between males and females, are subsumed under a general formulation for sex-ratio biases in structured populations. I found that the evolutionarily stable strategy sex ratio (males:females) for diploids is 1 - rho m:1 - rho f, where rho m is the regression coefficient of relatedness of the controlling genotypes on males competing for mates, rho f is the regression of controlling genotypes on females that compete for a fixed, limiting resource, and there is no inbreeding. For inbreeding and no competition among females, the evolutionarily stable strategy is 1 - rho m:1 + rho mf, where rho mf is the regression of controlling genotypes on females' mates.     

Rhomboid function in the midline of the Drosophila CNS

Rhomboid (Rho), a cell surface, seven-transmembrane domain protein, participates in Spitz-dependent activation of the Drosophila EGF receptor (EGFR). By contrast to transient expression in other embryonic tissues, rho is expressed continuously in the embryonic and larval Midline Glia (MG) lineage and is required upstream of, or in parallel with, S, Spi, and EGFR to establish MG cell number. EGFR signaling is necessary for the expression of rho in the MG and sufficient to stimulate rho expression in additional MG progenitors. rho expression is required continuously from embryonic stage 9-17 to suppress apoptosis in the MG. Although rho misexpression can increase MG number through a non-cell autonomous mechanism, the pattern of normal rho expression suggests that it functions by enhancing autocrine or paracrine signaling among MG cells.