Exact sample size needed to detect dependence in 2 x 2 x 2 tables

In the Georgia Centenarian Study (Poon et al., Exceptional Longevity, 2006), centenarian cases and young controls are classified according to three categories (age, ethnic origin, and single nucleotide polymorphisms [SNPs] of candidate longevity genes), where each factor has two possible levels. Here we provide methodologies to determine the minimum sample size needed to detect dependence in 2 x 2 x 2 tables based on Fisher's exact test evaluated exactly or by Markov chain Monte Carlo (MCMC), assuming only the case total L and the control total N are known. While our MCMC method uses serial computing, parallel computing techniques are employed to solve the exact sample size problem. These tools will allow researchers to design efficient sampling strategies and to select informative SNPs. We apply our tools to 2 x 2 x 2 tables obtained from a pilot study of the Georgia Centenarians Study, and the sample size results provided important information for the subsequent major study. A comparison between the results of an exact method and those of a MCMC method showed that the MCMC method studied needed much less computation time on average (10.16 times faster on average for situations examined with S.E. = 2.60), but its sample size results were only valid as a rule for larger sample sizes (in the hundreds).     

Conditioning in 2 x 2 tables

Summary .  Two-by-two tables arise in a number of diverse settings in biomedical research, including analysis of data from a clinical trial with a binary outcome and gating methods in flow cytometry to separate antigen-specific immune responses from general immune responses. These applications offer interesting challenges concerning what we should really be conditioning on—the total number of events, the number of events in the control condition, etc. We give several biostatistics examples to illustrate the complexities of analyzing what appear to be simple data.     

Fundamental clusters in spatial 2x2 games

The notion of fundamental clusters is introduced, serving as a rule of thumb to characterize the statistical properties of the complex behaviour of cellular automata such as spatial 2 x 2 games. They represent the smallest cluster size determining the fate of the entire system. Checking simple growth criteria allows us to decide whether the cluster-individuals, e.g. some mutant family, are capable of surviving and invading a resident population. In biology, spatial 2 x 2 games have a broad spectrum of applications ranging from the evolution of cooperation and intraspecies competition to disease spread. This methodological study allows simple classifications and long-term predictions in various biological and social models to be made. For minimal neighbourhood types, we show that the intuitive candidate, a 3 x 3 cluster, turns out to be fundamental with certain weak limitations for the Moore neighbourhood but not for the Von Neumann neighbourhood. However, in the latter case, 2 x 2 clusters generally serve as reliable indicators to whether a strategy survives. Stochasticity is added to investigate the effects of varying fractions of one strategy present at initialization time and to discuss the rich dynamic properties in greater detail. Finally, we derive Liapunov exponents for the system and show that chaos reigns in a small region where the two strategies coexist in dynamical equilibrium.     

Reduced recombination and distorted segregation in a Solanum tuberosum (2x) x S. spegazzinii (2x) hybrid.

In this paper we describe the reduced recombination and distorted segregation in an interspecific hybrid between Solanum tuberosum and Solanum spegazzinii. To study these phenomena, a cross was made between a (di)haploid S. tuberosum, used as a female parent, and a diploid wild potato species, S. spegazzinii, used as a male parent. Next, a backcross (BC) population was made with F1 genotype 38 that was backcrossed to S. tuberosum. In the backcross, S. tuberosum was used as the male parent. RFLP linkage maps were made using the F1 and the BC populations, yielding linkage maps of the interspecific hybrid, S. spegazzinii, and S. tuberosum from which male and female linkage maps could be constructed. The computer program JOINMAP was used to construct and combine the separate linkage maps. Subsequently, the separate linkage maps were compared with each other, and reduced recombination was observed in the linkage maps of the male S. tuberosum and the interspecific hybrid. The reason for this reduced recombination is discussed. Another common feature in linkage maps is the observation of distorted segregation. The distorted segregation of alleles from the interspecific hybrid was studied in more detail in the BC population. Most of the distortion was probably caused by gamete selection, but for 3 loci, on chromosomes 2, 3, and 4, we found evidence for the presence of a strong selection force acting at the zygote level against homozygous genotypes.     

The analysis of stratified 2 x 2 contingency tables

We consider the problem of testing for independence against the consistent superiority of one treatment over another when the response variable is binary and is compared across two treatments in each of several strata. Specifically, we consider the randomized clinical trial setting. A number of issues arise in this context. First, should tables be combined if there are small or zero margins? Second, should one assume a common odds ratio across strata? Third, if the odds ratios differ across strata, then how does the standard test (based on a common odds ratio) perform? Fourth, are there other analyzes that are more appropriate for handling a situation in which the odds ratios may differ across strata? In addressing these issues we find that the frequently used Cochran–Mantel–Haenszel test may have a poor power profile, despite being optimal when the odds ratios are common. We develop novel tests that are analogous to the Smirnov, modified Smirnov, convex hull, and adaptive tests that have been proposed for ordered categorical data. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximally selected chi2 statistics for k x 2 tables

It is common in epidemiologic analyses to summarize continuous outcomes as falling above or below a threshold. With such a dichotomized outcome, the usual chi2 statistics for association or trend can be used to test for equality of proportions across strata of the study population. However, if the threshold is chosen to maximize the test statistic, the nominal chi2 reference distributions are incorrect. In this paper, the asymptotic distributions of maximally selected chi2 statistics for association and for trend for the k x 2 table are derived. The methodology is illustrated with data from an AIDS clinical trial. The results of simulation experiments that assess the accuracy of the asymptotic distributions in moderate sample sizes are also reported.     

Cytological basis of 2n pollen formation in a wide range of 2x, 4x, and 6x taxa from tuber-bearing Solanum species.

Cytological mechanisms of 2n pollen formation were investigated in microsporogenesis and sporad development in 2x, 4x, and 6x taxa of tuber-bearing Solanum species that were originally collected from Mexico, and Central and South America. Parallel spindles and tripoles were seen at anaphase II in the majority of genotypes observed. The occurrence of parallel spindles and tripoles were well correlated with the occurrence of dyads and triads at the sporad stage. Further, the frequency of 2n pollen predicted by frequencies of parallel spindles and tripoles and by frequencies of dyads and triads were also well correlated with observed 2n pollen frequencies. Premature cytokinesis at telophase I to prophase II was seen only in 2x and 4x species from series Conicibaccata. The frequency of premature cytokinesis corresponded well with the frequency of dyads leading to 2n pollen formation. It was concluded that the major mechanisms of 2n pollen formation in tuber-bearing Solanum species is parallel spindles.     

Comparison of the Giemsa C-banded karyotypes of the three subspecies ofPsathyrostachys fragilis,subspp.villosus (2x),secaliformis (2x, 4x), andfragilis (2x) (Poaceae), with notes on chromosome pairing

The karyotypes of diploidP. fragilis subsp.villosus (2n = 2x = 14) and tetraploid subsp.secaliformis (2n = 4x = 28) were studied by Giemsa C- and N-banding, and AgNO₃ staining and compared with the karyotype of subsp.fragilis (2x). The complements of subsp.villosus and subsp.fragilis were similar, with 8 metacentric and 6 SAT-chromosomes, one metacentric and two submetacentric pairs, with small to minute, polymorphic, heterochromatic satellites. The complement of subsp.secaliformis on the whole agreed with a doubling of the complement of diploidP. fragilis, suggesting autopolyploidy. Only the presence of 12 nucleoli in interphases identified 6 SAT-chromosome pairs. In subsp.villosus one or two extra micronucleoli indicated a chromosome pair with very low nucleolusforming activity, bringing the number of SAT-chromosome pairs to 4. This number may be a characteristc ofPsathyrostachys. Besides very small, inconsistently observed bands, the C-banding pattern consisted of 0–3 small bands per chromosome at intercalary and terminal locations, and at NORs. The level of banding pattern polymorphism was low, but enough to indicate that the taxa are outbreeders. Similarities in chromosome morphology and C-banding patterns identified homology of all chromosomes of subsp.villosus, but for 12 pairs only in subsp.secaliformis. Between plants, reliable identification of homology and homoeology (subsp.secaliformis) was possible only for the SAT-chromosomes and the shortest metacentrics. Chromocentres were very small and the amount of constitutive heterochromatin was low. N-banding stained chromosomes uniformly. The basic karyotypes of theP. fragilis taxa were similar to those ofP. juncea, P. lanuginosa, andP. stoloniformis supporting a close relationship and the presence of a common genome, N. NORs had different nucleolus-forming activities. Meiotic analysis demonstrated a high level of bivalent pairing in the three taxa. A chromosomal rearrangement was suggested in subsp.villosus. The low multivalent frequency in subsp.secaliformis indicates the presence of a pairing regulation mechanism. The majority of chiasmata were interstitial. Pollen grain size discriminated between diploid and tetraploid taxa. The existence of a diploid cytotype of subsp.secaliformis is supported by pollen measurements of herbarium material.     

2 x 2 kappa coefficients: measures of agreement or association

Two general but different contexts in which kappa might be used are defined: agreement and association. Two models, one for agreement and one for utility of association, are defined yielding different kappa coefficients and different sampling theory. Asymptotic results are derived for both models. Small-sample evaluations are presented for the model for agreement.     

Elucidating the Foundations of Statistical Inference with 2 x 2 Tables

To many, the foundations of statistical inference are cryptic and irrelevant to routine statistical practice. The analysis of 2 x 2 contingency tables, omnipresent in the scientific literature, is a case in point. Fisher''s exact test is routinely used even though it has been fraught with controversy for over 70 years. The problem, not widely acknowledged, is that several different p-values can be associated with a single table, making scientific inference inconsistent. The root cause of this controversy lies in the table''s origins and the manner in which nuisance parameters are eliminated. However, fundamental statistical principles (e.g., sufficiency, ancillarity, conditionality, and likelihood) can shed light on the controversy and guide our approach in using this test. In this paper, we use these fundamental principles to show how much information is lost when the tables origins are ignored and when various approaches are used to eliminate unknown nuisance parameters. We present novel likelihood contours to aid in the visualization of information loss and show that the information loss is often virtually non-existent. We find that problems arising from the discreteness of the sample space are exacerbated by p-value-based inference. Accordingly, methods that are less sensitive to this discreteness - likelihood ratios, posterior probabilities and mid-p-values - lead to more consistent inferences.     

A new approach for obtaining rapid uniformity in rice (Oryza sativa L.) via a 3x x 2x cross

A triploid (2n = 3x = 36) rice plant was obtained by screening a twin seedling population in which each seed germinated to two or three sprouts that were then crossed with diploid plants. One diploid plant was chosen among the various F(1) progenies and developed into an F (2) population via self-pollination. Compared with the control variety Shanyou 63, this F (2) population had a stable agronomical performance in field trials, as confirmed by the F-test. The stability of the F (2) population was further substantiated by molecular analysis with simple sequence repeat markers. Specifically, of 160 markers assayed, 37 (covering all 12 chromosomes) were polymorphic between the parental lines. Testing the F (1) hybrid individually with these markers showed that each PCR product had only a single band instead of two bands from each parent. The bands were identical to either maternal (23 markers) or paternal (eight markers) bands or distinct from both parents (six markers). The amplified bands of all 60 randomly selected F (2) plants were uniform and identical to those of the F (1) hybrid. These results suggest that the F (1) plant is a non-segregating hybrid and that a stable F (2) population was obtained. This novel system provides an efficient means for shortening the cycle of hybrid rice seed production.     

Analyzing k (2 x 2) tables under cluster sampling

In many data analytic applications, such as ophthalmologic, longitudinal, or periodontal studies, multiple observations are recorded over several sites (or timepoints) within the same subject, bringing about dependence between measurements. This correlation, in turn, precludes the use of standard statistical methods that assume independence between outcome measurements. For example, the Mantel-Haenszel statistic, used to assess association between a binary outcome and a binary exposure while adjusting for a categorical covariate, does not follow the usual chi-squared distribution under the null hypothesis when there is correlation between observations. A modified Mantel-Haenszel procedure, which makes adjustment for dependence, is proposed. No particular correlation structure is assumed for responses within a cluster. This closed-form adjustment stems from Liang and Zeger's (1986, Biometrika 73, 13-22) generalized estimating equations approach for clustered data. The difference between this tabular (i.e., noniterative) technique and many earlier tabular methods is that the current method allows for consideration of site-specific exposure and covariate information. An example from a periodontal research study illustrates application of the method.     

The 2 x 2 matched-pairs trial: exact unconditional design and analysis

An exact unconditional method for the design and analysis of the 2 x 2 matched-pairs trial is presented. Unlike the exact conditional method, which is based on only the number of discordant pairs, the exact unconditional method uses the total number N of sampled pairs. This unconditional test, based on a simple Z statistic, yields sample sizes that are generally smaller than those produced by the exact conditional test for the cases tabulated herein, namely one-sided alpha = .01, .025, and .05 along with 80% and 90% power. Moreover, it is found to be uniformly more powerful than the latter for all the combinations of parameters considered in this paper, namely alpha = .01, .025, and .05, and N = 10(1)200. The method is illustrated by assessing various design options of an in vitro study of the effectiveness of antimicrobial agents. Some numerical examples of matching efficiency are also given.     

Generalized Mantel-Haenszel estimators for K 2 x J tables

Mickey and Elashoff (1985, Biometrics 41, 623-635) gave an extension of Mantel-Haenszel estimation to log-linear models for 2 x J x K tables. Their extension yields two generalizations of the Mantel-Haenszel odds ratio estimator to K 2 x J tables. This paper provides variance and covariance estimators for these generalized Mantel-Haenszel estimators that are dually consistent (i.e., consistent in both large strata and sparse data), and presents comparisons of the efficiency of the generalized Mantel-Haenszel estimators.     

The occurrence and frequency of 2n pollen in 2x, 4x,and 6x wild,tuber-bearing Solanum species from Mexico,and Central and South America

Summary The occurrence of 2n pollen-producing plants was investigated in 187 plant introductions (PIs) of 38 wild species of tuber-bearing Solanum. These 2x, 4x, and 6x species are from Mexico, and Central and South America. The determination of 2n pollen-producing plants was conducted using acetocarmine glycerol. Plants with more than 1% large-size pollen were regarded as 2n pollen-producing plants. 2n pollen-producing plants were identified in the following species: 10 out of 12 Mexican 2x species, seven of nine South American 2x species, seven of seven Mexican and Central American 4x species, five of five South American 4x species, and five of five Mexican 6x species. The frequency of 2n pollen-producing plants varied among species at the same ploidy level, but the range of frequency, generally between 2 and 10% among species, was similar over different ploidy levels. The general occurrence of 2n pollen in both 2x and polyploid species, which are evolutionarily related, is evidence that the mode of polyploidization in tuber-bearing Solanums is sexual polyploidization. Furthermore, the frequencies of 2n pollen-producing plants in autogamous disomic polyploid species were not markably different from those of their related diploid species. It is thought that the frequent occurrence of 2n gametes with autogamy tends to disturb the fertility and consequently reduce fitness of polyploids. Thus, we propose that the breeding behavior of polyploids and the occurrence of 2n gametes may be genetically balanced in order to conserve high fitness in polyploid species in tuberbearing Solanum.Paper No. 3114 from the Laboratory of Genetics. Research supported by the College of Agriculture and Life Sciences; International Potato Center; USDA, SEA, CGRO 84-CRCR-1-1389; and Frito Lay, Inc.     

Somatic hybrids obtained by fusion betweenPoncirus trifoliata (2x) andFortunella hindsii (4x) protoplasts

Somatic hybrids were obtained by the symmetric fusion of embryogenic callus cells from tetraploid Mame kumquat [Fortunella hindsii (Champ.) Swing.] and mesophyll cells from diploid trifoliate orange [Poncirus trifoliata (L.) Raf.]. Southern blot analysis of three regenerants revealed that they carried specific rDNA fragments from both fusion partners, thereby confirming their hybridity. In contrast, mitochondrial DNA (mtDNA) and chloroblast DNA (cpDNA) were unidirectionally transmitted from the callus parent without any evidence of recombination. No differences in the restriction fragment patterns of rDNA, mtDNA or cpDNA could be detected among the regenerants. Flow cytometry showed that two regenerants were hexaploids, as expected, but that one was pentaploid, probably due to elimination of chromosomes prior to the regeneration of this plant.Abbreviations MS Murashige and Skoog (1962) - DIG digoxigenin - AMPPD 3-(2-spiroadamantane)-4-methoxy-4-(3-phosphoryloxy)-phenyl-1,2-dioxetane - FACS fluorescence-activated cell sorter     

Bayesian inference for a correlated 2 x 2 table with a structural zero

We develop a new Bayesian approach to interval estimation for both the risk difference and the risk ratio for a 2 x 2 table with a structural zero using Markov chain Monte Carlo (MCMC) methods. We also derive a normal approximation for the risk difference and a gamma approximation for the risk ratio. We then compare the coverage and interval width of our new intervals to the score-based intervals over various parameter and sample-size configurations. Finally, we consider a Bayesian method for sample-size determination.