Comparing three methods for variance estimation with duplicated high density oligonucleotide arrays

Microarray experiments are being increasingly used in molecular biology. A common task is to detect genes with differential expression across two experimental conditions, such as two different tissues or the same tissue at two time points of biological development. To take proper account of statistical variability, some statistical approaches based on the t-statistic have been proposed. In constructing the t-statistic, one needs to estimate the variance of gene expression levels. With a small number of replicated array experiments, the variance estimation can be challenging. For instance, although the sample variance is unbiased, it may have large variability, leading to a large mean squared error. For duplicated array experiments, a new approach based on simple averaging has recently been proposed in the literature. Here we consider two more general approaches based on nonparametric smoothing. Our goal is to assess the performance of each method empirically. The three methods are applied to a colon cancer data set containing 2,000 genes. Using two arrays, we compare the variance estimates obtained from the three methods. We also consider their impact on the t-statistics. Our results indicate that the three methods give variance estimates close to each other. Due to its simplicity and generality, we recommend the use of the smoothed sample variance for data with a small number of replicates. Electronic Publication     

On the empirical choice of the time window for restricted mean survival time

The t-year mean survival or restricted mean survival time (RMST) has been used as an appealing summary of the survival distribution within a time window [0, t]. RMST is the patient's life expectancy until time t and can be estimated nonparametrically by the area under the Kaplan-Meier curve up to t. In a comparative study, the difference or ratio of two RMSTs has been utilized to quantify the between-group-difference as a clinically interpretable alternative summary to the hazard ratio. The choice of the time window [0, t] may be prespecified at the design stage of the study based on clinical considerations. On the other hand, after the survival data have been collected, the choice of time point t could be data-dependent. The standard inferential procedures for the corresponding RMST, which is also data-dependent, ignore this subtle yet important issue. In this paper, we clarify how to make inference about a random “parameter.” Moreover, we demonstrate that under a rather mild condition on the censoring distribution, one can make inference about the RMST up to t, where t is less than or even equal to the largest follow-up time (either observed or censored) in the study. This finding reduces the subjectivity of the choice of t empirically. The proposal is illustrated with the survival data from a primary biliary cirrhosis study, and its finite sample properties are investigated via an extensive simulation study.     

Robustness of Selection Procedures

In the article Bechhofers Indifference-zone formulation for selecting the t populations with the t highest means is considered in a set of non-normal distributions. Selection rules based on the sample mean, the 10% and the 20% trimmed means, two estimators proposed by Tiku (1981) for valuating the smallest and highest accepted sample values higher, the sample median and a linear combination of quantile estimators, two adaptive procedures and a ranksum procedure are investigated in a large scale simulation experiment in respect of their robustness against deviations from an assumed distribution. Robustness is understood as a small percentage of the difference β_A-β between the actual probability of incorrect selection β_A and the nominal β-value. We obtained a relatively good robustness for the classical sample mean selection rule and useful derivations for the employment of other selection rules in an area of practical importance.     

An Approximate Solution to the Behrens-Fisher Problem

An approximate and practical solution is proposed for the Behrens-Fisher problem. This solution is compared to the solutions considered by Mehta and Srinivasan (1970) and Welch's (1937) approximate t-test in terms of the stability of the size and magnitude of the power. It is shown that the stability of the size of the new test is better than that of Welch's t when at least one of the sample sizes is small. When the sample sizes are moderately large or large the sizes and powers of all the recommended tests are almost the same.     

Mixed Moment Estimation in Hermite Distribution and its Comparison with Other Estimation Procedures from Sample Size Considerations

In this paper (1) expressions (correct to n^?2 terms) for biases, variances, and covariances of the estimators a and b of Hermite distribution with probability generating function Exp[a(t–1) + b(t–1)] are obtained for two mixed moment estimates; (2) for the biases and variance-covariances, approximate regions of the parameter space (a>0, b>0) have been outlined where a sample of size 100 can be considered as “safe” in the sense that contribution of second order terms in them is 5% of that from the first order term; (3) comparison of the biases and variance-covariances of these two sets of estimators are made with those for the moment estimators, maximum likelihood estimates and the even point estimators for a sample of size 100 using the terms up to order n^?2; (4) the comparisons based on n^?2 terms in (3) have not only provided information on the estimation procedures included in the Hermite distribution, but also demonstrated the importance of higher order terms in the sampling properties of the various alternative techniques for the Hermite distribution.     

Recombination within mouse <Emphasis Type="Italic">t</Emphasis> haplotypes has replaced significant segments of <Emphasis Type="Italic">t</Emphasis>-specific DNA

Previous studies on the fourth inversion of the t complex, In17(4), suggest that loci near the center of this inversion have been subjected to segmental recombination during the past 1–2 million years. We have used a combination of PCR-based restriction site (PBR) analysis and DNA sequencing to perform a high-resolution analysis of a 2-million base pair (Mbp) segment in the middle of In17(4). We examined 21 restriction sites that are polymorphic between t haplotypes and their wild-type homologs, over nine distinct loci. In addition, we examined several other polymorphic sites through DNA sequence analysis of two of these nine loci. We analyzed several haplotypes in this way, including the “complete” t haplotypes t ^w2 , t ⁰ , t ^w32 , t ^w71 , and t ^w75 . We show that only t ^w32 is a true “complete” t haplotype; the remaining four t haplotypes have segments of wild-type DNA ranging from less than 100 bp to 2 Mbp. The sizes of these wild-type DNA segments are consistent with their being generated by gene-conversion events. The 2-Mbp segment is located in a region that may contain the t-complex distorter gene Tcd2. One of the nine loci examined in this study is Fgd2, a gene that has been proposed to encode Tcd2. Sequencing and PBR data show that at least a portion of the Fgd2 gene has been converted to the wild-type within t ^w71 and t ^w75 mice.     

Comparing short protein substructures by a method based on backbone torsion angles

An efficient algorithm was characterized that determines the similarity in main chain conformation between short protein substructures. The algorithm computes Δt, the root mean square difference in ? and ψ torsion angles over a small number of amino acids (typically 3–5). Using this algorithm, large number of protein substrates comparisons were feasible. The parameter Δt was sensitive to variations in local protein conformation, and it correlates with Δr, the root mean square deviation in atomic coordinates. Values for Δt were obtained that define similarity thresholds, which determine whether two substructure are considered structurally similar. To set a lower bound on the similarity threshold, we estimated the component of Δt due to measurement noise fromcomparisons of independently refined coordinates of the same protein. A sample distribution of Δt from nonhomologous protein comparisons identified an upper bound on the similarity threshold, one that refrains from incorporating large numbers of nonmatching comparisons large numbers of nonmatching comparisons. Unlike methods based on Cα atoms alone, Δt was sensitive to rotations in the peptide plane, shown to occur in several proteins. Comparisons of homologus proteins by Δt showed that the active site torsion angles are highly conserved. The Δt method was applied to the α-chain of human hemoglobin, where it readily demonstrated the local differences in the structures of different ligation states.     

Conditional Rejection Probabilities of Student's t‐test and Design Adaptations

For clinical trials with interim analyses conditional rejection probabilities play an important role when stochastic curtailment or design adaptations are performed. The conditional rejection probability gives the conditional probability to finally reject the null hypothesis given the interim data. It is computed either under the null or the alternative hypothesis. We investigate the properties of the conditional rejection probability for the one sided, one sample t‐test and show that it can be non monotone in the interim mean of the data and non monotone in the non‐centrality parameter for the alternative. We give several proposals how to implement design adaptations (that are based on the conditional rejection probability) for the t‐test and give a numerical example. Additionally, the conditional rejection probability given the interim t‐statistic is investigated. It does not depend on the unknown σ and can be used in stochastic curtailment procedures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transmission Ratio Distortion,Sterility, and Control of the t-Complex Function in Sperm

Mouse t-complex located on chromosome 17 contains genes affecting only male fertility. Some genes of this complex are recessive lethals; nonetheless, the high frequency of the t-complex carriers in a population is maintained due to a mechanism referred to as transmission ratio distortion (TRD), i.e., after crosses with wild-type females, males heterozygous for the t-complex transmit the t-bearing chromosome to nearly all their offspring, which suggests that the t-complex genes control sperm function. Analysis of this phenomenon shows that the resultant TRD is determined by the ratio between the distorter genes (Tcd) and a responder gene (Tcr) located within the t-complex region. Many authors believe that two to six distorter genes currently known have an additive effect. A genetic model of the non-Mendelian inheritance in the progeny of heterozygous male mice specifically explains sterility of animals carrying the t-complex with complementary lethal genes. The model suggests that some distorter gene products interacting with the responder gene have a selective effect on motility of both mutant and wild-type sperm. Insufficient sperm motility and/or their unsuccessful capacitation result in poor if any fertilization. Information on the t-complex genes is necessary for understanding the biological mechanisms of male sterility and may be used in medical practice.     

LOW FREQUENCY OF t HAPLOTYPES IN NATURAL POPULATIONS OF HOUSE MICE (MUS MUSCULUS DOMESTICUS)

t haplotypes are a naturally occurring, autosomal, meiotic-drive system found on chromosome 17 of the house mouse. They show non-Mendelian transmission from heterozygous +/t males, such that 90% or more of the male's offspring inherit the t-bearing chromosome. Although they are expected to become rapidly fixed, surveys of natural populations typically report low overall frequencies of only ~15–25% +/t heterozygotes. Generally, such studies of t haplotypes in wild populations have sampled only small numbers of individuals due to the need to genotype mice by breeding, thus we have conducted a large survey of wild mice, Mus musculus domesticus, using DNA markers to examine the frequency and distribution of t haplotypes in natural populations. The overall frequency of +/t heterozygotes from our entire sample was 0.062, which is much lower than all previous estimates of t haplotype frequency. t haplotypes were patchily distributed and rare, and were present in only 46% of the populations we sampled. There were no significant sex-specific differences in the frequency of t haplotypes. Our data suggest that the frequency of +/t heterozygotes in independent populations varies with respect to population size and stability: t haplotypes were at low frequency in all large, relatively persistent populations, whereas they were at more variable, and often higher, frequencies in small, temporally unstable populations. The extinction and recolonization of many of the smaller populations may contribute to the greater variation in t haplotype frequency observed, and small populations may be important reservoirs of t haplotypes in the wild. The highest frequencies of t haplotypes were obtained from populations with semilethal, or complementing lethal, t haplotypes, where t/t homozygous mice were present.     

Optimizing resolution in multidimensional NMR by three-way decomposition

Resolution depends on the number of points sampled in a FID; in indirectly detected dimensions it is an important determinant of the total experiment time. Based on the high redundancy present in NMR data, we propose the following timesaving scheme for three-dimensional spectra. An extensive grid of discrete t1- and t2-values is used, which increases resolution while preserving the spectral width. Total experiment time is reduced by avoiding the recording of t3-FIDs for selected pairs of t1 and t2; typically the recording is omitted for about 75% of the (t1,t2) combinations. These data sets are referred to as sparse, and post-experimental processing making optimal use of spectral redundancy provides the missing, non-recorded data. We have previously shown that three-way decomposition (TWD) within the MUNIN approach provides a practical way to process dense NMR data sets. Here, a novel TWD algorithm [Ibraghimov, (2002) Numer. Linear Algebra Appl. 9, 551–565] is used to complement a sparselyrecorded time-domain data set by providing the missing FIDs for all (t1,t2) combinations omitted in the experiment. A necessary condition is that for each t1-value at least a few FIDs are recorded, and similar for each t2-value. The method is demonstrated on non-uniformly sampled ¹⁵N-NOESY-HSQC data sets recorded for the 14 kD protein azurin. The spectra obtained by TWD, reconstruction and ordinary transform to frequency-domain are, in spite of the large number of signals and the high dynamic range typical for NOESYs, highly similar to a corresponding reference spectrum, for which all (t1,t2) combinations were recorded.     

Almost-one Parameter Hypothesis of Individuality in Height Growth

Thus far an individual height growth curve h_ij(t) of the i-th person in the j-th period, t being his (or her) age, has been studied as a function of t associated with its velocity curve. In this note we introduce a natural scale X(t) in place of t, which linearizes this personal curve and facilitates its analysis, in the sense that this equation of growth contains apparently two personal parameters for one period but one of them plays an essential role. The effectiveness of this approach will be seen in four figures.     

Effects of chimaerism for two T/t complex mutations on sperm function

Mouse chimaeras produced by aggregation of embryos heterozygous for two different recessive mutations at the T/t complex have been analyzed by breeding to explore the basis for the phenomena of male transmission ratio distortion and sterility associated with these genes. Whereas males of genotype t^w2/t^w5 are always sterile, male chimaeras of the type +/t^w2 ? +/t^w5 are normally fertile; furthermore, they transmit each t mutation to the same very high extent seen in ordinary (+/t) heterozygotes. Since spermatogenic cells derived from either the +/t^w2 and +/t^w5 genotypes thus function quite independently of one another in mosaic testes, it can be concluded that sterility, and presumably distorted transmission ratio as well, depends on specific interactions between T/t alleles in diploid spermatogenic cells or their individual meiotic descendants.     

Slow Absorption of Conjugated Linoleic Acid in Rat Intestines,and Similar Absorption Rates of 9c,11t-Conjugated Linoleic Acid and 10t,12c-Conjugated Linoleic Acid

We have previously shown that the 9c,11t-conjugated linoleic acid (CLA) concentration was always significantly higher than the 10t,12c-CLA concentration following the administration of these compounds to mice and rats, and considered that structural differences between the conjugated double bonds in these isomers affected absorption in the small intestine. This study investigates the absorption of CLA in the rat intestine by a lipid absorption assay of lymph from the thoracic duct. In Study 1, we used safflower oil and a triacylglycerol form of CLA (CLA-TG), while in Study 2, we used 9c,11t-CLA and 10t,12c-CLA. The cumulative recovery of CLA was lower than that of linoleic acid until two hours after sample administration. There was no difference in the extent of lymphatic recovery of 9c,11t-CLA and 10t,12c-CLA after the administration of CLA-TG, 9c,11t-CLA, and 10t,12c-CLA to the rats, suggesting that geometrical and positional isomerism of the conjugated double bonds did not influence the absorption.     

An integrated biogeochemical and economic analysis of bioenergy crops in the Midwestern United States

This study integrates a biophysical model with a county‐specific economic analysis of breakeven prices of bioenergy crop production to assess the biophysical and economic potential of biofuel production in the Midwestern United States. The bioenergy crops considered in this study include a genotype of Miscanthus, Miscanthus×giganteus, and the Cave‐in‐Rock breed of switchgrass (Panicum virgatum). The estimated average peak biomass yield for miscanthus in the Midwestern states ranges between 7 and 48 metric tons dry matter per hectare per year ( t DM ha^?1 yr^?1), while that for switchgrass is between 10 and 16 t DM ha^?1 yr^?1. With the exception of Minnesota and Wisconsin, where miscanthus yields are likely to be low due to cold soil temperatures, the yield of miscanthus is on average more than two times higher than yield of switchgrass. We find that the breakeven price, which includes the cost of producing the crop and the opportunity cost of land, of producing miscanthus ranges from $53 t^?1 DM in Missouri to $153 t^?1 DM in Minnesota in the low‐cost scenario. Corresponding costs for switchgrass are $88 t^?1 DM in Missouri to $144 t^?1 DM in Minnesota. In the high‐cost scenario, the lowest cost for miscanthus is $85 t^?1 DM and for switchgrass is $118 t^?1 DM, both in Missouri. These two scenarios differ in their assumptions about ease of establishing the perennial crops, nutrient requirements and harvesting costs and losses. The differences in the breakeven prices across states and across crops are mainly driven by bioenergy and row crop yields per hectare. Our results suggest that while high yields per unit of land of bioenergy crops are critical for the competitiveness of bioenergy feedstocks, the yields of the row crops they seek to displace are also an important consideration. Even high yielding crops, such as miscanthus, are likely to be economically attractive only in some locations in the Midwest given the high yields of corn and soybean in the region.     

Intra-H-2 recombination in t haplotypes shows a hot spot and close linkage of l tw5 to H-2K

The embryonic lethal mutation in the t ^w5 haplotype is known to map near the H-2K region of the mouse major histocompatibility complex. Additional data obtained by classical genetic methods demonstrate that the t ^w5 lethal gene is effectively inseparable from H-2K. No recombinants were found between H-2K and t ^w5 in a sample representing over 1200 mice. On a statistical basis t ^w5 must be less than 250 kb from the H-2K gene. In the course of these mapping studies we obtained a set of 11 intra-H-2 recombinants. We have analyzed these and three others derived from another experiment to define their breakpoints as precisely as possible. Southern blot analysis with molecular probes to the D, S, I, and K regions of the H-2 complex defines seven recombinations between the D and S regions, two between S and I, none within the I region, and five events between I and K. The last category was studied in finer detail by developing unique copy probes to the I-K boundary region. Two of the five events occurred within probably less than 6 kb of each other: these two recombinants define the centromeric limit of the location of the t ^w5 gene within the H-2K region. The other three I-K recombinants occurred in at least two other nearby locations. Altogether at least three, and probably all five I-K recombinants fall within a 45 kb recombinational hot spot recently identified in Mus musculus castaneus.     

High-resolution mapping of sperm function defects in the t complex fourth inversion

Structural variants of the mouse Chr 17-specific t complex, known as t haplotypes, express factors that alter the ability of sperm to carry out their roles in the normal fertilization process. In previous studies of males carrying heterospecific combinations of the t complex, we discovered a unique M. spretus/t haplotype phenotype of male sterility. In additional studies with mice carrying a series of M. spretus–M. m. domesticus recombinant Chr 17 homologs and a complete t haplotype (S-+/t), we monitored physiological aspects of sperm function to map a locus (Hst6) responsible for expression of the t-specific ``curlicue' sperm flagellar curvature phenotype to 1 cM within the fourth inversion of the t complex. In the present report, we quantitatively analyze the in vitro capability of sperm from mice with similar S-+/t Chr 17 genotypes to fertilize zona pellucida-free mouse eggs. The results identify a locus, Stop1, mapping distal to Pim1, with acute effects on the ability of sperm to penetrate the oolemma. The data suggest that Stop1 is a complex locus consisting of at least two genetic elements, a proximal one overlapping the Hst6 locus, and another, distal to the Hst6 locus. Further quantitative analyses of the ``curlicue' phenotype produced by sperm derived from these same animals indicate that expression of this chronic flagellar curvature phenotype also derives from at least two elements, both mapping within the Hst6 locus. Thus, these studies provide higher resolution mapping of the molecular basis of t haplotype-specific sperm dysfunction emanating from In(17)4. Received: 22 May 1998 / Accepted: 17 June 1998     

A Simple Survival Model for Combined Action of Two Damaging Agents

The aim of this study was mathematical formalization of the development of two damage processes in response to the combined action of two different damaging agents. The model allows to obtain the parametric family of distributions for the life-span when the organism's death is due to joint effect of two damage processes. Differing by their probabilistic meaning the notions of a priori (M (t)) and a posteriori (M(y)) conditional mean contributions of each damage process to the final effect of combined injury are introduced. The formulas permitting to compute the M and M values on the basis of survival data parametric analysis are given.     

Two-Stage Procedures for Comparing Treatments with a Control: Elimination at the First Stage and Estimation at the Second Stage

We consider the problem of comparing a set of p₁ test treatments with a control treatment. This is to be accomplished in two stages as follows: In the first stage, N₁ observations are allocated among the p₁ treatments and the control, and the subset selection procedure of Gupta and Sobel (1958) is employed to eliminate “inferior” treatments. In the second stage, N₂ observations are allocated among the (randomly) selected subset of p₂(≤p₁) treatments and the control, and joint confidence interval estimates of the treatment versus control differences are calculated using Dunnett's (1955) procedure. Here both N₁ and N₂ are assumed to be fixed in advance, and the so-called square root rule is used to allocate observations among the treatments and the control in each stage. Dunnett's procedure is applied using two different types of estimates of the treatment versus control mean differences: The unpooled estimates are based on only the data obtained in the second stage, while the pooled estimates are based on the data obtained in both stages. The procedure based on unpooled estimates uses the critical point from a p₂-variate Student t-distribution, while that based on pooled estimates uses the critical point from a p₁-variate Student t-distribution. The two procedures and a composite of the two are compared via Monte Carlo simulation. It is shown that the expected value of p₂ determines which procedure yields shorter confidence intervals on the average. Extensions of the procedures to the case of unequal sample sizes are given. Applicability of the proposed two-stage procedures to a drug screening problem is discussed.     

A Test of the Equivalence of Four Random Number Generators – A Simulation Study

Four pseudo random number generators were tested for equivalence. Using two different seed values for each generator, the robustness of a one-sample t-statistic was assessed under a stationary auto-regressive process. The degree of autocorrelation, sample size and significance level were varied for the eight sets of random numbers generated. Results showed an interaction effect between random number generator and alpha level of the t-distribution, suggesting that simulation results could depend on the random number generator selected.