Small Number Of Replicates Research Articles

Statistically Consistent Identification of Differentially Expressed Genes in DNA Chip Data Over the Whole Expression Range

It is a well known problem that standard techniques for analysing DNA chip data misspecify genes. In particular, genes that are confirmed to be active, often do not show up as potential candidates. This is possibly due to non-homogeneous distributions of expression levels over the whole expression range. We introduce a method that allows the detection of genes based on a self-adaptive threshold. The threshold is determined for equally-populated expression bands by assuming a normal distribution of logarithms of expression level ratios. By specifying a significance level, the threshold is set according to 'local' expression statistics within a band. We call this method the relative variance method (RVM). We derive a test statistic for the RVM and compare it with other methods. On this statistical basis, we show that RVM is a complementary approach to the t-test, significance analysis of microarrays (SAM) or empirical Bayes analysis of microarrays (EBAM). The RVM should be particularly useful for experiments with small sample size. Using a clinical dataset, we demonstrate that the RVM can correctly identify known marker genes, which are not found by the t-test, SAM or EBAM. In situations with limited sample material and small number of replicates, as is often the case in clinical datasets, use of the proposed RVM provides a higher reliability of potential candidate genes.

A Bootstrap Test for the Analysis of Microarray Experiments with a Very Small Number of Replications

In microarray studies it is common that the number of replications (i.e. the sample size) is small and that the distribution of expression values differs from normality. In this situation, permutation and bootstrap tests may be appropriate for the identification of differentially expressed genes. However, unlike bootstrap tests, permutation tests are not suitable for very small sample sizes, such as three per group. A variety of different bootstrap tests exists. For example, it is possible to adjust the data to have a common mean before the bootstrap samples are drawn. For small significance levels, which can occur when a large number of genes is investigated, the original bootstrap test, as well as a bootstrap test suggested for the Behrens-Fisher problem, have no power in cases of very small sample sizes. In contrast, the modified test based on adjusted data is powerful. Using a Monte Carlo simulation study, we demonstrate that the difference in power can be huge. In addition, the different tests are illustrated using microarray data.

Rank-invariant resampling based estimation of false discovery rate for analysis of small sample microarray data

BackgroundThe evaluation of statistical significance has become a critical process in identifying differentially expressed genes in microarray studies. Classical p-value adjustment methods for multiple comparisons such as family-wise error rate (FWER) have been found to be too conservative in analyzing large-screening microarray data, and the False Discovery Rate (FDR), the expected proportion of false positives among all positives, has been recently suggested as an alternative for controlling false positives. Several statistical approaches have been used to estimate and control FDR, but these may not provide reliable FDR estimation when applied to microarray data sets with a small number of replicates.ResultsWe propose a rank-invariant resampling (RIR) based approach to FDR evaluation. Our proposed method generates a biologically relevant null distribution, which maintains similar variability to observed microarray data. We compare the performance of our RIR-based FDR estimation with that of four other popular methods. Our approach outperforms the other methods both in simulated and real microarray data.ConclusionWe found that the SAM's random shuffling and SPLOSH approaches were liberal and the other two theoretical methods were too conservative while our RIR approach provided more accurate FDR estimation than the other approaches.

Probe rank approaches for gene selection in oligonucleotide arrays with a small number of replicates

One major area of interest in analyzing oligonucleotide gene array data is identifying differentially expressed genes. A challenge to biostatisticians is to develop an approach to summarizing probe-level information that adequately reflects the true expression level while accounting for probe variation, chip variation and interaction effects. Various statistical tools, such as MAS and RMA, have been developed to address this issue. In these approaches, the probe level expression data are summarized into gene level data, which are then used for downstream statistical analysis. Since probe variation is often larger than chip variation and there is also a potential interaction effect between probe affinity and treatment effect, strategies such as a gene level analysis, may not be optimal. In this study, we propose a procedure to analyze probe level data for selecting differentially expressed genes under two treatment conditions (groups) with a small number of replicates. The probe level discrepancy between two groups can be measured by a difference of the percentiles of probe perfect-match (PM) ranks or of probe PM weighted ranks. The difference is then compared with a pre-specified threshold to determine differentially expressed genes. The probe level approach takes into account non-homogenous treatment effects and reduces possible cross-hybridization effects across a set of probes. The proposed approach is compared with MAS and RMA using two benchmark gene array datasets. Positive predictivity and sensitivity are used for evaluation. Results show the proposed approach has higher positive predictivity and higher sensitivity. Available on request from the authors. dtchen@uab.edu.

Biological detection of low radiation doses by combining results of two microarray analysis methods.

The accurate determination of the biological effects of low doses of pollutants is a major public health challenge. DNA microarrays are a powerful tool for investigating small intracellular changes. However, the inherent low reliability of this technique, the small number of replicates and the lack of suitable statistical methods for the analysis of such a large number of attributes (genes) impair accurate data interpretation. To overcome this problem, we combined results of two independent analysis methods (ANOVA and RELIEF). We applied this analysis protocol to compare gene expression patterns in Saccharomyces cerevisiae growing in the absence and continuous presence of varying low doses of radiation. Global distribution analysis highlights the importance of mitochondrial membrane functions in the response. We demonstrate that microarrays detect cellular changes induced by irradiation at doses that are 1000-fold lower than the minimal dose associated with mutagenic effects.

A novel Mixture Model Method for identification of differentially expressed genes from DNA microarray data

BackgroundThe main goal in analyzing microarray data is to determine the genes that are differentially expressed across two types of tissue samples or samples obtained under two experimental conditions. Mixture model method (MMM hereafter) is a nonparametric statistical method often used for microarray processing applications, but is known to over-fit the data if the number of replicates is small. In addition, the results of the MMM may not be repeatable when dealing with a small number of replicates. In this paper, we propose a new version of MMM to ensure the repeatability of the results in different runs, and reduce the sensitivity of the results on the parameters.ResultsThe proposed technique is applied to the two different data sets: Leukaemia data set and a data set that examines the effects of low phosphate diet on regular and Hyp mice. In each study, the proposed algorithm successfully selects genes closely related to the disease state that are verified by biological information.ConclusionThe results indicate 100% repeatability in all runs, and exhibit very little sensitivity on the choice of parameters. In addition, the evaluation of the applied method on the Leukaemia data set shows 12% improvement compared to the MMM in detecting the biologically-identified 50 expressed genes by Thomas et al. The results witness to the successful performance of the proposed algorithm in quantitative pathogenesis of diseases and comparative evaluation of treatment methods.

Test of Significant Differences with a priori Probability in Microarray Experiments

A traditional method for comparing two expression levels of genes in microarray experiments is the two-sample t-test. Because of the difficulty in using a large number of microarrays, an alternative method is required which can provide a reliable judgment of the comparison from a small number of replicates, even from a single pair of control and treatment. We present a method for detecting the changes in the gene expression levels under two different conditions in microarray experiments. Our method targets a single experiment for each condition, while retaining the statistical advantages of the t-test. The new proposals are: 1) standard deviation (SD) estimates of the expression levels which are an indicator for significant differences are given a priori as a function of the expression levels; 2) the limit of detection (LOD) for the expression levels is used to eliminate the majority of genes expressed at extremely low levels. The a priori SD estimates are obtained from six replicates under a fixed condition and are shown to be the approximate, but proper description of the expression uncertainty covering diverse conditions (e.g., different samples (human and rat) and different DNA chips). The LOD is defined as three times blank SD according to the IUPAC recommendation. A cell line (HL60) which will undergo macrophage differentiation on treatment with 12-O-tetradecanoylphorbol 13-acetate (TPA) is taken as an example. Our method is compared with the t-test for the data on duplicate TPA experiments and the former alone is evaluated with the data on a single TPA experiment. The errors from sample preparation and instrumental analysis are discussed.

A mixture model approach to detecting differentially expressed genes with microarray data.

An exciting biological advancement over the past few years is the use of microarray technologies to measure simultaneously the expression levels of thousands of genes. The bottleneck now is how to extract useful information from the resulting large amounts of data. An important and common task in analyzing microarray data is to identify genes with altered expression under two experimental conditions. We propose a nonparametric statistical approach, called the mixture model method (MMM), to handle the problem when there are a small number of replicates under each experimental condition. Specifically, we propose estimating the distributions of a t -type test statistic and its null statistic using finite normal mixture models. A comparison of these two distributions by means of a likelihood ratio test, or simply using the tail distribution of the null statistic, can identify genes with significantly changed expression. Several methods are proposed to effectively control the false positives. The methodology is applied to a data set containing expression levels of 1,176 genes of rats with and without pneumococcal middle ear infection.

The new method ‘micromapping’, a means to study species-specific associations and exclusions of ectomycorrhizae

Ectomycorrhizae (ECM) are obligate symbiotic associations between higher fungi and most tree species of the temperate and boreal forests, and of some tree families in tropical areas. As the anatomical features of these symbiotic organs are very diverse and suggested to improve tree growth differently efficient, their frequency and natural distribution in the soil, as well as the differentiation and amount of their substrate exploiting extramatrical mycelia, are of special ecological interest. The soil with its heterogeneous assemblage of micro-niches certainly provokes ectomycorrhizal fungi to compete for such microsites. We therefore applied the method ‘micromapping’ to record the ECM in their natural position with the following question in mind: Do indicators exist for an exclusion of or an association with other ectomycorrhizal species or not? Thoroughly excavated and carefully cleaned ectomycorrhizae of the OF horizon of a Picea abies stand were drawn in their natural position on perspex plates of 5 × 5 cm mapping area (McMp) with ink of different colours. They were afterwards removed and specified. Following scanning of the McMp, a special computer program was applied to analyse their distribution. The spatial relations of the ECM were calculated according to the ‘growing grid method’. The preliminary results suggest that the ECM of Russula ochroleuca and Piceirhiza internicrassihyphis show no common occurrence within short distances. This possibly applies also for Russula ochroleuca in comparison to Piceirhiza cinnbadiosimilis, for Elaphomyces granulatus in comparison to Xerocomus badius, and Lactarius decipiens in comparison to Piceirhiza cinnbadiosimilis. Cortinarius obtusus with Piceirhiza internicrassihyphis, and Piceirhiza internicrassihyphis with Xerocomus badius, indicate, however, rather high values of common occurrence. Due to the small number of replications, the standard deviations are high. More detailed investigations are therefore necessary before definite conclusions can be made. This method, however, apparently provides a useful tool to analyse spatial relations of ECM in the soil. Possible reasons for exclusions and associations of ECM are briefly discussed.

PRECISÃO EXPERIMENTAL E TAMANHO DA ÁREA DE EXPERIMENTOS DE CAMPO COM FRUTEIRAS E OUTRAS PLANTAS PERENES ARBÓREAS EM FUNÇÃO DA UNIDADE EXPERIMENTAL E DO NÚMERO DE REPETIÇÕES

Um problema dos experimentos de campo com plantas perenes e frutíferas arbóreas é o tamanho da área, devido ao porte das plantas que normalmente exigem largos espaçamentos. É muito freqüente, nesses experimentos, o uso de parcelas grandes, em detrimento do número de repetições, com a justificativa de diminuir a área experimental, a mão-de-obra e o conseqüente custo da pesquisa. Essa prática, contudo, traz prejuízos à precisão das estimativas dos parâmetros e à aplicação eficiente de testes estatísticos. Este trabalho foi realizado com o objetivo de mostrar que o aumento do número de repetições com o uso de parcelas pequenas aumenta a precisão dos experimentos, das estimativas do erro experimental e dos efeitos de tratamentos, favorece a detecção de diferenças significativas entre os tratamentos e contribui para diminuir a área experimental. Desenvolveu-se um procedimento que associa o tamanho da unidade experimental ao número de repetições, pela minimização da variância da média de cada tratamento, que permite o uso de maior número de repetições, para aumentar a precisão dos testes, obter maior uniformidade no experimento e melhorar a qualidade das pesquisas. Os resultados obtidos permitiram observar que o uso de parcelas pequenas favorece o aumento do número de repetições, permite obter melhores estimativas do erro experimental, dos efeitos de tratamentos e dos parâmetros, além de dar mais eficiência aos testes estatísticos a serem aplicados aos dados. Observou-se, também, diminuição substancial do número de plantas necessárias aos experimentos e do tamanho da área experimental.

Variance and efficiency of the combined estimator in incomplete block designs of use in forest genetics: a numerical study

The efficiency of combined interblock-intrablock and intrablock analysis for the estimation of treatment contrasts in alpha designs is compared using Monte-Carlo simulation. The combined estimator considers treatments and replications as fixed effects and blocks as random effects, whereas the intrablock estimator considers treatments, replications, and blocks as fixed effects. The variances of the estimators are used as the criterion for comparison. The combined estimator yields more accurate estimates than the intrablock estimator when the ratio of the block to the error variance is small, especially for designs with the fewest degrees of freedom. The accuracy of both estimators is similar when the ratio of variances is large. The variance of the combined estimator is very close to that of the best linear unbiased estimator except for designs with small number of replicates and families or provenances. Approximations commonly used for the variance of the combined estimator when variances of the random effects are unknown are studied. The downward or negative bias in the estimates of the variance given by the standard approximation used in statistical packages is largest under the conditions in which the combined estimator is more efficient than the intrablock estimator. Estimates of the relative efficiency of combined estimators have an upward bias that can exceed 10% of the true value in small- and middle-sized designs with two or three replicates. In designs with four or more replicates, often used in forest genetics, the bias is negligible.

Experimental competition release in a community of web‐weaving spiders

Spiders are generalist predators, abundant in terrestrial ecosystems, and hence good candidates for interspecific competitive constraints. Contrary to these premises, evidence of interspecific competition in web‐weaving spiders is ambiguous, and new experiments on natural communities are interesting. This paper reports on an experiment designed to test competitive effects in a natural community of web‐weaving spiders, accomplished by selective removal of each of the three locally dominant species: Araneus marmoreus, Argiope bruennichi, and Agelena labyrinthi‐ca. No statistically significant difference was found among control plots and the plots where one of the three species had been removed, for any of the variables measured, but the small number of replicates limited statistic power. Considering only effect direction, and not statistical significance, these removal experiments supported competitive effects on community structure, habitat overlap, and spider body size. On the other hand, effects were contrary to competitive expectations as regards abundance and habitat breadth. These results are in line with the findings of the other removal experiments conducted so far on spiders, that uncovered small intra‐ and interspecific competitive effects for some variables but not for others. So far, there is no evidence that interspecific competition may affect spider communities, although a low‐level competition remains possible, but difficult to test statistically for its evolutionary consequence.

Local adjustment method for field experiments. 2. Application in experiments with and without replications

Presented are the results of an application of the local adjustment Field-method in the analysis of the real field experiments without replications and in experiments with pseudo- and random replications. It was found that the local adjustment allowed a more precise estimation of the real values of genotypes in all kinds of experiments over a wide interval of spatial heterogeneity. The results obtained indicate that by simply using up to 5 random replications the decrease of the spatial effects is inadequate for fields with medium and high soil heterogeneity or with unsuccessful randomization. The Field-method which has sufficient precision in evaluating spatial heterogeneity, considerably increases the reliability of estimates in experiments even with a small number of replications or with unsuccessful randomization. Therefore, for all field experiments that have fulfilled the necessary requirements the Field-method is recommended as an obligatory statistical procedure for a prior data analysis as it allows an accurate estimation of the real character expression of genotypes, as well as the random variance in experiments without replication and the parameters of ANOVA in ones with pseudo- and random replications.

A proposal for using previous experience in designing microbiological sampling plans based on variables

A microbiological sampling system on the basis of variables is presented which requires only small numbers of replicates (n greater than or equal to 2). The system uses previous data on standard deviations of numbers of micro-organisms in batches and is particularly useful for in-plant situations. The discriminating power of the system is comparable to that of the current International Committee on Microbiological Specifications for Foods sampling schemes but uses more replicates.

On the precision of the plaque count

A simple method of measuring “procedural error” of plaque counts of antibody-secreting cells which affects counts in addition to the Poissonian error inherent in counting is proposed and evaluated. The two components are easily combined to give an expression for the variance of a single count. We have examined two sets of duplicate counts of antihapten plaques, the first scored by a modification of the original method, and the second by the original procedure, and have found them to be comparable in precision. We have reexamined published data of an earlier analysis of anti-red cell plaques and the half-leaf method of assaying plant viruses. The procedural error for the former was slightly larger and for the latter 60 times as large as for our data sets. Since procedural error is a function both of the method and the skill of those handling it, we recommend its estimation for individual laboratories, and have outlined a simple method that achieves this end without special effort. The knowledge of procedural error, incorporated into a formula for the variance of a single count, leads to a considerable long-term improvement in the economy of experimentation involving counts. Instead of placing confidence in fairly unstable variance estimates from small numbers of replicates in a given experiment, one may rely on documented experience of relevant history. In practical terms, a simple z-test statistic can be computed which allows one to judge differences between a single “control” and a single “treatment” count.

A pilot Monte Carlo study of two sequential estimation procedures based on generalized U-statistics

Recently two sequential estimation procedures based on generalized U-statistics have appeared in the statistical literature [Williams and Sen (1973, 1974)]. One of these procedures concerns the multi-sample problem of estimating a vector of parameters when the total sample size is fixed. The other procedure concerns the multi-sample problem of constructing a confidence ellipsoid of bounded maximum width for a vector of parameters. To supplement the asymptotic theory discussed in these earlier papers, a Monte Carlo study investigating the efficiency of these procedures for moderate sample sizes would be useful. This paper describes a preliminary Monte Carlo study utilizing a small number of replications and performed to provide information for the design of a more extensive study.

Confidence Interval Lengths for Small Numbers of Replicates.

Confidence Interval Lengths for Small Numbers of Replicates.