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Journal articles on the topic 'Unequal sample size problem'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Klaassen, Chris A. J. "Dixie cups: sampling with replacement from a finite population." Journal of Applied Probability 31, no. 4 (1994): 940–48. http://dx.doi.org/10.2307/3215319.

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At which (random) sample size will every population element have been drawn at least m times? This special coupon collector's problem is often referred to as the Dixie cup problem. Some asymptotic properties of the Dixie cup problem with unequal sampling probabilities are described.

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2

Klaassen, Chris A. J. "Dixie cups: sampling with replacement from a finite population." Journal of Applied Probability 31, no. 04 (1994): 940–48. http://dx.doi.org/10.1017/s0021900200099472.

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At which (random) sample size will every population element have been drawn at least m times? This special coupon collector's problem is often referred to as the Dixie cup problem. Some asymptotic properties of the Dixie cup problem with unequal sampling probabilities are described.

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3

Chakraborty, R. "A class of population genetic questions formulated as the generalized occupancy problem." Genetics 134, no. 3 (1993): 953–58. http://dx.doi.org/10.1093/genetics/134.3.953.

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Abstract In categorical genetic data analysis when the sampling units are classified into an arbitrary number of distinct classes, sometimes the sample size may not be large enough to apply large sample approximations for hypothesis testing purposes. Exact sampling distributions of several statistics are derived here, using combinatorial approaches parallel to the classical occupancy problem to help overcome this difficulty. Since the multinomial probabilities can be unequal, this situation is described as a generalized occupancy problem. The sampling properties derived are used to examine nonrandomness of occurrence of mutagen-induced mutations across loci, to devise tests of Hardy-Weinberg proportions of genotype frequencies in the presence of a large number of alleles, and to provide a global test of gametic phase disequilibrium of several restriction site polymorphisms.

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4

Guo, Jiin-Huarng, Hubert J. Chen, and Wei-Ming Luh. "Optimal Sample Sizes for Testing the Equivalence of Two Means." Methodology 15, no. 3 (2019): 128–36. http://dx.doi.org/10.1027/1614-2241/a000171.

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Abstract. Equivalence tests (also known as similarity or parity tests) have become more and more popular in addition to equality tests. However, in testing the equivalence of two population means, approximate sample sizes developed using conventional techniques found in the literature on this topic have usually been under-valued as having less statistical power than is required. In this paper, the authors first address the reason for this problem and then provide a solution using an exhaustive local search algorithm to find the optimal sample size. The proposed method is not only accurate but is also flexible so that unequal variances or sampling unit costs for different groups can be considered using different sample size allocations. Figures and a numerical example are presented to demonstrate various configurations. An R Shiny App is also available for easy use ( https://optimal-sample-size.shinyapps.io/equivalence-of-means/ ).

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Kumar, Narinder, Amar Nath Gill, and Gobind P. Mehta. "Selection Procedures for Location Parameters Based on Two-Sample U-Statistics." Calcutta Statistical Association Bulletin 42, no. 3-4 (1992): 201–20. http://dx.doi.org/10.1177/0008068319920305.

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Let π1, ... , πk be k independent populations and let Fi ( x)= F( x - θi) be the absolutely continuous cumulative distribution function (cdf) of the i-th population indexed by the location parameter θi; i=1,,.... k. A class of subset selection procedures based on sub-sample extrema for unequal sample sizes is proposed for the problem of selecting a subset from ( π1, .... πk) which contains the population with largest location parameter. The proposed subset selection procedures are then compared with the subset selection procedures of Hsu (1981) in the sense of Pitman ARE (asymptotic relative efficiency). It is shown that these procedures can approximately be implemented with the help of existing tables and sample size sufficient for their implementation, based on simulation results, is discussed. AMS (1980) Subject Classification: Primary 62F07; Secondary 62H10

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Guo, Jiin-Huarng, and Wei-Ming Luh. "Approximate Sample Size Formulas for Testing Group Mean Differences When Variances Are Unequal in One-Way ANOVA." Educational and Psychological Measurement 68, no. 6 (2008): 959–71. http://dx.doi.org/10.1177/0013164408318759.

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This study proposes an approach for determining appropriate sample size for Welch's F test when unequal variances are expected. Given a certain maximum deviation in population means and using the quantile of F and t distributions, there is no need to specify a noncentrality parameter and it is easy to estimate the approximate sample size needed for heterogeneous one-way ANOVA. The theoretical results are validated by a comparison to the results from a Monte Carlo simulation. Simulation results for the empirical power indicate that the sample size needed by the proposed formulas can almost always achieve the desired power level when Welch's F test is applied to data that are conditionally nonnormal and heterogeneous. Two illustrative examples of the use of the proposed procedure are given to calculate balanced and optimal sample sizes, respectively. Moreover, three sample size tables for two-, four-, and six-group problems are provided, respectively, for practitioners.

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Huillet, Thierry E. "Partitioning Problems Arising From Independent Shifted-Geometric and Exponential Samples With Unequal Intensities." International Journal of Statistics and Probability 8, no. 6 (2019): 31. http://dx.doi.org/10.5539/ijsp.v8n6p31.

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Two problems dealing with the random skewed splitting of some population into J different types are considered. In a first discrete setup, the sizes of the sub-populations come from independent shifted-geometric with unequal characteristics. Various J &rarr; &infin; asymptotics of the induced occupancies are investigated: the total population size, the number of unfilled types, the index of consecutive filled types, the maximum number of individuals in some state and the index of the type(s) achieving this maximum. Equivalently, this problem is amenable to the classical one of assigning indistinguishable particles (Bosons) at J sites, in some random allocation problem. In a second parallel setup in the continuum, we consider a large population of say J &lsquo;stars&rsquo;, the intensities of which have independent exponential distributions with unequal inverse temperatures. Stars are being observed only if their intensities exceed some threshold value. Depending on the choice of the inverse temperatures, we investigate the energy partitioning among stars, the total energy emitted by the observed stars, the number of the observable stars and the energy and index of the star emitting the most.

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Kim, Tae-Hoon, Min-Chul Kang, Ga-Bin Jung, Dong Soo Kim, and Cheol-Woong Yang. "Novel Method for Preparing Transmission Electron Microscopy Samples of Micrometer-Sized Powder Particles by Using Focused Ion Beam." Microscopy and Microanalysis 23, no. 5 (2017): 1055–60. http://dx.doi.org/10.1017/s1431927617012557.

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AbstractThe preparation of transmission electron microscopy (TEM) samples from powders is quite difficult and challenging. For powders with particles in the 1–5 μm size range, it is especially difficult to select an adequate sample preparation technique. Epoxy is commonly used to bind powder, but drawbacks, such as differential milling originating from unequal milling rates between the epoxy and powder, remain. We propose a new, simple method for preparing TEM samples. This method is especially useful for powders with particles in the 1–5 μm size range that are vulnerable to oxidation. The method uses solder as an embedding agent together with focused ion beam (FIB) milling. The powder was embedded in low-temperature solder using a conventional hot-mounting instrument. Subsequently, FIB was used to fabricate thin TEM samples via the lift-out technique. The solder proved to be more effective than epoxy in producing thin TEM samples with large areas. The problem of differential milling was mitigated, and the solder binder was more stable than epoxy under an electron beam. This methodology can be applied for preparing TEM samples from various powders that are either vulnerable to oxidation or composed of high atomic number elements.

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9

Chiang, Chieh, and Chin-Fu Hsiao. "Use of interval estimations in design and evaluation of multiregional clinical trials with continuous outcomes." Statistical Methods in Medical Research 28, no. 7 (2018): 2179–95. http://dx.doi.org/10.1177/0962280217751277.

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Multiregional clinical trials have been accepted in recent years as a useful means of accelerating the development of new drugs and abridging their approval time. The statistical properties of multiregional clinical trials are being widely discussed. In practice, variance of a continuous response may be different from region to region, but it leads to the assessment of the efficacy response falling into a Behrens–Fisher problem—there is no exact testing or interval estimator for mean difference with unequal variances. As a solution, this study applies interval estimations of the efficacy response based on Howe’s, Cochran–Cox’s, and Satterthwaite’s approximations, which have been shown to have well-controlled type I error rates. However, the traditional sample size determination cannot be applied to the interval estimators. The sample size determination to achieve a desired power based on these interval estimators is then presented. Moreover, the consistency criteria suggested by the Japanese Ministry of Health, Labour and Welfare guidance to decide whether the overall results from the multiregional clinical trial obtained via the proposed interval estimation were also applied. A real example is used to illustrate the proposed method. The results of simulation studies indicate that the proposed method can correctly determine the required sample size and evaluate the assurance probability of the consistency criteria.

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Slinker, B. K., and S. A. Glantz. "Multiple linear regression is a useful alternative to traditional analyses of variance." American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 255, no. 3 (1988): R353—R367. http://dx.doi.org/10.1152/ajpregu.1988.255.3.r353.

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Physiologists often wish to compare the effects of several different treatments on a continuous variable of interest, which requires an analysis of variance. Analysis of variance, as presented in most statistics texts, generally requires that there be no missing data and often that each sample group be the same size. Unfortunately, this requirement is rarely satisfied, and investigators are confronted with the problem of how to analyze data that do not strictly fit the traditional analysis of variance paradigm. One can avoid these pitfalls by recasting the analysis of variance as a multiple linear regression problem. When there are no missing data, the results of a traditional analysis of variance and the corresponding multiple regression problem are identical; when the sample sizes are unequal or there are missing data, one can use a regression formulation to analyze data that cannot be easily handled in a traditional analysis of variance paradigm and thus overcome a practical computational limitation of traditional analysis of variance. In addition to overcoming practical limitations of traditional analysis of variance, the multiple linear regression approach is more efficient because in one run of a statistics routine, not only is the analysis of variance done but also one obtains estimates of the size of the treatment effects (as opposed to just an indication of whether such effects are present or not), and many of the pairwise multiple comparisons are done (they are equivalent to t tests for significance of the regression parameter estimates). Finally, interaction between the different treatment factors is easier to interpret than it is in traditional analysis of variance.

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Daneshgaran, Fred, Marina Mondin, and Khashayar Olia. "Quantization of high dimensional Gaussian vector using permutation modulation with application to information reconciliation in continuous variable QKD." International Journal of Quantum Information 15, no. 08 (2017): 1740028. http://dx.doi.org/10.1142/s0219749917400287.

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This paper is focused on the problem of Information Reconciliation (IR) for continuous variable Quantum Key Distribution (QKD). The main problem is quantization and assignment of labels to the samples of the Gaussian variables observed at Alice and Bob. Trouble is that most of the samples, assuming that the Gaussian variable is zero mean which is de-facto the case, tend to have small magnitudes and are easily disturbed by noise. Transmission over longer and longer distances increases the losses corresponding to a lower effective Signal-to-Noise Ratio (SNR) exasperating the problem. Quantization over higher dimensions is advantageous since it allows for fractional bit per sample accuracy which may be needed at very low SNR conditions whereby the achievable secret key rate is significantly less than one bit per sample. In this paper, we propose to use Permutation Modulation (PM) for quantization of Gaussian vectors potentially containing thousands of samples. PM is applied to the magnitudes of the Gaussian samples and we explore the dependence of the sign error probability on the magnitude of the samples. At very low SNR, we may transmit the entire label of the PM code from Bob to Alice in Reverse Reconciliation (RR) over public channel. The side information extracted from this label can then be used by Alice to characterize the sign error probability of her individual samples. Forward Error Correction (FEC) coding can be used by Bob on each subset of samples with similar sign error probability to aid Alice in error correction. This can be done for different subsets of samples with similar sign error probabilities leading to an Unequal Error Protection (UEP) coding paradigm.

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Clémençon, Stephan, Patrice Bertail, Emilie Chautru, and Guillaume Papa. "Optimal survey schemes for stochastic gradient descent with applications to M-estimation." ESAIM: Probability and Statistics 23 (2019): 310–37. http://dx.doi.org/10.1051/ps/2018021.

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Iterative stochastic approximation methods are widely used to solve M-estimation problems, in the context of predictive learning in particular. In certain situations that shall be undoubtedly more and more common in the Big Data era, the datasets available are so massive that computing statistics over the full sample is hardly feasible, if not unfeasible. A natural and popular approach to gradient descent in this context consists in substituting the “full data” statistics with their counterparts based on subsamples picked at random of manageable size. It is the main purpose of this paper to investigate the impact of survey sampling with unequal inclusion probabilities on stochastic gradient descent-based M-estimation methods. Precisely, we prove that, in presence of some a priori information, one may significantly increase statistical accuracy in terms of limit variance, when choosing appropriate first order inclusion probabilities. These results are described by asymptotic theorems and are also supported by illustrative numerical experiments.

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Pervez, Sara, and Khalid M. Iraqi. "Gender Discrimination – Prevailing State In Pakistan." Pakistan Journal of Gender Studies 16, no. 1 (2018): 153–70. http://dx.doi.org/10.46568/pjgs.v16i1.121.

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Gender inequality is defined as unequal or unjust treatment because of someone’s gender. Generally, females are the victims of such discrimination.. Inequality in terms of gender represents a significant social problem in Pakistan as well as throughout the world. The fact that women receive fewer privileges in terms of economic benefits and education has become a worldwide issue. Even in Pakistan there is a huge gender gap in terms of allocation of economic benefits and education. Not only that, females encounter discrimination in all other areas of life and face violence which has been mentioned in the study. The case study of different women has been used for this study. A sample size of 15 respondents was taken. A structured questionnaire of 23 questions was prepared. It was found that in Pakistan, the violence against women takes place in many forms such as honour killings, acid attacks, early marriages, human trafficking, rapes, sexual harassment at workplace etc. In addition to that, the Islamic concept of gender equality has been explained in order to examine the Islamic practices that take place in Pakistan in terms of gender equality. The data has been collected through secondary sources as well as primary including the interviews of the victims of gender violence, various research journals, scholarly articles, research papers etc

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14

Kundt, G. "An Alternative Proposal for “Mixed Randomization” by Schulz and Grimes." Methods of Information in Medicine 44, no. 04 (2005): 572–76. http://dx.doi.org/10.1055/s-0038-1634009.

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Summary Objective: Randomization is an important part of clinical trials. Using permuted-block randomization for forcing equal group sizes potentially harms the unpredictability of treatment assignments. This can allow bias to creep into a trial. As an alternative, Schulz and Grimes suggest a “Mixed randomization” scheme which introduces more complexity to realize randomization. The objective of our research was to work out a model for randomization which is easier to handle than “Mixed randomization”, with an equal level of performance in unpredictability and balance. Methods: We analyzed a “Mixed randomization” procedure regarding the degree of unpredictability and balancing power and compared performance using permuted-block randomization with very large block size in a worst case scenario. Our work was done by the application of Blackwell-Hodges model for evaluation of the unpredictability of treatment assignments. Results: Regarding unpredictability, performance of permuted-block randomization with block size b = 36 was very similar to that of “Mixed randomization”. Regarding balancing power it was more favourable than “Mixed randomization”. Conclusion: Results of Schulz and Grimes are very important as they emphasized that mildly unequal sample sizes of therapy groups don’t cause problems. But the suggested scheme of “Mixed randomization” to a large extent adds complexity and we do not believe that this proposal is very feasible. Basically, we rather recommend the use of only one restricted randomization procedure in the best way. This can be permuted-block randomization with optimum choice of a large block size.

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15

Smith, Bruce R., Christophe M. Herbinger, and Heather R. Merry. "Accurate Partition of Individuals Into Full-Sib Families From Genetic Data Without Parental Information." Genetics 158, no. 3 (2001): 1329–38. http://dx.doi.org/10.1093/genetics/158.3.1329.

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Abstract Two Markov chain Monte Carlo algorithms are proposed that allow the partitioning of individuals into full-sib groups using single-locus genetic marker data when no parental information is available. These algorithms present a method of moving through the sibship configuration space and locating the configuration that maximizes an overall score on the basis of pairwise likelihood ratios of being full-sib or unrelated or maximizes the full joint likelihood of the proposed family structure. Using these methods, up to 757 out of 759 Atlantic salmon were correctly classified into 12 full-sib families of unequal size using four microsatellite markers. Large-scale simulations were performed to assess the sensitivity of the procedures to the number of loci and number of alleles per locus, the allelic distribution type, the distribution of families, and the independent knowledge of population allelic frequencies. The number of loci and the number of alleles per locus had the most impact on accuracy. Very good accuracy can be obtained with as few as four loci when they have at least eight alleles. Accuracy decreases when using allelic frequencies estimated in small target samples with skewed family distributions with the pairwise likelihood approach. We present an iterative approach that partly corrects that problem. The full likelihood approach is less sensitive to the precision of allelic frequencies estimates but did not perform as well with the large data set or when little information was available (e.g., four loci with four alleles).

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Mehrotra, P. C., A. K. Srivastava, and K. K. Tyagi. "On Unequal Cluster Sampling for Fixed Sample Size." Statistician 36, no. 4 (1987): 385. http://dx.doi.org/10.2307/2348836.

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17

Zeng, Z. B. "Correcting the bias of Wright's estimates of the number of genes affecting a quantitative character: a further improved method." Genetics 131, no. 4 (1992): 987–1001. http://dx.doi.org/10.1093/genetics/131.4.987.

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Abstract Wright's method of estimating the number of genes contributing to the difference in a quantitative character between two populations involves observing the means and variances of the two parental populations and their hybrid populations. Although simple, Wright's method provides seriously biased estimates, largely due to linkage and unequal effects of alleles. A method is suggested to evaluate the bias of Wright's estimate, which relies on estimation of the mean recombination frequency between a pair of loci and a composite parameter of variability of allelic effects and frequencies among loci. Assuming that the loci are uniformly distributed in the genome, the mean recombination frequency can be calculated for some organisms. Theoretical analysis and an analysis of the Drosophila data on distributions of effects of P element inserts on bristle numbers indicate that the value of the composite parameter is likely to be about three or larger for many quantitative characters. There are, however, some serious problems with the current method, such as the irregular behavior of the statistic and large sampling variances of estimates. Because of that, the method is generally not recommended for use unless several favorable conditions are met. These conditions are: the two parental populations are many phenotypic standard deviations apart, linkage is not tight, and the sample size is very large. An example is given on the fruit weight of tomato from a cross with parental populations differing in means by more than 14 phenotypic standard deviations. It is estimated that the number of loci which account for 95% of the genic variance in the F2 population is 16, with a 95% confidence interval of 7-28, and the effect of the leading locus is 13% of the parental difference, with 95% confidence interval 8.5-25.7%.

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Schouten, Hubert J. A. "Sample size formula with a continuous outcome for unequal group sizes and unequal variances." Statistics in Medicine 18, no. 1 (1999): 87–91. http://dx.doi.org/10.1002/(sici)1097-0258(19990115)18:1<87::aid-sim958>3.0.co;2-k.

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Luh, Wei-Ming, and Jiin-Huarng Guo. "Approximate sample size formulas for the two-sample trimmed mean test with unequal variances." British Journal of Mathematical and Statistical Psychology 60, no. 1 (2007): 137–46. http://dx.doi.org/10.1348/000711006x100491.

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20

Ruvuna, Francis. "Unequal Center Sizes, Sample Size, and Power in Multicenter Clinical Trials." Drug Information Journal 38, no. 4 (2004): 387–94. http://dx.doi.org/10.1177/009286150403800409.

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21

Halabi, Susan, and Bahadur Singh. "Sample size determination for comparing several survival curves with unequal allocations." Statistics in Medicine 23, no. 11 (2004): 1793–815. http://dx.doi.org/10.1002/sim.1771.

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22

Tse, Siu-Keung. "A comparison of procedures for general contrasts -unequal sample size case." Communications in Statistics - Theory and Methods 18, no. 2 (1989): 613–32. http://dx.doi.org/10.1080/03610928908829922.

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23

Jain, R. K. "Unequal sample size allocation to optimal design for binomial logistic models." Statistische Hefte 28, no. 1 (1987): 285–90. http://dx.doi.org/10.1007/bf02932608.

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Hsieh, F. Y. "A simple method of sample size calculation for unequal-sample-size designs that use the logrank ort-test." Statistics in Medicine 6, no. 5 (1987): 577–81. http://dx.doi.org/10.1002/sim.4780060506.

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Wang, Shizhen S. "On the Statistical Testing Methods for Single Laboratory Validation of Qualitative Microbiological Assays with an Unpaired Design." Journal of AOAC INTERNATIONAL 103, no. 5 (2020): 1426–34. http://dx.doi.org/10.1093/jaoacint/qsaa038.

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Abstract Background There exists several statistical methods for detecting a difference of detection rates between alternative and reference qualitative microbiological assays in a single laboratory validation study with an unpaired design. Objective We compared performance of eight methods including Fisher’s exact test, unequal variance two-sample t-test, Wilcoxon rank-sum test, z-test, and methods based on Wilson confidence intervals, complementary log-log regression, Firth’s logistic regression, and ordinary logistic regression. Method We first compared the minimum detectable difference in the proportion of detections between the alternative and reference methods among these statistical methods for a varied number of test portions. We then compared power and size of test of these methods using simulated data. Results Firth’s logistic regression and the unequal variance two-sample t-test had the lowest minimum detectable difference and highest power. None of these statistical methods had an estimated size of test always within a 95% confidence interval of the nominal value 0.05 with small numbers of test portions (n = 12, 20, 30). Fisher’s exact test, the Wilcoxon rank-sum test, and the z-test were conservative even with a moderately large number of test portions (n = 40), while Firth’s logistic regression and the unequal variance two-sample t-test had a size of test closer to 0.05 than other methods. Conclusions Firth's logistic regression and the unequal variance two-sample t-test are better choices than other competing methods. Highlights We recommend the unequal variance two-sample t-test over Firth’s logistic regression because the unequal variance two-sample t-test is better known and easier to use. We provide an example using real data.

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Kikvidze, Z., and J. Moya-Laraño. "Unexpected failures of recommended tests in basic statistical analyses of ecological data." Web Ecology 8, no. 1 (2008): 67–73. http://dx.doi.org/10.5194/we-8-67-2008.

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Abstract. Ecologists, when analyzing the output of simple experiments, often have to compare statistical samples that simultaneously are of uneven size, unequal variance and distribute non-normally. Although there are special tests designed to address each of these unsuitable characteristics, it is unclear how their combination affects the tests. Here we compare the performance of recommended tests using generated data sets that simulate statistical samples typical in ecological research. We measured rates of type I and II errors, and found that common parametric tests such as ANOVA are quite robust to non-normality, uneven sample size, unequal variance, and their effect combined. ANOVA and randomization tests produced very similar results. At the same time, the t-test for unequal variance unexpectedly lost power with samples of uneven size. Also, non-parametric tests were strongly affected by unequal variance in large samples, yet non-parametric tests could complement parametric tests when testing samples of uneven size. Thus, we demonstrate that the robustness of each kind of test strongly depends on the combination of parameters (distribution, sample size, equality of variances). We conclude that manuals should be revised to offer more elaborate instructions for applying specific statistical tests.

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Becker, Gilbert. "Correcting the Point-Biserial Correlation for Attenuation Owing to Unequal Sample Size." Journal of Experimental Education 55, no. 1 (1986): 5–8. http://dx.doi.org/10.1080/00220973.1986.10806427.

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Kristunas, Caroline, Tom Morris, and Laura Gray. "Unequal cluster sizes in stepped-wedge cluster randomised trials: a systematic review." BMJ Open 7, no. 11 (2017): e017151. http://dx.doi.org/10.1136/bmjopen-2017-017151.

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ObjectivesTo investigate the extent to which cluster sizes vary in stepped-wedge cluster randomised trials (SW-CRT) and whether any variability is accounted for during the sample size calculation and analysis of these trials.SettingAny, not limited to healthcare settings.ParticipantsAny taking part in an SW-CRT published up to March 2016.Primary and secondary outcome measuresThe primary outcome is the variability in cluster sizes, measured by the coefficient of variation (CV) in cluster size. Secondary outcomes include the difference between the cluster sizes assumed during the sample size calculation and those observed during the trial, any reported variability in cluster sizes and whether the methods of sample size calculation and methods of analysis accounted for any variability in cluster sizes.ResultsOf the 101 included SW-CRTs, 48% mentioned that the included clusters were known to vary in size, yet only 13% of these accounted for this during the calculation of the sample size. However, 69% of the trials did use a method of analysis appropriate for when clusters vary in size. Full trial reports were available for 53 trials. The CV was calculated for 23 of these: the median CV was 0.41 (IQR: 0.22–0.52). Actual cluster sizes could be compared with those assumed during the sample size calculation for 14 (26%) of the trial reports; the cluster sizes were between 29% and 480% of that which had been assumed.ConclusionsCluster sizes often vary in SW-CRTs. Reporting of SW-CRTs also remains suboptimal. The effect of unequal cluster sizes on the statistical power of SW-CRTs needs further exploration and methods appropriate to studies with unequal cluster sizes need to be employed.

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Hinkle, Joshua C., David Weisburd, Christine Famega, and Justin Ready. "The Problem Is Not Just Sample Size." Evaluation Review 37, no. 3-4 (2013): 213–38. http://dx.doi.org/10.1177/0193841x13519799.

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Armstrong, Richard A. "Is there a large sample size problem?" Ophthalmic and Physiological Optics 39, no. 3 (2019): 129–30. http://dx.doi.org/10.1111/opo.12618.

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Das, Priyam, Christine B. Peterson, Kim-Anh Do, Rehan Akbani, and Veerabhadran Baladandayuthapani. "NExUS: Bayesian simultaneous network estimation across unequal sample sizes." Bioinformatics 36, no. 3 (2019): 798–804. http://dx.doi.org/10.1093/bioinformatics/btz636.

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Abstract Motivation Network-based analyses of high-throughput genomics data provide a holistic, systems-level understanding of various biological mechanisms for a common population. However, when estimating multiple networks across heterogeneous sub-populations, varying sample sizes pose a challenge in the estimation and inference, as network differences may be driven by differences in power. We are particularly interested in addressing this challenge in the context of proteomic networks for related cancers, as the number of subjects available for rare cancer (sub-)types is often limited. Results We develop NExUS (Network Estimation across Unequal Sample sizes), a Bayesian method that enables joint learning of multiple networks while avoiding artefactual relationship between sample size and network sparsity. We demonstrate through simulations that NExUS outperforms existing network estimation methods in this context, and apply it to learn network similarity and shared pathway activity for groups of cancers with related origins represented in The Cancer Genome Atlas (TCGA) proteomic data. Availability and implementation The NExUS source code is freely available for download at https://github.com/priyamdas2/NExUS. Supplementary information Supplementary data are available at Bioinformatics online.

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Ruvuna, Francis. "Erratum to: Unequal Center Sizes, Sample Size, and Power in Multicenter Clinical Trials." Drug Information Journal 39, no. 1 (2005): 62. http://dx.doi.org/10.1177/009286150503900108.

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33

Bruvold, Norman T., and Shanker Ganesan. "A sample size selection procedure for pairwise comparisons of means with unequal variances." Communications in Statistics - Theory and Methods 17, no. 5 (1988): 1497–505. http://dx.doi.org/10.1080/03610928808829694.

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34

Ramsey, Philip H., and Patricia P. Ramsey. "Power of pairwise comparisons in the equal variance and unequal sample size case." British Journal of Mathematical and Statistical Psychology 61, no. 1 (2008): 115–31. http://dx.doi.org/10.1348/000711006x153051.

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Kössler, Wolfgang. "Rank tests in the two-sample scale problem with unequal and unknown locations." Statistical Papers 40, no. 1 (1999): 13–35. http://dx.doi.org/10.1007/bf02927108.

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Kumar, A. "The Sample Size." Journal of Universal College of Medical Sciences 2, no. 1 (2014): 45–47. http://dx.doi.org/10.3126/jucms.v2i1.10493.

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Finding an "appropriate sample size" has been the most basic and foremost problem; a research worker is always faced with, in all sampling based analytical researches. This is so, since a very large sized sample results to unnecessary wastage of resources, while a very small sized sample may affect adversely the accuracy of sample estimates and thus in turn losing the very efficacy of selected sampling plan. The present paper attempts to highlight the main determinant factors and the analytical approach towards estimation ofrequired sample size, along with a few illustrations. DOI: http://dx.doi.org/10.3126/jucms.v2i1.10493 Journal of Universal College of Medical Sciences (2014) Vol.2(1): 45-47

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Zouein, P. P., H. Harmanani, and A. Hajar. "Genetic Algorithm for Solving Site Layout Problem with Unequal-Size and Constrained Facilities." Journal of Computing in Civil Engineering 16, no. 2 (2002): 143–51. http://dx.doi.org/10.1061/(asce)0887-3801(2002)16:2(143).

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PAULI, HANS-CHRISTIAN, and ASMITA MUKHERJEE. "ON THE SIZE OF HADRONS." International Journal of Modern Physics A 16, no. 26 (2001): 4351–64. http://dx.doi.org/10.1142/s0217751x01005407.

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The form factor and the mean-square radius of the pion are calculated analytically from a parametrized form of a [Formula: see text] wave function. The numerical wave function was obtained previously by solving numerically an eigenvalue equation for the pion in a particular model. The analytical formulas are of more general interest than just being valid for the pion and can be generalized to the case with unequal quark masses. Two different parametrizations are investigated. Because of the highly relativistic problem, noticeable deviations from a nonrelativistic formula are obtained.

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Xu, Li-Wen. "MANOVA for Nested Designs with Unequal Cell Sizes and Unequal Cell Covariance Matrices." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/649202.

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We propose and study parametric bootstrap (PB) tests for heteroscedastic two-factor MANOVA with nested designs. For the problem of testing “main effects” of both factors, we develop a flexible test based on a parametric bootstrap approach. The PB test is shown to be invariant under affine-transformations. Moreover, the PB test does not depend on the chosen weights used to define the parameters uniquely. The proposed test is compared with the approximate HotellingT2(AHT) test by the simulations. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations and generally outperforms the AHT test in terms of controlling the nominal size. For the heteroscedastic cases, the PB test outperforms the AHT test in terms of power. In addition, the PB test does not lose too much power when the homogeneity assumption is actually valid.

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Wilson, Fiona. "May the best man win." Equality, Diversity and Inclusion: An International Journal 33, no. 4 (2014): 361–71. http://dx.doi.org/10.1108/edi-11-2013-0095.

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Purpose – The purpose of this paper is to examine how female bank lenders are locked into a position of disadvantage in a UK bank. The work of Bourdieu is used to explore women's position of disadvantage and inequality. As Bourdieu helps us predict, the women are symbolically constructed as different, and face different problems to men. Women's social capital is not perceived as the same as men's. Design/methodology/approach – The research method involved preliminary research interviews with seven key senior staff in the bank followed by focus group discussions with 35 male and female bank loan officers on bank premises within a nine-month period. Six focus groups were held – three with men and three with women in four British cities – London, Manchester, Bristol and Edinburgh. All the interviews were tape-recorded and analysed. The participants were told that the discussion was completely confidential, and that we were interested in the role gender played in entrepreneurial and corporate life. Findings were verified by taking them back to a selection of those who had been involved in the focus groups. Findings – The findings show how the power dynamics are played out within the immediate workplace environment and influenced by the wider macro systems of society. The women differed in their views as to whether gender mattered. Despite the evident inequities these women face, some wished to deny or resist being seen as unequal or wanted to acknowledge inequity. The paper explains how and why this might be the case. Research limitations/implications – The research is limited by its sample size to 35 bank loan officers. Practical implications – The paper demonstrates some of the difficulties faced by those who wish to implement equality of opportunity in the face of women's denial of inequality. Social implications – The paper clearly illustrates the difficulties and challenges faced by female bank loan officers in banking. Originality/value – This paper discusses the subjective experience of equality, inequality and exclusion among female bankers showing how they are not a homogenous group, as they say they experience equality/inequality differently. These women face ideological dilemmas that are not widely discussed in the research literature. It is very unlikely that as a divided, heterogeneous group who find themselves in a very small minority in this bank, that greater equality for them is likely to come about.

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Biyani, S. H., and Chand K. Midha. "A Generalized Formula for Variance in Unequal Probability Sampling." Calcutta Statistical Association Bulletin 48, no. 3-4 (1998): 241–44. http://dx.doi.org/10.1177/0008068319980312.

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A variance formula is given for a general class of estimators for any fixed sample size design. A formula for an unbiased estimator of the variance is also provided. These results generalize the Yates-Grundy variance formula of the Horvitz-Thompson estimator.

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Dai, Li Ping, Qiang Qiang Yan, and Pu Gao. "The Study on Numerical Calculation of Unequal Potential Grounding Parameters of Large Hydropower Plant." Applied Mechanics and Materials 521 (February 2014): 379–88. http://dx.doi.org/10.4028/www.scientific.net/amm.521.379.

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To solve the problem of grounding parameters of large hydropower, which is in a special soil environment, we deduced unequal potential calculation formula ,based on the model of unequal potential in ground grid and combined with the boundary element method and node voltage method. Through example calculations, the paper analyzes the impact of the ground grid size, soil resistivity and conductor permeability on unequal potential and proves the accuracy of the results, compared with CDEGS software. This method can be used in grounding calculation and analysis of large hydropower stations.

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43

Thigpen, Charles C. "A Sample-Size Problem in Simple Linear Regression." American Statistician 41, no. 3 (1987): 214. http://dx.doi.org/10.2307/2685107.

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Thigpen, Charles C. "A Sample-Size Problem in Simple Linear Regression." American Statistician 41, no. 3 (1987): 214–15. http://dx.doi.org/10.1080/00031305.1987.10475483.

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45

Bacchetti, Peter. "Small sample size is not the real problem." Nature Reviews Neuroscience 14, no. 8 (2013): 585. http://dx.doi.org/10.1038/nrn3475-c3.

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46

Orme, John G., and Walter W. Hudson. "The problem of sample size estimation: Confidence intervals." Social Work Research 19, no. 2 (1995): 121–27. http://dx.doi.org/10.1093/swr/19.2.121.

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Kuo, Bor-Chen, and Kuang-Yu Chang. "Feature Extractions for Small Sample Size Classification Problem." IEEE Transactions on Geoscience and Remote Sensing 45, no. 3 (2007): 756–64. http://dx.doi.org/10.1109/tgrs.2006.885074.

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Bassiakos, Yiannis C., and Panos C. Katerelos. "Sample Size Calculation for the Therapeutic Equivalence Problem." Communications in Statistics - Simulation and Computation 35, no. 4 (2006): 1019–26. http://dx.doi.org/10.1080/03610910600880559.

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Wołynski, Waldemar. "Minimal Sample Size in the Group Classification Problem." Journal of Classification 22, no. 1 (2005): 49–58. http://dx.doi.org/10.1007/s00357-005-0005-8.

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Sengupta, Samindranath, and Erkki P. Liski. "Plot Sampling with Plots of Unequal Sizes." Calcutta Statistical Association Bulletin 48, no. 3-4 (1998): 221–28. http://dx.doi.org/10.1177/0008068319980309.

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Assuming that the trees in the forests are randomly distributed, it is demonstrated that plot sampling with unequal sized plots generally leads to a more precise estimate of the tree parameter totals than that with plots of equal sizes. The optimUlll choice of individual plot sizes has been discussed for a given (i) total plot size and (ii) cost (with an assumed cost structure). The problem of variance estimation when the plots are of unequal sizes has also been addressed .

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