Impossible Inferences: A Note on Inconsistency in Business Statistics

BackgroundIn systematic reviews and meta-analysis, researchers often pool the results of the sample mean and standard deviation from a set of similar clinical trials. A number of the trials, however, reported the study using the median, the minimum and maximum values, and/or the first and third quartiles. Hence, in order to combine results, one may have to estimate the sample mean and standard deviation for such trials.MethodsIn this paper, we propose to improve the existing literature in several directions. First, we show that the sample standard deviation estimation in Hozo et al.’s method (BMC Med Res Methodol 5:13, 2005) has some serious limitations and is always less satisfactory in practice. Inspired by this, we propose a new estimation method by incorporating the sample size. Second, we systematically study the sample mean and standard deviation estimation problem under several other interesting settings where the interquartile range is also available for the trials.ResultsWe demonstrate the performance of the proposed methods through simulation studies for the three frequently encountered scenarios, respectively. For the first two scenarios, our method greatly improves existing methods and provides a nearly unbiased estimate of the true sample standard deviation for normal data and a slightly biased estimate for skewed data. For the third scenario, our method still performs very well for both normal data and skewed data. Furthermore, we compare the estimators of the sample mean and standard deviation under all three scenarios and present some suggestions on which scenario is preferred in real-world applications.ConclusionsIn this paper, we discuss different approximation methods in the estimation of the sample mean and standard deviation and propose some new estimation methods to improve the existing literature. We conclude our work with a summary table (an Excel spread sheet including all formulas) that serves as a comprehensive guidance for performing meta-analysis in different situations.Electronic supplementary materialThe online version of this article (doi:10.1186/1471-2288-14-135) contains supplementary material, which is available to authorized users.

Hypothesis tests

A method is presented which allows correlation coefficients to be corrected for restriction of range when only the proportion of the population selected is known. The method assumes that the population distribution is normal and that selection has resulted in single truncation. A table is given which provides the ratio of the population and sample standard deviations, the truncation point, and the ordinate in N(0, 1), corresponding to the proportion of the population selected. Correlations can be corrected for the effects of direct or indirect selection by substitution of tabled values in traditional correction formulae. Further, the population mean or standard deviation can be estimated if the censored sample mean or standard deviation is known.

Empirical models are often used in EMP assessments of electronic equipment. Two such sets of models previously developed for prediction of EMP failure levels of discrete silicon bipolar semiconductor devices are evaluated. The sample mean and standard deviation of the ratio of failure model calculated to experimentally measured damage constant for each model and device type form the basis of the evaluation. The sample mean is used to evaluate the predictive accuracy of each model. The sample standard deviation is used to develop error bounds and to evaluate the recommended hierarchy of use of each of the model sets. The computerized data base SUPERSAP2 is the source of device numbers and experimental damage constants. Model input parameters are device electrical characteristics extracted from SUPERSAP2, vendor catalogs, and D.A.T.A. books.

Confidence limits on the failure rate and a rapid projection nomogram for the lognormal distribution

<title>ABSTRACT</title> <p>Fuel use and resupply provisions are vital to all military combat operations, and any significant reduction in the amount of fuel required to sustain the force becomes a large tactical advantage for commanders Developers have been seeking methods that offer even small gains in fuel economy. Small gains for a fleet of thousands of vehicles translate into fewer fuel convoys to theater and large costs savings over time. The challenge to the tester and evaluator is to determine if these small advances are relevant or merely normally expected test variation in the acquired fuel consumption parameters. All too often only the mean fuel economy parameters are compared with and without the new equipment or process without considering test variances inherent in collecting the parametric data. The resulting analysis may then be seriously flawed. Hypothesis testing is a useful statistical method for comparing two sets of test data (sample means and standard deviations) to determine if there is a statistically significant difference between the two sets. Often the two sets of data are made up of small sample sizes (5 test trials are very typical for sets of fuel consumption data). Therefore, for purposes of this discussion, we will consider only hypothesis tests for the differences between two sample means for small (the number of samples is less than 30) sample sizes. Several examples of fictional test data will be subjected to hypothesis testing to show the value of such an approach.</p>

Many randomized trials measure means and standard deviations of anesthesia recovery time (e.g., times to tracheal extubation). We show how to use generalized pivotal methods to compare the probabilities of exceeding a tolerance limit (e.g., > 15min, prolonged times to tracheal extubation). The topic isimportant because the economic benefits of faster anesthesia emergence depend onreducing variability, not means, especially prevention of very long recovery times. Generalized pivotal methods are applied using computer simulation (e.g., using two Excel formulas for one group and three formulas for two group comparisons). The endpoint for each study with two groups is the ratio between groups of the probabilities of times exceeding a threshold or the ratio of the standard deviations. Confidence intervals and variances for the incremental risk ratio of the exceedance probabilities and for ratios of standard deviations are calculated using studies' sample sizes, sample means in the time scale of recovery times, and sample standard deviations in the time scale. Ratios are combined among studies using the DerSimonian-Laird estimate of the heterogeneity variance estimate, with Knapp-Hartung adjustment for the relatively small (N = 15) numbers of studies in the meta-analysis. We show larger absolute variability among studies' results when analyzed based on exceedance probabilities rather than standard deviations. Therefore, if an investigator's primary goal is to quantify reductions in the variability of recovery times (e.g., times until patients are ready for post-anesthesia care unit discharge), we recommend analyzing the standard deviations. When exceedance probabilities themselves are relevant, they can be analyzed from the original studies' summary measures.

Mental status examinations of individuals applying for disability are most often authorized as "2-hour" evaluations and are to include a complete clinical and employment history, assessment of neurocognitive functions, an estimate of general level of intelligence, psychological and social adjustment, functional abilities, and a complete multipage report of the results. In the interest of meeting these demands we have been using the Nevada brief cognitive assessment instrument (NBCAI) to rapidly estimate verbal intelligence, and we have adopted the Saint Louis University Mental Status Examination (SLUMS) to investigate neurocognitive functions. Areas of correspondence and differences between these screens are presented along with an explanation of why execution of both screens could be useful in addressing psychological questions concerning disability. The two instruments failed to correlate significantly in a group of patients undergoing pre-surgical evaluation likely because this sample was more highly educated and more homogeneous than the sample of disability referrals. For example, although the ages of the samples were close (disability sample mean and standard deviation age: 43.88 and 11.35 years; pre-surgical sample mean and standard deviation age: 46.14 and 2.84 years), they differed considerably in level of education (disability sample mean and standard deviation: 11.03 and 2.15; pre-surgical mean and standard deviation: 15.22 and 2.84). The pre-surgical group had mean performances close to the ceiling levels of both instruments.

Estimating the population variance, standard deviation, and coefficient of variation: Sample size and accuracy

One-year of hourly average air pollutant concentration (APC) observations, including CO, NO, NO2, O3, PM10, and SO2 was used to examine the effects of autocorrelation on the assumption of Central Limit Theorem (CLT) by calculating the confidence intervals of the data that were known to be dependent. Monte Carlo sampling was used to draw random samples of various sizes from the population (1000 groups of each size), and the sample means and standard deviation of these observed means were then evaluated. Even with small sample sizes, the average of all the means in each group and the observed standard deviation of the means were found to closely approximate the means of the overall population and the standard deviation predicted by CLT, respectively. Moreover, the above consistency was closely related to coefficient of variation of the population rather than to the degree of long-range-dependence. These results were used to interpret why the right-skewed frequency distribution observed in the mutually dependent air quality data could be accurately described using the lognormal model derived from the CLT. The link between the lognormality and multifractality characteristics in APC time series was also discussed.

Simulated ordered powder mixture

The era of big data is coming, and evidence-based medicine is attracting increasing attention to improve decision making in medical practice via integrating evidence from well designed and conducted clinical research. Meta-analysis is a statistical technique widely used in evidence-based medicine for analytically combining the findings from independent clinical trials to provide an overall estimation of a treatment effectiveness. The sample mean and standard deviation are two commonly used statistics in meta-analysis but some trials use the median, the minimum and maximum values, or sometimes the first and third quartiles to report the results. Thus, to pool results in a consistent format, researchers need to transform those information back to the sample mean and standard deviation. In this article, we investigate the optimal estimation of the sample mean for meta-analysis from both theoretical and empirical perspectives. A major drawback in the literature is that the sample size, needless to say its importance, is either ignored or used in a stepwise but somewhat arbitrary manner, e.g. the famous method proposed by Hozo etal. We solve this issue by incorporating the sample size in a smoothly changing weight in the estimators to reach the optimal estimation. Our proposed estimators not only improve the existing ones significantly but also share the same virtue of the simplicity. The real data application indicates that our proposed estimators are capable to serve as "rules of thumb" and will be widely applied in evidence-based medicine.

This study aimed to explore the fetal heart rate trend in the week before birth, the transition from fetal to neonatal heart rate, and the neonatal heart rate trend in the week after birth in preterm neonates admitted to a tertiary care hospital, considering maternal and neonatal factors.A retrospective cohort study was conducted, including neonates born between 24 and 34 weeks of gestation. Baseline heart rate, average deceleration capacity, standard deviation, skewness, and sample entropy were assessed using interrupted time series and difference-in-differences analyses. Subgroup analyses were performed according to gestational age at birth, sex, birth weight, mode of delivery, Apgar score at 5 minutes, umbilical cord pH, and neonatal medication.The fetal and neonatal heart rate of 123 patients was analyzed. After birth, step change of baseline (2.23 bpm, p < 0.05), average deceleration capacity (0.43 bpm, p < 0.001), and skewness (0.47 nu, p < 0.001) increased, while sample entropy (-0.68 bits, p < 0.001) and standard deviation (-1.15 bpm, p < 0.001) decreased. Postnatally, baseline increased in linear slope and decreased in quadratic slope (both p < 0.001). Average deceleration capacity decreased in linear slope (p < 0.001). Sample entropy and standard deviation increased in linear slopes (both p < 0.001). Skewness increased in quadratic slope (p < 0.05). Subgroup analyses revealed that delivery mode, medication, and birth weight modulated these trends.This study provides unique insights into heart rate frequency and variability trends during the period around preterm birth. It highlights the dynamic physiological adaptation that occurs during the transition from intrauterine to extrauterine life in preterm infants and may help inform future research on fetal and neonatal monitoring and clinical management. · Heart rate frequency and variability dynamics were assessed around preterm birth in a NICU cohort.. · After birth, heart rate frequency increased while variability measures decreased.. · Heart rate frequency and variability differed by gestational age, birth weight, medication, and delivery mode..

This paper examines the problem of testing hypotheses about two population means. Most testing hypotheses about two population means, based on the assumption that each random sample drawn from a normal population. In this paper studied the problem of testing hypotheses about two population means are not using the assumption of normality and homogeneity. The method used to test hypotheses about two population means are the bootstrap method. The basic idea of the bootstrap method of repeated sampling with replacement from the sample data. Replication of samples used to determine the point estimator of Achieved Significance Level (ASL). Performance of the bootstrap method was tested using simulated data. The test results show that the bootstrap method to test the hypothesis of two population means well. Furthermore bootstrap method implemented on real data.

Impact of Poverty and Sickle Cell Disease on the Health-Related Quality of Life of Children.