Interval estimation of the mean difference in the analysis of over‐dispersed count data

This paper focuses on the development and study of the confidence interval procedures for mean difference between two treatments in the analysis of over‐dispersed count data in order to measure the efficacy of the experimental treatment over the standard treatment in clinical trials. In this study, two simple methods are proposed. One is based on a sandwich estimator of the variance of the regression estimator using the generalized estimating equations (GEEs) approach of Zeger and Liang (1986) and the other is based on an estimator of the variance of a ratio estimator (1977). We also develop three other procedures following the procedures studied by Newcombe (1998) and the procedure studied by Beal (1987). As assessed by Monte Carlo simulations, all the procedures have reasonably well coverage properties. Moreover, the interval procedure based on GEEs outperforms other interval procedures in the sense that it maintains the coverage very close to the nominal coverage level and that it has the shortest interval length, a satisfactory location property, and a very simple form, which can be easily implemented in the applied fields. Illustrative applications in the biological studies for these confidence interval procedures are also presented.     

Relative Efficiency of Estimators of the Mean of a Normal Distribution when Coefficient of Variation is Known

A biased but simple and consistent estimator of the parameter ? has been obtained for the normal distribution N(?, a?₂), ?>0 where a is a known constant. It is shown that the estimator is more efficient than the sample mean or any suitably chosen constant multiple of the sample standard deviation. It is also proved to be more efficient than the mimumum variance unbiased estimator among a typical class of unbiased estimators derived by RASUL KHAN (1968).     

Statistical characteristics of spontaneous and evoked unit activity in the cat superior colliculus

Spontaneous and evoked unit activity of the superior colliculus was studied in cats immobilized with gallamine or with the spinal cord divided (encephalé isolé). Statistical parameters of unit activity were calculated in light and darkness and in the presence of a moving photic stimulus. Spontaneously active neurons were divided into four main groups: 1) with a mean interspike interval of about 300 msec, coefficient of variation 1.9, and with two modes in the autocorrelation histogram; 2) with similar statistical parameters but distinguished by correlation with saccadic eye movements; 3) histogram of interspike intervals exponential in type, autocorrelation histogram flat, mean interval and standard deviation about 100 msec; 4) with symmetrical distribution of interspike intervals, mean interval about 50 msec, coefficient of variation 0.4–0.5. Spontaneous activity in light and darkness differed only in the neurons of the second group. Comparison of the statistical parameters of the spontaneous and evoked activity shows that encoding of the stimulus could be carried out by neurons of groups 3 and 4. In some cases the mode of the histogram of interspike intervals and the standard deviation changed significantly and activity appeared regularly in the autocorrelation histogram.     

Quantifying Proportional Variability

Real quantities can undergo such a wide variety of dynamics that the mean is often a meaningless reference point for measuring variability. Despite their widespread application, techniques like the Coefficient of Variation are not truly proportional and exhibit pathological properties. The non-parametric measure Proportional Variability (PV) [1] resolves these issues and provides a robust way to summarize and compare variation in quantities exhibiting diverse dynamical behaviour. Instead of being based on deviation from an average value, variation is simply quantified by comparing the numbers to each other, requiring no assumptions about central tendency or underlying statistical distributions. While PV has been introduced before and has already been applied in various contexts to population dynamics, here we present a deeper analysis of this new measure, derive analytical expressions for the PV of several general distributions and present new comparisons with the Coefficient of Variation, demonstrating cases in which PV is the more favorable measure. We show that PV provides an easily interpretable approach for measuring and comparing variation that can be generally applied throughout the sciences, from contexts ranging from stock market stability to climate variation.     

Statistical methods suitable for the analysis of plant tissue culture data

Statistical analyses are an essential part of biological research. Statistical methods are available to biological researchers that range from very simple to extremely complex. Therefore, caution should be used when selecting a statistical method. When possible it is best to avoid complicated statistical procedures that are difficult to interpret and may hinder the researcher's ability to make treatment comparisons. Instead a method should be chosen that compliments a logical and practical treatment design. Statistics should be used as a tool to compare treatments of interest and should not dictate the treatments. Experimental designs should take into account the eventual analysis, otherwise one could conceive of a design that could not be analyzed or, when analyzed, would not answer the desired questions. Therefore, time should be spent before conducting an experiment to plan an experimental design and analysis that best compliments the treatment scheme and questions to be answered. The purpose of this paper is to present examples of experimental designs, means separation procedures, data transformations and presentation methods suitable for plant cell and tissue culture data.Abbreviations ANOVA analysis of variance - BA benzyladenine - CV coefficient of variation - DF degrees of freedom - IAA indole-3-acetic acid - IBA indole-3-butyric acid - LOF lack-of-fit - MSE mean square error - P-ITB phenyl indole-3-thiolobutyrate - S standard deviation - SE standard error of the mean - TDZ thidiazuron     

Variability in mammalian dentitions: size-related bias in the coefficient of variation

The coefficient of variation has been used in many evolutionary studies. However, a strong negative correlation between this index and size may artificially inflate the apparent variability of small traits. This is most pronounced when variables whose size differs by more than an order of magnitude are compared or when the index is applied to variables whose size is within an order of magnitude of their measurement error. When this is likely to affect conclusions, other indexes of variability should be considered. One alternative is to use the standard deviation of log-transformed data; however, this index is sometimes still correlated with variable size so care should be exercised in its use. Another alternative is to regress the standard deviation onto mean variable size; however, this method may also be misleading if variables are not randomly distributed about the regression line. As an example of the effect of bias in the coefficient of variation, previous studies of mammalian dental variability profiles were re-evaluated. It was found that variation among teeth is relatively homogenous, both within and among species. The exception is that the canines of some mammalian species have variability that is considerably higher than would be expected from tooth size. Previous explanations of variability patterns that invoked developmental fields are incompatible with the new data.     

Tooth Size Variation in Pinniped Dentitions

It is contentious whether size variation among mammalian teeth is heterogeneous or homogeneous, whether the coefficient of variation is reliable, and whether the standard deviation of log-transformed data and the residual of standard deviation on mean variable size are useful replacements for the coefficient of variation. Most studies of tooth size variation have been on mammals with complex-crowned teeth, with relatively little attention paid to taxa with simple-crowned teeth, such as Pinnipedia. To fill this gap in knowledge and to resolve the existing controversies, we explored the variation of linear size variables (length and width) for all teeth from complete permanent dentitions of four pinniped species, two phocids (Histriophoca fasciata, Phoca largha) and two otariids (Callorhinus ursinus, Eumetopias jubatus). Size variation among these teeth was mostly heterogeneous both along the toothrow and among species. The incisors, canines, and mesial and distal postcanines were often relatively highly variable. The levels of overall dental size variation ranged from relatively low as in land carnivorans (Phoca largha and both otariids) to high (Histriophoca fasciata). Sexual size dimorphism varied among teeth and among species, with teeth being, on average, larger in males than in females. This dimorphism was more pronounced, and the canines were larger and more dimorphic relative to other teeth in the otariids than in the phocids. The coefficient of variation quantified variation reliably in most cases. The standard deviation of log-transformed data was redundant with the coefficient of variation. The residual of standard deviation on mean variable size was inaccurate when size variation was considerably heterogeneous among the compared variables, and was incomparable between species and between sexes. The existing hypotheses invoking developmental fields, occlusal complexity, and the relative timing of tooth formation and sexually dimorphic hormonal activity do not adequately explain the differential size variation along the pinniped toothrow.     

Vagal modulation of the heart and central hemodynamics during handgrip exercise

Blood pressure and continuous electrocardiogram recordings were obtained from 12 participants during spontaneous breathing (SB1), dynamic handgrip exercise at 20% (HG(20)) of maximal voluntary contraction (MVC), and spontaneous breathing (SB2) and dynamic handgrip exercise at 60% (HG(60)) of MVC. Repeated-measures ANOVAs were used to examine the effects of the exercise conditions on mean arterial pressure (MAP), on mean standard deviation (SDNN), and on the coefficient of variation of R-R intervals. The mean R-R interval responded to exercise in an intensity-dependent manner. SDNN decreased with exercise but was not intensity dependent. Coefficient of variation decreased during HG(20), and MAP increased following HG(60). These data are consistent with the notion that changes in cardiovascular function with low-intensity exercise are primarily mediated by parasympathetic withdrawal, and as exercise intensity increases, additional cardiovascular reactivity is mediated by increased sympathetic outflow. The change in the coefficient of variation from rest to exercise was unique in comparison to the changes in SDNN, and this merits further investigation.     

Effect of covariate omission in Weibull accelerated failure time model: A caution

The accelerated failure time model is presented as an alternative to the proportional hazard model in the analysis of survival data. We investigate the effect of covariates omission in the case of applying a Weibull accelerated failure time model. In an uncensored setting, the asymptotic bias of the treatment effect is theoretically zero when important covariates are omitted; however, the asymptotic variance estimator of the treatment effect could be biased and then the size of the Wald test for the treatment effect is likely to exceed the nominal level. In some cases, the test size could be more than twice the nominal level. In a simulation study, in both censored and uncensored settings, Type I error for the test of the treatment effect was likely inflated when the prognostic covariates are omitted. This work remarks the careless use of the accelerated failure time model. We recommend the use of the robust sandwich variance estimator in order to avoid the inflation of the Type I error in the accelerated failure time model, although the robust variance is not commonly used in the survival data analyses.     

A Monte Carlo simulation of the determination of mean particle volume using the Cavalieri estimator

BACKGROUND: A common morphometric problem is the determination of an estimate of the size of biological particles obtained from measurements made on a sample of profiles observed in sections. Results are reported typically in terms of mean caliper diameter or mean volume of the particle. METHODS: We have investigated the use of the Cavalieri estimator for obtaining estimates of mean particle volume using a Monte Carlo simulation. Samples of spherical and ellipsoidal particles were generated by computer and serially sectioned at a fixed mean thickness with a small, imposed random variation. The area of each profile was determined and the volume of the particle was calculated according to the Cavalieri estimator. The influence on the estimate of the mean particle volume and its 95% confidence interval was evaluated for several variables: the shape of the particles, the standard deviation of the particle volume in the population, the section thickness, and the standard deviation of the section thickness. RESULTS: The results obtained with the Cavalieri estimator correspond favorably with those obtained with previously reported alternative methods. This leads to a recommendation for the strong consideration for the use of the Cavalieri estimator in cases in which it is technically feasible to obtain at least three sections through the individual particles. Graphs are provided, which relate the confidence interval for the mean volume to the number of particles measured.     

Optimal sampling and estimation strategies under the linear model

In some cases model-based and model-assisted inferences canlead to very different estimators. These two paradigms are notso different if we search for an optimal strategy rather thanjust an optimal estimator, a strategy being a pair composedof a sampling design and an estimator. We show that, under alinear model, the optimal model-assisted strategy consists ofa balanced sampling design with inclusion probabilities thatare proportional to the standard deviations of the errors ofthe model and the Horvitz–Thompson estimator. If the heteroscedasticityof the model is 'fully explainable’ by the auxiliary variables,then this strategy is also optimal in a model-based sense. Moreover,under balanced sampling and with inclusion probabilities thatare proportional to the standard deviation of the model, thebest linear unbiased estimator and the Horvitz–Thompsonestimator are equal. Finally, it is possible to construct asingle estimator for both the design and model variance. Theinference can thus be valid under the sampling design and underthe model.     

On the covariate-adjusted estimation for an overall treatment difference with data from a randomized comparative clinical trial

To estimate an overall treatment difference with data from a randomized comparative clinical study, baseline covariates are often utilized to increase the estimation precision. Using the standard analysis of covariance technique for making inferences about such an average treatment difference may not be appropriate, especially when the fitted model is nonlinear. On the other hand, the novel augmentation procedure recently studied, for example, by Zhang and others (2008. Improving efficiency of inferences in randomized clinical trials using auxiliary covariates. Biometrics 64, 707-715) is quite flexible. However, in general, it is not clear how to select covariates for augmentation effectively. An overly adjusted estimator may inflate the variance and in some cases be biased. Furthermore, the results from the standard inference procedure by ignoring the sampling variation from the variable selection process may not be valid. In this paper, we first propose an estimation procedure, which augments the simple treatment contrast estimator directly with covariates. The new proposal is asymptotically equivalent to the aforementioned augmentation method. To select covariates, we utilize the standard lasso procedure. Furthermore, to make valid inference from the resulting lasso-type estimator, a cross validation method is used. The validity of the new proposal is justified theoretically and empirically. We illustrate the procedure extensively with a well-known primary biliary cirrhosis clinical trial data set.     

Unbiased estimation of gene diversity in samples containing related individuals: exact variance and arbitrary ploidy

Gene diversity, a commonly used measure of genetic variation, evaluates the proportion of heterozygous individuals expected at a locus in a population, under the assumption of Hardy-Weinberg equilibrium. When using the standard estimator of gene diversity, the inclusion of related or inbred individuals in a sample produces a downward bias. Here, we extend a recently developed estimator shown to be unbiased in a diploid autosomal sample that includes known related or inbred individuals to the general case of arbitrary ploidy. We derive an exact formula for the variance of the new estimator, H, and present an approximation to facilitate evaluation of the variance when each individual is related to at most one other individual in a sample. When examining samples from the human X chromosome, which represent a mixture of haploid and diploid individuals, we find that H performs favorably compared to the standard estimator, both in theoretical computations of mean squared error and in data analysis. We thus propose that H is a useful tool in characterizing gene diversity in samples of arbitrary ploidy that contain related or inbred individuals.     

Evaluating the mean frequency of responding cells in limiting dilution assays

Summary The validity of limiting dilution assays can be compromised or negated by the use of statistical methodology which does not consider all issues surrounding the biological process. This study critically evaluates statistical methods for estimating the mean frequency of responding cells in multiple sample limiting dilution assays. We show that methods that pool limiting dilution assay data, or samples, are unable to estimate the variance appropriately. In addition, we use Monte Carlo simulations to evaluate an unweighted mean of the maximum likelihood estimator, an unweighted mean based on the jackknife estimator, and a log transform of the maximum likelihood estimator. For small culture replicate size, the log transform outperforms both unweighted mean procedures. For moderate culture replicate size, the unweighted mean based on the jackknife produces the most acceptable results. This study also addresses the important issue of experimental design in multiple sample limiting dilution assays. In particular, we demonstrate that optimization of multiple sample limiting dilution assays is achieved by increasing the number of biological samples at the expense of repeat cultures.     

The arithmetic mean of ratio-normalized experiments overestimates the true E/C ratio

Biological assays often suffer from large systematic variation between sets of experiments. This variation is sometimes countered by normalizing the results of an "exposed" (E) experiment to that of a simultaneously performed "control" (C). We demonstrate that the arithmetic mean of such ratios overestimates the "true" E/C ratio. Fortunately, the overestimation may be calculated from experimentally accessible information, and it is generally possible to correct for this factor using formulas presented in this paper. We have studied the impact of this effect on a set of studies in the bioelectromagnetics literature and find that, although most results are weakened by the correction, few are significantly altered. Some of the papers used for our literature study are controversial; we believe that the present study may strengthen the quoted results by removing doubts about the statistical treatment of E/C ratios. Both false positives and negatives are possible if the proper correction is not made to the arithmetic mean of a set of E/C data. Realistic examples of erroneous statistical conclusions demonstrate that this is a real concern for E/C data which are marginal in both magnitude (mean < 2) and variance (standard deviation > 0.5).     

A mixed model-based variance estimator for marginal model analyses of cluster randomized trials

Generalized estimating equations (GEE) are used in the analysis of cluster randomized trials (CRTs) because: 1) the resulting intervention effect estimate has the desired marginal or population-averaged interpretation, and 2) most statistical packages contain programs for GEE. However, GEE tends to underestimate the standard error of the intervention effect estimate in CRTs. In contrast, penalized quasi-likelihood (PQL) estimates the standard error of the intervention effect in CRTs much better than GEE but is used less frequently because: 1) it generates an intervention effect estimate with a conditional, or cluster-specific, interpretation, and 2) PQL is not a part of most statistical packages. We propose taking the variance estimator from PQL and re-expressing it as a sandwich-type estimator that could be easily incorporated into existing GEE packages, thereby making GEE useful for the analysis of CRTs. Using numerical examples and data from an actual CRT, we compare the performance of this variance estimator to others proposed in the literature, and we find that our variance estimator performs as well as or better than its competitors.     

Weighted Multivariate Regression Estimates Solved by Random Effects Approach

The paper deals with the random effects model, where the expectation vector and the covariance matrix of the effect influencing the population are to be estimated. The iterated estimator of expectation vector is derived, based on the invariant estimator of the combined covariance matrix, and some of its statistical properties are shown.     

Estimating the size of the dividing stem cell pool after allogeneic bone marrow transplantation

Currently the problem of estimation of the number of pluripotent stem cells reconstituting marrow grafts following bone marrow transplantation, or studies looking at questions of clonal dominance in hematopoietic cell populations, rely on indirect measurement and a simple application of the formula for the sampling variation of a binomial proportion. This approach, from a statistical viewpoint, can be seen to be flawed. It is very easily remedied though and only requires appropriate use of variance stabilizing transformations. These lead to a very simple estimator for the number of hematopoietic stem cells involved in repopulating the marrow and require little in the way of additional calculation. We give the distribution theory for this estimator as well as simple approximations for practical application. As an illustration we rework data recently gathered to address the question as to whether or not reconstitution of marrow grafts in the clinical setting is oligoclonal.     

Variance estimation for systematic designs in spatial surveys

Summary In spatial surveys for estimating the density of objects in a survey region, systematic designs will generally yield lower variance than random designs. However, estimating the systematic variance is well known to be a difficult problem. Existing methods tend to overestimate the variance, so although the variance is genuinely reduced, it is over‐reported, and the gain from the more efficient design is lost. The current approaches to estimating a systematic variance for spatial surveys are to approximate the systematic design by a random design, or approximate it by a stratified design. Previous work has shown that approximation by a random design can perform very poorly, while approximation by a stratified design is an improvement but can still be severely biased in some situations. We develop a new estimator based on modeling the encounter process over space. The new “striplet” estimator has negligible bias and excellent precision in a wide range of simulation scenarios, including strip‐sampling, distance‐sampling, and quadrat‐sampling surveys, and including populations that are highly trended or have strong aggregation of objects. We apply the new estimator to survey data for the spotted hyena (Crocuta crocuta) in the Serengeti National Park, Tanzania, and find that the reported coefficient of variation for estimated density is 20% using approximation by a random design, 17% using approximation by a stratified design, and 11% using the new striplet estimator. This large reduction in reported variance is verified by simulation.