Calculation Of Standard Errors Research Articles

Using the hyperbolic distribution to estimate the precision of size distribution percentiles of fluvial gravels

The hyperbolic distribution is fitted to published grain-size data collected from the surface of two gravel-bed rivers (the Fraser and Mamquam River data sets of Rice and Church). Our parametric approach enables calculation of standard errors for estimates of percentiles, as an alternative to the use of the bootstrap for this purpose. For estimation, we have used the statistical package R and the advantages of this software for this type of analysis are highlighted in this paper.

Fisher Information Matrix of the Dirichlet-multinomial Distribution

In this paper we derive explicit expressions for the elements of the exact Fisher information matrix of the Dirichlet-multinomial distribution. We show that exact calculation is based on the beta-binomial probability function rather than that of the Dirichlet-multinomial and this makes the exact calculation quite easy. The exact results are expected to be useful for the calculation of standard errors of the maximum likelihood estimates of the beta-binomial parameters and those of the Dirichlet-multinomial parameters for data that arise in practice in toxicology and other similar fields. Standard errors of the maximum likelihood estimates of the beta-binomial parameters and those of the Dirichlet-multinomial parameters, based on the exact and the asymptotic Fisher information matrix based on the Dirichlet distribution, are obtained for a set of data from Haseman and Soares (1976), a dataset from Mosimann (1962) and a more recent dataset from Chen, Kodell, Howe and Gaylor (1991). There is substantial difference between the standard errors of the estimates based on the exact Fisher information matrix and those based on the asymptotic Fisher information matrix.

Standard errors and covariance matrices for smoothed rank estimators

A 'pseudo-Bayesian' interpretation of standard errors yields a natural induced smoothing of statistical estimating functions. When applied to rank estimation, the lack of smoothness which prevents standard error estimation is remedied. Efficiency and robustness are preserved, while the smoothed estimation has excellent computational properties. In particular, convergence of the iterative equation for standard error is fast, and standard error calculation becomes asymptotically a one-step procedure. This property also extends to covariance matrix calculation for rank estimates in multi-parameter problems. Examples, and some simple explanations, are given.

Evaluating Disease Management Program Effectiveness

Disease management (DM) program evaluations are somewhat limited in scope because of typically small sample sizes comprising important subsets of the treated population. Identifying subsets of the data that have differing results from the aggregate of the whole program can lend insight into where, when, and how the program achieves its results. Additionally, there is a very limited set of classical tools available for the smaller sample sizes typically encountered in DM. Without readily available standard error and confidence interval (CI) calculations, the analyst may be fooled by specious details. A method called the ‘bootstrap’ is introduced as a suitable technique for allowing DM program evaluators to use a broader array of quantities of interest and to extend inferences to the population based on results achieved in the program. The bootstrap uses the power of modern computers to generate many random samples from a given data set, allowing the use of repeated samples’ statistic (e.g. mean, proportion, and median). Using a congestive heart failure (CHF) program as an example, the bootstrap technique is used to extend a DM program evaluation beyond questions addressed using classical statistical inference: (i) how much of a median cost decrease can be expected as a result of the program?; (ii) did the program impact the highest and lowest costing members equally; and (iii) how much of a decrease in the proportion of patients experiencing a hospitalization can be expected as a result of the program? The potential advantages of the bootstrap technique in DM program evaluation were clearly illustrated using this small CHF program example. A more robust understanding of program impact is possible when more tools and methods are available to the evaluator. This is particularly the case in DM, which is inherently biased in case-mix (e.g. strive to enroll sickest first), often has skewed distributions or outliers, and may suffer from small sample sizes. The bootstrap technique creates distributions that allow for a more accurate method of drawing statistical inferences of a population. Moreover, since classical statistical inference techniques were designed specifically for parametric statistics (i.e. assuming a normal distribution), the bootstrap can be used for measures that have no convenient statistical formulae. Additionally, CIs can be defined around this statistic, making it a viable option for evaluating DM program effectiveness.

Building an identifiable latent class model with covariate effects on underlying and measured variables

In recent years, latent class models have proven useful for analyzing relationships between measured multiple indicators and covariates of interest. Such models summarize shared features of the multiple indicators as an underlying categorical variable, and the indicators' substantive associations with predictors are built directly and indirectly in unique model parameters. In this paper, we provide a detailed study on the theory and application of building models that allow mediated relationships between primary predictors and latent class membership, but that also allow direct effects of secondary covariates on the indicators themselves. Theory for model identification is developed. We detail an Expectation-Maximization algorithm for parameter estimation, standard error calculation, and convergent properties. Comparison of the proposed model with models underlying existing latent class modeling software is provided. A detailed analysis of how visual impairments affect older persons' functioning requiring distance vision is used for illustration.

The development and evaluation of a software prototype for computer-adaptive testing

This paper presents ongoing research at the University of Hertfordshire on the use of computer-adaptive tests (CATs) in Higher Education. A software prototype based on Item Response Theory has been developed and is described here. This application was designed to estimate the level of proficiency in English for those students whose first language is not English. Academic staff and students evaluated the prototype introduced here and we summarise their attitude to the user interface and to pedagogical aspects of the prototype. We provide evidence that learners are not disadvantaged by the CAT approach, based on a comparison of performance between CAT and computer-based tests. A group of international students also took part in a focus group session after using the software. During this session, students discussed issues related to computer-adaptive tests, ranging from their perception that very easy tests are “meaningless” to their insights into the fairness of such computer-assisted assessments. In addition, this paper outlines how our current work will be developed further by implementing multimedia resources, developing more subjective tests and adding a stop condition associated with the calculation of standard error. Finally, the benefits and potential limitations of this method of assessment are also presented here.

Assessing the informational value of parameter estimates in cognitive models.

Mathematical models of cognition often contain unknown parameters whose values are estimated from the data. A question that generally receives little attention is how informative such estimates are. In a maximum likelihood framework, standard errors provide a measure of informativeness. Here, a standard error is interpreted as the standard deviation of the distribution of parameter estimates over multiple samples. A drawback to this interpretation is that the assumptions that are required for the maximum likelihood framework are very difficult to test and are not always met. However, at least in the cognitive science community, it appears to be not well known that standard error calculation also yields interpretable intervals outside the typical maximum likelihood framework. We describe and motivate this procedure and, in combination with graphical methods, apply it to two recent models of categorization: ALCOVE (Kruschke, 1992) and the exemplar-based random walk model (Nosofsky & Palmeri, 1997). The applications reveal aspects of these models that were not hitherto known and bring a mix of bad and good news concerning estimation of these models.

A Note on Inference of Trait Associations with SNP Haplotypes and Other Attributes in Generalized Linear Models

Recently, Lake et al. [Human Heredity 2003;55:56–65] have proposed an approach based on the EM algorithm for maximum-likelihood inference of trait associations with haplotypes and environmental cofactors in generalized linear models. In this short report, we describe an extension to accommodate missing SNP genotype information. We also discuss differences in the calculation of standard errors between their implementation and our own. Finally, we present results indicating that inference is robust to low levels of dependence between haplotypes and nongenetic factors, but that biased inference can result when there is moderate to strong dependence. Overall, the method is found to perform well in the models we considered.

Model-Free Impulse Responses

This paper introduces methods for computing impulse response functions that do not require specification and estimation of the unknown dynamic multivariate system itself. The central idea behind these methods is to estimate flexible local projections at each period of interest rather than extrapolating into increasingly distant horizons from a given model, as it is usually done in vector autoregressions (VAR). The advantages of local projections are numerous: (1) they can be estimated by simple regression techniques with standard regression packages; (2) they are more robust to misspecification; (3) standard error calculation is direct; and (4) they easily accommodate experimentation with highly non-linear and flexible specifications that may be impractical in a multivariate context. Therefore, these methods are a natural alternative to estimating impulse responses from VARs. An application to a simple, closed-economy monetary model suggests that the output loss and inflation effects of an interest rate shock depend on the stage of the business cycle.

A visual method for indexing muskrat populations

Abstract The native range for muskrats (Ondatra zibethicus) includes much of North America, but they also have been introduced beyond their native range, including into the Fall River, California, where they have come into conflict with human interests. An easily applied method to assess their abundance is an important need for their management. We developed a muskrat visual index (MVI) to provide the information necessary to address this need. Observations were made at randomly located sites along the river. The number of muskrats observed during a 45 min period was recorded during the late afternoon peak activity time at each site on multiple days. The mean number observed over sites was calculated for each day. The index was the mean of the daily means. These design and measurement methods present valuable advantages over most traditional muskrat indexing methods in this environment. Traditional methods usually involve counting burrows or houses. However, in a relatively stable environment such as along the Fall River, muskat burrows and houses tend to be long-lasting structures, making acute changes in population difficult to detect by these methods. Examining these structures for activity can be time-consuming and labor-intensive. Of particular importance, the statistical properties inherent to the MVI data structure permit calculation of standard errors, confidence intervals and statistical tests allowing quantitative comparisons among MVI values. Development of a management program for muskrats on the Fall River will require understanding of muskrat population fluctuations and densities, as well as knowledge of the effectiveness (short- and long-term) of control strategies. Here we develop a useful method, derive its statistical properties, and present baseline information for managing muskrats along the Fall River.

Modified Vestibulectomy Is Inadequate Treatment for Secondary Vulvar Vestibulitis

Surgical resection of the vestibule is the treatment of choice when medical therapy of vulvar vestibulitis fails. We wished to determine whether a limited form of vestibulectomy could adequately treat the symptoms of vulvar vestibulitis limited to the posterior vestibule. A series of 27 patients with vulvar vestibulitis limited to the posterior vestibule were treated with modified vestibulectomy by the lead author in a private practice setting. Results were evaluated by calculation of standard error of the mean and 95% confidence intervals where appropriate. The procedure was associated with no surgical complications. Fourteen (52%; 95% CI 33.1, 70.7) of patients attained an immediate complete response with follow-up of up to 14 years. Five (19%; 95% CI 3.8, 33.2) of patients had no improvement in symptoms following surgery. Symptoms recurred in 8 (30%; 95% CI 12.5, 46) of patients within 2 years of treatment. Recurrent pain required repeat surgery or topical therapy to attain a satisfactory result. Modif...

Stagewise estimation for regression analysis when independent variables are latent variables

Psychological research often deals with psychological constructs that cannot be directly measured. Thus independent variables of regression analysis for an observable dependent variable are sometimes latent variables (factors) that are defined independently of the dependent variable. In this study we pointed out the problem associated with the use of factor analysis for the combined set of dependent variable and independent variables in such a cases; that is, the derived factors are different from those originally intended, and the true regression parameters cannot be reproduced. We proposed a stagewise estimation method to solve the problem. This method estimates parameters of measurement equation in the first stage, and then estimates parameters of structural equation in the second stage. Our proposed method enables calculation of standard errors of estimators using Bootstrapping method. Numerical studies showed that the proposed method improves the estimation efficiency over the conventional methods, and provides estimates which are robust with respect to misspecification of model.

On the applicability of regenerative simulation in Markov chain Monte Carlo

We consider the central limit theorem and the calculation of asymptotic standard errors for the ergodic averages constructed in Markov chain Monte Carlo. Chan & Geyer (1994) established a central limit theorem for ergodic averages by assuming that the underlying Markov chain is geometrically ergodic and that a simple moment condition is satisfied. While it is relatively straightforward to check Chan & Geyer's conditions, their theorem does not lead to a consistent and easily computed estimate of the variance of the asymptotic normal distribution. Conversely, Mykland et al. (1995) discuss the use of regeneration to establish an alternative central limit theorem with the advantage that a simple, consistent estimator of the asymptotic variance is readily available. However, their result assumes a pair of unwieldy moment conditions whose verification is difficult in practice. In this paper, we show that the conditions of Chan & Geyer's theorem are sufficient to establish the central limit theorem of Mykland et al. This result, in conjunction with other recent developments, should pave the way for more widespread use of the regenerative method in Markov chain Monte Carlo. Our results are illustrated in the context of the slice sampler.

Comment on “Estimating the Extent of Racially Polarized Voting in Multicandidate Contests” by Bernard Grofman and Michael Migalski

Double regression figures prominently in the analysis of racially polarized voting. Grofman and Migalski attempt three extensions of this technique: the application to multimember districts, the calculation of standard errors for the parameters of interest, and validation through comparisons with seemingly unrelated regression (SUR). All three extensions fail. The first is based on an arithmetical error. The second is based on an incomplete specification of the underlying statistical model. The third is based on a mistaken application of SUR in a context in which it is guaranteed to yield the same results as the ordinary least squares estimates of double regression.

Structuring and Measuring the Size of Business Markets

A review of business or ‘industrial’ markets research and how it differs from consumer research. Covers: a brief historical background to business research; how to define business markets; business market structures; business universes, sampling and statistics; collecting usage information; response rates and dealing with non-response; standard error calculations.

Polylink: to support two-point linkage analysis in autotetraploids.

Polylink runs under Microsoft Windows (95 or later). It performs various calculations that are useful for investigating two-point linkage analysis for autopolyploids, based on the random chromosome pairing model. These include calculation of offspring phenotypic probabilities as functions of the recombination fraction, calculation of theoretical standard errors for the maximum likelihood estimator of and numerical computation of maximum likelihood estimates. It also includes simulation facilities. Polylink is free and available from Xiangming Xu via email

A computationally efficient method for obtaining standard error estimates for the promax and related solutions.

A computationally efficient algorithm for computing the asymptotic standard errors for the promax factor solution is proposed. The algorithm covers promax rotation with or without row normalization of the pre-rotated factor matrix. It also covers situations with either even- or odd-powered promax targets. With some modifications, the algorithm applies to Procrustean rotations with fixed or random independent targets. Simulation results show that the algorithm provides reasonable approximate standard errors for the promax solution with N = 200. In a real data example, the numerical results of the standard error computation using the proposed algorithm match those of an existing method based largely on the augmented information approach. The reasons why the proposed algorithm is more efficient than the augmented information approach are discussed.

Statistics for clinicians: 4: Basic concepts of statistical reasoning: hypothesis tests and the t-test.

In the previous article in this series, 1 we introduced the concept of sampling variability, and showed how it underlies the calculation and interpretation of standard errors and confidence intervals (CIs). An example that was used to illustrate these ideas was the comparison of verbal IQ at age 5 years between a group of children who were born with extremely low birthweight (< 1000 g) and another group of children of birthweight 1000‐1500 g (see Fig. 3 of the previous article). Although there was a mean difference of 5.5 IQ points (94.7 vs 100.2, more than one-third of a standard deviation) between the two groups, 95% CIs for the two mean values overlapped quite substantially. A natural question is whether the observed difference reflects a real underlying difference or not. The overlapping CIs perhaps suggest that the answer is ‘no’, but the reader should be warned that examining overlap of CIs is not equivalent to formal hypothesis testing of differences between groups, which is the traditional approach to this sort of question. (We shall return later to the relationship between hypothesis testing and confidence intervals.) This article explains the basic logic of hypothesis tests and in particular discusses the t-test, one of the more common statistical tests, encountered in about one-third of original articles in the Journal of Paediatrics and Child Health. 2

Estimating and testing bioconcentration factors

Bioconcentration factors (BCFs) are commonly calculated to represent the equilibrium concentration of a substance in an organism relative to environmental concentrations of the same substance. The BCF is derived from parameters estimated in uptake and elimination experiments and is presented as a single value without error estimates or confidence intervals. However, it is desirable to know the variability/precision of the BCF estimate and to statistically compare BCFs among experimental conditions. In this study, the calculation of standard errors and confidence intervals for BCFs is presented. In addition, a statistical method for formally comparing the BCFs derived under two or more experimental conditions is discussed. These methods are illustrated using data from a study of DDT-exposed Hyalella azteca and Diporeia spp.

Evaluating screening questionnaires using receiver Operating Characteristic (ROC) curves from two‐phase (double) samples

AbstractThe characteristics of psychiatric screening tests (for example, sensitivity, specificity, and AUC – the area under an ROC curve) are frequently assessed using data arising from two‐phase samples. Too often, however, the statistical methods that are used are incorrect. They do not appropriately account for the sampling design. Valid methods for the estimate of sensitivity, specificity and, in particular, the AUC, together with its standard error, are discussed in detail and a Stata macro for the implementation of these methods is provided. Simple weighting procedures are used to correct for verification biases arising from the two‐phase design, together with bootstrap or jackknife sampling for the calculation of valid standard errors. Copyright © 2000 Whurr Publishers Ltd.