Confidence Limits for the Ratio of Two Binomial Proportions Based on Likelihood Scores: Non-Iterative Method

The interval estimation of the ratio of two binomial proportions based on the score statistic is superior over other methods. Iterative algorithms for calculating the approximate confidence interval have been provided by, e.g., KOOPMAN (1984, Biometrics 40:513–517) and GART and NAM (1988a, Biometrics 44:323–338). This note presents the analytical solutions for upper and lower confidence limits in a closed form and gives examples for numerical illustration. The non-iterative method is generally more desirable than the iterative method.     

Evaluation of Mite-Away-II™ for fall control of <Emphasis Type="Italic">Varroa destructor</Emphasis> (Acari: Varroidae) in colonies of the honey bee <Emphasis Type="Italic">Apis mellifera</Emphasis> (Hymenoptera: Apidae) in the northeastern USA

Mite-Away II™, a recently-registered product with a proprietary formulation of formic acid, was evaluated under field conditions in commercial apiaries in upstate New York (USA) for the fall control of Varroa destructor Anderson & Trueman in colonies of the honey bee, Apis mellifera L. Ambient temperatures during the treatment period were in the lower half of the range recommended on the label, but were typical for early fall in upstate New York. Average mite mortality was 60.2 ± 2.2% in the Mite-Away II group and 23.3 ± 2.6% in the untreated control group. These means were significantly different from each other, but the level of control was only moderate. These results demonstrate that Mite-Away II may not always provide an adequate level of control even when the temperature at the time of application falls within the recommended range stated on the product’s label. To make the best use of temperature-sensitive products, I suggest that the current, single-value, economic treatment threshold be replaced with an economic treatment range. The limits for this range are specified by two pest density values. The lower limit is the usual pest density that triggers a treatment. The upper limit is the maximum pest density that one can expect to reduce to a level below the lower limit given the temperatures expected during the treatment period. When the actual pest density exceeds the upper limit, the product should not be recommended; or, a warning should be included indicating that acceptable control may not be achieved.     

Epizootic cutaneous papillomatosis in roach Rutilus rutilus: sex and size dependence,seasonal occurrence and between-population differences

Validation of a single round PCR-based assay to confirm as Myxobolus cerebralis myxospores obtained from pepsin-trypsin digest preparations is described. The assay is a modification of a PCR assay published previously, based on the amplification of a segment of the gene encoding the 18S ribosomal subunit of M. cerebralis. The sensitivity, specificity and upper and lower detection limits were determined using known M. cerebralis and non-M. cerebralis myxospores and M. cerebralis-free fish. The sensitivity of PCR confirmation was 100% (95% confidence interval of 83.2-100%). The specificity was 100% (95% confidence interval of 87.2-100%). The upper detection limit was approximately 100,000 myxospores per reaction; the lower detection limit was approximately 50 myxospores per reaction. Given the high sensitivity and specificity of the assay, substitution of this assay for histologic confirmation of M. cerebralis infection is encouraged.     

Responses to climate by tree-ring widths and maximum latewood densities of two Abies species at upper and lower altitudinal distribution limits in central Japan

This study examined the effects of climate on tree-ring widths and maximum latewood densities of Abies veitchii and Abies mariesii at the upper and lower distribution limits in central Japan. A. veitchii and A. mariesii dominated at the lower and upper parts of the subalpine zone, respectively. Residual chronologies of tree-ring width and maximum latewood density were developed for the two Abies species at the upper and lower distribution limits, and were compared with monthly mean temperatures and monthly sums of precipitation. Tree-ring widths of the two Abies species at the upper and lower distribution limits positively correlated with temperatures during the beginning of the dormant season and during the growing season of the current year, except for A. veitchii at the lower distribution limit, which showed no positive correlation with temperature. Maximum latewood densities of the two Abies species at the upper and lower distribution limits positively and negatively correlated with temperatures and precipitation, respectively, during the growing season of the current year. Therefore, tree-ring widths and maximum latewood densities of the two Abies species were sensitive to low temperature, except for the tree-ring width of A. veitchii at the lower distribution limit with the warmest thermal conditions along the altitude. Global warming is suggested to affect maximum latewood densities and tree-ring widths of the two Abies species along the altitude.     

A confidence interval for the unknown value of the regressor with given expectation of the regressand in model I of linear regression

In the case of model I of linear regression there is derived a confidence interval for that x_o where the “true line” will reach a given value y_o. The interval can be given by the intersections between the line y = y_o and the hyperbolas providing pointwise confidence intervals of the expectations of y.     

A new method for choosing sample size for confidence interval-based inferences

Scientists often need to test hypotheses and construct corresponding confidence intervals. In designing a study to test a particular null hypothesis, traditional methods lead to a sample size large enough to provide sufficient statistical power. In contrast, traditional methods based on constructing a confidence interval lead to a sample size likely to control the width of the interval. With either approach, a sample size so large as to waste resources or introduce ethical concerns is undesirable. This work was motivated by the concern that existing sample size methods often make it difficult for scientists to achieve their actual goals. We focus on situations which involve a fixed, unknown scalar parameter representing the true state of nature. The width of the confidence interval is defined as the difference between the (random) upper and lower bounds. An event width is said to occur if the observed confidence interval width is less than a fixed constant chosen a priori. An event validity is said to occur if the parameter of interest is contained between the observed upper and lower confidence interval bounds. An event rejection is said to occur if the confidence interval excludes the null value of the parameter. In our opinion, scientists often implicitly seek to have all three occur: width, validity, and rejection. New results illustrate that neglecting rejection or width (and less so validity) often provides a sample size with a low probability of the simultaneous occurrence of all three events. We recommend considering all three events simultaneously when choosing a criterion for determining a sample size. We provide new theoretical results for any scalar (mean) parameter in a general linear model with Gaussian errors and fixed predictors. Convenient computational forms are included, as well as numerical examples to illustrate our methods.     

Confidence intervals for directly standardized rates using mid‐p gamma intervals

Directly standardized rates continue to be an integral tool for presenting rates for diseases that are highly dependent on age, such as cancer. Statistically, these rates are modeled as a weighted sum of Poisson random variables. This is a difficult statistical problem, because there are k observed Poisson variables and k unknown means. The gamma confidence interval has been shown through simulations to have at least nominal coverage in all simulated scenarios, but it can be overly conservative. Previous modifications to that method have closer to nominal coverage on average, but they do not achieve the nominal coverage bound in all situations. Further, those modifications are not central intervals, and the upper coverage error rate can be substantially more than half the nominal error. Here we apply a mid‐p modification to the gamma confidence interval. Typical mid‐p methods forsake guaranteed coverage to get coverage that is sometimes higher and sometimes lower than the nominal coverage rate, depending on the values of the parameters. The mid‐p gamma interval does not have guaranteed coverage in all situations; however, in the (not rare) situations where the gamma method is overly conservative, the mid‐p gamma interval often has at least nominal coverage. The mid‐p gamma interval is especially appropriate when one wants a central interval, since simulations show that in many situations both the upper and lower coverage error rates are on average less than or equal to half the nominal error rate.     

Variation transmission model for setting acceptance criteria in a multi-staged pharmaceutical manufacturing process

Pharmaceutical manufacturing processes consist of a series of stages (e.g., reaction, workup, isolation) to generate the active pharmaceutical ingredient (API). Outputs at intermediate stages (in-process control) and API need to be controlled within acceptance criteria to assure final drug product quality. In this paper, two methods based on tolerance interval to derive such acceptance criteria will be evaluated. The first method is serial worst case (SWC), an industry risk minimization strategy, wherein input materials and process parameters of a stage are fixed at their worst-case settings to calculate the maximum level expected from the stage. This maximum output then becomes input to the next stage wherein process parameters are again fixed at worst-case setting. The procedure is serially repeated throughout the process until the final stage. The calculated limits using SWC can be artificially high and may not reflect the actual process performance. The second method is the variation transmission (VT) using autoregressive model, wherein variation transmitted up to a stage is estimated by accounting for the recursive structure of the errors at each stage. Computer simulations at varying extent of variation transmission and process stage variability are performed. For the scenarios tested, VT method is demonstrated to better maintain the simulated confidence level and more precisely estimate the true proportion parameter than SWC. Real data examples are also presented that corroborate the findings from the simulation. Overall, VT is recommended for setting acceptance criteria in a multi-staged pharmaceutical manufacturing process.     

The utilization of interphase cytogenetic analysis for the detection of mosaicism

This study describes a method for defining mosaic aneuploidy by interphase cytogenetics based on statistical limits established from control specimens. Fluorescence in situ hybridization (FISH) has been used to detect the number of copies of specific chromosomes in interphase nuclei from placental tissues of diploid controls and mosaic placentas. FISH was performed using probes D7Z1/D7Z2, D9Z1, D10Z1, and D18Z1, all purchased from Oncor, Inc. Statistical analysis of data obtained from diploid controls was used to determine the one-sided upper reference limit and corresponding 95% confidence interval for the proportion of cells with one and three signals for each of the probes used. The one-sided upper reference limits established the lower levels of monosomy and trisomy detectable using each of the four probes. These statistical parameters were then used to interpret the results obtained by FISH applied to the study of term placentas for the confirmation of prenatally diagnosed chromosomal mosaicism.     

COMPUTATION OF MODEL-DEPENDENT TOLERANCE BANDS FOR AMBULATORILY MONITORED BLOOD PRESSURE

The construction of time-specified reference limits requires systematic sampling in clinical health, particularly for those variables characterized by a circadian rhythm of large amplitude, as it is the case for blood pressure (BP). For the detection of false negatives, tolerance intervals (limits that will include at least a specified proportion of the population with a stated confidence) are important and should substitute when possible for prediction limits. We have previously described a nonparametric method for the computation of model-independent tolerance intervals that are constructed by first dividing the sampling range in several time spans in which no appreciable changes in population characteristics (namely, mean and variance) take place. The tolerance interval is then computed for each of the time spans. The limits thus computed, as well as results of any comparison of a given individual's profile against such tolerance intervals, are highly dependent on the sampling scheme of both the reference individuals and the test subject. To avoid this problem, we have developed an alternative method that allows the computation of model-dependent tolerance bands for hybrid time series. Assuming that a set X of longitudinal series monitored from a given group of reference individuals can be fitted with the same individual model, a population model C(X,t) can be also determined, as well as the deviation S(X,t) of each individual curve from the population model. The tolerance band will then have the form C(X,t) ± kS(X,t), where k is here estimated following a nonparametric approach based on bootstrap techniques. Alternatively, two different values of k can be estimated (for the lower and upper limits of the tolerance interval, respectively) in cases for which we cannot assume symmetry. The method is generally applicable for any population model describing the reference population (including the fit of multiple significant components, nonsinusoidal waveforms, and/or trends). The method was used to establish time-specified tolerance bands for time series of blood pressure monitored automatically in healthy individuals of both genders. Model-dependent intervals are preferred to the model-independent limits when reliance on a specified sampling rate needs to be avoided. These limits may serve for an objective and positive definition of health, for the screening and diagnosis of disease, and for gauging the subject's response to treatment. (Chronobiology International, 17(4), 567–582, 2000)     

Confidence bands for low-dose risk estimation with quantal response data

We study the use of simultaneous confidence bands for low-dose risk estimation with quantal response data, and derive methods for estimating simultaneous upper confidence limits on predicted extra risk under a multistage model. By inverting the upper bands on extra risk, we obtain simultaneous lower bounds on the benchmark dose (BMD). Monte Carlo evaluations explore characteristics of the simultaneous limits under this setting, and a suite of actual data sets are used to compare existing methods for placing lower limits on the BMD.     

Unbiased Confidence Intervals for the Odds Ratio of Two Independent Binomial Samples with Application to Case–Control Data

The problem of confidence interval construction for the odds ratio of two independent binomial samples is considered. Two methods of eliminating the nuisance parameter from the exact likelihood, conditioning and maximization, are described. A conditionally exact tail method exists by putting together upper and lower bounds. A shorter interval can be obtained by simultaneous consideration of both tails. We present here new methods that extend the tail and simultaneous approaches to the maximized likelihood. The methods are unbiased and applicable to case-control data, for which the odds ratio is important. The confidence interval procedures are compared unconditionally for small sample sizes in terms of their expected length and coverage probability. A Bayesian confidence interval method and a large-sample chi2 procedure are included in the comparisons.     

Exact one-sided confidence bounds for the risk ratio in 2 x 2 tables with structural zero

This paper examines exact one-sided confidence limits for the risk ratio in a 2 x 2 table with structural zero. Starting with four approximate lower and upper limits, we adjust each using the algorithm of Buehler (1957) to arrive at lower (upper) limits that have exact coverage properties and are as large (small) as possible subject to coverage, as well as an ordering, constraint. Different Buehler limits are compared by their mean size, since all are exact in their coverage. Buehler limits based on the signed root likelihood ratio statistic are found to have the best performance and recommended for practical use.     

Herbivory and pollen limitation at the upper elevational range limit of two forest understory plants of eastern North America

Studies of species' range limits focus most often on abiotic factors, although the strength of biotic interactions might also vary along environmental gradients and have strong demographic effects. For example, pollinator abundance might decrease at range limits due to harsh environmental conditions, and reduced plant density can reduce attractiveness to pollinators and increase or decrease herbivory. We tested for variation in the strength of pollen limitation and herbivory by ungulates along a gradient leading to the upper elevational range limits of Trillium erectum (Melanthiaceae) and Erythronium americanum (Liliaceae) in Mont Mégantic National Park, Québec, Canada. In T. erectum, pollen limitation was higher at the range limit, but seed set decreased only slightly with elevation and only in one of two years. In contrast, herbivory of T. erectum increased from <10% at low elevations to >60% at the upper elevational range limit. In E. americanum, we found no evidence of pollen limitation despite a significant decrease in seed set with elevation, and herbivory was low across the entire gradient. Overall, our results demonstrate the potential for relatively strong negative interactions (herbivory) and weak positive interactions (pollination) at plant range edges, although this was clearly species specific. To the extent that these interactions have important demographic consequences—highly likely for herbivory on Trillium, based on previous studies—such interactions might play a role in determining plant species' range limits along putatively climatic gradients.     

Stepwise Procedures under Unknown Variances for Toxicological Evaluation

Toxicological study is of practical importance in modern drug development. Proper statistical methodologies for toxicological evaluation of new developed drugs are undoubtedly necessary. In toxicological studies, it is practically desirable for a method to not declare the safety of a developed drug at a higher dosage prior to the declaration of the safety at lower dosages. Hsu and Berger 's stepwise confidence interval method was recently proposed for this purpose. Unfortunately, their procedure necessitates the homogeneity of variances among dosages, which is seldom satisfied in practice. In this article, via the application of the Stein 's two‐stage sampling method, we propose a stepwise confidence interval procedure for the same task without the homoscedasticity restriction. In addition, our procedure is shown to control its family‐wise type I error rate at the pre‐chosen nominal level. A simulation study will be conducted to compare our method, Hsu and Berger 's stepwise confidence interval method, and a single stage stepwise testing procedure based on Welch 's approximation. Our procedure is empirically shown to outperform Hsu and Berger 's procedure under heteroscedasticity and perform similarly with Welch 's procedure. An example will be used to illustrate our method.     

Confidence Intervals for the Risk Ratio in Cohort Studies under Inverse Sampling

Three simple interval estimates for the risk ratio in inverse sampling are considered. The first two interval estimates are derived on the basis of Fieller's Theorem and the delta method with the logarithmic transformation, respectively. The third interval estimate is derived on the basis of an F-test statistic proposed by BENNETT (1981) for testing equal probabilities of a disease between two comparison groups when the disease is rare. To evaluate the performance of these three methods, a Monte Carlo simulation is used to compare the actual coverage probability with the nominal confidence level for each method and to estimate the expected length of the corresponding confidence interval in a variety of situations. On the basis of the results found in the simulation, we have concluded that the method with the logarithmic transformation is either equivalent to or better than the other two methods for all situations considered here.     

An Improvement of the Westlake Symmetric Confidence Interval

A common statistical method for assessing bioequivalence of two formulations of a chemical substance is the symmetric confidence interval of WESTLAKE (1972). As mentioned by WEST -LAKE (1981) and SCHUIRMAN (1981) a more powerful method consists of two one-sided t-tests. An (1-α)-confidence interval consistent with the two one-sided t-tests procedure is given by [min(α, 0), max (0, b)] where [a, b] is the conventional (1–2α)-confidence interval of the t-test. This “central” confidence interval is always a strict subset of the symmetric confidence interval and thus has more power in proving bioequivalence. The central confidence interval has properties comparable with those of the conventional one-sided confidence intervals.     

Lower Confidence Limits for the Positive Linear Combination of two Variances

An alternative method for determining the approximate lower confidence limits for the positive linear combination of two variances based on an approach similar to BULMER (1957) has been proposed. The probability coverage of the proposed alternative limits has been compared with the other existing methods.     

Evaluation of an immunomagnetic separation assay in combination with cultivation to improve Legionella pneumophila serogroup 1 recovery from environmental samples

Aims: Legionella isolation from environmental samples is often difficult because of the presence of heterotrophic‐associated bacteria that frequently overgrow when using standard culture (ISO 11731, 1998; NF T90‐431, 2003) methods. To improve Legionella pneumophila recovery from complex water samples (water from cooling towers, biofilms), we evaluated an immunomagnetic separation (IMS) assay using a monoclonal antibody raised against the lipopolysaccharide of Leg. pneumophila sg1 in combination with culture. Methods and Results: This study was conducted on 51 environmental specimens. The comparison between IMS‐culture and standard culture (ISO 11731, 1998; NF T90‐431, 2003) methods was made using ISO 17994, 2004 criteria for establishing equivalence between microbiological methods based on the upper and lower (X_H and X_L) values of the relative difference (95% confidence limit) and D as maximum acceptable deviation (value of the confidence limit). Conclusions: We found that the average performance of IMS culture was higher than the reference method. Significance and Impact of the Study: Thus, this IMS‐culture assay is particularly well adapted to the detection of Leg. pneumophila sg1 in environmental samples with high levels of interfering microflora.     

Diminished Confidence Levels in the Interval Estimation of an Unknown Regressor in Model I of Linear Regression

For the model y=β₀ + β₁ x + e (model I of linear regression) in the literature confidence estimators of an unknown position x₀ are given at which either the expectation of y is given (see FIELLER, 1944; FINNEY, 1952), or realizations of y are given (see GRAYBILL, 1961). These confidence regions with level 1—α need not be intervals. The occurrence of interval shape is a random event. Its probability is equal to the power of the t test for the examination of the hypothesis H: β₁ = 0. The papers mentioned above claim to provide confidence intervals with level 1 ? α. But because of the restriction of (1 —α)—confidence regions to intervals the true confidence probability is the conditional probability W_c: W_c = P (the confidence region covers x₀| the region has interval shape). Here this conditional probability is shown to be less than 1 —α. Evidence on the possible deviations from 1 —α has been obtained by simulations.