Nonparametric Ranked-set Sampling Confidence Intervals for Quantiles of a Finite Population

Ranked-set sampling from a finite population is considered in this paper. Three sampling protocols are described, and procedures for constructing nonparametric confidence intervals for a population quantile are developed. Algorithms for computing coverage probabilities for these confidence intervals are presented, and the use of interpolated confidence intervals is recommended as a means to approximately achieve coverage probabilities that cannot be achieved exactly. A simulation study based on finite populations of sizes 20, 30, 40, and 50 shows that the three sampling protocols follow a strict ordering in terms of the average lengths of the confidence intervals they produce. This study also shows that all three ranked-set sampling protocols tend to produce confidence intervals shorter than those produced by simple random sampling, with the difference being substantial for two of the protocols. The interpolated confidence intervals are shown to achieve coverage probabilities quite close to their nominal levels. Rankings done according to a highly correlated concomitant variable are shown to reduce the level of the confidence intervals only minimally. An example to illustrate the construction of confidence intervals according to this methodology is provided.     

Modelling temporal and spatial variability in tag reporting-rates for Newfoundland cod (<Emphasis Type="Italic">Gadus morhua</Emphasis>)

Mark–recapture experiments can be used to estimate the exploitation rate of a fishery; however, the estimate is influenced by the tag reporting-rate by the fishers. We present two methods to estimate the reporting rates in high/low reward ($100 and $10 CAD respectively) long-term cod tagging experiments. We fit two binomial logistic mixed-effect models, one with temporal auto-correlation in the reporting-rate year-effects and one with independent year-effects. We estimate reporting-rates separately for recreational and commercial fishers, and test for spatial variation using fixed-effects for spatial regions. Due to the complexity of the fishery, our models account for factors such as recapture-fishery type, fish-size and time-at-liberty. Our results indicate that the recreational fishers reporting-rate was constant at 0.51 across all regions and years. The commercial fishery showed more spatial and temporal variation, with the reporting-rates estimates lying between 0.67 and 0.87 for the independent year-effect model, and between 0.57 and 0.84 for the random walk model. Furthermore, we assessed the model performance as well as the coverage probability of nominal 95 % confidence intervals using simulations. We found that the models performed adequately; however, the nominal 95 % confidence intervals tended to be too narrow.     

Bootstrap confidence intervals for adaptive cluster sampling design based on Horvitz–Thompson type estimators

Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) introduced several bootstrap methods under adaptive cluster sampling using a Horvitz–Thompson type estimator. Using a simulation study, they showed that their proposed methods provide confidence intervals with highly understated coverage rates. In this article, we first show that their bootstrap methods provide biased bootstrap estimates. We then define two bootstrap methods, based on the method of Gross (Proceeding of the survey research methods section. American Statistical Association, Alexandria, VA, pp 181–184, 1980) and Bootstrap With Replacement, that provide unbiased bootstrap estimates of the population mean with bootstrap variances matching the corresponding unbiased variance estimator. Using a simulation study, we show that the bootstrap confidence intervals based on our proposed methods have better performance than those based on available bootstrap methods, in the sense of having coverage proportion closer to the nominal coverage level. We also compare the proposed intervals to empirical likelihood based intervals in small samples.     

Empirical likelihood confidence intervals for adaptive cluster sampling

Adaptive cluster sampling (ACS) is an efficient sampling design for estimating parameters of rare and clustered populations. It is widely used in ecological research. The modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators based on small samples under ACS have often highly skewed distributions. In such situations, confidence intervals based on traditional normal approximation can lead to unsatisfactory results, with poor coverage properties. Christman and Pontius (Biometrics 56:503–510, 2000) showed that bootstrap percentile methods are appropriate for constructing confidence intervals from the HH estimator. But Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) showed that bootstrap confidence intervals from the HT estimator are even worse than the normal approximation confidence intervals. In this article, we consider two pseudo empirical likelihood functions under the ACS design. One leads to the HH estimator and the other leads to a HT type estimator known as the Hájek estimator. Based on these two empirical likelihood functions, we derive confidence intervals for the population mean. Using a simulation study, we show that the confidence intervals obtained from the first EL function perform as good as the bootstrap confidence intervals from the HH estimator but the confidence intervals obtained from the second EL function perform much better than the bootstrap confidence intervals from the HT estimator, in terms of coverage rate.     

Comparing model averaging with other model selection strategies for benchmark dose estimation

Model averaging (MA) has been proposed as a method of accommodating model uncertainty when estimating risk. Although the use of MA is inherently appealing, little is known about its performance using general modeling conditions. We investigate the use of MA for estimating excess risk using a Monte Carlo simulation. Dichotomous response data are simulated under various assumed underlying dose–response curves, and nine dose–response models (from the USEPA Benchmark dose model suite) are fit to obtain both model specific and MA risk estimates. The benchmark dose estimates (BMDs) from the MA method, as well as estimates from other commonly selected models, e.g., best fitting model or the model resulting in the smallest BMD, are compared to the true benchmark dose value to better understand both bias and coverage behavior in the estimation procedure. The MA method has a small bias when estimating the BMD that is similar to the bias of BMD estimates derived from the assumed model. Further, when a broader range of models are included in the family of models considered in the MA process, the lower bound estimate provided coverage close to the nominal level, which is superior to the other strategies considered. This approach provides an alternative method for risk managers to estimate risk while incorporating model uncertainty.

Evaluation of variance approximation techniques for non-linear photosynthesis—irradiance models

Parameters derived from photosynthesis-irradiance (P-I) models, although often empirical in nature, are useful indicators of the photoadaptive state of phytoplankton in culture and in situ. However objective criteria for determining significant changes in P-I curves are rarely provided, because confidence intervals for parameters of non-linear models are not estimated easily. Examination of least-squares residuals in parameter space and Monte Carlo approaches have been used to estimate confidence regions around parameter values, but the computationally intensive nature of these methods has prevented their routine application. We present an alternative method of estimating confidence intervals for parameters of P-I curves that runs quickly on a microcomputer and is easily combined with common parameter-estimation routines. This algorithm was tested using a 3-parameter P-I model and curves describing a wide range of photoadaptive states, with different numbers of observations and different amounts of inherent variability. The method produced results comparable to the Monte Carlo technique. This analysis makes it possible to specify the sample size required to define parameters with acceptable confidence as a function of data variance and photoadaptive state. In most reasonable situations, 25 observations are sufficient.     

Continuous dose-response modeling and risk analysis with the gamma and reciprocal gamma distributions

Kodell and West (1993) describe two methods for calculating pointwise upper confidence limits on the risk function with normally distributed responses and using a certain definition of adverse quantitative effect. But Banga et al. (2000) have shown that these normal theory methods break down when applied to skew data. We accordingly develop a risk analysis model and associated likelihood-based methodology when the response follows either a gamma or reciprocal gamma distribution. The model supposes that the shape (index) parameter k of the response distribution is held fixed while the logarithm of the scale parameter is a linear model in terms of the dose level. Existence and uniqueness of the maximum likelihood estimates is established. Asymptotic likelihood-based upper and lower confidence limits on the risk are solutions of the Lagrange equations associated with a constrained optimization problem. Starting values for an iterative solution are obtained by replacing the Lagrange equations by the lowest order terms in their asymptotic expansions. Three methods are then compared for calculating confidence limits on the risk: (i) the aforementioned starting values (LRAL method), (ii) full iterative solution of the Lagrange equations (LREL method), and (iii) bounds obtained using approximate normality of the maximum likelihood estimates with standard errors derived from the information matrix (MLE method). Simulation is used to assess coverage probabilities for the resulting upper confidence limits when the log of the scale parameter is quadratic in the dose level. Results indicate that coverage for the MLE method can be off by as much as 15% points and converges very slowly to nominal coverage levels as the sample size increases. Coverage for the LRAL and LREL methods, on the other hand, is close to nominal levels unless (a) the sample size is small, say N < 25, (b) the index parameter is small, say k 1, and (c) the direction of adversity is to the left for the gamma distribution or to the right for the reciprocal gamma distribution.     

Comparison of distribution function estimators for forestry applications following variable probability sampling

Practical considerations often motivate employing variable probability sampling designs when estimating characteristics of forest populations. Three distribution function estimators, the Horvitz-Thompson estimator, a difference estimator, and a ratio estimator, are compared following variable probability sampling in which the inclusion probabilities are proportional to an auxiliary variable, X. Relative performance of the estimators is affected by several factors, including the distribution of the inclusion probabilities, the correlation () between X and the response Y, and the position along the distribution function being estimated. Both the ratio and difference estimators are superior to the Horvitz-Thompson estimator. The difference estimator gains better precision than the ratio estimator toward the upper portion of the distribution function, but the ratio estimator is superior toward the lower end of the distribution function. The point along the distribution function at which the difference estimator becomes more precise than the ratio estimator depends on the sampling design, as well as the coefficient of variation of X and . A simple confidence interval procedure provides close to nominal coverage for intervals constructed from both the difference and ratio estimators, with the exception that coverage may be poor for the lower tail of the distribution function when using the ratio estimator.     

Environmental policy and macroeconomic dynamics in a new Keynesian model

This paper studies the dynamic behavior of an economy under different environmental policy regimes in a New Keynesian model with nominal and real uncertainty. We find the following results: (i) an emissions cap policy is likely to dampen macroeconomic fluctuations; (ii) staggered price adjustment alters significantly the performance of the environmental policy regime put in place; (iii) the optimal environmental policy response to shocks is strongly influenced by the degree to which prices adjust and by the monetary policy reaction.     

Variances and variance estimators of the improved ratio estimators under adaptive cluster sampling

Although not design-unbiased, the ratio estimator is recognized as more efficient when a certain degree of correlation exists between the variable of primary interest and the auxiliary variable. Meanwhile, the Rao–Blackwell method is another commonly used procedure to improve estimation efficiency. Various improved ratio estimators under adaptive cluster sampling (ACS) that make use of the auxiliary information together with the Rao–Blackwellized univariate estimators have been proposed in past research studies. In this article, the variances and the associated variance estimators of these improved ratio estimators are proposed for a thorough framework of statistical inference under ACS. Performance of the proposed variance estimators is evaluated in terms of the absolute relative percentage bias and the empirical mean-squared error. As expected, results show that both the absolute relative percentage bias and the empirical mean-squared error decrease as the initial sample size increases for all the variance estimators. To evaluate the confidence intervals based on these variance estimators and the finite-population Central Limit Theorem, the coverage rate and the interval width are used. These confidence intervals suffer a disadvantage similar to that of the conventional ratio estimator. Hence, alternative confidence intervals based on a certain type of adjusted variance estimators are constructed and assessed in this article.     

Messung und Quantifizierung von Mineralölkohlenwasserstoffen

Investigations concerning the measurement and evaluation of mineral oil hydrocarbons using Fourier-transform-infrared (FT/IR-)-spectroscopy,¹H-Nuclear magnetic resonance (¹H-NMR)-spectroscopy and Capillary gas chromatography — Flame ionisation detection (GC-FID) are presented. By means of various mineral oils and three certified reference materials (CRM) all tested methods were within a ±7%-range to the mineral oil nominal value and the 95% confidence intervals of the CRM’s, respectively. The GC-FID evaluation could be done without calibration using an relative response ratio of mineral oil to an internal standard (n-tetracontane). A¹H-NMR-method was developed for the quantitative determination of mineral oil hydrocarbons, successfully applied down to 0.2 mg/ml. Due to the determination limit achieved, the¹H-NMR-spectroscopy gain in importance as a reference method for the analysis of mineral oils.     

The Use of Bayesian Model Averaging to Better Represent Uncertainty in Ecological Models

Abstract:   In conservation biology, uncertainty about the choice of a statistical model is rarely considered. Model-selection uncertainty occurs whenever one model is chosen over plausible alternative models to represent understanding about a process and to make predictions about future observations. The standard approach to representing prediction uncertainty involves the calculation of prediction (or confidence) intervals that incorporate uncertainty about parameter estimates contingent on the choice of a "best" model chosen to represent truth. However, this approach to prediction based on statistical models tends to ignore model-selection uncertainty, resulting in overconfident predictions. Bayesian model averaging (BMA) has been promoted in a range of disciplines as a simple means of incorporating model-selection uncertainty into statistical inference and prediction. Bayesian model averaging also provides a formal framework for incorporating prior knowledge about the process being modeled. We provide an example of the application of BMA in modeling and predicting the spatial distribution of an arboreal marsupial in the Eden region of southeastern Australia. Other approaches to estimating prediction uncertainty are discussed.     

Confidence intervals for the mean of the delta-lognormal distribution

Data that are skewed and contain a relatively high proportion of zeros can often be modelled using a delta-lognormal distribution. We consider three methods of calculating a 95% confidence interval for the mean of this distribution, and use simulation to compare the methods, across a range of realistic scenarios. The best method, in terms of coverage, is that based on the profile-likelihood. This gives error rates that are within 1% (lower limit) or 3% (upper limit) of the nominal level, unless the sample size is small and the level of skewness is moderate to high. Our results will also apply to the delta-lognormal linear model, when we wish to calculate a confidence interval for the expected value of the response variable, given the value of one or more explanatory variables. We illustrate the three methods using data on red cod densities, taken from a fisheries trawl survey in New Zealand.

Local complexity in patterns of canopy–benthos associations produces regional patterns across temperate Australasia

Across subtidal coasts of temperate Australasia, canopy–benthos associations are mostly understood from broadly defined studies of kelp forests within eastern Australia and north-eastern New Zealand. We tested the hypotheses that (1) benthic assemblages differ between monospecific stands of Ecklonia radiata and stands that comprise E. radiata mixed with other canopy-forming species, (2) patterns observed locally (i.e. within sites 1–10 km apart) match those observed among regions (>1,000 km apart) for which (3) eastern Australia and northern New Zealand are representative of other regions of temperate Australasia. Benthic assemblages almost always differed among monospecific, mixed, and open stands indicating that failure to distinguish between superficially similar habitats can lead to over-generalised conclusions about the ecology of kelp forests. Patterns of differences among stands did not change between western and southern Australia but differed from eastern Australia, and patterns from all regions were distinct from New Zealand (WA=SAEANZ). Whilst local patterns were complex, the major morphological groups that often characterise benthos (i.e. encrusting coralline algae and turf-forming algae) revealed patterns that could be related across space from local to regional scales. These findings demonstrate that knowledge about the configuration of canopy-forming species will improve confidence about the representativeness of results and that any local complexity need not impede searches for generality when the spatial limits of patterns are also understood.Communicated by M.S. Johnson, Crawley     

Animal abundance estimation in independent observer line transect surveys

The theory of conventional line transect surveys is based on an essential assumption that 100% detection of animals right on the transect lines can be achieved. When this assumption fails, independent observer line transect surveys are used. This paper proposes a general approach, based on a conditional likelihood, which can be carried out either parametrically or nonparametrically, to estimate the abundance of non-clustered biological populations using data collected from independent observer line transect surveys. A nonparametric estimator is specifically proposed which combines the conditional likelihood and the kernel smoothing method. It has the advantage that it allows the data themselves to dictate the form of the detection function, free of any subjective choice. The bias and the variance of the nonparametric estimator are given. Its asymptotic normality is established which enables construction of confidence intervals. A simulation study shows that the proposed estimator has good empirical performance, and the confidence intervals have good coverage accuracy.     

The influence of successional processes and disturbance on the structure of Tsuga canadensis forests

Old-growth forests are valuable sources of ecological, conservation, and management information, yet these ecosystems have received little study in New England, due in large part to their regional scarcity. To increase our understanding of the structures and processes common in these rare forests, we studied the abundance of downed coarse woody debris (CWD) and snags and live-tree size-class distributions in 16 old-growth hemlock forests in western Massachusetts. Old-growth stands were compared with eight adjacent second-growth hemlock forests to gain a better understanding of the structural differences between these two classes of forests resulting from contrasting histories. In addition, we used stand-level dendroecological reconstructions to investigate the linkages between disturbance history and old-growth forest structure using an information-theoretic model selection framework. Old-growth stands exhibit a much higher degree of structural complexity than second-growth forests. In particular, old-growth stands had larger overstory trees and greater volumes of downed coarse woody debris (135.2 vs. 33.2 m3/ha) and snags (21.2 vs. 10.7 m3/ha). Second-growth stands were characterized by either skewed unimodal or reverse-J shaped diameter distributions, while old-growth forests contained bell-shaped, skewed unimodal, rotated sigmoid, and reverse J-shaped distributions. The variation in structural attributes among old-growth stands, particularly the abundance of downed CWD, was closely related to disturbance history. In particular, old-growth stands experiencing moderate levels of canopy disturbance during the last century (1930s and 1980s) had greater accumulations of CWD, highlighting the importance of gap-scale disturbances in shaping the long-term development and structural characteristics of old-growth forests. These findings are important for the development of natural disturbance-based silvicultural systems that may be used to restore important forest characteristics lacking in New England second-growth stands by integrating structural legacies of disturbance (e.g., downed CWD) and resultant tree-size distribution patterns. This silvicultural approach would emulate the often episodic nature of CWD recruitment within old-growth forests.     

Bat Use of Remnant Old-Growth Redwood Stands

Most of the old-growth redwood ( Sequoia sempervirens ) in California has been cut; regenerating forests will probably never resemble those that were harvested, and what old growth remains on private land occurs in small, isolated remnant patches. The landscapes in which these stands occur differ so markedly from their original condition that their value as habitat to many species of wildlife, including bats, is unknown. Previous research in unfragmented redwood forests demonstrated that bats use basal hollows in old-growth redwoods as roosts. We sought to determine whether bats use similar old-growth trees as roosts when they occur in small, remnant patches of isolated old growth on commercial forest land. We compared bat occurrence in remnant and contiguous stands by collecting guano in traps suspended in hollows and by monitoring flight activity with ultrasonic bat detectors. Hollows in trees within the remnant stands had significantly more guano deposited per tree than the trees within the contiguous forest. The mean numbers of bat passes per night were statistically indistinguishable between the two treatments, although mean flight activity in the remnant stands was greater than in the contiguous forest. Bats frequently used basal hollows in small (<5 ha) stands of remnant old growth, which may be due to the closer proximity of remnants to stream courses, to their greater interface with edge where foraging success may be greater, or to the fact that the lower density of hollow-bearing trees in remnants than in contiguous forest favored greater use per tree. Significant use of small, residual old-growth redwood provides reason to maintain these remnants in managed landscapes as potentially important habitat for forest bats.     

Tree species diversity and its relationship to stand parameters and geomorphology features in the eastern Black Sea region forests of Turkey

We studied the effects of stand parameters (crown closure, basal area, stand volume, age, mean stand diameter number of trees, and heterogeneity index) and geomorphology features (elevation, aspect and slope) on tree species diversity in an example of untreated natural mixed forest stands in the eastern Black Sea region of Turkey. Tree species diversity and basal area heterogeneity in forest ecosystems are quantified using the Shannon-Weaver and Simpson indices. The relationship between tree species diversity basal area heterogeneity stand parameters and geomorphology features are examined using regression analysis. Our work revealed that the relationship between tree species diversity and stand parameters is loose with a correlation coefficient between 0.02 and 0.70. The correlation of basal area heterogeneity with stand parameters fluctuated between 0.004 and 0.77 (R2). According to our results, stands with higher tree species diversity are characterised by higher mean stand diameter number of diameter classes, basal area and lower homogeneity index value. Considering the effect of geomorphology features on tree species or basal area heterogeneity we found that all investigated relationships are loose with R < or = 0.24. A significant correlation was detected only between tree species diversity and aspect. Future work is required to verify the detected trends in behaviour of tree species diversity if it is to estimate from the usual forest stand parameters and topography characteristics.     

Quantifying parameter uncertainty in a coral reef model using Metropolis-Coupled Markov Chain Monte Carlo

Coral reefs are threatened ecosystems, so it is important to have predictive models of their dynamics. Most current models of coral reefs fall into two categories. The first is simple heuristic models which provide an abstract understanding of the possible behaviour of reefs in general, but do not describe real reefs. The second is complex simulations whose parameters are obtained from a range of sources such as literature estimates. We cannot estimate the parameters of these models from a single data set, and we have little idea of the uncertainty in their predictions.We have developed a compromise between these two extremes, which is complex enough to describe real reef data, but simple enough that we can estimate parameters for a specific reef from a time series. In previous work, we fitted this model to a long-term data set from Heron Island, Australia, using maximum likelihood methods. To evaluate predictions from this model, we need estimates of the uncertainty in our parameters. Here, we obtain such estimates using Bayesian Metropolis-Coupled Markov Chain Monte Carlo. We do this for versions of the model in which corals are aggregated into a single state variable (the three-state model), and in which corals are separated into four state variables (the six-state model), in order to determine the appropriate level of aggregation. We also estimate the posterior distribution of predicted trajectories in each case.In both cases, the fitted trajectories were close to the observed data, but we had doubts about the biological plausibility of some parameter estimates. We suggest that informative prior distributions incorporating expert knowledge may resolve this problem. In the six-state model, the posterior distribution of state frequencies after 40 years contained two divergent community types, one dominated by free space and soft corals, and one dominated by acroporid, pocilloporid, and massive corals. The three-state model predicts only a single community type. We conclude that the three-state model hides too much biological heterogeneity, but we need more data if we are to obtain reliable predictions from the six-state model. It is likely that there will be similarly large, but currently unevaluated, uncertainty in the predictions of other coral reef models, many of which are much more complex and harder to fit to real data.