Assessing the adequacy of variance function in heteroscedastic regression models

Heteroscedastic data arise in many applications. In heteroscedastic regression analysis, the variance is often modeled as a parametric function of the covariates or the regression mean. We propose a kernel-smoothing type nonparametric test for checking the adequacy of a given parametric variance structure. The test does not need to specify a parametric distribution for the random errors. It is shown that the test statistic has an asymptotical normal distribution under the null hypothesis and is powerful against a large class of alternatives. We suggest a simple bootstrap algorithm to approximate the distribution of the test statistic in finite sample size. Numerical simulations demonstrate the satisfactory performance of the proposed test. We also illustrate the application by the analysis of a radioimmunoassay data set.     

Differential measurement errors in zero-truncated regression models for count data

Measurement errors in covariates may result in biased estimates in regression analysis. Most methods to correct this bias assume nondifferential measurement errors-i.e., that measurement errors are independent of the response variable. However, in regression models for zero-truncated count data, the number of error-prone covariate measurements for a given observational unit can equal its response count, implying a situation of differential measurement errors. To address this challenge, we develop a modified conditional score approach to achieve consistent estimation. The proposed method represents a novel technique, with efficiency gains achieved by augmenting random errors, and performs well in a simulation study. The method is demonstrated in an ecology application.     

Ratio estimation with measurement error in the auxiliary variate

Summary .  With auxiliary information that is well correlated with the primary variable of interest, ratio estimation of the finite population total may be much more efficient than alternative estimators that do not make use of the auxiliary variate. The well-known properties of ratio estimators are perturbed when the auxiliary variate is measured with error. In this contribution we examine the effect of measurement error in the auxiliary variate on the design-based statistical properties of three common ratio estimators. We examine the case of systematic measurement error as well as measurement error that varies according to a fixed distribution. Aside from presenting expressions for the bias and variance of these estimators when they are contaminated with measurement error we provide numerical results based on a specific population. Under systematic measurement error, the biasing effect is asymmetric around zero, and precision may be improved or degraded depending on the magnitude of the error. Under variable measurement error, bias of the conventional ratio-of-means estimator increased slightly with increasing error dispersion, but far less than the increased bias of the conventional mean-of-ratios estimator. In similar fashion, the variance of the mean-of-ratios estimator incurs a greater loss of precision with increasing error dispersion compared with the other estimators we examine. Overall, the ratio-of-means estimator appears to be remarkably resistant to the effects of measurement error in the auxiliary variate.     

Method for estimating density of Cryphalus piceae (Ratz.) brood galleries using a regression model

Estimation of the severity of infestation by the bark beetle (Cryphalus piceae Ratz.) on silver fir (Abies alba Mill.) is commonly done using methods that lack satisfactory statistical foundations. The object of this study was to develop a statistical method for estimating stem infestation density which permits calculation of the estimation error and which also does not require debarking of the entire stem. The study was carried out in the near‐natural forests of the ?wi?tokrzyski National Park in central Poland. Two general patterns (A and B) of distribution of C. piceae brood galleries on stems of A. alba were observed. In pattern A, brood galleries occurred on the 1st 1‐m length of stem (measured from the base), and the probability of infestation on successive 1‐m lengths further up the tree was also very high (above 90%). In pattern B, brood galleries were absent from the 1st 1‐m length but the probability of infestation in successive 1‐m lengths increased as the distance from the base increased. In a statistical analysis of linear correlations between the numbers of C. piceae brood galleries in selected 1‐m‐long stem lengths and the total density of infestation, the most significant correlations for pattern A were found for the 1st, 2nd, 6th and 7th 1‐m lengths (counting from the base). Here, mean relative errors of estimation did not exceed 35%. For pattern B, the most significant correlations were found for the 8th, 10th, 11th, 12th, and 13th 1‐m lengths, where mean relative errors of estimation did not exceed 42%. The method described is only mildly invasive and therefore may be used in strictly protected areas of reserves and national parks.     

Likelihood-based genetic mark-recapture estimates when genotype samples are incomplete and contain typing errors

Genotypes produced from samples collected non-invasively in harsh field conditions often lack the full complement of data from the selected microsatellite loci. The application to genetic mark-recapture methodology in wildlife species can therefore be prone to misidentifications leading to both ‘true non-recaptures’ being falsely accepted as recaptures (Type I errors) and ‘true recaptures’ being undetected (Type II errors). Here we present a new likelihood method that allows every pairwise genotype comparison to be evaluated independently. We apply this method to determine the total number of recaptures by estimating and optimising the balance between Type I errors and Type II errors. We show through simulation that the standard error of recapture estimates can be minimised through our algorithms. Interestingly, the precision of our recapture estimates actually improved when we included individuals with missing genotypes, as this increased the number of pairwise comparisons potentially uncovering more recaptures. Simulations suggest that the method is tolerant to per locus error rates of up to 5% per locus and can theoretically work in datasets with as little as 60% of loci genotyped. Our methods can be implemented in datasets where standard mismatch analyses fail to distinguish recaptures. Finally, we show that by assigning a low Type I error rate to our matching algorithms we can generate a dataset of individuals of known capture histories that is suitable for the downstream analysis with traditional mark-recapture methods.     

Extreme regression

We develop a new method for describing patient characteristics associated with extreme good or poor outcome. We address the problem with a regression model composed of extrema (maximum and minimum) functions of the predictor variables. This class of models allows for simple regression function inversion and results in level sets of the regression function which can be expressed as interpretable Boolean combinations of decisions based on individual predictors. We develop an estimation algorithm and present clinical applications to symptoms data for patients with Hodgkin's disease and survival data for patients with multiple myeloma.     

Censored partial regression

In this work we study the effect of several covariates on a censored response variable with unknown probability distribution. A semiparametric model is proposed to consider situations where the functional form of the effect of one or more covariates is unknown, as is the case in the application presented in this work. We provide its estimation procedure and, in addition, a bootstrap technique to make inference on the parameters. A simulation study has been carried out to show the good performance of the proposed estimation process and to analyse the effect of the censorship. Finally, we present the results when the methodology is applied to AIDS diagnosed patients.     

Partial inverse regression

In regression with a vector of quantitative predictors, sufficientdimension reduction methods can effectively reduce the predictordimension, while preserving full regression information andassuming no parametric model. However, all current reductionmethods require the sample size n to be greater than the numberof predictors p. It is well known that partial least squarescan deal with problems with n < p. We first establisha link between partial least squares and sufficient dimensionreduction. Motivated by this link, we then propose a new dimensionreduction method, entitled partial inverse regression. We showthat its sample estimator is consistent, and that its performanceis similar to or superior to partial least squares when n < p,especially when the regression model is nonlinear or heteroscedastic.An example involving the spectroscopy analysis of biscuit doughis also given.