Approximate Standard Errors Research Articles

Precision of prediction in second‐order calibration, with focus on bilinear regression methods

AbstractWe consider calibration of hyphenated instruments with particular focus on determination of the unknown concentrations of new specimens. A hyphenated instrument generates for each specimen a two‐way array of data. These are assumed to depend on the concentrations through a bilinear regression model, where each constituent is characterized by a pair of profiles to be determined in the calibration. We discuss the problem of predicting the unknown concentrations in a new specimen, after calibration. We formulate three different predictor construction methods, a ‘naive’ method, a least squares method, and a refined version of the latter that takes account of the calibration uncertainty. We give formulae for the uncertainty of the predictors under white noise, when calibration can be seen as precise. We refine these formulae to allow for calibration uncertainty, in particular when calibration is carried out by the bilinear least squares (BLLS) method or the singular value decomposition (SVD) method proposed by Linder and Sundberg (Chemometrics Intell. Lab. Syst. 1998; 42: 159–178). By error propagation formulae and previous results on the precision of $\widehat{A}$ and $\widehat{B}$ we can obtain approximate standard errors for the predicted concentrations, according to each of the two estimation methods. The performance of the predictors and the precision formulae is illustrated on both real (fluorescence) and simulated data. Copyright © 2002 John Wiley & Sons, Ltd.

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.

Maximum likelihood computations for regression with measurement error

This paper presents a general computational method for maximum likelihood analysis for generalized regression with measurement error in a single explanatory variable. The method is the EM algorithm with Gauss–Hermite quadrature in the E-step. Although computationally intensive, this method provides maximum likelihood estimation under a broad range of distributional assumptions. This is important because maximum likelihood estimators can be more efficient than commonly used moment estimators and likelihood ratio tests and confidence intervals can be substantially superior to those based on asymptotic normality with approximate standard errors.

Local estimation of age-dependent variance components from longitudinal twin data.

In the study of longitudinal twin and family data, interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function. Various parametric models for covariance structures have been proposed but, e.g., in studies of children where growth spurts occur at various ages, it is difficult to a priori determine an appropriate parametric model for the covariance structure. In particular, there is a general lack of the visualization procedures, such as lowess, that are invaluable in the initial stages of constructing a parametric model for a mean function. Here we use kernel smoothing to modify a cross-sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data. The methods are proposed to be exploratory as an aid to parametric modeling rather than inferential, although approximate asymptotic standard errors are derived in the Appendix.

Approximate Standard Errors and Confidence Intervals for Indices of Positive and Negative Agreement

Indices of positive and negative agreement for observer reliability studies, in which neither observer can be regarded as the standard, have been proposed. In this article, it is demonstrated by means of an example and a small simulation study that a recently published method for constructing confidence intervals for these indices leads to intervals that are too wide. Appropriate asymptotic (i.e., large sample) variance estimates and confidence intervals for the positive and negative agreement indices are presented and compared with bootstrap confidence intervals. We also discuss an alternative method of interval estimation motivated from a Bayesian viewpoint. The asymptotic intervals performed adequately for sample sizes of 200 or more. For smaller samples, alternative confidence intervals such as bootstrap intervals or Bayesian intervals should be considered.

Uncertainty in Palaeoclimatic Reconstructions Based on Leaf Physiognomy

There are several methods of predicting terrestrial palaeoclimates from the size and shape of fossil leaves (foliar physiognomy). The assumptions and sources of uncertainty of these methods are considered and used to determine the true uncertainty. Their ability to predict mean annual temperature (MAT) is poor. The approximate standard errors for samples of living vegetation in North America are in the range of 1.7˚C to 2.5˚C, but the true uncertainty for fossil samples is higher. Specimens with very different physiognomy to typical specimens in the model have higher uncertainties. Besides these uncertainties, the processes of fossilisation, the allocation of specimens to taxa, and the effects of other factors on foliar physiognomy all increase the uncertainty of the predictions. Overall uncertainties in the predictions of MAT are equivalent to standard errors of about 3–5˚C depending on the nature of the fossil site and flora. Other factors affect foliar physiognomic predictions significantly because predicted MAT does not change as rapidly with altitude as true MAT, and floras from different parts of the world with similar temperatures give different temperature predictions. Mean annual temperature and one precipitation parameter (probably mean annual precipitation or the growing season precipitation) can be predicted more or less independently, although the predictions of precipitation are weak. Physiognomic signals for other climatic parameters are weak or apparently non-existent, and previously published predictions of past equability are primarily based on correlations with modern MAT, rather than physiognomy.

Standard Errors for the Eigenvalues in Second-Order Response Surface Models

In the interpretation of the geometry of second-order response surface models, standard errors and confidence intervals for the eigenvalues of the second-order coefficient matrix play an important role. In this article, we propose a new method for estimating the standard errors, and hence approximate confidence intervals, of these eigenvalues. The method is simple in both concept and execution. It involves the refitting of a full quadratic model after rotating the coordinate system to coincide with the canonical axes. The estimated standard errors of the pure quadratic terms from this refitting are then used as approximate standard errors of the eigenvalues. Because this approach is based on the canonical form, it is geometrically intuitive and easily taught. Our method is intended as a way for practitioners to get quick estimates of the standard errors of the eigenvalues. In our justification of the approach, we show that it is equivalent to using the delta method proposed by Carter, Chinchilli, and Camp...

Use of deterministic sampling for exploring likelihoods in linkage analysis for quantitative traits

Deterministic sampling was used to numerically evaluate the expected log-likelihood surfaces of QTL-marker linkage models in large pedigrees with simple structures. By calculating the expected values of likelihoods, questions of power of experimental designs, bias in parameter estimates, approximate lower-bound standard errors of estimates and correlations among estimates, and suitability of statistical models were addressed. Examples illustrated that bracket markers around the QTL approximately halved the standard error of the recombination fraction between the QTL and the marker, although they did not affect the standard error of the QTL's effect, that overestimation of the distance between the markers caused overestimation of the distance between the QTL and marker, that more parameters in the model did not affect the accuracy of parameter estimates, that there was a moderate positive correlation between the estimates of the QTL effect and its recombination distance from the marker, and that selective genotyping did not introduce bias into the estimates of the parameters. The method is suggested as a useful tool for exploring the power and accuracy of QTL linkage experiments, and the value of alternative statistical models, whenever the likelihood of the model can be written explictly.

Approximate maximum likelihood estimation in the presence of extra-poisson variation

A two-stage hierarchical model for analysis of discrete data with extra-Poisson variation is examined. The model consists of a Poisson distribution with a mixing lognormal distribution for the mean. A method of approximate maximum likelihood estimation of the parameters is proposed. The method uses the EM algorithm and approximations to facilitate its implementation are derived. Approximate standard errors of the estimates are provided and a numerical example is used to illustrate the method.

Statistical Analysis of Spectra from Electron Spectroscopy for Chemical Analysis

Electron spectroscopy for chemical analysis is a key technique in the study of modified material surfaces. Analysis of the resulting spectra consists in decomposing multiple peaks into subpeaks, whose functional form is known up to a few parameters. Statistical inference consists in estimating these parameters and derived quantities, such as peak area ratios, and in assessing the accuracies of these estimates. Purely likelihood-based peak decomposition is notorious for problems of identifiability. In practice, however, additional knowledge exists about some of the parameters and can, when incorporated in the model as an informative prior, produce unique decompositions. Modem tools for Bayesian statistics, such as profile diagnostics, Laplacian approximations of marginals and Markov chain algorithms for sampling from the posterior, can then be used to obtain inference reaching beyond point estimates and approximate standard errors.

Log-Linear Models for Survey Data with Non-Ignorable Non-Response

SUMMARY We demonstrate the feasibility of constructing, interpreting and fitting computable log-linear models to categorical survey data with arbitrary non-nested patterns of non-ignorable non-response. Under our approach, the non-response probability for each cell of the classification defined by the categories of interest is modelled separately from the classification probability itself, and we adopt a model formulation which allows the non-response model to depend on scores, discrete covariates, continuous covariates or a mixture of types of covariate. We obtain explicit expressions for the score and information functions generated by the observed data which allow us to compute approximate standard errors and test statistics based on these functions. Through illustrative examples we quantify the fact that inferences obtained from different non-ignorable non-response models can vary considerably. This lends support to the calls in the literature for caution in using such models.

Recovering extra-binomial variation

A hierarchical logit-normal model for analysis of binary data with extra-binomial variation is examined. A method of approximate maximum likelihood estimation of the parameters is proposed. The method uses the EM algorithm and approximations to facilitate its implementation are derived. Approximate standard errors of the estimates are provided and a numerical example is used to illustrate the method.

Geostatistical evaluation of satellite radar altimetry for high-resolution mapping of Lambert Glacier, Antarctica

The potential of satellite radar altimetry for high-resolution mapping of Antarctic ice streams is evaluated, using retracked and slope-corrected data from the Lambert Glacier and Amery Ice Shelf area, East Antarctica, acquired by Geosat during the Exact Repeat Mission (ERM), 1986–89. The map area includes lower Lambert Glacier north of 72.18°S, the southern Amery Ice Shelf, and the grounded inland ice sheet on both sides. The Geosat ERM altimetry is found to provide substantially more complete coverage than the 1978 Seasat altimetry, due to improved tracking. Variogram methods are used to estimate the noise levels in the data as a function of position throughout the map area. The spatial structure in the data is quantified by constructing experimental variograms using altimetry from the area of the grounding zone of Lambert Glacier, which is the area chiefly of interest in this topographically complex region. Kriging is employed to invert the along-track height measurements onto a fine-scale 3 km grid. The unsmoothed along-track Geosat ERM altimetry yields spatially continuous maps showing the main topographic features of lower Lambert Glacier, upper Amery Ice Shelf and the adjacent inland ice sheet. The probable position of the grounding line of Lambert Glacier is identified from a break in slope at the grounded ice/floating ice transition. The approximate standard error of the kriged map is inferred from the data noise levels.

Geostatistical evaluation of satellite radar altimetry for high-resolution mapping of Lambert Glacier, Antarctica

The potential of satellite radar altimetry for high-resolution mapping of Antarctic ice streams is evaluated, using retracked and slope-corrected data from the Lambert Glacier and Amery Ice Shelf area, East Antarctica, acquired by Geosat during the Exact Repeat Mission (ERM), 1986–89. The map area includes lower Lambert Glacier north of 72.18°S, the southern Amery Ice Shelf, and the grounded inland ice sheet on both sides. The Geosat ERM altimetry is found to provide substantially more complete coverage than the 1978 Seasat altimetry, due to improved tracking. Variogram methods are used to estimate the noise levels in the data as a function of position throughout the map area. The spatial structure in the data is quantified by constructing experimental variograms using altimetry from the area of the grounding zone of Lambert Glacier, which is the area chiefly of interest in this topographically complex region. Kriging is employed to invert the along-track height measurements onto a fine-scale 3 km grid. The unsmoothed along-track Geosat ERM altimetry yields spatially continuous maps showing the main topographic features of lower Lambert Glacier, upper Amery Ice Shelf and the adjacent inland ice sheet. The probable position of the grounding line of Lambert Glacier is identified from a break in slope at the grounded ice/floating ice transition. The approximate standard error of the kriged map is inferred from the data noise levels.

On the Performance of a Test for Coefficient of Variation

Approximate standard errors (s.e) of sample coefficient of variation (c.v.) are available in the literature (Kendall and Stuart, 1977; Serfting 1980). Using these s.e.'s a large sample test can be constructed to test for the specified value of population c.v. The test can also be justified on the principles of Wald test. These s.e.'s have been worked out assuming that the random variable has a distribution. Following Foutz and Srivastava (1977), we study the asymptotic performance of the test when the model is incorrect. Examples from realistic situations have been considered and approximate Babadur efficiencies have been worked out. Numerical results indicate that the test is fairly robust when the departure from the model is not too pronounced.

Estimation of Milk Protein Gene Frequencies in Crossbred Cattle by Maximum Likelihood

A maximum likelihood method is presented to estimate the fraction of animals misclassified and breed effects for milk protein gene frequencies based on crossbred data. A simulation study indicates that the method provides estimates of gene frequencies that agree closely with the true values. Gene frequencies in the Dutch Black and White and the Dutch Red and White crossbred populations, based on data on 10,151 and 580 animals respectively, were estimated. Dutch Friesian and Holstein-Friesian breeds differ in gene frequencies for β-casein and β-lactoglobulin. Estimates for fractions misclassified are zero for αs1-casein, .09 for β-casein and β-lactoglobulin, and .12 for κ-casein. Differences between Dutch Red and Whites and Red Holstein-Friesian breeds are small, and estimates for fractions misclassified are high but have high approximate standard errors. Compared with the Black and White breeds, the Red and Whites have a high κ-casein B gene frequency.

Standard Errors and Confidence Interval Estimators for Expected Selection Response

Expected selection response (R) is widely used in plant breeding to compare populations and selection schemes, but methods have not been described for estimating variances or confidence intervals of R, or, if they have, their validity is uncertain. In this paper, we describe an exact and approximate standard error of R and normal‐approximation intervals of R. We used simulation to test the validity of these intervals by comparing realized coverage probabilities to stated coverage probabilities for different experiment sizes, stated coverage probability values, and values of family‐mean heritability (H). Coverages of the normal‐approximation interval estimated using the exact standard error (interval R1) ranged from 0.874 to 0.922 and 0.932 to 0.966 for stated coverages of 0.90 and 0.95, respectively. Coverages of the normal‐approximation interval estimated using the approximate standard error (interval R2) ranged from 0.880 to 0.940 and 0.934 to 0.980 for stated coverages of 0.90 and 0.95, respectively. Both R1 and R2 are valid interval estimators for R if the usual assumptions of the analysis of variance are met.

Inference on system parameters in a model for interacting species

A system of p interacting species whose behaviour can be approximated by a Markovian model is considered. Estimates for system parameters are obtained by the method of moments, when the means, variances and covariances can be estimated from observed population sizes over a period of time. Further, approximate standard errors of these estimates are obtained using the δ-technique.

Additive and Nonadditive Genetic Variation for Production Traits in Canadian Holsteins

Additive and dominance genetic variances were estimated for milk and fat yield in Canadian Holsteins using derivative-free REML procedures. Traits were analyzed under two sire and dam models differing in assumptions on the data structure. Within herd, dam relationships were first defined to be hierarchical within sire to estimate the two genetic variances, assuming no maternal, epistatic genetic, or other biases on the estimates. The potential impact of some of these biases was estimated as the difference between the two parental variances under a cross-classified model. Heritabilities based on sire components from both models were statistically identical and were .41 for milk yield and .32 for fat yield. Dominance genetic variance was 25% of the total variation for fat yield but was unimportant for milk yield based on approximate standard errors. However, results from the cross-classified model indicated dam components were significantly greater by 4% than sire components for fat yield. Therefore, hierarchical estimates of dominance genetic variance for fat yield may be biased upward. Some form of nonadditive genetic variation for fat yield, other than that based on sire components, may require consideration in dairy cattle breeding programs.

An APL function for the asymptotic test of significance for Goodman and Kruskal’s gamma statistic

Goodman and Kruskal's (1954) gamma is a measure of association that is often used to measure the degree of association between two cross-elassified ordinal variables. Gamma is a symmetric measure of ordinal association that utilizes infonnation only from concordant and discordant (i.e., untied) object pairs; Somers' (1962) d, on the other hand, is an asymmetric measure of ordinal association that utilizes the infonnation from the concordant and discordant object pairs, but adjusts for additional object pairs that are tied on the dependent variable but not on the independent variable. Although these two measures examine somewhat different kinds of object pairs obtained from the two ordinal variables, they both estimate the degree to which knowledge of order on one variable predicts order on the other variable. Several APL routines exist to compute Goodman and Kruskal's (1954) symmetric measure of ordinal association, gamma, as well as Somers' (1962) asymmetric measures, dre , with row variable dependent, and den with column variable dependent (Berry & Mielke, 1986; Brennan, Nitz, & Daly, 1988). Unfortunately, none of these routines provides the probability value, under the null hypothesis, for any of the three measures. The function GAMMA is a noniterative APL function that calculates the three coefficients of ordinal association-gamma, drc> and der-the associated two-sided probability values, and the 95% and 99% confidence limits, under full multinomial sampling. The probability values and the confidence limits are computed using the asymptotic standard errors developed by Goodman and Kruskal (1963, 1972) and, specifically, do not use the approximate standard errors given in many textbooks. The approximate tests are based on the maximum standard error (Goodman & Kruskal, 1963), which yields a conservative test with a probability of falsely rejecting the null hypothesis known only to be less than some nominal value, or they assume unifonn marginals (Somers, 1980), which may yield inaccurate test results for cross classifications with severely skewed marginal distributions. Moreover, because the approximations introduced by Somers (1980) are based on null populations, they preclude the computation of confidence limits. Procedure. Consider a full multinomial sample of n discrete objects, which are cross classified by two ordi-