Equilibrating errors: reliable estimation of information transmission rates in biological systems with spectral analysis-based methods

Shannon’s seminal approach to estimating information capacity is widely used to quantify information processing by biological systems. However, the Shannon information theory, which is based on power spectrum estimation, necessarily contains two sources of error: time delay bias error and random error. These errors are particularly important for systems with relatively large time delay values and for responses of limited duration, as is often the case in experimental work. The window function type and size chosen, as well as the values of inherent delays cause changes in both the delay bias and random errors, with possibly strong effect on the estimates of system properties. Here, we investigated the properties of these errors using white-noise simulations and analysis of experimental photoreceptor responses to naturalistic and white-noise light contrasts. Photoreceptors were used from several insect species, each characterized by different visual performance, behavior, and ecology. We show that the effect of random error on the spectral estimates of photoreceptor performance (gain, coherence, signal-to-noise ratio, Shannon information rate) is opposite to that of the time delay bias error: the former overestimates information rate, while the latter underestimates it. We propose a new algorithm for reducing the impact of time delay bias error and random error, based on discovering, and then using that size of window, at which the absolute values of these errors are equal and opposite, thus cancelling each other, allowing minimally biased measurement of neural coding.     

Stability of hemoglobin mass over 100 days in active men

The purpose of this study was to investigate the suggestion in a recent meta-analysis that variability in hemoglobin mass increases when time between measurements increases from days to months. Hemoglobin mass of six active men was measured with the carbon monoxide method every 1-6 days for 100-114 days (42 +/- 3 measurements, mean +/- SD). Measurement error for each individual's series was estimated from the standard deviation of consecutive pairwise changes and compared with his total error (standard deviation of all values). Linear trends and periodicities in each series were quantified by regression and spectral analysis. Series with known random error and periodicity were also simulated and analyzed. There were clear differences in the pairwise error of measurement between subjects (range 1.4-2.7%). For five men, there was little difference between the total and pairwise errors; their mean ratio (1.06, 90% confidence limits 0.96-1.17) was less than ratios for simulated sinusoidal series with random error of 2%, amplitude of 2%, and periods of 20-100 days (ratios 1.13-1.21). Spectral analysis clearly revealed such periodicities in the simulated series but not in the series of these subjects. The sixth man, who had donated blood 12 days before commencing measurements, showed errors, trend, and periodicity consistent with gradual restoration of hemoglobin mass. Measurement error of hemoglobin mass does not increase over 100 days. Consequently, hemoglobin mass may be suitable for long-term monitoring of small changes that might occur with training or erythropoietin abuse, taking into consideration the small differences between athletes in errors and trends.     

Anatomy of emotion: a 3D study of facial mimicry

Alterations in facial motion severely impair the quality of life and social interaction of patients, and an objective grading of facial function is necessary. A method for the non-invasive detection of 3D facial movements was developed. Sequences of six standardized facial movements (maximum smile; free smile; surprise with closed mouth; surprise with open mouth; right side eye closure; left side eye closure) were recorded in 20 healthy young adults (10 men, 10 women) using an optoelectronic motion analyzer. For each subject, 21 cutaneous landmarks were identified by 2-mm reflective markers, and their 3D movements during each facial animation were computed. Three repetitions of each expression were recorded (within-session error), and four separate sessions were used (between-session error). To assess the within-session error, the technical error of the measurement (random error, TEM) was computed separately for each sex, movement and landmark. To assess the between-session repeatability, the standard deviation among the mean displacements of each landmark (four independent sessions) was computed for each movement. TEM for the single landmarks ranged between 0.3 and 9.42 mm (intrasession error). The sex- and movement-related differences were statistically significant (two-way analysis of variance, p=0.003 for sex comparison, p=0.009 for the six movements, p<0.001 for the sex x movement interaction). Among four different (independent) sessions, the left eye closure had the worst repeatability, the right eye closure had the best one; the differences among various movements were statistically significant (one-way analysis of variance, p=0.041). In conclusion, the current protocol demonstrated a sufficient repeatability for a future clinical application. Great care should be taken to assure a consistent marker positioning in all the subjects.     

Filtered kriging for spatial data with heterogeneous measurement error variances

When predicting values for the measurement-error-free component of an observed spatial process, it is generally assumed that the process has a common measurement error variance. However, it is often the case that each measurement in a spatial data set has a known, site-specific measurement error variance, rendering the observed process nonstationary. We present a simple approach for estimating the semivariogram of the unobservable measurement-error-free process using a bias adjustment of the classical semivariogram formula. We then develop a new kriging predictor that filters the measurement errors. For scenarios where each site's measurement error variance is a function of the process of interest, we recommend an approach that also uses a variance-stabilizing transformation. The properties of the heterogeneous variance measurement-error-filtered kriging (HFK) predictor and variance-stabilized HFK predictor, and the improvement of these approaches over standard measurement-error-filtered kriging are demonstrated using simulation. The approach is illustrated with climate model output from the Hudson Strait area in northern Canada. In the illustration, locations with high or low measurement error variances are appropriately down- or upweighted in the prediction of the underlying process, yielding a realistically smooth picture of the phenomenon of interest.     

Estimation of the commercial height of trees with laser meter: a viable alternative for forest management in the Brazilian Amazon

Commercial height of the tree is a key variable for estimating the wood stock in tropical forests managed for timber production purposes. Most available measurement devices suffer limitations in this type of forest, promoting low precision measurements with high variation errors. The laser meter device appears as a viable alternative, as in addition to using trigonometric principles, it is not necessary that the device is close to the eyes of the meter to carry out the measurement. The device can be used to measure commercial height of trees on flat or sloping terrain, at different distances from the tree. However, there are no studies evaluating the precision of this device. The objective of this study was to determine the precision of the laser meter method for estimating the commercial height of trees, as compared to the actual measurement in a tropical forest in the Brazilian Amazon. Measurements were made on 300 trees with commercial height between 7 and 14 m. Actual commercial heights were measured with graduated ruler. Applied tests were paired t test, graphical analysis of residuals and calculations of bias statistics, mean absolute deviation, standard deviation of differences, and coefficient of determination (R²). Paired t test indicated that the mean of the heights measured by the laser meter is statistically equal to that of the graduated ruler. Measurements with laser meter did not show bias and had mean error of 0.0745. The standard deviation of the differences indicated dispersion of errors of 0.97, equal to that shown in the graduated rule. Laser meter presents an alternative method for estimating the commercial height of trees in tropical forest in the Brazilian Amazon. There was no tendency to underestimate or overestimate the commercial heights of trees. Use of the laser meter is potentially of use for measuring the commercial height of trees in tropical forests.     

The use of sequential MR image sets for determining tibiofemoral motion: reliability of coordinate systems and accuracy of motion tracking algorithm

The use of magnetic resonance imaging has been proposed by many investigators for establishment of joint reference systems and kinematic tracking of musculoskeletal joints. In this study, the intraobserver and interobserver reliability of a strategy to establish anatomic reference systems using manually selected fiducial points were quantified for seven sets of MR images of the human knee joint. The standard error of the measurement of the intraobserver and interobserver errors were less than 2.6 degrees, and 1.2 mm for relative tibiofemoral orientation and displacement, respectively. An automated motion tracking algorithm was also validated with a controlled motion experiment in a cadaveric knee joint. The controlled displacements and rotations prescribed in our motion tracking validation were highly correlated to those predicted (Pearson's correlation = 0.99, RMS errors = 0.39 mm, 0.38 degree). Finally, the system for anatomic reference system definition and motion tracking was demonstrated with a set of MR images of in vivo passive flexion in the human knee.     

Joint models for a primary endpoint and multiple longitudinal covariate processes

Summary .   Joint models are formulated to investigate the association between a primary endpoint and features of multiple longitudinal processes. In particular, the subject-specific random effects in a multivariate linear random-effects model for multiple longitudinal processes are predictors in a generalized linear model for primary endpoints. Li, Zhang, and Davidian (2004, Biometrics 60 , 1–7) proposed an estimation procedure that makes no distributional assumption on the random effects but assumes independent within-subject measurement errors in the longitudinal covariate process. Based on an asymptotic bias analysis, we found that their estimators can be biased when random effects do not fully explain the within-subject correlations among longitudinal covariate measurements. Specifically, the existing procedure is fairly sensitive to the independent measurement error assumption. To overcome this limitation, we propose new estimation procedures that require neither a distributional or covariance structural assumption on covariate random effects nor an independence assumption on within-subject measurement errors. These new procedures are more flexible, readily cover scenarios that have multivariate longitudinal covariate processes, and can be implemented using available software. Through simulations and an analysis of data from a hypertension study, we evaluate and illustrate the numerical performances of the new estimators.     

Bias and error in using survey records for ponderosa pine landscape restoration

Aim Public land survey records are commonly used to reconstruct historical forest structure over large landscapes. Reconstruction studies have been criticized for using absolute measures of forest attributes, such as density and basal area, because of potential selection bias by surveyors and unknown measurement error. Current methods to identify bias are based upon statistical techniques whose assumptions may be violated for survey data. Our goals were to identify and directly estimate common sources of bias and error, and to test the accuracy of statistical methods to identify them. Location Forests in the western USA: Mogollon Plateau, Arizona; Blue Mountains, Oregon; Front Range, Colorado. Methods We quantified both selection bias and measurement error for survey data in three ponderosa pine landscapes by directly comparing measurements of bearing trees in survey notes with remeasurements of bearing trees at survey corners (384 corners and 812 trees evaluated). Results Selection bias was low in all areas and there was little variability among surveyors. Surveyors selected the closest tree to the corner 95% to 98% of the time, and hence bias may have limited impacts on reconstruction studies. Bourdo’s methods were able to successfully detect presence or absence of bias most of the time, but do not measure the rate of bias. Recording and omission errors were common but highly variable among surveyors. Measurements for bearing trees made by surveyors were generally accurate. Most bearings were less than 5° in error and most distances were within 5% of our remeasurements. Many, but not all, surveyors in the western USA probably estimated diameter of bearing trees at stump height (0.3 m). These estimates deviated from reconstructed diameters by a mean absolute error of 7.0 to 10.6 cm. Main conclusions Direct comparison of survey data at relocated corners is the only method that can determine if bias and error are meaningful. Data from relocated trees show that biased selection of trees is not likely to be an important source of error. Many surveyor errors would have no impact on reconstruction studies, but omission errors have the potential to have a large impact on results. We suggest how to reduce potential errors through data screening.     

Estimation of a Nonlinear Density-Dependence Parameter for Wild Turkey

Abstract: Although previous research and theory has suggested that wild turkey (Meleagris gallopavo) populations may be subject to some form of density dependence, there has been no effort to estimate and incorporate a density-dependence parameter into wild turkey population models. To estimate a functional relationship for density dependence in wild turkey, we analyzed a set of harvest-index time series from 11 state wildlife agencies. We tested for lagged correlations between annual harvest indices using partial autocorrelation analysis. We assessed the ability of the density-dependent theta-Ricker model to explain harvest indices over time relative to exponential or random walk growth models. We tested the homogeneity of the density-dependence parameter estimates (θ) from 3 different harvest indices (spring harvest no. reported harvest/effort, survey harvest/effort) and calculated a weighted average based on each estimate's variance and its estimated covariance with the other indices. To estimate the potential bias in parameter estimates from measurement error, we conducted a simulation study using the theta-Ricker with known values and lognormally distributed measurement error. Partial autocorrelation function analysis indicated that harvest indices were significantly correlated only with their value at the previous time step. The theta-Ricker model performed better than the exponential growth or random walk models for all 3 indices. Simulation of known parameters and measurement error indicated a strong positive upward bias in the density-dependent parameter estimate, with increasing measurement error. The average density-dependence estimate, corrected for measurement error ranged 0.25 ≤ θC ≤ 0.49, depending on the amount of measurement error and assumed spring harvest rate. We infer that density dependence is nonlinear in wild turkey, where growth rates are maximized at 39-42% of carrying capacity. The annual yield produced by density-dependent population growth will tend to be less than that caused by extrinsic environmental factors. This study indicates that both density-dependent and density-independent processes are important to wild turkey population growth, and we make initial suggestions on incorporating both into harvest management strategies.     

Bias and error in estimates of mean shape in geometric morphometrics

Sampling experiments were performed to investigate mean square error and bias in estimates of mean shape produced by different geometric morphometric methods. The experiments use the isotropic error model, which assumes equal and independent variation at each landmark. The case of three landmarks in the plane (i.e., triangles) was emphasized because it could be investigated systematically and the results displayed on the printed page. The amount of error in the estimates was displayed as RMSE surfaces over the space of all possible configurations of three landmarks. Patterns of bias were shown as vector fields over this same space. Experiments were also performed using particular combinations of four or more landmarks in both two and three dimensions.It was found that the generalized Procrustes analysis method produced estimates with the least error and no pattern of bias. Averages of Bookstein shape coordinates performed well if the longest edge was used as the baseline. The method of moments (Stoyan, 1990, Model. Biomet. J. 32, 843) used in EDMA (Lele, 1993, Math. Geol. 25, 573) exhibits larger errors. When variation is not small, it also shows a pattern of bias for isosceles triangles with one side much shorter than the other two and for triangles whose vertices are approximately collinear causing them to resemble their own reflections. Similar problems were found for the log-distance method of Rao and Suryawanshi (1996, Proc. Nat. Acad. Sci. 95, 4121). These results and their implications for the application of different geometric morphometric methods are discussed.     

Empirical Bayes estimation of random effects parameters in mixed effects logistic regression models

We extend an approach for estimating random effects parameters under a random intercept and slope logistic regression model to include standard errors, thereby including confidence intervals. The procedure entails numerical integration to yield posterior empirical Bayes (EB) estimates of random effects parameters and their corresponding posterior standard errors. We incorporate an adjustment of the standard error due to Kass and Steffey (KS; 1989, Journal of the American Statistical Association 84, 717-726) to account for the variability in estimating the variance component of the random effects distribution. In assessing health care providers with respect to adult pneumonia mortality, comparisons are made with the penalized quasi-likelihood (PQL) approximation approach of Breslow and Clayton (1993, Journal of the American Statistical Association 88, 9-25) and a Bayesian approach. To make comparisons with an EB method previously reported in the literature, we apply these approaches to crossover trials data previously analyzed with the estimating equations EB approach of Waclawiw and Liang (1994, Statistics in Medicine 13, 541-551). We also perform simulations to compare the proposed KS and PQL approaches. These two approaches lead to EB estimates of random effects parameters with similar asymptotic bias. However, for many clusters with small cluster size, the proposed KS approach does better than the PQL procedures in terms of coverage of nominal 95% confidence intervals for random effects estimates. For large cluster sizes and a few clusters, the PQL approach performs better than the KS adjustment. These simulation results agree somewhat with those of the data analyses.     

Reliability of Confidence Interval Estimators under Various Nested Design Parental Sample Sizes

The coverage probabilities of several confidence limit estimators of genetic parameters, obtained from North Carolina I designs, were assessed by means of Monte Carlo simulations. The reliability of the estimators was compared under three different parental sample sizes. The coverage of confidence intervals set on the Normal distribution, and using standard errors either computed by the “delta” method or derived using an approximation for the variance of a variance component estimated by means of a linear combination of mean squares, was affected by the number of males and females included in the experiment. The “delta” method was found to provide reliable standard errors of the genetic parameters only when at least 48 males were each mated to six different females randomly selected from the reference population. Formulae are provided for obtaining “delta” method standard errors, and appropriate statistical software procedures are discussed. The error rates of confidence limits based on the Normal distribution and using standard errors obtained by an approximation for the variance of a variance component varied widely. The coverage of F-distribution confidence intervals for heritability estimates was not significantly affected by parental sample size and consistently provided a mean coverage near the stated coverage. For small parental sample sizes, confidence intervals for heritability estimates should be based on the F-distribution.     

Estimates of absolute cause-specific risk in cohort studies

In this paper we study methods for estimating the absolute risk of an event c1 in a time interval [t1, t2], given that the individual is at risk at t1 and given the presence of competing risks. We discuss some advantages of absolute risk for measuring the prognosis of an individual patient and some difficulties of interpretation for comparing two treatment groups. We also discuss the importance of the concept of absolute risk in evaluating public health measures to prevent disease. Variance calculations permit one to gauge the relative importance of random and systematic errors in estimating absolute risk. Efficiency calculations were also performed to determine how much precision is lost in estimating absolute risk with a nonparametric approach or with a flexible piecewise exponential model rather than a simple exponential model, and other calculations indicate the extent of bias that arises with the simple exponential model when that model is invalid. Such calculations suggest that the more flexible models will be useful in practice. Simulations confirm that asymptotic methods yield reliable variance estimates and confidence interval coverages in samples of practical size.     

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.     

Modeling observation error and its effects in a random walk/extinction model

This paper examines the consequences of observation errors for the "random walk with drift", a model that incorporates density independence and is frequently used in population viability analysis. Exact expressions are given for biases in estimates of the mean, variance and growth parameters under very general models for the observation errors. For other quantities, such as the finite rate of increase, and probabilities about population size in the future we provide and evaluate approximate expressions. These expressions explain the biases induced by observation error without relying exclusively on simulations, and also suggest ways to correct for observation error. A secondary contribution is a careful discussion of observation error models, presented in terms of either log-abundance or abundance. This discussion recognizes that the bias and variance in observation errors may change over time, the result of changing sampling effort or dependence on the underlying population being sampled.     

Measurement error: Inter-and Intraobserver Variability. An Empiric Study

The intra- and inter-observer measurement error variability was studied using univariate and multivariate statistical tests. Eleven skeletal variables of four individuals each in four Primate species were measured ten times by three different researchers, using six different tools. An average measurement error of 0.52 mm. was obtained. Univariate statistics showed significant differences among reseachers. A multivariate discriminant analysis could also discriminate them. The measurement error may be either systematic or random, and depends not only on the researcher, but also on the tool used, the variable measured, and on the magnitude of the variable. The technique of Measurement Replication is proposed in order to reduce the measurement error, specially when compairing small samples or when trying to find small average differences between populations. The replication technique also reduces the standard deviation of the population sample.     

Is cross-validation valid for small-sample microarray classification?

MOTIVATION: Microarray classification typically possesses two striking attributes: (1) classifier design and error estimation are based on remarkably small samples and (2) cross-validation error estimation is employed in the majority of the papers. Thus, it is necessary to have a quantifiable understanding of the behavior of cross-validation in the context of very small samples. RESULTS: An extensive simulation study has been performed comparing cross-validation, resubstitution and bootstrap estimation for three popular classification rules-linear discriminant analysis, 3-nearest-neighbor and decision trees (CART)-using both synthetic and real breast-cancer patient data. Comparison is via the distribution of differences between the estimated and true errors. Various statistics for the deviation distribution have been computed: mean (for estimator bias), variance (for estimator precision), root-mean square error (for composition of bias and variance) and quartile ranges, including outlier behavior. In general, while cross-validation error estimation is much less biased than resubstitution, it displays excessive variance, which makes individual estimates unreliable for small samples. Bootstrap methods provide improved performance relative to variance, but at a high computational cost and often with increased bias (albeit, much less than with resubstitution).     

EM-REML estimation of covariance parameters in Gaussian mixed models for longitudinal data analysis

This paper presents procedures for implementing the EM algorithm to compute REML estimates of variance covariance components in Gaussian mixed models for longitudinal data analysis. The class of models considered includes random coefficient factors, stationary time processes and measurement errors. The EM algorithm allows separation of the computations pertaining to parameters involved in the random coefficient factors from those pertaining to the time processes and errors. The procedures are illustrated with Pothoff and Roy''s data example on growth measurements taken on 11 girls and 16 boys at four ages. Several variants and extensions are discussed.     

How long is a piece of Strix? Methods in measuring and measuring the measurers

An experiment to quantify intra- and interobserver error in anatomical measurements found that interobserver measurements can vary by over 14% of mean specimen length; disparity in measurement increases logarithmically with the number of contributors; instructions did not reduce variation or measurement disparity; scale of the specimen influenced the precision of measurement (relative error increasing with specimen size); different methods of taking a measurement yielded different results, although they did not differ in terms of precision, and topographical complexity of the elements being considered may potentially influence error (error increasing with complexity). These results highlight concerns about introduction of noise and potential bias that should be taken into account when compiling composite datasets and meta-analyses.     

Seemingly unrelated measurement error models, with application to nutritional epidemiology

Motivated by an important biomarker study in nutritional epidemiology, we consider the combination of the linear mixed measurement error model and the linear seemingly unrelated regression model, hence Seemingly Unrelated Measurement Error Models. In our context, we have data on protein intake and energy (caloric) intake from both a food frequency questionnaire (FFQ) and a biomarker, and wish to understand the measurement error properties of the FFQ for each nutrient. Our idea is to develop separate marginal mixed measurement error models for each nutrient, and then combine them into a larger multivariate measurement error model: the two measurement error models are seemingly unrelated because they concern different nutrients, but aspects of each model are highly correlated. As in any seemingly unrelated regression context, the hope is to achieve gains in statistical efficiency compared to fitting each model separately. We show that if we employ a "full" model (fully parameterized), the combination of the two measurement error models leads to no gain over considering each model separately. However, there is also a scientifically motivated "reduced" model that sets certain parameters in the "full" model equal to zero, and for which the combination of the two measurement error models leads to considerable gain over considering each model separately, e.g., 40% decrease in standard errors. We use the Akaike information criterion to distinguish between the two possibilities, and show that the resulting estimates achieve major gains in efficiency. We also describe theoretical and serious practical problems with the Bayes information criterion in this context.