Growth models and the expected distribution of fluctuating asymmetry

Multiplicative error accounts for much of the size-scaling and leptokurtosis in fluctuating asymmetry. It arises when growth involves the addition of tissue to that which is already present. Such errors are lognormally distributed. The distribution of the difference between two lognormal variates is leptokurtic. If those two variates are correlated, then the asymmetry variance will scale with size. Inert tissues typically exhibit additive error and have a gamma distribution. Although their asymmetry variance does not exhibit size-scaling, the distribution of the difference between two gamma variates is nevertheless leptokurtic. Measurement error is also additive, but has a normal distribution. Thus, the measurement of fluctuating asymmetry may involve the mixing of additive and multiplicative error. When errors are multiplicative, we recommend computing log  E ( l ) − log  E ( r ), the difference between the logarithms of the expected values of left and right sides, even when size-scaling is not obvious. If l and r are lognormally distributed, and measurement error is nil, the resulting distribution will be normal, and multiplicative error will not confound size-related changes in asymmetry. When errors are additive, such a transformation to remove size-scaling is unnecessary. Nevertheless, the distribution of l  −  r may still be leptokurtic.  © 2003 The Linnean Society of London, Biological Journal of the Linnean Society , 2003, 80, 57–65.     

No relationship between fluctuating asymmetry and fitness in Lychnis viscaria

Several earlier studies have indicated a negative relationship between fluctuating asymmetry (FA) and fitness. We tested this assumption by investigating the association between petal asymmetry and several fitness-related characters among natural and common garden populations of Lychnis viscaria. Neither seed set, germination percentage nor the growth rate of seedlings were related to the level of flower asymmetry either among natural populations or in common garden conditions. The only significant association found was a positive connection between petal asymmetry and seed mass measured from natural populations. Thus, in contrary to many earlier published reports, we did not find any evidence for a negative relationship between FA and fitness even if we controlled for measurement error, we had adequate sample size and we measured these characters in two environments. This suggests that FA is not consistently related to individual quality and fitness.     

Inference about misclassification probabilities from repeated binary responses

Repeated binary responses provide efficient information for two purposes: (1) estimating two misclassification (false-positive and false-negative error) probabilities and (2) testing the hypothesis that either is zero in a reliability study. We focus on the assessment of reliability of a diagnostic test when there is no gold standard. This paper uses a latent class model and illustrates some of its properties. In addition, application to data containing variation among individuals is considered. We apply this model to the serological data on the MNSs blood group of atomic bomb survivors and their children. The results provide valuable information for examining measurement reliability.     

Linear measurement error models with restricted sampling

The relationship between nutrient consumption and chronic disease risk is the focus of a large number of epidemiological studies where food frequency questionnaires (FFQ) and food records are commonly used to assess dietary intake. However, these self-assessment tools are known to involve substantial random error for most nutrients, and probably important systematic error as well. Study subject selection in dietary intervention studies is sometimes conducted in two stages. At the first stage, FFQ-measured dietary intakes are observed and at the second stage another instrument, such as a 4-day food record, is administered only to participants who have fulfilled a prespecified criterion that is based on the baseline FFQ-measured dietary intake (e.g., only those reporting percent energy intake from fat above a prespecified quantity). Performing analysis without adjusting for this truncated sample design and for the measurement error in the nutrient consumption assessments will usually provide biased estimates for the population parameters. In this work we provide a general statistical analysis technique for such data with the classical additive measurement error that corrects for the two sources of bias. The proposed technique is based on multiple imputation for longitudinal data. Results of a simulation study along with a sensitivity analysis are presented, showing the performance of the proposed method under a simple linear regression model.     

Modelling developmental instability in relation to individual fitness: a fully Bayesian latent variable model approach

Since the influential paper by Palmer and Strobeck in 1986, the statistical analysis of fluctuating asymmetry and developmental stability has received much attention. Most studies deal with one of the following four difficulties: (i) correcting for bias in asymmetry estimates due to measurement error; (ii) quantifying sampling error in the estimation of individual developmental stability using individual asymmetry; (iii) the detection of directional asymmetry and antisymmetry; and (iv) combining data from several traits. Yet, few studies have focused on statistical properties of estimating a relationship between individual developmental stability and other factors (e.g. fitness). In this paper I introduce a fully Bayesian model in which the unobservable individual developmental stability is treated as a latent variable. The latter is then related to individual fitness. I show by means of the analysis of simulated data that this approach has several advantages over traditional techniques. First, the method provides unbiased (but slightly less accurate) estimates of slopes between developmental stability and fitness taking all sources of error into account. Secondly, it allows proper investigation of non‐linear associations. Finally, the model allows unbiased estimation of unobserved fitness of individuals that have been measured on left and right side.     

Segmented regression in the presence of covariate measurement error in main study/validation study designs

This article considers the problem of segmented regression in the presence of covariate measurement error in main study/validation study designs. First, we derive a closed and interpretable form for the full likelihood. After that, we use the likelihood results to compute the bias of the estimated changepoint in the case when the measurement error is ignored. We find the direction of the bias in the estimated changepoint to be determined by the design distribution of the observed covariates, and the bias can be in either direction. We apply the methodology to data from a nutritional study that investigates the relation between dietary folate and blood serum homocysteine levels and find that the analysis that ignores covariate measurement error would have indicated a much higher minimum daily dietary folate intake requirement than is obtained in the analysis that takes covariate measurement error into account.     

Correction for Covariate Measurement Error in Nonparametric Longitudinal Regression

Summary We introduce a correction for covariate measurement error in nonparametric regression applied to longitudinal binary data arising from a study on human sleep. The data have been surveyed to investigate the association of some hormonal levels and the probability of being asleep. The hormonal effect is modeled flexibly while we account for the error‐prone measurement of its concentration in the blood and the longitudinal character of the data. We present a fully Bayesian treatment utilizing Markov chain Monte Carlo inference techniques, and also introduce block updating to improve sampling and computational performance in the binary case. Our model is partly inspired by the relevance vector machine with radial basis functions, where usually very few basis functions are automatically selected for fitting the data. In the proposed approach, we implement such data‐driven complexity regulation by adopting the idea of Bayesian model averaging. Besides the general theory and the detailed sampling scheme, we also provide a simulation study for the Gaussian and the binary cases by comparing our method to the naive analysis ignoring measurement error. The results demonstrate a clear gain when using the proposed correction method, particularly for the Gaussian case with medium and large measurement error variances, even if the covariate model is misspecified.     

Intra- and interrater reliability of the establishment of one repetition maximum on squat and seated knee extension

The purpose of the present study was to develop a systematic procedure for the establishment of 1 repetition maximum (1RM) in order to describe an easily accessible test procedure that is applicable for physical therapists and athletic trainers who manage strength training for healthy individuals and patients. Another purpose was to investigate the intra- and interrater reliability of 1RM of squat on 1 leg and seated knee extension on 1 leg. Estimates of leg strength and ratings of perceived exertion formed the basis of the amount of load selected. The reliability of the procedure was assessed by a test-retest design. One RM was established for 16 and 27 healthy individuals, for squat and knee extension, respectively. The intrarater reliability of 1RM of squat on 1 leg was questionable (intraclass correlation [ICC] 0.64, measurement error 13.1 kg). The interrater reliability of 1RM of squat on 1 leg was clinically acceptable (ICC 0.94, measurement error 5.2 kg). The intrarater and interrater reliability of 1RM of seated knee extension on 1 leg was clinically acceptable (ICC 0.90, measurement error 5.1 kg and ICC 0.96, measurement error 3.2 kg, respectively). In conclusion, both exercises can be used to determine the load in exercise programs. In addition, seated knee extension may be used to evaluate strength. In contrast, squat on 1 leg is more uncertain to use at assessments between different days, which may be due to the complexity of this exercise. The test, performed in the described manner, is suitable for physical therapists, athletic trainers, and strength and conditioning coaches in clinical practice working with strength training and rehabilitation.     

Optimal allocation of replicates for measurement evaluation studies

Optimal experimental design is important for the efficient use of modern highthroughput technologies such as microarrays and proteomics. Multiple factors including the reliability of measurement system, which itself must be estimated from prior experimental work, could influence design decisions. In this study, we describe how the optimal number of replicate measures （technical replicates） for each biological sample （biological replicate） can be determined. Different allocations of biological and technical replicates were evaluated by minimizing the variance of the ratio of technical variance （measurement error） to the total variance （sum of sampling error and measurement error）. We demonstrate that if the number of biological replicates and the number of technical replicates per biological sample are variable, while the total number of available measures is fixed, then the optimal allocation of replicates for measurement evaluation experiments requires two technical replicates for each biological replicate. Therefore, it is recommended to use two technical replicates for each biological replicate if the goal is to evaluate the reproducibility of measurements.     

On latent-variable model misspecification in structural measurement error models for binary response

Summary .  We consider structural measurement error models for a binary response. We show that likelihood-based estimators obtained from fitting structural measurement error models with pooled binary responses can be far more robust to covariate measurement error in the presence of latent-variable model misspecification than the corresponding estimators from individual responses. Furthermore, despite the loss in information, pooling can provide improved parameter estimators in terms of mean-squared error. Based on these and other findings, we create a new diagnostic method to detect latent-variable model misspecification in structural measurement error models with individual binary response. We use simulation and data from the Framingham Heart Study to illustrate our methods.     

Measurement error caused by spatial misalignment in environmental epidemiology

In many environmental epidemiology studies, the locations and/or times of exposure measurements and health assessments do not match. In such settings, health effects analyses often use the predictions from an exposure model as a covariate in a regression model. Such exposure predictions contain some measurement error as the predicted values do not equal the true exposures. We provide a framework for spatial measurement error modeling, showing that smoothing induces a Berkson-type measurement error with nondiagonal error structure. From this viewpoint, we review the existing approaches to estimation in a linear regression health model, including direct use of the spatial predictions and exposure simulation, and explore some modified approaches, including Bayesian models and out-of-sample regression calibration, motivated by measurement error principles. We then extend this work to the generalized linear model framework for health outcomes. Based on analytical considerations and simulation results, we compare the performance of all these approaches under several spatial models for exposure. Our comparisons underscore several important points. First, exposure simulation can perform very poorly under certain realistic scenarios. Second, the relative performance of the different methods depends on the nature of the underlying exposure surface. Third, traditional measurement error concepts can help to explain the relative practical performance of the different methods. We apply the methods to data on the association between levels of particulate matter and birth weight in the greater Boston area.     

Accuracy and power in fluctuating asymmetry studies: effects of sample size and number of within-subject repeats

This paper presents results from simulations investigating the effect of sample size, number of within-subject repeats and relative degree of measurement error on the power and accuracy of test for fluctuating asymmetry (FA). These data confirm that sampling variation of population-level FA-estimates is large and that high sample size is required to obtain reasonably high power when testing for FA or comparing FA levels between populations. The results also clearly show that increasing the number of within-subject repeats can dramatically increase accuracy and power when measurement error is relatively high.     

Intra- and inter-rater reliability of electrical impedance myography using adhesive electrodes in healthy volunteers

In spite of the growing use of the electrical impedance myography (EIM) measures for clinical assessment and follow-up of diseased muscle tissue, reliability studies are scarce. We evaluate the reliability of the (EIM) technique using four adhesive electrodes over the muscle of interest.Intra- and inter-rater reliability was studied within the same session and between sessions. Thirty-one healthy and volunteer subjects aged between 20 and 26 years were recruited. Phase angle, reactance and resistance were assessed for each EIM measurement. Intraclass correlation coefficient (ICC) was used to determine the relative reliability. Absolute reliability was expressed as the standard error of measurement and the minimum detectable change.Relative reliability within the same session and between sessions for the EIM technique was excellent (ICCs > 0.9) concerning both intra- and inter-rater reliability, except for the component reactance. The absolute reliability was very high for the three EIM components.EIM measures using four adhesive electrodes over the area of interest is a reliable technique to assess muscle tissue status. This study confirms that these measurement results barely vary depending on the examiner and the moment. The present study also confirms phase angle as the least affected EIM component by examiner and evaluation moment.     

A New 3-Dimensional Dynamic Quantitative Analysis System of Facial Motion: An Establishment and Reliability Test

This study aimed to establish a 3-dimensional dynamic quantitative facial motion analysis system, and then determine its accuracy and test-retest reliability. The system could automatically reconstruct the motion of the observational points. Standardized T-shaped rod and L-shaped rods were used to evaluate the static and dynamic accuracy of the system. Nineteen healthy volunteers were recruited to test the reliability of the system. The average static distance error measurement was 0.19 mm, and the average angular error was 0.29°. The measuring results decreased with the increase of distance between the cameras and objects, 80 cm of which was considered to be optimal. It took only 58 seconds to perform the full facial measurement process. The average intra-class correlation coefficient for distance measurement and angular measurement was 0.973 and 0.794 respectively. The results demonstrated that we successfully established a practical 3-dimensional dynamic quantitative analysis system that is accurate and reliable enough to meet both clinical and research needs.     

Developmental Instability and Sexual Selection in Stag Beetles from Chernobyl and a Control Area

Radioactivity may have both phenotypic and genetic effects through its disruption of physiological processes and mutations. I studied the size and asymmetry of secondary sexual and ordinary morphological characters of stag beetles ( Lucanus cervus ) in two areas in the Ukraine: near Chernobyl, where levels of radiation are high, and in a control area with low background radiation. Developmental instability of morphological characters was estimated from the degree of fluctuating asymmetry using the restricted maximum likelihood parameter estimation (REML) method to partition measurement error from asymmetry. The degree of asymmetry estimated from unsigned differences in size of right and left secondary sexual character was similar to estimates based on the REML method. Beetles from the contaminated area had a significantly elevated level of fluctuating asymmetry in the secondary sexual character compared with animals from the control area. Male stag beetles found with a female had significantly lower asymmetry than males found alone. While mated males did not differ in asymmetry between areas, unmated males from Chernobyl were much more asymmetric than unmated males from the control area. These findings provide evidence for radiation disrupting developmental homeostasis, and thereby affecting the mating status of free-living beetles.     

Ordinary least squares regression is indicated for studies of allometry

When it comes to fitting simple allometric slopes through measurement data, evolutionary biologists have been torn between regression methods. On the one hand, there is the ordinary least squares (OLS) regression, which is commonly used across many disciplines of biology to fit lines through data, but which has a reputation for underestimating slopes when measurement error is present. On the other hand, there is the reduced major axis (RMA) regression, which is often recommended as a substitute for OLS regression in studies of allometry, but which has several weaknesses of its own. Here, we review statistical theory as it applies to evolutionary biology and studies of allometry. We point out that the concerns that arise from measurement error for OLS regression are small and straightforward to deal with, whereas RMA has several key properties that make it unfit for use in the field of allometry. The recommended approach for researchers interested in allometry is to use OLS regression on measurements taken with low (but realistically achievable) measurement error. If measurement error is unavoidable and relatively large, it is preferable to correct for slope attenuation rather than to turn to RMA regression, or to take the expected amount of attenuation into account when interpreting the data.     

Unbiased estimation of individual asymmetry

The importance of measurement error (ME) for the estimation of population level fluctuating asymmetry (FA) has long been recognized. At the individual level, however, this aspect has been studied in less detail. Recently, it has been shown that the random slopes of a mixed regression model can estimate individual asymmetry levels that are unbiased with respect to ME. Yet, recent studies have shown that such estimates may fail to reflect heterogeneity in these effects. In this note I show that this is not the case for the estimation of individual asymmetry. The random slopes adequately reflect between‐individual heterogeneity in the underlying developmental instability. Increased levels of ME resulted in, on average, lower estimates of individual asymmetry relative to the traditional unsigned asymmetry. This well‐known shrinkage effect in Bayesian analysis adequately corrected for ME and heterogeneity in ME resulting in unbiased estimates of individual asymmetry that were more closely correlated with the true underlying asymmetry.     

Fluctuating asymmetry in relation to indices of quality and fitness in the damselfly, Enallagmaebrium (Hagen)

Fluctuating asymmetry (FA) refers to random deviations from symmetry of otherwise bilaterally symmetric traits. Researchers have hypothesized that FA should be inversely related to individual quality or fitness. In this study, we tested for FA-quality and FA-fitness relations in the damselfly, Enallagmaebrium (Hagen). We used wet mass of an individual as a measure of its quality and longevity as a measure of its fitness. Contrary to predictions, we found no relation between FA and quality or fitness, even after we controlled for possible confounding factors, such as measurement error and inadequate sample 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.     

Measurement error in heat tolerance assays

Biological measurements inherently involve some measurement error (ME), which is a major concern because measurement accuracy (how closely a measurement reproduces the true value of the attribute being measured) and statistical power steadily decrease with increasing ME. However, ME has been largely overlooked in the thermal biology literature, which can be explained by the fact that thermotolerance estimates often involve the collapse or death of the tested individuals and measurements cannot be repeated in vivo with the same specimen under identical conditions. Here we assess inter- and intra-researcher (test-retest) reliability of heat tolerance measured as knockdown time from digital recordings of Drosophila subobscura flies individually assayed in vials with a dynamic method. We provide a summary of various estimators used to compute measurement reliability (the degree to which the measurement is affected by ME) together with their statistical properties. Our results indicate that the estimation of heat knockdown time has poor reliability: intra-researcher ME=29% and inter-researcher ME=34%. This difference is attributed to lack of ‘accurateness’ (the difference in the marginal distributions of the measurements taken by the two researchers) because measurement imprecision was essentially the same in both estimates (27%). In view of our results we conclude that thermal biologists should report the reliability of thermotolerance estimates and, when necessary, adopt some straightforward guidelines suggested here to improve measurement reliability.