On rethinking allometry: which regression model to use?

Recently the methods of major axis analysis and reduced major axis analysis have frequently been used in allometric studies as a complement to linear regression analysis. This paper gives a comparison, based on some reasonable assumptions, of these three methods. It is shown how the choice of method should depend on (i) which a priori assumptions that are present and (ii) the purpose of the analysis. Neither major axis analysis nor reduced major axis analysis are appropriate for estimating regression lines, but both methods are valuable when estimating a structural relation. Major axis analysis gives consistent estimates of the coefficients when the two error variances are equal, while reduced major axis analysis gives consistent estimates when the relative sizes of the two error variances are equal.None of the methods have universal application. The three methods give rise to the same conclusions only in data-sets with very high correlation coefficients     

Alternative statistical analyses for micropropagation: A practical case of proliferation and rooting phases in <Emphasis Type="Italic">Viburnum opulus</Emphasis>

Summary The kinds of data obtained in micropropagation studies are very often problematic, since they do not follow continous distribution and observations through culture vessels complicate measurement. Accordingly, standard analyses are often used, leading to misinterpretation of results. In this paper, we present a study of Viburnum opulus micropropagation using planned contrasts and fitting regression models in generalized linear models as an alternative statistical analysis of micropropagation results, and compare the results with that of traditional ANOVA. The advantages and possibilities of the alternative data analyses in plant tissue culture are discussed.     

Interpreting the evolutionary regression: the interplay between observational and biological errors in phylogenetic comparative studies

Regressions of biological variables across species are rarely perfect. Usually, there are residual deviations from the estimated model relationship, and such deviations commonly show a pattern of phylogenetic correlations indicating that they have biological causes. We discuss the origins and effects of phylogenetically correlated biological variation in regression studies. In particular, we discuss the interplay of biological deviations with deviations due to observational or measurement errors, which are also important in comparative studies based on estimated species means. We show how bias in estimated evolutionary regressions can arise from several sources, including phylogenetic inertia and either observational or biological error in the predictor variables. We show how all these biases can be estimated and corrected for in the presence of phylogenetic correlations. We present general formulas for incorporating measurement error in linear models with correlated data. We also show how alternative regression models, such as major axis and reduced major axis regression, which are often recommended when there is error in predictor variables, are strongly biased when there is biological variation in any part of the model. We argue that such methods should never be used to estimate evolutionary or allometric regression slopes.     

Linear relations in biomechanics: the statistics of scaling functions

The problem of fitting a linear relation to a bivariate data cluster obtained from morphometric measurement or from experiment is formulated rigorously, and a family of solutions (the general structural relation , g.s.r.) is derived. The regression, reduced major axis and major axis models are special cases of this model; it permits a more realistic treatment of the errors in the variates, in particular when the errors are correlated, which is particularly important in the many biological situations in which the variates contain uncontrolled real variation in addition to measurement errors. The analysis is particularly directed to the testing of hypotheses about scaling relations derived from biomechanical theory. By making different assumptions about the configuration of the errors, the g.s.r. can also be used to test for transposition allometry and for the significance of an estimated or hypothesized gradient. The model generalizes simply to multivariate problems. Application is demonstrated with examples drawn from the study of bird flight mechanics. Finally, it is demonstrated that since observed quantities correspond to peaks of adaptation or of selective fitness, scaling relations are determined primarily by scale variation of constraints on adaptation and behaviour, and are the result of a variety of interacting factors rather than a response to a single selective force described by one simple hypothesis.     

Basicranial axis length v. skull length in analysis of carnivore skull shape

Skull length is the measurement most commonly used as a standard against which other aspects of cranial morphology are compared to derive an index of relative size or proportions. However, skull length is composed of two different functional components, facial skull and cerebral skull, which vary independently and have different scaling relationships with body size. An analysis of carnivore skull shape with measurements standardized against basicranium length produced very different results than an analysis using skull length as the standard. For example, expressions of relative size of cranial measurements were reduced by 13% in mustelids and increased by 20% in canids, reflecting removal of jaw length (short in mustelids and long in canids) from the comparative standard (basicranial axis length). Cranial measurements scale with higher allometric exponents against basicranial axis length than against skull length.     

Residual analysis for linear mixed models

Residuals are frequently used to evaluate the validity of the assumptions of statistical models and may also be employed as tools for model selection. For standard (normal) linear models, for example, residuals are used to verify homoscedasticity, linearity of effects, presence of outliers, normality and independence of the errors. Similar uses may be envisaged for three types of residuals that emerge from the fitting of linear mixed models. We review some of the residual analysis techniques that have been used in this context and propose a standardization of the conditional residual useful to identify outlying observations and clusters. We illustrate the procedures with a practical example.     

Genomic-assisted prediction of genetic value with semiparametric procedures

Semiparametric procedures for prediction of total genetic value for quantitative traits, which make use of phenotypic and genomic data simultaneously, are presented. The methods focus on the treatment of massive information provided by, e.g., single-nucleotide polymorphisms. It is argued that standard parametric methods for quantitative genetic analysis cannot handle the multiplicity of potential interactions arising in models with, e.g., hundreds of thousands of markers, and that most of the assumptions required for an orthogonal decomposition of variance are violated in artificial and natural populations. This makes nonparametric procedures attractive. Kernel regression and reproducing kernel Hilbert spaces regression procedures are embedded into standard mixed-effects linear models, retaining additive genetic effects under multivariate normality for operational reasons. Inferential procedures are presented, and some extensions are suggested. An example is presented, illustrating the potential of the methodology. Implementations can be carried out after modification of standard software developed by animal breeders for likelihood-based or Bayesian analysis.     

Single-particle tracking of oriC-GFP fluorescent spots during chromosome segregation in Escherichia coli

DNA regions close to the origin of replication were visualized by the green fluorescent protein (GFP)-Lac repressor/lac operator system. The number of oriC-GFP fluorescent spots per cell and per nucleoid in batch-cultured cells corresponded to the theoretical DNA replication pattern. A similar pattern was observed in cells growing on microscope slides used for time-lapse experiments. The trajectories of 124 oriC-GFP spots were monitored by time-lapse microscopy of 31 cells at time intervals of 1, 2, and 3 min. Spot positions were determined along the short and long axis of cells. The lengthwise movement of spots was corrected for cell elongation. The step sizes of the spots showed a Gaussian distribution with a standard deviation of approximately 110 nm. Plots of the mean square displacement versus time indicated a free diffusion regime for spot movement along the long axis of the cell, with a diffusion coefficient of 4.3+/-2.6x10(-5) microm2/s. Spot movement along the short axis showed confinement in a region of the diameter of the nucleoid ( approximately 800 nm) with an effective diffusion coefficient of 2.9+/-1.7x10(-5) microm2/s. Confidence levels for the mean square displacement analysis were obtained from numerical simulations. We conclude from the analysis that within the experimental accuracy--the limits of which are indicated and discussed--there is no evidence that spot segregation requires any other mechanism than that of cell (length) growth.     

When Should Epidemiologic Regressions Use Random Coefficients?

Regression models with random coefficients arise naturally in both frequentist and Bayesian approaches to estimation problems. They are becoming widely available in standard computer packages under the headings of generalized linear mixed models, hierarchical models, and multilevel models. I here argue that such models offer a more scientifically defensible framework for epidemiologic analysis than the fixed-effects models now prevalent in epidemiology. The argument invokes an antiparsimony principle attributed to L. J. Savage, which is that models should be rich enough to reflect the complexity of the relations under study. It also invokes the countervailing principle that you cannot estimate anything if you try to estimate everything (often used to justify parsimony). Regression with random coefficients offers a rational compromise between these principles as well as an alternative to analyses based on standard variable-selection algorithms and their attendant distortion of uncertainty assessments. These points are illustrated with an analysis of data on diet, nutrition, and breast cancer.     

Robustness to failure of assumptions of tests for a common slope amongst several allometric lines--a simulation study

In allometry, researchers are commonly interested in estimating the slope of the major axis or standardized major axis (methods of bivariate line fitting related to principal components analysis). This study considers the robustness of two tests for a common slope amongst several axes. It is of particular interest to measure the robustness of these tests to slight violations of assumptions that may not be readily detected in sample datasets. Type I error is estimated in simulations of data generated with varying levels of nonnormality, heteroscedasticity and nonlinearity. The assumption failures introduced in simulations were difficult to detect in a moderately sized dataset, with an expert panel only able to correct detect assumption violations 34-45% of the time. While the common slope tests were robust to nonnormal and heteroscedastic errors from the line, Type I error was inflated if the two variables were related in a slightly nonlinear fashion. Similar results were also observed for the linear regression case. The common slope tests were more liberal when the simulated data had greater nonlinearity, and this effect was more evident when the underlying distribution had longer tails than the normal. This result raises concerns for common slopes testing, as slight nonlinearities such as those in simulations are often undetectable in moderately sized datasets. Consequently, practitioners should take care in checking for nonlinearity and interpreting the results of a test for common slope. This work has implications for the robustness of inference in linear models in general.     

Principal components analysis and trend surface analysis of a small-scale pattern in a transition mire

Principal components analysis and trend surface analysis have been applied to a transition mire with the aim to characterize the vegetation pattern and reveal the major trends of variation. The first three PCA axes were ecologically interpretable, viz. the 1 st and 2nd as a complex soil moisture gradient and the 3rd axis as a gradient in the amount of peat in the soil. The ecological interpretability of the 1st axis of PCA after VARIMAX rotation, is unclear because some outlier samples caused a reorientation of the axis. TSA appeared to be useful for the clarification of joint patterns of species groups, which were major contributors to ordination axes in terms of component loadings. The smooth effect of TSA was briefly discussed in connection with the influence of extremes upon the outcoming trend structure. The use of four-variable TSA including a time series is emphasized for the study of spatial-temporal relations and ecological succession.     

Ratios as a size adjustment in morphometrics

Simple ratios in which a measurement variable is divided by a size variable are commonly used but known to be inadequate for eliminating size correlations from morphometric data. Deficiencies in the simple ratio can be alleviated by incorporating regression coefficients describing the bivariate relationship between the measurement and size variables. Recommendations have included: 1) subtracting the regression intercept to force the bivariate relationship through the origin (intercept-adjusted ratios); 2) exponentiating either the measurement or the size variable using an allometry coefficient to achieve linearity (allometrically adjusted ratios); or 3) both subtracting the intercept and exponentiating (fully adjusted ratios). These three strategies for deriving size-adjusted ratios imply different data models for describing the bivariate relationship between the measurement and size variables (i.e., the linear, simple allometric, and full allometric models, respectively). Algebraic rearrangement of the equation associated with each data model leads to a correctly formulated adjusted ratio whose expected value is constant (i.e., size correlation is eliminated). Alternatively, simple algebra can be used to derive an expected value function for assessing whether any proposed ratio formula is effective in eliminating size correlations. Some published ratio adjustments were incorrectly formulated as indicated by expected values that remain a function of size after ratio transformation. Regression coefficients incorporated into adjusted ratios must be estimated using least-squares regression of the measurement variable on the size variable. Use of parameters estimated by any other regression technique (e.g., major axis or reduced major axis) results in residual correlations between size and the adjusted measurement variable. Correctly formulated adjusted ratios, whose parameters are estimated by least-squares methods, do control for size correlations. The size-adjusted results are similar to those based on analysis of least-squares residuals from the regression of the measurement on the size variable. However, adjusted ratios introduce size-related changes in distributional characteristics (variances) that differentially alter relationships among animals in different size classes. © 1993 Wiley-Liss, Inc.     

A new method for estimating joint parameters from motion data

Joint centers and axes of rotation (joint parameters) are central to all branches of movement analysis. In gait analysis, the standard protocol used to determine hip and knee joint parameters is prone to errors arising from palpation, anthropometric regression equations, and misplaced alignment devices. Several alternative methods have been proposed, but to date none have been shown to be accurate and reliable enough for use in the clinical setting. This article describes a new method for joint parameter estimation. The new method can be summarized as follows: (i) the motions of two adjacent segments spanning a single joint are tracked, (ii) the axis of rotation between every pair of observed segment configurations is computed, (iii) the most likely intersection of all axes (effective joint center) and most likely orientation of the axes (effective joint axis) is found. Initial validation of the method was conducted on a hinged mechanical analog and a single healthy adult subject. For the analog, the center was found to be within 3.8 mm of the geometric center and 2.0 degrees of the geometric axis (standard deviation). For the adult subject, hip centers varied on the order of 1-3 mm, knee centers by 3-9 mm, and knee axes by 2.0 degrees. The results suggest that the new method is an objective, precise, and practical alternative to the standard clinical approach.     

Bivariate random‐effects meta‐analysis models for diagnostic test accuracy studies using arcsine‐based transformations

Diagnostic or screening tests are widely used in medical fields to classify patients according to their disease status. Several statistical models for meta‐analysis of diagnostic test accuracy studies have been developed to synthesize test sensitivity and specificity of a diagnostic test of interest. Because of the correlation between test sensitivity and specificity, modeling the two measures using a bivariate model is recommended. In this paper, we extend the current standard bivariate linear mixed model (LMM) by proposing two variance‐stabilizing transformations: the arcsine square root and the Freeman–Tukey double arcsine transformation. We compared the performance of the proposed methods with the standard method through simulations using several performance measures. The simulation results showed that our proposed methods performed better than the standard LMM in terms of bias, root mean square error, and coverage probability in most of the scenarios, even when data were generated assuming the standard LMM. We also illustrated the methods using two real data sets.     

A simple method to fit geometric series and broken stick models in community ecology and island biogeography

Species abundance distributions are widely used in explaining natural communities, their natural evolution and the impacts of environmental disturbance. A commonly used approach is that of rank-abundance distributions. Favored, biologically founded models are the geometric series (GS) and the broken stick (BS) model. Comparing observed abundance distributions with those predicted by models is an extremely time-consuming task. Also, using goodness-of-fit tests for frequency distributions (like Chi-square or Kolmogorov–Smirnov tests) to compare observed with expected frequencies is problematic because the best way to calculate expected frequencies may be controversial. More important, the Chi-square test may prove if an observed distribution statistically differs from a model, but does not allow the investigator to choose among competing models from which the observed distribution does not differ. Both models can be easily tested by regression analysis. In GS, if a log scale is used for abundance, the species exactly fall along a straight line. The BS distribution shows up as nearly linear when a log scale is used for the rank axis. Regression analysis is proposed here as a simpler and more efficient method to fit the GS and BS models. Also, regression analysis (1) does not suffer from assumptions related to Chi-square tests; (2) obviates the need to establish expected frequencies, and (3) offers the possibility to choose the best fit among competing models. A possible extension of abundance-rank analysis to species richness on islands is also proposed as a method to discriminate between relict and equilibrial models. Examples of application to field data are also presented.     

A simple method to fit geometric series and broken stick models in community ecology and island biogeography

Species abundance distributions are widely used in explaining natural communities, their natural evolution and the impacts of environmental disturbance. A commonly used approach is that of rank-abundance distributions. Favored, biologically founded models are the geometric series (GS) and the broken stick (BS) model. Comparing observed abundance distributions with those predicted by models is an extremely time-consuming task. Also, using goodness-of-fit tests for frequency distributions (like Chi-square or Kolmogorov–Smirnov tests) to compare observed with expected frequencies is problematic because the best way to calculate expected frequencies may be controversial. More important, the Chi-square test may prove if an observed distribution statistically differs from a model, but does not allow the investigator to choose among competing models from which the observed distribution does not differ. Both models can be easily tested by regression analysis. In GS, if a log scale is used for abundance, the species exactly fall along a straight line. The BS distribution shows up as nearly linear when a log scale is used for the rank axis. Regression analysis is proposed here as a simpler and more efficient method to fit the GS and BS models. Also, regression analysis (1) does not suffer from assumptions related to Chi-square tests; (2) obviates the need to establish expected frequencies, and (3) offers the possibility to choose the best fit among competing models. A possible extension of abundance-rank analysis to species richness on islands is also proposed as a method to discriminate between relict and equilibrial models. Examples of application to field data are also presented.