Application of multilevel models to morphometric data. Part 1. Linear models and hypothesis testing.

Morphometric data usually have a hierarchical structure (i.e., cells are nested within patients), which should be taken into consideration in the analysis. In the recent years, special methods of handling hierarchical data, called multilevel models (MM), as well as corresponding software have received considerable development. However, there has been no application of these methods to morphometric data yet. In this paper we report our first experience of analyzing karyometric data by means of MLwiN - a dedicated program for multilevel modeling. Our data were obtained from 34 follicular adenomas and 44 follicular carcinomas of the thyroid. We show examples of fitting and interpreting MM of different complexity, and draw a number of interesting conclusions about the differences in nuclear morphology between follicular thyroid adenomas and carcinomas. We also demonstrate substantial advantages of multilevel models over conventional, single-level statistics, which have been adopted previously to analyze karyometric data. In addition, some theoretical issues related to MM as well as major statistical software for MM are briefly reviewed.     

Predicting renal graft failure using multivariate longitudinal profiles

Patients who have undergone renal transplantation are monitored longitudinally at irregular time intervals over 10 years or more. This yields a set of biochemical and physiological markers containing valuable information to anticipate a failure of the graft. A general linear, generalized linear, or nonlinear mixed model is used to describe the longitudinal profile of each marker. To account for the correlation between markers, the univariate mixed models are combined into a multivariate mixed model (MMM) by specifying a joint distribution for the random effects. Due to the high number of markers, a pairwise modeling strategy, where all possible pairs of bivariate mixed models are fitted, is used to obtain parameter estimates for the MMM. These estimates are used in a Bayes rule to obtain, at each point in time, the prognosis for long-term success of the transplant. It is shown that allowing the markers to be correlated can improve this prognosis.     

Generating Correlation Matrices Based on the Boundaries of Their Coefficients

Correlation coefficients among multiple variables are commonly described in the form of matrices. Applications of such correlation matrices can be found in many fields, such as finance, engineering, statistics, and medicine. This article proposes an efficient way to sequentially obtain the theoretical bounds of correlation coefficients together with an algorithm to generate n n correlation matrices using any bounded random variables. Interestingly, the correlation matrices generated by this method using uniform random variables as an example produce more extreme relationships among the variables than other methods, which might be useful for modeling complex biological systems where rare cases are very important.     

Cellular automaton simulation of tumour growth -- equivocal relationships between simulation parameters and morphologic pattern features.

OBJECTIVE: To develop an interpretation procedure which estimates simulation parameters (tumour cell motility, tumour cell adhesion, autocrine and paracrine growth control, stroma destruction) of simulated patterns solely based on morphometric features of the morphologic pattern. METHODS: A cellular automaton computer simulation program was developed which produces morphologic patterns by growth of a seed of tumour cells. At the beginning of each simulation run certain simulation parameters are assigned to the tumour cells. After the run has been completed, the resulting pattern is evaluated by a set of morphometric features. Simulation parameters and resulting morphometric features of 27,800 simulations were stored in a database and were used for the evaluation of potential relationships. RESULTS: Correlation analysis showed highly significant correlations between morphometric features on the one hand and the preset simulation parameters (tumour cell motility, tumour cell adhesion, autocrine and paracrine growth control, stroma destruction) on the other. Correlation coefficients, however, varied from 0.72 to 0.99. When only one simulation parameter varied while all others were kept constant, morphometric features yielded a highly reliable estimate of the particular simulation parameter. When variability was extended to 4 simulation parameters, morphometric features were less effective in estimating the setting of the parameters. Though in all patterns tested several possible simulation parameter constellations could be ruled out, morphometric features were usually compatible with more than one set of simulation parameters thus preventing a straightforward interpretation. CONCLUSIONS: Though simulation parameters significantly and reproducibly influence the resulting morphologic pattern as characterized by morphometric features, estimates of the simulation parameters based on morphometric features yield equivocal results.     

Analysis of functional abilities for elderly Danish twins using GEE models.

In this work we present a new method for genetic analysis of twin data which is based on generalized estimating equations and allows for analysis of various response types (e.g., continuous, binary, counts) combined with estimation of residual correlations. The new approach allows for control of covariates of any kind (e.g., continuous, counts) by modeling the dependence of mean and variance on background variables. The proposed method was applied to identify the covariates that have a significant influence on elderly people's functional abilities, and find the estimates for the correlation coefficients of residuals for MZ and DZ twins in a sample of 2401 Danish twin 75 years of age or older. The bootstrap method was used to obtain standard errors for correlation coefficients. It was shown, that the chosen covariates have similar effects on MZ and DZ twins, and that the residual correlation in MZ twins is significantly higher than in DZ twins, which indicates that genetic factors play an etiological role in the determination of physical status of elderly people, controlled for 10 background variables.     

Multiple mapping method: a novel approach to the sequence-to-structure alignment problem in comparative protein structure modeling

A major bottleneck in comparative protein structure modeling is the quality of input alignment between the target sequence and the template structure. A number of alignment methods are available, but none of these techniques produce consistently good solutions for all cases. Alignments produced by alternative methods may be superior in certain segments but inferior in others when compared to each other; therefore, an accurate solution often requires an optimal combination of them. To address this problem, we have developed a new approach, Multiple Mapping Method (MMM). The algorithm first identifies the alternatively aligned regions from a set of input alignments. These alternatively aligned segments are scored using a composite scoring function, which determines their fitness within the structural environment of the template. The best scoring regions from a set of alternative segments are combined with the core part of the alignments to produce the final MMM alignment. The algorithm was tested on a dataset of 1400 protein pairs using 11 combinations of two to four alignment methods. In all cases MMM showed statistically significant improvement by reducing alignment errors in the range of 3 to 17%. MMM also compared favorably over two alignment meta-servers. The algorithm is computationally efficient; therefore, it is a suitable tool for genome scale modeling studies.     

MMM: a sequence-to-structure alignment protocol

MOTIVATION: Accurate alignment of a target sequence to a template structure continues to be a bottleneck in producing good quality comparative protein structure models. RESULTS: Multiple Mapping Method (MMM) is a comparative protein structure modeling server with an emphasis on a novel alignment optimization protocol. MMM takes inputs from five profile-to-profile based alignment methods. The alternatively aligned regions from the input alignment set are combined according to their fit in the structural environment of the template structure. The resulting, optimally spliced MMM alignment is used as input to an automated comparative modeling module to produce a full atom model. AVAILABILITY: The MMM server is freely accessible at http://www.fiserlab.org/servers/mmm     

Distinct evolutionary patterns of morphometric sperm traits in passerine birds

The striking diversity of sperm shape across the animal kingdom is still poorly understood. Postcopulatory sexual selection is an important factor driving the evolution of sperm size and shape. Interestingly, morphometric sperm traits, such as the length of the head, midpiece and flagellum, exhibit a strong positive phenotypic correlation across species. Here we used recently developed comparative methods to investigate how such phenotypic correlations between morphometric sperm traits may evolve. We compare allometric relationships and evolutionary trajectories of three morphometric sperm traits (length of head, midpiece and flagellum) in passerine birds. We show that these traits exhibit strong phenotypic correlations but that allometry varies across families. In addition, the evolutionary trajectories of the midpiece and flagellum are similar while the trajectory for head length differs. We discuss our findings in the light of three scenarios accounting for correlated trait evolution: (i) genetic correlation; (ii) concerted response to selection acting simultaneously on different traits; and (iii) phenotypic correlation between traits driven by mechanistic constraints owing to selection on sperm performance. Our results suggest that concerted response to selection is the most likely explanation for the phenotypic correlation between morphometric sperm traits.     

Morphological disparity in theropod jaws: comparing discrete characters and geometric morphometrics

Disparity, the diversity of form and function of organisms, can be assessed from cladistic or phenetic characters, and from discrete characters or continuous characters such as landmarks, outlines, or ratios. But do these different methods of assessing disparity provide comparable results? Here we provide evidence that all metrics correlate significantly with each other and capture similar patterns of morphological variation. We compare three methods of capturing morphological disparity (discrete characters, geometric morphometric outlines and geometric morphometric landmarks) in coelurosaurian dinosaurs. We standardize our study by focusing all our metrics on the mandible, so avoiding the risk of confounding disparity methods with anatomical coverage of the taxa. The correlation is strongest between the two geometric morphometric methods, and weaker between the morphometric methods and the discrete characters. By using phylogenetic simulations of discrete character and geometric morphometric data sets, we show that the strength of these correlations is significantly greater than expected from the evolution of random data under Brownian motion. All disparity metrics confirm that Maniraptoriformes had the highest disparity of all coelurosaurians, and omnivores and herbivores had higher disparity than carnivores.     

Postnatal development of the dentate gyrus: a karyometric and topographic study.

We have analysed the postnatal development of the nuclear sizes of the granular cells of the dentate gyrus in 5- to 190-day-old male mice. The study was performed in three topographic levels: rostral, intermediate and caudal. Three subdivisions were analysed in each level: suprapyramidal blade, infrapyramidal blade and the transition between them, the angular zone. Additionally, each of these subdivisions was measured in its external and internal layer, separately. Three gradients of postnatal karyometric development can be described: external-to-internal, suprapyramidal-to-infrapyramidal, and caudal-to-rostral, indicating that the external, suprapyramidal and caudal cells show higher karyometric sizes than the other subdivisions. These gradients are related to the ontogenetic gradients of these neurons.     

Comparative protein structure modeling by combining multiple templates and optimizing sequence-to-structure alignments

MOTIVATION: Two major bottlenecks in advancing comparative protein structure modeling are the efficient combination of multiple template structures and the generation of a correct input target-template alignment. RESULTS: A novel method, Multiple Mapping Method with Multiple Templates (M4T) is introduced that implements an algorithm to automatically select and combine Multiple Template structures (MT) and an alignment optimization protocol (Multiple Mapping Method, MMM). The MT module of M4T selects and combines multiple template structures through an iterative clustering approach that takes into account the 'unique' contribution of each template, their sequence similarity among themselves and to the target sequence, and their experimental resolution. MMM is a sequence-to-structure alignment method that optimally combines alternatively aligned regions according to their fit in the structural environment of the template structure. The resulting M4T alignment is used as input to a comparative modeling module. The performance of M4T has been benchmarked on CASP6 comparative modeling target sequences and on a larger independent test set, and showed favorable performance to current state of the art methods.     

HOW SIMILAR ARE GENETIC CORRELATION STRUCTURES? DATA FROM MICE AND RATS

An integral assumption of many models of morphometric evolution is the equality of the genetic variance-covariance structure across evolutionary time. To examine this assumption, the quantitative-genetic aspects of morphometric form are examined for eight pelvic traits in laboratory rats (Rattus norvegicus) and random-bred ICR mice (Mus musculus). In both species, all traits are significantly heritable, and there are significant phenotypic and genetic correlations among traits, although environmental correlations among the eight traits are low. The size relations among the pelvic variables are isometric. Three matrix-permutation tests are used to examine similarity of phenotypic, genetic, and environmental covariance and correlation matrices within and between species. Independent patterns of morphometric covariation and correlation arise from genetic and environmental effects within each species and from environmental effects between species. The patterns of phenotypic and genetic covariation and correlation are similar within each species, and the phenotypic and genetic correlations are also similar between these species. However, genetic covariance matrices show no significant statistical association between species. It is suggested that the assumption of equality of genetic variance-covariance structures across divergent taxa should be approached with caution.     

Information theory reveals large-scale synchronisation of statistical correlations in eukaryote genomes

We study short-range correlations in DNA sequences with methods from information theory and statistics. We find a persisting degree of identity between the correlation patterns of different chromosomes of a species. Except for the case of human and chimpanzee inter-species differences in this correlation pattern allow robust species distinction: in a clustering tree based upon the correlation curves on the level of individual chromosomes distinct clusters for the individual species are found. This capacity of distinguishing species persists, even when the length of the underlying sequences is drastically reduced. In comparison to the standard tool for studying symbol correlations in DNA sequences, namely the mutual information function, we find that an autoregressive model for higher order Markov processes significantly improves species distinction due to an implicit subtraction of random background.     

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.     

Influence of staining and sampling procedures on goat sperm morphometry using the Sperm Class Analyzer

Computer-assisted sperm morphometry analysis (ASMA) has improved the assessment of sperm morphology, but the results depend on the use of adequate sampling and staining procedures of spermatozoa from individual species. In this study, the Sperm Class Analyzer ASMA system was used for the morphometric analysis of goat sperm heads. Semen samples, obtained from four bucks, were used to evaluate the influence of three staining procedures (Diff-Quik, Hemacolor and Harris' Haematoxylin) on the accuracy of image processing and sperm morphometry, the effect of the number of cells analysed and the repeatability of the method. These experiments were performed to obtain objective, accurate and reliable sperm morphometric measurements of goat spermatozoa. Diff-Quik and Harris' Haematoxylin were significantly (p<0.05) more accurate than Hemacolor. However, Diff-Quik obtained the highest proportion of correctly analysed sperm heads (86.06%) and the lowest coefficients of variation on the image processing and morphometric measurements. The staining methods affected significantly the sperm dimensions (p<0.001) with increased values from Diff-Quik than Hemacolor and Harris' Haematoxylin, respectively (Diff-Quik>Hemacolor>Harris' Haematoxylin). No differences in morphometric parameters were found when 100, 150, 175 or 200 spermatozoa were analysed. The repeatability of results obtained was very high since no differences were found when measuring the same sperm on multiple attempts. In conclusion, to obtain objective, accurate and repeatable sperm morphometric measurements by the Sperm Class Analyzer system in goats, the analysis of 100 spermatozoa from slides which have been previously stained with Diff-Quik is recommended.     

Modified nonparametric approaches to detecting differentially expressed genes in replicated microarray experiments

MOTIVATION: An important goal in analyzing microarray data is to determine which genes are differentially expressed across two kinds of tissue samples or samples obtained under two experimental conditions. Various parametric tests, such as the two-sample t-test, have been used, but their possibly too strong parametric assumptions or large sample justifications may not hold in practice. As alternatives, a class of three nonparametric statistical methods, including the empirical Bayes method of Efron et al. (2001), the significance analysis of microarray (SAM) method of Tusher et al. (2001) and the mixture model method (MMM) of Pan et al. (2001), have been proposed. All the three methods depend on constructing a test statistic and a so-called null statistic such that the null statistic's distribution can be used to approximate the null distribution of the test statistic. However, relatively little effort has been directed toward assessment of the performance or the underlying assumptions of the methods in constructing such test and null statistics. RESULTS: We point out a problem of a current method to construct the test and null statistics, which may lead to largely inflated Type I errors (i.e. false positives). We also propose two modifications that overcome the problem. In the context of MMM, the improved performance of the modified methods is demonstrated using simulated data. In addition, our numerical results also provide evidence to support the utility and effectiveness of MMM.     

Spatial and Space-Time Correlations in Systems of Subpopulations with Genetic Drift and Migration

The geographic distribution of genetic variation is an important theoretical and experimental component of population genetics. Previous characterizations of genetic structure of populations have used measures of spatial variance and spatial correlations. Yet a full understanding of the causes and consequences of spatial structure requires complete characterization of the underlying space-time system. This paper examines important interactions between processes and spatial structure in systems of subpopulations with migration and drift, by analyzing correlations of gene frequencies over space and time. We develop methods for studying important features of the complete set of space-time correlations of gene frequencies for the first time in population genetics. These methods also provide a new alternative for studying the purely spatial correlations and the variance, for models with general spatial dimensionalities and migration patterns. These results are obtained by employing theorems, previously unused in population genetics, for space-time autoregressive (STAR) stochastic spatial time series. We include results on systems with subpopulation interactions that have time delay lags (temporal orders) greater than one. We use the space-time correlation structure to develop novel estimators for migration rates that are based on space-time data (samples collected over space and time) rather than on purely spatial data, for real systems. We examine the space-time and spatial correlations for some specific stepping stone migration models. One focus is on the effects of anisotropic migration rates. Partial space-time correlation coefficients can be used for identifying migration patterns. Using STAR models, the spatial, space-time, and partial space-time correlations together provide a framework with an unprecedented level of detail for characterizing, predicting and contrasting space-time theoretical distributions of gene frequencies, and for identifying features such as the pattern of migration and estimating migration rates in experimental studies of genetic variation over space and time.