Comparative quantitative studies of mental and physical development of children up to age 3

Biomathematical-methodical aspects are exposed of quantitative recording of mean growth courses which will be taken as basis for comparisons between different sample curves. In the paper the body length growth process of children in the age period from 0 to 3 years is considered. There are samples of children growing up under different social conditions and data of corresponding samples collected about 20 years ago. After remarks on the correlation between physical and psychical development of infants several possibilities for representing mean body length growth curves are discussed. The advantages of empirical regression are explained as the best suited method for a modelfree data evaluation. The nonparametric location trend test of Cox and Stuart is used for statistical proving of global difference or parallelism in the course of 2 mean growth curves. Level differences in the global sense between 2 compared curves may be tested by constructing a Scheffé confidence region for a properly defined constrast. Local level differences can be proved by the t-test for those pairs of corresponding mean values for which the conditions of applying this test are fulfilled. The calculation of a curve-related normal belt as a succession of 95%-tolerance regions (without confidence probability) is demonstrated by an example.     

Functional regression analysis of fluorescence curves

Summary .  Fluorescence curves are useful for monitoring changes in photosynthesis activity. Various summary measures have been used to quantify differences among fluorescence curves corresponding to different treatments, but these approaches may forfeit valuable information. As each individual fluorescence curve is a functional observation, it is natural to consider a functional regression model. The proposed model consists of a nonparametric component capturing the general form of the curves and a semiparametric component describing the differences among treatments and allowing comparisons of treatments. Several graphical model-checking approaches are introduced. Both approximate, asymptotic confidence intervals as well as simulation-based confidence intervals are available. Analysis of data from a crop experiment using the proposed model shows that the salient features in the fluorescence curves are captured adequately. The proposed functional regression model is useful for analysis of high throughput fluorescence curve data from regular monitoring or screening of plant growth.     

An algol program for the computation of empiric regressions

An explanation is given about the meaning of empirical regression and on the domain of application of this biomathematical-statistical procedure. It may be helpful in data handling after the measurements and in a first stage of data processing especially if there is a large amount of datas. An empirical regression can provide the basis for a functional relationship analysis by giving hints for the choice of empirical mathematical functions. This will be useful and necessary in such cases where the measured values have a greater dispersion and one wants to get an analytical expression for the course of measured points. In the appendix a program listing of the ALGOL-program for empirical regression is presented. Detailed remarks are made in the text concerning the program structure, the data input and output resp. the program control parameters to enable the biological or medical user to adapt the program to their special problems without the help by a mathematician, and neither with deeper knowledge of mathematics nor with detailed insight to computer technical aspects of data processing.     

Prediction of Growth: A Hierarchical Bayesian Approach

A nonparametric hierarchical growth curve model is proposed. Different levels in the model hierarchy are intended to correspond to different sources of variation in an individual's growth. The nonparametric character of the model offers considerable flexibility in fitting the growth curves to empirical data. Here the emphasis is on prediction, and for this purpose the adopted Bayesian inferential approach seems particularly natural and efficient. A Markov chain Carlo method is used to perform the numerical computations. As an illustration of the techniques, we consider the growth of children, during their first two years.     

Case studies of the use of the locally adjusted polynomial regression

Firstly the operation mode of the procedures of local adjusted polynomial regression is explained by an artificial example of data. After that, the properties of these nonparametric estimation procedures of unknown continuous functions from measured data are demonstrated by some examples arising in experimental examinations. Especially the aspect is discussed how far the main task of separating the deterministic component from the random one in the course of measured values may be fulfilled by using different degrees of polynomials in connection with different values of the smoothing parameter.     

Comparing the areas under two or more correlated receiver operating characteristic curves: a nonparametric approach

Methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed. When a test is based on an observed variable that lies on a continuous or graded scale, an assessment of the overall value of the test can be made through the use of a receiver operating characteristic (ROC) curve. The curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates. When two or more empirical curves are constructed based on tests performed on the same individuals, statistical analysis on differences between curves must take into account the correlated nature of the data. This paper presents a nonparametric approach to the analysis of areas under correlated ROC curves, by using the theory on generalized U-statistics to generate an estimated covariance matrix.     

Estimation of Sample Size for Reference Interval Studies

The currently used criterion for sample size calculation in a reference interval study is not well stated and leads to imprecise control of the ratio in question. We propose a generalization of the criterion used to determine sufficient sample size in reference interval studies. The generalization allows better estimation of the required sample size when the reference interval estimation will be using a power transformation or is nonparametric. Bootstrap methods are presented to estimate sample sizes required by the generalized criterion. Simulation of several distributions both symmetric and positively skewed is presented to compare the sample size estimators. The new method is illustrated on a data set of plasma glucose values from a 50‐g oral glucose tolerance test. It is seen that the sample sizes calculated from the generalized criterion leads to more reliable control of the desired ratio.     

Distribution-free R sample tests for the hypothesis of parallelism of response profiles

In this paper two nonparametric tests are given for testing the nullhypothesis of parallelity of response curves in r samples. The first procedure is done by a permutation test, whose practical applicability is ensured by a FORTRAN-subroutine available from the author. As the computational work and time grows rapidly with sample sizes, a Monte-Carlo-solution is optionally given by the same procedure. Alternatively an asymptotically distribution-free test based on quadratic forms is constructed for large sample situations.     

Why water-soluble, compact, globular proteins have similar specific enthalpies of unfolding at 110 degrees C.

The changes in free energy, enthalpy, and entropy of unfolding have been measured for many water-soluble, compact, globular proteins by a number of workers. In principle, a wide range in stability could be achieved by proteins, as measured by the free energy of unfolding; in practice, evolution only allows a narrow range in this quantity. Proteins are only marginally stable at room temperature for many possible reasons, including ensuring that folding is reversible and polypeptide chains are not trapped in incorrectly folded structures. Many of these proteins have approximately the same values of enthalpy of unfolding around 110 degrees C. We show here that this arises because the change in entropy of unfolding at room temperature and the change in heat capacity on unfolding, which governs the temperature variation of the enthalpy and entropy, both vary with the magnitude of the hydrophobic effect in the protein. As all these proteins have evolved to achieve similar stabilities at room temperature, the enthalpy of unfolding will also vary with the size of the hydrophobic effect in the protein. A consequence of this is that curves of the specific unfolding enthalpy against temperature for different proteins intersect around 110 degrees C. A similar conclusion, on the basis of similar melting points rather than similar free energies of unfolding, has been reached independently by Baldwin and Muller (R. L. Baldwin, personal communication).     

A new index of interactivity in parasite communities.

A new index of interactivity which allows objective evaluation and comparison of interactivity in communities between different host species is presented. The index is derived from the equations for species-accumulation curves generated using non-linear regression (with the Levenberg-Marquardt algorithm) of sample infracommunity richness data. It is advantageous in that it requires only presence/absence data to calculate, is applicable to all parasite taxa (including asexual species), is largely independent of sample size and allows objective comparison of parasite communities while correcting for differences in total richness. Iterative randomisation of infracommunity richness values to generate a mean value for the index avoids spurious results which may be generated by heterogeneity in infracommunity richness and the variation this produces in the non-linear regression results.     

Biological arches and changes to the curvilinear form of the aging maxilla

The facial skeleton can be conceptualized as a series of arches aligned along a vertical axis, with regional differences in shape. Previous work suggested that the maxilla undergoes differential growth with time. Because these arch forms resemble geometric forms, it may be possible to measure changes to their shape with the aging process. A contour analysis of the aging maxilla was undertaken. Computed tomographic data were assembled retrospectively. Only men were studied, and they were divided into two groups: young (aged 18 to 24 years; n = 6) and old (aged 40 to 66 years; n = 6). The computed tomography data were reconstructed into three-dimensional images and underwent standardization for comparison purposes. An axial view of the maxilla along the Frankfort horizontal was used for analysis. To quantify changes between curves, measurements were made of the vertical distance from eight equidistant points along each curve perimeter to a standardized baseline. Average arch forms for young and old men were then generated for comparison purposes. Each of the eight points along the older maxillary arch form existed further from the baseline than the younger arch form. This difference reached statistical significance at three of the eight points measured along the two curves. The greatest difference occurred at point 8 (p = .0006), which was at the medial maxilla near the nasomaxillary junction. The results of the study suggest that the actual contour of the maxilla undergoes changes as a result of the aging process. Mathematically defined idealized skeletal contours could help guide reconstructive surgery and aesthetic augmentation of the facial skeleton in the future. In addition, certain soft tissues of the face and torso seem to share a common shape (an alpha contour). This shape may also exhibit idealized forms open to quantitative analysis. The skeletal arch form and alpha contour most likely exhibit sexual dimorphism and will require further investigation. The ubiquity of the arch form in nature and possible implications for growth and development of the human facial skeleton are also discussed.     

Estimating the species accumulation curve using mixtures

As a significant tool in ecological studies, the species accumulation curve or the collector's curve is the graph of the expected number of detected species as a function of sampling effort. The problem of estimating the species accumulation curve based on an empirical data set arising from quadrat sampling is studied in a nonparametric binomial mixture model. It will be shown that estimating the species accumulation curve not only is independent of the unknown number of species but also includes estimating the number of species as a limiting case. For the purpose of interpolation, moment-based estimators, associated with asymptotic confidence intervals, are developed from several points of view. A likelihood-based procedure is developed for the purpose of extrapolation, associated with bootstrap confidence intervals. The proposed methods are illustrated by ecological data sets.     

A New Nonparametric Approach to the Comparison of K Independent Samples of Response Curves II: A K Sample Generalization of the FRIEDMAN Test

A K sample generalization of the FRIEDMAN test (1937) is introduced which can be used as a nonparametric procedure for testing the homogeneity of the profiles of K independent samples of response curves measured at T identical points of time. While a similar procedure in LEHMACHER & WALL (1978), section 3, is based on T combined tests, each of them at level a/T, here a finite and asymptotic test is presented which is based on a single test statistic. The application of the new multivariate test is illustrated by the same numerical example as in LEHMACHER & WALL (1978). The properties of this test are discussed and compared with the combined test mentioned above.     

Reliability of Bayesian posterior probabilities and bootstrap frequencies in phylogenetics

Many empirical studies have revealed considerable differences between nonparametric bootstrapping and Bayesian posterior probabilities in terms of the support values for branches, despite claimed predictions about their approximate equivalence. We investigated this problem by simulating data, which were then analyzed by maximum likelihood bootstrapping and Bayesian phylogenetic analysis using identical models and reoptimization of parameter values. We show that Bayesian posterior probabilities are significantly higher than corresponding nonparametric bootstrap frequencies for true clades, but also that erroneous conclusions will be made more often. These errors are strongly accentuated when the models used for analyses are underparameterized. When data are analyzed under the correct model, nonparametric bootstrapping is conservative. Bayesian posterior probabilities are also conservative in this respect, but less so.     

On Sample Size of the Kruskal–Wallis Test with Application to a Mouse Peritoneal Cavity Study

Summary As the nonparametric generalization of the one‐way analysis of variance model, the Kruskal–Wallis test applies when the goal is to test the difference between multiple samples and the underlying population distributions are nonnormal or unknown. Although the Kruskal–Wallis test has been widely used for data analysis, power and sample size methods for this test have been investigated to a much lesser extent. This article proposes new power and sample size calculation methods for the Kruskal–Wallis test based on the pilot study in either a completely nonparametric model or a semiparametric location model. No assumption is made on the shape of the underlying population distributions. Simulation results show that, in terms of sample size calculation for the Kruskal–Wallis test, the proposed methods are more reliable and preferable to some more traditional methods. A mouse peritoneal cavity study is used to demonstrate the application of the methods.     

Comparison of normal distribution–based and nonparametric decision limits on the GH‐2000 score for detecting growth hormone misuse (doping) in sport

This paper is motivated by the GH‐2000 biomarker test, though the discussion is applicable to other diagnostic tests. The GH‐2000 biomarker test has been developed as a powerful technique to detect growth hormone misuse by athletes, based on the GH‐2000 score. Decision limits on the GH‐2000 score have been developed and incorporated into the guidelines of the World Anti‐Doping Agency (WADA). These decision limits are constructed, however, under the assumption that the GH‐2000 score follows a normal distribution. As it is difficult to affirm the normality of a distribution based on a finite sample, nonparametric decision limits, readily available in the statistical literature, are viable alternatives. In this paper, we compare the normal distribution–based and nonparametric decision limits. We show that the decision limit based on the normal distribution may deviate significantly from the nominal confidence level or nominal FPR when the distribution of the GH‐2000 score departs only slightly from the normal distribution. While a nonparametric decision limit does not assume any specific distribution of the GH‐2000 score and always guarantees the nominal confidence level and FPR, it requires a much larger sample size than the normal distribution–based decision limit. Due to the stringent FPR of the GH‐2000 biomarker test used by WADA, the sample sizes currently available are much too small, and it will take many years of testing to have the minimum sample size required, in order to use the nonparametric decision limits. Large sample theory about the normal distribution–based and nonparametric decision limits is also developed in this paper to help understanding their behaviours when the sample size is large.     

Iwao’s patchiness regression through the origin: biological importance and efficiency of sampling applications

Iwao’s mean crowding-mean density relation can be treated both as a linear function describing the biological characteristics of a species at a population level, or a regression model fitted to empirical data (Iwao’s patchiness regression). In this latter form its parameters are commonly used to construct sampling plans for insect pests, which are characteristically patchily distributed or overdispersed. It is shown in this paper that modifying both the linear function and statistical model to force the intercept or lower functional limit through the origin results in more intuitive biological interpretation of parameters and better sampling economy. Firstly, forcing the function through the origin has the effect of ensuring that zero crowding occurs when zero individuals occupy a patch. Secondly, it ensures that negative values of the intercept, which do not yield an intuitive biological interpretation, will not arise. It is shown analytically that sequential sampling plans based on regression through the origin should be more efficient compared to plans based on conventional regression. For two overdispersed data sets, through-origin based plans collected a significantly lower sample size during validation than plans based on conventional regression, but the improvement in sampling efficiency was not large enough to be of practical benefit. No difference in sample size was observed when through-origin and conventional regression based plans were validated using underdispersed data. A field researcher wishing to adopt a through-origin form of Iwao’s regression for the biological reasons outlined above can therefore be confident that their sampling strategies will not be affected by doing so.     

Arteriolar and systemic autoregulatory responses during the development of hypertension in the spontaneously hypertensive rat

Pressure-flow curves were constructed to determine whether acute autoregulation in rat skeletal muscle was altered during the development of hypertension in the spontaneously hypertensive rat (SHR). Under chloralose:urethane anesthesia, hindlimb blood flow and pressure, plus diameter changes of gracilis muscle arterioles, were simultaneously measured in the 6- and 9-week Wistar-Kyoto (WKY) and SHR. Femoral blood flow was measured by electromagnetic flowmetry and hindlimb pressure controlled with an hydraulic occluder. Arteriolar diameters were measured using image shearing techniques. Acute autoregulatory capacity was assessed by comparing the closed-loop gain and the regression lines over the regulated and passive pressure ranges of the pressure-flow curves. The lower pressure limit of autoregulation (LPLAR) shifted upward as the blood pressure increased in the SHR with age; it did not shift in the WKY. Resting hindlimb flow, elevated in the SHR at 6 weeks, was also elevated at the LPLAR. At 9 weeks hindlimb blood flow was comparable in the WKY and SHR. As blood pressure was increased autoregulation was accompanied by vasoconstriction of gracilis arterioles. However, neither the gain of the autoregulatory system nor the regression lines describing the pressure-flow curves were different between the hypertensive and normotensive animals at either age. These results indicate that the acute autoregulatory response mechanism was not affected by the developing hypertension in the SHR, and is consistent with a structural basis for the chronic maintenance of the elevated peripheral vascular resistance.     

Inherent bias toward the null hypothesis in conventional multipoint nonparametric linkage analysis

Traditional nonparametric "multipoint" statistical procedures have been developed for assigning allele-sharing values at a locus of interest to pairs of relatives for linkage studies. These procedures attempt to accommodate a lack of informativity, nongenotyped loci, missing data, and related issues concerning the genetic markers used in a linkage study. However, such procedures often cannot overcome these phenomena in compelling ways and, as a result, assign relevant relative pairs allele-sharing values that are "expected" for those pairs. The practice of assigning expected allele-sharing values to relative pairs in the face of a lack of explicit allele-transmission information can bias traditional nonparametric linkage test statistics toward the null hypothesis of no locus effect. This bias is due to the use of expected values, rather than to a lack of information about actual allele sharing at relevant marker loci. The bias will vary from study to study on the basis of the DNA markers, sample size, relative-pair types, and pedigree structures used, but it can be extremely pronounced and could contribute to a lack of consistent success in the application of traditional nonparametric linkage analyses to complex human traits and diseases. There are several potential ways to overcome this problem, but their foundations deserve greater research. We expose many of the issues concerning allele sharing with data from a large affected-sibling-pair study investigating the genetic basis of autism.