Comparison of frequency distributions of doubled haploid and single seed descent lines in barley

Summary Both doubled haploid (DH) and single seed descent (SSD) methods were used to derive homozygous lines from two crosses of barley. The frequency distributions of grain yield, heading date, and plant height of the DH and SSD lines were compared by the Mann-Whitney U test, Kolmogorov-Smirnov twosample test and Wald-Wolfowitz runs test. It was found that the DH lines distributed in the same manner as the SSD lines with respect to the three characters. The results indicated that although the SSD method had more opportunity for recombination than the DH method, it did not produce a sample of recombinants which differed significantly from the DH sample; thus both methods were equally efficient for use in deriving homozygous lines from F₁ hybrids in a relatively short time.Contribution no. 455 Charlottetown Research Station, Agriculture Canada, P.E.I. (Canada)     

Quantifying change in distributions: a new departure index that detects,measures and describes change in distributions from population structures,size-classes and other ordered data

Many statistics are available to compare distributions. Some are limited to nominal data while others, such as skew, Kullback-Leibler, Kolmogorov-Smirnov and the Gini coefficient, are useful for providing information about ordered distributions. While many of these tests are useful for determining properties of data in histograms, there has not been a test until now that allows for the detection of differences between distributions, describes the difference and is sensitive to the location of the departures. Such a test could be critical for comparing pre-and post-event distributions, such as a change in the distribution of biomass due to fire, for example, or for comparing data from different locations, such as soil size distributions, and even for evaluating economic disparity or examining differences in age demographics. We present a new statistic, a departure index, which allows a test distribution to be compared with any reference distribution. The resulting index contains information about the location, magnitude and direction of departure from the reference distribution to the test distribution. The departure index in turn provides a standardized response range that allows for a comparison of results from different analyses. A case study of actual fire data demonstrates the sensitivity and range of the test.     

A Simple Method to Obtain all the Sample-Values and their Probabilities of the Wald-Wolfowitz Test Statistics in the Existence of Ties

In order to carry out non-conservative tests in the general two-sample problem with ties, we want to know all possible sample-values of the used test statistics and their occurrence probabilities as well. But this knowledge can be acquired only after very protracted attempts. In the present paper we depict a simple technique for obtaining that without any exertion in the case of the Wald-Wolfowitz test statistic. With that, we then are able to lead the Wald-Wolfowitz test easily and effortlessly in any manner conservative or non-conservative and in the existence of any number and any length of ties.     

Parameter-free testing of the shape of a probability distribution

The Kolmogorov-Smirnov test determines the consistency of empirical data with a particular probability distribution. Often, parameters in the distribution are unknown, and have to be estimated from the data. In this case, the Kolmogorov-Smirnov test depends on the form of the particular probability distribution under consideration, even when the estimated parameter-values are used within the distribution. In the present work, we address a less specific problem: to determine the consistency of data with a given functional form of a probability distribution (for example the normal distribution), without enquiring into values of unknown parameters in the distribution. For a wide class of distributions, we present a direct method for determining whether empirical data are consistent with a given functional form of the probability distribution. This utilizes a transformation of the data. If the data are from the class of distributions considered here, the transformation leads to an empirical distribution with no unknown parameters, and hence is susceptible to a standard Kolmogorov-Smirnov test. We give some general analytical results for some of the distributions from the class of distributions considered here. The significance level and power of the tests introduced in this work are estimated from simulations. Some biological applications of the method are given.     

Note on the side test of Griffith and Horn

The side test, developed by Griffith and Horn (1966), as a means for determining the serial correlation of events is shown to hold only when it is in agreement with the Wald-Wolfowitz runs test (1940).     

On the WILCOXON-MANN-WHITNEY-Test as Nonparametric Analogue and Extension of t-Test

In the present paper some basic aspects of the WILCOXON -MANN -WHITNEY -test under various assumptions concerning the underlying distributions are studied. Starting with a formal analogy to Student's t-test its specific sensibility is worked out and the connexion to the problem of testing differences in location is discussed in detail as well as to the model with ordered alternatives. Further a counter example is given showing that the common verbal formulation the WILCOXON -MANN -WHITNEY -test being a test on ‘differences in distribution’ in general is misleading. As an example of an extended application of the WILCOXON -MANN -WHITNEY -statistic the test of POTT-HOFF (1963) on differences in medians of symmetric distributions is treated together with a discussion on suitable variance estimators to guarantee appropriate asymptotic distribution.     

Comparison of leaf morphology among submersed species of Myriophyllum (Haloragaceae) from different habitats and geographical distributions

An image analysis system was used to calculate the surface area and volume of the submerged leaves of seven species of Myriophyllum. We separated the species into two categories based on their distribution and habitat differences and compared their surface area, volume, dry mass, specific leaf area, and surface to volume ratios. The geographic location and habitat of the species explained the greatest percentage of total variance for these variables when compared by analysis of variance. A leaf shape analysis was performed by regressing surface area with volume for each species. Interspecific comparisons of leaf shape were made using analysis of covariance. Fundamental interspecific differences in submerged leaf shape, associated with differences in their geographic distributions and habitats, are explained as adaptations for different nutrient uptake regimes.     

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.     

Distribution of the intensity-characteristic parameters of cat rapidly adapting mechanoreceptive fibers

Modeling population responses of nerve fibers requires statistical characterization of fiber-response properties. The rate/intensity characteristics of cat rapidly adapting (RA) fibers were fitted by four-parameter, piece-wise linear functions using nonlinear regression (n = 14; R2 > 0.958). The parameters were tested against the null hypothesis that they are log normally distributed. The test fail to reject this hypothesis (Kolmogorov-Smirnov p>0.477). However, a significant statistical difference was found between the specific lognormal distributions obtained from monkey (Johnson, J Neurophysiol 37: 48-72, 1974) and cat for all four parameters (Kolmogorov-Smirnov, p<0.0075, p<0.05, p<0.0001, p<0.00007). Although the stimulus contactor size was not the same in monkey and cat studies, the differences between monkey and cat fibers are attributed to anatomical differences in the glabrous sin of both species. Modeling studies suggest that the absolute firing thresholds of RA fibers have a right-skewed distribution because of the anatomical constraints present in both species' skin. Meissner corpuscles, which are the sensory end-organs of RA fibers, are likely to be found deeper in the skin within dermal papilla, therefore, the thresholds can be elevated. However, the thresholds are bounded at lower end, probably due to the epidermal junction that acts as a superficial mechanical barrier for these corpuscles.     

On an Unbiased Variance Estimator for the WILCOXON-MANN-WHITNEY-Statistic Based on Ranks

For some applications of the WILCOXON-MANN-WHITNEY-statistic its variance has to be estimated. So e.g. for the test of POTTHOFF (1963) to detect differences in medians of two symmetric distributions as well as for the computation of approximate, confidence bounds for the probability P(X₁ ≥ X₂), cf. GOVINDARAJULU (1968). In the present paper an easy to compute variance estimator is proposed which as only information uses the ranks of the data with the additional property that it is unbiased for the finite variance. Because of its invariance under any monotone transformation of the data its applicability is not confined to quantitative data. The estimator may be applied to ordinal data just as well. Some properties are discussed and a numerical example is given.     

Application of the Kolmogorov-Smirnov test to seasonal phenomena may be inappropriate

The Kolmogorov-Smirnov one-sample test is sometimes used to test seasonality in births or other annually cyclic phenomena. However, it is shown that the maximum deviation from expected (D) may differ by a factor of almost two if the cycle is initiated in different months. Thus, different results are possible from the same data. Unless there is a logical initiation point in a cycle, it is probably better to use other nonparametric statistical tests.     

Scalability Properties of the S-Distribution

Previous work has shown that the S-distribution is a valuable tool for data analysis and for the classification of continuous and discrete distribution functions. The distribution has four parameters: one determines its location, one is related to the variance, and two control its shape. The distributional structure allows symmetry as well as skewness to the right or the left. This offers great flexibility and, among other analyses, facilitates the simultaneous investigation of random variables whose distributions differ in shape. The present paper demonstrates that mean, variance, and quantiles of any S-distribution can be computed with simple algebraic operations from corresponding properties of a standard S-distribution, sSd , which is characterized by only the two shape parameters. This scalability is comparable with the use of z-scores when dealing with normal distributions.     

Statistical analyses support power law distributions found in neuronal avalanches

The size distribution of neuronal avalanches in cortical networks has been reported to follow a power law distribution with exponent close to -1.5, which is a reflection of long-range spatial correlations in spontaneous neuronal activity. However, identifying power law scaling in empirical data can be difficult and sometimes controversial. In the present study, we tested the power law hypothesis for neuronal avalanches by using more stringent statistical analyses. In particular, we performed the following steps: (i) analysis of finite-size scaling to identify scale-free dynamics in neuronal avalanches, (ii) model parameter estimation to determine the specific exponent of the power law, and (iii) comparison of the power law to alternative model distributions. Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect). This scale-free dynamics suggests the power law as a model for the distribution of avalanche sizes. Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports. Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test. Both the power law distribution without and with exponential cut-off provided significantly better fits to the cluster size distributions in neuronal avalanches than the exponential, the lognormal and the gamma distribution. In summary, our findings strongly support the power law scaling in neuronal avalanches, providing further evidence for critical state dynamics in superficial layers of cortex.     

Estimation of the Power of the Kruskal-Wallis Test

Power calculations of a statistical test require that the underlying population distribution(s) be completely specified. Statisticians, in practice, may not have complete knowledge of the entire nature of the underlying distribution(s) and are at a loss for calculating the exact power of the test. In such cases, an estimate of the power would provide a suitable substitute. In this paper, we are interested in estimating the power of the Kruskal-Wallis one-way analysis of variance by ranks test for a location shift. We investigated an extension of a data-based power estimation method presented by Collings and Hamilton (1988), which requires no prior knowledge of the underlying population distributions other than necessary to perform the Kruskal-Wallis test for a location shift. This method utilizes bootstrapping techniques to produce a power estimate based on the empirical cumulative distribution functions of the sample data. We performed a simulation study of the extended power estimator under the conditions of k = 3 and k = 5 samples of equal sizes m = 10 and m = 20, with four underlying continuous distributions that possessed various location configurations. Our simulation study demonstates that the Extended Average × & Y power estimation method is a reliable estimator of the power of the Kruskal-Wallis test for k = 3 samples, and a more conservative to a mild overestimator of the true power for k = 5 samples.     

On the use of the Kolmogorov-Smirnov statistical test for immunofluorescence histogram comparison

BACKGROUND: The problem considered is the quantitative comparison of immunofluorescence frequency distributions in order to detect their differences of biological significance, i.e., to evaluate the potential positivity of a cell sample with respect to negative control cells. The Kolmogorov-Smirnov (KS) statistical test, proposed in the literature for this purpose, is examined and discussed through its application to a set of experimental measurements. It is shown that even differences due to the stain procedure or to instrumental biases may be considered significant by the test implemented in the standard form. METHODS: In order to ensure valid results, it is necessary to take into account the various sources of variation in the specific experimental context. A procedure is proposed that uses the KS statistics as a reference for determining an appropriate estimate of the overall variability in the control data. This estimate is derived from the comparisons of the cumulative distributions associated with repeated measurements of the negative cell sample. RESULTS AND CONCLUSIONS: The KS-related index thus defined provides a tool for assessing the potential positivity of a cell sample, since it allows to distinguish between statistical and biological significance of the difference between the histogram to be tested and the set of control data. In particular, if a cell sample is not included in the control variability, either a positive cell subpopulation is present, or all cells are positive. Instead, for a sample included in the control variability, the difference will be not biologically meaningful, even if statistically significant. Moreover, when a purely positive control sample is also available, it is possible to derive a measure of the precision at which a true biological positivity can be detected. Finally, since the index is not absolute, but relative to the features of the instrumentation, of the antibodies and of the fluorochromes used, it represents a quantitative measure of the stability and reproducibility of the measurement process and could be used for quality control of flow cytometric experiments in immunofluorescence.     

Disentangling the influences of mean body size and size structure on ecosystem functioning: an example of nutrient recycling by a non‐native crayfish

Body size is a fundamental functional trait that can be used to forecast individuals' responses to environmental change and their contribution to ecosystem functioning. However, information on the mean and variation of size distributions often confound one another when relating body size to aggregate functioning. Given that size‐based metrics are used as indicators of ecosystem status, it is important to identify the specific aspects of size distributions that mediate ecosystem functioning. Our goal was to simultaneously account for the mean, variance, and shape of size distributions when relating body size to aggregate ecosystem functioning. We take advantage of habitat‐specific differences in size distributions to estimate nutrient recycling by a non‐native crayfish using mean‐field and variance‐incorporating approaches. Crayfishes often substantially influence ecosystem functioning through their omnivorous role in aquatic food webs. As predicted from Jensen's inequality, considering only the mean body size of crayfish overestimated aggregate effects on ecosystem functioning. This bias declined with mean body size such that mean‐field and variance‐incorporating estimates of ecosystem functioning were similar for samples at mean body sizes >7.5 g. At low mean body size, mean‐field bias in ecosystem functioning mismatch predictions from Jensen's inequality, likely because of the increasing skewness of the size distribution. Our findings support the prediction that variance around the mean can alter the relationship between body size and ecosystem functioning, especially at low mean body size. However, methods to account for mean‐field bias performed poorly in samples with highly skewed distributions, indicating that changes in the shape of the distribution, in addition to the variance, may confound mean‐based estimates of ecosystem functioning. Given that many biological functions scale allometrically, explicitly defining and experimentally or statistically isolating the effects of the mean, variance, and shape of size distributions is necessary to begin generalizing relationships between animal body size and ecosystem functioning.     

Frequency responses of cat rapidly adapting mechanoreceptive fibers

The frequency responses of 11 rapidly adapting (RA) fibers in cat were studied by representing the average firing rate as a function of sinusoidal stimulus amplitude and stimulus frequency. Specifically, rate-intensity functions at different stimulation frequencies were fitted by four-parameter (a0, a1, a2, a3), piece-wise linear functions using nonlinear regression (n = 59; R2 > 0.877). Rate-intensity functions at intermediate frequencies were found by linear interpolation. The result of this analysis is rate-amplitude-frequency functions plotted as two-dimensional surfaces. The surfaces consist of five regions separated and sufficiently defined by four space curves. At 14 different frequencies, the statistical distribution of each rate-intensity-function parameter could be approximated by a particular lognormal distribution (n = 56; R2 > 0.796). The Kolmogorov-Smirnov test fails to reject this hypothesis for each combination of frequency and parameter (56 tests; p > 0.39). Therefore, at a given frequency, the variation of the parameters can be represented by lognormal distributions with specific means and standard deviations. Responses of six RA fibers, which are different from the data-set used for modeling, were compared with the stochastic model at different frequencies. The parameters of those fibers were tested against the null hypotheses that they were sampled from the particular parameter distributions dictated by the model. The Kolmogorov-Smirnov test fails to reject all the hypotheses at the alpha = 0.05 level (44 tests). At the alpha = 0.10 level, only a few test parameters were found to be departing from the model (a0 and a1 at 5 Hz; a2 at 20 Hz; a2 and a3 at 50 Hz). The remaining test parameters could be accurately described by the model. Having confirmed the validity of the model, the logarithmic means and the logarithmic standard deviations of the lognormally distributed rate-intensity-function parameters were estimated in the frequency range of 4-200 Hz. The rate-amplitude-frequency surfaces sampled from the established stochastic model completely characterize the rate responses of RA fibers to sinusoidal stimuli and are superior to tuning curves which require selecting criterion responses. The current rate-response model is promising for future computational work, especially on population modeling.     

Estimating length frequency distributions of large reef fish underwater

We describe the training of divers to recognise and remove bias in estimating lengths of fish underwater. Divers were asked to allocate objects, from a population (N=50) with a known length frequency distribution, to ten 100 mm size classes. Observed and expected distributions were then compared and the divers informed of their errors. Training continued until divers consistently produced length frequency distributions that were not significantly different from the expected distribution (=0.8) by the one sample Kolmogorov-Smirnov (K-S) test. Divers were trained in five trials, but after six months they had lost all their ability and had to be retrained. Three trained divers observing the same population of the large reef fish Plectropomus leopardus (Serranidae) produced length frequency distributions that were not significantly different (P>0.1) on 67% of occasions. Data collected by divers can be used to detect small but real differences in length frequency distributions of populations when analysed using the two sample K-S test. We suggest a means of determining within site variation in length frequency relative to between site variation.     

Estimates of a Treatment Effect Based on Censored Data Rank Tests

Linear rank test statistics are applied to the problem of estimating a treatment effect if two sets of censored failure time data are compared and the distributions of the log-failure times of the two samples are assumed to differ only in location. Rank tests for this accelerated failure time model are reviewed and Hodges-Lehmann type estimates for the shift parameter are proposed. Properties of these estimates are investigated, computational aspects are discussed and an example is presented.