Properties and Application of the Location-Test V (n,k)

The present paper is concerned with the properties of a test statistic V(n, k) to test location differences in the one-sample case with known hypothetical distribution G(x). The test is similar to the WILCOXON two-sample statistic after replacement of the second sample by quantiles of the hypothetical distribution. A comparison with the exact distribution of V(n, k) shows that an approximation by means of the normal distribution provides good results even for small sample sizes. The V-test is unbiased against one-tailed alternatives and it is consistent with a restriction which is hardly relevant in practical applications. With regard to the application we are interested especially in the power and robustness against extreme observations for small sample size n. It is shown that in a normal distribution with known standard deviation V(n, k) is more powerful than STUDENT's t for small n and more robust in the sense considered here. The test statistic is based on grouping of the observations into classes of equal expected frequency. A generalization to arbitrary classes provides an essential extension of applicability such as to discrete distributions and to situations where only relative frequencies of G(x) in fixed classes are known.     

Estimating the Order of a Sequence of Physiological Changes from Incomplete Data

The problem considered in this paper is the estimation of the underlying order of m physiological changes from incomplete observations. The observations are incomplete in the sense that, for each individual in the sample, only the identity, but not the order, of the first k < m changes are observed, with k varying between individuals. An application to some data on physiological changes in fish undergoing sex change is presented.     

A Note on Estimation of Several Missing Values in Two Way Classified Data with Many Observations Per Cell

In this paper an attempt has been made to reduce the computational complexities involved in estimation of several missing values. As a result it has been shown that one can estimate m missing values by developing only k (≤m) linear equations, where m and k are respectively the number of missing values and missing cells. The procedure is also illustrated with the help of a numerical example.     

A Report on a Monte Carlo Simulation of the Sampling Distribution of the Incidence Test and its Two Related Tests

A large-scale (5000 throws) Monte Carlo simulation experiment was carried out to study the nature of the sampling distributions of the incidence test and its two related tests, the FRIEDMAN test and the dual test, and to evaluate the goodness of the proposed gamma approximation relative to the conventional chi-square approximation. For all three tests, on the basis of the experimental results obtained for k = 3, 4, 5, and N = k(1) (120/k), the gamma approximation should be preferred over the chi-square. Exact companion tabulations show that use of the chi-square approximation entails an appreciable conservative bias. Extensive listings are provided of the moments of the experimental distributions of the three test statistics, and of the observed distributions of the corresponding gamma probabilities. The simulation experiment was run on a UNIVAC 1108 large-scale computer system.     

Simultaneous Comparisons of Multiple Treatments to Two (or More) Control

DUNNETT (1955) developed a procedure simultaneously comparing k treatments to one control with an exact overall type I error of α when all sampling distributions are normal. Sometimes it is desirable to compare k treatments to m≧2 controls, in particular to two controls. For instance, several new therapies (e.g., pain relievers) could be compared to two standard therapies (e.g., Aspirin and Tylenol). Alternatively, a standard therapy could be very expensive, difficult to apply and/or have bad side effects, making it useful to compare each new therapy to both standard therapy and no therapy (Placebo). Dunnett's method is expanded here to give comparisons of mean values for k treatments to mean values for m≧2 controls at an exact overall type I error of α when all sampling distributions are normal. Tabled values needed to make exact simultaneous comparisons at α = .05 are given for m = 2. An application is made to an example from the literature.     

Comparing Power Behaviours for Some Goodness-of-Fit Test Statistics in Sparse Multinomials

This paper is concerned with the power behaviour of four goodness-of-fit test statistics in sparse multinomials with k cells. Most previous work has been concerned only with both Pearson's X² and the likelihood ratio test statistics. We consider in this study, two additional test statistics, namely, the Cressie-Read test statistic – I(2/3) and the modified Freeman-Tukey test (FT) statistic. Because k ≥ 10 in this study, a Monte Carlo procedure based on 1000 simulated samples is used to estimate the powers for the four test statistics. Alternatives on various line segments are employed. Results suggest that none of the test statistics completely dominate the other and that the choice of which test to use depends on the nature of the alternative hypothesis. These results are consistent with those obtained by West and Kempthorne (1972), although, the Pearson's χ² test statistic may be preferred because of its closer approximation to the χ² distribution in terms of the attained α levels.     

Estimating the power of the two-sample Wilcoxon test for location shift

Traditional methods for calculating the power of a statistical test for location shift require knowledge of the shape of the underlying probability distribution. The distribution shape, however, may be unknown. This paper describes a bootstrap method for using observed data (or pilot data) to approximate the power. No assumptions need be made about the shape of the underlying continuous probability distribution. Simulation evidence shows that, when applied to the Wilcoxon two-sample test for location shift, the suggested method is reliable. The evidence also shows that it is more accurate than a benchmark traditional approach. The bootstrap method is applied to a real-data example. The analysis demonstrates how the method can be used to determine sample sizes and how to choose the more powerful of two alternative tests for location shift.     

The size distribution of insertions and deletions in human and rodent pseudogenes suggests the logarithmic gap penalty for sequence alignment

The size distributions of deletions, insertions, and indels (i.e., insertions or deletions) were studied, using 78 human processed pseudogenes and other published data sets. The following results were obtained: (1) Deletions occur more frequently than do insertions in sequence evolution; none of the pseudogenes studied shows significantly more insertions than deletions. (2) Empirically, the size distributions of deletions, insertions, and indels can be described well by a power law, i.e., f _k = Ck ^–b , where f _k is the frequency of deletion, insertion, or indel with gap length k, b is the power parameter, and C is the normalization factor. (3) The estimates of b for deletions and insertions from the same data set are approximately equal to each other, indicating that the size distributions for deletions and insertions are approximately identical. (4) The variation in the estimates of b among various data sets is small, indicating that the effect of local structure exists but only plays a secondary role in the size distribution of deletions and insertions. (5) The linear gap penalty, which is most commonly used in sequence alignment, is not supported by our analysis; rather, the power law for the size distribution of indels suggests that an appropriate gap penalty is w _k = a + b ln k, where a is the gap creation cost and blnk is the gap extension cost. (6) The higher frequency of deletion over insertion suggests that the gap creation cost of insertion (a _i ) should be larger than that of deletion (a _d ); that is, a _i – a _d = In R, where R is the frequency ratio of deletions to insertions. Correspondence to: W.-H. Li     

A comparative analysis on the distribution of nymphal populations of some leaf- and planthoppers on rice plant

The frequency distribution of the number of nymphs per hill of rice plant were analyzed for three species of rice leaf- and planthoppers, Nilaparvata lugens, Delphacodes striatella and Nephotettix cincticeps, based on the sampling data obtained during their last generations in the paddy field. For every species concerned, individual distributions were proved to be contagious and to fit well to the negative binomial distribution. Further, it was found that the value of negative binomial parameter k is so stable for same species that a single value of k is applicable for a series of counts with different means, whereas that k differs remarkably among different species: if the reciprocal of the weighted estimate of common k which is an adequate index measuring degree of contagiousness of the distribution, is compared among different species, it is higher in the order of Nilaparvata, Delphacodes and Nephotettix. The ecological and practical implication of constancy and heterogeneity within and among species was discussed respectively as to the value of parameter k.     

Evaluation of Root-n Bandwidth Selectors for Kernel Density Estimation

ABSTRACT The kernel density estimator is used commonly for estimating animal utilization distributions from location data. This technique requires estimation of a bandwidth, for which ecologists often use least-squares cross-validation (LSCV). However, LSCV has large variance and a tendency to under-smooth data, and it fails to generate a bandwidth estimate in some situations. We compared performance of 2 new bandwidth estimators (root-n) versus that of LSCV using simulated data and location data from sharp-shinned hawks (Accipter striatus) and red wolves (Canis rufus). With simulated data containing no repeat locations, LSCV often produced a better fit between estimated and true utilization distributions than did root-n estimators on a case-by-case basis. On average, LSCV also provided lower positive relative error in home-range areas with small sample sizes of simulated data. However, root-n estimators tended to produce a better fit than LSCV on average because of extremely poor estimates generated on occasion by LSCV. Furthermore, the relative performance of LSCV decreased substantially as the number of repeat locations in the data increased. Root-n estimators also generally provided a better fit between utilization distributions generated from subsamples of hawk data and the local densities of locations from the full data sets. Least-squares cross-validation generated more unrealistically disjointed estimates of home ranges using real location data from red wolf packs. Most importantly, LSCV failed to generate home-range estimates for >20% of red wolf packs due to presence of repeat locations. We conclude that root-n estimators are superior to LSCV for larger data sets with repeat locations or other extreme clumping of data. In contrast, LSCV may be superior where the primary interest is in generating animal home ranges (rather than the utilization distribution) and data sets are small with limited clumping of locations.     

Use of known Coefficients of Variation in the Estimation of the Common mean of two Normal Populations

The problem of estimating the common mean of two normal populations N(?, a1?2) and N(?, a2?2) where the coefficients of variation of two populations respectively, are known constants, on the basis of two independent random samples, one from each population, is considered. The minimum mean square estimator is proposed. It is also shown that the proposed estimator is Best Asymptotic Normal (BAN) estimator. It is pointed out that the result can be generalized to k population problem. It is remarked that the same method works, also for the problem of estimating the common standard deviation of k normal populations when coefficients of variation are known.     

Pairwise Comparisons Using Trimmed Means or M-Estimators when Working with Dependent Groups

For J dependent groups, let θ_j, j = 1, …, J, be some measure of location associated with the jth group. A common goal is computing confidence intervals for the pairwise differences, θ_j — θ_k, j < k, such that the simultaneous probability coverage is 1 — α. If means are used, it is well known that slight departures from normality (as measured by the Kolmogorov distance) toward a heavy-tailed distribution can substantially inflate the standard error of the sample mean, which in turn can result in relatively low power. Also, when distributions differ in shape, or when sampling from skewed distributions with relatively light tails, practical problems arise when the goal is to obtain confidence intervals with simultaneous probability coverage reasonably close to the nominal level. Extant theoretical and simulation results suggest replacing means with trimmed means. The Tukey-McLaughlin method is easily adapted to the problem at hand via the Bonferroni inequality, but this paper illustrates that practical concerns remain. Here, the main result is that the percentile t bootstrap method, used in conjunction with trimmed means, gives improved probability coverage and substantially better power. A method based on a one-step M-estimator is also considered but found to be less satisfactory.     

A Comparative Statistical Analysis of Genetic Distances. I - Estimation of Distances and of their Variances: Distribution of the Estimators

We present a Monte-Carlo simulation analysis of the statistical properties of absolute genetic distance and of Nei's minimum and standard genetic distances. The estimation of distances (bias) and of their variances is analysed as well as the distributions of distance and variance estimators, taking into account both gamete and locus samplings. Both of Nei's statistics are non-linear when distances are small and consequently the distributions of their estimators are extremely asymmetrical. It is difficult to find theoretical laws that fit such asymmetrical distributions. Absolute genetic distance is linear and its distributions are better fit by a normal distribution. When distances are medium or large, minimum distance and absolute distance distributions are close to a normal distribution, but those of the standard distance can never be considered as normal. For large distances the jack-knife estimator of the standard distance variance is bad; another standard distance estimator is suggested. Absolute distance, which has the best mathematical properties, is particularly interesting for small distances if the gamete sample size is large, even when the number of loci is small. When both distance and gamete sample size are small, this statistic is biased.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Distribution-free simultaneous tests for location–scale and Lehmann alternative in two-sample problem

The paper deals with the classical two-sample testing problem for the equality of two populations, one of the most fundamental problems in biomedical experiments and case–control studies. The most familiar alternatives are the difference in location parameters or the difference in scale parameters or in both the parameters of the population density. All the tests designed for classical location or scale or location–scale alternatives assume that there is no change in the shape of the distribution. Some authors also consider the Lehmann-type alternative that addresses the change in shape. Two-sample tests under Lehmann alternative assume that the location and scale parameters are invariant. In real life, when a shift in the distribution occurs, one or more of the location, scale, and shape parameters may change simultaneously. We refer to change of one or more of the three parameters as a versatile alternative. Noting the dearth of literature for the equality two populations against such versatile alternative, we introduce two distribution-free tests based on the Euclidean and Mahalanobis distance. We obtain the asymptotic distributions of the two test statistics and study asymptotic power. We also discuss approximating p-values of the proposed tests in real applications with small samples. We compare the power performance of the two tests with several popular existing distribution-free tests against various fixed alternatives using Monte Carlo. We provide two illustrations based on biomedical experiments. Unlike existing tests which are suitable only in certain situations, proposed tests offer very good power in almost all types of shifts.     

A Comparison of Robust Estimators in the Presence of Arbitrarily Right Censored Data

Robust estimation of a location parameter is considered when the data from an unknown symmetric population are subject to arbitrary right-censorship. Comparisons are made between various M-estimators, several L-estimators (trimmed means), and the Kaplan-Meier median. Ten sampling distributions, two uniform censoring distributions, and three sample sizes are examined. A Cauchy censoring distribution is also considered when the sample size is equal to twenty for each of the ten sampling distributions. Performance is based on the estimated mean square error.     

Proposal of k Sample Tests for Bivariate Censored Data for Nondecreasing Ordered Alternatives

Tests are introduced which are designed to test for a nondecreasing ordered alternative among the survival functions of k populations consisting of multiple observations on each subject. Some of the observations could be right censored. A simulation study is conducted comparing the proposed tests on the basis of estimated power when the underlying distributions are multivariate normal. Equal sample sizes of 20 with 25% censoring, and 40 with both 25% and 50% censoring are considered for 3 and 4 populations. All of the tests hold their α‐values well. A recommendation is made as to the best overall test for the situations considered.     

The Sensitivity of the Stationary Age Class Distribution o Changes in the Life History Components

The influence of the various components of the life history on the stationary age class distribution of a population was studied by applying the Leslie matrix model and then considering the single age class frequencies as well as the average age and the cumulative age class distribution in the stationary state. Since, for fixed survival probabilities, the effects of the (age specific) fecundities were shown to be already expressed by the (ultimate) growth rate, this investigation was concentrated on the influence of the survival probabilities and the growth rate. After having supplemented some useful details concerning the dependence of the growth rate on the life history and introduction of the shift concept as an interpretation of cumulative distributions, the mathematical results could be given a biologically intelligible meaning: An increase in the growth rate for fixed survival probabilities is followed by a shift of all age class frequencies to earlier ages. Whereas age specific changes in fecundity do not result in age specific changes in the stationary age class distribution, increases in survival up to and including age k, say, do exhibit such changes in that they cause a shift of age class frequencies at least from those ages exceeding k to earlier ages.     

On Combination of Ratio and PPS Estimators

AGARWAL and KUMAR (1980) proposed an estimator, combining ratio and pps estimators of population mean and proved that the proposed estimator would always be better (in minimum mean square error sense) than the pps estimator or the ratio estimator under pps sampling scheme for optimum value of constant k (parameter). The optimum value of k is rarely known in practice, hence the alternative is to replace k from the sample-values. In this paper, an estimator depending on estimated optimum value of k based on sample-values, under pps sampling scheme is proposed and studied.     

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.