A Mixed Model for Binary Data: The Modified X2-test

In this paper, a statistical model for clinical trials is presented for the special situation that a varying and unstructered number of binary responses is obtained from each subject. The assumptions of the model are the following: 1.) For each subject there is a (constant) individual Bernoulli parameter determining the distribution of the binary responses of this subject. 2.) The Bernoulli parameters associated with the subjects are realizations of independent random variables with distributions P_g in treatment group g(g = 1, 2, …, G). 3.) Given the value of the Bernoulli parameter, the observations are stochastically independent within each subject. Under these assumptions, a test statistic is derived to test the hypothesis H₀:E(P₁) = E(P₂) = … = E(P_G). It is proven and demonstrated by simulations, that the test statistic asymptotically (i.e. for a large number of subjects) follows the X²-distribution.     

On the KOLMOGOROV-SMIRNOV Test for the POISSON Distribution with Unknown Mean

The standard tables for the KOLMOGOROV -SMIRNOV test are valid only in the case of testing whether a set of observations is from a completely specified cumulative distribution, F₀(X), with all parameters known. If the parameters are unknown and must be estimated from the sample, then the tables are not valid. A table is given in this paper for use with the KOLMOGOROV -SMIRNOV statistic in the case of testing whether a set of observations is from the POISSON distribution with an unknown mean that must be estimated from the sample. The table is obtained from a Monte Carlo calculation.     

Molecular assays of pollen use consistently reflect pollinator visitation patterns in a system of flowering plants

Determining how pollinators visit plants vs. how they carry and transfer pollen is an ongoing project in pollination ecology. The current tools for identifying the pollens that bees carry have different strengths and weaknesses when used for ecological inference. In this study we use three methods to better understand a system of congeneric, coflowering plants in the genus Clarkia and their bee pollinators: observations of plant–pollinator contact in the field, and two different molecular methods to estimate the relative abundance of each Clarkia pollen in samples collected from pollinators. We use these methods to investigate if observations of plant–pollinator contact in the field correspond to the pollen bees carry; if individual bees carry Clarkia pollens in predictable ways, based on previous knowledge of their foraging behaviors; and how the three approaches differ for understanding plant–pollinator interactions. We find that observations of plant–pollinator contact are generally predictive of the pollens that bees carry while foraging, and network topologies using the three different methods are statistically indistinguishable from each other. Results from molecular pollen analysis also show that while bees can carry multiple species of Clarkia at the same time, they often carry one species of pollen. Our work contributes to the growing body of literature aimed at resolving how pollinators use floral resources. We suggest our novel relative amplicon quantification method as another tool in the developing molecular ecology and pollination biology toolbox.     

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.     

Randomization Analysis of Incomplete Data in Some Basic Designs

The first two randomization moments of W = Tt.S.S./(Tt.S.S. + Error S.S.) are derived in case of (a) a completely randomized design with one observation missing using (I) Yates method of fitting constants and (II) the available observations only and in case of (b) a randomized block design in which one treatment is tested twice by mistake in a block and another treatment is not tested at all in that block using (I) all the available observations and (II) the data after deleting the observation due to the treatment tested by mistake in place of another treatment in a block. It is concluded that in each of the two cases for a completely randomized design, the F-test is unbiased whereas in each of the two cases for a randomized block design the F-test is in general not unbiased. However, if one treatment is tested twice by mistake in randomly chosen plots of some block, the F-test is unbiased in both cases for a randomized block design. For a completely randomized design the F-test is found to be a good approximation to the corresponding randomized test if N ≧ 80 in case (I) whereas in case (II) this approximation is of the same accuracy as that of case (I) if N ≧ 90. For a randomized block design it is seen that in case (I), the F-test provides a good approximation to the corresponding randomization test if v≧r≧7 and in case (II) this approximation is of the same accuracy if r≧6 and r(v-1)≧45. The analysis of several uniformity trial data shows that for values of “N” in the neighbourhood of 50 the agreement of the first two moments of “W” under the randomization theory and normal theory are reasonably close in all the four cases considered.     

Some Tests on a Single Change in Independent Copies of Short Sequences: A Cancer Diagnostic Study

Testing a single change in mean or variance when the observations are n independent short sequences has not been fully researched in the literature. Two statistical tests, max t and modified likelihood radtio, are used in this study. Power comparisons and examples are also given. As expected the modified likelihood ratio test yields the largest power. However, with the assumptions we have in the examples, the max t might be the optimal method to use. The max t may prove useful in assessing departures from the steady state in which the data are short sequences of diagnostic information.     

A discrete,stochastic model of colonial phytoplankton population size structure: Development and application to in situ imaging-in-flow cytometer observations of Dinobryon

The scientific community lacks models for the dynamic changes in population size structure that occur in colonial phytoplankton. This is surprising, as size is a key trait affecting many aspects of phytoplankton ecology, and colonial forms are very common. We aim to fill this gap with a new discrete, stochastic model of dynamic changes in phytoplankton colonies' population size structure. We use the colonial phytoplankton Dinobryon as a proof-of-concept organism. The model includes four stochastic functions—division, stomatocyst production, colony breakage, and colony loss—to determine Dinobryon population size structure and populations counts. Although the functions presented here are tailored to Dinobryon, the model is readily adaptable to represent other colonial taxa. We demonstrate how fitting our model to in situ observations of colony population size structure can provide a powerful approach to explore colony size dynamics. Here, we have (1) collected high-frequency in situ observations of Dinobryon in Lac (Lake) Montjoie (Quebec, Canada) in 2013 with a moored Imaging FlowCytobot (IFCB) and (2) fit the model to those observations with a genetic algorithm solver that extracts parameter estimates for each of the four stochastic functions. As an example of the power of this model-data integration, we also highlight ecological insights into Dinobryon colony size and stomatocyst production. The Dinobryon population was enriched in larger, flagellate-rich colonies near bloom initiation and shifted to smaller and emptier colonies toward bloom decline.     

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.     

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.     

Primate origins: Lessons from a neotropical marsupial

The didelphid Caluromys shows evolutionary convergence towards prosimians in having a relatively large brain, large eyes, small litters, slow development, and agile locomotion. The selection pressures that favored the emergence of primate-like traits in Caluromys from a generalized didelphid ancestor may be analogous to the selection pressures favoring the initial divergence of primates from a primitive nonprimate ancestor, and thus Caluromys provides an independent test of the arboreal hypothesis (Smith: Annual Report of the Board of Regents of the Smithsonian Institution 1912:553–572, 1913), the visual predation hypothesis (Cartmill: The Functional and Evolutionary Biology of Primates, pp. 97–122, 1972), and the angiosperm exploitation hypothesis (Sussman: American Journal of Primatology, in press) of primate origins. Quantitative data on free-ranging C. derbianus in Costa Rica demonstrate that it is highly arboreal, uses visually directed predation to capture arthropod prey, and makes extensive use of terminal branch foraging, where it feeds on small angiosperm products. These observations are consistent with predictions from each model of primate origins, thus suggesting that the hypotheses are not mutually exclusive but are interdependent. The initial divergence of primates probably involved exploitation of the rich angiosperm products and associated insects found in fine terminal branches; visually directed predation may have evolved as an efficient method of insect capture in the terminal branch milieu.     

Median Tests for Censored Survival Data; a Contingency Table Approach

Summary The median failure time is often utilized to summarize survival data because it has a more straightforward interpretation for investigators in practice than the popular hazard function. However, existing methods for comparing median failure times for censored survival data either require estimation of the probability density function or involve complicated formulas to calculate the variance of the estimates. In this article, we modify a K ‐sample median test for censored survival data ( Brookmeyer and Crowley, 1982 , Journal of the American Statistical Association 77, 433–440) through a simple contingency table approach where each cell counts the number of observations in each sample that are greater than the pooled median or vice versa. Under censoring, this approach would generate noninteger entries for the cells in the contingency table. We propose to construct a weighted asymptotic test statistic that aggregates dependent χ² ‐statistics formed at the nearest integer points to the original noninteger entries. We show that this statistic follows approximately a χ² ‐distribution with k? 1 degrees of freedom. For a small sample case, we propose a test statistic based on combined p ‐values from Fisher’s exact tests, which follows a χ² ‐distribution with 2 degrees of freedom. Simulation studies are performed to show that the proposed method provides reasonable type I error probabilities and powers. The proposed method is illustrated with two real datasets from phase III breast cancer clinical trials.     

Two-Stage Procedures for Comparing Treatments with a Control: Elimination at the First Stage and Estimation at the Second Stage

We consider the problem of comparing a set of p₁ test treatments with a control treatment. This is to be accomplished in two stages as follows: In the first stage, N₁ observations are allocated among the p₁ treatments and the control, and the subset selection procedure of Gupta and Sobel (1958) is employed to eliminate “inferior” treatments. In the second stage, N₂ observations are allocated among the (randomly) selected subset of p₂(≤p₁) treatments and the control, and joint confidence interval estimates of the treatment versus control differences are calculated using Dunnett's (1955) procedure. Here both N₁ and N₂ are assumed to be fixed in advance, and the so-called square root rule is used to allocate observations among the treatments and the control in each stage. Dunnett's procedure is applied using two different types of estimates of the treatment versus control mean differences: The unpooled estimates are based on only the data obtained in the second stage, while the pooled estimates are based on the data obtained in both stages. The procedure based on unpooled estimates uses the critical point from a p₂-variate Student t-distribution, while that based on pooled estimates uses the critical point from a p₁-variate Student t-distribution. The two procedures and a composite of the two are compared via Monte Carlo simulation. It is shown that the expected value of p₂ determines which procedure yields shorter confidence intervals on the average. Extensions of the procedures to the case of unequal sample sizes are given. Applicability of the proposed two-stage procedures to a drug screening problem is discussed.     

Testing multiple hypotheses for the maintenance of male homosexual copulatory behaviour in flour beetles

Diverse animal groups exhibit homosexual interactions, yet the evolutionary maintenance of such behaviours remains enigmatic as they do not directly increase reproductive success by generating progeny. Here, we use Tribolium castaneum flour beetles, which exhibit frequent male homosexual copulations, to empirically test several hypotheses for the maintenance of such behaviours: (1) establishing social dominance; (2) practice for future heterosexual encounters; and (3) indirect sperm translocation. We found no evidence that Tribolium males use homosexual copulations either to establish dominance or to practice behaviours that increase their subsequent heterosexual reproductive performance. Our results provide limited support for the hypothesis of indirect sperm translocation: when males from two genetic strains mated with females immediately following a homosexual copulation, females produced progeny sired not only by the directly mating male, but also by that male’s homosexual partner. However, this phenomenon was detected in only 7% of homosexual pairs, and in each case such indirectly sired progeny accounted for < 0.5% of females’ total progeny. Direct observations indicated that mounting males often released spermatophores during homosexual copulations. These observations suggest that homosexual copulations may be a behavioural mechanism that allows males to expel older, potentially low‐quality sperm. Additional work is needed to test this new hypothesis, and to determine whether sperm release during homosexual copulations occurs in other groups.     

Application of two tests of multivariate discordancy to fisheries data sets

The generalized (Mahalanobis) distance and multivariate kurtosis are two powerful tests of multivariate discordancies (outliers). Unlike the generalized distance test, the multivariate kurtosis test has not been applied as a test of discordancy to fisheries data heretofore. We applied both tests, along with published algorithms for identifying suspected causal variable(s) of discordant observations, to two fisheries data sets from Lake Erie: total length, mass, and age from 1,234 burbot, Lota lota; and 22 combinations of unique subsets of 10 morphometrics taken from 119 yellow perch, Perca flavescens. For the burbot data set, the generalized distance test identified six discordant observations and the multivariate kurtosis test identified 24 discordant observations. In contrast with the multivariate tests, the univariate generalized distance test identified no discordancies when applied separately to each variable. Removing discordancies had a substantial effect on length-versus-mass regression equations. For 500-mm burbot, the percent difference in estimated mass after removing discordancies in our study was greater than the percent difference in masses estimated for burbot of the same length in lakes that differed substantially in productivity. The number of discordant yellow perch detected ranged from 0 to 2 with the multivariate generalized distance test and from 6 to 11 with the multivariate kurtosis test. With the kurtosis test, 108 yellow perch (90.7%) were identified as discordant in zero to two combinations, and five (4.2%) were identified as discordant in either all or 21 of the 22 combinations. The relationship among the variables included in each combination determined which variables were identified as causal. The generalized distance test identified between zero and six discordancies when applied separately to each variable. Removing the discordancies found in at least one-half of the combinations (k = 5) had a marked effect on a principal components analysis. In particular, the percent of the total variation explained by second and third principal components, which explain shape, increased by 52 and 44% respectively when the discordancies were removed. Multivariate applications of the tests have numerous ecological advantages over univariate applications, including improved management of fish stocks and interpretation of multivariate morphometric data.     

Testing Equality of Regressions When Error Variances are Heterogeneous to Study the Production Differentials of Rice in Bangladesh

Method to test the equality of p sets of regression coefficients in presence of heterogeneous error variances is suggested. The sets of regression coefficients are obtained from a sample where the sample observations are classified into p classes. The test statistic to test the significance of the regression is also suggested. As an application of the method, the equality of production functions of boro rice produced in different districts of Bangladesh in two different periods is tested.     

A New Statistical Goodness-of-Fit Test Based on Graphical Representation

A statistical goodness-of-fit test, based on representing the sample observations by linked vectors, is developed. The direction and the length of the linked vectors are defined as functions of the expected values of the order statistics and sample order statistics, respectively. The underlying method can be used to test distributional assumptions for any location-scale family. A test statistic Q_n is introduced and some of its properties are studied. It is shown that the proposed test can be generalized to test if two or more independent samples come from the same distribution. The test procedure provides a graphical method of identifying the true distribution when the null hypothesis is rejected.     

Using Discrete Distributions to Compute Powers for Goodness-of-Fit Test Statistics in a One-Way Multinomial Setting

Both the Bionomial and Poisson distributions are employed in this study to compute approximate powers for goodness-of-fit test statistics. The procedure adopted involves simulating 1000 samples from each of these distributions. These samples are then employed to compute both the randomized nominal critical values and estimated powers. The type I error rates returned from the use of the randomized critical levels Cα fall within the acceptable regions. We illustrate the use of the procedure with the Pearson's X² test statistic and show that this can readily be extended to any of the other well known goodness-of-fit test statistics.     

On Testing Equality of Means of Correlated Variates with Missing Data on One Response

A statistic is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one of the variates. The distribution of the statistic is approximated by a normal distribution under the null hypothesis. The empirical powers of the statistic are computed and compared with those of the conventional paired t and the other known statistics. The power comparisons support the use of the proposed test.     

Analysis of Proportions in One- or Two-Way Tables with Spoiled Data

This paper discusses the analysis of data on proportions in contingency tables. The X² ‘index of dispersion test’ (e.g. Fisher, 1954) is developed in these situations and compared with the use of the ‘logit’ transformation. An example using Osborn's (1979) data is given, illustrating the estimation of one or more missing observations.     

Larger agglutinated foraminifera of McMurdo Sound, Antarctica: Are Astrammina rara and Notodendrodes antarctikos allogromiids incognito?

Explorers Cove, an embayment on the oligotrophic western side of McMurdo Sound, Antarctica, is noted for physical and biological similarities to the deep sea. This locality is an important site for foraminiferal biologists because large, deep-sea-like agglutinated taxa are abundant at depths accessible to divers. Using microdissection methods, we found that many of the giant, single-chambered species from this locality display an allogromiid internal organization (i.e., a monothalamous cell body encased by an organic wall or “theca”). Another characteristic feature of these species is the presence of a distinct space separating the cell body and agglutinated test. We used high voltage transmission electron microscopy of thick sections to examine the theca of Astrammina rara, a dominant Explorers Cove species possessing a simple spherical test morphology, and Notodendrodes antarctikos, a distinctive arborescent species. We found that the simple gross morphology of A. rara's agglutinated test belies the complex ultrastructure of its allogromiid-like theca, and that the complex dendritic morphology of N. antarctikos' agglutinated test contradicts the simple ultrastructure of its theca. Additional field and laboratory observations on A. rara revealed that the agglutinated test can be a transient structure. Our observations raise questions regarding the taxonomy and phylogeny of unilocular agglutinated foraminifera and other primitive granuloreticuloseans. The results are also discussed in terms of test function in larger agglutinated foraminifera and the ecology of this important taxon.