Simultaneous One-Sided Pairwise Comparisons in a Two-Way Design

Cheung and Chan (1996) provided a procedure for simultaneous two-sided pairwise comparisons of treatment means in a two-way design. Expanding on this earlier work, we develop a procedure where all such comparisons are instead one-sided. The new procedure is appropriate in situations where the treatment means are a priori ordered. The overall type I error rate for the simultaneous inferences is controlled at a designated level. Tables of upper percentage points of the test statistic are given to facilitate the implementation of our method. The application of the testing procedure is illustrated with an example from a medical study.     

Application of maximum likelihood paired comparison ranking to estimation of a linear dominance hierarchy in animal societies

Dominance hierarchies have been widely used for describing the outcome of competitive interactions in an animal group. We present a procedure for estimating the linear dominance hierarchy. The procedure uses the statistical method of paired comparisons, assuming weak stochastic transitivity to model interactions within a linear dominance hierarchy. The linear dominance hierarchy is estimated using a maximum likelihood ranking procedure. This method allows unequal numbers of encounters between pairs and does not require all pairs to have observed encounters. The method is illustrated by application to behavioural data from a group of 10 baboons (Papio cynocephalus anubis).     

Multiple Comparisons of Polynomial Distributions

Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X²-statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many-one tests and TUKEY'S range test (cf. MILLER, 1981). The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H₀. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979). The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3-tables the limit distribution of the test statistic under H₀ is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni-Holm procedure. Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X²-variables applicable to simultaneous X²-tests.     

Nonparametric All‐Pairs Multiple Comparisons

Nonparametric all‐pairs multiple comparisons based on pairwise rankings can be performed in the one‐way design with the Steel‐Dwass procedure. To apply this test, Wilcoxon's rank sum statistic is calculated for all pairs of groups; the maximum of the rank sums is the test statistic. We provide exact calculations of the asymptotic critical values (and P‐values, respectively) even for unbalanced designs. We recommend this asymptotic method whenever large sample sizes are present. For small sample sizes we recommend the use of the new statistic according to Baumgartner , Weiss , and Schindler (1998, Biometrics 54 , 1129–1135) instead of Wilcoxon's rank sum for the multiple comparisons. We show that the resultant procedure can be less conservative and, according to simulation results, more powerful than the original Steel‐Dwass procedure. We illustrate the methods with a practical data set.     

Comparison of t-tests for differences in sexual dimorphism between populations

A t-test that can be used for evaluating the significance of differences in metric sexual dimorphism between populations is derived directly from mathematical considerations of the differences between distributions. It is compared with the t-test derived by Relethford and Hodges (1985), which was based upon linear regression with sex as a dummy variable. Both are determined to be mathematically equivalent, though the one derived here is more similar in form to traditional t-tests of differences and therefore may be simpler to employ. Both tests require only summary statistics for comparisons between populations and comparisons between generations within populations.     

Gender determination by body weight and linear measurements in American and Chilean flamingos,previously surgically sexed: Within-sex comparison to greater flamingo measurements

Gender was determined by laparoscopic visualization of the gonads for 38 adult American flamingos (Phoenicopterus ruber ruber L.) and 36 adult Chilean flamingos (P. chilensis L.). Concomitant body weight (kg) and linear measurements (mm) of the culmen (bill), tarsus (tarsometatarsus), middle toe, and wing were taken. Statistical comparisons of body weight and linear measurements for male vs. female were made for each species. Also, the same-sex statistical comparisons were made between these two species, and between each of these two species and with data for greater flamingos (P. r. roseas L.) from a previous publication. As previously published for greater flamingos, an overlap between sexes existed in all measurements with males on average larger than females for both American and Chilean flamingos. However, Students' t-test indicated a significant sexual difference for all measurements between males and females of each species except for culmen length in Chilean flamingos. Students' t-test also indicated a significant difference when species were compared (Chilean vs. greater, and American vs. Chilean) and subspecies (American vs. greater) were compared for most of the 5 measurements. Thus, despite limitations imposed by between-sex overlap, weights and linear measurements, especially tarsus, middle toe, and wing length, appear to be useful in determining an individual's gender when species or subspecies identification is considered.     

Monte Carlo Comparison of Multiple Comparison Procedures

Seven procedures of multiple comparisons: Tukey, Scheffé, Bonferroni, Studentized Maximum Modulus, Duncan, Newman-Keuls and F are compared with respect to the probability of the correct decision. Monte Carlo simulation shows that there is no the best procedure. AMS 1985 Subject Classification: 62 J 15.     

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.     

Exact Multiple Comparisons of Three or More Regression Lines: Pairwise Comparisons and Comparisons with a Control

The problem of finding exact simultaneous confidence bounds for differences in regression models for k groups via the union‐intersection method is considered. The error terms are taken to be iid normal random variables. Under an assumption slightly more general than having identical design matrices for each of the k groups, it is shown that an existing probability point for the multivariate studentized range can be used to find the necessary probability point for pairwise comparisons of regression models. The resulting methods can be used with simple or multiple regression. Under a weaker assumption on the k design matrices that allows more observations to be taken from the control group than from the k‐1 treatment groups, a method is developed for computing exact probability points for comparing the simple linear regression models of the k‐1 groups to that of the control. Within a class of designs, the optimal design for comparisons with a control takes the square root of (k‐1) times as many observations from the control than from each treatment group. The simultaneous confidence bounds for all pairwise differences and for comparisons with a control are much narrower than Spurrier's intervals for all contrasts of k regression lines.     

On the Numerical Availability of Multiple Comparison Procedures

In the past many multiple comparison procedure were difficult to perform. Usually, such procedures can be traced back to studentized multiple contrast tests. Numerical difficulties restricted the use of the exact procedures to simple, commonly balanced, designs. Conservative approximations or simulation based approaches have been used in the general cases. However, new efforts and results in the past few years have led to fast and efficient computations of the underlying multidimensional integrals. Inferences for any finite set of linear functions of normal means are now numerically feasible. These include all‐pairwise comparisons, comparisons with a control (including dose‐response contrasts), multiple comparison with the best, etc. The article applies the numerical progress on multiple comparisons procedures for common balanced and unbalanced designs within the general linear model.     

Evolutionary divergence of mitochondrial DNA from Paramecium aurelia

Summary Mitochondrial (mt) DNA from four sibling species within the Paramecium aurelia complex, including stocks of different geographic origin and mutants, were analyzed using four 6-bp recognition site and one 4-bp recognition site endonucleases and the sequence divergence was estimated using Upholt's (1977) statistical procedure. All four species were readily distinguishable regardless of the restriction endonuclease employed. With intraspecies comparisons, no differences were observed which could be accounted for on the basis of geographic origin. Except for species 4, each stock (and mutant) gave a species-specific fragment pattern. For species 4, while the patterns were distinct from the other species, two species-specific type of patterns were found, designated A and B. The sequence divergence between these was estimated to be between 1 and 2 percent. With interspecies comparisons, the sequence divergence ranged from 3.9 to 10.3% with the greatest divergence being between species 1 and 4, and the least between species 1 and 5. The similarity between species 1 and 5 is in accord with other criteria for interspecies comparisons. The degree of sequence divergence measured here in Paramecium mt DNA is well within the range reported for rodents and primates. All four species mt DNA were cleaved to many DNA fragments by DPN II, an enzyme which recognizes non-methylated sites, and not by DPNI, the methyl-site specific counterpart of DPN II, suggesting that mt DNA from Paramecium aurelia is not appreciably methylated, if at all.     

Discriminant functions for linear comparisons of means when covariance matrices are unequal

Situations exist, as in the biological example of discriminant analysis for natural hybridization, cited in the text, where (a) not all populations have equal variances, and (b) comparisions based on single degrees of freedom must be planned. This paper presents a statistical methodology of estimating discriminant functions for linear comparisons among k(<2) multivariate normal populations, and of testing their significance, when these populations have unequal covariance matrices.     

Generalized Maximum Range Tests for Pairwise Comparisons of Several Populations

A multiple comparison procedure (MCP) is proposed for the comparison of all pairs of several independent samples. This MCP is essentially the closed procedure with union-intersection tests based on given single tests Q_ij for the minimal hypotheses H_ij. In such cases where the α-levels of the nominal tests associated with the MCP can be exhausted, this MCP has a uniformly higher all pair power than any refined Bonferroni test using the same Q_ij. Two different general algorithms are described in section 3. A probability inequality for ranges of i.i.d. random variables which is useful for some algorithms is proved in section 4. Section 5 contains the application to independent normally distributed estimates and section 6 the comparisons of polynomial distributions by multivariate ranges. Further applications are possible. Tables of the 0.05-bounds for the tests of section 5 and 6 are enclosed.     

Hommel's procedure in linear time

Hommel's and Hochberg's procedures for familywise error control are both derived as shortcuts in a closed testing procedure with the Simes local test. Hommel's shortcut is exact but takes quadratic time in the number of hypotheses. Hochberg's shortcut takes only linear time after the P‐values are sorted, but is conservative. In this paper, we present an exact shortcut in linear time on sorted P‐values, combining the strengths of both procedures. The novel shortcut also applies to a robust variant of Hommel's procedure that does not require the assumption of the Simes inequality.     

Inferences Concerning a Mean Vector when the Variables are Grouped into Subsets

In experiments involving multivariate measurements biological considerations often indicate a grouping of the variables into subsets such that all the variables in a subset are equally important and the subsets may be arranged in a decreasing order of importance (JENSEN, 1972). This paper deals with a method of analysis appropriate in such cases, to solve multivariate problems such as profile analysis, comparison of two mean vectors, or univariate problems such as analysis of SCHEFFÉ'S mixed effects model with interaction (SCHEFFÉ, 1959; MILLER, 1966), usually treated using HOTELLING'S T². An invariance structure underlying this procedure is examined and used to obtain noncentrality parameters of the power function, and derive some of its properties. The multiple comparisons procedure in MUDHOLKAR and SUBBAIAH (1976) is extended and illustrated in terms of the data from a clinical trial.     

Variation In Surface Proteins In Tetrahymena*

SYNOPSIS. Surface proteins of Tetrahymena were identified by lactoperoxidase iodination, and comparisons were made between a number of strains and species within the genus. an adequate procedure for strain comparisons was found to be solubilization of whole cells following iodination, separation of total cell protein using polyacrylamide gel electrophoresis, and identification of surface proteins by autoradiography of dried gels. the results obtained in the present study show the existence of both interspecific and intraspecific variation in surface proteins of Tetrahymena, but the differences tend to be small within species and large between species. the relation of these cell surface fingerprints to the present taxonomic designations within the genus is discussed. Questions are raised about the functional significance of these surface proteins.     

Hair removal methods: A comparative study for dermoscopy images

Removal and restoration of hair and hair-like regions within skin lesion images is needed so features within lesions can be more effectively analyzed for benign lesions, cancerous lesions, and for cancer discrimination. This paper refers to “melanoma texture” as a rationale for supporting the need for the proposed hair detection and repair techniques, which incompletely represents why hair removal is an important operation for skin lesion analysis. A comparative study of the state-of-the-art hair-repaired methods with a novel algorithm is also proposed by morphological and fast marching schemes. The hair-repaired techniques are evaluated in terms of computational, performance and tumor-disturb patterns (TDP) aspects. The comparisons have been done among (i) linear interpolation, inpainting by (ii) non-linear partial differential equation (PDE) and (iii) exemplar-based repairing techniques. The performance analysis of hair detection quality, was based on the evaluation of the hair detection error (HDE), quantified by statistical metrics and manually used to determine the hair lines from a dermatologist as the ground truth. The results are presented on a set of 100 dermoscopic images. For the two characteristics measured in the experiments the best method is the fast marching hair removal algorithm (HDE: 2.98%, TDP: 4.21%). This proposed algorithm repaired the texture of the melanoma, which becomes consistent with human vision. The comparisons results obtained, indicate that hair-repairing algorithm based on the fast marching method achieve an accurate result.     

Modeling of microbial substrate conversion,growth and product formation in a recycling fermentor

Paracoccus denitrificans and Bacillus licheniformis were grown in a carbon- and energy source-limited recycling fermentor with 100% biomass feedback. Experimental data for biomass accumulation and product formation as well as rates of carbon dioxide evolution and oxygen consumption were used in a parameter optimization procedure. This procedure was applied on a model which describes biomass growth as a linear function of the substrate consumption rate and the rate of product formation as a linear function of the biomass growth rate. The fitting procedure yielded two growth domains for P. denitrificans. In the first domain the values for the maximal growth yield and the maintenance coefficient were identical to those found in a series of chemostat experiments. The second domain could be described best with linear biomass increase, which is equal to a constant growth yield. Experimental data of a protease producing B. licheniformis also yielded two growth domains via the fitting procedure. Again, in the first domain, maximal growth yield and maintenance requirements were not significantly different from those derived from a series of chemostat experiments. Domain 2 behaviour was different from that observed with P. denitrificans. Product formation halts and more glucose becomes available for biomass formation, and consequently the specific growth rate increases in the shift from domain 1 to 2. It is concluded that for many industrial production processes, it is important to select organisms on the basis of a low maintenance coefficient and a high basic production of the desired product. It seems less important that the maximal production becomes optimized, which is the basis of most selection procedures.     

The use of polynomial regression analysis with indicator variables for interpretation of mercury in fish data

Mercury levels in fish in reservoirs and natural lakes have been monitored on a regular basis since 1978 at the La Grande hydroelectric complex located in the James Bay region of Québec, Canada. The main analytical tools historically used were analysis of covariance (ANCOVA), linear regression of the mercury-to-length relationship and Student-Newman-Keuls (SNK) multiple comparisons of mean mercury levels. Inadequacy of linear regression (mercury-to-length relationships are often curvilinear) and difficulties in comparing mean mercury levels when regressions differ lead us to use polynomial regression with indicator variables.For comparisons between years, polynomial regression models relate mercury levels to length (L), length squared (L²), binary (dummy) indicator variables (B_n), each representing a sampled year, and the products of each of these explanatory variables (L × B₁, L² × B₁, L × B₂, etc.). Optimal transformations of the mercury levels (for normality and homogeneity) were found by the Box-Cox procedure. The models so obtained formed a partially nested series corresponding to four situations: (a) all years are well represented by a single polynomial model; (b) the year-models are of the same shape, but the means may differ; (c) the means are the same, but the year-models differ in shape; (d) both the means and shapes may differ among years. Since year-specific models came from the general one, rigorous statistical comparisons are possible between models.Polynomial regression with indicator variables allows rigorous statistical comparisons of mercury-to-length relationships among years, even when the shape of the relationships differ. It is simple to obtain accurate estimates of mercury levels at standardized length, and multiple comparisons of these estimations are simple to perform. The method can also be applied to spatial analysis (comparison of sampling stations), or to the comparison of different biological forms of the same species (dwarf and normal lake whitefish).     

Leaf area estimation of sunflower leaves from simple linear measurements

Simple, accurate, and non-destructive methods for determining leaf area (LA) of plants are important for many experimental comparisons. Determining the individual LA of sunflower (Helianthus annuus L.) involves measurements of leaf parameters such as length (L) and width (W), or some combinations of these parameters. Two field experiments were carried out during 2003 and 2004 to compare predictive equations of sunflower LAs using simple linear measurements. Regression analyses of LA vs. L and W revealed several equations that could be used for estimating the area of individual sunflower leaves. A linear equation having W² as the independent variable provided the most accurate estimate (r ² = 0.98, MSE = 985) of sunflower LA. Validation of the equation having W² of leaves measured in the 2004 experiment showed that the correlation between calculated and measured areas was very high.