John H. Skillings

On the Use of a Friedman-Type Statistic in Balanced and Unbalanced Block Designs

Distribution-free tests and multiple comparison procedures are considered for randomized block designs. A general Friedman-type statistic is defined that is usable in almost any block design with arbitrary missing data structure. For balanced incomplete block designs a multiple comparison procedure and exact tables are given for a statistic equivalent to one proposed by Durbin (1951). Finally, for designs with missing observations on only one treatment, we present exact tables for the test statistic, and we consider a multiple comparison procedure.

A Friedman-Type Rank Test for Main Effects in a Two-Factor ANOVA

Abstract A distribution-free test is considered for testing main effects in a two-factor ANOVA. Large-sample properties (cell sizes tending to infinity) including efficiencies and the limiting distribution of the test statistic are obtained. For proportional frequencies a simple sum of squares formula is obtained for the test statistic and multiple comparison procedures are given. For general designs the statistic is expressed using generalized inverses. Some tables for the null distribution of the statistic are also given.

Acute toxicity of copper to three life stages of Chironomus tentans as affected by water hardness-alkalinity

Abstract The acute toxicity of copper to first-and fourth-instar larvae of the midge Chironomus tentans was determined in soft, medium and hard water (43, 110 and 172 mg litre −1 as CaCO 3 total hardness, respectively). Percentage hatch of C. tentans eggs as influenced by copper concentration was studied in soft and hard water. Fourth instars were 12–27 times more resistant to copper stress than were first instars. Eggs were much more resistant to copper than either larval stage. Water hardness-alkalinity had a significant effect on copper toxicity to both larval instars. The concentration of copper necessary to immobilise 50% of the test population after 96 h (96-h EC 50 ) was not significantly different in soft and medium water (16·7 and 36·5 μg Cu litre −1 , respectively) for first-instar larvae, but both were significantly less than in hard water (98·2 μg Cu litre −1 ). Mean 96-h EC 50 s for fourth-instar larvae were not significantly different in medium (997·5 μg Cu litre −1 ) and hard (1184·6 μg Cu litre −1 ) water, but both were significantly higher than in soft (211·8 μg Cu litre −1 ) water. The influence of hardness-alkalinity on hatching success was less apparent. Percentage hatch in soft water was significantly reduced only at 5000 μg C litre −1 , while in hard water, percentage hatch was not affected even at that copper concentration. If sensitivity to copper is representative of sensitivity to other chemicals, the practice of using fourth-instar larvae in toxicity tests may grossly underestimate the sensitivity of the species to toxicants.

Filtration and phototactic behavior as indices of chronic copper stress inDaphnia magna straus

Filtration rate and negative phototactic behavior ofDaphnia magna were evaluated as potential predictors of the chronic no-effect copper concentration. The effects of copper on filtration and phototactic behavior were compared to the effects of chronic copper exposure on survivorship, number of juvenile molts, age at reproductive maturity, and neonate body length. Animals exposed to copper concentrations ⩾20μg/L exhibited reductions in filtration rate, negative phototaxis, body length of neonates and survival time. Animals exposed to 10μg/L exhibited a reduction in filtration rate, negative phototaxis and body length of neonates, but not in survival time. Number of juvenile molts, age at reproductive maturity, and mean brood size each responded erratically to copper exposure and are poor indices of copper stress. Since phototactic behavior, filtration rate, and neonate body length were all reduced at copper concentrations which did not reduce longevity or reproduction, bioassays in which only the latter are examined may underestimate the toxicity of copper and other chemicals. This may occur if bioassays are conducted for the frequently chosen 21-day interval.

Nonparametric Stepwise Multiple Comparison Procedures

Abstract Nonparametric multiple comparisons are discussed, with particular emphasis given to stepwise procedures. Nonparametric analogs of the stepwise all-subset procedure of Einot and Gabriel are presented, along with an ad hoc nonparametric analog of the Newman—Keuls procedure. These new procedures are compared among themselves and with nonstepwise procedures based on Type I error levels and comparisonwise power. It is shown that these stepwise nonparametric procedures control Type I error levels, and that they have superior pairwise power compared to the commonly used nonstepwise procedures.

Distribution-Free Tests for Ordered Alternatives in a Randomized Block Design

Abstract In this article we consider a class of test statistics that are useful for testing against ordered alternatives in balanced or unbalanced block designs. The approach consists of forming a statistic on each block and then combining the block statistics as a weighted sum to obtain one overall test statistic. Large-sample properties such as asymptotic normality and asymptotic relative efficiency as the number of blocks tends to infinity are considered. Throughout the article special attention is given to a Jonckheere-type statistic. Some consideration is given to the selection of the optimal weighting constants for the Jonckheere statistic.

Testing for ordered alternatives by combining independent distribution-free block statistics

A class of distribution-free tests for ordered alternatives in a block design is presented. On each block a distribution-free statistic is selected, and a weighted sum of these statistics is then formed. A procedure for selecting the weighting constants which maximize the asymptotic relative efficiency is presented. The improvement in the efficiency of the weighted sum of block statistics over the unweighted sum is considered and proves interesting. Some attention is also given to the idea of using different types of statistics on different blocks.

Distribution-Free Tests for Main Effects in Multifactor Designs

Abstract This article concerns distribution-free tests for main effects in M ≥ 1 factor designs when the factors are crossed. A single test statistic using Wilcoxon type scores is introduced that can be used for designs with any value of M. We illustrate that the properties of the test statistic, such as limiting distribution and asymptotic relative efficiency (ARE), can be obtained from known properties of test statistics proposed for two-factor designs. Large-sample properties are found based on cell sizes tending to infinity and based on the number of levels of the other factors tending to infinity. It is also shown that the Kruskal–Wallis and Friedman test statistics are special cases.

Nonparametric approaches to testing and multiple comparisons in a one-way anova

Two questions of interest involving nonparametric multiple comparisons are considered. The first question concerns whether it is appropriate to use a multiple comparison procedure as a test of the equality of k treatments, and if it is, which procedure performs best as a test. Our results show that for smaller k values some multiple comparison procedures perform well as tests. The second question concerns whether a joint ranking or a separate ranking multiple comparison procedure performs better as a test and as a device for treatment separation. We find that the joint ranking procedure does slightly better as a test, but for treatment separation the answer depends on the situation.

On the Null Distribution of Jonckheere's Statistic Used in Two-Way Models for Ordered Alternatives

Jonckheere (1954) proposed a test for ordered alternatives in a design with k treatments and b levels of another factor (blocks). In this paper tables are given for the null distribution of Jonckheeres statistic. Exact probabilities and critical values are given for α-levels near .2, .l, .05, .01 and .001, where k = 2(1)6 and b = 2(1)5. Tables appear for designs in which there are an equal number of observations in treatments within a block, with cell sample sizes ranging from one to five.