Robustness of Multiple Comparison Procedures: Treatment versus Control

Computer simulation techniques were used to investigate the Type I and Type II error rates of one parametric (Dunnett) and two nonparametric multiple comparison procedures for comparing treatments with a control under nonnormality and variance homogeneity. It was found that Dunnett's procedure is quite robust with respect to violations of the normality assumption. Power comparisons show that for small sample sizes Dunnett's procedure is superior to the nonparametric procedures also in non-normal cases, but for larger sample sizes the multiple analogue to Wilcoxon and Kruskal-Wallis rank statistics are superior to Dunnett's procedure in all considered nonnormal cases. Further investigations under nonnormality and variance heterogeneity show robustness properties with respect to the risks of first kind and power comparisons yield similar results as in the equal variance case.     

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.     

The Exact Distribution of the Anderson-Kannemann Statistic

The Anderson-Kannemann test is a rank test for treatment effects in a randomized block design with K treatments and N blocks. In this paper, an algorithm for computing the exact distribution of the Anderson-Kannemann test statistic under the null hypothesis is deduced. Then, the exact distribution is compared with the asymptotic χ²-distribution, and it is shown that the exact distribution is approximated fairly well by the asymptotic distribution. Tables of the exact distribution are given for K = 3, N = 3(1)15; K = 4, N = 3(1)11; K = 5, N = 3(1)7; and K = 6, N = 3(1)5.     

Family outcrossing rates and neighborhood floral density in natural populations of swamp milkweed (Asclepias incarnata): potential statistical artifacts

 To evaluate how environmental and genetic factors influence mating-system evolution, accurate estimates of outcrossing rates of individual plants (families) are required. Using isozyme markers, we observed wide variation in family outcrossing rates in three natural populations of Asclepias incarnata using three statistical methods: (1) a multilocus maximum-likelihood procedure (t _m); (2) a multilocus method-of-moments procedure (t _a); and (3) a direct comparison of progeny phenotypes against maternal phenotypes (t _d). Neighborhood floral-display size was positively correlated with t _a in one population, but showed no relationship with any of the other estimates of outcrossing for any population. Monte-Carlo simulations revealed that statistical variation associated with these estimation procedures can be large enough to explain all of the observed variation in outcrossing. We also found that significant, spurious correlations with neighborhood floral display could arise, on average, 7% of the time by chance alone. Our observations suggest that it is difficult to obtain accurate estimates of outcrossing in naturally pollinated plants using the estimation procedures currently available. Moreover, we caution that attempts to interpret observed variation in family outcrossing estimates by observing variation in ecological parameters could be misleading. Received: 28 September 1998 / Accepted: 27 October 1998     

Testing for the Equality of the Variance-Covariance Matrices of Two Jointly Normal Vector Variables

Assume a joint 2P-variate normal distribution for the p-component vectors x and y with unknown mean and unknown variance-covariance matrices Σ_xx and Σ_yy respectively, with Cov (x , y )=Σ_xy. No assumptions are made about the nature of Σ_xy. The likelihood ratio method is investigated to test the hypothesis that Σ_xx = Σ_yy. A method of numerical solution to the likelihood equations in the restricted parameter case is given when p=2, and approximate solutions are suggested for p>2.     

Selecting the Best Population,Provided It Is Better than a Control: The Unequal Variance Case

Bechhofer and Turnbull (1978) proposed two procedures to compare k normal means with a standard and the procedures guarantee that (1) with probability at least P₀* (specified), no category is selected when the best experimental category is sufficiently worst than the standard, and (2) with probability at least P₁* (specified), the best experimental category is selected when it is sufficiently better than the second best and the standard. For the case of common known variance, they studied a single-stage procedure. For the case of common unknown variance, they studied a two-stage procedure. Under the same formulation of Bechhofer and Turnbull (1978) and for the same selection goals (1) and (2) described above, Wilcox (1984a) proposed a procedure to the case of unknown and unequal variances, and supplied a table of the necessary constants to implement the procedure. This paper considers the case of unknown and unequal variances for the same formulation of Bechhofer and Turnbull, and Wilcox, but assumes that μ₀ is an unknown control. A two-stage procedure is proposed to solve the problem. A lower bound of the probability of a correct selection is derived and it takes the same form as the double integral appeared in Rinott (1978) which was used for the lower bound of the probability of a correct selection for a different selection goal.     

Chemical and histochemical studies of normal and diseased gastrointestinal tract. IV. A comparison of histochemical and chemical methods for the estimation of side chain O-acylated sialic acids

Summary Statistically significant correlations were obtained between a chemical assay for the proportion of colonic epithelial glycoprotein sialic acids with side chainO-acyl substituents and two histochemical methods, the PBT-KOH-PAS sequence (r _s=0.7485 forN=31,P=0.01, one-sided test) and the PAPT-KOH-Bh-PAS procedure (r _s=0.7024 forN=34). A positive correlation (r _s=0.8654 forN=30,P=0.01) was also obtained between the results of the two histochemical procedures. It is concluded that, on average, histochemical observations are a reliable semiquantitative comparative method for the estimation of side chainO-acetylated sialic acids.     

Photosynthetic Parameters at the Vegetative Stage and during Grain Development of Two Hexaploid Wheat Cultivars Differing in Salt Tolerance

Response of two spring wheat (Triticum aestivum L.) cultivars, salt tolerant SARC-I and salt sensitive Potohar, to different concentrations of NaCl was examined under glasshouse conditions. Eighteen-day-old plants of both the lines grown in sand culture were irrigated with 0 (control), 80, 160 or 240 mM NaCl in full strength Hoagland's nutrient solution. Shoot fresh and dry masses, and leaf area per plant of SARC-I at the vegetative stage, were significantly greater than those of cv. Potohar at higher salt concentrations, however, relative growth rate (RGR) of cv. Potohar was significantly higher than that of SARC-I. SARC-I had higher net photosynthetic rate (P_N), stomatal conductance (g_s) and transpiration rate (E) than cv. Potohar at the vegetative stage, but the cultivars did not differ significantly in water-use efficiency (P_N/E), intrinsic water use efficiency (P_N/g_s), and intercellular/ambient CO₂ concentration ratio. At the grain development stage, SARC-I had significantly higher P_N and g_s in the flag leaf than cv. Potohar under salinity. SARC-I was superior to cv. Potohar with respect to number of grains per spike, number of grains per spikelet, mean grain mass, and grain yield per plant at all NaCl concentrations.     

Many‐to‐one Comparisons in Stratified Designs

Cheung and Holland (1992) extended Dunnett's procedure for comparing all active treatments with a control simultaneously within each of r groups while maintaining the Type I error rate at some designated level α allowing different sample sizes for each of the group‐treatment categories. This paper shows that exact percentage points can be easily calculated with current available statistical software (SAS). This procedure is compared to resampling techniques and a Bonferroni corrected Dunnett‐within‐group procedure by means of a simulation study.     

Statistical Analysis of Joint Effects of Major Genes and Polygenes in Quantitative Genetics

In this paper we analyze a quantitative genetic character which is controlled by both major genes and polygenes. Assuming that there are no epistatic effects, no linkage and no genetic-environmental interactions, we follow TAN and CHANG (1972) to derive the probability distributions for segregating populations. The numbers of major genes and polygenes, and the additive and dominance effects of major genes and polygenes are then estimated by using the procedures developed in TAN and CHANG (1972) and the POWELL -FLETCHER search procedure for maximum values. In this paper, we consider the case involving data from P₁, P₂, F₂, B₁ (Backcross to P₁) and B₂ (Backcross to P₂) as this type of experiment is common in practical applications. The analyses are applied to a simulated model generated by using binomial, multinomial and normal variables and to the data of an experiment on kernel weight of sorghum plant provided to the authors by Professor GEORGE H. L. LIANG of Kansas State University. The analysis of these data indicate clearly that the method derived in this paper is useful and desirable.     

Effective number of breeding adults in Oregon spotted frogs (<Emphasis Type="Italic">Rana</Emphasis><Emphasis Type="Italic">pretiosa</Emphasis>): genetic estimates at two life stages

We used genetic methods to estimate the effective number of breeders (N _b) in a population of Rana pretiosa, an imperiled amphibian in western North America. Microsatellite data was gathered from large samples of adults, eggs, and juveniles collected in 2006. We wished to determine where in the life cycle the greatest reductions in N _b occur, and to compare genetic estimates of N _b to an egg mass count estimate of the number of breeding adults. We predicted that N _b estimated at the metamorph stage would be reduced by increased variance in family size due to egg mass mortality. Contrary to our prediction, estimates of N _b at the egg and metamorph stages were similar. Thus, we found no evidence of inflated variance in family size between the two stages. If our results for this population are typical for R. pretiosa, then increased variance in family size during the egg to metamorph stage may not be a strong factor in reducing the effective population sizes (N _e) relative to the census sizes (N) in this species.     

On the difference between gauss-markov and least squares estimates

For the usual full rank univariate least squares regression model y = XB + e, E(e) = 0, E(ee) = A, the equality of the estimates occurs when B-B^* = (XA^?1X)^?1XA^-1y-(XX)^?1Xy = 0. A necessary and sufficient condition for this equality is that A has some N - k + 1 roots equal where N is the rank of A and k is the rank of X.     

Sieving alive or after fixation: effects of sieving procedure on macrobenthic diversity,density and community structure

Although combining datasets is often needed to unravel large-scale or long-term patterns in benthos ecology, this is frequently hampered by differences in technical design of the individual studies. One element that often vary among macrobenthic studies is the sieving procedure: sieving alive versus sieving after fixation. This study therefore aimed at the qualification and quantification of the impact of sieving procedure, using a 1 mm mesh sized sieve, at three levels of ecological organisation: (1) diversity, (2) species and taxon density, and (3) community structure. To include a maximum suite of macrobenthic species and to evaluate the community-specific effects, the impact of sieving procedure was investigated within four widely spread macrobenthos communities in the Belgian part of the North Sea. Sieving alive negatively impacted all tested diversity measures (S, N₁, N₂, N_∞, H′, ES ₁₀₀ and J′): community-dependent relative losses of up to 35% were observed. However, most trends were ambiguous and statistically non-significant. Community- and taxon-dependent impacts were detected at the level of density. Mainly polychaetes were found to be negatively impacted by sieving alive (relative losses maximum 81%): especially small, interstitial polychaetes (e.g. Hesionura elongata and Spio filicornis) tend to actively escape from the sieve (relative loss up to 100%). Next to size, also behaviour, the presence of head appendages, the depth of the sampling stations and sampling season are believed to influence the sieving procedure impact. While detailed community composition was impacted (ANOSIM dissimilarity: maximum 85%), no major impact on the differentiation between the investigated communities was detected. The present study thus demonstrated that combining data, retrieved with a different sieving procedure can be useful, but its reliability will mainly depend on the type of questions one wants to answer. In all cases caution at all levels of ecological organisation is advised.     

On the Probabilities of Obtaining Negative Estimates of Genetic Variance Components

This paper derives the probabilities of obtaining negative estimates of additive and dominance genetic variances when one uses the traditional weighted least square method for estimating genetic variances as given in MATHER and JINKS (1971). The model considered involves P₁, P₂, F₂, B₁ (Backcross to P₁) and B₂ (Backcross to P₂). The results are derived under the ordinary assumptions as made in the genetic literatures. It is shown that unless the genetic effects are very large and environmental effects small, the probabilities of obtaining negative estimates of additive and dominance variances are in general quite large.     

Synthesis of N-acridinyl-N'-alkylguanidines: dramatic influence of amine to guanidine replacement on the physicochemical properties

Transformation of aminoacridines into N-acridinyl-N′-alkylguanidines is described. The chosen procedure allows introduction of pendent substituents (exemplified by N,N-dimethylaminopropyl chain) into key acridinyl thioureas, thus opening the way to structural diversity. Spectroscopic study and pK_a determination show that the presence of the strongly basic guanidine has a dramatic influence on the ionization of the acridine nucleus by lowering the pk_a value down to 4.49.     

Genetic correlation between the ages of menarche and menopause

Using mostly prospective menstrual data from mothers and daughters in the Tremin Trust Menstrual Reproductive History Program, this study produces the first estimates of the genetic correlation between the ages of menarche and menopause. I carried out two separate analyses. Standard regression analysis of 21 mother/daughter dyads with natural menopause yielded a nonsignificant negative mean genetic correlation of r _A =−0.139±1.268. Survival analysis/maximum likelihood estimation on a dataset which included an additional 85 dyads with censored observations on daughters (N=106) yielded a nonsignificant positive genetic correlation of r _A =0.613±0.587 (p=0.1492). Although a substantial non-zero correlation cannot be ruled out, these estimates suggest there is no genetic correlation between menarche and menopause, in agreement with previous phenotypic findings. As inconclusive as they may be, these estimates also contribute to our understanding of the nature of selection working on the reproductive life span of human females.     

A weighted FDR procedure under discrete and heterogeneous null distributions

Multiple testing (MT) with false discovery rate (FDR) control has been widely conducted in the “discrete paradigm” where p-values have discrete and heterogeneous null distributions. However, in this scenario existing FDR procedures often lose some power and may yield unreliable inference, and for this scenario there does not seem to be an FDR procedure that partitions hypotheses into groups, employs data-adaptive weights and is nonasymptotically conservative. We propose a weighted p-value-based FDR procedure, “weighted FDR (wFDR) procedure” for short, for MT in the discrete paradigm that efficiently adapts to both heterogeneity and discreteness of p-value distributions. We theoretically justify the nonasymptotic conservativeness of the wFDR procedure under independence, and show via simulation studies that, for MT based on p-values of binomial test or Fisher's exact test, it is more powerful than six other procedures. The wFDR procedure is applied to two examples based on discrete data, a drug safety study, and a differential methylation study, where it makes more discoveries than two existing methods.     

On weighted Hochberg procedures

We consider different ways of constructing weighted Hochberg-typestep-up multiple test procedures including closed proceduresbased on weighted Simes tests and their conservative step-upshort-cuts, and step-up counterparts of two weighted Holm procedures.It is shown that the step-up counterparts have some seriouspitfalls such as lack of familywise error rate control and lackof monotonicity in rejection decisions in terms of p-values.Therefore an exact closed procedure appears to be the best alternative,its only drawback being lack of simple stepwise structure. Aconservative step-up short-cut to the closed procedure may beused instead, but with accompanying loss of power. Simulationsare used to study the familywise error rate and power propertiesof the competing procedures for independent and correlated p-values.Although many of the results of this paper are negative, theyare useful in highlighting the need for caution when procedureswith similar pitfalls may be used.     

Gram-Schmidt Orthogonalization of Multinormal Variates: Applications in Genetics

The computation of an N-variate normal density function requires the inversion of an N × N co-variance matrix. Furthermore, if each mean depends on u unobservable factors, a mixture of u^N N-variate normal densities must be computed, making likelihood calculations impractical even for moderate N. The Gram-Schmidt orthogonalization process is used to express a multinormal density as a product of univariate normal densities. When the pattern of the correlation matrix is taken into account the formulas may be considerably simplified. In some cases each of the orthogonal variates can be written as a linear combination of only a few of the original variates. Such results are crucial for applications of multinormal distributions and of mixtures of multinormal distributions. An intraclass correlation model and a genetic variance components model applicable to family data are discussed as examples.     

Construction of Equi-replicated Balanced Block Designs with Block Sizes Exceeding the Number of Treatments

In this note it is shown that the block design with incidence matrix Ñ = [NN… N], where N = c_1hN_h + c_oh (11′–N_h). c_oh and c_1h are any non-negative integers and N_h,h = 1, 2,…,p, are incidence matrices of balanced incomplete block designs with the same number of treatments t, is a balanced block design with the block sizes exceeding the number of treatments. In derivation the matrix M₀, introduced by CALIński (1971) is utilized.