Multivariate Homogeneity Testing Using an Extended Concept of Nearest Neighbors

Given independent multivariate random samples {X_ij: j = 1, …, n_i} from F_i, for i = 1,2, a test is desired for H₀: F₁ = F₂ against general alternatives. Consider the k · (n₁ + n₂) possible ways of choosing one observation from the combined samples and then one of its k nearest neighbors, and let S_k be the proportion of these choices in which the point and neighbor are in the same sample. Schilling (1986) proposed S_k as a test statistic, but did not indicate how to determine k. We suggest as test statistic W = N Σ kS_k, which we show is equivalent to a sum of N Wilcoxon rank sums, and also to a sum of two two-sample U-statistics of degrees (1, 2) and (2, 1). Simulation with multivariate normal data suggests that our test is generally more powerful than Schilling's test using k = 1, 2, or 3. We illustrate its use with Fisher's iris data.     

On a Characterization of the Exponential Distribution by Weak Homoscedasticity of Record Values

A sequence {X_n, n≤1} of independent and identically distributed random values with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j ≤ max (X₁, X₂, …, X_j?1). We define L(n)=min.(j!j>L(n?1.), X_j<X_L(n?1)), with L(0) = 1. Let Z_n=X_L(n)? X_L(n?1), n ≤ 1. We will show that the conditional variance of Z_n given X_L(n?1)=x does not depend on × if and only if F(x) is exponential.     

An Unbalanced Two-way Model With Random Effects Having Unequal Variances

Consider the model y_ijk=u ± a_i ± b_i ± c_ij ± e_ijk i=1, 2,…, t; j=1, 2,…b; k=1, 2,…,n_ij where μ is a constant and a_i, b_i, c_ij are distributed independently and normally with zero means and variances Δ₂ Δ2/bd_ij and δ₂ respectively. It is assumed that d_i's, and d_ij's are known (positive) constants (for all i and j). In this paper procedures for estimating the variance components (Δ₂, Δ²_b and Δ²_a) and for testing the hypothesis H_oc:Δ²_c/Δ² = y₃ and H_oa:Δ²_b/Δ² = y₄ (where y₂, y₃, and y₄, are specified constants) are presented. A generalization for the mixed model case is discussed in the last section.     

Estimation of a Conditional Mean in a Linear Regression Model

Consider the two linear regression models of Y_ij on X_ij, namely Y_ij = β_io + β_il X_ij + ε_ij,j = 1,2,…,n_i, i = 1,2, where ε_ij are assumed to be normally distributed with zero mean and common unknown variance σ². The estimated value of a mean of Y₁ for a given value of X₁ is made to depend on a preliminary test of significance of the hypothesis β₁₁ = β₂₁. The bias and the mean square error of the estimator for the conditional mean of Y₁ are given. The relative efficiency of the estimator to the usual estimator is computed and is used to determine a proper choice of the significance level of the preliminary test.     

The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

The Point Evaluation Method for Approximating Function Means,Variances and Covariances

A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X₁, X₂, …, X_n which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2ⁿ points given by X₁=μ₁±₁, X₂ = μ₂ ±₂, …, X_n = = μ_n μ_n, where μ_i is the mean and σ_i the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2ⁿ points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.     

On the Use of Ranks in Combined Estimates in Quantal Bioassays

If {U_i} i = 1, …, k is a sequence of binary responses of n_i subjects at each of k successive dose levels x_i, there is the problem of the statistical treatment of the observed proportions P_i = U_i/ni when neither the probit nor the logit transformation may be assumed. This paper considers the use of the midranks of the responses for point and interval estimation of relative potency in the case of parallel line assay in particular. More generally the problem of combining the results of several independent estimates using ranks is discussed. Several examples illustrate the method.     

An Unbalanced Nested Model with Random Effects Having Unequal Variances

Consider the model Y_ijk=μ + a_i + b_ij + e_ijk (i=1, 2,…, t; j=1,2,…, B_i; k=1,2…,n_ij), where μ is a constant and a¹,b_ij and e_ijk are distributed independently and normally with zero means and variances σ²_ad_ij and σ², respectively, where it is assumed that the d_i's and d_ij's are known. In this paper procedures for estimating the variance components (σ², σ²_a and σ²_b) and for testing the hypothesis σ²_b = 0 and σ²_a = 0 are presented. In the last section the mixed model y_ijk, where x_ijkkm are known constants and β_m's are unknown fixed effects (m = 1, 2,…,p), is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the β_m's.     

Note on Confidence Regions in Multiple Assays

This note is in continuation of the author's results on ’classical‘ confidence regions for relative potencies in multiple assays (Bennett, 1987). It is shown by the use of a Bonferroni inequality (e.g. Miller, ch. 2, 1980) that approximate confidence regions for the relative potencies {M_i} i = 1, …, p may also be obtained directly. A comparison with the classical regions is made for the case: p = 2 using Finney's example (1978, ch. 2).     

Synthesis of mixed N,N′-(Ar,Ar′-diaryl)iminoisoindolines for applications in palladacycle formations

The synthesis of N,N′-(Ar,Ar′-diaryl)iminoisoindolines containing different aryl groups bound to the two nitrogen atoms is described. The iminoisoindolines were obtained by a three component, one-pot reaction of phthalaldehyde with 1 equivalent p-NO₂-aniline and 1 equivalent p-R-aniline, where R = H, Me, MeO or ⁱPr, resulting in formation of non-symmetrically substituted (mixed) iminoisoindolines, 1-p-nitrophenylimino-2-p-R-phenylisoindoline (R = H (1), Me (2), MeO (3), and ⁱPr (4)), as analytically pure precipitates requiring no further purification. Only one isomer precipitates from solution wherein the nitro group resides exclusively at the imine position while the more electron donating substituent ends up on the isoindoline ring position. Further reaction with Pd(OAc)₂ in dichloromethane at room temperature results in formation of six-membered [C,N] dinuclear cyclopalladated complexes with the general formula [(Ar,Ar′-diaryliminoisoindoline)Pd{μ-OAc}]₂.     

Characterization of the WEIBULL Distribution by Properties of Order Statistics

Let X_1:n, X_2:n, X_3:n…, X_n:n be the order statistics of n independent random variables with the common (absolutely continuous strictly increasing) distribution function F. The main results given in this article are:

• 1 For any fixed r and two distinct numbers s₁ and s₂ (1<r<s₁<s₂≦n) the distributions of V_i and W_i (defined in (1.11) and (1.12) are identical for i = 1,2 iff F(x) is WEIBULL (1.2).
• 2 The statistics D₁ and D₂ (as in (1.8) and (1.9)) are independent iff F(x) is WEIBULL (1.2).
• 3 The statistics U_i (1≦j≦n?1) and X_i:n (i ≦ j) are independent iff F(x) is WEIBULL (1.2).
• 4 Let X, X₁, X₂, …, X_k be random variables such that

These conditions are necessary and sufficient for F(x) to be WEIBULL .     

F-tests for Hypotheses with Block Matrices and Under Conditions of Orthogonality in the General Multivariate Gauss-Markoff Model

The multivariate general Gauss-Markoff (MGM) model (U, XB, ∑?σ²V) when the matrices V ≥ 0 and ∑ > 0 are known and the scalar σ² > 0 is unknown, is considered. The present paper is a continuation of two earlier works (Oktaba, 1988a, b). If XB = X₁Σ + X₂Δ, then the F-test for verification the hypothesis WΣA = 0 is presented. Moreover, under conditions of orthogonality the decomposition of the matrix S_A (?BCA)′L^?(?BCA) into the sum of s = r(L) matrices is given, where ?BCA is the estimator of the parametric estimable functions ?BCA, Cov (?BCA) = A′ ∑?σ²L = ?C₄?′, B? = (X′T^?X)^?X′T^?U, C₄ = (X′T^?X)^?M, where M = M′ is any arbitrary matrix such that R(X) ? R(T), T=V+XMX′; T^? is any c-inverse. R(A) is the linear space generated by the colums of A. Then under additional assumption on normality of U the statistics F for testing ?BA = 0 is deduced. Under conditions of normality of U and decomposition of S_A, the statistics F₁, …, F_s for the hypotheses jⁱ BA = 0 (i = 1,…, s) are established.     

Asymmetric epoxidation of cinnamyl alcohol with optically active titanium complexes

A family of titanium(IV) alkoxo compounds [{Ti(O‐i‐Pr)₂(OR)₂}₂] 1–4 prepared by alcohol exchange of Ti(O‐i‐Pr)₄ and a chiral higher‐boiling alcohol [ROH = 1,2:3,4‐di‐O‐isopropylidene‐α‐d ‐galactopyranose, 1,2:5,6‐di‐O‐isopropylidene‐α‐d ‐glucofuranose, (1R,2S,5R)‐(?)‐menthol, (1S‐endo)‐(?)‐borneol, (1S,2R,5S)‐(+)‐menthol, and (+)‐borneol] has been tested to evaluate their catalytic activity and stereoselectivity in the asymmetric epoxidation of cinnamyl alcohol. © 2005 Wiley‐Liss, Inc. Chirality     

Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields

Summary Two types of neuronal lateral inhibition in one-dimensional fields of receptors and neurons are considered. The first type, which has been demonstrated in the eye of Limulus, is called subtractive inhibition (SI): it assumes that neuronal activity depends on the difference between the total excitation and inhibition. The second type is called shunting inhibition (SHI): it assumes that inhibitory influences cause a shunting of a portion of the excitation-produced depolarizing current. Consideration of the shunting model is dictated by its considerable physiological plausibility. The actions of SI and SHI, examined for a variety of coupling conditions and time-stationary positive inputs, are shown to be markedly different. The results indicate that SI is most suited for obtaining (1) a linearity between input and output, (2) a contrasting effect that does not depend on the presence of input discontinuities, and (3) contrasting whose degree is independent of input amplitude. SI is especially useful if coupling coefficients can be varied to accommodate the various input form functions or if, for fixed coupling coefficients, the class of input form functions is limited. On the other hand SHI appears most suited for obtaining (1) a nonlinear input-output relation, (2) a relative contrasting only of discontinuities, and (3) a dependence of the contrasting upon input amplitude.List of Main Symbols a coupling coefficient for neighboring units, also called coupling amplitude - V _j output of receptor number j - i _j generator current of neuron number j - g inhibitory function for subtractive inhibition - h inhibitory function for shunting inhibition - v ₂/v ₁ [applies to two-unit case] - N _k neuron number k - I _k total source current produced by excitatory influences on N _k - G _k conductance for source current not shunted (with shunting inhibition) - i portion of source current shunted as a result of inhibition - m number of inhibitory influences [in Eq. (1)] - G _kj conductance of inhibitory shunt path j for neuron N _k - q number of receptors - n number of neurons - R _j receptor number j - x distance - y(x) input stimulus to receptors - y _j =y(x _j ) input stimulus to receptor R _j - v _j v_j for v _j 0, zero otherwise - a _kj G _kj /v _j , inhibitory coupling coefficient for forward shunting inhibition [refer to Eq. (2)] - b _kj excitatory coupling coefficient for contribution to source current of neuron N _k by receptor R _j [refer to Eq. (3)] - i _j i _j for i _j 0, zero otherwise - c _kj G _kj /i _j , inhibitory coupling coefficient for backward shunting inhibition [refer to Eq. (4)] - â _kj inhibitory coupling coefficient for forward subtractive inhibition [refer to Eq. (5)] - _kj inhibitory coupling coefficient for backward subtractive inhibition [refer to Eq. (6)] - y(x _j )=Af(x _j ) sensory input function - A input amplitude - f(x _j ) sensory input form function, also called a sensory image - i(x _j ) generator current output of neuron Nj which is located at x=x _j - y (y ₁, y ₂, ..., y _n), a column vector - i (i ₁, i ₂, ..., i _n), a column vector, also called generator current configuration - a an n by n matrix having a _kj as the term in the k-th row, j-th column - U the unit matrix - d ¦k-j¦, separation between neurons N _k and N _j - a a _kj for d=1, called coupling amplitude - SI subtractive inhibition - SHI shunting inhibition - FSI forward subtractive inhibition - BSI backward subtractive inhibition - FSHI forward shunting inhibition - BSHI backward shunting inhibition - s i/i ₅₁ = (s ₁, s ₂, ..., s _n), normalized generator current vector, also called normalized generator current configuration - s _j i _j/i ₅₁, normalized generator current of neuron N _j - f(x) continuous input form function of which f(x _j ) is a sampled version - ^p f(x)/x ^p p-th order derivative of f(x)     

Various X2-Tests of Homogeneity: Some Comments

Methodological issues in the analysis of incidence rates or prevalence proportions for count data, presented in a form of a sequence of 2×2 tables, corresponding to levels (strata) of a specified variable (risk factor) X, are discussed. Suppose λ_1i and λ_2i are the incidence rates of an event D in the ith stratum for populations 1 and 2, respectively. The homogeneity (null) hypothesis is formulated in the form: H_0:λ1i=λ_2i for all i (i = 1, 2, …, I). Three X²-tests for H₀ and their theoretical bases are discussed: X_Total² which is sensitive to alternatives HA :λ1i± λ2i for at least some i; X_Comb² which is sensitive to alternatives H A : λ1iλ2i² or < λ_2i but not both for all i; and X_Diff² which is sensitive to alternatives H_A:λ1i>λ2i³ for some i and λ_1i′ < λ_2i′ for some i′ (i≠i′). These statistics satisfy the relation X_Total² = X_Comb² + X_Diff². Also, X′²-statistic for pooled data is calculated, which in conjunction with X_Comb² can serve for detecting confounding. Although most of these techniques are known, they are rather scattered in the literature, and not always considered jointly, as it is emphasized in the present paper. It is hoped that these comments will be helpful to biostatisticians as well as to epidemiologists and medical researchers in the analysis of mortality and morbidity data. For illustration, two examples with large sets of epidemiological data are given.     

Confidence intervals for demographic projections based on products of random matrices

This work is concerned with the growth of age-structured populations whose vital rates vary stochastically in time and with the provision of confidence intervals. In this paper a model Y_{t + 1}(ω) = X_{t + 1}(ω)Y_t(ω) is considered, where Y_t is the (column) vector of the numbers of individuals in each age class at time t, X is a matrix of vital rates, and ω refers to a particular realization of the process that produces the vital rates. It is assumed that {X_i} is a stationary sequence of random matrices with nonnegative elements and that there is an integer n₀ such that any product X_{j + n0} ··· X_{j + 1}X_j has all its elements positive with probability one. Then, under mild additional conditions, strong laws of large numbers and central limit results are obtained for the logarithms of the components of Y_t. Large-sample estimators of the parameters in these limit results are derived. From these, confidence intervals on population growth and growth rates can be constructed. Various finite-sample estimators are studied numerically. The estimators are then used to study the growth of the striped bass population breeding in the Potomac River of the eastern United States.     

Equi-replicated balanced block designs generated by a balanced matrix

In this paper it is shown that if N= \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \sum \limits_{i = 1}^{S_h} $\end{document} c_ihN_ih, where c_ih are some non-negative integer numbers and N_ih are such incidence matrices that A_h = \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \sum \limits_{i = 1}^{S_h} $\end{document} i N_ih is a balanced matrix defined by SHAH (1959), for h = 1, 2,…, p, then a block design with an incidence matrix Ñ = [N, N,…,N] is an equi-replicated balanced block design. Here the balance of a block design is defined in terms of the matrix M₀ introduced by CALI?SKI (1971).     

Using a Normal Approximation to Test for the Binomial Distribution

The commonly used method to test for the binomial distribution is the x²-test. In this paper, we introduce an alternative method to test for the binomial distribution. Suppose N is the number of sample groups with n individuals each, x_ij is the jth sample in ith group, a Bernoulli variable with parameter and VVI=s²/[m(1 - m)/n]. Then it is well know that the asymptotic distribution of the statistic (N - 1) VVI is x²(N - 1) under the hypothesis p₁ = p₂ = … = p_N. Here we find that VVI has an asymptotic normal distribution N(1, 2(1 - 1/n)/(N - 1)). Unlike the x²-statistic, the variance of the normal test statistic is a function of n. This method is convenient in detecting spatial patterns and dispersion in the study of diseased organisms (e.g., plants) in field samples.     

Approximate Confidence Intervals for Contrasts in the Balanced Three Factor Mixed Model

When conducting a statistical analysis of data from a designed experiment, an investigator is often interested in confidence intervals for contrasts of the fixed effects. If the analysis involves a mixed linear model, exact confidence intervals for contrasts of the fixed effects are not always available. In such cases, confidence intervals with approximate coverage probabilities must be used. As will be shown, this problem may be generalized to that of constructing a confidence interval for the parameter μ, where X is a normal random variable with mean μ and variance ∑ a_qθ_q, where a₁…,a_Q are known constants, U_q = n_qS/θ_q is a chi-squared random variable with n_q degrees of freedom, for each q = 1,…, Q, and X,U₁,…, U_Q are mutually independent. In this paper, we consider the case where Q = 3 and a₃ ≤0.