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.     

Remarks upon Empirical Regression Belt

In the presented paper the method of the empirical regression belt is demonstrated. An empirical regression curve r(x), which is determined by the realizations (measured points) (x₁, y₁), i = 1,…., n of a continuous two-dimensional random variable (X, Y), is enclosed by a belt, the local width of which varies dependent on local frequency and variance of the measured points. This empirical regression belt yields certain information for evaluating the empirical regression curve, providing a useful basis for the biomathematical forming of a model. By giving three examples derived from morphometrics the authors discuss important qualities of the empirical regression belt.     

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.     

Confidence Estimation of an Unknown Component of the Vector of Regressor Variables in Multivariate Linear Regression

The model used in this paper is Y = Xβ, where with unknown x₀. Estimators of x₀ are derived by putting β_mx₀ =β_m+1 regarding β_m+1 as a new unknown parameter. Formally we use the model Y = X₁β₊ + e where β′₊ = (β₀, …β_m+1 and Then β_m+1/ β_m is a point estimator of x₀. Assuming normality for e and taking the random variable z=β_mx₀?β_m+1 we get a t-distributed variable and finally a confidence estimator of x₀. The formulas are applied in dose response relations in antibiotic assays refering to a standard. Now we can take into account not only the dependence on the dose/concentration but also on the position on the test agar plate where the test solution is filled in. As a consequence the confidence interval of the unknown dose/concentration x₀ becomes shorter and by it the statements more precise.     

Optimal Fixing of the Bandwidth-Parameter for the Empirical Regression1,2

The estimator ?₀(x) of the regression r(x) = E (Y | × = x) from measured points (x_i, y_i), i = 1(1) n, of a continuous two-dimensional random variable (X, Y) with unknown continuous density function f(x, y) and with moments up to the second order can be made with the help of a density estimation f?₀(x, y) (see e.g. SCHMERLING and PEIL, 1980). Here f?₀(x, y) still contains free parameters (so-called band-width-parameters), the values of which have to be optimally fixed in the concrete case. This fixing can be done by using a modification of the maximum-likelihood principle including jackknife techniques. The parameter values can be also found from the estimators for r(x). Here the cross-validation principle can be applied. Some numerical aspects of these possibilities for optimally fixing the bandwidth-parameter are discussed by means of examples. If ?₀(x) is used as a smoothing operator for time series the optimal choice of the parameter values is dependent on the purpose of application of the smoothed time series. The fixing will then be done by considering the so-called filter-characteristic of ?C₀(x).     

Use of Transformed Auxiliary Variable in Estimating the Finite Population Mean

For estimating the finite population mean Y- of the study character y, an estimator using a transformed auxiliary variable has been defined. The bias and mean-squared error (MSE) of the proposed estimator have been obtained. The regions of preference have been obtained under which it is better than usual unbiased estimator y-, the ratio estimator y-_R = y-X-/x-, Sisodia and Dwivedi (1981) estimator y-_s = y-(X- + C_x)/(x- + C_x) and Singh and Kakran (1993) estimator y-_k = y[X- + β₂(x)]/[x- + β₂(x)]. An empirical study has been carried out to demonstrate the superiority of the suggested estimator over the others.     

The Mean of the Inverse of a Punctured Normal Distribution and Its Application

The fundamental properties of a punctured normal distribution are studied. The results are applied to three issues concerning X/Y where X and Y are independent normal random variables with means μ_X and μ_Y respectively. First, estimation of μ_X/μ_Y as a surrogate for E(X/Y) is justified, then the reason for preference of a weighted average, over an arithmetic average, as an estimator of μ_X/μ_Y is given. Finally, an approximate confidence interval for μ_X/μ_Y is provided. A grain yield data set is used to illustrate the results. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multivariate Probability in Terms of Marginal Probability and Correlation Coefficient

This paper is motivated by a practical problem relating to student performance in a number of subjects of equal standing. Its mathematical formulation is to find an approximation to a multivariate probability of the form Pr {X₁⩾a, X₂⩾a, …, X_N⩾a} for arbitrary a and N, in terms of p = Pr {X₁⩾a} and q = Corr (X_i, X_j), i≠j, where X_i, i = 1, …, N are exchangeable random variables with mean 0 and variance unity.     

On Admissible Estimators in a Linear Model

Let Y be observable random vector such that EY=Xβ and D(Y)=ρ²V. Linear estimation of a parameter p′β under the squared loss is considered. RAO, 1976 and 1979, obtained a necessary and sufficient condition for admissibility of an estimator t′Y in the case X=I. This result will be extended for arbitrary X. AMS 1970 subject classifications. Primary 62J05; secondary 62C15.     

Lysine Requirements of Rats of Various Body Weights

The lysine requirements of rats of various body weights were estimated using the feeding and isotope tests.

The regression equation obtained by the feeding test was Y= 1.03 – 0.58 log X. Where Y is lysine percentage of the diet and X is the mean of initial and final body weights (g) of rats achieving optimal growth gains during the feeding period.

The regression equation obtained by the isotope test was 7=0.90 – 0.49 log X, where Y and X are lysine percentage in the diet and body weights (g) of rats achieving optimal growth gains at the injection time respectively.     

On a Class of Almost Unbiased Ratio Estimators

A class of almost unbiased ratio estimators for population mean σ is derived by weighting sample σ = (1/n) σ y_i, ratio estimators σ and an estimator, σ (y_i/x_i). It is shown that NIETO DE PASCUAL (1961) estimator is a particular member of the class and an optimum estimator in the class (in the minimum variance sense) is identified. The results are illustrated through two numerical examples.     

Optimization of polylactic-co-glycolic acid nanoparticles containing itraconazole using 2<Superscript>3</Superscript> factorial design

This study investigated the utility of a 2³ factorial design and optimization process for polylactic-co-glycolic acid (PLGA) nanoparticles containing itraconazole with 5 replicates at the center of the design. Nanoparticles were prepared by solvent displacement technique with PLGAX ₁ (10, 100 mg/mL), benzyl benzoateX ₂ (5, 20 μg/mL), and itraconazoleX ₃ (200, 1800 μg/mL). Particle size (Y ₁), the amount of itraconazole entrapped in the nanoparticles (Y ₂), and encapsulation efficiency (Y ₃) were used as responses. A validated statistical model having significant coefficient figures (P<.001) for the particle size (Y ₁), the amount of itraconazole entrapped in the nanoparticles (Y ₂), and encapsulation efficiency (Y ₃) as function of the PLGA (X ₁), benzyl benzoate (X ₂), and itraconazole (X ₃) were developed: Y₁=373.75+66.54X₁+52.09X₂+105.06X₃−4.73X₁X₂+46.30X₁X₃; Y₂=472.93+73.45X₁+ 169.06X₂+333.03X₃+62.40X₁X₃+141.49X₂X₃; Y₃= 57.36+6.53X₁+15.52X₂−12.59X₃+1.01X₁X₃+ 1.73X₂X₃.X ₁,X ₂, andX ₃ had a significant effect (P<.001) onY ₁,Y ₂, andY ₃. The particle size, the amount of itraconazole entrapped in the nanoparticles, and the encapsulation efficiency of the 4 formulas were in agreement with the predictions obtained from the models (P<.05). An overlay plot for the 3 responses shows the boundary in whichY ₁ shows the boundary in which a number of combinations of concentration of PLGA, benzyl benzoate, and itraconazole will result in a satisfactory process. Using the desirability approach with the same constraints, the solution composition having the highest overall desirability (D=0.769) was 10 mg/mL of PLGA, 16.94 μg/mL of benzyl benzoate, and 1001.01 μg/mL of itraconazole. This approach allowed the selection of the optimum formulation ingredients for PLGA nanoparticles containing itraconazole of 500 μg/mL.     

Testing Hypotheses about Regression Parameters,When the Error Term Is Heteroscedastic

The paper considers methods for testing H₀: β₁ = … = β_p = 0, where β₁, … ,β_p are the slope parameters in a linear regression model with an emphasis on p = 2. It is known that even when the usual error term is normal, but heteroscedastic, control over the probability of a type I error can be poor when using the conventional F test in conjunction with the least squares estimator. When the error term is nonnormal, the situation gets worse. Another practical problem is that power can be poor under even slight departures from normality. Liu and Singh (1997) describe a general bootstrap method for making inferences about parameters in a multivariate setting that is based on the general notion of depth. This paper studies the small-sample properties of their method when applied to the problem at hand. It is found that there is a practical advantage to using Tukey's depth versus the Mahalanobis depth when using a particular robust estimator. When using the ordinary least squares estimator, the method improves upon the conventional F test, but practical problems remain when the sample size is less than 60. In simulations, using Tukey's depth with the robust estimator gave the best results, in terms of type I errors, among the five methods studied.     

Effect of Intracluster Correlation on the R-Square Statistic

R²-statistic is a popular and very widely used statistic in regression analysis to estimate the square multiple correlation (SMC), ρ², between a response variable Y and p predictor variables, _X1, …, _Xp. Numerous articles are available in the statistical literature on the properties of R² as an estimator of ρ² when the observations are uncorrelated. However, relatively little is known about the behavior of R² when the available observations are correlated such as the data that result from complex sampling schemes. In this paper, we study the behavior R² in the presence of two-stage sampling data. An approximate expressions for the variance and the bias of R² in the presence of two-stage cluster sampling data with positive intracluster correlation (ρ*) are obtained. It is evident from these formulas and from a simulation study that R² is a poor estimator of ρ² except when ρ* is small. As such, we consider several alternative estimators of ρ² and evaluate their theoretical properties and finite sample performance using a simulation study.     

Formulation of Dacarbazine-loaded Cubosomes—Part II: Influence of Process Parameters

The purpose of this study was to investigate the combined influence of three-level, three-factor variables on the formulation of dacarbazine (a water-soluble drug) loaded cubosomes. Box–Behnken design was used to obtain a second-order polynomial equation with interaction terms to predict response values. In this study, the selected and coded variables X ₁, X ₂, and X ₃ representing the amount of monoolein, polymer, and drug as the independent variables, respectively. Fifteen runs of experiments were conducted, and the particle size (Y ₁) and encapsulation efficiency (Y ₂) were evaluated as dependent variables. We performed multiple regression to establish a full-model second-order polynomial equation relating independent and dependent variables. A second-order polynomial regression model was constructed for Y ₁ and confirmed by performing checkpoint analysis. The optimization process and Pareto charts were obtained automatically, and they predicted the levels of independent coded variables X ₁, X ₂, and X ₃ (−1, 0.53485, and −1, respectively) and minimized Y ₁ while maximizing Y ₂. These corresponded to a cubosome formulation made from 100 mg of monoolein, 107 mg of polymer, and 2 mg with average diameter of 104.7 nm and an encapsulation efficiency of 6.9%. The Box–Behnken design proved to be a useful tool to optimize the particle size of these drug-loaded cubosomes. For encapsulation efficiency (Y ₂), further studies are needed to identify appropriate regression model.     

A new chromosomal sex-determining mechanism inMicrotus arvalis pallas

The karyotype in the rodentMicrotus arvalis comprises 21 autosome pairs and two heterosome pairs of the X₁X₂Y₁Y₂/X₁X₁X₂X₂ type. The occurrence of multiple sex chromosomes is thought to be due to a translocation of one arm of a metacentric autosome to the Y chromosome. This translocation would result in an additional acrocentric sex chromosome confined to the(heterogametic) male line, i.e., a Y₂. The original metacentric chromosome thereby turns into an X₂. Because of the translocation mentioned, a trivalent figure of the Y₁Y₂X₂ type occurs in the first meiotic metaphase in the male.     

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.     

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.     

A Distribution-Free Two-Sample Equivalence Test Allowing for Tied Observations

A new testing procedure is derived which enables to assess the equivalence of two arbitrary noncontinuous distribution functions from which unrelated samples are taken as the data to be analyzed. The equivalence region is defined to consist of all pairs (F,G) of distribution functions such that for independent X ∼ F, Y ∼ G the conditional probability of {X > Y} given {X ≠ Y} lies in some short interval around 1/2. The test rejects the null hypothesis of nonequivalence if and only if the standardized distance between the U-statistics estimator of P[X > Y ∣ X ≠ Y] and the center of the equivalence interval (1/2 — ε₁, 1/2 + ε₂) does not exceed a critical upper bound which has to be computed as the α-quantile of a χ²-distribution with one degree of freedom and a random noncentrality parameter proportional to the squared length of that interval. The test is shown to maintain the asymptotic significance level under very weak regularity conditions. Results of an extensive simulation study suggest that its level properties are very satisfactory in small samples as well. The power turns out to be inversely related to the rate P[X = Y] of ties between observations from different samples.     

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.