Estimation of a Conditional Mean in a Linear Regression Model after a Preliminary Test on Regression Coefficient

Consider the two linear regression models of Y_ij on X_ij, namely Y_ij = β_io + β_ij, X_ij + E_ij = 1, 2,…, n_i, i = 1, 2, where E_ij are assumed to be normally distributed with zero mean and common unknown variance σ². The problem of estimating the conditional mean of Y₁ for a given value of X₁ is considered when it is a priori suspected that β₁₀ = β₂₀ and β₁₁ = β₂₁. The preliminary test estimator is proposed. The exact expressions for the bias and the mean square error of the estimator are derived. The relative efficiency of the new estimator to the usual least square estimator based on the first regression alone is computed and is used to determine the appropriate value of the significance level of the preliminary test β₁₀ = β₂₀ and β₁₁ = β₂₁.     

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.     

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.     

A New Approach to Equivalence Assessment in Standard Comparative Bioavailability Trials by Means of the Mann-Whitney Statistic

By a suitable transformation of the pairs of observations obtained in the successive periods of the trial, bioequivalence assessment in a standard comparative bioavailability study reduces to testing for equivalence of two continuous distributions from which unrelated samples are available. Let the two distribution functions be given by F(x) = P[X ≤ x], G(y) = P[Y ≤ y] with (X, Y) denoting an independent pair of real-valued random variables. An intuitively appealing way of putting the notion of equivalence of F and G into nonparametric terms can be based on the distance of the functional P[X > Y] from the value it takes if F and G coincide. This leads to the problem of testing the null hypothesis H_o P[X > Y] ≤ 1/2 - ε₁ or P[X > Y] ≥ 1/2 + ε₂ versus H₁ : 1/2 ? ε₁ < P[X > Y] < 1/2 + ∈₂, with sufficiently small ε₁, ε₂ ∈ (0, 1/2). The testing procedure we derive for (₀, H₁) and propose to term Mann-Whitney test for equivalence, consists of carrying out in terms of the U-statistics estimator of P[X > Y] the uniformly most powerful level a test for an interval hypothesis about the mean of a Gaussian distribution with fixed variance. The test is shown to be asymptotically distribution-free with respect to the significance level. In addition, results of an extensive simulation study are presented which suggest that the new test controls the level even with sample sizes as small as 10. For normally distributed data, the loss in power as against the optimal parametric procedure is found to be almost as small as in comparisons between the Mann-Whitney and the t-statistic in the conventional one or two-sided setting, provided the power of the parametric test does not fall short of 80%.     

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.     

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.     

Response surface methodology for optimization and characterization of limonene-based coenzyme Q10 self-nanoemulsified capsule dosage form

The aim of this study was to systematically obtain a model of factors that would yield an optimized self-nanoemulsified capsule dosage form (SNCDF) of a highly lipophilic model compound, Coenzyme Q10 (CoQ). Independent variables such as amount of R-(+)-limonene (X ₁), surfactant (X ₂), and cosurfactant (X ₃), were optimized using a 3-factor, 3-level Box-Behnken statistical design. The dependent variables selected were cumulative percentage of drug released after 5 minutes (Y ₁) with constraints on drug release in 15 minutes (Y ₂), turbidity (Y ₃), particle size (Y ₄), and zeta potential (Y ₅). A mathematical relationship obtained,Y ₁=78.503+6.058X ₁ +13.738X ₂+5.986X ₃−25.831X ₁ ² +9.12X ₁X₂−26.03X ₁X₃−38.67X ₂ ² +11.02X ₂X₃−15.55X ₃ ³ (r ²=0.97), explained the main and quadratic effects, and the interaction of factors that affected the drug release. Response surface methodology (RSM) predicted the levels of factorsX ₁,X ₂, andX ₃ (0.0344, 0.216, and 0.240, respectively), for a maximized response ofY ₁ with constraints of >90% release onY ₂. The observed and predicted values ofY ₁ were in close agreement. In conclusion, the Box-Behnken experimental design allowed us to obtain SNCDF with rapid (>90%) drug release within 5 minutes with desirable properties of low turbidity and particle size.     

P2Y purinoceptors induce changes in intracellular calcium in acinar cells of rat lacrimal glands

Adenosine 5′-triphosphate (ATP) is an extracellular signal that regulates various cellular functions. Cellular secretory activities are enhanced by ATP as well as by cholinergic and adrenergic stimuli. The present study aimed to determine which purinoceptors play a role in ATP-induced changes in the intracellular concentration of calcium ions ([Ca²⁺]_i) and in the fine structure of acinar cells of rat lacrimal glands. ATP induced exocytotic structures, vacuolation and an increase in [Ca²⁺]_i in acinar cells. The removal of extracellular Ca²⁺ or the use of Ca²⁺ channel blockers partially inhibited the ATP-induced [Ca²⁺]_i increase. U73122 (an antagonist of PLC) and heparin (an antagonist of IP₃ receptors) did not completely inhibit the ATP-induced [Ca²⁺]_i increase. P1 purinoceptor agonists did not induce any changes in [Ca²⁺]_i, whereas suramin (an antagonist of P2 receptors) completely inhibited ATP-induced changes in [Ca²⁺]_i. A P2Y receptor agonist, 2-MeSATP, induced a strong increase in [Ca²⁺]_i, although UTP (a P2Y_2,4,6 receptor agonist) had no effect, and reactive blue 2 (a P2Y receptor antagonist) resulted in partial inhibition. The potency order of ATP analogs (2-MeSATP > ATP ⋙ UTP) suggested that P2Y₁ played a significant role in the cellular response to ATP. BzATP (a P2X₇ receptor agonist) induced a small increase in [Ca²⁺]_i, but α,β-meATP (a P2X_1,3 receptor agonist) had no effect. RT-PCR indicated that P2X_2,3,4,5,6,7 and P2Y_1,2,4,12,14 are expressed in acinar cells. In conclusion, the response of acinar cells to ATP is mediated by P2Y (especially P2Y₁) as well as by P2X purinoceptors.     

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.     

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.     

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.     

The Nature of Conformational Correlations in α- and β-Anomers of Nucleic Acid Components in Aqueous Solution by Nuclear Magnetic Resonance

Abstract

The solution distribution of combinations of the sugar ring puckering domains, C2′endo(S), C3′endo(N), and C4′-C5′ rotamers, +sc(g⁺), ap(t), -sc(g^?), in α and β-anomers in ribo- and deoxyribo- pyrimidine nucleic acid components can be determined from vicinal coupling constants (M. Remin, J. Biomol. Str. Dyn. 2, 211 (1984). A general correlation pattern with a conformational constant λ, reflecting an intrinsic physical property of the sugar - side chain ensemble, is developed and expressed in terms of four principles:

I) The +sc rotamer contributes to the C3′endo population to a higher extent (1 - Y_t) than to C2′endo,(l-Y_t-Y_g-/X_s).

II) The ap rotamer contributes to both C2′endo and C3′endo populations to the same extent (Y_t).

III) The—sc rotamer contributes only to the C2′endo population, (Y_g-/X_s).

IV) The molar fractions X_s, Y_t and Y_g- of conformations C2′endo, ap and—sc, respectively, are strongly correlated, λ = (Y_g-/X_s)/Y_t ≈ 0.5, and therefore Y_t is a basic variable parameter which determines all others in the correlation pattern.

In α-anomers, regardless of the type and conformation of the sugar ring and base, the molar fraction Y_t = 0.37 ± 0.02. This finding means that different α-anomers show one correlation pattern free of the influence of the base. In β-anomers, structure and conformation of the base are important factors which modulate (through Y_t) the correlation pattern, conserving its fundamental features. Y_t is considerably increased by a syn-oriented pyrimidine base, but decreases when the base is anti. The transition from anti to syn orientation of the base is followed by destabilization of (C2′endo, +sc) in favor of (C3′endo, ap). The principles of conformational correlations rationalize a variety of correlations observed in the past.     

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.     

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.     

Activation of protein kinase C inhibits ATP-induced [Ca2+]i elevation in rat osteoblastic cells: Selective effects on P2Y and P2U signaling pathways

Extracellular ATP elicits transient elevation of cytosolic free Ca²⁺ concentration ([Ca²⁺]_i) in osteoblasts through interaction with more than one subtype of cell surface P₂-purinoceptor. Elevation of [Ca²⁺]_i arises, at least in part, by release of Ca²⁺ from intracellular stores. In the present study, we investigated the possible roles of protein kinase C (PKC) in regulating these signaling pathways. [Ca²⁺]_i of indo-1-loaded UMR-106 osteoblastic cells was monitored by spectrofluorimetry. In the absence of extracellular Ca²⁺, ATP (100 μM) induced transient elevation of [Ca²⁺]_i to a peak 57 ± 7 nM above basal levels (31 ± 2 nM, means ± S. E. M., n = 25). Exposure of cells to the PKC activator 12-O-tetradecanoyl-β-phorbol 13-acetate (TPA, 100 nM) for 2 min significantly reduced the amplitude of the ATP response to 13 ± 4 nM (n = 11), without altering basal [Ca²⁺]_i. Inhibition was half-maximal at approximately 1 nM TPA. The Ca²⁺ response to ATP was also inhibited by the PKC activators 1,2-dioctanoyl-sn-glycerol or 4β-phorbol 12, 13-dibutyrate, but not by the control compounds 4α-phorbol or 4α-phorbol 12, 13-didecanoate. Furthermore, exposure of cells to the protein kinase inhibitors H-7 or staurosporine for 10 min significantly attenuated the inhibitory effect of TPA. However, these protein kinase inhibitors did not prolong the [Ca²⁺]_i response to ATP alone, indicating that activation of PKC does not account for the transient nature of this response. When the effects of other nucleotides were examined, TPA was found to cause significantly greater inhibition of the response to the P_2Y-receptor agonists, ADP and 2-methylthioATP, than the response to the P_2U-receptor agonist, UTP. These data indicate that activation of PKC selectively inhibits the P_2Y signaling pathway in osteoblastic cells. In vivo, endocrine or paracrine factors, acting through PKC, may regulate the responsiveness of osteoblasts to extracellular nucleotides. © 1995 Wiley-Liss, Inc.     

Extracellular NAD induces a rise in [Ca]i in activated human monocytes via engagement of P2Y1 and P2Y11 receptors

Extracellular nicotinamide adenine dinucleotide (NAD⁺) is known to increase the intracellular calcium concentration [Ca²⁺]_i in different cell types and by various mechanisms. Here we show that NAD⁺ triggers a transient rise in [Ca²⁺]_i in human monocytes activated with lipopolysaccharide (LPS), which is caused by a release of Ca²⁺ from IP₃-responsive intracellular stores and an influx of extracellular Ca²⁺. By the use of P2 receptor-selective agonists and antagonists we demonstrate that P2 receptors play a role in the NAD⁺-induced calcium response in activated monocytes. Of the two subclasses of P2 receptors (P2X and P2Y) the P2Y receptors were considered the most likely candidates, since they share calcium signaling properties with NAD⁺. The identification of P2Y₁ and P2Y₁₁ as receptor subtypes responsible for the NAD⁺-triggered increase in [Ca²⁺]_i was supported by several lines of evidence. First, specific P2Y₁ and P2Y₁₁ receptor antagonists inhibited the NAD⁺-induced increase in [Ca²⁺]_i. Second, NAD⁺ was shown to potently induce calcium signals in cells transfected with either subtype, whereas untransfected cells were unresponsive. Third, NAD⁺ caused an increase in [cAMP]_i, prevented by the P2Y₁₁ receptor-specific antagonist NF157.     

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.     

The Statistical Properties of Genetic Absolute Distance

The statistical properties of one estimator of absolute genetic distance (1/2) ^K∑_i=1 |pxi-yr|, tween two populations X and Y, are presented. It is shown that using this distance in small samples can be misleading particularly when populations are close to each other.     

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.     

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.