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Orthogonal‐array composite designs (OACDs) and orthogonal‐uniform composite designs (OUCDs) are orthogonal composite designs that combine two‐level full or fractional factorial and three‐level orthogonal‐array/uniform designs for estimation of the linear, bilinear, and quadratic effects in a second‐order response surface model. In this study, the effects of missing one observation in the various design portions (factorial (f) axial (a) and center (c)), on the precision of parameter estimates, prediction variance and design efficiency of OACDs and OUCDs for 5 ≤ k ≤ 9 factors at different values of α (the distance of a non‐zero co‐ordinate in an additional design point from the center) are evaluated. The results showed that missing a factorial and an axial point have adverse effect on the precision of parameter estimates of OACDs and OUCDs, while missing a center point has little effect. Missing an axial point caused the highest effect on the prediction variance and design efficiencies. The FDS plots showed OACDs to be better designs for k ≤ 7 and OUCDs for k = 8 and 9 factors. [ABSTRACT FROM AUTHOR]

In this paper, we show that each non-degenerate Lucas sequence contains only finitely many terms which can be written as products of double factorials or as sums or differences of double factorials. In particular, we find all the solutions to the products of double factorials which are sums or differences of two Fibonacci numbers, and Fibonacci numbers which are sums or differences of two double factorials. [ABSTRACT FROM AUTHOR]

This study proposes a procedure for substantive dimensionality estimation in the presence of wording effects, the inconsistent response to regular and reversed self-report items. The procedure developed consists of subtracting an approximate estimate of the wording effects variance from the sample correlation matrix and then estimating the substantive dimensionality on the residual correlation matrix. This is achieved by estimating a random intercept factor with unit loadings for all the regular and unrecoded reversed items. The accuracy of the procedure was evaluated through an extensive simulation study that manipulated nine relevant variables and employed the exploratory graph analysis (EGA) and parallel analysis (PA) retention methods. The results indicated that combining the proposed procedure with EGA or PA achieved high accuracy in estimating the substantive latent dimensionality, but that EGA was superior. Additionally, the present findings shed light on the complex ways that wording effects impact the dimensionality estimates when the response bias in the data is ignored. A tutorial on substantive dimensionality estimation with the R package EGAnet is offered, as well as practical guidelines for applied researchers. [ABSTRACT FROM AUTHOR]

The present paper is dedicated to the examination of maximum and minimum results based on Green's functions via delta fractional differences for a class of fractional boundary problems. For such a purpose, we built the corresponding Green's functions based on the falling factorial functions. In addition, using the constructed Green's function, the positivity of the function and its corresponding delta function are presented. We also verified the occurrence of two distinct functions with the same Green's function. The maximality and minimality of the Green's function show a good qualitative agreement. Finally, we considered some special examples to explain the obtained results. [ABSTRACT FROM AUTHOR]

We study the structure of Jacobians of geometrically reduced curves over arbitrary (i.e., not necessarily perfect) fields. We show that, while such a group scheme cannot in general be decomposed into an affine and an Abelian part as over perfect fields, several important structural results for these group schemes nevertheless have close analoga over imperfect fields. We apply our results to prove two conjectures due to Bosch-Lütkebohmert-Raynaud about the existence of Néron models and Néron lft-models over excellent Dedekind schemes in the special case of Jacobians of geometrically reduced curves. Finally, we prove some existence results for semi-factorial models and related objects for general geometrically integral curves in the local case. [ABSTRACT FROM AUTHOR]

The w-operation, a "smoother" variant of the classic t-operation on the set of ideals of a commutative ring R, has recently attracted significant attention. We study when R's monoid of w-ideals is "factorial" in some sense. For example, we show that every proper w-ideal of R has a unique up to order irredundant w-factorization into w-unfactorable w-ideals if and only if R is a finite direct product of Krull domains and principal ideal rings, if and only if every w-ideal is w-quasiprincipal. We prove several analogous results regarding other kinds of "w-ideal factoriality" and generalize these results via semistar operations. [ABSTRACT FROM AUTHOR]

Copyright of Environmental & Social Management Journal / Revista de Gestão Social e Ambiental is the property of Environmental & Social Management Journal and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The role of disability in producing disadvantage in employers' hiring assessments was explored in a factorial survey, where a random sample of Norwegian employers (n = 1341) evaluated fictional job-seeker profiles. The results revealed that including an impairment description in a job-seeker profile significantly decreased the likelihood that employers would want to hire a candidate. The degree of disadvantage varied with the type of impairment. Being eligible for a wage subsidy scheme improved employers' assessments of candidates while including information about other types of support measures did not. Furthermore, when an impairment description was introduced into a job-seeker profile, other crucial characteristics of the job seeker lost some or all of their impact on employers' assessment scores. These findings are interpreted as disability becoming a 'master status' when employers make hiring assessments. [ABSTRACT FROM AUTHOR]

An order of addition experiment is an experiment to study how the order of addition of components affect the results, with the objective of predicting and determining the optimal order of addition of components. Order of addition experiment are also commonly used in the drug combination therapy, where experimenting with all drugs combinations is unaffordable. To solve this problem, we constructed a new design table, two-level component factorial design table (TLCF), which combine the component orthogonal array design table and the two-level partial factorial design table by matrix product. TLCF can explore the order and dosage effect of components on the results and can greatly reduce the number of experiments. We also prove that the relative D-efficiency of the TLCF can reach 100% and solve an explicit expression for the D-efficiency of the full design. In the simulation experiment, we compare the D-efficiency of the TLCF with the random design table to prove the superiority of TLCF. Finally, we give a treatment plan for the combination of three drugs for glioblastoma based on the TLCF, which provides a new perspective for the precision treatment of patients. [ABSTRACT FROM AUTHOR]

Saturated orthogonal array is an important type of fractional factorial designs in experimental design. On the basis of guaranteeing orthogonality, it maximizes the number of factors and saves the cost of experiments. Therefore, it is increasingly used in practical experiment. For quantitative factors, the minimum β-aberration criterion is suitable for selecting good designs under a polynomial model. However, construction methods of designs with less β-aberration have not been fully investigated. In this article, we apply the Rao-Hamming method to construct a class of three-level saturated orthogonal arrays, and perform a special level permutation on the designs of the Rao-Hamming construction, which makes the resultant designs mirror-symmetric ones. Furthermore, we analyze the distribution of elements in each row and among all possible pairs of rows of the mirror-symmetric designs, and discuss some properties of the β-wordlength pattern for three-level mirror-symmetric designs constructed by the Rao-Hamming method. In addition, we provide the explicit expressions of β3 and β4 for three-level mirror-symmetric designs, which can reduce the computational complexity of the β-wordlength pattern, and provide an effective idea for finding designs under the minimum β-aberration criterion. Finally, numerical examples are used to illustrate and verify the results. [ABSTRACT FROM AUTHOR]

Objectives Methods Results Conclusions This study examined the factorial invariance of the factor structure of the Wechsler Intelligence Scale for Children, Fifth Edition (WISC‐V) across the UK, US and Australia & New Zealand (A&NZ). The factorial equivalence of cognitive assessments should be demonstrated before assuming cross‐culture generalizability and interpretations of score comparisons.Data were obtained from the UK, US and A&NZ normative standardizations of the WISC‐V. The samples consisted of 415 UK, 2200 US and 528 A&NZ children, aged 6–16. Confirmatory factor analysis was applied separately in each sample to establish the baseline model. Next, tests of factorial invariance were undertaken using the recommended hierarchical approach, firstly across the UK and A&NZ samples and then across the UK and US samples.The five‐factor first‐order scoring model was found to be excellent fit across all three samples independently. Strict factorial invariance of the WISC‐V was demonstrated firstly across the UK and A&NZ and secondly the UK and US nationally representative standardization samples. Comparison of latent means found small but significant differences in female children across the UK and A&NZ samples.Consistent with previous research, these results demonstrate the generality of the WISC‐V factor structure across the UK, US and A&NZ. Furthermore, as the WISC‐V factor structure aligns with the Cattell–Horn–Carroll (CHC) model of cognitive abilities, the results add further support to the cross‐cultural generalizability of the CHC model. Small but significant differences in latent factor scores found across samples support the development and use of local normative data. [ABSTRACT FROM AUTHOR]

The development and evaluation of novel treatment combinations is a key component of modern clinical research. The primary goals of factorial clinical trials of treatment combinations range from the estimation of intervention-specific effects, or the discovery of potential synergies, to the identification of combinations with the highest response probabilities. Most factorial studies use balanced or block randomization, with an equal number of patients assigned to each treatment combination, irrespective of the specific goals of the trial. Here, we introduce a class of Bayesian response-adaptive designs for factorial clinical trials with binary outcomes. The study design was developed using Bayesian decision-theoretic arguments and adapts the randomization probabilities to treatment combinations during the enrollment period based on the available data. Our approach enables the investigator to specify a utility function representative of the aims of the trial, and the Bayesian response-adaptive randomization algorithm aims to maximize this utility function. We considered several utility functions and factorial designs tailored to them. Then, we conducted a comparative simulation study to illustrate relevant differences of key operating characteristics across the resulting designs. We also investigated the asymptotic behavior of the proposed adaptive designs. We also used data summaries from three recent factorial trials in perioperative care, smoking cessation, and infectious disease prevention to define realistic simulation scenarios and illustrate advantages of the introduced trial designs compared to other study designs. [ABSTRACT FROM AUTHOR]

We deal with a population model with renewal growth of individuals under mild (geometric) catastrophes. We study the model in a discrete-time setup and consider both an early arrival system and late arrival system with delayed access. Our main objective is to derive the time-dependent solution of the population size distribution at an arbitrary point of time using supplementary variables and some properties of difference equations. Further, we obtain the time-dependent factorial moments and generate certain numerical results to study the system characteristics. [ABSTRACT FROM AUTHOR]

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete "stick breaking") subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior. [ABSTRACT FROM AUTHOR]

Many industrial experiments involve factors with levels more difficult to change or control than others, which leads to the development of two-level fractional factorial split-plot (FFSP) designs. Recently, mixed-level FFSP designs were proposed due to the requirement of different-level factors. In this paper, we generalize the Bayesian optimal criterion for mixed two- and four-level FFSP designs, and then provide Bayesian minimum aberration (MA) criterion to rank FFSP designs. Bayesian MA criterion can give a natural ordering for the effects involving two-level factors and three components of a four-level factor. We also discuss the relationship between the Bayesian optimal and Bayesian MA criteria. Furthermore, we consider the designs with both qualitative and quantitative factors. [ABSTRACT FROM AUTHOR]

The Surányi–Hickerson conjecture is a long-standing unsolved problem of Diophantine equations. This conjecture states that all the solutions to ℓ 1 ! ⋯ ℓ m ! = k ! with k - ℓ m ≥ 2 are (ℓ 1 , ... , ℓ m ; k) = (6 , 7 ; 10) , (3 , 5 , 7 ; 10) , (2 , 5 , 14 ; 16) and (2, 3, 3, 7; 9). In this paper, we generalize the Surányi–Hickerson conjecture to ℓ 1 ! ⋯ ℓ m ! = k 1 ! ⋯ k n ! . We say that a solution (ℓ 1 , ... , ℓ m ; k 1 , ... , k n) is trivial if there exists a pair (i, j) such that | ℓ i - k j | = 1 . As in the Surányi–Hickerson conjecture, we give theoretical and computational results. In particular, we suggest that all non-trivial solutions to the equation ℓ 1 ! ℓ 2 = k 1 ! k 2 ! are (ℓ 1 , ℓ 2 ; k 1 , k 2) = (7 , 13 ; 4 , 15) , (14, 62; 7, 66) and (22, 54; 18, 57). [ABSTRACT FROM AUTHOR]

The multivariate coefficient of variation (MCV) is an attractive and easy‐to‐interpret effect size for the dispersion in multivariate data. Recently, the first inference methods for the MCV were proposed for general factorial designs. However, the inference methods are primarily derived for one special MCV variant while there are several reasonable proposals. Moreover, when rejecting a global null hypothesis, a more in‐depth analysis is of interest to find the significant contrasts of MCV. This paper concerns extending the nonparametric permutation procedure to the other MCV variants and a max‐type test for post hoc analysis. To improve the small sample performance of the latter, we suggest a novel bootstrap strategy and prove its asymptotic validity. The actual performance of all proposed tests is compared in an extensive simulation study and illustrated by real data analysis. All methods are implemented in the R package GFDmcv, available on CRAN. [ABSTRACT FROM AUTHOR]

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset S of Z. Bhargava's factorials k ! S are invariants, constructed using the notion of p -orderings of S where p is a prime. This paper defines b -orderings of any nonempty subset S of Z for all integers b ≥ 2 , as well as "extreme" cases b = 1 and b = 0. It defines generalized factorials k ! S , T and generalized binomial coefficients ( k + ℓ k ) S , T as nonnegative integers, for all nonempty S and allowing only b in T ⊆ N. It computes b -ordering invariants when S is Z and when S is the set of all primes. [ABSTRACT FROM AUTHOR]

For a multi-dimensional measure of positive youth development (PYD), its factor structure should be invariant across groups and over time. This study examined the factorial validity of the 44-item short form of the "Chinese Positive Youth Development Scale" (CPYDS-SF) that assesses 15 dimensions of PYD attributes. Using two waves of longitudinal data with a one-year interval in between, this study examined the factor structure of the scale and whether the structure is invariant between gender groups and across time. The data were collected from 3,328 adolescents at Wave 1 and 3,638 adolescents at Wave 2, with a matched sample of 2,905 adolescents (mean age = 12.57 ± 0.72 at Wave 1; 49.54% girls). Confirmatory factor analysis revealed that the 15-factor structure fitted the data well. The findings of invariance tests further supported this structure's invariance across gender and time, indicating a stable factor structure of CPYDS-SF among Chinese adolescents. These findings suggest that CPYDS-SF can be used to examine gender differences and the longitudinal development of PYD qualities among Chinese adolescents. [ABSTRACT FROM AUTHOR]

We study configuration spaces |$C(n; p, q)$| of |$n$| ordered unit squares in a |$p$| by |$q$| rectangle. Our goal is to estimate the |$j$| th Betti number for large |$n$|⁠ , |$j$|⁠ , |$p$|⁠ , and |$q$|⁠. We consider sequences of area-normalized coordinates, where |$\left (\frac {n}{pq}, \frac {j}{pq}\right)$| converges as |$n$|⁠ , |$j$|⁠ , |$p$|⁠ , and |$q$| approach infinity. For every sequence that converges to a point in the "feasible region" in the |$(x,y)$| -plane identified in [ 3 ], we show that the factorial growth rate of the Betti numbers is the same as the factorial growth rate of |$n!$|⁠. This implies that (1) the Betti numbers are vastly larger than for the configuration space of |$n$| ordered points in the plane, which have the factorial growth rate of |$j!$|⁠ , and (2) every point in the feasible region is eventually in the homological liquid regime. [ABSTRACT FROM AUTHOR]

We introduce the edge Schur functions E λ that are defined as a generating series over edge labeled tableaux. We formulate E λ as the partition function for a solvable lattice model, which we use to show they are symmetric polynomials and derive a Cauchy-type identity with factorial Schur polynomials. Finally, we give a crystal structure on edge labeled tableau to give a positive Schur polynomial expansion of E λ and show it intertwines with an uncrowding algorithm. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic impacted the mental health of more citizens globally than any previous modern viral outbreak. In response to the psychological challenges associated with COVID-19, the COVID Stress Scales (CSS) were developed to assess the presence and severity of COVID-related distress. The initial North American validation study of the CSS identified that the scale comprised five factors: danger and contamination fears, fear of socioeconomic consequences, xenophobia, checking and reassurance seeking, and traumatic stress symptoms. The CSS have since been validated across a multitude of international populations. However, findings support a five- and six-factor model. Methodological issues make interpreting most studies supporting a five-factor model challenging. The purpose of this study was to re-evaluate the factor structure of the CSS using data from North American samples, to assess for potential factorial invariance, and compare these results to crosscultural findings. Multiple confirmatory factor analyses (mCFA) were conducted across 28 different groups (e.g., age, ethnicity/race, sex) from two large independent North American samples from 2020 (n = 6827) and 2021 (n = 5787), assessing the fit indices of the five-, six-, and alternative-factor model of the CSS. The current results provide evidence for factorial invariance of the sixfactor model of the CSS across different North American demographics and highlight potential challenges in interpreting the results of studies that have supported a five-factor model of the CSS. [ABSTRACT FROM AUTHOR]

The planting of sorghum hybrids for grazing is an alternative for forage production in the state of Tocantins. The objective of this work was to evaluate the morphogenesis characteristics of sorghum hybrids for grazing and to identify the plant density that promotes better yield. The experimental design was in complete randomized blocks, in a 2 x 4 factorial arrangement, with four replications, totaling 32 experimental units. The first factor represented two sorghum hybrids (BRS 800 and CMSXS 766), while the second factor included four row spacing levels (15, 30, 45 and 60 cm). The variables analyzed were green forage accumulation, leaf elongation rate, stem elongation rate, number of live leaves per tiller, leaf appearance rate, length end of leaves and phyllochron. At the end of the study, it was observed that the sorghum hybrid for grazing CMSXS 766 used in the spacing of 15 cm between rows (800,000 thousand plants per hectare) is the most suitable to achieve higher productivity under the imposed conditions. The sorghum hybrids BRS 800 and CMSXS 766 are viable in the State of Tocantins and can be used up to the third regrowth. [ABSTRACT FROM AUTHOR]

This study aimed to determine the role of water hyacinth (Eichhornia crassipes Mart.) organic matter on water retention at pF 2.54 and pF 4.2, available water content, and bulk density (BD). The research method used was a factorial randomized block design. The first factor was the sorghum variety, which consisted of 2 levels, namely, V1 (variety) and V2 (Kawali). The second factor was the dose of water hyacinth organic matter, which consisted of 5 levels, namely P0 (without the application of water hyacinth organic matter), P1 (10 tons.ha-1), P2 (20 tons.ha-1), P3 (30 tons.ha-1), and P4 (40 tons.ha-1). Water retention was measured at pF 2.54 and pF 4.2 using a pressure plate apparatus and pressure membrane apparatus, respectively, and BD was measured using the ring method. The results showed that the treatment of organic matter application and varieties had a significant effect on water retention at pF 2.54, available water content, and BD with the best treatment of 40 tons of organic matter application and Kawali variety, but not affecting water retention at pF 4.2. There was no interaction between water hyacinth organic matter and sorghum varieties to improve water retention and BD. The highest water retention at pF 2.54 was 32.24% (P4), and the highest available water content was 11.67% (P4) but not significantly different from P3 (11.55%); the lowest BD was 1.11 g.cm-3 (P4) but not significantly different from P3 (1.13 g.cm-3). A very strong positive linear and negative linear correlation exists between available water content and BD with plant height, panicle length, and sorghum yield. [ABSTRACT FROM AUTHOR]

In the paper, with the aid of the Faà di Bruno formula, in terms of the central factorial numbers and the Stirling numbers of the second kinds, the authors derive several series expansions for any positive integer powers of the sinc and sinhc functions, discover several closed-form expressions for partial Bell polynomials of all derivatives of the sinc function, establish several series expansions for any real powers of the sinc and sinhc functions, and present several identities for central factorial numbers of the second kind and for the Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

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Background: Though the Spanish version of the Coping Strategies Inventory (CSI) is frequently used, it has not been subjected to any significant re-evaluation from a psychometric perspective. Method: We analysed CSI data that was focused on an academic stress situation, using a university sample of 874 participants, 50% of each gender. We conducted reliability, confirmatory factor, and factorial invariance across gender analyses. Results: The first-order factorial structure was confirmed, showing an adequate fit for the eight-factor coping strategies model with good reliability indices. Neither the second-order (problem-centred vs. emotion-centred) nor third-order (engagement vs. disengagement) strategies showed an adequate fit. Factorial invariance for gender was confirmed. Conclusions: The theoretical and applied implications of the results are discussed. [ABSTRACT FROM AUTHOR]

This study aimed to develop the scoring functions for the recently developed value assessment framework (VAF) for China, which comprises 12 attributes. We implemented a factorial survey among Chinese healthcare stakeholders from July to September 2022. A total of 240 hypothetical drug value profiles described by the VAF were grouped into 60 blocks and randomly assigned to respondents. Each respondent was assigned with 1 block, each presented in 3 disease scenarios of different levels of severity. For each profile, respondents were asked to assess the drug's value on a scale from 0 (lowest) to 10 (highest) and make 1 of the 3 insurance recommendations: cover, to be negotiated for coverage, or reject. Linear and logistic mixed-effects models were used to develop scoring functions for aggregating the value attributes. A total of 365 respondents participated in the survey. 3968 responses from 331 respondents were included in the analysis. Most of the included respondents were under 45 (n = 256, 77.3%), females (n = 208, 62.8%), living in urban areas (n = 296, 89.4%), and with a bachelor's degree or higher (n = 303, 91.5%). Health benefits and safety carried more weights than other attributes in the scoring functions across disease scenarios. The value and probability of entering negotiation or receiving insurance coverage for the attribute profiles for severe/critical disease were higher than for mild/moderate disease. The scoring functions of the VAF can be used to assess the value of a drug and its probability of entering negotiation or receiving insurance coverage in China. • Through qualitative interviews and document analysis, 12 attributes grouped in 8 categories were identified and included in a value assessment framework for China. The value assessment framework comprises severity of disease, health benefits, safety, economic impact, innovation, organizational impact, health equity, and quality of evidence. • We conducted a factorial survey in which 240 value profiles with 10 attributes experimentally varying in their levels were randomly assigned to 365 healthcare stakeholders in China to assess the value of hypothetical drugs and make insurance coverage recommendations. • The scoring functions can be used to estimate the value of a drug and the probability of its entering negotiation or receiving coverage in China in different disease scenarios. In the scoring functions for severe/critical disease or mild/moderate disease, health benefits and safety carried higher weights than other attributes. The value and probability of entering negotiation or receiving insurance coverage are higher for attribute profiles for severe/critical disease than for mild/moderate disease. [ABSTRACT FROM AUTHOR]

A semi-industrial scale hydrocyclone with a 250 mm internal diameter was used to concentrate medium-grade celestine ore (75%–85% celestine) from the Montevive deposit of Granada (Spain) using a dense ferrosilicon (FeSi) medium. For this purpose, a Box–Behnken factorial design (BBD) was carried out, with the response variable being the Sr concentration measured by X-ray fluorescence (XRF), as well as the concentration of celestine measured by X-ray diffraction (XRD) of the mineral collected from the under (sunk) stream of the hydrocyclone. The experimental factors to be optimised were the density of the medium in the mixing tank (water, FeSi, and feed mineral) varying from 2.7 to 2.9 kg/L, the hydrocyclone inlet pressure from 0.8 to 1.2 bar, and the hydrocyclone inclination (from 15° to 25° from the horizontal). The range of densities of the dense medium to be tested was determined from previous sink–float experiments using medium-grade ore, in which the distribution of mineral phases with different particle size fractions was determined. To evaluate the separation behaviour, the following parameters were considered: the enrichment ratio (E), the tailings discarding ratio (R), and the mineral processing recovery (ε). From the factorial design and the response surface, the optimum parameters maximising celestine concentration in the under stream (78%), were determined. These optimised parameters were: a density of 2.75 kg/L for the dense medium, an inlet pressure of 1.05 bar, and a hydrocyclone inclination varying from 18° to 20°. Under these conditions, a 94% recovery of celestine (68% Sr) can be achieved. These results show that medium-grade celestine ore, accumulated in mine tailings dumps, can be effectively concentrated using DMS hydrocyclones and that the operating parameters can be optimised using a factorial experiment design. This study can contribute to reducing overexploitation of strategic mineral resources, avoiding blasting and environmentally damaging clearing, by applying a simple and sustainable technique. [ABSTRACT FROM AUTHOR]

The statistical design for order‐of‐addition (OofA) experiments has received much recent interest as its potential in determining the optimal sequence of multiple components, for example, the optimal sequence of drug administration for disease treatment. The traditional OofA experiments focus mainly on the sequence effects of components, that is, the experimenters fix the factor level of each component and observe how the response is affected by varying the sequences of components. However, the components may also have factorial effects in that changing their factor levels in a given sequence can affect the response. In view of this, we consider the design problem for OofA experiments where each component is experimented at two levels. A systematic method is given to construct OofA designs that jointly considers the sequence design and factorial design for all components. By appropriately choosing the sequence and factorial designs, we show that the combination of the two parts results in a balanced design with an economical run size. Moreover, the constructed designs enjoy a number of optimality properties such as D$$ D $$‐, A$$ A $$‐ and E$$ E $$‐optimalities under some empirical models. The design method proposed can be extended to some other practical situations like the number of process variables is different from the number of components, and OofA experiments with multilevel components. [ABSTRACT FROM AUTHOR]

Multivariate repeated measures data naturally arise in clinical trials and other fields such as biomedical science, public health, agriculture, social science and so on. For data of this type, the classical approach is to conduct multivariate analysis of variance (MANOVA) based on Wilks' Lambda and other multivariate statistics, which require the assumptions of multivariate normality and homogeneity of within-cell covariance matrices. However, data being analyzed nowadays show marked departure from multivariate normality and homogeneity. This paper proposes a finite-sample test by modifying the sums of squares matrices to make them insensitive to the heterogeneity in MANOVA. The proposed test is invariant to affine transformation and robust against nonnormality. The proposed method can be used in various experimental designs, for example, factorial design and crossover design. Under various simulation settings, the proposed method outperforms the classical Doubly Multivariate Model and Multivariate Mixed Model proposed elsewhere, especially for unbalanced sample sizes with heteroscedasticity. The applications of the proposed method are illustrated with ophthalmology data in factorial and crossover designs. The proposed method successfully identified and validated a significant main effect and demonstrated that univariate analysis could be oversensitive to small but clinically unimportant interactions. [ABSTRACT FROM AUTHOR]

Randomized block factorial experiments are widely used in industrial engineering, clinical trials, and social science. Researchers often use a linear model and analysis of covariance to analyze experimental results; however, limited studies have addressed the validity and robustness of the resulting inferences because assumptions for a linear model might not be justified by randomization in randomized block factorial experiments. In this article, we establish a new finite population joint central limit theorem for usual (unadjusted) factorial effect estimators in randomized block 2 K factorial experiments. Our theorem is obtained under a randomization-based inference framework, making use of an extension of the vector form of the Wald–Wolfowitz–Hoeffding theorem for a linear rank statistic. It is robust to model misspecification, numbers of blocks, block sizes, and propensity scores across blocks. To improve the estimation and inference efficiency, we propose four covariate adjustment methods. We show that under mild conditions, the resulting covariate-adjusted factorial effect estimators are consistent, jointly asymptotically normal, and generally more efficient than the unadjusted estimator. In addition, we propose Neyman-type conservative estimators for the asymptotic covariances to facilitate valid inferences. Simulation studies and a clinical trial data analysis demonstrate the benefits of the covariate adjustment methods. for this article are available online. [ABSTRACT FROM AUTHOR]

We prove that the sequence 1!, 2!, 3!, . . . produces at least (√ 2+o(1))√p distinct residues modulo prime p. Moreover, the factorials within an interval I⊆ {0, 1, . . ., p - 1} of length N > p7/8+ε produce at least (1 + o(1)) √p distinct residues modulo p. As a corollary, we prove that every non-zero residue class can be expressed as a product of seven factorials n1! ... n7! modulo p, where ni = O(p6/7+ε) for all i = 1, . . ., 7, which provides a polynomial improvement upon the preceding results. [ABSTRACT FROM AUTHOR]

The Brief Self-Control Scale (BSCS) is a 13-item personality measure capturing how people differ in their capacity to exert self-control. Although the BSCS was originally regarded as a one-dimensional scale, subsequent psychometric studies have provided support for the empirical distinction of two and four interrelated but distinct components of self-control. Using a large sample of Spanish adults (n = 1,558; 914 female, 58.7%), we performed a comprehensive data-driven comparison of the most well-established item-level latent structures for the BSCS. Results showed that the differentiation between general self-discipline and impulse control offered a better fit to the observed data than did the unidimensional representation of self-control. This two-dimensional structure for the BSCS scores was also supported in terms of its internal consistency, measurement invariance across gender and age groups, and meaningful correlations with wellbeing-related indicators and Big Five personality traits. Plausible implications of these findings are discussed. [ABSTRACT FROM AUTHOR]

Background: The continual development and implementation of effective digital interventions is one important strategy that may serve to bridge the well-known treatment gap related to problematic alcohol use. Research suggests that clinician guidance, provided in different ways during the digital intervention (i.e., written weekly messages, phone calls etc.), can boost intervention engagement and effects. Digital psychological self-care (DPSC) is a new delivery format wherein an unguided digital intervention is provided within the framework of a structured care process that includes initial clinical assessment and follow-up interviews. In a recent feasibility study, a DPSC intervention for problematic alcohol use, ALVA, provided without any extra guidance, was found safe and credible and to have promising within-group effects on alcohol consumption. The aim of the current study is to gather information on the effects and efficiency of different forms of guidance added to ALVA, in order to optimize the intervention. Methods: This protocol describes a randomized factorial trial where the effects of two different ways of providing guidance (mid-treatment interview, weekly written messages, respectively) in DPSC for problematic alcohol use are investigated. Optimization criteria will be applied to the results regarding how effective the intervention is at reducing alcohol consumption measured by the number of standard drinks per week together with the clinician time spent on guidance. Discussion: This study will investigate the added benefit of different forms of guidance to DPSC for problematic alcohol use. These added effects will be compared to the added cost of guidance, according to pre-defined optimization criteria. Trial registration: Clinicaltrials.gov: NCT05649982. Registered on 06 December 2022. Prospectively registered. [ABSTRACT FROM AUTHOR]

This research was carried out in Santo Domingo de los Tsáchilas with the objective of evaluating effect of different biostimulants on banana (Musa AAB) var. Dominican Harton through drench application to increase root development; for this purpose, 10 treatments were established including the witness and in turn resulted from the factorial interaction A (Biostimulants) and B (Dose): Treatments T1 (Biosolar, 7,5 ml), T2 (Biosolar, 5,0 ml), T3 (Biosolar, 2,5 ml), T4 (Seaweed extract, 7,5 ml), T5 (Seaweed extract, 5,0 ml), T6 (Seaweed extract, 2,5 ml), T7 (Vital Humus, 7,5 ml), T8 (Vital Humus, 5,0 ml), T9 (Vital Humus, 2,5 ml), T10 (witness) implemented in a completely randomized experimental design (DCA) with factorial arrangement AxB+N, the variables evaluated were:, number of roots at harvest and root fresh weight at harvest. The results showed that Treatment 4 (Seaweed Extract at high dosage) (7,5 ml), achieved a higher number of roots (60,75) per plantain (Musa AAB) var. Dominican Harton. [ABSTRACT FROM AUTHOR]

This research aimed to develop miconazole-based microemulsions using oleic acid as a natural lipophilic phase and a stabilizer mixture comprising Tween 20 and PEG 400 to solubilize miconazole as an antifungal agent known for its activity in oral candidiasis and to improve its bioavailability. The formulation and preparation process was combined with a mathematical approach using a 23-full factorial plan. Fluid and gel-like microemulsions were obtained and analyzed considering pH, conductivity, and refractive index, followed by extensive analyses focused on droplet size, zeta potential, rheological behavior, and goniometry. In vitro release tests were performed to assess their biopharmaceutical characteristics. Independent variables coded X1-Oleic acid (%, w/w), X2-Tween 20 (%, w/w), and X3-PEG 400 (%, w/w) were analyzed in relationship with three main outputs like mean droplet size, work of adhesion, and diffusion coefficient by combining statistical tools with response surface methodology. The microemulsion containing miconazole base–2%, oleic acid–5%, Tween 20–40%, PEG 400–20%, and water–33% exhibited a mean droplet size of 119.6 nm, a work of adhesion of 71.98 mN/m, a diffusion coefficient of 2.11·10−5 cm2/s, and together with remarked attributes of two gel-like systems formulated with higher oil concentrations, modeled the final optimization step of microemulsions as potential systems for buccal delivery. [ABSTRACT FROM AUTHOR]

Follow-up experimental designs are frequently employed in a wide range of scientific studies and industries. Lower bounds of average mixture discrepancy for row augmented designs are determined in this paper utilizing the level permutation technique, which is also used to analyse the uniformity of row augmented designs with asymmetrical factorials. Between the row augmented designs and its initial designs or follow-up designs, quantitative relationships of lower bounds of average mixture discrepancy are constructed. Additionally, a proposal is made for the construction algorithm of asymmetrical uniform row augmented designs. Numerical examples indicate that the asymmetrical uniform row augmented designs can be constructed with high efficiency and recommended for use in practice. [ABSTRACT FROM AUTHOR]

This paper deals with the first study on the construct validity of the GRS 2 Rating Scale for Teachers in Spain, on a sample of 348 subjects. A confirmatory factorial study, CFA, was carried out using robust least squares estimation procedures (WLSMV). A sixfactor structure has been selected that explains 66% of the variance. The fit indices of the model are satisfactory (CFI, .93; TLI, .93). Convergent validity (AVE, .66) and composite reliability (between .87 and .96) are adequate. This scale is the first validated in Spain and provides a relevant measure for the identification of high ability students. [ABSTRACT FROM AUTHOR]

Barren sites that lack soil are exposed to some of the harshest elements, which include high temperatures, solar radiation, wind, extreme temperature changes, and low soil moisture and nutrient conditions. An ecological restoration experiment was conducted using three site-preparation treatments, straw (S), Meri-Crusher (MC), and coarse woody debris (CWD), in a site-/no site-preparation 2 × 2 × 2 factorial on sites that had been barren for 25 years. In addition, four early successional deciduous species, white birch (WB, Betula papyrifera Marshall), gray birch (GB, Betula populifolia Marshall), green alder (GA, Alnus viridis Vill. subsp. crispa Ait), and speckled alder (SA, Alnus incana L. subsp. rugosa Du Roi), were examined. The two- and three-way interactions were almost all magnitude effects and not rank changes. Gray birch had the greatest overall first-year height growth, followed by GA, SA, and WB, with 12.1, 9.7, 9.6, and 5.6 cm, respectively. Straw doubled first-year growth, while CWD and MC increased first-year height growth by 43 and 31%, respectively. Straw's ability to retain moisture in the dry summer provided the greatest benefit. In the second year, GA had the greatest height growth, followed by SA, GB, and WB, with 42.5, 30.5, 13.4, and 13.0 cm, respectively. Alders form symbiotic relationships with N-fixing bacteria and, although this was observed in some first-year roots, they did not fully express this advantage at these severely degraded sites until the second year, which allowed them to surpass birches in growth. Site-preparation treatments furthered their height growth affect, with S, and CWD doubling second-year height growth and MC, with an increase of 25%. Alders and birches had, on average, three and one stems, respectively, and the mean stem number of alders increased under S and CWD. After two years, overall stem dry mass had very large genus and species differences with GA, SA, GB, and WB, with 58.4, 30.3, 5.4, and 4.0 g, respectively. The N-fixing ability of alders under these conditions resulted in a 13-fold stem dry mass production increase compared with birches. Straw tripled, CWD doubled, and MC increased stem dry mass by 40%. For WB, site-preparation combinations had an additive effect, whereas GB, GA, and SA had several combined site-preparation treatments showing synergistic results, which were greater than the additive effects of single treatments. Under the control (no site prep.), second-year stem dry masses for WB, GB, GA, and SA were 0.7, 1.4, 17.8, and 0.5 g, respectively. Under the three combined treatments, MC × S × CWD, WB, GB, GA, and SA had 6.6, 12.3, 115.7, and 70.6 g stem dry masses, respectively. SA is ecologically a lowland species, hence the low 0.5 g under the control; however, the result under the three combined treatments demonstrates their combined effectiveness on these barren sites. Green alder seems to be the best adapted to the sites, having the greatest stem dry mass under control, although that was considerably magnified under the site-preparation treatments. This study using combinations of treatments with these early successional species introduces a novel research concept, and similar studies in the literature are currently lacking, creating an opportunity for future exploration. [ABSTRACT FROM AUTHOR]

Given the various sources and consequences of opioid use disorder (OUD) stigma, the mechanisms by which stigma occurs becomes more important. Educational deficits regarding OUD as a disease and treatment influence OUD stigma. The purpose of the study was to develop, pilot test and psychometrically evaluate the Opioid-Knowledge (O-Know) test in a community setting. A social media strategy was used to recruit 294 participants, in 2019, from a midwestern region experiencing an HIV epidemic, stemming from opioid and injection drug use. The Factor analysis used a tetrachoric correlation matrix, with principal axis factor extraction and Promax rotation. Horn's parallel analysis supported the factorial validity of a two-factor model representing OUD as a Disease (Cronbach's alpha =.57) and OUD Treatment and Recovery (Cronbach's alpha =.62). The reliability estimates were deemed adequate given the intended group-level uses for the instrument. Convergent construct validity was partially supported via the OUD Treatment and Recovery factor's negative correlation with personal opioid stigmatization (B = −.29, p =.001). OUD research must produce measures that convert observations to empirical generalizations. The psychometric analysis of the O-Know scale demonstrated progress in reducing measurement error to facilitate that conversion. [ABSTRACT FROM AUTHOR]

This article provides an overview of response surface designs (RSM) commonly used in research projects. The three most common RSM designs discussed are central composite, Box-Behnken, and optimal design. Central composite designs estimate main effects, two-factor interactions, and quadratic effects, while Box-Behnken designs only estimate main effects and quadratic terms. Optimal designs are customizable and can include multilinear constraints and higher-order polynomials. Each design has its advantages and disadvantages, and researchers should choose the design that best fits their specific needs. The article emphasizes the importance of replicates and conducting confirmation runs to ensure reproducibility of results. [Extracted from the article]

For various arithmetic functions f: N - R, the behavior of f(n!) and that of Σn≤Nf(n!) can be intriguing. For instance, for some functions f, we have f(n!) = Σk≤nf(k), for others, we have f(n!) = ΣpͰnf(p) (where the sum runs over all the primes p ≤ n). Also, for some f, their minimum order coincides with limn→∞ f(n!), for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of f(n!) and of Σn≤Nf(n!) for a wide variety of arithmetical functions f. In particular, letting d(n) and σ(n) stand respectively for the number of positive divisors of n and the sum of the positive divisors of n, we obtain new accurate asymptotic expansions for d(n!) and σ(n!). Furthermore, setting σ1(n): = max{d|n:d≤√n} and observing that no one has yet obtained an asymptotic value for Σn≤Nσ(n) as N → →, we show how one can obtain the asymptotic value of Σn≤Nσ1(n!). [ABSTRACT FROM AUTHOR]

Various creative multi-response optimisation approaches have been developed in the literature. Most of these researches are based on the normality assumption of the response distribution. However, this assumption does not necessarily hold in some real cases, such as non-normal multiple responses. Also, the reproducibility of optimisation results does not hold in some practical applications due to the variability of predicted responses associated with model uncertainty. In this paper, a novel approach is proposed to address the issues for non-normal multi-response optimisation. The proposed method not only identifies significant effects for each response by incorporating factorial effect principles into the framework of the Bayesian generalised linear models (GLMs) but also takes into account the model uncertainty and the variability of predicted responses by using the Bayesian sampling technique and Pareto optimal strategy. Furthermore, the optimal parameter settings are found by using grey incidence analysis (GIA). Besides, two examples are used to illustrate the effectiveness of the proposed method. The results show that the proposed method not only effectively identify significant factors but also find more satisfactory parameter settings when the reliability and reproducibility of optimisation results are considered simultaneously. [ABSTRACT FROM AUTHOR]

Background: Few large randomized controlled trials provide strong evidence to guide surgical repair of primary rhegmatogenous retinal detachment (RRD) repair. The purpose of this factorial, single-blind, randomized controlled trial is to analyze and compare the surgical outcomes, functional visual outcomes, complications, and quality of life associated with RRD repair using (A) pars plana vitrectomy only (PPV) or PPV with scleral buckle (PPV-SB) and (B) sulfur hexafluoride gas (SF6) or perfluoropropane gas (C3F8) tamponade. Methods: Eligible patients with moderately complex RRD will be randomized 1:1 to PPV or PPV-SB and 1:1 to SF6 or C3F8 gas tamponade. Approximately 560 patients will be recruited to be able to detect a difference of around 10% in SSAS rate between the groups. Patients will be followed using multimodal imaging and quality of life questionnaires after the surgical repair until 1 year postoperative. The primary outcome will be a single-surgery anatomic success (SSAS), defined as the absence of reoperation for recurrent RRD in the operating room. Secondary outcomes will be pinhole visual acuity (PHVA) at 8–10 weeks and 6 months, final best-corrected visual acuity (BCVA), final retina status (i.e., attached or detached), time to onset of RRD recurrence, severity and number of complications, and questionnaire results. Discussion: This will be the first 2 × 2 factorial RCT examining repair techniques in primary RRD. It will also be the first RCT to compare gas tamponade between the two most common agents. Notably, it will be adequately powered to detect a clinically significant effect size. The use of multimodal imaging will also be a novel aspect of this study, allowing us to compare head-to-head the impact of adding an SB to the retina's recovery after RRD repair and of differing gas tamponades. Until now, the treatment of RRD has been largely guided by pragmatic retrospective cohort studies. There is a lack of strong evidence guiding therapeutic decisions and this trial will address (1) whether supplemental SB is justified and (2) whether longer duration gas tamponade with C3F8 is necessary. Trial registration: ClinicalTrials.gov NCT05863312. Registered on 18 May 2023. [ABSTRACT FROM AUTHOR]

Importance: Transparent reporting of randomized trials is essential to facilitate critical appraisal and interpretation of results. Factorial trials, in which 2 or more interventions are assessed in the same set of participants, have unique methodological considerations. However, reporting of factorial trials is suboptimal. Objective: To develop a consensus-based extension to the Consolidated Standards of Reporting Trials (CONSORT) 2010 Statement for factorial trials. Design: Using the Enhancing the Quality and Transparency of Health Research (EQUATOR) methodological framework, the CONSORT extension for factorial trials was developed by (1) generating a list of reporting recommendations for factorial trials using a scoping review of methodological articles identified using a MEDLINE search (from inception to May 2019) and supplemented with relevant articles from the personal collections of the authors; (2) a 3-round Delphi survey between January and June 2022 to identify additional items and assess the importance of each item, completed by 104 panelists from 14 countries; and (3) a hybrid consensus meeting attended by 15 panelists to finalize the selection and wording of items for the checklist. Findings: This CONSORT extension for factorial trials modifies 16 of the 37 items in the CONSORT 2010 checklist and adds 1 new item. The rationale for the importance of each item is provided. Key recommendations are (1) the reason for using a factorial design should be reported, including whether an interaction is hypothesized, (2) the treatment groups that form the main comparisons should be clearly identified, and (3) for each main comparison, the estimated interaction effect and its precision should be reported. Conclusions and Relevance: This extension of the CONSORT 2010 Statement provides guidance on the reporting of factorial randomized trials and should facilitate greater understanding of and transparency in their reporting. This article provides an extension of the Consolidated Standards of Reporting Trials 2010 checklist for the reporting of factorial trials. [ABSTRACT FROM AUTHOR]