Two-way Layout Research Articles

Order Restricted Randomized Block Designs

Field experiments are run under two competing objectives, high precision and minimal cost. The precision can be increased either by using sound experimentation techniques that account for the sources of variation with reasonable statistical models or by increasing the sample size. Large sample sizes usually increase the cost of the experiment and may not be feasible. This paper uses order restricted randomized designs (ORRD) to increase the precision while keeping the sample size and cost of the experiment minimal. The ORRD described here starts with a randomized block design but adds a second layer of blocking by ranking plots within each block. This creates a two-way lay-out, blocks and ranking groups, and uses a restricted randomization to improve the precision of estimation of the treatment parameters. Ranking groups create a correlation structure for within-block units. The restricted randomization uses this correlation structure to reduce the error variance of the experiment. The paper computes the expected mean square for each source of variation in the ORRD design under a suitable linear model. It also provides approximate F-tests for treatment and ranking group effects. The efficiency of the ORRD is investigated through empirical power studies. Finally, an example based on a uniformity field trial illustrates the use of the method in a split-plot experiment. Supplementary material to this paper is provided online.

Optical frequency transfer with below 10−21 uncertainty using a DFB–laser-based fiber Brillouin amplifier

Exploiting the outstanding performance of optical atomic clocks for improved timekeeping, relativistic geodesy, and fundamental physics beyond the standard model demands comparing distant state-of-the-art optical clocks. Interferometric optical fiber links have been demonstrated as an eminent method for such frequency comparisons over distances up to thousands of kilometers. However, for such distances, the optical fiber attenuation mandates signal amplification. Fiber Brillouin amplification (FBA) has been proven as an efficient amplification technique for coherent frequency transfer. Demonstrated FBA schemes have been designed based on costly narrow-linewidth pump lasers and analog pump-to-signal phase locking schemes. Furthermore, the high pump power requirement of these FBAs hinders the integration of FBA-based frequency dissemination on fiber connections for shared telecommunication signals in the C-band. In this paper, we propose and experimentally demonstrate a novel FBA module (FBAM) employing cost-effective distributed feedback (DFB) pump lasers assisted by a digital phase locking scheme based on a field programmable gated array. The new FBAM is compact, cost-effective, and directly applicable to different bands, which opens up new opportunities to establish a frequency metrology infrastructure within existing telecommunication fiber networks. Additionally, the small-footprint of the DFB-FBAM allows for frequent amplification stages with lower pump power to reach continental scale optical metrology links with an optimized signal-to-noise ratio. We characterized the DFB-FBAM’s frequency transfer uncertainty using a two-way layout over an in-lab 100 km long optical fiber link and reach a fractional frequency instability of 9.3 × 10−22 at a 10 ks integration time. The DFB-FBAM characterizations show uncertainty contributions of (−2.1 ± 3.3) × 10−22 and below for averaging times >100 ks. For the first time, we assess the temporal Brillouin frequency shift variations in an underground fiber link and implement a scheme to track these changes in a remote FBAM.

Some inferential questions in regard to analysing two-way Layouts and associated linear model theory and practice

In analysing a well known data set from the literature which can be thought of as a two-way layout it transpires that a robust adaptive regression approach for identifying outliers fails to be sensitive enough to detect the possible interchange of two observations. On the other hand if one takes the classical approach of diagnostic checking one may also stop too early and be satisfied with a model that falls short of a more detailed analysis that takes account of heteroscedasticity in the data. An exact F-test for heteroscedasticity in the two way layout is compared with various more general tests proposed by Shukla. In conclusion it is noted that when modelling the particular form of heteroscedasticity countenanced here, the estimated column effects are unchanged from those estimated from the model assuming homogeneous error variance structure. It is only the estimated variances of these column effects that changes.

Auxiliary Stopping Area Layout Method for High-Speed Maglev Operated Bidirectionally on Single Track

Auxiliary stopping area (ASA) is the necessary emergency facility for train safety of the normal high-speed maglev. The study addresses the ASA layout problem of the high-speed maglev operated bidirectionally on single track. First, an optimization model of the ASA layout for unidirectional double-track lines considering train safety, operation efficiency, and construction cost is established, and two basic methods of the ASA layout are investigated based on the distance demand characteristic of ASAs. Then, the ASA layout problem of bidirectional single-track lines is analyzed, and an ASA two-way coordination layout algorithm (ASA-TWCLA) is proposed. Finally, a numerical experiment is carried out. The results suggest that under the premise of train safety and operation efficiency, compared with using the basic methods separately on the two directions, adopting the ASA-TWCLA algorithm can obtain a more economical ASA layout scheme for the same scenario.

Two-way layout factorial experiments of spatial point pattern responses in mineral flotation

Factorial experiments are well-understood when the given observations are outcomes of random variables. However, when we observe spatial point patterns in each combination of factors cells, the methodology is much less developed. Motivated by a real problem of locations of bubbles in a mineral flotation experiment where the interest is analysing if the spatial distribution might be affected by frother concentrations and volumetric airflow rates, we develop an approach for statistical testing of two-way factorial experiments for spatial point patterns. We describe the point patterns through the K-function, a second-order summary statistic, and develop a set of new Fisher-based statistics using weighted means. For inference by Monte Carlo, we use random permutations of weighted residuals depending on the null hypothesis. We conduct simulation experiments to demonstrate the performance of the new test statistics and present the results of the real problem.

Detection of Hidden Additivity and Inference Under Model Uncertainty for Unreplicated Factorial Studies via Bayesian Model Selection and Averaging

The two-way unreplicated layout remains a popular study design in the physical sciences. However, detection of statistical interaction and subsequent inference has been problematic in this class of designs. First, lack of replication precludes inclusion of standard interaction parameters. Second, while several restricted forms of interaction have been considered, existing approaches focus primarily on accept/reject decisions with respect to the presence of interaction. Approaches to estimate cell means and error variance are lacking when the possibility of interaction exists. For these reasons, we propose model selection and averaging-based approaches to facilitate statistical inference when the presence of interaction is uncertain. Hidden additivity, a recently proposed and intuitive form of interaction, is used to accommodate latent group-based nonadditive effects. The approaches are fully Bayesian and use the Zellner–Siow formulation of the mixture g-prior. The method is illustrated on three empirical datasets and simulated data. The estimates from the model averaging approach are compared with a customized regularization approach which shrinks interaction effects toward the additive model. The study concludes that Bayesian model selection is a fruitful approach to detect hidden additivity, and model averaging allows for inference on quantities of interest under model uncertainty with respect to interaction effects within the two-way unreplicated design.

Closed Testing Procedures for All Pairwise Comparisons of Main Effects in the Two-Way Layout with Replications

Closed Testing Procedures for All Pairwise Comparisons of Main Effects in the Two-Way Layout with Replications

Generalized prediction intervals for treatment effects in random-effects models.

This article derives generalized prediction intervals for random effects in linear random-effects models. For balanced and unbalanced data in two-way layouts, models are considered with and without interaction. Coverage of the proposed generalized prediction intervals was estimated in a simulation study based on an agricultural field experiment. Generalized prediction intervals were compared with prediction intervals based on the restricted maximum likelihood (REML) procedure and the approximate methods of Satterthwaite and Kenward and Roger. The simulation study showed that coverage of generalized prediction intervals was closer to the nominal level 0.95 than coverage of prediction intervals based on the REML procedure.

A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests

There are several non‐functional‐form‐based interaction tests for testing additivity in unreplicated two‐way layouts. The distinctness of the various tests in this setting is somewhat unfortunate, as no single test can detect all patterns of possible interaction, and the tests are sensitive to a particular pattern of interaction. Therefore, it is reasonable to combine multiple interaction tests for testing additivity. The main problem is how we can combine multiple dependent tests into a single test such that it has an acceptable performance in terms of controlling the type I error rate and maximizing the power. In this paper, we review six methods for combining interaction tests which utilize six different non‐functional‐form‐based interaction tests. We also propose two novel combination methods based on the Jacobi polynomial expansion and the Gaussian copula. We first compare these eight combination methods in terms of controlling the type I error rate via a simulation study. We then compare the six interaction tests and four combination methods in terms of power. The simulation results show that our proposed combination method that is based on the Jacobi polynomial expansion has the best performance in terms of controlling the type I error rate and maximizing the power. Therefore, we recommend this combination method as a test procedure for testing additivity. The major contribution of this work is to provide researchers with a testing approach which leverages many existing methods to detect disparate forms of non‐additivity. This alleviates the need for researchers to naively choose among highly specialized methods by instead allowing for a general approach that captures a number of interaction effects with reasonable power.

Comparison of nonparametric analysis of variance methods: A vote for van der Waerden

ABSTRACTFor two-way layouts in a between-subjects analysis of variance design, the parametric F-test is compared with seven nonparametric methods: rank transform (RT), inverse normal transform (INT), aligned rank transform (ART), a combination of ART and INT, Puri & Sen's L statistic, Van der Waerden, and Akritas and Brunners ANOVA-type statistics (ATS). The type I error rates and the power are computed for 16 normal and nonnormal distributions, with and without homogeneity of variances, for balanced and unbalanced designs as well as for several models including the null and the full model. The aim of this study is to identify a method that is applicable without too much testing for all the attributes of the plot. The Van der Waerden test shows the overall best performance though there are some situations in which it is disappointing. The Puri & Sen's and the ATS tests show generally very low power. These two and the other methods cannot keep the type I error rate under control in too many situations. Especially in the case of lognormal distributions, the use of any of the rank-based procedures can be dangerous for cell sizes above 10. As already shown by many other authors, nonnormal distributions do not violate the parametric F-test, but unequal variances do, and heterogeneity of variances leads to an inflated error rate more or less also for the nonparametric methods. Finally, it should be noted that some procedures show rising error rates with increasing cell sizes, the ART, especially for discrete variables, and the RT, Puri & Sen, and the ATS in the cases of heteroscedasticity.

Pairwise comparisons of treatments with ordinal responses in a two-way setting

ABSTRACTIn clinical studies, pairwise comparisons are frequently performed to examine differences in efficacy between treatments. The statistical methods of pairwise comparisons are available when treatment responses are measured on an ordinal scale. The Wilcoxon–Mann–Whitney test and the latent normal model are popular examples. However, these procedures cannot be used to compare treatments in parallel groups (a two-way design) when overall type I error must be controlled. In this paper, we explore statistical approaches to the pairwise testing of treatments that satisfy the requirements of a two-way layout. The results of our simulation indicate that the latent normal approach is superior to the Wilcoxon–Mann–Whitney test. Clinical examples are used to illustrate our suggested testing methods.

The aligned rank transform and discrete variables: A warning

ABSTRACTFor two-way layouts in a between subjects ANOVA design the aligned rank transform (ART) is compared with the parametric F-test as well as six other nonparametric methods: rank transform (RT), inverse normal transform (INT), a combination of ART and INT, Puri & Sen's L statistic, van der Waerden and Akritas & Brunners ATS. The type I error rates are computed for the uniform and the exponential distributions, both as continuous and in several variations as discrete distribution. The computations had been performed for balanced and unbalanced designs as well as for several effect models. The aim of this study is to analyze the impact of discrete distributions on the error rate. And it is shown that this scaling impact is restricted to the ART- as well as the combination of ART- and INT-method. There are two effects: first with increasing cell counts their error rates rise beyond any acceptable limit up to 20 percent and more. And secondly their rates rise when the number of distinct values of the dependent variable decreases. This behavior is more severe for underlying exponential distributions than for uniform distributions. Therefore there is a recommendation not to apply the ART if the mean cell frequencies exceed 10.

Factors promoting sense of coherence among university students in urban areas of Japan: individual-level social capital, self-efficacy, and mental health.

Sense of coherence (SOC) is a concept that helps to explain the relation between personal intentionality as psychosocial factors and health-related behaviors. Thus, it is essential to enhance SOC when encouraging a healthy lifestyle. However, the factors that promote SOC have not been fully investigated among university students. The objective of this study was to clarify the general resistance resources (GRRs) that may promote the development of the SOC among university students. Therefore, we examined the relationship between SOC and social capital (SC), self-efficacy, and mental health. Participants included 443 students from nine academic departments at eight universities in the Kanto or Kinki metropolitan areas of Japan. Participants completed an anonymous questionnaire. Individual-level cognitive and structural SC, generalized self-efficacy, mental health inventory (from SF-36v2), and SOC were measured. Confirmatory factor analysis using structural equation modeling was conducted to verify the factor structure of the SOC-13 scale. Stepwise multiple regression analysis and two-way layout analysis of variance were performed with SOC as the dependent variable. The factor structure of SOC indicated the optimal model fit in the second-order three-factor model of the 12 items. SOC was predicted by five variables: age, cognitive SC, structural SC, mental health, and self-efficacy. For students from urban areas, SOC was predicted by the interaction between cognitive and structural SC. SOC was significantly related to cognitive SC, structural SC, and self-efficacy as well as mental health in university students from urban areas. Furthermore, the combination of higher-level cognitive SC and higher-level structural SC exerted an inhibitory influence on SOC among students who previously and currently live in urban areas. Therefore, the findings indicated that both cognitive and structural SC as well as self-efficacy may act as GRRs that promote the development of SOC, and similarly, good mental health may promote a strong SOC.

Vector-IM-based assessment of alternative framing systems under bi-directional ground-motion

This paper examines the seismic performance of steel buildings with alternative framing systems subjected to bi-directional ground-motion. Peak drifts of one-way (perimeter framing) and two-way (space framing) systems are assessed by means of scalar and vector-valued probabilistic methods. Extensive non-linear response history analyses over idealized 3D structures representing 6- and 9-storey buildings are performed under pairs of linearly scaled ground-motions. Both far-field and near-field non-pulselike acceleration series are considered. The spectral acceleration of the geometric mean of the two horizontal components (Sa,GM) is taken as the primary intensity measure (IM) while four other ground-motion parameters are employed to construct IM-vectors including: the spectral acceleration ratio (RT3,T1), the spectral shape parameter (Np), and two frequency content parameters (Tm and To). This paper shows that incorporating the vector 〈Sa,GM,Np〉 into the assessment of bi-directionally loaded 3D buildings yields up to 40% lower conditional standard deviations than a purely scalar formulation at large drift levels while the vector 〈Sa,GM,RT3,T1〉 is more efficient at smaller drifts. The effects of alternative framing systems on structural fragilities are found to differ depending on the number of storeys. For 6-storey structures, consistently higher capacities are observed in two-way layouts with respect to one-way systems but they are associated with increasing variabilities at larger demand levels. Conversely, the 9-storey two-way building experiences 5% lower mean capacities than its one-way counterpart. Finally, drift hazard curves are calculated by combining the building fragilities with idealized ground-motion hazard estimates. The results indicate that one-way buildings experience consistently lower drift exceedance rates regardless of the ground-motion type, especially for drift levels larger than 2% although the differences are larger for the 9-storey frames in comparison with their 6-storey counterparts. This study represents a first attempt to implement vector-valued analysis in the context of bi-directionally loaded structures and its results constitute an important step towards discerning the most favourable framing system at different seismic performance levels.

A method for testing interaction in unreplicated two-way tables: using all pairwise interaction contrasts

Several methods exist for testing interaction in unreplicated two-way layouts. Some are based on specifying a functional form for the interaction term and perform well provided that the functional form is appropriate. Other methods do not require such a functional form to be specified but only test for the presence of non-additivity and do not provide a suitable estimate of error variance for a non-additive model. This paper presents a method for testing for interaction in unreplicated two-way tables that is based on testing all pairwise interaction contrasts. This method (i) is easy to implement, (ii) does not assume a functional form for the interaction term, (iii) can find a sub-table of data which may be free from interaction and to base the estimate of unknown error variance, and (iv) can be used for incomplete two-way layouts. The proposed method is illustrated using examples and its power is investigated via simulation studies. Simulation results show that the proposed method is competitive with existing methods for testing for interaction in unreplicated two-way layouts.

Parametric and nonparametric bootstrap methods for general MANOVA

We develop parametric and nonparametric bootstrap methods for multi-factor multivariate data, without assuming normality, and allowing for covariance matrices that are heterogeneous between groups. The newly proposed, general procedure includes several situations as special cases, such as the multivariate Behrens–Fisher problem, the multivariate one-way layout, as well as crossed and hierarchically nested two-way layouts. We derive the asymptotic distribution of the bootstrap tests for general factorial designs and evaluate their performance in an extensive comparative simulation study. For moderate sample sizes, the bootstrap approach provides an improvement to existing methods in particular for situations with nonnormal data and heterogeneous covariance matrices in unbalanced designs. For balanced designs, less computationally intensive alternatives based on approximate sampling distributions of multivariate tests can be recommended.

CIBIC Plus-J Assessment Using a Videotaped Method in Alzheimer’s Disease Patients

Background/Aims: CIBIC plus-J is the Japanese language version equivalent to CIBIC plus. Variability of CIBIC plus-J arises among raters in accordance with their experience and their memories of patients’ conditions at baseline. Therefore, in a multicenter trial of Alzheimer’s disease, CIBIC plus-J interviews with Alzheimer’s disease patients were videotaped, and the tapes were assessed by central raters as a means to improve the reliability of CIBIC plus-J assessment. Methods: Two of eight central raters were randomly selected and independently assessed the CIBIC plus-J of each patient. Results: CIBIC plus-J of 41 patients was assessed. The agreement rate between the two raters was 46.3% (19/41), when two raters assessed the CIBIC plus-J of the same patient. However, when considering disagreement between adjacent points as ‘agree’, the agreement rate was 97.6% (40/41). Although the kappa coefficients contained coincidence, simple and quadratic weighted kappa coefficients [95% confidential interval (CI)] were 0.226 (0.066–0.386) and 0.633 (0.507–0.759), respectively, and when considering disagreement between adjacent points as ‘agree’, the agreement kappa was 0.896 (0.752–1.041). The interclass coefficient from the two-way layout model was 0.639. Conclusion: The reliability of the CIBIC plus-J assessment with the videotaped method was acceptable.

Maximum likelihood degree of variance component models

Most statistical software packages implement numerical strategies for computation of maximum likelihood estimates in random effects models. Little is known, however, about the algebraic complexity of this problem. For the one-way layout with random effects and unbalanced group sizes, we give formulas for the algebraic degree of the likelihood equations as well as the equations for restricted maximum likelihood estimation. In particular, the latter approach is shown to be algebraically less complex. The formulas are obtained by studying a univariate rational equation whose solutions correspond to the solutions of the likelihood equations. Applying techniques from computational algebra, we also show that balanced two-way layouts with or without interaction have likelihood equations of degree four. Our work suggests that algebraic methods allow one to reliably find global optima of likelihood functions of linear mixed models with a small number of variance components.

Location and Magnitude Estimation of the 9 October 2006 Korean Nuclear Explosion Using the Southern Great Basin Digital Seismic Network as a Large-Aperture Array

The Southern Great Basin Digital Seismic Network (SGBDSN) has been designed for monitoring high-frequency (1–40-Hz) local events at and near Yucca Mountain, Nevada, the designated site for the national high-level nuclear-waste repository. We find that the network is also effective as a large-aperture teleseismic array for monitoring events in and close to North Korea, the recent location of an underground nuclear test that occurred on 9 October 2006, 01:35:28 coordinated universal time (UTC), National Earthquake Information Center (NEIC) m b 4.3. We explain this by (1) low ambient noise, (2) energy-efficient propagation paths (the nuclear explosion and deep earthquakes on nearly the same ray path show dominant frequencies between 0.9 and 2.5 Hz), and (3) coherent signals across the SGBDSN. The network, when used as an array, provides a particularly good beam signal-to-noise ratio (SNR) for the nuclear explosion. Estimated beam SNR is 20 dB at frequencies between 0.9–2.5 Hz. Between January 1996 and December 2006, 58% (11 of 19) of the events with 7.1> m b>3.3 located within 300 km of the North Korean nuclear explosion by NEIC are considered large enough to be confidently picked by an SGBDSN analyst. The first-arrival of the North Korean event itself is apparent on 25 of the 29 SGBDSN unfiltered recordings. The direct P phase is confidently identified using slowness and back azimuth estimated with cross-correlation and frequency-wavenumber methods. Static time corrections for beamforming are estimated using 11 deep earthquakes within 300 km of the nuclear explosion. With the statistical regression model named two-way layout, we estimate the event first-arrival time offset and the mean station effects. The resulting relative time delays are used to calibrate the nuclear explosion single-array location to the NEIC location, using the delays derived from 11 earthquakes. With magnitude substituted for time, the same method is used to estimate magnitude corrections. The SGBDSN magnitude estimate is 4.3 using the Veith–Clawson body-wave magnitude formula.

Nonparametric multiple comparison procedures for unbalanced two-way layouts

In this paper, we consider nonparametric multiple comparison procedures for unbalanced two-way factorial designs under a pure nonparametric framework. For multiple comparisons of treatments versus a control concerning the main effects or the simple factor effects, the limiting distribution of the associated rank statistics is proven to satisfy the multivariate totally positive of order two condition. Hence, asymptotically the proposed Hochberg procedure strongly controls the familywise type I error rate for the simultaneous testing of the individual hypotheses. In addition, we propose to employ Shaffer's modified version of Holm's stepdown procedure to perform simultaneous tests on all pairwise comparisons regarding the main or simple factor effects and to perform simultaneous tests on all interaction effects. The logical constraints in the corresponding hypothesis families are utilized to sharpen the rejective thresholds and improve the power of the tests.