On screening samples in the laboratory and factors in factorial investigations

Procedures such as group testing and fractional replication can greatly reduce the workload. The former is especially efficient when the presence of a characteristic in samples is low. Fractional replication can greatly reduce the amount of experimentation when some of the parameters associated with a factorial are nonexistent or negligible. Supersaturated fractional replication can be used to efficiently screen factors when it is suspected that they may not affect a process but it is desired to check this suspicion. There are many types of grouping and pooling material in an investigation. Some of these are discussed as are the interrelationships among the above procedures.     

Bayes factors for goodness of fit testing

We propose the use of the generalized fractional Bayes factor for testing fit in multinomial models. This is a non-asymptotic method that can be used to quantify the evidence for or against a sub-model. We give expressions for the generalized fractional Bayes factor and we study its properties. In particular, we show that the generalized fractional Bayes factor has better properties than the fractional Bayes factor.     

Designing fractional factorial split-plot experiments with few whole-plot factors

Summary.  When it is impractical to perform the experimental runs of a fractional factorial design in a completely random order, restrictions on the randomization can be imposed. The resulting design is said to have a split-plot, or nested, error structure. Similarly to fractional factorials, fractional factorial split-plot designs can be ranked by using the aberration criterion. Techniques that generate the required designs systematically presuppose unreplicated settings of the whole-plot factors. We use a cheese-making experiment to demonstrate the practical relevance of designs with replicated settings of these factors. We create such designs by splitting the whole plots according to one or more subplot effects. We develop a systematic method to generate the required designs and we use the method to create a table of designs that is likely to be useful in practice.     

Parameter estimation for fractional birth and fractional death processes

The fractional birth and the fractional death processes are more desirable in practice than their classical counterparts as they naturally provide greater flexibility in modeling growing and decreasing systems. In this paper, we propose formal parameter estimation procedures for the fractional Yule, the fractional linear death, and the fractional sublinear death processes. The methods use all available data possible, are computationally simple and asymptotically unbiased. The procedures exploited the natural structure of the random inter-birth and inter-death times that are known to be independent but are not identically distributed. We also showed how these methods can be applied to certain models with more general birth and death rates. The computational tests showed favorable results for our proposed methods even with relatively small sample sizes. The proposed methods are also illustrated using the branching times of the plethodontid salamanders data of (Syst. Zool. 28:579–599, 1979).     

Alternative Bayes factors for model selection

Several alternative Bayes factors have been recently proposed in order to solve the problem of the extreme sensitivity of the Bayes factor to the priors of models under comparison. Specifically, the impossibility of using the Bayes factor with standard noninformative priors for model comparison has led to the introduction of new automatic criteria, such as the posterior Bayes factor (Aitkin 1991), the intrinsic Bayes factors (Berger and Pericchi 1996b) and the fractional Bayes factor (O'Hagan 1995). We derive some interesting properties of the fractional Bayes factor that provide justifications for its use additional to the ones given by O'Hagan. We further argue that the use of the fractional Bayes factor, originally introduced to cope with improper priors, is also useful in a robust analysis. Finally, using usual classes of priors, we compare several alternative Bayes factors for the problem of testing the point null hypothesis in the univariate normal model.     

On the Use of Semi-folding in Regular Blocked Two-level Factorial Designs

In this article, we consider experimental situations where a blocked regular two-level fractional factorial initial design is used. We investigate the use of the semi-fold technique as a follow-up strategy for de-aliasing effects that are confounded in the initial design as well as an alternative method for constructing blocked fractional factorial designs. A construction method is suggested based on the full foldover technique and sufficient conditions are obtained when the semi-fold yields as many estimable effects as the full foldover.     

Fractional Brownian motion and long term clinical trial recruitment

Prediction of recruitment in clinical trials has been a challenging task. Many methods have been studied, including models based on Poisson process and its large sample approximation by Brownian motion (BM); however, when the independent incremental structure is violated for BM model, we could use fractional Brownian motion to model and approximate the underlying Poisson processes with random rates. In this paper, fractional Brownian motion (FBM) is considered for such conditions and compared to BM model with illustrated examples from different trials and simulations.     

Statistics Problems with Simple Numbers

This article concerns the construction of simple numerical illustrations of statistical techniques for use in introductory classes. Minimizing the amount of calculation facilitates checking, promotes reliability, is quicker, and reinforces the confidence of the student. Methods are described for generating (a) samples of size 3 upwards with integer-valued means and standard deviations, and (b) simple linear regressions with integer-valued intercepts and integer-valued or simple fractional slopes. Extensions to give exact pooled standard deviations in the two-sample problem and simple exact fractional correlation coefficients are also indicated. Further statistical procedures amenable to the same general approach are listed.     

Some new augmented fractional Box–Behnken designs

The augmented Box–Behnken designs are used in the situations in which Box–Behnken designs (BBDs) could not estimate the response surface model due to the presence of third-order terms in the response surface models. These designs are too large for experimental use. Usually experimenters prefer small response surface designs in order to save time, cost, and resources; therefore, using combinations of fractional BBD points, factorial design points, axial design points, and complementary design points, we augment these designs and develop new third-order response surface designs known as augmented fractional BBDs (AFBBDs). These AFBBDs have less design points and are more efficient than augmented BBDs.     

Objective Bayesian Testing of a Poisson Mean

We consider testing hypotheses about a single Poisson mean. When prior information is not available, use of objective priors is of interest. We provide intrinsic priors based on the arithmetic intrinsic and fractional Bayes factors, and evaluate their characteristics.     

On Semiparametric Testing of I(d) by FEXP Models

To statistically test the assumption that a scalar-valued process is I(d), this article proposes a test statistic based on a broadband semiparametric method by use of fractional exponential (FEXP) models. The test is semiparametric in terms of flexibility in modeling spectral density. We establish the asymptotic behavior and report finite-sample performance by a Monte Carlo study.     

Sampling methods and sensitivity analysis for large parameter sets

Models with large parameter (i.e., hundreds or thousands of parameters) often behave as if they depend upon only a few parameters, with the rest having comparatively little influence. One challenge of sensitivity analysis with such models is screening parameters to identify the influential ones, and then characterizing their influences.

Large models often require significant computing resources to evaluate their output, and so a good screening mechanism should be efficient: it should minimize the number of times a model must be exercised. This paper describes an efficient procedure to perform sensitivity analysis on deterministic models with specified ranges or probability distributions for each parameter.

It is based on repeated exercising of the model, which can be treated as a black box. Statistical checks can ensure that the screening identified parameters that account for the bulk of the model variation. Subsequent sensitivity analysis can use the screening information to reduce the investment required to characterize the influence of influential and other parameters.

The procedure exploits simplifications in the dependence of a model output on model inputs. It works best where a small number of parameters are much more influential than all the rest. The method is much more sensitive to the number of influential parameters than to the total number of parameters. It is most effective when linear or quadratic effects dominate higher order effects and complex interactions.

The paper presents a set of M athematica functions that can be used to create a variety of types of experimental designs useful for sensitivity analysis, including simple random, latin hypercube and fractional factorial sampling. Each sampling method can use discretization, folding, grouping and replication to create composite designs. These techniques have beencombined in a composite approach called Iterated Fractional Factorial Design (IFFD).

The procedure is applied to model of nuclear fuel waste disposal, and to simplified example models to demonstrate the concepts involved.     

Identification of fractional differencing autoregressive models †

This paper proposes an identification method to fractional differencing autoregressive models, and this method gives a consistent estimator for fractional differencing order and efficient estimates for parameters in fractional differencing autoregressive models.     

Estimation of the Hurst parameter in some fractional processes

We propose to estimate the Hurst parameter involved in fractional processes via a method based on the Karhunen–Loève expansion of a Gaussian process. We specifically investigate the cases of the fractional Brownian motion, the fractional Ornstein–Uhlenbeck family and the fractional Brownian bridge. We numerically compare our results with the ones obtained by the maximum-likelihood method, which show the validity of our proposal.     

A comparison between several adjustment models to simulated teletraffic data

In this paper we assess the suitability of the fractional Brownian motion and the Markovian modulated Poisson process to represent modern teletraffic data when we use them in dimensioning and quality of service studies. The procedure consists in adjusting a model of each family to a stream of data simulated with a physical model that emulates the real flow of data. Then, both models are profusely studied in order to establish which properties of the original data are captured.     

Analysis of variance of balanced fractional 2n factorial designs of resolution 2l+1

By use of the algebraic structure of the triangular multidimensional partially balanced association scheme, we present the analysis of variance and the hypotheses testing of a balanced fractional 2ⁿfactorial design of resolution 2l+1, which is derived from a balanced array of strength 2l.     

Parametric fractional imputation for nonignorable missing data

Parameter estimation with missing data is a frequently encountered problem in statistics. Imputation is often used to facilitate the parameter estimation by simply applying the complete-sample estimators to the imputed dataset.In this article, we consider the problem of parameter estimation with nonignorable missing data using the approach of parametric fractional imputation proposed by Kim (2011). Using the fractional weights, the E-step of the EM algorithm can be approximated by the weighted mean of the imputed data likelihood where the fractional weights are computed from the current value of the parameter estimates. Calibration fractional imputation is also considered as a way for improving the Monte Carlo approximation in the fractional imputation. Variance estimation is also discussed. Results from two simulation studies are presented to compare the proposed method with the existing methods. A real data example from the Korea Labor and Income Panel Survey (KLIPS) is also presented.     

On sum composition of fractional factorial designs

Combinatorial extension and composition methods have been extensively used in the construction of block designs. One of the composition methods, namely the direct product or Kronecker product method was utilized by Chakravarti [1956] to produce certain types of fractional factorial designs. The present paper shows how the direct sum operation can be utilized in obtaining from initial fractional factorial designs for two separate symmetrical factorials a fractional factorial design for the corresponding asymmetrical factorial. Specifically, we provide some results which are useful in the construction of non-singular fractional factorial designs via the direct sum composition method. In addition a modified direct sum method is discussed and the consequences of imposing orthogonality are explored.     

Constructing non-regular robust parameter designs

In recent years there has been considerable attention paid to robust parameter design as a strategy for variance reduction. Of particular concern is the selection of a good experimental plan in light of the two different types of factors in the experiment (control and noise) and the asymmetric manner in which effects of the same order are treated. Recent work has focussed on the selection of regular fractional factorial designs in this setting. In this article, we consider the construction and selection of optimal non-regular experiment plans for robust parameter design. Our approach defines the word-length pattern for non-regular fractional factorial designs with two different types of factors which allows for the choice of optimal design to emphasize the estimation of the effects of interest. We use this new word-length pattern to rank non-regular robust parameter designs. We show that one can easily find minimum aberration robust parameter designs from existing orthogonal arrays. The methodology is demonstrated by finding optimal assignments for control and noise factors for 12, 16 and 20-run orthogonal arrays.