On a Heuristic Analysis of Highly Fractionated 2n Factorial Experiments

The paper deals with a method for the analysis of highly fractionated factorial designs proposed by Raghavarao and Altan (2003). We show that the method will find "active" factors with almost any set of random numbers. Once that an alias set is found active, Raghavarao and Altan (2003) claim that their method can resolve the alias structure of the design and identify which of several confounded effects is active. We show that their method cannot do that. The error in Raghavarao and Altan's (2003) arguments lies in the fact that they treat a set of highly dependent (sometimes even identical) F-statistics as if they were independent. --Fractional factorial designs,half-normal plot,heuristic arguments,active effects,alias set

Exact optimal designs for weighted least squares analysis with correlated errors

In the common linear and quadratic regression model with an autoregressive error structure exact D-optimal designs for weighted least squares analysis are determined. It is demonstrated that for highly correlated observations the D-optimal design is close to the equally spaced design. Moreover, the equally spaced design is usually very efficient, even for moderate sizes of the correlation, while the D-optimal design obtained under the assumptions of independent observations yields a substantial loss in efficiency. We also consider the problem of designing experiments for weighted least squares estimation of the slope in a linear regression and compare the exact D-optimal designs for weighted and ordinary least squares analysis. --Autoregressive errors,linear regression,quadratic regression,exact D-optimal designs,estimation of the slope,generalized MANOVA

On Nearly Balanced Designs for Sensory Trials

In sensory experiments, often designs are used that are balanced for carryover effects. It is hoped that this controls for possible carryover effects, like, e.g., a lingering taste of the products. Proper randomization is essential to guarantee the usual model assumption of independent identically distributed (i.i.d.) errors. We consider a randomization procedure that permutes treatment labels and assessors. This restricted randomization leaves the neighbour structure unchanged and validates the assumption of i.i.d. errors if the design used is a Generalized Youden Design (GYD). However, the use of a neighbour balanced GYD may require too many assessors. The question arises, whether nearly balanced designs may be used without grossly violating the validity of the analysis. We therefore do a simulation study to assess the properties (under this restricted randomization) of nearly balanced designs like, e.g., the ones proposed by Périnel and Pag?s (2004, Food Quality and Preference 15, 439?446). We observe that, if there are no carryover effects, the variance estimates for treatment contrasts are not significantly biased whenever we use designs that are nearly GYD. Additionally, designs that are nearly carryover balanced still produce conservative variance estimates, even in the presence of large carryover effects. In all, ?nearly neighbour balanced nearly GYD? as proposed by Périnel and Pag?s (2004) appear to be useful in experimental situations where the use of GYD is too restrictive. It should be stressed, however, that these results are true only if randomization is used as a protection against effects unaccounted for in the statistical model. --carryover balance,nearly balanced designs,randomization,validity

Optimal designs for an interference model

Kunert and Martin (2000) determined optimal and efficient block designs in a model for field trials with interference effects, for block sizes up to 4. In this paper we use Kushner's method (Kushner, 1997) of finding optimal approximate designs to extend the work of Kunert and Martin (2000) to optimal designs with five or more plots per block. We give an overall upper bound a*t,b,k for the trace of the information matrix of any design and show that an universally optimal approximate design will have all its sequences from merely four different equivalence classes. We further determine the efficiency of a binary type I orthogonal array under the general p-criterion. We find that these designs achieve high efficiencies of more than 0:94. --

On repeated difference testing

If the number of assessors in a difference test is not large enough to ensure the desired power of the testing procedure, then it is often advised to use assessors repeatedly. That is, each assessor performs the testing not just once but several times. There is a discussion going on, how results of a repeated difference testing are to be analysed. The present paper (as was to be expected) supports the point of view expressed in Kunert and Meyners (1999). It also tries to generalise their approach such that we get confidence limits. While the exposition concentrates on the triangle test, the approach is also applicable for other difference testing procedures (e.g. pairwise difference test, duo-trio test)

Unreplicated fractional factorials, analysis with the half-normal plot and randomization of the run order

There is an ongoing discussion whether it is wise to randomize the run order of a factorial experiment if there is concern about a possible time trend in the experiment. It can be argued that a randomized order is not very effective because the trend inflates the error. Some authors even criticize that a randomized order will normally not be orthogonal to trend, they claim that therefore there will be bias under the randomized order. On the other hand, a systematic order will only be useful if the true trend is behaving as is predicted by the model. The present paper investigates the properties of different run order strategies in a simulation study with unreplicated factorial designs. We check to which extend the presence of a time trend might inflate the probability of false rejection of a true nullhypothesis, and we compare the power of significance tests based on the half-normal plot under the various run order concepts

On MSE-optimal crossover designs

In crossover designs, each subject receives a series of treatments one after the other. Most papers on optimal crossover designs consider an estimate which is corrected for carryover effects. We look at the estimate for direct effects of treatment, which is not corrected for carryover effects. If there are carryover effects, this estimate will be biased. We try to find a design that minimizes the mean square error, that is the sum of the squared bias and the variance. It turns out that the designs which are optimal for the corrected estimate are highly efficient for the uncorrected estimate

Optimal designs for an interference model

Kunert and Martin (2000) determined optimal and efficient block designs in a model for fi eld trials with interference effects, for block sizes up to 4. In this paper we use Kushner's method (Kushner, 1997) of fi nding optimal approximate designs to extend the work of Kunert and Martin (2000) to optimal designs with five or more plots per block. We give an overall upper bound a_(t,b,k) for the trace of the information matrix of any design and show that an universally optimal approximate design will have all its sequences from merely four di fferent equivalence classes. We further determine the efficiency of a binary type I orthogonal array under the general phi_p-criterion. We find that these designs achieve high efficiencies of more than 0.94

On a heuristic analysis of highly fractionated 2 n factorial experiments

The paper deals with a method for the analysis of highly fractionated factorial designs proposed by Raghavarao and Altan (2003). We show that the method will find "active" factors with almost any set of random numbers. Once that an alias set is found active, Raghavarao and Altan (2003) claim that their method can resolve the alias structure of the design and identify which of several confounded effects is active. We show that their method cannot do that. The error in Raghavarao and Altan's (2003) arguments lies in the fact that they treat a set of highly dependent (sometimes even identical) F-statistics as if they were independent

On Nearly Balanced Designs for Sensory Trials

In sensory experiments, often designs are used that are balanced for carryover effects. It is hoped that this controls for possible carryover effects, like, e.g., a lingering taste of the products. Proper randomization is essential to guarantee the usual model assumption of independent identically distributed (i.i.d.) errors. We consider a randomization procedure that permutes treatment labels and assessors. This restricted randomization leaves the neighbour structure unchanged and validates the assumption of i.i.d. errors if the design used is a Generalized Youden Design (GYD). However, the use of a neighbour balanced GYD may require too many assessors. The question arises, whether nearly balanced designs may be used without grossly violating the validity of the analysis. We therefore do a simulation study to assess the properties (under this restricted randomization) of nearly balanced designs like, e.g., the ones proposed by Périnel and Pagès (2004, Food Quality and Preference 15, 439–446). We observe that, if there are no carryover effects, the variance estimates for treatment contrasts are not significantly biased whenever we use designs that are nearly GYD. Additionally, designs that are nearly carryover balanced still produce conservative variance estimates, even in the presence of large carryover effects. In all, ”nearly neighbour balanced nearly GYD” as proposed by P´erinel and Pagès (2004) appear to be useful in experimental situations where the use of GYD is too restrictive. It should be stressed, however, that these results are true only if randomization is used as a protection against effects unaccounted for in the statistical model