A Study on Trend-Free Designs

Abstract

The data generated from designed experiments are used to draw valid inferences about the population. Heterogeneity in the experimental material is the major source of variability to be reckoned within the statistical designing of scientific experiments. Occasionally, one can find a certain factor (called nuisance factor) that though not of interest to the experimenter, does contribute significantly to the variability in the experimental material. Various levels of this factor are used for blocking. In experimental situations with only one nuisance factor, block designs are used. These designs are useful in controlling the heterogeneity of the experimental units and it is ascribed to between blocks variability. Much of the literature is available on block designs viz. randomized complete block designs, balanced incomplete block designs, partially incomplete block designs, variance balanced block designs, etc. But many times experimental situations arise in which the response is dependent on the spatial or temporal position of the experimental units within a block and thus trend in the experimental units become another important nuisance factor. In such situations, a common polynomial trend of a specified degree over units within experimental units may be appropriately assumed. One way to account for the presence of trends is to use the analysis of covariance, treating trend values as covariates. However, one may think of suitable designs, in which treatment effects are orthogonal to trend effects, in the sense that analysis of the design could be done in usual manner, as if no trend effects were present. Such designs may be called trend-free designs. When experimental units within blocks, in block designs, exhibit a trend we use trend-free block designs. The meaning of trend-free block design is to assign treatments to plots within blocks so that the known properties of ordinary analysis of variance for treatment and block sum of squares are preserved and variation due to trend effect is removed from the error sum of square. Such an arrangement is called as trend-free block design. The work on trend-free block designs in proper block settings under homoscedastic model is available in the literature. There, however do occur experimental situations where block designs with unequal block sizes and/ or with unequal replications are to be used. For example, non-proper block design setting occurs while experimenting with natural blocks such as littermates (animal experiments), trusses per blossom (horticultural experiments), family sizes as blocks (psychological experiments), batches of test material (industrial experiments), etc. Experimenting on hilly areas, wastelands or salinity in field experiments may also force the experimenter to have blocks of unequal sizes. It is also known that in the class of binary block designs with unequal replications under non-proper settings, binary variance balanced block (BBB) designs are the most efficient designs for estimating all possible elementary contrasts among treatments. In variance balanced block designs, generally it is assumed that intrablock variances are constant. Through empirical investigations, however, it has been shown that intra-block variances are proportional to non-negative real power of block sizes. However, the work on trend-free block designs for heteroscedastic model under non-proper block design settings and for nested balanced incomplete block designs could not be traced from the available literature. This investigation, therefore, deals with the trend-free block designs under heteroscedastic set up when intra-block variances are proportional to non-negative real power of block sizes. Further, there do occur experimental situations in which one or more factors are nested within the blocking factor. In such situations nested block designs and nested balanced incomplete block designs are quite useful. Such designs may also have trend-effect at sub block or block level. Similar to block designs, experimental units in block design for a diallel cross experiments may be subject to trend-effect over space or time. Thus, trend-free nested balanced incomplete block designs and trend-free block design for a diallel cross experiments have been studied. Many times it may not be possible to convert every block design to trend-free block design, we go for linear trend-free design because using linear trend-free designs eliminates much of the trend. Sometimes, it is not possible to make the design linear trend-free or trend-free and this provides a motivation to go for nearly linear trend-free designs. Thus nearly linear trend-free designs have also been investigated. In Chapter I, various experimental situations have been described in which the trend may exist in non-proper block designs and NBIB designs. Some examples have been illustrated for better understanding of trend effect in complete randomized, randomized block and in factorial designs. In Chapter II, a necessary and sufficient condition for a block design to be trend free block design under heteroscedastic set up when intra-block variances are proportional to non-negative real power of block sizes is obtained. Using the condition catalogues of trend-free BBB designs of Type alpha, both under homoscedastic (aloha = 0) and heteroscedastic model (for alpha = 1, 2, 3), is prepared. Heteroscedasticity of the model increases as value of alpha increases. Catalogues of trend-free balanced incomplete block (BIB) designs with replications, r <= 20 and two associate class partially balanced incomplete block (PBIB) designs are also prepared. Sometimes it may not be possible to convert every design to trend-free design then linear trend-free designs are given. Further, nearly linear trend-free designs are identified when it is not possible to obtain even linear trend-free designs. Chapter III deals with nested balanced incomplete block (NBIB) designs and block design for a diallel cross experiments. A necessary and sufficient condition for a NBIB design to be trend-free block design at sub-block level is obtained. Catalogues of trend-free/ linear trend-free NBIB designs at sub-block levels, of NBIB designs given by given by Morgan, Preece and Rees (2001), are prepared. NBIB designs with sub-block size 2 and designs for complete diallel cross experiments have a one-to-one correspondence. Utilizing this relationship a catalogue of trend-free optimal block designs for a diallel cross experiments with number of inbred lines, p<=30 is prepared.