Preface This book is a small addition to a number of excellent books already available on design of experiments .The book is primarily addressed to the beginners-students at graduate and post-graduate levels and research workers from diversified segments of agricultural sciences. The book consists of 11 chapters. Chapter-1 is related to statistical results and completely randomized design for agricultural field experiments. An example is provided to make the concepts clear. Chapter-2 is devoted to randomized block design used for local control, and replication in agricultural experiments. Further, randomized block design with one missing value is dealt herewith. Moreover, statistical analysis of two-way classified data with m observations per cell was given. Chapter-3 is devoted to factorial experiments in randomized block design involving two or more factors. Complete confounding and partial confounding have been illustrated with examples. 3 factors each at 3 levels have also been included.Mixed factorial with 2 factors, each at 3 levels and 3rd factor at 2 levels has been analytically explained with an example. Chapter-4 involves split-plot design and strip-plot design. Chapter-5 illustrates split-split plot design with an example. Chapter-6 deals with Latin Square design. To illustrate the results, an example has been provided. Chapter-7 touches incomplete block design meant for plant breeding trials. To illustrate balanced incomplete block design and partially incomplete block designs, two examples explaining the computational procedures are given. Chapter-8 deals with transformation of data which is applied when the given data is not normally distributed. Examples are provided to make clear the transformations. Chapter-9 deals with series of experiments with same set of treatments over different seasons or different places. Examples have been given to illustrate the computational procedures. Chapter-10 deals with Analysis of co-variance technique with example. Chapter-11 involves long-term experiments where same set of treatments are continued over the same site without re- randomization. Examples have been provided to make clear the concepts. The book in a sense is exhaustive and complete by itself. Further, the authors would welcome the comments and suggestions for improvement of the book. This book is a sequel to senior author’s earlier book viz. "Statistical Methods for Agricultural Field Experiments" by Vijay Katyal and B.Gangwar.

Normal distribution :- A random variable x is said to be normally distributed with population mean u and standard deviation σ if it assumes the following distribution

Introduction In any experiment, for instance, a varietal trial or a treatment trial, the main purpose of the experimental design would be to detect the differences among the varieties or treatments. This assumes, as an obvious prerequisite that the experimental units, such as plots or animals, be as alike as possible so that differences observed in responses, such as yield in varietal trial, could be attributable to the varietal or treatment differences. But more often than not, this condition is not realized to a satisfactory degree in practice due to heterogeneity present in experimental units. In a field experiment, for instance, a big piece of land might be quite heterogeneous regarding soil fertility, moisture content, salinity, etc; so that experimental plots marked on this land would differ widely among themselves, though plots nearer to each other would be more alike or homogeneous than the plots scattered distant apart; also in the case of animal experiments, animals forming the experimental units might be heterogeneous regarding their ages , weights, genetic composition, etc. With such experimental units in the experiment, the variability produced in the observations will be comparatively high so that only large treatment differences are detectable, since small differences are obsecured by large error variability. The randomized complete block design (RCBD), is perhaps the most widely used design of all experimental designs, mainly for its simplicity, flexibility and validity. The main objective of the design is either to control, remove or reduce the variability due to heterogeneity in experimental material, which is quite irrelevant for treatment comparisons. In CRD, the error variability as estimated from the residual sum of squares will be considerably high. The RCBD is a device introduced in this design for separating out the variability due to heterogeneity among experimental units from the variability due to all extraneous sources, thus purifying in one sense. The design is especially suitable for field experiments where the number of treatments is not large and the experimental area has a predictable productivity gradient.The primary distinguishing feature of the RCBD is the presence of blocks of equal size, each of which contains all the treatments.

Introduction In many of the experimental designs considered previously like CRD and RCBD ,the main feature was to compare varieties,methods of cultivation,different kinds of fertilizers, etc. In general, in such experiments, only one factor is allowed to vary at a time, while all other factors known to affect the response are generally controlled. Such experiments are called ‘Simple’, or ‘Classical’ experiments. Biological organisms are simultaneously exposed to many growth factors during their lifetime. Because an organism’s response to any simple factor may vary with the level of the other factors, simple factor experiments are often criticized for their narrowness. Indeed, the result of a single-factor experiment is, strictly speaking, applicable only to the particular level in which the other factors were maintained in the trial. But there are numerous occasions, especially in agriculture and animal experiments, where the main interest of an investigator does not lie with the effect of a single factor at a time but lies with simultaneous influence of many factors. For example, the interest of an investigator may not be in merely knowing the effect of spacing on the yield of a certain crop, but his interest would lie in finding out the simultaneous effect of both spacing and method of fertilizer application, that is, whether along with spacing, split dose of fertilizer at periodical intervals would be more beneficial than applying full recommended dose at one time. In order to answer this and many other questions, there are some special experiments called factorial experiments.

Introduction The split-plot design is specifically suited for a two-factor experiment that has more treatments than can be accommodated by a RCBD. In a split-plot design,one of the factors is assigned to the main plot. The assigned factor is called the main-plot factor. The main plot is divied into subplots to which the second factor ,called the subplot factor, is assigned. Thus, each mainplot becomes a block for the subplot treatments (i.e the levels of the subplot factor). The split-plot design is profitably employed under the following situations. An experimenter may have already possessed some knowledge about the main effect of a certain factor so that his interest lies in estimating the interaction effect of this factor with some other more important factor in whose main effect also he is equally interested. In either case he wants to estimate the latter two effects with greater precision for which he is prepared to sacrifice the precision in estimating the first mainfactor. The second common type of situation when the split-plot design is automatically suggestive is the difficulties in the execution of other design. For instance, if ploughing is one of the factors of interest,then one cannot have different depths of ploughing in different plots scattered randomly apart. Similarly, if different levels of irrigation are chosen,then random distribution of these levels in an ordinary experiment may involve difficulties in the practical execution of such plans. This difficulty may,however, be offset partly by having strips of plots (in one unit of block,say), all treated alike ;receiving the same level of irrigation with different depths of ploughing; or different levels of irrigation with same depth of ploughing. Different levels of irrigation or different depths of ploughing can,however,be randomly distributed over such strips. The third situation in which the experimenter may adopt a split-plot design is this. A particular factor (factors) under investigation exhibits large treatment differences which can be easily detected even in bigger plots,while a second factor (factors) requires comparatively smaller plots to detect smaller treatment differences with higher precision. Then the first factor will be tested in whole plots and the second factor whose treatment differences are small will be tested in small plots or subplots.

The split-split plot design is an extension of the split-plot design to accommodate a third factor .It is uniquely suited for a three –factor experiment where three different levels of precision are desired for the various effects. Each level of precision is assigned to the effects associated with each of the three factors. This design is characterized by two important features : There are three plot sizes corresponding to the three factors, namely, the largest plot (main plot) for the main-plot factor, the intermediate plot (sub-plot) for the subplot factor, and the smallest plot (sub-sub plot ) for the sub-subplot factor. There are three levels of precision, with the main-plot factor receiving the lowest degree of precision and the sub-subplot factor receiving the highest degree of precision. The method of analysis is entirely analogous to that for the case of 2 factors. The analysis consists of 3 parts, between whole plots, between split plots within whole plots,between split-split plots within split plots, each part being put on the basis of a split-split plot in order that the parts may be combined into one complete analysis.

Introduction As a chief feature of RCBD, formation of blocks is a necessity to achieve homogeneity of plots within each block so that an irrevelant source of variation existing to the error component could be removed for purposes of treatment comparisons. But quite frequently, in agricultural experiments, the fertility gradient varies in all directions so that the plots within a block in RCBD could still be heterogeneous. In order to control or remove this additional source of variation, one can think of extending the concept of blocking yet in another direction , orthogonal to the first direction, leading to a new design called Latin Square Design(LSD). In otherwords, if the blocking concept is one-dimensional in RCBD, it is two-dimensional in LSD. As a guiding principle for formation of two-dimensional blocks, it may be stated that each row-block and each column–block should remove as much variability as is possible from total variability so that error component becomes smaller. The chief restrictive features of LSD are:

Complete block designs become less efficient as the number of treatments increases,primarily because block size increases proportionally with the number of treatments,and the homogeneity of experimental plots within a large block is difficult to maintain.That is , the experimental error of a complete block design is generally expected to increase with the number of treatments. An alternative set of designs for single –factor experiments having a large number of treatments is the incomplete block design.As the name implies, each block in an incomplete block design does not contain all treatments and a reasonably small block size can be maintained even if the number of treatments is large.With smaller blocks,the homogeneity of experimental units in the same block is easier to maintain and a higher degree of precision can generally be expected.

Square-Root Transformation for Counts stabilizes the variance more effectively. Counts of rare events, such as number of defects or of accidents tend to be distributed approximately in poisson fashion.

In many experimental situations, it becomes necessary to repeat an experiment over time ( for a number of seasons or years ) and / or over space ( a number of places). This repetition ( or replication ) of the experiment broadens the scope of the experiment in the sense that our recommendations will be applicable for a number of seasons /or a number of places . A single experiment performed in one place for only one season will provide recommendations for that place and for that season . It may not be applicable to other places or for other seasons . In the case of agricultural experiments , for example , there may be present treatment x place interaction and/ or treatment x season interaction . So the results of a series of experiments , performed over different places for different seasons with the same set of treatments,will have wider applicability . We shall consider the simplest case of repetition of experiments of identical structure over a number of places ( it is the same for a number of seasons ). Let us consider randomized block experiments with t treatments in r blocks and conducted in p places. The analysis of the experiment in a place is based on the linear model . Before attempting a combined analysis for the p experiments ,it is necessary to perform the analysis for the p places separately and interpret the results separately .It may be of interest to find out whether differences among the treatments are the same in the different places so that a ‘ best ‘ treatment may be recommended for all the places, or whether different treatments are to be recommended for different places . The model for this combined analysis may be written as

This is an extension of the analysis of variance to cover the case where observations are taken on more than one variable from each experimental unit. Interest, however, centres on one of these (y, called the dependent variable) and the question is whether the variation of the dependent variable over the classes is due to class effects or due to its dependence on the other variables, (x’s, called the independent or concomitant variables ), which also vary from class to class. The analysis of covariance controls the experimental error by taking ito consideration the dependence of y on x.

The experimental design and analysis discussed in previous chapters refer essentially to experiments on annual crops. Even when these experiments are repeated in different years, this is done at a new site each year .The use of new crop production practices will generally change the important physical and biological factors of the environment. This process of change can take many years. Generally, the productivity of some types of technology must be evaluated by a series of trials repeated over time. Such trials are generally referred to as long-term experiments. Long-term experiments are those which are continued on the same set of plots over a series of years with a pre-planned sequence of treatments or crops or both. Treatments may be applied every year or periodically. The same crop may be repeated year after year or once planted may continue for several years as perennial trees ,or a sequence of crops may be grown in rotation as for instance rice-rice or rice-wheat (1 year sequence with 2 crops per year ) or sorghum,groundnut,cotton and wheat, each crop cycle being completed in four years. The main features of long-term experiments are as follows :

References Federer, W.T. 1968 Experimental Design Theory and Application. Oxford & IBH Publishing co. New Delhi. Gomez, Kwanchai A and Gomez Arturo A. 1984. Statistical Procedures For Agricultural Research John Wiley and Sons. New York. Goon, A.M., Gupta, M.K. and Das Gupta, B. 1976. Fundamentals of Statistics. Volume two. The World Press Private Ltd. Calcutta.