Factorial Treatment Combinations Research Articles

Boron, Calcium, and Potassium Interactions in Spanish Peanuts

AbstractExperiments were conducted in the field and greenhouse to evaluate interactions between application of B, Ca (gypsum) and K upon yield and quality of ‘Comet’ variety Spanish peanuts (Arachis hypogaea). In the greenhouse, using Eufaula fine sand, peanuts received factorial treatment combinations of three rates of B (0, 0.25, and 0.50 ppm), three rates of Ca (0, 88, and 176 ppm as gypsum), and three rates of K (25, 50, and 100 ppm). Factorial treatment combinations of three rates of B (0, 0.56, and 1.12 kg/ha), and three rates of Ca (0, 65, and 130 kg as gypsum/ha) were applied under field conditions at three locations.Under greenhouse conditions, significant interactions between the effects of B and Ca on growth and quality were observed. High Ca levels in the absence of applied B resulted in increased yields, lower seed quality (percent sound mature kernels), and extremely large increases in amounts of internal damage (due to B deficiency). No significant difference in yield response to B and Ca applications were found under field conditions. Peanuts showed a significant B by K interaction. Application of B with rates of potassium resulted in small increases in yield, except at the highest B and K levels where yields were reduced.

Response Surface Fitting Using a Generalization of the Bradley-Terry Paired Comparison Model

SUMMARY A paired comparison technique is presented for fitting response surfaces. This is especially useful when subjective responses are involved, where it is often difficult to justify the basic assumptions of the classical procedure. This report discusses aspects of the estimation of parameters and their properties, tests of relevant hypotheses and the selection of experimental designs. The method is applied to an example in food testing. PAIRED comparison techniques have been used for many years for the subjective comparison of several treatments (David, 1963), and extensions are available for the comparison of factorial treatment combinations. This report presents a further extension of the paired comparison method, the analogue of multiple regression analysis. Although it is possible to formulate a general paired comparison regression model this report will concentrate on the analogue of multiple linear regression mainly because of its application to response surface fitting. In the field of sensory assessment, it is frequently required to determine the effect of varying product composition. The classical response surface approach is useful here, despite many of its basic assumptions being invalid in the sensory situation, but occasionally it becomes virtually impossible to implement the classical method, particularly when carry-over and sensory organ fatigue effects are present. The degree to which these affect the results depends on the sense used, taste and smell being usually the more prone. Classical response surface designs, even with blocking, frequently require more units than may be reliably assessed in one sensory assessment session. Further, the assessors may not, due to the above effects-and others-be able reliably to score their impressions on a continuous scale. If any of these factors is present, the method of paired comparisons becomes extremely useful. A number of models have been proposed for the analysis of paired comparison data in the analogue of one-way analysis of variance. The specific analogue one-way ANOVA paired comparison model which this paper adapts to analogue multiple regression is that due to Rao and Kupper (1967), itself a generalization of the Bradley-Terry model (1952). A parallel generalization of the Bradley-Terry model, due to Davidson (1970), could also be extended for the present purpose in an exactly similar manner.

Attracting Honeybees to Alfalfa with Citral, Geraniol and Anise

SummaryWhen plots of alfalfa (Medicago sativa) were sprayed with citral, geraniol, anise, and sucrose in factorial treatment-combinations, plots treated with two or three scents in sucrose solution were more attractive to honeybees than plots treated with only one scent. Geraniol was generally more attractive than citral when it was applied in a solution containing 5 or 10% sucrose, but less attractive than citral when applied in water. The synergistic effect between citral and geraniol was no greater than that between citral and anise or geraniol and anise.

A Comparison of Injected Anhydrous Ammonia into Bermudagrass Sod Compared to Topdressed Applications of Urea and Ammonium Nitrate

AbstractSoil fertility experiments, primarily concerned with the evaluation of anhydrous NH3 as a source of N for bermudagrass, Cynodon dactylon (L.), when applied with the rolling coulter‐type applicator, were conducted on Parsons silt loam and Port sandy loam soils. Two factorial treatment combinations were used. The first consisted of four rates of N applied as (NH2)2CO, NH4NO3 and anhydrous NH3. The second consisted of complete factorial combinations of four rates of N as anhydrous NH3 and three spacings of the NH3 applicator knife.In general, yields were equal for all sources and all clippings at lower rates of N application. At the higher rates of N application, anhydrous NH3 produced lower yields in the first clipping and higher yields in succeeding clippings. This lag in response from anhydrous NH3 was attributed to sod burn from escaping ammonia. Reducing the applicator knife spacing tended to increase NH3 retention, decrease the sod burn, and increase the NH3 efficiency at the high rates of N application.

Incomplete Split Plot Designs

A split plot design for the factorial treatment combinations of factors A and B with t and s levels respectively, consists of wholeplots made up of s subplots or experimental units with each level of A applied to r wholeplots and the levels of B applied to the s subplots within each wholeplot. Often this process increases the amount of information for the estimates of the main effect B and the interaction A X B at the expense of information for the main effect A. There may arise situations where the number of subplots in each wholeplot is restricted to say, kc < s. An incomplete split plot design can now be set up with each level of factor A applied to r wholeplots and the s levels of factor B arranged in a balanced incomplete block design within each level of factor A by considering the wholeplots as blocks. That is, on a particular level of factor A each level of factor B occurs R times and each pair of levels of factor B occur in the same wholeplot X times, where R = rk/s and X = R(k l)/(s 1). Here r must be chosen so that the design exists and in some cases this may be so large that it will be impractical. Throughout this paper the wholeplot design will be considered as a conmpletely randomized design. However, it will sometimes be possible to have block designs for the wholeplots, but these will not be considered here.

Growth and chemical composition of Townsville lucerne (Stylosanthes humilis). 2. Chemical composition, with special reference to cations, as affected by the principal constituent elements of molybdenized superphosphate

The chemical composition of Stylosanthes humilis H.B.K. in response to a factorial combination of phosphorus, sulphur, molybdenum, and calcium carbonate treatments was studied with special reference to cation composition and protein synthesis. The aim was to assess the potential of S. humilis to adapt to a wide range of nutrient treatments. Protein synthesis was enhanced by high phosphorus-high sulphur treatments, and was accompanied by a low soluble : residual nitrogen ratio. This occurred despite the development of an apparent potassium shortage at this treatment combination. High phosphorus-high sulphur plants had low potassium : sodium ratios, and, although large, their potassium content was below critical levels (9-14 m.-equiv. per 100g). Potassium was partially substituted for by sodium and to a lesser degree by magnesium under these conditions. Although high protein levels were achieved, the full potential for protein synthesis did not seem to be attained by reason of the potassium shortage which developed. S. humilis was able to grow and fix significant quantities of nitrogen that bore a close relation at all levels to the wide range in chemical composition and yield response that developed with treatment.

The factorial combination of treatments in rotation experiments

The paper deals with points arising in the design of factorial schemes for incorporating fertilizer and other treatments in rotation experiments. These schemes make extensive use of fractional replication and confounding, including split-plot confounding. They are complicated by the need to apply some treatments cumulatively and others with levels changed between plots in successive cycles so as to minimize errors due to constant plot differences.Design keys are used to identify the treatment main effects and rotation differences with particular plot contrasts. The keys exhibit the structure of a design more compactly and clearly than a complete enumeration of identities and confounded contrasts and enable treatment combinations to be generated systematically.

The effect of land treatment in the preceding wet season on yield of linseed in the Ord River Valley

Two experiments were carried out at Kimberley Research Station, W.A., in 1960-61 and 1961-62, to test the effects of treatment of land in the wet season prior to growing a dry season irrigated linseed crop. In both experiments, the lowest linseed yields were obtained after a resting fallow, when the land remained uncultivated over the whole wet season, and high yields were obtained after a clean fallow, involving repeated cultivations. However, in the second experiment, equally high yields were obtained after a single early wet season ploughing. Factorial combinations of herbicide and nitrogen fertilizer treatments were superimposed. From the interactions it was concluded that, in the first experiment, the beneficial effects of wet season cultivation were the result of increased available soil nitrogen supply, and, in the second experiment, of increased nitrogen supply and reduction in weed infestation.

Applications of the calculus of factorial arrangements: I. Block and direct product designs

Abstract : This paper deals with some applications of a general theory for the analysis of factorial experiments as reported by the authors in the June 1962 issue of the Annals of Mathematical Statistics. General expressions are given for the usual quantities associated with the analysis of variance for the cases where simple treatments or factorial treatment-combinations are applied to Randomized Blocks, Balanced Incomplete Blocks, Group Divisible designs, and a wide class of Kronecker Product designs. The main point of the new theory is that, for a wide class of the more practical designs, the complete analysis can be carried out almost by inspection of the normal equations, with no requirement for inverting the normal equations. The complete version of this paper is published in BIOMETRIKA, Vol. 50, Parts 1 and 2, June 1963.

Factorial Treatments in Rectangular Lattice Designs

Abstract This paper reports how factorial treatment combinations may be used in rectangular lattice designs. The analysis of variance is obtained both with and without recovery of inter-block information. Explicit formulae are presented for the variances and covariances of the treatment effects as well as single degree-of-freedom comparisons. In fact, this paper extends all the uses of factorial treatment combinations in complete block designs to rectangular lattice designs.

Factorial Treatments in Rectangular Lattice Designs

Abstract This paper reports how factorial treatment combinations may be used in rectangular lattice designs. The analysis of variance is obtained both with and without recovery of inter-block information. Explicit formulae are presented for the variances and covariances of the treatment effects as well as single degree-of-freedom comparisons. In fact, this paper extends all the uses of factorial treatment combinations in complete block designs to rectangular lattice designs.

Examples of Intra-Block Analysis for Factorials in Group Divisible, Partially Balanced, Incomplete Block Designs

Incomplete block designs were developed to accommodate experiments wherein it is necessary that the lumber of experimental units per block be less than the number of 'treatments. These designs are used extensively in many fields of research. Factorial treatment combinations were introduced so that the effects of several variables, together with their interactions, may be studied in a single experimeent. Factorial treatment combinations are being used in many fields of research. Kramer [1] and Kramer and Bradley [2] have presented the theory allowing one to combine these two important concepts, incomplete blocks and factorials, to increase the utility of both. Partially balanced incomplete block designs were first discussed by Bose and Nair [8] and these desigfns have proved to be very useful. Bose and Shimamoto [9] later classified partially balanced incomplete block designs and, since then, Bose, Clatworthy, and Shrikhande [3] have provided a catalogue of such designs with two associate classes. Designs are given in the catalogue for block sizes 3 < k < 10 and replications r < 10 and when E1 and E2 , efficiency factors that will be further defined later, are not too different. Group divisible designs form a large and important class of partially balanced, incomplete block designs with two associate classes. In the catalogue, the parameters for each design, the association matrices, and block layouts are given. Clatworthy [4] has considered designs with k = 2 and r < 10

Intra-Block Analysis for Factorials in Two-Associate Class Group Divisible Designs

Group divisible incomplete block designs form an important class of incomplete block designs useful in a wide variety of experimental situations. Their properties, construction, and analysis have been thoroughly discussed in statistical literature, and we cite only several recent references [1], [2], and [3] to work of Bose and his co-workers dealing with partially balanced designs with two associate classes with which we shall be concerned. The utility of incomplete block designs would be increased with means of incorporating factorial treatment combinations in them. The use of factorials is widespread and stimulated by the concepts of confounding, partial confounding, and fractional replication. A mathematical summary on factorials is given by Kempthorne [4]. Kramer and Bradley [5] considered factorials in near-balance incomplete block designs, and here we generalize to the wider class of group divisible designs with two associate classes. Harshbarger [6] used a $2^3$ factorial in a Latinized rectangular lattice and this seems to be the first use of a factorial in a partially balanced incomplete block design. We obtain the intra-block analysis of variance for two-associate class group divisible designs with the adjusted treatment sum of squares in a modified form that more clearly indicates the structure of that quantity. Factorial treatment combinations are then identified with basic treatments through the association scheme of a design. This identification is effected in such a way that the factors are divided into two groups. For example, the design for 18 treatments (see [2], Design S60), divisible into six groups of three, in blocks of six, treatments replicated five times, can be adapted to a $6 \times 3$ factorial scheme; by regarding the six groups as made up of a $2 \times 3$ classification, the same design can be used for a $2 \times 3^2$ factorial scheme. Single-degree-of-freedom comparisons are obtainable in much the usual way and use of fractional replication, essentially within the groups of factors, is possible. The analyses for factorials depend on the estimators of basic treatment effects. We are not concerned with the construction of two-associate class group divisible designs and all known such designs for which $r \leqq 10, 3 \leqq k \leqq 10$, where $r$ is the number of replications and $k$ is the number of plots per block, are given in [7].

A Rankit Analysis of Paired Comparisons for Measuring the Effect of Sprays on Flavor

Experiments on consumer preference have taken many forms. In that known as paired comparisons, the several treatments or stimuli are compared in pairs, each treatment appearing with every other treatment in the same pair. It is an especially appropriate design for testing the effect upon a food of chemically different pesticides. These may produce qualitative differences in flavor which the subject finds easier to compare in pairs. Usually he is asked to report a simple preference for one of the two samples in each pair. Scheffe [1952] has extended this procedure by asking each subject to note the degree of his preference within each pair as well as its direction. A recent experiment of this type concerned the relative palatability of apples sprayed with two different insecticides and with two different fungicides in a 2 X 2 factorial design. In its analysis, several statistical methods were to be compared in respect to their sensitivity, consistency and computational requirements. Among them were the Mosteller [1951a,b] modification of the original proposal by Thurstone, the Bradley-Terry [1952a,b; 1953] analysis, and the Scheffe [1952] technique. In the course of the study, a new approach was developed which shows particular promise. It is based upon the mean normal deviate, tabled by Fisher and Yates [1953] and called the by Ipsen and Jerne [1944]. It answered several questions which we could not otherwise resolve so readily, if at all. We will describe this rankit analysis in its present application and compare it with alternative procedures. The experiment had several objectives. One was to arrange the treatments on a linear scale, spaced so as to reflect the average degree of preference expressed by the tasters. Criteria were needed for judging the significance on this scale of effects associated with factorial combinations of the individual treatments. The scale had to be validated by a test of its additivity or subtractivity. If, for example, B was preferred to