Post-Stratified Inference for Pairwise Treatment Comparisons in Randomized Block Designs

One of the most popular designs in experimental work is the randomized block. These designs can be put into three broad classes viz. complete block design, balanced incomplete block design, and the partially balanced incomplete block design. These designs are all special cases of the general two way classification with unequal numbers in the subclasses, but since the analysis of this general classification is quite complex, these special cases have evolved which are adequate to fit most needs and the analysis of these special designs is relatively easy. [1], [2], [6], [8]. However, most of the block designs considered to date have one feature in common--they require each block to contain an equal number of experimental units. The exceptions are given in [9], [10], where designs are considered in which the number of experimental units in blocks differ by one. The purpose of this paper is to extend the randomized block design to include the case where all blocks do not contain the same number of experimental units. We have called this the staircase design. Suppose an experimenter, wishing to run an experiment using $N$ treatments, decides to use a randomized block design, but after arranging his material into homogeneous groups he finds that he has blocks available which have varying number of experimental units. The experimenter has various courses open to him: (1) If enough blocks are available with $N$ or more experimental units he can discard the extra units in these blocks, discard all the blocks which have less than $N$ units, and use a randomized complete block design; (2) He can discard units in the blocks until he has enough units and blocks for a balanced incomplete block or a partially balanced incomplete block design; (3) He can use all the experimental units and use the staircase design proposed in this paper. For example, if an experimenter has $N$ treatments with which he wishes to experiment using a randomized block design, and if he has blocks of unequal size, then he must rank his $N$ treatments in the order of their importance, i.e., $T_1, T_2, \\cdots, T_N,$ where he considers $T_1$ the most important and $T_N$ the least important. Now suppose he has at his disposal $b_1$ blocks which each contain $N$ experimental units. Then all $N$ treatments are randomized in each of the $b_1$ blocks. Suppose further that he has $b_2$ blocks which each contain $N_1$ experimental units $(N_1 < N).$ Then the first $N_1$ treatments are arranged at random in each of the $b_2$ blocks. This process is continued until all the blocks are used. A particular example where this would be useful is an experiment involving animals as experimental units where a block consists of litter mates. Let us suppose that we have two litters of size seven, three of size five, and one of size four. Using the staircase design we can include seven treatments and still have the four we are most interested in replicated six times.

Durian is a typical tropical plant that has high economic value to increase farmers' income, foreign exchange, and agribusiness needs. This study aims to determine the effect and interaction between grafting methods and the age of different rootstocks on the success of durian propagation. This study used a factorial randomized block design (RBD). The first factor consisted of 2 levels and the second factor consisted of 3 levels which were repeated 4 times. Based on the factorial randomized completely block design (RCBD) experimental design, there were 6 treatment combinations with 4 repetitions, so that 24 experimental units were obtained, in each experimental unit there were 6 durian plants and 3 samples were taken, so that the total number of plants planted was 144 durians. plant. Treatment of different grafting methods had a significant effect on the parameters of shoot emergence and leaf emergence where the insertion grafting method gave the best results when shoots appeared at 21.25HSG and leaves appeared at 28,25 HSG. Treatment of differences in the age of the rootstock had a significant effect on the parameters of the percentage of live grafting, the percentage of finished grafting, shoot length, number of leaves, leaf length, leaf width, and leaf area at B3 level (5 months old) with the best value of 85 respectively. 39%, 72.84%, 13.01cm, 4.83 strands, 11.92 cm, 4.52 cm and 4.52 cm2.

Design and analysis of efficacy evaluation trials

Factorial experiment is an experiment where is in a condition (experiment unit) were attempted simultaneously from several single experiment. Two-factor factorial experiment with the basic design CRBD (Completely Randomized Block Design) is used to assess the interaction of genotype and environment on multi-location trials. The analysis can be applied in multi-location trials is AMMI analysis (additive main effects and multiplicative interaction). AMMI analysis in the calculations using analysis of variance in a factorial experiment to test the effect of the interaction and Principal Component Analysis (PCA) to elucidate the effect of the interaction with the interpretation of the results using the biplot-AMMI. Based on research with seven genotypes of rice (S382b-2-2-3, 3-2-3-1 S2389d-, S24871-65-4, S2824-1d-6, S2945f-59, Poso, and C22) and four locations (Sukamandi 94, Batang 94, Taman Bogo 94, and Garut 94) there is the influence of genotype, location, and interaction with genotype and location on rice production. Obtained three Principal Component Interactions (KUI1, KUI2 and KUI3) with the contribution of diversity respectively 78.29%, 13.94% and 7.77%. Interpretation of the AMMI Biplot is obtained genotype 1 (S382b-2-2-3) very suitable to be planted in a location 4 (Garut 94), genotype 2 (S2389d-3-2-3-1) very suitable to be planted in a location 3 (Taman Bogo 94), genotype 3 (S24871-65-4) is more suitable to be planted in locations 2 (Batang 94), genotype 4 (S2824-1d-6) are very suitable to be planted in a location 4 (Garut 94), genotype 5 (S2945f-59) is more suitable to be planted in locations 2 (Batang 94), genotype 6 (Poso) very suitable to be planted in a location 1 (Sukamandi 94) and genotype 7 (C22) is very suitable to be planted in locations 2 (Batang 94). Keywords: Factorial Experiment, CRBD, AMMI, Analysis of Variance, PCA, Biplot

Sample size and genetic divergence: a principal component analysis for soybean traits

Variability among experimental plots may be a relevant problem in field genotype experiments, especially when a large number of entries are involved. Four field trials on 24 durum wheat genotypes were conducted in 2013/14 in order to evaluate the efficiency of Incomplete Block, Alpha and Augmented designs in comparison with the traditional Randomized Complete Block Design (RCBD). The results showed that the RCBD can be replaced by an Alpha design, which provides better control of variability among the experimental units when the number of treatments to be tested in an experiment exceeds twenty. The ranking of the genotypes across the four designs was not constant.

A field experiment was carried out during the agricultural season (2021-2022) in Al-Samawah district in the Um Al-Akf region, which is (5 km from the center of Al-Muthanna Governorate). The aim of the study was to investigate the effect of tillage systems and boron spraying on some physical properties of the soil and growth and yield of wheat crop .The experiment was carried out according to the Split Block Design method, using the randomized complete block design (R.C.B.D), and with three replications. The experiment contained (36) experimental units, each area of which was 4.0 m2.Each experimental unit contained (10) lines, the Distance between one line and another was (20) cm. The experiment included two factors, the first was the tillage systems (no tillage ; reduce tillage at a depth of (10 cm) and conventional tillage at a depth of (20 cm) which occupied the vertical distribution. The second factor was the spraying of boron (0, 50, 100 and 150) mg B L-1, which occupied the horizontal distribution. Boron was sprayed at the stage of 75% flowering, and on 18/11/2021 wheat cultivar (Ibaa 99) was planted. The results of the study indicated that the conventional tillage system was significantly superior in reducing the value of bulk density, as it gave the lowest average of 1.41 mg m-3, and was also superior in increasing porosity by giving it the highest average of 46.64 %. While the no-till system was significantly superior in the characteristics of plant height, number of tillers, number of grains per spike, and grain yield, as it gave the highest averages (79.66 cm, 279.1 tillers m-2, 55.95 grains spike-1, and 3.540 ton h-1) respectively. While the Reduce tillage system was significantly superior. in the characteristic of harvest index by giving it the highest average amounted to 43.57%. As for the effect of boron, it was significant in most of the characteristics of the yield and its components, as the concentration of 150 mg B L-1 was significantly superior in the characteristic of the number of grains per spike 60.66 grains spike-1, the weight of 1000 grain 36.30 g, and the yield 3.498 ton h-1, the treatment of overlapping (Reduce tillage x 100 mg B L-1) recorded the highest average weight of 1000 grains was 38.03g

The agricultural field experiments play an important role for selection and screening of varieties. In Pakistan, these experiments are usually conducted in randomized complete block design (RCBD). It is very difficult to maintain the homogeneity condition when number of varieties exceeds eight or ten in a block. Due to this reason, in scientifically advance countries, RCBD are criticized for large field experiments for large number of varieties and varietal trials are conducted using generalized lattice design called Alpha designs. Nine maize varietal trials were conducted at Maize and Millet Research Institute (MMRI), Yousafwala (Sahiwal, Pakistan) during the year 2004-07 using alpha design. Data from each trial was analyzed using Alpha Lattice and RCBD and the efficiency was calculated using Yates [30] method. The average calculated efficiency was 114% in favour of Alpha Lattice. The detailed examination suggested that alpha lattice design is more effective. Ranking of genotypes in all trials were performed using both designs and differences were observed in ranks. Since the changeover to alpha lattice design did not require major inputs or changes in present field layout except changes in the randomization plan and decreasing the size of block therefore use of alpha lattice design for varietal trials in Pakistan in different crops like maize, wheat, rice etc., is strongly recommended.

The study was conducted in one of the fields affiliated to the Agricultural Research Station at the University of Basra, Karmat Ali site. The agricultural field experiment was carried out by dividing the field land according to the Randomized Complete Block Design (RCBD) in a one-time split plots design, where the main plots represent smoothing machines and sub-plots, the levels of organic residues and three replicators. The soil was plowed with a moldboard plow at a depth of 25-30 cm, and the organic residues were mixed with the soil by use of secondary tillage equipment. While the area of the experimental unit was (150 * 250) cm2, a distance of 2m was left between the experimental sectors and units to ensure that there was no overlap between the treatments. Seeds of oats were planted for all experimental units after the implementation of the treatments and at a seeding rate according to the recommendation followed in the region. Soil characteristics are measured for depths (0-15, 15-30) cm and for two periods (beginning and end of the plant growing season), and plant growth and yield indicators are measured at the end of the growing season. The aim of the research is to study the effect of adding levels of manure residues mixed with the surface layer in different ways on soil characteristics at the beginning and end of the oat crop growing season. The results of the experiment indicate the superiority of the spring cultivator and the percentage of adding organic residues (40 tons. Ha-1) over the rest of the machines and fertilization treatments by achieving the lowest apparent density at the beginning and end of the oats growing season, which amounted to 0.992 and 1.113 Mg m-3, respectively. In addition, they have achieved the highest the percentage of total porosity was 62.56 and 57.97 %, respectively. It also gave the highest weighted average diameter, which reached 0.859 and 0.688 mm, respectively. In addition, it gave the lowest soil penetration resistance, which reached 0.887 and 1.130 MPa, respectively.

Rumen’s animal waste is usually abundant in quantity. The rumen is not only a store house of microbes that assist in the digestion of roughage in ruminant animals but it is also an important source of nutrient for crop plant. This study aimed to determine the effect of cow’s rumen waste on eggplant crop production. The study consisted of a single factor, the application of cow’s rumen contents, with 6 treatment levels, namely: R0 = Control (without giving cow's rumen); R1 = 5 tons cow’s rumen per hectare (3.6 kg per plot); R2 = 10 tons of cow's rumen per hectare (7.2 kg per plot); R3 = 15 tons of cow's rumen per hectare (10.8 kg per plot); R4 = 20 tons of cow's rumen per hectare (14.4 kg per plot); R5 = 25 tons of cow’s rumen per hectare (18 kg per plot); and R6 = 30 tons of cow’s rumen per hectare (21.6 kg per plot). The study was based on Complete Randomized Block Design (CRBD), which consisted of three blocks, so overall consisted of 21 experimental units. The results showed that the application of cow's rumen waste gave a very significant effect on the weight of eggplant fresh fruit, fruit diameter, fruit number, and gave significant effect on fruit length. The application of cow’s rumen with a dose of 25 tons per hectare (18 kg per plot) resulted in the highest agronomic results for eggplant production, which was 28.85 tons per hectare, or equivalent to 20.77 kg per plot. But economically, rumen dose of 15 tons per hectare (10.8 kg per plot) has given the best results, with production per hectare of 21:39 tons, or equivalent to 15.4 kg per plot.

A class of locally most powerful linear rank tests for no treatment effect for various randomized block designs are presented. Closed form expressions of the test statistic are derived for the complete block design, the balanced incomplete block design and a special design in which there are missing observations on one treatment only. In addition, a test statistic is UCLiVCU LUL LUC unbalanced incomplete block design. The maximin efficiency robust test is given for a randomized block design* This test maximizes the minimum Pitman efficiency over a class of possible error distributions. Multiple comparison procedures are presented

A field experiment was conducted during the agricultural season 2023-2024 inside the greenhouse at the Second Agricultural Research Station of the College of Agriculture/AlMuthanna University in the Al-Bandar region. The aim of this study was to evaluate the response of eggplant (variety Barcelona hybrid) to fractionate of potassium fertilizer and humic acid on the phenotypic and nutritional qualities. Potassium was added according to the (N:120 P2O5:160 K:120) fertilizer recommendation and the fractionation required was as P1: Without fertilization, P2: Full recommendation for potassium in one does, P3: Full recommendation for potassium in two doses, P4: half recommendation for potassium and in one does, P5: half of the recommendation for fertilizer and in two doses. Humic acid was added by spraying on the plant and soil in three doses and the concentrations were as (0, 5, 10 ml liter-1 of water) and symbolized by the symbols (H1, H2, and H3) after three weeks for planting and sprayed on plants in the experimental unit. A factorial experiment was conducted using Random Complete Block Design (RCBD) with three replications and the indicators of study were averaged and compared using the Least Significant Difference (LSD) test at a probability level of 0.05. The results showed that a significant superiority of the full recommendation treatment of potassium in two doses and the treatment of spraying with humic at a concentration of 10 ml liter-1 separately in all the studied traits, and the plants ofthe interaction treatment (H3P3) in the characteristic of plant height, chlorophyll content (SPAD readings), Normalized Difference Vegetation Index (NDVI), and content of elements N, P and K in the plant, where it reached to 102.330 cm, 74.000, 0.901, 5.500%, 0.450%, and 3.780%, respectively

Several inputs are needed to increase the production of soybean in Ultisol. The Application of organic soil amendment will increase carbon, nutrients, and water stored in the soil. Besides, the soil structure becomes stable so that plants will be more tolerant of climate change. The higher organic C in the soil, the more fertile soils, the plants grow healthier and fertile, and then the emission of CO2 gas into the air can be suppressed. Therefore, this study aims to determine soil respiration (CO2) due to the application of in situ soil amendment and phosphate fertilizer. This research was conducted at the research station of BPTP Natar, South Lampung, from July to November 2019, arranged with random complete block design in two factorials. The first factor is the source of phosphate fertilizer is without P fertilizer, TSP fertilizer 200 kg ha-1, and rock phosphate 5 tons ha-1, and the second factor is the application of soil in situ amendments. i.e.: without the soil amendment, rice husk biochar 5 tons ha-1 (B1), organonitrofos compost 10 tons ha-1, and cow dung manure 10 tons ha-1. All experimental units were fertilized with urea at a dose of 50 kg ha-1 and KCl at a dose of 200 kg ha-1. The result showed that the highest soil respiration was obtained in the plot treated with rock phosphate and cow dung manure at the single factor. The combination of rock phosphate and cow dung manure resulted in the highest of soil respiration.

An experiment was conducted to determine water requirements of coriander Unapal Precoso related to fresh foliage yield and dry matter. Plants were sown in pots with 6 kg of soil and water depths of 140, 160, 200 (witness proposed by Vallejo y Estrada, 2004), 240 and 280 mm were applied. A random complete block design was used with 5 treatments and 4 replications, with 36 plants per experimental unit. Daily evapotranspiration measurements were made. Production of foliage and biomass was evaluated 35 days after sowing. Increased production of foliage was obtained with 200 mm of water depth, value associated with greater efficiency in water use (0.64 grams of dry matter per liter of water applied). The crop Kc was determined: 0.83 at the stage of germination, 1.12 in linear growth stage and 1.40 on the stage of forming flower stems and harvesting.

In the RCBD, the block system causes, for most applications, strong correlation between responses from experimental units in the same block. Therefore, the usual model with the usual assumptions about zi j. (i.i.d. N(0,σ2)) is not realistic. In this paper we study the consequences of more realistic assumptions about zi j. and verify that the usual F test, for testing treatment effects, can be used even if there is a constant covariance between responses from experimental units in the same block. When the variances associated with each treatment are not the same we give an approximate solution for testing treatment effects.