quasi-Latin Square Research Articles

Quasi-Latin designs

This paper gives a general method for constructing quasi-Latin square, quasi-Latin rectangle and extended quasi-Latin rectangle designs for symmetric factorial experiments. Two further methods are given for parameter values satisfying certain conditions. The construction of designs for a range of numbers of rows and columns is discussed so that the different construction techniques are covered. For some row and column combinations, different designs are compared. The construction of designs with rows and columns that are nested or contiguous is also discussed.

Grain Processing, Forage-to-Concentrate Ratio, and Forage Length Effects on Ruminal Nitrogen Degradation and Flows of Amino Acids to the Duodenum

The objectives of this study were to evaluate effects of dietary factors that alter ruminal fermentability on rumen N degradation, microbial protein synthesis, duodenal flows, and digestibility of amino acids (AA) in the intestines and the total tract. The experiment was a double 4×4 quasi-Latin square with a 23 factorial arrangement of treatments. The dietary factors were extent of barley grain processing, coarse (processing index; PI = 75.5%) or flat (PI = 60.2%); forage-to-concentrate (F:C) ratio, low (35:65) or high (55:45) on a DM basis; and forage particle length (FPL), long (7.59mm) or short (6.08mm). Eight lactating cows with ruminal and duodenal cannulas were offered ad libitum access to a total mixed diet. There were no significant interactions between dietary treatments for ruminal N degradation or its duodenal flow and digestibility in the intestines. Passage of microbial protein to the duodenum was improved with increased F:C ratio of the diet but was not affected by grain processing or FPL. Ruminal digestibility of N was increased with increased F:C ratio (49 vs. 60%) and with reduced FPL (59 vs. 50%). Increased grain processing improved N digestibility both in the intestine (15%) and in the total tract (8%). Reduction in the FPL of the diets reduced intestinal N digestion by 14% without affecting the N digestion in the total tract. Increased extent of grain processing tended to enhance duodenal flows of AA. In contrast, reducing FPL lowered flows of dietary AA to the duodenum because of lowered flows of feed plus endogenous N. Increased F:C ratio of the diet did not change the flow of total AA, but there was a reduced flow of dietary AA and increased flow of microbial AA. Flows of several individual AA were increased by feeding flatly rolled barley with limited effects of F:C ratio or FPL. An interaction between grain processing and FPL was detected for flows of some AA. Diets formulated with flatly rolled barley plus long FPL increased Arg, Thr, Asp, Glu, Ser, Tyr, and nonessential AA (NEAA) by more than 24%, compared with other combinations of grain processing and FPL. Digestibility of essential AA (EAA) in the intestine (68%) was higher than that of NEAA (63%), but digestion of total AA (65%) was similar to that of total N (66%). Digestibilities of individual AA in the intestine ranged from 46 to 77% and were generally improved with increased grain processing. However, effects of F:C ratio or FPL on digestion of AA were limited. These results indicate that manipulation of dairy cow diets can improve ruminal N degradation, microbial protein synthesis, flows of AA to the duodenum, and intestinal digestibility of AA. Combining dietary factors can be more beneficial than changing individual dietary factors for improving the delivery of AA to the small intestine.

Barley Processing, Forage:Concentrate, and Forage Length Effects on Chewing and Digesta Passage in Lactating Cows

Dietary factors that alter fermentability, NDF content, or particle size of the diet were evaluated for their effects on chewing behavior and distribution and passage of feed particles in the digestive tract of dairy cows. A double 4 × 4 quasi-Latin square design with a 23 factorial arrangement of treatments was used. The dietary factors were: extent of barley grain processing, coarse (1.60mm) or flat (1.36mm); forage-to-concentrate ratio (F:C), low (35:65) or high (55:45) (dry matter basis); and forage particle length, long (7.59mm) or short (6.08mm). Eight lactating cows with ruminal and duodenal cannulas were offered ad libitum access to total mixed diets. Chewing time, expressed as minutes per day or per kilogram of dry matter or neutral detergent fiber (NDF), was increased with high F:C diets due to increased eating and ruminating times but was decreased when expressed per kilogram of NDF intake from forage. The influence of forage particle length or grain processing on chewing activity was less pronounced than F:C ratio. Chewing activity was positively correlated to proportion of long forage particles in the diet but not to particle length of the diets. Influence of feed particle size on particle size distribution in different sites of the digestive tract was minimal. Particle size distributions of duodenal digesta and feces differed; the proportion of particles retained on the 3.35- or 1.18-mm screens was higher, but proportion of particles that passed through the 1.18-mm screen was lower in duodenal digesta than in feces. Relationships between chewing activities and ruminal pH or fractional passage rate of rumen contents were not significant. These results indicate that particle size of barley-based diets was not a reliable indicator of chewing activity. Forage particle size and NDF content of the diets were more reliable indicators of chewing activity than was the NDF content of forage. Fecal particle size was not an appropriate means of estimating the size of particles exiting the reticulorumen, at least for barley-based diets. Breakdown of coarse particles was necessary, but not a rate-limiting step for particles exiting the rumen. Passage rate of the rumen contents was affected by numerous factors including chewing activity.

Effects of Grain Processing, Forage to Concentrate Ratio, and Forage Particle Size on Rumen pH and Digestion by Dairy Cows

Dietary factors that alter the intake of effective fiber were evaluated for their effects on rumen fermentation, digestion, and milk production using a double 4 x 4 quasi-Latin square design with a 23 factorial arrangement of treatments. The dietary factors were extent of barley grain processing, coarse (1.60mm) or flat (1.36mm); forage-to-concentrate (F:C) ratio, low (35:65) or high (55:45) (dry matter basis); and forage particle length, long (7.59mm) or short (6.08mm). Eight lactating cows with ruminal and duodenal cannulas were offered ad libitum access to a total mixed diet and milked twice daily. Dry matter intake was increased by increasing the extent of grain processing. Mean rumen pH was lower for cows fed flatly rolled barley than for cows fed coarsely rolled barley, whereas F:C ratio or forage particle size had no effect on rumen pH. Rumen pH was not correlated with effective NDF intake but tended to be correlated with digestibility of starch in the rumen. Total tract digestibilities of dry matter, organic matter, starch, and neutral detergent fiber were increased by feeding flatly rolled barley or low F:C ratio diets. Milk yield and milk protein content were higher in cows fed flatly rolled barley or low F:C ratio diets. Milk fat content tended to increase with high F:C ratio or long forage particle length but was reduced by feeding flatly rolled barley. In this study, extent of grain processing and intake of ruminal available starch were the most influential factors affecting milk production. Reducing the ratio of F:C improved total digestion and actual milk production. Forage particle length had minimal impact on digestibility and milk production.

Effect of dietary factors on distribution and chemical composition of liquid- or solid-associated bacterial populations in the rumen of dairy cows.

Effects of dietary factors including kernel thickness of processed barley grain, ratio of forage to concentrate, and forage particle length on chemical composition of bacteria, bacterial colonization of feed particles and distribution in the rumen, and duodenal flow of bacteria in dairy cows were evaluated. The experiment was designed as a double 4 x 4 quasi-Latin square with a 2 x 2 x 2 factorial arrangement of treatments using eight lactating cows with ruminal and duodenal cannulas. Barley grain was steam-rolled to two thicknesses: coarse (1.60 mm) or flat (1.36 mm); ratio of forage:concentrate was low (35:65) or high (55:45) (DM basis); and forage particle length was long (7.59 mm) or short (6.08 mm). Cows were offered ad libitum access to a total mixed diet. Chemical composition was different (P < 0.01) between liquid-associated bacteria (LAB) and solid-associated bacteria (SAB). Reduced barley thickness increased (P < 0.05) N content and 15N enrichment, but a high ratio of forage:concentrate decreased (P < 0.01) 15N enrichment of both the LAB and SAB. Significant differences between AA composition of the LAB and SAB were observed for 15 out of 17 AA studied. Bacterial colonization was linearly increased (P < 0.01) from about 5 to 70% as particle length of rumen contents was reduced from 3.35 mm to 0.15 mm (sieve size). The degree of colonization on each fraction of the rumen particulate matter was only affected (P < 0.10) by the ratio of forage:concentrate, with consistently higher (P < 0.10) bacterial colonization noted for high than for low forage:concentrate diets. Of the total bacterial mass within the rumen, less than 20% was associated with the liquid and over 70% was associated with the small particles that passed through the 0.6-mm sieve. Although the bacterial pool in the rumen was lower (P < 0.04) when flatly rolled barley rather than coarsely rolled barley was fed, bacterial flow to the duodenum was greater (P < 0.10) with increasing ratio offorage:concentrate. The present results confirm the differences of chemical composition and biomass for LAB and SAB. Manipulation of dietary factors such as ratio of forage:concentrate have the potential to alter bacterial colonization of rumen particles and the relative proportion of LAB to SAB, which were positively correlated to bacterial flow to the duodenum.

Systematic Row-And-Column Designs Balanced for Low Order Polynomial Interactions between Rows and Columns

SUMMARY Systematic row-and-column designs intended to accommodate low order polynomial rows × columns interaction effects are discussed. Latin square, Graeco-Latin square and quasi-Latin square designs of size 2N × 2M and size 9 × 9 are examined by using plot pseudofactor methods. Latin square and Graeco-Latin square designs of size 5 × 5 and 7 × 7 are examined by using a published technique for the design of quantitative factorial experiments with a prime number of factor levels. A 6 × 6 Latin square design is constructed by trial and error. Finally, the control of positional effects by fitted quadratic response surfaces is discussed.

Effects of isoacids, urea and sulfur on ruminal fermentation in sheep fed pineapple tops

Eight Tabasco rams were used in a 2 3 factorial crossover experiment conducted in two 4 × 4 quasi-Latin squares to study effects of supplementary isoacids, urea and sulfur on ruminal fermentation of pineapple tops. ‘Isoacids’ (ISO) refers to an equal-weight mixture of isobutyrate, 2-methylbutyrate, isovalerate and valerate. Treatments in g kg BW −1 day −1 were: ISO, 0.07 and 0.14; urea, 0 and 0.43; sulfur (S), 0 and 0.086. After each weekly experimental period, the rate of ruminal acetate production was taken as a measure of ruminal fermentation. There was significant interaction only between ISO and S. Acetate production ranged from 0.17 to 0.44 mol h −1 per sheep for the eight treatment combinations. Acetate was increased ( P < 0.05) by 0.17 mol h −1 per sheep, or 81%, when ISO were fed at the high level in diets supplemented with S. When S was added to diets containing a high level of ISO, acetate production increased ( P < 0.01) by 0.14 mol h −1 per sheep, or 56%. When urea was added to high ISO and S, acetate production was 33% higher (0.33 vs. 0.44 mol h −1 per sheep). Optimum N:S ratio was 5:1, and this apparently was more important than absolute concentration of rumen ammonia. Ruminal fermentation of pineapple tops appears to be improved by supplementing the diet with 0.14 g ISO, 0.43 g urea and 0.086 g S per kg BW of sheep.

BASIC algorithms for random sampling and treatment randomization

Five BASIC programs to select random samples from populations or to randomize treatments are presented. Program 1 is used to obtain randomization of any number of treatments in an equal number of positions or test units for any number of replicates. Program 2 produces latin squares of any size for treatment randomization. Program 3 is used to obtain a specific number of randomly selected samples from a population without replacement. Program 4 produces quasi-latin squares that have treatments repeated equally in all rows and columns, with identical treatments either spaced or not. Program 5 can be used with any size grid to place 3–100 treatments in equal proportions and with spacing of identical treatments. Both programs 4 and 5 allow for horizontal and vertical separation between identical treatments at sampling places while still retaining the quality of randomness. These programs should facilitate random sampling and randomization procedures which are required to correctly analyze experiments by the methods of statistical probability.

Discussion of Professor Bartlett's Paper

Journal of the Royal Statistical Society: Series B (Methodological)Volume 40, Issue 2 p. 158-174 DiscussionFree Access Discussion of Professor Bartlett's Paper First published: 1978 https://doi.org/10.1111/j.2517-6161.1978.tb01658.xAboutPDF ToolsExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat References in the Discussion Bellman, R. A. (1960). Introduction to Matrix Analysis. New York: McGraw Hill. Berenblut, I. I. and Webb, G. I. (1974). Experimental design in the presence of autocorrelated errors. Biometrika, 61, 427– 437. Besag, J. (1975). Statistical analysis of non-lattice data. The Statistician, 24, 179– 195. Besag, J. (1977). Errors-in-variables estimation for Gaussian lattice schemes. J. R. Statist. Soc. B, 39, 73– 78. Booth, N. B. and Smith, A. F. M. (1976). Batch acceptance schemes based on an autoregressive prior. Biometrika, 63, 133– 136. Duby, C., Guyon, X. and Prum, B. (1977). The precision of different experimental designs for a random field. Biometrika, 64, 59– 66. Dyke, G. V. (1964). Restricted randomization for blocks of sixteen plots. J. Agric. Sci., 62, 215– 217. Fairfield Smith, H. (1938). An empirical law describing heterogeneity in the yields of agricultural crops. J. Agric. Sci., 28, 1– 23. Fasoulas, A. (1977). Field designs for genotypic evaluation and selection. Publication No. 7, Dept. of Gen. Plant Breeding, Aristotelian University of Thessaloniki, Greece. Federer, W. T. and Schlottfeldt, C. S. (1954). The use of covariance to control gradients in experiments. Biometrics, 10, 282– 290. Freeman, G. H. (1963). The combined effect of environmental and plant variation. Biometrics, 19, 273– 277. Grundy, P. M. and Healy, M. J. R. (1950). Restricted randomization and quasi-Latin squares. J. R. Statist. Soc. B, 12, 286– 291. Lindley, D. V. and Smith, A. F. M. (1972). Bayes estimates for the linear model (with Discussion). J. R. Statist. Soc. B, 34, 1– 41. Martin, R. J. (1977). Spatial models with applications in sampling and experimental design. Ph.D. Thesis, University of London. Matérn, B. (1970). Performance of various designs of field experiments as a factor influencing plot yields. In Proceedings of 3rd Conference of the Advisory Group of Forest Statisticians, pp. 119– 128. Paris: Institut National de la Recherche Agronomique. Matérn, B. (1972). Performance of various designs of field experiments when applied in random fields. I.U.F.R.O., 3rd Conference of the Advisory Group of Forest Statisticians, pp. 72– 73. Paris: Institut National de la Recherche Agronomique. Nelder, J. A. (1962). New kinds of systematic designs for spacing experiments. Biometrics, 18, 283– 307. Neyman, J. (1929). The theoretical basis of different methods of testing cereals. Part II. The method of parabolic curves. In Pedigree Seed Cultures, No. 4. Warsaw: K. Buszcynski & Sons Ltd. Olesen, K. (1976). A completely balanced polycross design. Euphytica, 25, 485– 488. Olesen, K. and Olesen, O. J. (1973). A polycross pattern formula. Euphytica, 22, 500– 502. Ord, J. K. (1975). Estimation methods for models of spatial interaction. J. Amer. Statist. Ass., 70, 120– 126. Pearce, S. C. (1976). An examination of Fairfield Smith's law of environmental variation. J. Agric. Sci., 87, 21– 24. Pearce, S. C. (1978). The control of environmental variation in some West Indian maize experiments. Trop. Agric., 55 (in press). Rives, M. (1969). Description de l'hétérogénéité des champs d'expérience. Biom. Z., 11, 113– 122. Särndal, C.-E. (1978). Design-based and model-based inference in survey sampling. Scand. J. Statist., 5, 27– 52. Seeley, J. (1971). Quadratic subspaces and completeness. Ann. Math. Statist., 42, 710– 721. Smith, A. F. M. (1973). Comment on a paper by Efron and Morris. J. R. Statist. Soc. B, 35, 410– 412. Yates, F. (1936). A new method of arranging variety trials involving a large number of varieties. J. Agric. Sci., 26, 426– 455. Yates, F. (1970). Experimental Designs. London: Griffin. Volume40, Issue21978Pages 158-174 This article also appears in:Discussion Papers ReferencesRelatedInformation

Effects of Mowing and Raking on Collembola

On a patch of prairie near Northfield, Minnesota, four treatment factors (early mowing and raking, late mowing, raking, and fertilizing) were each applied or not applied to plots of a quasi—Latin square. Five epigeic collembolan species (Entomobrya bicolor, Orchesella ainslei, Tomocerus flavescens, Lepidocyrtus cyaneus, Isotoma viridis) were shown to have noncongruent but overlappihg distributions on the plots. Early mowing and raking raised the number of E. bicolor, Arion fasciatus (a slug), Labidostommid mite, Lycosid spiders, and spiders in general, O. ainslei and L. cyaneus decreased, as did a species of Oribatid mite (family Belbidae). Late mowing decreased the numbers of E. bicolor, O. ainslei, I. viridis, and harvestmen but increased the number of spiders. An interaction between early mowing and raking and late mowing resulted in a decrease in E. bicolor and an increase in O. ainslei. Regression analysis indicated that slope (topography) was a good predictor of the numbers of E. bicolor. O. ainslei, and T. flavescens, positively correlated for the former, negatively for the latter two species. The number of I. viridis was positively correlated with the density of other species. The number of L. cyaneus had a high, but nonsignificant, correlation with the productivity biomass ratio. Total collembola had a significant negative correlation with standing crop (but the multiple correlation coefficient was not significant). It was hypothesized that the five collembola are competitors whose areas of residence expand and contract with seasonal climatic change because of competitive interaction and changes in their microclimate and environment.

Peat-sand substrates for plants grown in containers

The N, P, K, and Ca fertilizer requirements of a 75% sphagnum peat 25% sand mixture were studied using 34 quasi-latin square designs with tomato and antirrhinum as test plants.

Changes in Cryptozoan Numbers Due to Systematic Variation of Covering Boards

The area beneath small square boards and the boards themselves were varied in a systematic factorial experiment placed in a quasi—Latin square on a patch of prairie near Northfield, Minnesota, in the summer of 1961. As compared to small boards (8 x 8 in.), large boards (12 x 12 in.) had per board 5 more animals, one more kind, .5 more of a collembolan, .5 more Entomobrya color (collembola), 1.8 more Lasius niger (ant) and .9 more common crickets. White boards as compared with black boards had per board one more kind and 4.6 more Gastrocopta armifera (snail). The number of animals under boards on peaks was less per board than the number under boards on depressions by 1 spider, 1 lycosid, .25 orobatid mites, 1 beetle, 1.6 crickets, 2.3 Lasius niger and –.5 kinds. Redistribution of E. bicolor beneath the boards had no discernible effect on the numbers of this or any other species. Soaking of the underboard area was also without effect. Diversity (Margalef index) was significantly greater on peaks than on depressions. Regression analysis showed that numbers of collembola were significantly predicted by diversity and were correlated with numbers of kinds of other animals and the number of other animals. Few interactions of factors were observed.

A General Computer Programme for the Analysis of Factorial Experiments

A general programme for the analysis of factorial experiments is described. The programme was written in Extended Mercury Autocode for the I.C.T. Orion computer at Rothamsted. It can deal with experiments with up to seven factors, including those with complete or partial confounding and multiple splits, arranged in randomised blocks, Latin or quasi-Latin squares, or criss-cross designs. Partially confounded contrasts have to be specified by the user, but for the commoner types of design the correct arrangement of the analysis of variance is provided automatically. Properly adjusted tables of treatment means are produced and covariance and missing values are provided for. An outline of the least-square theory of confounded designs is appended. Improvements incorporated in a revised version of the programme are described in an addendum.

Restricted randomization for blocks of sixteen plots

Grundy &amp; Healy (1950) developed a method of ‘restricted randomization’ applicable to factorial experiments of 2n or 3n type laid out either in quasi-Latin squares of size 8 × 8 or 9 × 9, or in blocks of 8 or 9 plots. This method of randomization, while in no way affecting the validity of the normal analysis, prevents the appearance of designs with obviously undesirable features. Grundy &amp; Healy gave an example of a quasi-Latin square inwhich the main effect of one factor corresponded to the pattern:

The Analysis of a 3 x 6 Experiment Arranged in a Quasi-Latin Square

The Analysis of a 3 x 6 Experiment Arranged in a Quasi-Latin Square

Restricted Randomization and Quasi-Latin Squares

Restricted Randomization and Quasi-Latin Squares

The Biological Assay of Penicillin by a Modified Plate Method

Summary: The general form of assay for streptomycin described by Brownlee, Delves, Dorman, Green, Grenfell, Johnson &amp; Smith (1948), based on an 8 × 8 quasi-Latin square layout of cavities on a large agar-covered glass plate, in place of Petri dishes, can be adapted to penicillin. A new design, using the principle of double confounding, allows the estimation by the usual four-point assay, of seven (instead of three) unknown solutions per plate of 64 holes. In routine use this assay gives fiducial limits (P = 0·95) of about ± 9·6%, including dilution errors. The assay procedure previously described in this Journal (Brownlee et al. 1948) for streptomycin, is immediately adaptable to penicillin. For dealing with large numbers of fermenter samples in particular, we have found modifications desirable, which we describe in the present paper.

The Biological Assay of Streptomycin by a Modified Cylinder Plate Method

SUMMARY: The use of plate glass sheets carrying twenty-four cylinders in place of Petri dishes for the biological assay of streptomycin by the cylinder-plate method initially led to large assay errors. The substitution of cavities for cylinders, of a single layer of agar for two layers, one sterile and one inoculated, of the four-point assay design for ‘reading off the curve’, and of an optical projection method of measurement diminished the fiducial limits of error of an assay to 80-125% (P = 0·95). Over the range of concentrations 0·1-100·0 units streptomycin/ml. the deviations from linearity of the graph of zone diameter against the logarithm of the concentration were very small. Variations in thickness of the agar layer and in the intensity of inoculation had a marked effect on zone diameter. An 8×8 Latin square lay-out reduced the internal error of an assay to 91-110% (P = 0·95), and with this increased accuracy a bias in the results was revealed arising from the time interval between filling successive cavities on the plate. This time effect was eliminated by the adoption of an appropriate order of filling, with a quasi-Latin square lay-out. The form of the assay finally adopted had fiducial limits of error, estimated both internally and externally, of about 95-105% (P = 0·95).