Mathematics
Randomized Block Design
Randomized block design is a method used in experimental design to reduce variability and increase the precision of an experiment. It involves grouping experimental units into blocks based on similar characteristics, and then randomly assigning treatments within each block. This helps control for potential sources of variation, such as environmental factors or individual differences, leading to more reliable results.
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9 Key excerpts on "Randomized Block Design"
eBook - PDF- Rudolf J. Freund, William J. Wilson(Authors)
- 2003(Publication Date)
- Academic Press(Publisher)
As previously noted, this analysis assumes that there are no interactions among any of the sources of variation. It could, for example, be argued that different types of music may have different effects at different times of day, which constitutes an interaction that would invalidate this analysis. Unfortunately, there is no test for the hypothesis of no interaction using these data. The relative efficiency of the Latin square design may be computed using the approach outlined for the Randomized Block Design. These efficiencies can specify the gain in efficiency due to blocking by rows, or columns, or the entire Latin square. For details, see Kuehl (2000). ■ 10.5 Other Designs The randomized block and Latin square designs use the principle of blocking for the purpose of increasing the precision of the analysis. Other, more complex designs, such as the Graeco–Latin square design can be used to eliminate more than two blocking factors (see, for example, Montgomery, 1984, Chapter 5). Experimental designs can be used to accommodate more complex exper-imental situations such as factorial experiments. That is, the treatments in a Randomized Block Design may consist of all factor level combinations of a factorial experiment. This application is presented at the beginning of this section. Another application of experimental design occurs when experimental units contain subunits that are then used as observations. This so-called nested design is outlined later in this section. A factorial experiment requiring 10.5 Other Designs 481 experimental units or blocks of different sizes for different factors, called a “split plot” design, is also outlined in this section, along with other considera-tions of experimental design. Factorial Experiments in a Randomized Block Design At this point it may be difficult to differentiate the analysis of a Randomized Block Design and a factorial experiment because they both result in the same partitioning of the sum of squares.
eBook - PDF- John Lawson(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
CHAPTER 4 Randomized Block Designs 4.1 Introduction In order to eliminate as much of the natural variation as possible and increase the sensitivity of experiments, it would be advisable to choose the experimen-tal units for a study to be as homogeneous as possible. In mathematical terms this would reduce the variance, σ 2 , of the experimental error and increase the power for detecting treatment factor effects. On the other hand, most exper-imenters would like the conclusions of their work to have wide applicability. Consider the following example. An experimenter would like to compare sev-eral methods of aerobic exercise to see how they affect the stress and anxiety level of experimental subjects. Since there is wide variability in stress and anxiety levels in the general population, as measured by standardized test scores, it would be difficult to see any difference among various methods of exercise unless the subjects recruited to the study were a homogeneous group each similar in their level of stress. However, the experimenter would like to make general conclusions from his study to people of all stress levels in the general population. Blocking can be used in this situation to achieve both objectives. Blocking is the second technique that falls in the category of error control defined in Section 1.4. In a Randomized Block Design, a group of heterogeneous exper-imental units is used so that the conclusions can be more general; however, these heterogeneous experimental units are grouped into homogeneous sub-groups before they are randomly assigned to treatment factor levels. The act of grouping the experimental units together in homogeneous groups is called blocking.
eBook - ePubDesign of Experiments
A Realistic Approach
- Virgil L. Anderson, Robert A. McLean(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
Chapter 5 RANDOMIZED COMPLETE BLOCK DESIGN (RCBD)Basically, the randomized complete block design is a group of completely randomized designs (usually, however, each CRD has only one experimental unit per treatment combination). Ordinarily each member of the group contains a sufficient number of homogeneous experimental units to accommodate a complete set of treatment combinations. This collection of experimental units is referred to as a block . One major reason for blocking is that the experimenter does not have a sufficient number of homogeneous experimental units available to run a completely randomized design with several observations per treatment combination.A realistic situation where blocking would be required would be in an industrial type experiment where there is only sufficient time to run one set of treatment combinations per day. In such an experiment one would be blocking on time and the number of days that the experiment is carried out would be the number of blocks. This type of experiment points out another reason for blocking which is the technique that is utilized to expand the inference space. Even if it was possible to run a large number of each treatment combination on one day the experimenter might still choose to block over time so that he could make inferences over time rather than to just one particular day.If an experimenter were to run a RCBD and analyze it as a CRD, then any effects which should have been attributed to blocks would end up in the error term of the model. Thus another major reason for blocking is to remove a source of variation from the error. It should be observed that if the experimental unit variation among blocks is not larger than the variation within blocks, there is no reason to run a RCBD rather than a CRD.The group of blocks, which may be random or fixed, make up the design called the randomized complete block design (RCBD). These blocks are called “complete” because all treatments or treatment combinations appear in each block. As a result of the construction of this design, there is one restriction on the randomization in that the randomization of treatments onto the experimental units is carried out within each block separately, not over all the experimental units at one time as with a CRD.
eBook - PDFDesign and Analysis of Experiments
Classical and Regression Approaches with SAS
- Leonard C. Onyiah(Author)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
Designs with Randomization Restriction 279 This clearly indicates that in a very simple manner, paired comparison elim-inated the effect of a factor in which we had little interest, namely, the blocks, thus enabling us to study the differences in treatments (sole materials). In the design of experiments, one strategy used to advantage when the experi-mental material to be employed in our study is the concern, is to group our experimental units into homogeneous units (called blocks) and compare sev-eral treatments within the blocks. By doing so, we eliminate the differences between the blocks from our experimental error and increase the precision of our result. The other advantages of this design are that there are no restric-tions on the number of blocks that could be employed, or in the number of replications to be used in an experiment, except those imposed by resource constraints. Moreover, the method of analysis is relatively simple. Soil types and fertilizers are good examples of agricultural experiments that benefit from blocking. In manufacturing, experimental materials may come from different batches of raw materials, each of which could be treated as a block to remove differences in batches from the experimental error. Further, when the per-formance of each full replicate of an experiment is carried out on a different day or carried out by a different group of workers, each replicate could be regarded as a block to eliminate the effect of days or groups on the responses. In experiments involving humans and animals, some peculiarities of the units used in the experiment may dictate that membership of a block be based on age, sex, weight, or some other characteristics of the individuals. As already mentioned, the overall aim is to choose a block so that the units in each of the blocks are more homogeneous than the whole aggregate of the units to which the treatments are to be applied.
eBook - PDFAgricultural Field Experiments
Design and Analysis
- Roger G. Petersen(Author)
- 1994(Publication Date)
- CRC Press(Publisher)
There are also some disadvantages to this design: Basic Experimental Designs 51 I. Missing data can cause some difficulty in the analysis. One or two missing plots can be handled fairly easily but numerous missing data can cause real problems. 2. Assignment of treatments by mistake to plots in the wrong block can lead to problems in the analysis. 3. The design is less efficient than others in the presence of more than one source of unwanted variation. 4. If the plots are uniform, the RBD is less efficient than the CRD. The Randomized Block Design has a number of uses: 1. It can be used to eliminate one source of unwanted variation. Often, it provides satisfactory precision without the need for a more com-plex design. 2. It provides unbiased estimates of the means of the blocking factor. Hence these means can be estimated using the Randomized Block Design. 3.3.3 Randomization In constructing a Randomized Block Design, the plots are grouped into blocks. Normally, the number of plots in each block is equal to the number of treatments, while the number of blocks is equal to the number of replica-tions per treatment. That is, in the usual Randomized Block Design each treatment occurs once, and only once, in each block. Once the blocks have been formed, the treatments are assigned at random to the plots within the blocks. Randomization may be done either by lot or by using a random number table as described for the CRD. A separate randomization is used in each block. 3.3.4 Analysis The data analysis proceeds in much the same way as for the completely randomized design: 1. Construct a table of totals and means. 2. Compute the entries in an ANOVA table. 3. Compute a CV. 4. Conduct significance tests. 5. Compute means and standard errors. To illustrate the analysis suppose that we have a RBD with r blocks and p treatments. Let Y;i represent the yield of the j-th treatment in the i-th block.
- Roger E Millsap, Alberto Maydeu-Olivares, Roger E Millsap, Alberto Maydeu-Olivares(Authors)
- 2009(Publication Date)
- SAGE Publications Ltd(Publisher)
The design described next, a Randomized Block Design, enables a researcher to isolate and remove one source of variation among participants that ordinarily would be included in the error effects of the F statistic. As a result, the Randomized Block Design is usually more powerful than the completely randomized design. Randomized Block Design The Randomized Block Design can be thought of as an extension of a dependent samples t -statistic design for the case in which the treatment has two or more levels. The layout for a Randomized Block Design with p = 3 levels of treatment A and n = 10 blocks is shown in Figure 2.9. Comparison of the layout in this figure with that in Figure 2.5 for a dependent samples t -statistic design reveals that they are the same except that the Randomized Block Design has three treatment levels. A block can contain a single participant who is observed under all p treatment levels or p participants who are similar with respect to a variable that is positively correlated with the dependent variable. If each block contains one participant, the order in which the treat-ment levels are administered is randomized independently for each block, assuming that the nature of the treatment and the research hypothesis permit this. If a block contains p matched participants, the participants in each block are randomly assigned to the treatment levels. The use of repeated measures or matched participants does not affect the statistical analysis. However, the alternative procedures do affect the interpretation of the results. For example, the results of an experiment with repeated measures generalize to a population of participants who have been exposed to all of the treatment levels. The results of an experiment with matched participants generalize to a population of participants who have been exposed to only one treatment level. Some writers reserve the designation Randomized Block Design for this latter case.
- Alan G. Clewer, David H. Scarisbrick(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Chapter 10 The Randomised Block Design 10.1 INTRODUCTION The main disadvantage of the completely randomised design (CRD) is revealed if it is used to assess treatment effects on an experimental site which is not uniform. Experimental units (plots) treated alike may have a large variation in their yields because plots having the same treatments can be widely scattered. The estimate of the error variance may be so great that large differences between treatment means may not be declared significant as it could be argued that observed differences are really due to random background variation. A particular treatment can be assigned by chance to plots in the most fertile part of the experimental site. A false conclusion may then be made about the response to this treatment. These disadvantages can mainly be overcome by using the principle of blocking. For example, if it is suspected there is systematic variation in soil conditions across the site due to a gradual increase in pH or soil fertility, you can allow for this by dividing the experimental area into blocks at right angles to the gradient. The treatments would then be randomly assigned to plots within each block. In a greenhouse there is likely to be a light gradient so blocks could consist of rows of pots such that each row has a different light level. In order to compare five treatments (T 1, T 2 … T 5) using six replications per treatment, you could form six blocks each consisting of five equal sized plots. The blocks would be separated so that environmental conditions vary between them. The five treatments would be randomly assigned to the five plots within each block. Figure 10.1 shows a typical layout for two of the blocks. The spaces between the plots are to allow access and to reduce the possibility of neighbouring plots affecting each other
eBook - PDF- Douglas C. Montgomery(Author)
- 2019(Publication Date)
- Wiley(Publisher)
We would like to make the experimental error as small as possible; that is, we would like to remove the variability between coupons from the experimental error. A design that would accomplish this requires the experimenter to test each tip once on each of four coupons. This design, shown in Table 4.1, is called a randomized complete block design (RCBD). The word “complete” indicates that each block (coupon) contains all the treatments (tips). By using this design, the blocks, or coupons, form a more homogeneous experimental unit on which to compare the tips. Effectively, this design strategy improves the accuracy of the comparisons among tips by eliminating the variability among the coupons. Within a block, the order in which the four tips are tested is randomly determined. Notice the similarity of this design problem to the paired t-test of Section 2.5.1. The randomized complete block design is a generalization of that concept. The RCBD is one of the most widely used experimental designs. Situations for which the RCBD is appropriate are numerous. Units of test equipment or machinery are often different in their operating characteristics and would be a typical blocking factor. Batches of raw material, people, and time are also common nuisance sources of variability in an experiment that can be systematically controlled through blocking. 1 Blocking may also be useful in situations that do not necessarily involve nuisance factors. For example, suppose that a chemical engineer is interested in the effect of catalyst feed rate on the viscosity of a polymer. She knows that there are several factors, such as raw material source, temperature, operator, and raw material purity that are very difficult to control in the full-scale process. Therefore, she decides to test the catalyst feed rate factor in blocks, where each block consists of some combination of these uncontrollable factors.
eBook - PDFDesign of Experiments
An Introduction Based on Linear Models
- Max Morris(Author)
- 2010(Publication Date)
- Chapman and Hall/CRC(Publisher)
In a Randomized Complete Block Design, treatments are randomly applied to units within each block, but these random assignments cannot be made in the same manner as in a CRD because they are restricted to balance across the units within each block. So, for example, after dividing a batch of paint into five quantities, additive 1 might be applied to the first such unit selected at random, but after this assignment is made, none of the remaining quantities from that batch could be used with additive 1. Note that if treatments are sequentially applied to units randomly selected from those still available in the block, the final assignment is automatic, that is, completely determined after the other four units have been allocated. 4.1.1 Example: structural reinforcement bars Kocaoz, Samaranayake, and Nanni (2005) performed a laboratory experiment to compare the effects of four coatings on the tensile strength of steel reinforce-ment bars of the type used in concrete structures. Three of the coatings were formed from a common matrix of Engineering Thermoplastic Polyurethane (ETPU), embedded with glass fibers, carbon fibers or aramid fibers, respec-tively; the fourth coating consisted of ETPU only (i.e., no added fibers) and served as an experimental control. The N = 32 specimens (coated bars) were: “ . . . prepared in eight groups of four, with each bar type represented in each of the eight groups. The groups act as the ‘blocks’ in a randomized complete block ...design, thus adjusting for systematic trends in environmental factors or testing conditions across time. The bars with(in) each group were prepared in random order ... ” The prepared bars were tested (destructively) for strength in a set-up re-quiring each bar to be anchored in a pipe filled with grout. The bars from a given block were tested together: “ Since all four bars in a group were tested within a short period of time (1h) it is assumed that the test conditions within a group were similar.
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