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If it only means order and all the cows start lactating at the same time it might mean the same. But if some of the cows are done in the spring and others are done in the fall or summer, then the period effect has more meaning than simply the order. Although this represents order it may also involve other effects you need to be aware of this. A Case 3 approach involves estimating separate period effects within each square.
Chapter 12 Block Designs
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To simplify things, we will assume that you have one main blocking factor that you want to balance over. The first step of implementing blocking is deciding what variables you need to balance across your treatment groups. Here are some examples of what your blocking factor might look like. So even after I account for Irrigation and Fertilizer treatments, observations within a plot will be more similar to each other than observations in two different plots. In general, we are faced with a situation where the number of treatments is specified, and the block size, or number of experimental units per block (k) is given. This is usually a constraint given from the experimental situation.
What is a Randomized Block Experiment?
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These blocks are just different patches of land, and each block is partitioned into four plots. Then we randomly assign which variety goes into which plot in each block. You will note that variety A appears once in each block, as does each of the other varieties. Often there are covariates in the experimental units that are known to affect the response variable and must be taken into account. Ideally an experimenter can group the experimental units into blocks where the within block variance is small, but the block to block variability is large. For example, in testing a drug to prevent heart disease, we know that gender, age, and exercise levels play a large role.
5 - What do you do if you have more than 2 blocking factors?
If the only covariate we care about is the most deeply nested effect, then we can do the usual analysis and recognize the p-value for the blocking variable is nonsense, and we don’t care about it. Note that the least squares means for treatments when using PROC Mixed, correspond to the combined intra- and inter-block estimates of the treatment effects. Since the first three columns contain some pairs more than once, let's try columns 1, 2, and now we need a third...how about the fourth column. If you look at all possible combinations in each row, each treatment pair occurs only one time. Together, you can see that going down the columns every pairwise sequence occurs twice, AB, BC, CA, AC, BA, CB going down the columns.
This is a simple extension of the basic model that we had looked at earlier. The row and column and treatment all have the same parameters, the same effects that we had in the single Latin square. In a Latin square, the error is a combination of any interactions that might exist and experimental error. The Latin Square Design gets its name from the fact that we can write it as a square with Latin letters to correspond to the treatments. The treatment factor levels are the Latin letters in the Latin square design. The number of rows and columns has to correspond to the number of treatment levels.
With our first cow, during the first period, we give it a treatment or diet and we measure the yield. Obviously, you don't have any carryover effects here because it is the first period. However, what if the treatment they were first given was a really bad treatment? In fact in this experiment the diet A consisted of only roughage, so, the cow's health might in fact deteriorate as a result of this treatment. This carryover would hurt the second treatment if the washout period isn't long enough. The measurement at this point is a direct reflection of treatment B but may also have some influence from the previous treatment, treatment A.
Although the sex of the patient is not the main focus of the experiment—the effect of the drug is—it is possible that the sex of the individual will affect the amount of weight lost. Blocking is one of those concepts that can be difficult to grasp even if you have already been exposed to it once or twice. Because the specific details of how blocking is implemented can vary a lot from one experiment to another. For that reason, we will start off our discussion of blocking by focusing on the main goal of blocking and leave the specific implementation details for later.
We will try to account for these nuisance factors in our model and analysis. The single design we looked at so far is the completely randomized design (CRD) where we only have a single factor. In the CRD setting we simply randomly assign the treatments to the available experimental units in our experiment. There are plenty of experimental designs where we have levels of treatments nested within each other for practical reasons. The literature often gives the example of an agriculture experiment where we investigate the effect of irrigation and fertilizer on the yield of a crop.
Choose your blocking factor(s)
This type of design is called a Randomized Complete Block Design (RCBD) because each block contains all possible levels of the factor of primary interest. Furthermore, as mentioned early, researchers have to decide how many blocks should there be, once you have selected the blocking variable. We want to carefully consider whether the blocks are homogeneous. In the case of driving experience as a blocking variable, are three groups sufficient? Can we reasonably believe that seasoned drivers are more similar to each other than they are to those with intermediate or little driving experience? If the blocks aren't homogeneous, their variability will not be less than that of the entire sample.
To conduct this experiment we assign the tips to an experimental unit; that is, to a test specimen (called a coupon), which is a piece of metal on which the tip is tested. Often in medical studies, the blocking factor used is the type of institution. Sometimes several sources of variation are combined to define the block, so the block becomes an aggregate variable.
The choice of case depends on how you need to conduct the experiment. If you are simply replicating the experiment with the same row and column levels, you are in Case 1. If you are changing one or the other of the row or column factors, using different machines or operators, then you are in Case 2. If both of the block factors have levels that differ across the replicates, then you are in Case 3.
Suppose engineers at a semiconductor manufacturing facility want to test whether different wafer implant material dosages have a significant effect on resistivity measurements after a diffusion process taking place in a furnace. They have four different dosages they want to try and enough experimental wafers from the same lot to run three wafers at each of the dosages. A design with the parameters of the extension of an affine plane, i.e., a 3-(n2 + 1, n + 1, 1) design, is called a finite inversive plane, or Möbius plane, of order n. In that sense, Latin Square designs are useful building blocksof more complex designs, see for example Kuehl (2000). Notice that in our block level, there is no p-value to assess if the blocks are different. So our analysis respects that blocks are present, but does not attempt any statistical analyses on them.
Blocking involves grouping experimental units based on levels of the nuisance variable to control for its influence. Randomization helps distribute the effects of nuisance variables evenly across treatment groups. A special case is the so-calledLatin Square design where we have two blockfactors and one treatment factor having \(g\) levels each (yes, all of them!).Hence, this is a very restrictive assumption. In a Latin Square design, eachtreatment (Latin letters) appears exactly once in each row and once ineach column. A Latin Square design blocks on both rows and columnssimultaneously. If we ignore the columns of a Latin Square designs, the rows form anRCBD; if we ignore the rows, the columns form an RCBD.
The smallest crossover design which allows you to have each treatment occurring in each period would be a single Latin square. Many times there are nuisance factors that are unknown and uncontrollable (sometimes called a “lurking” variable). We always randomize so that every experimental unit has an equal chance of being assigned to a given treatment. Randomization is our insurance against a systematic bias due to a nuisance factor. A nuisance factor is a factor that has some effect on the response, but is of no interest to the experimenter; however, the variability it transmits to the response needs to be minimized or explained. We will talk about treatment factors, which we are interested in, and blocking factors, which we are not interested in.
Suppose you have a green house study where you have rooms where you can apply a temperature treatment, within the room you have four tables and can apply a light treatment to each table. Finally within each table you can have four trays where can apply a soil treatment to each tray. This is a continuation of the split-plot design and by extending the nesting we can develop split-split-plot and split-split-split-plot designs. Because this is an orthogonal design, the sums of squares doesn’t change regardless of which order we add the factors, but if we remove one or two observations, they would. For an odd number of treatments, e.g. 3, 5, 7, etc., it requires two orthogonal Latin squares in order to achieve this level of balance.
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