Complete block designs, randomized

The randomized paired comparison design discussed in the previous section separates the effect of a qualitative factor, fruit, from the effect of a quantitative factor, temperature (see Section 1.2). The randomized complete block design discussed in this section allows us to investigate more than one purely qualitative variable and to estimate their quantitative effects. 

Suppose the researcher involved with the sodium ion concentration study of Sections 15.1 and 15.2 becomes interested in the wine-making process we have been discussing in Sections 15.3 and 15.4. In particular, let us assume the researcher is interested in determining the effects on the percent alcohol response of adding 10 milligrams of three different univalent cations (Li, Na, and K ) and 10 milligrams 

In the classical statistical literature, one of the two qualitative factors is referred to as the treatments and the other qualitative factor is referred to as the blocks . Hence, the term block designs . In some studies, one of the qualitative factors might be correlated with time, or might even be the factor time itself by carrying out the complete set of experiments in groups (or blocks ) based on this factor, estimated time effects can be removed and the treatment effects can be revealed 

Randomized complete block design for determining the effects of univalent and divalent cations on a wine-making system. 

Experiment Univalent cation Divalent cation % Alcohol 

Because qualitative factors are not continuous, we cannot use a linear model such as yu = / + S,jc,x + /J2x2l + ru to describe the behavior of this system. For example, if x, were to represent the factor univalent cation , what value would x, take when Li was used Or Na Or K There is no rational basis for assigning numerical values to xt, so we must abandon the familiar linear models containing continuous (quantitative) factors. 

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Figure 15.17 Factor combinations for the randomized complete block design investigating two qualitative factors, type of univalent cation and type of divalent cation. Each factor combination is replicated.

Note that for the randomized complete block design, there is no covariance between the effects of the univalent cations and the effects of the divalent cations. 

The randomized complete block design has provided a sensitive way of viewing the data from this set of experiments involving two qualitative factors. The linear model using dummy variables ignores much of the variation in the data by again focusing on pairwise differences associated with the different discrete levels of the factors of interest. 

Because of tradition, the model of Equation 15.32 is seldom used for the randomized complete block design. Instead, a somewhat different but essentially equivalent model is used ... 

The remaining parameters of Equation 15.36 (y and x ) are calculated from Equations 15.40 and 15.41. Thus, if the model of Equation 15.32 has been used to treat the data from a randomized complete block design, the results may be readily converted to the form of Equation 15.36. 

Design a randomized complete block design to help answer the question raised by the sports enthusiast. What are the blocks and treatments in your design ... 

What is the relationship - Youden square designs Latin square designs balanced incomplete block designs randomized complete block designs ... 

To screen out the effects of systematic errors, the effects of factor-level variations are researched in each block by random order. This is the origin of the term randomized complete-block design. These blocks originate from studies in agronomy, for in it there appeared the most drastic case of inequality of agricultural lots where... 

Field Plot Technique. Herbicide treatments were made to plots measuring 3.1 to 6.1 by 13.2 to 18.3 m arranged in a randomized complete block design with 3 replications. Herbicides were applied with tractor-mounted compressed air or C0 backpack sprayers delivering 187 to 280 L/ha at 139 to 207 kPa. Herbicides were uniformly mixed with the top 8 to 10 cm of soil with 2 pass incorporation. 

Another way to guard against any possible bias due to the effect of ETHNIC is to carry out the experiment Ijy the randomized complete block design (RCBD), as shown in Table 4. Here, each of the three ethnic groups constitutes a block and receives all of the three treatments in random order. 

TABLE 4 Randomized Complete Block Design (RCBD) ... 

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