Randomized paired comparison designs

In the previous section, the discussion following Equation 12.15 suggested that certain experimental designs could be used to decrease the uncertainty associated with a parameter estimate by minimizing the effect of uncontrolled factors. In this section we discuss one of these experimental designs, the randomized paired comparison design. 

But what if we did not know that sugar content was an important factor, or what if we were unable to measure the level of sugar content in each fruit We would then not have been able to include this factor in our model and the confidence in the temperature effect could not have been improved. 

Randomized paired comparison experimental design for determining the effect of temperature on a wine-making system. 

Up to this point, the randomized paired comparison design has not offered any great improvement over the completely randomized design. However, the fact that each fruit has been investigated at a pair of temperatures allows us to carry out a different type of data treatment based upon a series of paired comparisons. 

We realize that there are a number of factors in addition to temperature that influence the % alcohol response of the wine sugar content, pressure, magnesium concentration in the fruit, phosphate concentration in the fruit, presence of natural bacteria, etc. Although we strive to keep as many of these factors as controlled and therefore as constant as we can (e.g., pressure), we have no control over many of the other factors, especially those associated with the fruit (see Section 1.2). However, even though we do not have control over these factors, it is nonetheless reasonable to expect that whatever the % alcohol response is at 23° C, the % alcohol response at 27 °C should increase for each of the fruits in our study if temperature has a significant effect. That is, if we are willing to make the assumption that there are no interactions between the factor of interest to us (temperature) and the other factors that influence the system, the differences in responses at 27 °C and 23° C should be about the same for each pair of experiments carried out on the same fruit. 

In the previous section, the discussion following Equation 15.15 suggested that 

In the completely randomized design, a different fruit was randomly assigned a temperature, either 23°C or 27°C. Let us consider now a different experimental design. We will still employ the same number of experiments (20), but we will use only half as many fruit types, assigning each fruit type to both temperatures. Thus, each fruit will be involved in a pair of experiments, one experiment at 23°C and the 

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Figure 15.13 Factor combinations for a randomized paired comparison design investigating the effect of temperature. Fruit number is an arbitrarily assigned, qualitative factor. Numbers beside factor combinations indicate the time order in which experiments are run.

Thus, the randomized paired comparison design has allowed a more sensifive way of viewing our data, a view that ignores much of the variation caused by the use of different fruits and focuses on pairwise differences associated with the single factor of interest, temperature. 

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. 

Figure 15.16 Sums of squares and degrees of freedom tree for the randomized paired comparison design.

Suppose someone comes to you with the hypothesis that shoes worn on the right foot receive more wear than shoes worn on the left foot. Design a randomized paired comparison experiment to test this hypothesis. How might it differ from a completely randomized design ... 

Figure 13 illustrates the comparison between the experimental xsans(F) data for a pair of systems that are labeled by Graessley et al. [28] as H38/D25 and H25/D38 and the BLCT xsansCF) as calculated from Eq. 37 for the set of caa. sbej ab> and y specified above. The notation H38/D25, for example, denotes the isotopic blend of hydrogenated (H) and deuterated (D) polybutadiene random copolymers, and the numbers indicate the fractions of 1,2 units in each of the blend components. The experimental points [28] are designated in Fig. 13 by circles and triangles, whereas the fines represent the theoretical fits. Figure 13 demonstrates that the theory reproduces the overall values of the X parameters within the experimental error bars of 1.5 x 10 [28]. 

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