For example, in applying a treatment, nuisance factors might be the specific operator who prepared the treatment, the time of day the experiment was run, and the room temperature. All experiments have nuisance factors. The experimenter will typically need to spend some time deciding which nuisance factors are important enough to keep track of or control, if possible, during the experiment.
When we can control nuisance factors, an important technique known as blocking can be used to reduce or eliminate the contribution to experimental error contributed by nuisance factors. The basic concept is to create homogeneous blocks in which the nuisance factors are held constant and the factor of interest is allowed to vary. Within blocks, it is possible to assess the effect of different levels of the factor of interest without having to worry about variations due to changes of the block factors, which are accounted for in the analysis.
A nuisance factor is used as a blocking factor if every level of the primary factor occurs the same number of times with each level of the nuisance factor.
The analysis of the experiment will focus on the effect of varying levels of the primary factor within each block of the experiment. The general rule is: "Block what you can, randomize what you cannot. Randomization is then used to reduce the contaminating effects of the remaining nuisance variables. When this is not possible, proper blocking , replication, and randomization allow for the careful conduct of designed experiments.
Outline the methodology for designing experiments in terms of comparison, randomization, replication, blocking , orthogonality, and factorial experiments Experiments exercises c Does this study make use of blocking? If so, what is the blocking variable? Statistical Graphics They include plots such as scatter plots , histograms, probability plots, residual plots, box plots, block plots and bi-plots.
Random Sampling For example, while surveying households within a city, we might choose to select city blocks and then interview every household within the selected blocks , rather than interview random households spread out over the entire city. ANOVA Design The protocol's description of the assignment mechanism should include a specification of the structure of the treatments and of any blocking.
More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks. We are primarily interested in testing the equality of treatment means, but now we have the ability to remove the variability associated with the nuisance factor the blocks through the grouping of the experimental units prior to having assigned the treatments. In the RCBD we have one run of each treatment in each block. In some disciplines, each block is called an experiment because a copy of the entire experiment is in the block but in statistics, we call the block to be a replicate.
This is a matter of scientific jargon, the design and analysis of the study is an RCBD in both cases. This is just an extension of the model we had in the one-way case. The algebra of the sum of squares falls out in this way. We can partition the effects into three parts: sum of squares due to treatments, sum of squares due to the blocks and the sum of squares due to error. The partitioning of the variation of the sum of squares and the corresponding partitioning of the degrees of freedom provides the basis for our orthogonal analysis of variance.
In Table 4. We obtain the Mean Square values by dividing the sum of squares by the degrees of freedom. Then, under the null hypothesis of no treatment effect, the ratio of the mean square for treatments to the error mean square is an F statistic that is used to test the hypothesis of equal treatment means. The Analysis of Variance table shows three degrees of freedom for Tip three for Coupon, and the error degrees of freedom is nine. The ratio of mean squares of treatment over error gives us an F ratio that is equal to Our 2-way analysis also provides a test for the block factor, Coupon.
So, there is a large amount of variation in hardness between the pieces of metal. This is why we used specimen or coupon as our blocking factor. We expected in advance that it would account for a large amount of variation.
By including block in the model and in the analysis, we removed this large portion of the variation, such that the residual error is quite small. By including a block factor in the model, the error variance is reduced, and the test on treatments is more powerful. The test on the block factor is typically not of interest except to confirm that you used a good blocking factor.
The results are summarized by the table of means given below. This isn't quite fair because we did in fact block, but putting the data into one-way analysis we see the same variation due to tip, which is 3. So we are explaining the same amount of variation due to the tip. That has not changed.
But now we have 12 degrees of freedom for error because we have not blocked and the sum of squares for error is much larger than it was before, thus our F -test is 1. If we hadn't blocked the experiment our error would be much larger and in fact, we would not even show a significant difference among these tips.
This provides a good illustration of the benefit of blocking to reduce error. The RCBD utilizes an additive model — one in which there is no interaction between treatments and blocks. The error term in a randomized complete block model reflects how the treatment effect varies from one block to another. Both the treatments and blocks can be looked at as random effects rather than fixed effects, if the levels were selected at random from a population of possible treatments or blocks.
We consider this case later, but it does not change the test for a treatment effect. What are the consequences of not blocking if we should have? Generally the unexplained error in the model will be larger, and therefore the test of the treatment effect less powerful.
How to determine the sample size in the RCBD?
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