A researcher conducted an experiment to examine the efficiency of four types of fungal sprays (T1,T2,T3,T4) in controlling fungal rot on blueberries. Four adjacent rows of blueberries are available, each with 21 plants. Sprays can be applied to individual blueberry plants. The outcome / response variable is the proportion of blueberries with rot.
For the following design, specify experimental unit, blocking factor, and number of replications.
Each row is divided into 3 plots of 7 plants each. The sprays are randomly allocated to plots within each row.
Answer
Experimental unit = plot of 7 plants
Blocking factor is row
Number of replications is 3
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I'm not sure if I totally understand blocking.
In Q6, there are 4 treatments.
There are also 4 adjacent rows, each row has 21 plants.
In part (b) of the question, each row is divided into 3 plots (of 7 plants each).
The treatments are randomly allocated to plots within each row.
If I'm reading it correctly, there are 4 treatments, but each block has only 3 spaces; therefore it's not possible to allocate all treatments to each block.
The answer says the blocking factor is row, but I understood each block should be a complete replication of the set of treatments.
??
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Hi Graham,
The bit that confused me more about this was where the 3 replications came from, bu then I wrote this answer and figured it out :) It is late at night and I probably should have read this awhile ago! My reading of this is that by blocking each of 4 rows into 3 blocks you're left with 12 "units" to assign treatments to. If you start at row 1, randomly assign a treatment, move to row 2 randomly assign another treatment (without replacement) etc. Then start again at Row 1 with the same procedure, means you get the 4 treatments assigned 3 times each. My understanding is that blocking is a way of compensating for potential variation across your field that might influence your results but that you're not interested in. So by splitting your field up into smaller units this way and undertaking some sort of random assignment to blocks you might account for some of this impact. In this way you get your row blocking factor and your 3 replications because T1 is allocated 3 times across the field. I think this is it! I don't think it is necessary that each block is a complete replication of treatments? But I could be wrong. The agricultural experiments mess with me because it is so far outside of my field of expertise, I have to try and think how it relates to something I know and not easy!
Cheers,
Kate
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Having worked in agricultural research, perhaps I should weigh in with my humble opinion :-)
Q6 (b) is a rather bad example. In fact, by splitting each row into three plots, and then randomly assigning four treatments, until you have used up all 12 plots, you essentially get a fully randomised design, not a randomised blocks design. By taking care to assign the four treatments of replication 1 to the first row plus one plot, then replication 2 to the remaining two plots of row 2 plus the next two of row 3, etc., you could conceivably get a randomised blocks design, but it's messy.
In randomised blocks, each block should be laid out in one part of the field. A block doesn't have to be a uniform shape, but in practice, it normally is. The main thing is that the plots making up a block should be grouped together in one area, to reduce the effects of fertility trends (soil heterogeneity). Graham is correct: In a (simple) randomised blocks design, the number of replications normally equals the number of blocks - at least, for the purposes of this course. You can theoretically have as many treatments per replication (block) as you like, although you shouldn't - see below.
I say "normally", since the rule of thumb in field experiments is that a block should not exceed 16 experimental units (plots) in size, or the advantages of local control of soil heterogeneity are lost. This means that for experiments with more that 16 treatments, which is quite possible in factorial designs, the use of ordinary randomised blocks designs runs into limitations. With Latin squares, the situation is even worse.
For experiments with more that 16 treatments (or factor combinations), the researcher has to resort to more sophisticated experimental designs, in which several blocks per replication are employed. These so-called incomplete blocks designs use deliberate confounding of certain treatment effects with block effects. Higher-order interactions of factorial designs, which are generally of less interest to the researcher, are typically chosen for such confounding, and are dealt with in the analysis. Here one also ventures into the area of so-called graeco-latin squares, and lattice designs, typically used in variety trials, where a large number of cultivars may be planted in a single trial. Clearly, with varieties, you are not dealing with a factorial design, and so-called quasi-factors are employed in the design. It is even possible to have single-replication designs, in which higher-order interactions are used to estimate error in the ANOVA. Such experimental designs are clearly beyond the scope of this course, and I mention these just as a matter of interest.
Not surprisingly, much of the pioneering work in experimental design and analysis has come from agricultural research. For those interested in this field (pardon the pun), the statistical package GENSTAT, developed at the Rothamsted agricultural research station in the UK, where Sir Ronald Fisher worked (with his associate Frank Yates) and developed ANOVA, is one of the few packages which makes a clear distinction between the blocking structure of an experiment and its treatment structure, and allows straightforward analysis of these fairly complex incomplete blocks designs. It was a nightmare in the days of hand calculations. SAS also has origins in agricultural research. The classic text to consult is Cochran and Cox's Experimental Designs.
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