Math 558 Lecture #13
Blocking in design of experiments refers to dividing experimental units into homogeneous groups. This means that the experimental units within a group should share almost the same characteristics.The blocking process starts in the beginning of the experiment.
As an example consider agricultural field experiments. The neighbouring plots in the field are alike in terms of soil quality, sunlight, water access. These plots are grouped together to achieve homogeneous blocks. Usually the blocks are kept of the same size.
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Randomized Block Designs
In randomized block designs the experimental units are divided into groups each of which contains a single replication. If each block contains the experimental unit equal or greater to the number of treatments to be tested with each treatment occurring at least once in each block, the design is called Randomized Complete Block design.If we have t treatments and b blocks, then the total number of experimental units are b × t. The treatments are applied randomly to the experimental units in each block.
Blocking Why do we block?
Blocking results in smaller experimental error and more precise tests and estimates as compared to the randomized designs with no blocking.
We can use as many treatments and as many replications as our resources allow.
The statistical analysis is straight forward. We will see some examples in our coming lectures.
Types of blocks 1
The commonly used types of the blocks are
Natural Discrete Divisions
These are natural divisions among experimental units.
In experiments on new born animals, litters make natural blocks. In experiments involving people, sexes can make natural blocks.
In industrial processes batches of certain material can make natural process.
Example 4.1( Text ) (Insect repellent) Midges in Scotland are a severe irritant in July and August. A researcher wants to try out some insect repellents, which are applied to people’s skin. Twelve people volunteer for the experiment.
1textbook pg 53
Types of Blocks Natural Discrete Divisions
The researcher decides to use people as a blocks due to inherent differences in their attractiveness to midges. What can be the possible experimental units?
Example 4.2 (Text)(Irrigated rice field) Consider a rice field with 32 plots in a rice paddy to be used for an experiment.Rice crop requires well planned irrigation.For the field under consideration a main irrigation channel is branched of to sub-channels each supplying water to a strips of the field. . These strips, or “irrigation groupings”, are natural blocks.
Types of Blocks Continuous gradients
Continuous gradients Sometimes the variables under consideration are spread overlarge spaces or long time periods. In such cases the underlying continuity needs to taken under consideration to define the blocks.
For some field experiments there are no natural boundaries to help the experimenter to define the blocks so we can use the groups of units closer to one another for blocking .
For experiments with people or animals, the variables like age, weight, height, or state of health can be grouped in blocks which are the ranges of measurements.
Example 4.5 (Text)(Field trial) The plots in an agricultural field trial may cover quite a large area, encompassing changes in fertility. Sometimes it is possible to form natural blocks by marking out a stony area, a shady area and so on. More often it is simply assumed that plots close to each other are more likely to respond similarly than plots far apart, so small compact areas are chosen as blocks.
Types of Blocks
Choice of blocking for trial management
Sometimes the management and planning considerations force differences between the plots. In lab experiments the technicians in charge of experimental runs can be considered as blocks. Similarly in clinical trials doctors or nurses can be taken as blocks.
Example 4.6 Text (Citrus orchards) Similarly, citrus orchards are planted with the trees in a rectangular grid. The space between rows is bigger than the space between columns, so that lorries can drive along the rows for operations such as applying pesticides or harvesting. Therefore, both contiguity and management considerations suggest that rows should be blocks.
How to Block
How to block:
All blocks should have same number of plots. Number of plots
determine the size of the blocks.
The blocks should be large enough to allow each treatment to occur atleast once in each block. If each treatment occurs atleast once in each block, such a design is called complete block design
Example 4.7 Text (Piglets) If the experimental units are piglets then litters are natural blocks. Litters are not all of the same size, typically being in the range 8–12, depending on the breed. It would be sensible to use only some fixed number, say nine, of piglets from each litter. Then you need an objective rule for which piglets to choose from the larger litters, such the heaviest piglets. Alternatively, if larger blocks are needed, use only those litters large enough to give, say, ten piglets.
In real life situations the above two blocking requirements may be difficult to meet. In that case we should try our best to have same sized blocks (blocks with same number of plots). There is a lot literature to deal with the situations when the blocks are of same size but they are not complete.
It is always preferable to use natural blocks if they exists.However, natural blocks have limit on their size. Therefore we may be able to satisfy the first requirement but not the second.
Further, while managing clinical trials and lab experiments the origin of management differences should inform the decision of blocking. Management blocks can be chosen to satisfy both the conditions.
Randomized block designs in R
Randomized block designs can be easily created using the function design.rcbd from the package agricolae (de Mendiburu, 2012b) . By default this function labels the experimental units as ,” and uses integers for the block numbers. The seed argument in the function call is for the randomization, and running the code with the same seed will result in the same randomized list.
Randomized block designs in R
treat<-c(1,2,3,4) outdesign <- design.rcbd(treat, 4, seed = 10) rcb <- outdesign$book levels(rcb$block) <- c("carnation", "daisy", "rose", "tulip") rcb Example Lawson page 117 An experiment was conducted to test the effect of different water solutions on the life of cut flowers. The treatment factor was the liquid to fill the vase. The levels were: 1. Tap water 2. Tap water with one spoonful of sugar added 3. Tap water with one cup of carbonated water 4. Tap water with one cup of 7-up The experimental units were single flowers and the response was the time in days until the flower wilted. The experimenter wanted the conclusions of her study to apply to many types of flowers, so she used an RCB design. The natural blocks were the flowers. 2. Carnation Randomized Complete Block Design block treat carnation 1 carnation 3 carnation 4 carnation 2 daisy 4 daisy 3 daisy 1 daisy 2 rose 3 rose 2 rose 4 rose 1 tulip 2 tulip 1 tulip 4 tulip 3 程序代写 CS代考 加微信: cscodehelp QQ: 2235208643 Email: email@example.com