Math 558 Lecture #3
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Experimental units can be referred to as plots. In this course we will denote the plots in general,by lower case Greek letters, i.e. ¦Á, ¦Â, ¦Ã. The set of plots in an experiment will be denoted by ¦¸.The total number of plots will be denoted by N.
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The general treatments will be denoted by lower-case Latin letters such as i, j, k. The set of treatments in an experiment will be denoted by ¦Ó. The total number of treatments will be denoted by t.
More Terminology, Notations and Examples
Note that while designing an experiment we need to provide a precise description of a treatment. By precise description we mean
the ingredients that combine to form a treatment, like fertilizer types and wheat varieties with the specific names of the ingredients used. Example: Rye-grass experiment
The time and the quantity administered.
Once we specify the treatments with all the description we can associate a symbols like treatment A, treatment B etc.
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Definition 1
Treatment structure is an arrangement of treatments from the set of treatments ¦Ó according to the plots.
new treatments plus control
All combinations of two factors
Example: Rye grass experiment
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Definition 2
Plot structure means meaningful ways of dividing up the set ¦¸ of plots, ignoring the treatments.
Example: blocking
Definition 3
The design is allocation of treatments to plots.
Definition 4
The plan or layout is the execution of the design to actual plots.
Bailey pg. 13
Allocation of treatments to plots
Allocation of treatments to plots can be seen as a function that maps plots to treatments. That is T : ¦¸ ¡ú ¦Ó such that the plot ¦Ø gets the treatment T(¦Ø)
Completely randomized designs
In a completely randomized designs the treatments are applied to the plots randomly. The plot arrangement is unstructured. More specifically, if a treatment has to be applied to four plots, each plot has equal chance of receiving the treatment. If a treatment i is applied to ri plots we say that the treatment i is replicated ri times. If we have t treatments in total then i = 1, 2, …t. All these treatment replications fill N plots. Therefore
¡Æri = r1 +r2 +r3…..rt = N i=1
Completely randomized designs
Any number of treatments and replications can be used.
Statistical analysis is easy due to the simple layout. Completely randomized designs are used where
plots are approximately alike (difference in plots do not make considerable contribution to to the experimental error)
we may not be able to obtain response on all the plots
small experiments for which complex designs can be too costly
Completely randomized designs are mostly used in lab experiments (engineering, cookery) where a material or combination of materials are divided into batches to which treatments are applied. These designs can be used in the field experiments only if the number of plots is small.
How to randomize1 Number the plots 1,2,..N
Allocate treatment 1 to plots 1, 2…r1, treatment 2 to the plots r1 +1,r1 +2,…..r1 +r2 andsoon
Apply random permutation to the design
Example: Suppose there are three baking times 15 minutes(A), 16 minutes(B) and 17 minutes(C) that a chef wants to chose from to bake her cookies. She prepares the dough, divides it in 13 batches and wants to apply baking times (treatments in this case) randomly to her cookies. This how we can proceed.
Label the batches(plots) from 1..13
Initially allocate treatment A to plots 1 to 5, treatment B to plots 6 to 9 and treatment C to plots 10 to 13.
Randomize 1Bailey pg 19
Step 1:Allocate treatments to the plots
Plots 1 2 3 4 5 6 7 8 9 10 11 12 13 Treatments A A A A A B B B B C C C C
Step 2: Randomize2
1 2 3 4 5 6 7 8 9 10 11 12 13 6 2 8 11 13 1 12 5 7 4 9 3 10
Step 3: CRD
Plots 1 2 3 4 5 6 7 8 9 10 11 12 13 Treatments B A C C B A B A C C A B A
2you can use random.org to randomize your list of plots
Potato Scab Disease3
An experiment was performed to see the affects of sulphur on reducing scab disease on potatoes. Sulphur increases the acidity in the soil which helps cure the disease. The purpose of the experiment was to choose the right amount of sulphur per acre. The amounts of sulphur used were 0, 300, 600 and 1200 per acre. The applications were applied in Fall and Spring. There were seven treatments in all: F3, F6, F12, S3, S6, S12, 0. The response to be analysed was scab index which is the surface area of the potato that is infected by the scab. It is obtained by randomly selecting 100 potatoes, grading each on a scale of 0 to 100 infected, and taking the average.
3Cox and Cochran pg 97
Potato Scab Disease
The experimental design is given below.
F39 012 S618 F1210 S624 S1217 S330 F616 010 S612 F39 F124 S39 018 S1217 S619 030 F125 026 F34
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