- Blocking is a procedure under which experimental units are grouped into "blocks" that are expected to be as alike as possible within themselves but may be consistently different from each other.
- The blocks should then remove a possible element of systematic variation so that the residual mean square in the usual analysis of variance truly estimates just experimental error and is not inflated by such a source of consistent variation.
- Comparisons between treatment means are then more precise.
- For example, in an industrial experiment that takes some time to perform, the blocks might be days, or shifts, or parts of days, chosen so that every treatment can be examined (at least) once in each block using the same machinery.
Explain what is meant by the main effect of a factor and the interaction between two factors in a 2x2 factorial experiment.
- The main effect of a factor in a 2x2 experiment is the difference between the results with the factor at its high level and those with it at its low level; thus, for factor A, it is given by ab + a – (b + (1)) [an average difference might be used, i.e. with a divisor of 2].
- Similarly, for B it is given by ab + b – (a + (1)).
- The remaining independent comparison that is possible is ab + (1) – (a + b).
- By rearranging this as (ab – b) – (a – (1)), it can be seen to measure the difference between the "responses" to factor A at the high level of B and those at the low level of B.
- Equivalently, the roles of A and B can be interchanged throughout this.
- It is called the interaction between A and B.
Explain what is meant by a central composite design and mention briefly the advantages of building up the design sequentially.
- In response surface analysis, when an experimental region has been identified whose centre (coded O) is thought to be near the maximum (or minimum) response, and k factors are being studied, a central composite design is suitable for fitting a quadratic model which allows the turning point to be located.
- It consists of a 2k factorial (or fractional factorial), at points coded ±1 equidistant from O, together with several centre points (0, 0, …, 0) and a set of "axial" points (±α, 0, …, 0), (0, ±α, 0, …, 0), …, (0, …, ±α).
- A central composite design can be built up from the first-order 2k design by adding central and axial points.
- Blocking may be used to eliminate any changes in experimental conditions between the first-order design and later additions; it is possible to arrange for block parameters to be estimated independently of model parameters.
Discuss briefly the advantages and disadvantages of a Latin square design.
- Combinations of all cars with all roads and all drivers would require 4×4×4 = 64 runs.
- The Latin square scheme, in 16 runs, allows orthogonal comparisons of the three factors, on the assumption that there are no interactions.
- A 4×4 square has only 6 degrees of freedom for residual, and often that would not be enough to give a reliable estimate of σ square ; here, however, the estimate is quite small, so a useful analysis has resulted.
- Using two squares would give ample degrees of freedom for F and t tests.
Explain the principles that should be followed when selecting a particular Latin square of the appropriate size for use in an experiment.
- There are four "standard" 4×4 squares (letters in alphabetical order in first row and in first column), one of which must be chosen at random.
- The rows of this square are then permuted at random, as are the columns, to give a randomised design.
- The letters A, B, C, D are then allocated at random to the "treatments" (drivers).
- This gives a random choice from all possible 4×4 squares.
There is a rising trend for star college athletes to turn professional without finishing there degrees. A study is performed to assess whether reading an article about professional salaries has an impact on such decisions. Randomisation can be used to split the subjects into two groups, and those in one group given the article before answering questions. How can a block design be incorporated into the design of this experiment?
- The subjects can be split into two blocks, underclass and upperclass, before using randomisation to assign some to read the article before questioning.
- With this design, the impact of the salary article on freshmen and sophomores can be distinguished from the impact on juniors and seniors.
- Similarly, blocking can be used to separately analyse men and women, those with high GPAs and those with low GPAs, those in different sports, those with different majors, ad so on.
- Completely Randomised Design
- Block Design
