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MANE 6313

Week 8, Module E

Student Learning Outcome

  • Select an appropriate experimental design with one or more factors,
  • Select an appropriate model with one or more factors,
  • Evaluate statistical analyses of experimental designs,
  • Assess the model adequacy of any experimental design, and
  • Interpret model results.

Module Learning Outcome

Creating designs with more than two blocks.

Constructing the \(2^k\) design in 4 blocks

  • You must define 2 linear combinations \(L_1\) and \(L_2\)

  • For each treatment effect, we construct the ordered pair \((L_1\mbox{ mod }2,L_2\mbox{ mod }2)\)

  • This will result in four blocks having values (0,0), (0,1), (1,0) and (1,1)

  • Confounding scheme is more complicated (discussed later in module)

  • Work a \(2^4\) example in 4 blocks

Table 7.9 (10th edition)

Table 7.9 (textbook)

Four Block Example

  • Use ABD and ABC as block generators (not those in Table 7.9)

Design Matrix by hand

Four Block Example, continued

Block Assignments by Hand

Generalized Interactions

  • Examining the example problem, we see three degrees of freedom for blocks (4-1). One degree can be associated with \(ABD\) and the second with \(ABC\). There is a missing effect.

  • There is a generalized interaction that occurs when two linear combinations are used.

  • \(GI=ABD(ABC)=A^2B^2CD=CD\)

  • Care must be exercised in selecting \(L_1\) and \(L_2\) because the GI might contain important information.

Confounding the \(2^k\) Design in \(2^p\) Blocks

  • Select \(p\) independent effects to be confounded.

  • Independent effects means that no effect chosen is the generalized interaction of the others.

  • The \(p\) defining contrasts \(L_1,L_2,\ldots,L_p\) are used to define the \(2^p\) blocks

  • In addition, there are exactly \(2^p-p-1\) other effects confounded with these blocks.

  • Often people look to tables such as Table 7.9 on page 271 for helping in selecting the effects used to generate blocks