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MANE 6313

Week 5, Module A

Student Learning Outcome

Analyze simple comparative experiments and experiments with a single factor.

Module Learning Outcome

Evaluate randomized block designs.

Chapter 4 Terminology

  • Nuisance Factor is a design factor that probably has an effect on the response, but we are not interested in its effect.

  • Some nuisance factors are unknown and uncontrollable. These variables are said to be "lurking"

  • The only protection against unknown and uncontrollable nuisance factors is randomization

  • If the factor is known and uncontrollable we can compensate by using analysis of covariance

  • If the factor is known and controllable, we can use blocking.

Chapter 4 Terminology, continued

  • Blocking allows the systematic elimination of a nuisance variables effect upon the statistical comparisons.

  • A Block is a portion of the experimental material that is more homogeneous than the aggregate (total).

  • If not recognized and accounted for, the variability associated with blocks will be grouped with the random error and result in an insensitive test.

Randomized Complete Block Design

  • New model:

$$ y_{ij}=\mu+\tau_i+\beta_j+\varepsilon_{ij}\;\left{\begin{array}{l} i=1,2,\ldots,a\j=1,2,\ldots,b\ \end{array}\right. $$ Note that \(\sum_{i=1}^a\tau_i=0\) and \(\sum_{j=1}^b\beta_j=0\)

ANOVA Table

Source of Variation Sum of Squares Degrees of Freedom Mean Square F
Factor \(SS_{\mbox{Factor}}\) \(a-1\) \(\frac{SS_{\mbox{Factor}}}{a-1}\) \(F_0=\frac{MS_{\mbox{Factor}}}{MS_E}\)
Blocks \(SS_{\mbox{Blocks}}\) \(b-1\) \(\frac{SS_{\mbox{Blocks}}}{b-1}\)
Error \(SS_E\) \((a-1)(b-1)\) \(\frac{SS_E}{(a-1)(b-1)}\)
Total \(SS_T\) \(N-1\)

SS formula for CRBD

\[ \begin{aligned} SS_T&=&\sum_{i=1}^a\sum_{j=1}^by^2_{ij}-\frac{y^2_{..}}{N}\\ SS_{\mbox{Treatments}}&=&\frac{1}{b}\sum_{i=1}^ay^2_{i.}-\frac{y^2_{..}}{N}\\ SS_{\mbox{Blocks}}&=&\frac{1}{a}\sum_{i=1}^by^2_{.j}-\frac{y^2_{..}}{N}\\ SS_e&=&SS_T-SS_{\mbox{Treatments}}-SS_{\mbox{Blocks}}\end{aligned} \]

More Details

  • If the fixed effects model is used, multiple comparisons can be made.

  • Blocking is a restriction on randomization. In a completely randomized design, there are \(an\) experimental units and treatments are randomly applied to each experimental unit

  • A completely randomized block design, we have \(b\) blocks and randomization happens within each block (not experimental unit).