Chapter 4 Randomized Blocks, Latin Squares, and Related Designs

1
Chapter 4 Randomized Blocks, Latin Squares, and
Related Designs
2
4.1 The Randomized Complete Block Design

• Nuisance factor a design factor that probably
  has an effect on the response, but we are not
  interested in that factor.
• If the nuisance variable is known and
  controllable, we use blocking
• If the nuisance factor is known and
  uncontrollable, sometimes we can use the analysis
  of covariance (see Chapter 14) to remove the
  effect of the nuisance factor from the analysis

3

• If the nuisance factor is unknown and
  uncontrollable (a lurking variable), we hope
  that randomization balances out its impact across
  the experiment
• Sometimes several sources of variability are
  combined in a block, so the block becomes an
  aggregate variable

4

• We wish to determine whether 4 different tips
  produce different (mean) hardness reading on a
  Rockwell hardness tester
• Assignment of the tips to an experimental unit
  that is, a test coupon
• Structure of a completely randomized experiment
• The test coupons are a source of nuisance
  variability
• Alternatively, the experimenter may want to test
  the tips across coupons of various hardness
  levels
• The need for blocking

5

• To conduct this experiment as a RCBD, assign all
  4 tips to each coupon
• Each coupon is called a block that is, its a
  more homogenous experimental unit on which to
  test the tips
• Variability between blocks can be large,
  variability within a block should be relatively
  small
• In general, a block is a specific level of the
  nuisance factor
• A complete replicate of the basic experiment is
  conducted in each block
• A block represents a restriction on randomization
• All runs within a block are randomized

6

• Suppose that we use b 4 blocks
• Once again, we are interested in testing the
  equality of treatment means, but now we have to
  remove the variability associated with the
  nuisance factor (the blocks)

7

• Statistical Analysis of the RCBD
• Suppose that there are a treatments (factor
  levels) and b blocks
• A statistical model (effects model) for the RCBD
  is
• ? is an overall mean, ?i is the effect of the
  ith treatment, and ?j is the effect of the jth
  block
• ?ij NID(0,?2)

8

• Means model for the RCBD
• The relevant (fixed effects) hypotheses are
• An equivalent way for the above hypothesis
• Notations

9

• ANOVA partitioning of total variability

10

• SST SSTreatment SSBlocks SSE
• Total N ab observations, SST has N 1 degrees
  of freedom.
• a treatments and b blocks, SSTreatment and
  SSBlocks have a 1 and b 1 degrees of freedom.
• SSE has ab 1 (a 1) (b 1) (a 1)(b
  1) degrees of freedom.
• From Theorem 3.1, SSTreatment /?2, SSBlocks / ?2
  and SSE / ?2 are independently chi-square
  distributions.

11

• The expected values of mean squares
• For testing the equality of treatment means,

12

• The ANOVA table
• Another computing formulas

13

• Example 4.1
• 4.1.2 Model Adequacy Checking
• Residual Analysis
• Residual
• Basic residual plots indicate that normality,
  constant variance assumptions are satisfied
• No obvious problems with randomization

14
(No Transcript)
15
(No Transcript)
16

• Can also plot residuals versus the type of tip
  (residuals by factor) and versus the blocks. Also
  plot residuals v.s. the fitted values. Figure 4.5
  and 4.6 in Page 137
• These plots provide more information about the
  constant variance assumption, possible outliers
• 4.1.3 Some Other Aspects of the Randomized
  Complete Block Design
• The model for RCBD is complete additive.

17

• Interactions?
• For example
• The treatments and blocks are random.
• Choice of sample size
• Number of blocks ?, the number of replicates and
  the number of error degrees of freedom ?

18

• Estimating miss values
• Approximate analysis estimate the missing values
  and then do ANOVA.
• Assume the missing value is x. Minimize SSE to
  find x
• Table 4.8
• Exact analysis

19

• 4.1.4 Estimating Model Parameters and the General
  Regression Significance Test
• The linear statistical model
• The normal equations

20

• Under the constraints,
• the solution is
• and the fitted values,
• The sum of squares for fitting the full model
• The error sum of squares

21

• The sum of squares due to treatments

22
4.2 The Latin Square Design

• RCBD removes a known and controllable nuisance
  variable.
• Example the effects of five different
  formulations of a rocket propellant used in
  aircrew escape systems on the observed burning
  rate.
• Remove two nuisance factors batches of raw
  material and operators
• Latin square design rows and columns are
  orthogonal to treatments.

23

• The Latin square design is used to eliminate two
  nuisance sources, and allows blocking in two
  directions (rows and columns)
• Usually Latin Square is a p ? p squares, and each
  cell contains one of the p letters that
  corresponds to the treatments, and each letter
  occurs once and only once in each row and column.
  
• See Page 145

24

• The statistical (effects) model is
• yijk is the observation in the ith row and kth
  column for the jth treatment, ? is the overall
  mean, ?i is the ith row effect, ?j is the jth
  treatment effect, ?k is the kth column effect and
  ?ijk is the random error.
• This model is completely additive.
• Only two of three subscripts are needed to denote
  a particular observation.

25

• Sum of squares
• SST SSRows SSColumns SSTreatments SSE
  
• The degrees of freedom
• p2 1 p 1 p 1 p 1 (p 2)(p 1)
• The appropriate statistic for testing for no
  differences in treatment means is
• ANOVA table (Table 4-10) (Page 146)
• Example 4.3

26

• The residuals
• Table 4.13
• If one observation is missing,
• Replication of Latin Squares
• Three different cases
• See Table 4.14, 4.15 and 4.16
• Crossover design Pages 150 and 151

27
4.3 The Graeco-Latin Square Design

• Graeco-Latin square
• Two Latin Squares
• One is Greek letter and the other is Latin
  letter.
• Two Latin Squares are orthogonal
• Table 4.18
• Block in three directions
• Four factors (row, column, Latin letter and Greek
  letter)
• Each factor has p levels. Total p2 runs

28

• The statistical model
• yijkl is the observation in the ith row and lth
  column for Latin letter j, and Greek letter k
• ? is the overall mean, ?i is the ith row effect,
  ?j is the effect of Latin letter treatment j , ?k
  is the effect of Greek letter treatment k, ?l is
  the effect of column l.
• ANOVA table (Table 4.19)
• Under H0, the testing statistic is
  Fp-1,(p-3)(p-1) distribution.
• Example 4.4

29
4.4 Balance Incomplete Block Designs

• May not run all the treatment combinations in
  each block.
• Randomized incomplete block design (BIBD)
• Any two treatments appear together an equal
  number of times.
• There are a treatments and each block can hold
  exactly k (k lt a) treatments.
• For example A chemical process is a function of
  the type of catalyst employed. See Table 4.22

30

• 4.4.1 Statistical Analysis of the BIBD
• a treatments and b blocks. Each block contains k
  treatments, and each treatment occurs r times.
  There are N ar bk total observations. The
  number of times each pairs of treatments appears
  in the same block is
• The statistical model for the BIBD is

31

• The sum of squares

32

• The degree of freedom
• Treatments(adjusted) a 1
• Error N a b 1
• The testing statistic for testing equality of the
  treatment effects
• ANOVA table (see Table 4.23)
• Example 4.5

33

• 4.4.2 Least Squares Estimation of the Parameters
• The least squares normal equations
• Under the constrains,
• we have

34

• For the treatment effects,