Download Lecture 11. Experimental design. Blocking

Lecture 11. Experimental design. Blocking
Jesper Rydén
Matematiska institutionen, Uppsala universitet
[email protected]
Regression and Analysis of Variance • fall 2014
Blocking: An important technique in design of experiments
Earlier example of blocking encountered: the paired t-test.
Examples of blocks: people/operators, batches, time (days,
measurement occasions)
Randomized block design with four treatmentments A, B, C and D,
and three blocks. Randomization within blocks.
The model for a randomized block design
The response y for a randomized block design is a function of two
qualitative variables: blocks and treatments.
Example of model with four treatments A, B, C and D and three
blocks:
y = β0 + β1 x1 + β2 x2 + β3 x3 + β4 x4 + β5 x5 + where
x1
x1
x2
x2
x3
x3
x4
x4
x5
x5
=1
=0
=1
=0
=1
=0
=1
=0
=1
=0
if
if
if
if
if
if
if
if
if
if
measurement made in Block 2,
not;
measurement made in Block 3,
not;
treatment B is applied,
not;
treatment C is applied,
not;
treatment D is applied,
not.
How the randomized block design reduces noise
Interpretations of β parameters? Suppose we are interested in
predicting the average response for Treatment A in Block 1. For
such an observation
x1 = x2 = x3 = x4 = x5 = 0
and thus
y = β0 + .
Conclusion: β0 is the average response for Treatment A in Block 1.
Blackboard
ANOVA table: Randomized Complete Block Design
Block design ANOVA table – we have met before?
The two-factor factorial model, one observation per cell, looks
exactly like the randomized complete block model.
The experimental situations that lead to the two models are very
different:
Factorial model: All ab runs are made in random order.
Randomized block model: Randomization occurs only within the
block.
Remarks
Testing for block effect. Careful when testing for blocks using
(MSBlocks /MSE ), should be used informally. See comments by
Montgomery, referring to Box and Hunter.
When is blocking necessary? Suppose an experiment is
conducted as a randomized block design, and blocking was not
really necessary. Then:
ab observations and (a − 1)(b − 1) degrees of freedom for error.
If run as completely randomized single-factor design with b
replicates,
ab observations and a(b − 1) degrees of freedom for error.
When is blocking necessary?
Montgomery and Runger (2011):
As a general rule, when in doubt as to the importance of block
effects, the experimenter should block and gamble that the block
effect does exist.
If the experimenter is wrong, the slight loss in the degrees of
freedom for error will have a negligible effect, unless the number of
degrees of freedom is very small.
Example. Comparison of fertilizers
The effect of three fertilizers, A, B and C, is investigated. The
yield was measured at 12 test squares, more precisely 4 blocks with
3 test squares in each. Incomplete ANOVA table:
Source
Fertilizer
Block
Residual
Total
D.f.
S. Sq.
312
132
24
468
MS
Write down a model, complete the ANOVA table. Test hypotheses
of interest. Fertilizer A is a standard one, while B and C are new
options. Differences?
Blackboard
Blocking in two directions: Latin Square
Experiment: The effect of five different chemicals on a material
manufactured in a continous process (paper, textiles). The
response of interest is the strength of the material, collected on a
roll.
Blocking along the roll (variability over time).
Blocking across the roll.
Latin squares – Fisher, Sudoku. . .
Latin square
A 5 × 5 latin-square design:
Latin square: regression model
Consider a 3 × 3 latin square.
y = β0 + β1 x1 + β2 x2 + β3 x3 + β4 x4 +
|
{z
}
|
{z
}
Row differences
Column differences
β x +β x
| 5 5 {z 6 }6
Treatment differences
+
Latin square, cont.
Let β0 be the average response for treatment A in row 1,
column 1. Further:
β1
β2
β3
β4
β5
β6
=
=
=
=
=
=
difference
difference
difference
difference
difference
difference
between
between
between
between
between
between
rows 2 and 1,
rows 3 and 1,
columns 2 and 1,
columns 3 and 1,
treatments B and A,
treatments C and A.
For example, the model for the observation in row 2, column 3
implies
x1 = 1,
x2 = 0,
x3 = 0,
x4 = 1,
x5 = 0,
x6 = 0.
Latin Square: ANOVA table
Example. Latin Square
Gasoline consumption, four different gasolines A, B, C and D. Four
cars, four drivers.
D 19.5
B 18.0
A 18.1
C 20.1
B 16.8
A 17.9
D 21.0
D 19.2
A
C
B
B
19.8
17.9
17.5
17.0
C 19.2
D 17.7
B 17.2
A 18.5
Is there a difference between treatments? Between cars, drivers?
Example. Yield of grass.
The yield from two types of grass is investigated. The yield could
depend on the number of harvests per year: two, three or four
times. Moreover, the experiment is conducted in four blocks.
Block
1
2
3
4
Elephant grass
2
3
4
109 222 187
97 125 163
133 134 143
113 173 179
Guatemala
2
3
277 246
293 263
260 194
325 190
grass
4
252
181
224
248
Example, cont.
ANOVA table (incomplete):
Source
G: Grass type
H: Harvests/year
G*H (interaction)
Block
Error
Total
D.f.
S. Sq.
57526.04
225
18176.33
4478.46
12734.79
93140.63
M. Sq.
Write down the model, complete the ANOVA table, test relevant
hypotheses and make an interpretation of interaction. Which
combination of grass type and harvests/year is most benefitial for
the yield?