14 The accompanying data is from the article “Quick and Easy Analysis of
ExamplE 11.14 The accompanying data is from the article “Quick and Easy Analysis of
Unreplicated Factorials” (Technometrics, 1989: 469–473) . The four factors are
A5 acid strength, B 5 time, C 5 amount of acid, and D 5 temperature, and the response variable is the yield of isatin. The observations, in standard order, are .08, .04, .53, .43, .31, .09, .12, .36, .79, .68, .73, .08, .77, .38, .49, and .23. Table 11.14 displays the effect estimates as given in the article (which uses contrast8 rather than contrast16).
474 Chapter 11 Multifactor analysis of Variance
Table 11.14 Effect Estimates for Example 11.14 Effect
AD BD ABD CD ACD BCD ABCD
Figure 11.11 is a normal probability plot of the effect estimates. All points in the plot fall close to the same straight line, suggesting the complete absence of any effects (we will shortly give an example in which this is not the case).
Effect estimate 0.3 0.2 0.1 0.0
–0.1 –0.2 –0.3
z percentile
Figure 11.11 A normal probability plot of effect estimates from Example 11.14
n
Visual judgments of deviation from straightness in a normal probability plot are rather subjective. The article cited in Example 11.14 describes a more objective technique for identifying significant effects in an unreplicated experiment.
confounding
It is often not possible to carry out all 2 p experimental conditions of a 2 p factorial experiment in a homogeneous experimental environment. In such situations, it may
be possible to separate the experimental conditions into 2 r homogeneous blocks (r , p), so that there are 2 p2r experimental conditions in each block. The blocks may, for example, correspond to different laboratories, different time periods, or different operators or work crews. In the simplest case, p 5 3 and r 5 1, so that there are two blocks, with each block consisting of four of the eight experimental conditions.
As always, blocking is effective in reducing variation associated with extrane- ous sources. However, when the 2 p experimental conditions are placed in 2 r blocks,
the price paid for this blocking is that 2 r 2 1 of the factor effects cannot be esti- mated. This is because 2 r 2 1 factor effects (main effects andor interactions) are
mixed up, or confounded , with the block effects. The allocation of experimental conditions to blocks is then usually done so that only higher-level interactions are confounded, whereas main effects and low-order interactions remain estimable and hypotheses can be tested.
11.4 2 p Factorial experiments 475
To see how allocation to blocks is accomplished, consider first a 2 3 experiment
with two blocks (r 5 1) and four treatments per block. Suppose we select ABC as the effect to be confounded with blocks. Then any experimental condition having an odd number of letters in common with ABC, such as b (one letter) or abc (three letters), is placed in one block, whereas any condition having an even number of letters in common with ABC (where 0 is even) goes in the other block. Figure 11.12 shows this allocation of treatments to the two blocks.
Block 1
Block 2
(1), ab, ac, bc
a , b, c, abc
Figure 11.12 Confounding ABC in a 2 3 experiment
In the absence of replications, the data from such an experiment would usually
be analyzed by assuming that there were no two-factor interactions (additivity) and using SSE 5 SSAB 1 SSAC 1 SSBC with 3 df to test for the presence of main effects. Alternatively, a normal probability plot of effect contrasts or effect parameter estimates could be examined. Most frequently, though, there are replications when just three factors are being studied. Suppose there are u replicates, resulting in a total of 2 r ? u blocks in the experiment. Then after subtracting from SST all sums of squares associated with effects not confounded with blocks (computed using Yates’s method), the block sum of squares is computed using the 2 r ? u block totals and then subtracted to yield SSE (so there are 2 r ? u2 1 df for blocks).
ExamplE 11.15
The article “Factorial Experiments in Pilot Plant Studies” (Industrial and Eng.
Chemistry, 1951: 1300–1306) reports the results of an experiment to assess the effects of reactor temperature (A), gas throughput (B), and concentration of active constituent (C) on the strength of the product solution (measured in arbitrary units) in a recirculation unit. Two blocks were used, with the ABC effect confounded with blocks, and there were two replications, resulting in the data in Figure 11.13. The four block 3 replication totals are 288, 212, 88, and 220, with a grand total of 808, so
s288d 2 1 s212d 2 1 s88d 2 1 s220d 2 s808d 2
Figure 11.13 Data for Example 11.15
The other sums of squares are computed by Yates’s method using the eight experi- mental condition totals, resulting in the ANOVA table given as Table 11.15. By com-
parison with F .05,1,6 5 5.99, we conclude that only the main effects for A and C differ
significantly from zero (P-value , .05 for just f A and f C ).
476 Chapter 11 Multifactor analysis of Variance
Table 11.15 ANOVA Table for Example 11.15
Source of Variation
df Sum of Squares
Mean Square f
.64 AC 1 30.25 .093 BC 1 25 .077
confounding using More than two Blocks
In the case r 5 2 (four blocks), three effects are confounded with blocks. The experimenter first chooses two defining effects to be confounded. For example, in
a five- factor experiment (A, B, C, D, and E), the two three-factor interactions BCD and CDE might be chosen for confounding. The third effect confounded is then the generalized interaction of the two, obtained by writing the two chosen effects side by side and then cancelling any letters common to both: sBCDdsCDEd 5 BE. Notice that if ABC and CDE are chosen for confounding, their generalized interac- tion is sABCdsCDEd 5 ABDE, so that no main effects or two-factor interactions are confounded.
Once the two defining effects have been selected for confounding, one block consists of all treatment conditions having an even number of letters in common with both defining effects. The second block consists of all conditions having an even number of letters in common with the first defining contrast and an odd number of letters in common with the second contrast, and the third and fourth blocks consist of the “oddeven” and “oddodd” contrasts. In a five-factor experiment with defining effects ABC and CDE, this results in the allocation to blocks as shown in Figure 11.14 (with the number of letters in common with each defining contrast appearing beside each experimental condition).
abcde (3, 3) Figure 11.14 Four blocks in a 2 5 factorial experiment with defining effects ABC and CDE
abce
The block containing (1) is called the principal block . Once it has been con- structed, a second block can be obtained by selecting any experimental condition not in the principal block and obtaining its generalized interaction with every condition in the principal block. The other blocks are then constructed in the same way by
11.4 2 p Factorial experiments 477
first selecting a condition not in a block already constructed and finding generalized interactions with the principal block.
For experimental situations with p . 3, there is often no replication, so sums of squares associated with nonconfounded higher-order interactions are usually pooled to obtain an error sum of squares that can be used in the denominators of the various F statistics. All computations can again be carried out using Yates’s technique, with SSBl being the sum of sums of squares associated with confounded effects.
When r . 2, one first selects r defining effects to be confounded with blocks, making sure that no one of the effects chosen is the generalized interaction of any
other two selected. The additional 2 r 2 r2 1 effects confounded with the blocks are
then the generalized interactions of all effects in the defining set (including not only generalized interactions of pairs of effects but also of sets of three, four, and so on).