12 In an experiment to investigate the compressive strength properties of cement–soil
Example 11.12 In an experiment to investigate the compressive strength properties of cement–soil
mixtures, two different aging periods were used in combination with two different temperatures and two different soils. Two replications were made for each combi- nation of levels of the three factors, resulting in the following data:
The computed cell totals are x 111 . 5 884, x 211 . 5 1349, x 121 . 5 1037, x 221 . 5 1501,
x 112 . 5 819, x 212 . 5 1475, x 122 . 5 1123 , and x 222 . 5 1547 , so x ” 5 9735 . Then
aˆ 1 5 (884 2 1349 1 1037 2 1501 1 819 2 1475 1 1123 2 1547)16
5 2125.5625 5 2aˆ 2
gˆ AB 11 5 (884 2 1349 2 1037 1 1501 1 819 2 1475 2 1123 1 1547)16
5 214.5625 5 2gˆ AB AB 12 AB 5 2gˆ 21 5 gˆ 22
The other parameter estimates can be computed in the same manner.
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Analysis of a 2 3 Experiment Sums of squares for the various effects are easily
obtained from the parameter estimates. For example,
SSA 5 g g g g aˆ 2 i 2 5 4n g aˆ i 5 4n[aˆ 2 1 (2a 2 ] 5 8naˆ 1 2 ˆ 1 ) 1
SSAB 5 g g g g (gˆ AB ij ) 2
i j k l
5 2n g g (gˆ AB ) 2 5 2n[(gˆ AB ) 2 AB ij 2 1 (2gˆ 2 AB AB 11 2 11 ) 1 (2gˆ 11 ) 1 (gˆ 11 ) ]
i51 j51
5 8n(gˆ AB 11 ) 2
Since each estimate is a contrast in the cell totals multiplied by 1(8n) , each sum
of squares has the form (contrast) 2 (8n). Thus to compute the various sums of squares, we need to know the coefficients ( 11 or 21 ) of the appropriate contrasts. The signs
(1 or 2) on each x ijk in each effect contrast are most conveniently displayed in a
table. We will use the notation (1) for the experimental condition i 5 1, j 5 1, k 5 1, a for i 5 2, j 5 1, k 5 1 , ab for i 5 2, j 5 2, k 5 1 , and so on. If level 1 is thought of as “low” and level 2 as “high,” any letter that appears denotes a high level of the associated factor. Each column in Table 11.10 gives the signs for a particular
effect contrast in the x ijk ’s associated with the different experimental conditions.
11.4 2 p Factorial Experiments
Table 11.10 Signs for Computing Effect Contrasts Experimental Cell
Factorial Effect
In each of the first three columns, the sign is 1 if the corresponding factor is at the high level and 2 if it is at the low level. Every sign in the AB column is then
the “product” of the signs in the A and B columns, with (1)(1) 5 (2)(2) 5 1 and (1)(2) 5 (2)(1) 5 2 , and similarly for the AC and BC columns. Finally, the signs in the ABC column are the products of AB with C (or B with AC or A with BC). Thus, for example,
AC contrast 5 1 x 111 2x 211 1x 121 2x 221 2x 112 1x 212 2x 122 1x 222
Once the seven effect contrasts are computed, (effect contrast) 2
SS(effect) 5 8n
Software for doing the calculations required to analyze data from factorial exper- iments is widely available (e.g., Minitab). Alternatively, here is an efficient method for hand computation due to Yates. Write in a column the eight cell totals in the standard order, as given in the table of signs, and establish three additional columns. In each of these three columns, the first four entries are the sums of entries 1 and 2, 3 and 4, 5 and 6, and 7 and 8 of the previous columns. The last four entries are the differences between entries 2 and 1, 4 and 3, 6 and 5, and 8 and 7 of the previous column. The last
column then contains x and the seven effect contrasts in standard order. Squaring
each contrast and dividing by 8n then gives the seven sums of squares.
Example 11.13 Since n 5 2, 8n 5 16 . Yates’s method is illustrated in Table 11.11. (Example 11.12 continued)
Table 11.11 Yates’s Method of Computation
Treatment
Condition
x ijk 1 2 Effect Contrast
SS 5 (contrast) 2 16
a5x 211 1349
b5x 121 1037
ab 5 x 221 1501
ac 5 x 212 1475
bc 5 x 122 1123
abc 5 x 222 1547
CHAPTER 11 Multifactor Analysis of Variance
From the original data, g i g j g k g l x 2 ijkl 5 6,232,289, and x 2
16 so SST 5 6,232,289 2 5,923,139.06 5 309,149.94
SSE 5 SST 2 [SSA 1 c 1 SSABC] 5 309,149.94 2 292,036.42
5 17,113.52 The ANOVA calculations are summarized in Table 11.12.
Table 11.12 ANOVA Table for Example 11.13 Source of
f
Variation
df Sum of Squares
Mean Square
Figure 11.10 shows SAS output for this example. Only the P-values for age (A) and temperature (B) are less than .01, so only these effects are judged significant.
Analysis of Variance Procedure Dependent Variable: STRENGTH
DF Squares
Square
F Value
Corrected Total
R-Square
C.V. Root MSE
POWERUSE Mean
DF Anova SS Mean Square
F Value Pr . F
Figure 11.10 SAS output for strength data of Example 11.13
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