Question 9.16: A 2^5 factorial experiment was conducted to estimate the eff...
A 2^5 factorial experiment was conducted to estimate the effects of five factors on the quality of lightbulbs manufactured by a certain process. The factors were A: plant (1 or 2), B: machine type (low or high speed), C: shift (day or evening), D: lead wire material (standard or new), and E: method of loading materials into the assembler (manual or automatic). One replicate was obtained for each treatment. Table 9.9 presents the results. Compute estimates of the main effects and interactions, and their sums of squares. Assume that the third-, fourth-, and fifth-order interactions are negligible, and add their sums of squares to use as a substitute for an error sum of squares. Use this substitute to test hypotheses concerning the main effects and second-order interactions.
| TABLE 9.9 | |||||||
| Treatment | Outcome | Treatment | Outcome | Treatment | Outcome | Treatment | Outcome |
| 1 | 32.07 | d | 35.64 | e | 25.1 | de | 40.6 |
| a | 39.27 | ad | 35.91 | ae | 39.25 | ade | 37.57 |
| b | 34.81 | bd | 47.75 | be | 37.77 | bde | 47.22 |
| ab | 43.07 | abd | 51.47 | abe | 46.69 | abde | 56.87 |
| c | 31 .55 | cd | 33. 16 | ce | 32.55 | cde | 34.51 |
| ac | 36.51 | acd | 35.32 | ace | 32.56 | acde | 36.67 |
| bc | 28.8 | bcd | 48.26 | bce | 28.99 | bcde | 45.15 |
| abc | 43.05 | abcd | 53.28 | abce | 48.92 | abcde | 48.72 |
We compute the effects, using the rules for adding and subtracting observations given by the sign table, and the sums of squares, using Equation (9.46). See Table 9.10.
\text{Sum of squares for an effect}=\frac{K(\text{contrast})^2}{2^p} (9.46)
Note that none of the three-, four-, or five-way interactions are among the larger effects. If some of them were, it would not be wise to combine their sums of squares. As it is, we add the sums of squares of the three-, four-, and five-way interactions. The results are presented in the following output (from MINITAB).
Factorial Fit : Response versus A. B. C. D. E
Estimated Effects and Coefficients for Response (coded units)
\begin{array}{lrrrrr}\text{Term}& \text{Effect}& \text{Coef}& \text{SE Coef}& \text{T}& \text{P}\\\text{Constant}& & 39.658 & 0.5854 & 67.74 & 0.000 \\\text{A}& 6.325 & 3.163 & 0.5854 & 5.40 & 0.000 \\\text{B}& 9.536 & 4.768 & 0.5854 & 8.14 & 0.000\\ C & -2.066 & -1.033 & 0.5854 & -1.76 & 0.097 \\D & 6.696 & 3.348 & 0.5854 & 5.72 & 0.000 \\E & 0.576 & 0.288 & 0.5854 & 0.49 & 0.629 \\A \star B & 2.840 & 1.420 & 0.5854 & 2.43 & 0.027 \\A \star C & 0.183 & 0.091 & 0.5854 & 0.16 & 0.878 \\A \star D & -3.385 & -1.693 & 0.5854 & -2.89 & 0.011 \\A \star E & 0.595 & 0.298 & 0.5854 & 0.51 & 0.618 \\B \star C & -0.494 & -0.247 & 0.5854 & -0.42 & 0.679 \\B \star D & 4.131 & 2.066 & 0.5854 & 3.53 & 0.003 \\B \star E & 0.654 & 0.327 & 0.5854 & 0.56 & 0.584 \\C \star D & -0.179 & -0.089 & 0.5854 & -0.15 & 0.881 \\C \star E & -0.809 & -0.404 & 0.5854 & -0.69 & 0.500 \\D \star E & 0.239 & 0.119 & 0.5854 & 0.20 & 0.841\end{array}S = 3 .31179 R-Sq = 90 .89% R-Sq(adj) = 82 .34%
Analysis of Variance for Response (coded units )
\begin{array}{lrrrrrr}\text{Source}& \text{DF}& \text{Seq SS}& \text{Adj SS}& \text{Adj MS}& \text{F}& \text{P}\\\text{Main Effects}& 5 & 1443.1 & 1443.1 & 288.62 & 26.31 & 0.000 \\\text{2-Way Interactions}& 10 & 307.1 & 307.1 & 30.71 & 2.80 & 0.032 \\\text{Residual Error}& 16 & 175.5 & 175.5 & 10.97 & & \\\text{Tota1}& 31 & 1925.7 & & & &\end{array}The estimates have not changed for the main effects or two-way interactions. The residual error sum of squares ( 175.5) in the analysis of variance table is found by adding the sums of squares for all the higher-order interactions that were dropped from the model. The number of degrees of freedom ( 16) is equal to the sum of the degrees of freedom (one each) for the 16 higher-order interactions. There is no sum of squares for pure error (SSE), because there is only one replicate per treatment. The residual error sum of squares is used as a substitute for SSE to compute all the quantities that require an error sum of squares.
We conclude from the output that factors A, B, and Dare likely to affect the outcome. There seem to be interactions between some pairs of these factors as well. It might be appropriate to plan further experiments to focus on factors A, B, and D.
| TABLE 9.10 | |||||
| Term | Effect | Sum of Squares | Term | Effect | Sum of Squares |
| A | 6.33 | 320.05 | ABD | -0.29 | 0.67 |
| B | 9.54 | 727.52 | ABE | 0.76 | 4.59 |
| C | -2.07 | 34.16 | ACD | 0.11 | 0.088 |
| D | 6.7 | 358.72 | ACE | -0.69 | 3.75 |
| E | 0.58 | 2.66 | ADE | -0.45 | 1.6 |
| AB | 2.84 | 64.52 | BCD | 0.76 | 4.67 |
| AC | 0.18 | 0.27 | BCE | -0.82 | 5.43 |
| AD | -3.39 | 91.67 | BDE | -2.17 | 37.63 |
| AE | 0.6 | 2.83 | CDE | -1.25 | 12.48 |
| BC | -0.49 | 1.95 | ABCD | -2.83 | 63.96 |
| BD | 4.13 | 136.54 | ABCE | 0.39 | 1.22 |
| BE | 0.65 | 3.42 | ABDE | 0.22 | 0.37 |
| CD | -0.18 | 0.26 | ACDE | 0.18 | 0.24 |
| CE | -0.81 | 5.23 | BCDE | -0.25 | 0.52 |
| DE | 0.24 | 0.46 | ABCDE | -1.73 | 23.8 |
| ABC | 1.35 | 14.47 | |||