Question 9.16: A 2^5 factorial experiment was conducted to estimate the eff......

ISE Principles of Statistics for Engineers and Scientists [987713]

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 mate-rial (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 negligi-ble, 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	25.10	de	40.60
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	28.80	bcd	48.26	bce	28.99	bcde	45.15
abc	43.05	abcd	53.28	abce	48.92	abcde	48.72

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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.

Sum of squares for an effect = $\frac{K(contrast)^{2} }{2p}$ (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.

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	6.70	358.72	ACE	−0.69	3.75
E	0.58	2.66	ADE	−0.45	1.60
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	0.60	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	23.80
ABC	1.35	14.47

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Script File

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)

Term Effect Coef SE Coef T P
Constant 39.658 0.5854 67.74 0.000
A 6.325 3.163 0.5854 5.40 0.000
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*B 2.840 1.420 0.5854 2.43 0.027
A*C 0.183 0.091 0.5854 0.16 0.878
A*D -3.385 -1.693 0.5854 -2.89 0.011
A*E 0.595 0.298 0.5854 0.51 0.618
B*C -0.494 -0.247 0.5854 -0.42 0.679
B*D 4.131 2.066 0.5854 3.53 0.003
B*E 0.654 0.327 0.5854 0.56 0.584
C*D -0.179 -0.089 0.5854 - 0.15 0.881
C*E -0.809 -0.404 0.5854 -0.69 0.500
D*E 0.239 0.119 0.5854 0.20 0.841

S = 3.31179 R-Sq = 90.89% R-Sq(adj) = 82.34%

Analysis of Variance for Response (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 5 1443.1 1443.1 288.62 26.31 0.000
2-Way Interactions 10 307.1 307.1 30.71 2.80 0.032
Residual Error 16 175.5 175.5 10.97
Total 31 1925.7

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 D are 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.