Metal Bonding Experiment. Authors: E. Del Castillo, D. C. Montgomery and D. R. M

Question

Metal Bonding Experiment. Authors: E. Del Castillo, D. C. Montgomery and D. R. McCarville (1996), “Modified desirability functions for multiple response optimisation. Journal of Quality Technology, 28, pp. 337-345. The experiment below is on a melting bonding process and was designed to determine the effect of the flow rate, the flow temperature and the block temperature on the maximum temperature recorded at the middle of the weld
i   yi xi1   xi2 xi3 1 115 40   200 250 2 117 120   200 250 3 147 40 450 250 4 199 120   450 250 5 134 40   325 150 6   134 120   325 150 7 143 40 325 350 8 152 120 325 350 9 111 80 200 150 10 176 80   450 150 11   131 80 200 350 12     192 80 450 350 13 155 80 325 250 14 161 80 325 250 15 158 80   325 250 i   yi xi1   xi2 xi3 1 115 40   200 250 2 117 120   200 250 3 147 40 450 250 4 199 120   450 250 5 134 40   325 150 6   134 120   325 150 7 143 40 325 350 8 152 120 325 350 9 111 80 200 150 10 176 80   450 150 11   131 80 200 350 12     192 80 450 350 13 155 80 325 250 14 161 80 325 250 15 158 80   325 250
yi = Temperature recorded in the middle of the weld. xi1 = Flow rate. xi2 = Flow temperature in degrees Celsius. xi3 = Block temperature in degrees Celsius . i = 1 to n measurements. n = 15. yi = Temperature recorded in the middle of the weld. xi1 = Flow rate. xi2 = Flow temperature in degrees Celsius. xi3 = Block temperature in degrees Celsius . i = 1 to n measurements. n = 15.


Peak Height Experiment. Authors: Panagiotis Kakleas, Triantafyllos Kaloudis and Ewan MacArthur (2008), “Speeding up chemical analysis; an example of developing fast yet reliable methods for the determination of cylindrospermopsin in water, by using HPLC and ELISA”, Proceedings of the eRA 3 Conference, TEI of Piraeus, Aegina, 19-20 September, 2008.
yi = Peak height for detecting cylindrospermopsin in water measured by high pressure liquid chromatography (HPLC).
xi1 = Flow rate.
xi2 = Column temperature in degrees Celsius.
xi3 = Gradient time.
i = 1 to n measurements. n = 15. Peak Height Experiment. Authors: Panagiotis Kakleas, Triantafyllos Kaloudis and Ewan MacArthur (2008), “Speeding up chemical analysis; an example of developing fast yet reliable methods for the determination of cylindrospermopsin in water, by using HPLC and ELISA”, Proceedings of the eRA 3 Conference, TEI of Piraeus, Aegina, 19-20 September, 2008.
yi = Peak height for detecting cylindrospermopsin in water measured by high pressure liquid chromatography (HPLC).
xi1 = Flow rate.
xi2 = Column temperature in degrees Celsius.
xi3 = Gradient time.
i = 1 to n measurements. n = 15.

i yi xi1 xi2 xi3 1 2.088 1.3 40 10 2 2.084 1.3 40 10 3 1.435 2.1 30 10 4 2.835 0.5 50 10 5 1.809 1.3 30 15 6 2.637 0.5 40 15 7 1.65 2.1 40 5 8 2.296 1.3 50 5 9 2.146 1.3 40 10 10 1.626 2.1 40 15 11 2.752 0.5 40 5 12 2.626 0.5 30 10 13 2.07 1.3 30 5 14 2.218 1.3 50 15 15 1.763 2.1 50 10 i yi xi1 xi2 xi3 1 2.088 1.3 40 10 2 2.084 1.3 40 10 3 1.435 2.1 30 10 4 2.835 0.5 50 10 5 1.809 1.3 30 15 6 2.637 0.5 40 15 7 1.65 2.1 40 5 8 2.296 1.3 50 5 9 2.146 1.3 40 10 10 1.626 2.1 40 15 11 2.752 0.5 40 5 12 2.626 0.5 30 10 13 2.07 1.3 30 5 14 2.218 1.3 50 15 15 1.763 2.1 50 10
QUESTION 2 Calculate 97.5% confidence intervals for all the estimated parameters in the second order response surface model. Based on these intervals which one of the following statements is correct: VariableX2 is important in controlling the response but only via its interaction with variable X3- o variable X2 is important in controlling the response but only via its interaction with variable X1. Variable X2 is important, but its effect on the response is limited to being linear in nature. None of these options. Variable X2 is not important in controlling the response. Variable X2 is important, and its effect on the response is non linear in nature.

Explanation / Answer

We can do in R

library(rsm)
data=read.csv("Boo.csv")
y=data[,2]
x1=data[,3]
x2=data[,4]
x3=data[,5]
rs=rsm(y ~ SO(x1, x2, x3), data = data)
summary(rs)

Call:
rsm(formula = y ~ SO(x1, x2, x3), data = data)

               Estimate Std. Error t value Pr(>|t|)
(Intercept) 28.99125000 55.56135906 0.5218 0.62410
x1           0.50625000 0.53705992 0.9426 0.38915
x2           0.09640000 0.19582006 0.4923 0.64337
x3           0.29100000 0.23952288 1.2149 0.27864
x1:x2        0.00250000 0.00077363 3.2315 0.02317 *
x1:x3        0.00056250 0.00096703 0.5817 0.58602
x2:x3       -0.00008000 0.00030945 -0.2585 0.80631
x1^2        -0.00789063 0.00251630 -3.1358 0.02579 *
x2^2        -0.00005600 0.00025767 -0.2173 0.83654
x3^2        -0.00046250 0.00040261 -1.1488 0.30263
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Multiple R-squared: 0.9694,    Adjusted R-squared: 0.9143
F-statistic: 17.6 on 9 and 5 DF, p-value: 0.002824

Analysis of Variance Table

Response: y
                Df Sum Sq Mean Sq F value    Pr(>F)
FO(x1, x2, x3)   3 8192.3 2730.75 45.6266 0.0004689
TWI(x1, x2, x3) 3 649.2 216.42 3.6160 0.1001556
PQ(x1, x2, x3)   3 638.6 212.86 3.5566 0.1028544
Residuals        5 299.3   59.85                
Lack of fit      3 281.3   93.75 10.4167 0.0888549
Pure error       2   18.0    9.00                

Stationary point of response surface:
        x1         x2         x3
-56.17091 -632.48491 335.13801

Eigenanalysis:
$values
[1] 0.0001386252 -0.0004519532 -0.0080957970

$vectors
           [,1]        [,2]        [,3]
x1 -0.154041062 -0.03577014 0.98741676
x2 -0.988044200 0.01197400 -0.15370518
x3 -0.006325272 -0.99928831 -0.03718697

variable x2 is important in controlling the response but only via the interaction with variable x1.

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