Question

4. In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C) and temperature (D). A single replicate of a 24 design was run, and the resulting are shown in the Table below a. Construct a normal probability plot of the effect estimates. Which factors appear to have large effects? b. Conduct an analysis of variance using the normal probability plot in part (a) for guidance in forming an error term. What are your conclusions! c. Write down a regression model relating yield to the important process variables. d. Analyze the residuals. Does your analysis indicate any potential problems? e. Can this design be collapsed into a 23 design with two raplicates f. Use the regression model to generate a response surface contour plot of yielc. Yield Actual Run Factor order A B CD (lba) High 18 A (h) 2.5 14 13 C (psi) D (deg 60 250 15 15 13 19 13 10 15

Answer 1

(a) The table below provided by Minitab shows effect estimates for the yield experiment in which a 24 factorial design with a single replicate is run, using factors As Time (-= 2.5 h, + = 3 h) B= Concentration (-=14%, +=18%) C-Pressure (-= 60 psi, + = 80 psi) D= Temperature (-= 225 C, +-250 Estimated Effects and Coefficients for y (coded units) Term Effect Coef Constant 17.375 4.500 2.250 0.500 0.250 2.000 1.000 3.250 1.625 -0.750 -0.375 A*B 4.250-2.125 4.000 2.000 0.250 0.125 0.000 0.000 0.000 0.000 1.000 0.500 0.750 0.375 A*B*D A*C*D -0.250 -0.125 -0.750 -0.375 B*C*D 1.000 0.500 A*B*C*D

To identify which effects are important in an unreplicated 2* factorial design, we can use a simple method by constructing a normal probability plot of effects. On the normal plot of effect s, effects that are negligible will fall approximately along a straight line (i.e., normally with mean zero), whereas effects that are important will deviate with nonzero means. The figure below provided by Minitab shows the normal probability plot of effect estimates from the yield experiment.

XI AD AD C 60 + U 50 30 10 AC 2.5 0.0 5.0 2.5 Effect From the normal probability plot of the effects, we see that factors A , C, and D, and the AC and AD interactions appear to have large effects because they deviate significantly from the other points that fall approximately along a straight line

(b) We use Minitab to perform the analysis of variance for the yield experiment as shown below, using the normal probability plot of the effects in part (a) as guidance in forming the mean square for residual error. The analysis of variance confirms our findings that the main effects of A, C and D, and the Ac and AD interactions are significant Factorial Fit: y versus A, C, D Estimated Effects and Coefficients for y (coded units) Term Effect Coef SE Coef Constant 17.375 0.3187 54.52 0.000 4.500 2.250 0.3187 7.06 0.000 2.000 1.000 0.3187 3.14 0.011 3.250 1.6250.3187 5.10 0.000 A*C -4.250 -2.125 0.3187 -6.67 0.000 A*D 4.000 2.000 0.3187 6.28 0.000

Analysis of Variance for y (coded units) Source DF Seq SS Adj Ss Adj MS Main Effects 3 139.250 139.250 46.4167 28.56 0.000 1 81.000 81.000 81.0000 49.85 0.000 1 16.000 16.000 16.0000 9.85 0.011 1 42.250 42.250 42.2500 26.00 0.000 2-Way Interactions2 136.250 136.250 68.1250 41.92 0.000 1 72.250 72.250 72.2500 44.46 0.000 A*D 1 64.000 64.000 64.0000 39.38 0.000 Residual Error 10 16.250 16.250 1. 6250 Lack of Fit 2 0.250 0.250 0.1250 0.06 0.940 Pure Error 8 16.000 16.000 2.0000 15 291.750 Total C) The analysis of a 2k factorial design assumes that the sampled observations are normally and independently distributed with equal variances for each factor level (treatment). To check these assumptions, we can use residual plots for the experiment. For a 2* factorial design, a residual is the difference between an individual observation and its fitted value, denoted as у-у е

To check the normality assumption, we can construct a normal probability plot of the residuals. If data points fall approximately along a straight line (i.e., the normal probability plot appears linear), it indicates that the observations have an approximately normal distribution. If data points have significant or systematic deviations from a straight line (i.e., the normal probability plot does not appear linear), it indicates that the observations do not have an approximately normal distribution. The figure below provided by Minitab shows a normal probability plot of the residuals from the yield experiment.

ormal Probabil (response is yield) 80 TH 70 U 50 H 40 -2 Residual

From Minitab's normal probability plot of the residuals, we see that the data points fall approximately along the straight line. Therefore, there is no indication of a problem with the normality assumption.

To check the assumption of equal variances, we can plot the residuals against the fitted values y If the variability in the residuals depends in some way on the values of y, we may conclude that the equal variances assumption is violated. The figure below provided by Minitab shows a plot of residuals versus the fitted values y from the yield experiment. Versus Fits (response is yield) 1.5 1.0 0.5 0.0 -0.5 1.0 1.5 2.0 10 12 14 16 18 20 24 26 Fitted Value

From Minitab's plot of the residuals versus y, we see that the variability in the residuals appears not to depend on the values of y. Therefore, we conclude that the assumption of equal variances is adequate. To check the assumption of equal variances, we can also plot the residuals against the levels of factors. If the variability in the residuals depends in some way on the values of a factor, we may conclude that the equal variances assumption is violated. The figure below provided by Minitalb shows a plot of residuals versus time (factor A) from the yield experiment.

Residuals Versus Time (response is yield) 1.5 1.0 0.5 O0.0 O-0.5 2.0 2.5h 3h A (Time) The plot of residuals versus time suggests that Time 2.5 h has greater variability in yield than Time 31h The figure below provided by Minitab shows a plot of residuals versus concentration (factor B) from the yield experiment.

The figure below provided by Minitab shows a plot of residuals versus pressure (factor C) from the yield experiment. eseu (response is yield) Residuals Versus Pressure 1.0 0.5 O0.0 -0.5 2.0 80psi 60psi C (Pressure) The plot of residuals versus pressure suggests that the two levels of pressure have similar variability in yield

Residuals Versus Concentration (response is yield) 1.5- 1.00 0.5 0.0 -0.5 1.00 2.0 L 14% 18% B (Concentration) The plot of residuals versus concentration suggests that the two levels of concentration have similar variability in yield.

(d) For the yield experiment, since the main effects of A. C.and D, and the Ac and AL interactions are significant, the original 24 design can be collapsed into a 23 design with two replicates using only the factors time (A). pressure (C), and temperature (D). The table below shows the average and range of yields at each run for the new 23 design. Note that in the new 23 design, design factors A, B, and C represent time, pressure, and temperature respectively Treatment Design FactorYield Combination B C RangeAverage 12.5 -1 17.0 -12 18.5 13 -1

15.0 ab 1 11.5 24.5 ac 18.0 bc abc 1 2 22.0 This 23 design sketch aids in data interpretation because the range of yields at each run is very small relative to the average.