Workshop28_2023

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University of Colorado, Boulder *

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3010

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Computer Science

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Jan 9, 2024

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pdf

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CHEN 3010 Applied Data Analysis Fall 2023 Workshop #28: Setting Up and Analyzing a Fractional Factorial Design Objective As the number of factors gets to 5 or 6 and above, it is rarely feasible to carry out a full factorial design; consequently, fractional factorial designs are used. These designs should not be chosen in a haphazard fashion because they will then typically yield unfavorable aliasing schemes. The best choices for designs have been worked out (see table at end of workshop). The purpose of this exercise is to have you set up a realistic fractional factorial design and analyze the results in typical fashion. Procedure 1. Open a Word document for your results from this workshop. Launch Excel with a new, blank worksheet. Enter your name(s) at the top of the worksheet. "Save As" [F12 key] this file immediately as an Excel workbook file named Workshop28.xlsx . 2. First, you will create a 2 5-2 ¼-fractional factorial design. You will start by creating a standard 2 3 full factorial design, and then augment it with two additional factors. Enter centered labels A , B and C in cells B5, C5 and D5, respectively. In cells B6 and B7, enter -1 and +1 respectively. Select these two cells and copy them to B8:B9. Select cells B6:B9 and copy them to B10:B13. This creates a standard A effect column with alternating signs. Enter -1 in C6 and C7, then +1 in C8 and C9. Select C6:C9 and copy them to C10:C13. This creates a B effect column with alternating pairs of signs. Enter -1 ’s in cells D6:D9 and +1 ’s in cells D10:D13. The creates the C effect column with two blocks of signs. You can use this same approach to generate any size full factorial design. In cell E5, enter a label, based on the generator, D=AB , and in cell F5, enter another label, E=AC . This indicates that the D and E main effects will be aliased (indistinguishable one from the other) with the AB and AC effects respectively. Since this is a Resolution III design, the main effects, A, B, C, D, and E, are not aliased with each other, but they are aliased with binary and higher effects. Therefore, we proceed with the analysis on the assumption that the main effects are the only significant effects. Anything neglected will be lumped into the error term in the analysis of variance. Using the typical multiplicative formulas, complete row 6, columns E and F, and then copy E6:F6 down to fill out the table. In column A, enter labels for the various treatments, based on the effect A through E columns (enter the lower-case letter if there is a +1 in the column).

CHEN 3010 Workshop 28 Page 2 Create column heading to the right of the effect E column for Response. Generally, experiments are run in a randomized order (Minitab randomizes the run order automatically unless you tell it not to). In the table below, you will find one replicate (n=1) of responses along with their respective treatment combinations. Enter these on your speadsheet in the appropriate order by matching treatment combinations. In cell G15, compute SST. A shortcut for this is =VAR.S(G6:G13)*(COUNT(G6:G13)-1) Enter a label SST in cell H15. In cell G16, compute the average of the responses. Enter a label ybar in cell H16. Transfer the labels in cells H15:H16 as names for cells G15:G16. Above the table, create labels and named cells for k , p , and n , and enter appropriate values in the named cells. Now, you’re read y to process and analyze the results. But, first, save your workbook. 3. Computing the contrasts, effects and component sum of squares. Enter labels Contrast , Effect and SS in cells A14:A16 respectively. In cell B14, enter the SUMPRODUCT formula for the A contrast. Make the reference to the response values absolute. The formula to implement for the Effect in cell B15 is   A A k p 1 Contrast Effect n 2 − − =  Note: for a full factorial, p = 0, and the formula simplifies to what you have seen before. And the formula to insert for the component sum of squares in cell B16 is   2 A A k p Contrast SS n 2 − =  Select cells B14:B16 and copy them across to complete the calculation of these values. Save your workbook. Copy your Excel table to your Word document. Treatment Combination Response be 27.17 bc 20.32 a 11.75 de 9.76 ace 21.09 cd 6.06 abcde 13.6 abd 7.67

CHEN 3010 Workshop 28 Page 3 4. Creating the ANOVA table to screen the effects. In cell A19, enter the label ANOVA Summary Table . In cells A20:G20, enter the labels Source , SS , d.o.f. , MS , F 0 , F , and P , respectively. Enter the headings A through E=AC downward below the Source heading, followed by Error and Total . Use a TRANSPOSE array formula to enter the component sum of squares values below the SS heading. For the Total SS entry, enter a pointer formula to your SST value. The Error SS value is the difference between the Total SS value and the sum of the component SS values. Each effect in the d.o.f. column has 1 degree of freedom. The Total d.o.f. entry is the number of responses minus 1. The Error d.o.f. is again the difference between Total and the sum of the effect d.o.f.’s. The entries in the MS column for the effects and Error are computed by dividing the SS values by their respective d.o.f. values. The entries in the F 0 column for the effects are computed by dividing the MS for the effect by the MS for Error. Make the reference to the MS Error absolute. Label a cell alpha and name an adjacent cell alpha . Enter 5% there. In the F column, enter a formula using the F.INV.RT function with arguments alpha , d.o.f. for the effect, and d.o.f. for Error (make absolute). Copy this down opposite the effects. In the P column, enter a formula using the F.DIST.RT function with arguments F 0 , d.o.f. for the effect, and d.o.f. for Error (make absolute). Copy this down opposite the effect. Highlight (with color) the rows in the ANOVA table of the effects that are significant. Which are these? Copy your ANOVA Summary Table to your Word document. Save your workbook. 5. Regression model, model predictions and residuals. In a set of cells, compute the coefficients of a regression model (reduced model, retaining only significant effects). Recall that the coefficients of the factors are the effects divided by two. Also, the intercept is the grand average, ybar . In the column to the right of the responses, compute a set of model predictions based on the factor levels and the coefficients. In the column to the right of the model predictions, compute a set of residuals. Copy the model predictions and residuals to your Word document. Launch Minitab. Copy the model predictions and the residuals from the spreadsheet to Minitab and create a probability plot and a scatter plot of residuals vs. model predictions. Copy these plots to your Word document. Comment on the plot on the plots in terms of model adequacy. Having screened the factors, it would be likely now to set up a full factorial experiment with replicates to get better information for a more detailed model. Save your workbook again. 6. Save your files. Submit your Word file Canvas. 7. On the pages that follow, for your reference, the application of Minitab to set up the design and analyze the results is shown. It isn’t required that you do this, but you might want to try it if you have time. Otherwise, you can use it as a reference.

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