4 Surface Roughness Consider the surface roughness experiment originally described in Exam-

EXAMPLE 14-4 Surface Roughness Consider the surface roughness experiment originally described in Exam-

  ple 14-2. This is a 2 3 factorial design in the factors feed rate ( ) A , depth of cut ( ) B , and tool angle

  () C , with n = 2 replicates. Table 14-16 presents the observed surface roughness data.

  The effect of A, for example, is

  A =

  [ + a ab ac abc − () 1 −−− 4 b c bc n ]

  22 27 23 30 16 20 21 18 −−

  Chapter 14Design of Experiments with Several Factors

  5"- t 14-16 Surface Roughness Data for Example 14-4

  Design Factors

  A B C Roughness

  and the sum of squares for A is found using Equation 14-15:

  SS A =

  contrast A 27

  It is easy to verify that the other effects are

  Examining the magnitude of the effects clearly shows that feed rate (factor

  A) is dominant, followed by depth of cut

  B and the AB interaction, although the interaction effect is relatively small. The analysis, summarized in Table 14-17, confi rms our interpretation of the effect estimates.

  The output from the computer software for this experiment is shown in Table 14-18. The upper portion of the table displays the effect estimates and regression coeffi cients for each factorial effect. To illustrate, for the main effect of

  2 feed, the computer output reports t 2 = 4.32 (with 8 degrees of freedom), and t = ( 4.32 ) 18.66 , which is approxi- mately equal to the F-ratio for feed reported in Table 14-18 ( F = 18.69 . This F-ratio has one numerator and 8 denomi- )

  nator degrees of freedom.

  The lower panel of the computer output in Table 14-18 is an analysis of variance summary focusing on the types of terms in the model. A regression model approach is used in the presentation. You might fi nd it helpful to review Section 12-2.2, particularly the material on the partial F-test. The row entitled “main effects’’ under source refers to the three main effects feed, depth, and angle, each having a single degree of freedom, giving the total 3 in the column headed “DF.’’ The column headed “Seq S’ (an abbreviation for sequential sum of squares) reports how much the model sum of squares increases when each group of terms is added to a model that contains the terms listed above the groups. The fi rst number in the “Seq S’ column presents the model sum of squares for fi tting a model having only the three main effects. The row labeled “2-Way Interactions’’ refers to AB, AC, and BC , and the sequential sum of squares reported here is the increase in the model sum of squares if the interaction terms are added to a model containing only the main effects. Similarly, the sequential sum of squares for the three-way interaction is the increase in the model sum of squares that results from adding the term ABC to a model containing all other effects.

  The column headed “Adj S’ (an abbreviation for adjusted sum of squares) reports how much the model sum of squares increases when each group of terms is added to a model that contains all the other terms. Now because any 2 k

  design with an equal number of replicates in each cell is an orthogonal design , the adjusted sum of squares equals the

  Section 14-52 k Factorial Designs

  5"- t 14-17 Analysis for the Surface Roughness Experiment

  Term

  Effect

  Coeffi cient

  SE Coeffi cient

  BC −0.6250

  Source of

  Sum of

  Degrees of

  Mean Square

  f 0 P-Value

  5"- t 14-18 Computer Analysis for the Surface Roughness Experiment in Example 14-4

  Estimated Effects and Coeffi cients for Roughness Term

  Effect

  Coef

  StDev Coef

  Analysis of Variance for Roughness Source

  Main effects

  2-Way interactions

  3-Way interactions

  Residual error

  Pure error

  Total

  Chapter 14Design of Experiments with Several Factors

  sequential sum of squares. Therefore, the F-tests for each row in the computer analysis of variance in Table 14-18 are testing the significance of each group of terms (main effects, two-factor interactions, and three-factor interactions) as if they were the last terms to be included in the model. Clearly, only the main effect terms are significant. The t-tests on the individual factor effects indicate that feed rate and depth of cut have large main effects, and there may be some mild interaction between these two factors. Therefore, the computer output agrees with the results given previously.

  Models and Residual Analysis We may obtain the residuals from a 2 k design by using the method demonstrated earlier for the

  2 design. As an example, consider the surface roughness experiment. The three largest effects are A, B, and the AB interaction. The regression model used to obtain the predicted values is

  Y = β 0 + 1 x 1 2 2 12 xx 12 e

  where x 1 represents factor A, x 2 represents factor B, and x x 12 represents the AB interaction. The regression coefficients β 1 , β 2 , and β 12 are estimated by one-half the corresponding effect

  estimates, and β 0 is the grand average. Thus,

  =. 11 0625 1 6875 +. x 1 0 8125 2 0 6875 xx 12

  Note that the regression coefficients are presented in the upper panel of Table 14-18. The pre- dicted values would be obtained by substituting the low and high levels of A and B into this equation. To illustrate this, at the treatment combination where A B , , and C are all at the low level, the predicted value is

  y ˆ = . 11 0625 1 6875 1 + ( −+ ) . 0 8125 1 ( ) . 0 6875 1 ( −−= )( 1 ) . 9 25

  Because the observed values at this run are 9 and 7, the residuals are 9 − . 9 25 = − 0.25 and

  7 − . 9 25 =− 2 25 . Residuals for the other 14 runs are obtained similarly.

  See a normal probability plot of the residuals in Fig. 14-22. Because the residuals lie approximately along a straight line, we do not suspect any problem with normality in the data.

  20 Normal probability

  FIGURE 14-22 Normal probability

  plot of residuals from the surface rough-

  –2.250 –1.653 – 0.917 –0.250 0.417

  ness experiment.

  Residual

  Section 14-52 k Factorial Designs

  There are no indications of severe outliers. It would also be helpful to plot the residuals versus the predicted values and against each of the factors A B , , and C.

  Projection of 2 k Designs

  Any 2 k design collapses or projects into another 2 design in fewer variables if one or more of the original factors are dropped. Sometimes this can provide additional insight into the

  k

  remaining factors. For example, consider the surface roughness experiment. Because factor

  C and all its interactions are negligible, we could eliminate factor C from the design. The result is to collapse the cube in Fig. 14-20 into a square in the A − plane; therefore, each of B the four runs in the new design has four replicates. In general, if we delete h factors so that

  k

  r r = − factors remain, the original 2 k h design with n replicates projects into a 2 design with

  n h 2 replicates.

  14-5.3 SINGLE REPLICATE OF THE 2 k DESIGN

  As the number of factors in a factorial experiment increases, the number of effects that can

  be estimated also increases. For example, a 2 4 experiment has 4 main effects, 6 two-factor interactions, 4 three-factor interactions, and 1 four-factor interaction, and a 2 6 experiment has

  6 main effects, 15 two-factor interactions, 20 three-factor interactions, 15 four-factor interac- tions, 6 five-factor interactions, and 1 six-factor interaction. In most situations, the sparsity of effects principle applies; that is, the system is usually dominated by the main effects and low-order interactions. The three-factor and higher order interactions are usually negligible. Therefore, when the number of factors is moderately large, say, k

  ≥ 4 or 5, a common practice

  is to run only a single replicate of the 2 design and then pool or combine the higher order interactions as an estimate of error. Sometimes a single replicate of a 2 k design is called an

  k

  unreplicated 2 k factorial design.

  When analyzing data from unreplicated factorial designs, occasionally real high-order interactions occur. The use of an error mean square obtained by pooling high-order interac- tions is inappropriate in these cases. A simple method of analysis can be used to overcome this problem. Construct a plot of the estimates of the effects on a normal probability scale. The

  effects that are negligible are normally distributed with mean zero and variance 2 σ and tend to

  fall along a straight line on this plot, whereas significant effects has nonzero means and will not lie along the straight line. We illustrate this method in Example 14-5.

  Example 14-5

  Plasma Etch An article in Solid State Technology [“Orthogonal Design for Process Optimiza- tion and Its Application in Plasma Etching” (May 1987, pp. 127–132)] describes the application

  of factorial designs in developing a nitride etch process on a single-wafer plasma etcher. The process uses C 2 F 6 as the

  reactant gas. It is possible to vary the gas flow, the power applied to the cathode, the pressure in the reactor chamber, and the spacing between the anode and the cathode (gap). Several response variables would usually be of interest in this process, but in this example, we concentrate on etch rate for silicon nitride.

  We use a single replicate of a 2 4 design to investigate this process. Because it is unlikely that the three- and four-

  factor interactions are significant, we tentatively plan to combine them as an estimate of error. The factor levels used in the design follow:

  Design Factor

  CF 26 Flow

  Power

  (cm)

  (mTorr)

  (SCCM)

  (w)

  Low (–)

  High (+)

  Chapter 14Design of Experiments with Several Factors

  Refer to Table 14-19 for the data from the 16 runs of the 2 4 design. Table 14-20 is the table of plus and minus signs for the 2 4 design. The signs in the columns of this table can be used to estimate the factor effects. For example, the

  estimate of factor A is

  A = [ a + ab ac abc ad abd acd abcd − () 1 −−− b c bc −− d bd − cd bcd

  = [ 669 + 650 642 635 749 868 860 729 − 550 604 633 601 1037 − 552 1075 1063 10 − ]

  8 =− 101 625 .

  Thus, the effect of increasing the gap between the anode and the cathode from 0.80 to 1.20 centimeters is to decrease the etch rate by 101.625 angstroms per minute.

  It is easy to verify (using computer software, for example) that the complete set of effect estimates is

  The normal probability plot of these effects from the plasma etch experiment is shown in Fig. 14-23. Clearly, the main effects of

  A and D and the AD interaction are signifi cant because they fall far from the line passing through the other

  points. The analysis, summarized in Table 14-21, confi rms these fi ndings. Notice that in the analysis of variance we have pooled the three- and four-factor interactions to form the error mean square. If the normal probability plot had indicated that any of these interactions were important, they would not have been included in the error term.

  TABLE t 14-19 The 2 4 Design for the Plasma Etch Experiment

  A B C D Etch Rate

  (Gap)

  (Pressure)

  (C 2 F 6 Flow)

  (Power)

  (Åmin)

  –1

  1 –1

  –1

  1 –1

  1 –1

  –1

  1 –1

  1 –1

  1 –1

  –1

  1 –1

  1 –1

  –1

  1 –1

  –1

  1 –1

  1 –1

  –1

  1 –1

  –1

  Section 14-52 k Factorial Designs

  TABLE t 14-20 Contrast Constants for the 2 4 Design

  A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD

  abcd + + TABLE t 14-21 Analysis for the Plasma Etch Experiment

  Term

  Effect

  Coeffi cient

  SE Coeffi cient

  t P -Value

  A -101.62

  11.28 -4.50 0.006

  B -1.62

  11.28 -0.07 0.945 C 7.37 3.69 11.28 0.33 0.757

  AB -7.88

  11.28 -0.35 0.741

  AC -24.88

  11.28 -1.10 0.321

  AD -153.62

  11.28 -6.81 0.001

  BC -43.87

  11.28 -1.94 0.109

  BD -0.63

  11.28 -0.03 0.979

  CD -2.13

  11.28 -0.09 0.929

  Source of Variation Sum of Squares Degrees of Freedom Mean Square

  f 0 P-Value

  Practical Interpretation: Because A = −101.625, the effect of increasing the gap between the cathode and anode is to decrease the etch rate. However, D = 306.125; thus, applying higher power levels increase the etch rate. Figure 14-24 is a plot of the AD interaction. This plot indicates that the effect of changing the gap width at low power

  settings is small but that increasing the gap at high power settings dramatically reduces the etch rate. High etch rates are obtained at high power settings and narrow gap widths.

  Chapter 14Design of Experiments with Several Factors

  D(Power) high = 325 w

  Normal probability 10 A

  D(Power) low = 275 w

  Etch rate Åmin 400

  1 AD

  –141.1 –64.5 12.1 88.8 165.4 242.0 318.6

  Low (0.80 cm)

  High (1.20 cm)

  Effect

  A (Gap)

  FIGURE 14-23 Normal probability plot of

  FIGURE 14-24 AD (gap-power) interaction from

  effects from the plasma etch experiment.

  the plasma etch experiment.

  The residuals from the experiment in Example 14-5 can be obtained from the regression model

  = 776 0625 . − ⎛

  For example, when both A and D are at the low level, the predicted value is

  ⎛ 101 625 . ⎞

  ⎛ 306 125 . ⎞

  ⎛⎛ 153 625 y . = 776 0625 . − ( −+ 1 ) ( −− 1 ) ⎞ ( −−= 1 )( ) 597

  and the four residuals at this treatment combination are

  e 1 = 550 − 597 =− 47 2 = 604 597 − = 7

  3 633 − 597 = 36 4 601 597 − = 4

  The residuals at the other three treatment combinations (A high, D low), (A low, D high), and (A high, D high) are obtained similarly. A normal probability plot of the residuals is shown in Fig. 14-25. The plot is satisfactory.

  20 Normal probability 10

  FIGURE 14-25 Normal probability

  plot of residuals from the plasma etch

  –72.50 –49.33 –26.17 –3.00

  experiment.

  Residual

  Section 14-52 k Factorial Designs

  14-5.4 ADDITION OF CENTER POINTS TO A 2 k DESIGN

  A potential concern in the use of two-level factorial designs is the assumption of linearity in the factor effects. Of course, perfect linearity is unnecessary, and the 2 k system works quite well even when the linearity assumption holds only approximately. However, a method of replicating certain points in the 2 k factorial provides protection against curvature and allows

  an independent estimate of error to be obtained. The method consists of adding center

  points to the 2 k design. These consist of n C replicates run at the point x i = 0 (i = 1, 2, . . . ,

  k). One important reason for adding the replicate runs at the design center is that center points do not affect the usual effects estimates in a 2 k design. We assume that the k factors are quantitative.

  To illustrate the approach, consider a 2 2 design with one observation at each of the facto- rial points (–, –), (+, –), (–, +), and (+, +) and n C observations at the center points (0, 0). Fig- ure 14-26 illustrates the situation. Let y F be the average of the four runs at the four factorial points, and let y C be the average of the n C run at the center point. If the difference y F − is y C

  small, the center points lie on or near the plane passing through the factorial points, and there

  is no curvature. On the other hand, if y F − is large, curvature is present. y C

  Similar to factorial effects, a test for curvature can be based on a F-statistic or an equiva-

  lent t-statistic. A single degree-of-freedom sum of squares for curvature is compared to MS E

  to produce the F-statistic. Alternatively, a t-statistic similar to the one used to compare two

  means can be computed. A coefficient for curvature is defined to be y F − , and σ is estimated y C

  by the square root of MS E . This leads to the following formulas:

  Curvature Sum of Squares and t-Statistic

  ( y F − C )

  SS Curvature =

  -s ttatistic for Curvature =

  FIGURE 14-26 A2 2 design with center points.

  Chapter 14Design of Experiments with Several Factors

  where, in general, n F is the number of factorial design points. The SS Curvature may be compared

  to the error mean square to produce the F-test for curvature. Notice that, similar to the test for other effects, the square of the t-statistic equals the F-statistic.

  When points are added to the center of the 2 k design, the model we may entertain is

  e

  Y = b 0 + ∑ x

  where the β jj are pure quadratic effects. The test for curvature actually tests the hypotheses

  k

  H 0 : β= jj 0 H 1 : β≠ jj 0

  j ∑ = 1 ∑ j =

  k

  Furthermore, if the factorial points in the design are unreplicated, we may use the n C center

  points to construct an estimate of error with n C – 1 degrees of freedom.

  Example 14-6

  Process Yield

  A chemical engineer is studying the percentage of conversion or yield of a process.

  There are two variables of interest, reaction time and reaction temperature. Because she is uncertain

  about the assumption of linearity over the region of exploration, the engineer decides to conduct a 2 2 design (with a single

  replicate of each factorial run) augmented with fi ve center points. The design and the yield data are shown in Fig. 14-27.

  Table 14-22 summarizes the analysis for this experiment. The mean square error is calculated from the center points as follows:

  ∑

  2 ( 5 y

  i − C )

  y

  ( i −. Center points 40 46 ) 0 .. 1720

  B = Temperature (C)

  FIGURE 14-27

  The 2 –1 2 design with fi ve center points

  for the process yield experiment in

  Example 14-6.

  A = Reaction time (min)

  TABLE t 14-22 Analysis for the Process Yield Experiment with Center Points

  Term

  Effect

  Coeffi cient

  SE Coeffi cient

  A 1.5500 0.7750 0.1037 7.47 0.002 B 0.6500

  AB −0.0500

  Source of Variation Sum of Squares Degrees of Freedom Mean Square f 0 P-Value

  A (Time)

  B (Temperature)

  AB 0.0025

  Curvature

  Error

  Total

  Section 14-52 k Factorial Designs

  The average of the points in the factorial portion of the design is y F =. 40 425, and the average of the points at the center

  is y C =. 40 46. The difference y F − = 40.425 – 40.46 = –0.035 appears to be small. The curvature sum of squares in y C

  the analysis of variance table is computed from Equation 14-19 as follows:

  nn 2

  FC y F − C

  ) = ( )( ) 45 ( −. 0 035 ) =. 0 0027

  SS Curvature =

  n F + C 45 + The coeffi cient for curvature is y y F − C = 0.035, and the t-statistic to test for curvature is

  Practical Interpretation: The analysis of variance indicates that both factors exhibit signifi cant main effects, that there is no interaction, and that there is no evidence of curvature in the response over the region of exploration. That is,

  the null hypothesis H k

  0 : ∑ j = 1 β= jj 0 cannot be rejected.