More Than Two Factors Designing experiments with two factors extends - - PowerPoint PPT Presentation

ST 370 Probability and Statistics for Engineers

More Than Two Factors Designing experiments with two factors extends easily to experiments with more factors: Choose a few (often 2 or 3) levels of each factor; All combinations of factor levels define the treatments. Choose the total number of runs, and block them if necessary. The analysis of the results is also an extension of the two-factor analysis, but it is complicated by the presence of high-order interactions.

1 / 14 More Than Two Factors General Factorial Experiments

ST 370 Probability and Statistics for Engineers

Example: Surface roughness in a metal-cutting operation Response: A measure of roughness; Factor A: Feed rate, 20 and 30 inches per minute (a = 2); Factor B: Depth of cut, 25 and 40 thousands of an inch (b = 2); Factor C: Tool angle, 15◦ and 25◦ (c = 2); Treatments: a × b × c = 8 treatments. Replication: n = 2 samples were run for each treatment; Design: completely randomized: 16 runs in random order.

2 / 14 More Than Two Factors General Factorial Experiments

ST 370 Probability and Statistics for Engineers

In R

rough <- read.csv("Data/Table-14-10.csv"); rough <- within(rough, {A <- factor(FeedRate); B <- factor(DepthOfCut); C <- factor(ToolAngle)}) library(gplots); qqnorm(aov(Roughness ~ A * B * C, rough), label = TRUE)

In this case, qqnorm() makes a half-normal plot of the factor effects and interactions. Substantial effects show up as extreme points; here,

• nly A is notable.

We follow up the graphical indications with hypothesis tests.

3 / 14 More Than Two Factors General Factorial Experiments

ST 370 Probability and Statistics for Engineers

Statistical model yi,j,k,l = µ + τi + βj + γk + (τβ)i,j + (τγ)i,k + (βγ)j,k + (τβγ)i,j,k + ǫi,j,k,l i = 1, 2, . . . , a j = 1, 2, . . . , b k = 1, 2, . . . , c l = 1, 2, . . . , n. Constraints In R, the first level of each factor is the baseline level. Any parameter with i = 1, j = 1, or k = 1 is constrained to be zero.

4 / 14 More Than Two Factors General Factorial Experiments

ST 370 Probability and Statistics for Engineers

Interpretation High-order interactions like (τβγ)i,j,k are hard to interpret. Recall that a two-factor interaction like (τβ)i,j measures the extent to which the level of factor B changes the effect of factor A... ...and, symmetrically, the extent to which the level of factor A changes the effect of factor B.

5 / 14 More Than Two Factors General Factorial Experiments

ST 370 Probability and Statistics for Engineers

The three-factor interaction (τβγ)i,j,k measures the extent to which the level of factor C changes the interaction between factors A and B... ...and, symmetrically, the extent to which the level of factor B changes the interaction between factors A and C... ...and the extent to which the level of factor A changes the interaction between factors B and C. Empirically it is often found, as here, that three-factor and higher-order interactions are negligible. When they are substantial, reporting the results requires careful explanation.

6 / 14 More Than Two Factors General Factorial Experiments

ST 370 Probability and Statistics for Engineers

In R

summary(aov(Roughness ~ A * B * C, rough))

Output

Df Sum Sq Mean Sq F value Pr(>F) A 1 45.56 45.56 18.692 0.00253 ** B 1 10.56 10.56 4.333 0.07093 . C 1 3.06 3.06 1.256 0.29485 A:B 1 7.56 7.56 3.103 0.11620 A:C 1 0.06 0.06 0.026 0.87675 B:C 1 1.56 1.56 0.641 0.44646 A:B:C 1 5.06 5.06 2.077 0.18751 Residuals 8 19.50 2.44

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

7 / 14 More Than Two Factors General Factorial Experiments

ST 370 Probability and Statistics for Engineers

Only A has a significant effect. However, B is close (P ≈ .07) and interestingly the next lowest P-value is for the A:B interaction. Further study of A and B is indicated. Interaction plot

with(rough, interaction.plot(A, B, Roughness))

The interaction plot suggests that when A (feed rate) is at its low level, B (depth of cut) has little effect on surface roughness, but when feed rate is at its high level, increasing the depth of cut from 0.025 inch to 0.040 inch increases the roughness.

8 / 14 More Than Two Factors General Factorial Experiments

ST 370 Probability and Statistics for Engineers

Two-level Designs Experiments in which four or more factors are varied are often designed with two levels of each factor. Example: Plasma etching Response: Etch rate (

• A /min)

Factor A: Anode-cathode gap, 0.8 cm and 1.2 cm; Factor B: Reactor chamber pressure, 450 mTorr and 550 mTorr; Factor C: C2F6 flow rate, 125 SCCM and 200 SCCM; Factor D: Power, 275 w and 325 w. Treatments: a × b × c × d = 16 treatments. Replication: n = 1; no replication; Design: completely randomized: 16 runs in random order.

9 / 14 More Than Two Factors Two-level Designs

ST 370 Probability and Statistics for Engineers

In R

plasma <- expand.grid(A = c("gapLo", "gapHi"), B = c("preLo", "preHi"), C = c("floLo", "floHi"), D = c("powLo", "powHi")) plasma$rate <- c( 550, 669, 604, 650, 633, 642, 601, 635, 1037, 749, 1052, 868, 1075, 860, 1063, 729) library(gplots) qqnorm(aov(rate ~ A * B * C * D, plasma), label = TRUE)

10 / 14 More Than Two Factors Two-level Designs

ST 370 Probability and Statistics for Engineers

The half-normal plot indicates that factors A, D, and their interaction may have substantial effects. Interaction plot

with(plasma, interaction.plot(A, D, rate))

In this unreplicated design, we can test hypotheses only by assuming that certain interactions are zero. Because the interaction plot showed large values only for main effects and one two-factor interaction, we assume that all higher-order interactions are zero.

11 / 14 More Than Two Factors Two-level Designs

ST 370 Probability and Statistics for Engineers

In R

summary(aov(rate ~ (A + B + C + D)^2, plasma))

Output

Df Sum Sq Mean Sq F value Pr(>F) A 1 41311 41311 20.276 0.00638 ** B 1 11 11 0.005 0.94539 C 1 218 218 0.107 0.75707 D 1 374850 374850 183.988 3.9e-05 *** A:B 1 248 248 0.122 0.74135 A:C 1 2475 2475 1.215 0.32058 A:D 1 94403 94403 46.336 0.00104 ** B:C 1 7700 7700 3.779 0.10950 B:D 1 2 2 0.001 0.97898 C:D 1 18 18 0.009 0.92864 Residuals 5 10187 2037

12 / 14 More Than Two Factors Two-level Designs

ST 370 Probability and Statistics for Engineers

The interaction plot suggests that A (gap) has little effect when D (power) is at its low level. We can test the null hypothesis that the effect is zero, but because this test was not pre-planned, we use Tukey’s HSD method:

TukeyHSD(aov(rate ~ A * D, plasma), "A:D")

Output

$‘A:D‘ diff lwr upr p adj gapHi:powLo-gapLo:powLo 52.00

• 35.52333

139.5233 0.3357729 gapLo:powHi-gapLo:powLo 459.75 372.22667 547.2733 0.0000000 gapHi:powHi-gapLo:powLo 204.50 116.97667 292.0233 0.0000803 gapLo:powHi-gapHi:powLo 407.75 320.22667 495.2733 0.0000001 gapHi:powHi-gapHi:powLo 152.50 64.97667 240.0233 0.0011462 gapHi:powHi-gapLo:powHi -255.25 -342.77333 -167.7267 0.0000086

13 / 14 More Than Two Factors Two-level Designs

ST 370 Probability and Statistics for Engineers

The first row tests the hypothesis that A has no effect when D is at its low level, and the hypothesis is not rejected (the confidence interval contains 0). All other effects are significant.

14 / 14 More Than Two Factors Two-level Designs