What If There Are More Than. Two Factor Levels?

What If There Are More Than Chapter 3 Two Factor Levels? Comparing more that two factor levels the analysis of variance ANOVA decomposition of total variability Statistical testing & analysis Checking assumptions, model validity Post-ANOVA testing of means ANOVA_EXAMPLE P60 Brainerd 1

What If There Are More Than Two Factor Levels? The t-test does not directly apply There are lots of practical situations where there are either more than two levels of interest, or there are several factors of simultaneous interest The analysis of variance (ANOVA) is the appropriate analysis engine for these types of experiments Chapter 3, textbook The ANOVA was developed by Fisher in the early 1920s, and initially applied to agricultural experiments Used extensively today for industrial experiments ANOVA_EXAMPLE P60 Brainerd 2

An Example (See pg. 60) Consider an investigation into the formulation of a new synthetic fiber that will be used to make cloth for shirts The response variable is tensile strength The experimenter wants to determine the best level of cotton (in wt %) to combine with the synthetics Cotton content can vary between 10 40 wt %; some nonlinearity in the response is anticipated The experimenter chooses 5 levels of cotton content ; 15, 20, 25, 30, and 35 wt % The experiment is replicated 5 times runs made in random order ANOVA_EXAMPLE P60 Brainerd 3

ANOVA: Design of Experiments Chapter 3 A A product development engineer engineer is is interested is interested investigating investigating the tensile the strength tensile of a strength new synthetic of a fiber new that synthetic will be used fiber to that make will men s be shirts. used The to engineer knows from previous experience that the strength is affected by the make men s shirts. The engineer knows from previous weight The engineer percent knows of cotton from used previous in the blend experience of materials that the for strength the fiber. is affected experience Furthermore, by weight he that percent suspects the of strength cotton that increasing used is affected the the blend cotton by of the content materials weight will for increase percent the fiber. the of cotton strength, used at least in initially. the blend He of also materials knows that for cotton the fiber. content Furthermore, should range he between Furthermore, suspects about that 10 he suspects percent increasing and that 40 increasing the percent cotton if the the content cotton final cloth content will is increase to will have increase other quality the strength, characteristics at at least initially. that are He desired. also knows The engineer that cotton decides content to test should specimens range between at five about levels 10 of percent cotton and weight 40 percent: if the 15 percent, final cloth 20 is percent, to have 25 percent, other quality 30 percent, characteristics and 35 percent. that are desired. The engineer decides to test specimens at five levels of cotton weight percent: 15 percent, 20 percent, 25 percent, 30 percent, and 35 percent. He also decides to test five specimens at each level of cotton content. ANOVA_EXAMPLE P60 Brainerd 4

ANOVA: Design of Experiments Chapter 3 H : µ = µ = µ =... = 0 1 2 3 H a : At least one µ i µ is different i Tensile Strength EXAMPLE PROBLEM A single-factor experiment with a = 5 levels of the factor and n = 5 replicates. The 25 runs should be made in random order. 15 20 25 30 35 Cotton % ANOVA_EXAMPLE P60 Brainerd 5

ANOVA: Design of Experiments Chapter 3 H : µ = µ = µ =... = 0 1 2 3 H a : At least one µ i µ is different i Tensile Strength ONE FROM EACH 15 20 25 30 35 Can we determine which is better? Cotton % ANOVA_EXAMPLE P60 Brainerd 6

ANOVA: Design of Experiments Chapter 3 H : µ = µ = µ =... = 0 1 2 3 µ i Tensile Strength H a : At least one µ i is different REPLICATION 15 20 25 30 35 Cotton % What does replication provide? ANOVA_EXAMPLE P60 Brainerd 7

Tensile Strength ANOVA: Design of Experiments Chapter 3 H : µ = µ = µ =... = 0 1 2 3 H a : At least one µ i µ is different i REPLICATION Effect If the sample mean is used to estimate the effect of a factor in the experiment, then replication permits the experimenter to obtain a more precise estimate of this effect. 15 20 25 30 35 Cotton % What else does replication provide? ANOVA_EXAMPLE P60 Brainerd 8

Tensile Strength ANOVA: Design of Experiments Chapter 3 H : µ = µ = µ =... = 0 1 2 3 H a : At least one µ is different REPLICATION Effect Error 2 σ σ x = estimate if error becomes a basic unit i of measurement n for determining whether observed differences in the data are really statistically 15 20 25 30 35 different. Cotton % What assumption does the error estimate depend upon? i µ ANOVA_EXAMPLE P60 Brainerd 9 i Allows the experimenter to obtain an estimate of the 2 experimental error. This

ANOVA: Design of Experiments Chapter 3 H : µ = µ = µ =... = 0 1 2 3 µ i Tensile Strength H a : Cotton % At least one µ 15 20 25 30 35 i is different REPLICATION Effect Error RANDOMIZATION Both the allocation of the experimental material and the order in which the individual runs or trials of the experiment are to be performed are randomly determined. ANOVA_EXAMPLE P60 Brainerd 10

ANOVA: Design of Experiments Chapter 3 Need to randomize run order COTTON% Experimental Run # 15 1 2 3 4 5 20 6 7 8 9 10 25 11 12 13 14 15 30 16 17 18 19 20 35 21 22 23 24 25 ANOVA_EXAMPLE P60 Brainerd 11

ANOVA: Design of Experiments Chapter 3 Test Seq Run # % 1 8 20 2 18 30 3 10 20 4 23 35 5 17 30 * * * 25 3 15 RANDOMIZE RUNS ANOVA_EXAMPLE P60 Brainerd 12

Tensile Strength ANOVA: Design of Experiments Chapter 3 H : µ = µ = µ =... = 0 1 2 3 H a : At least one µ i µ is different i REPLICATION Effect Error RANDOMIZATION Cotton % 15 20 25 30 35 What if the measurements varied widely because of human operators? ANOVA_EXAMPLE P60 Brainerd 13

ANOVA: Design of Experiments Chapter 3 PUSHUP EXAMPLE Test if one can do push ups better in the morning or afternoon. 20 DATA POINTS Select 40 people at random Is PM really better than AM? AM PM ANOVA_EXAMPLE P60 Brainerd 14

ANOVA: Design of Experiments Chapter 3 What if the PM group of 20 was in better shape then the AM group of 20? What if the test was conducted on a Monday morning? What if different people counted the push ups between AM and PM? Is PM really better than AM? AM PM ANOVA_EXAMPLE P60 Brainerd 15

ANOVA: Design of Experiments Chapter 3 Controllable Factors Input PROCESS Output Uncontrollable Factors ANOVA_EXAMPLE P60 Brainerd 16

ANOVA: Design of Experiments Chapter 3 BLOCKING Used to limit the uncontrollable factors Therefore increase precision PUSH UP EXAMPLE Paired Data = BLOCKING Have same person do AM and PM You are investigating AM vs PM not which group can do more pushups. Randomly Sort experiment by Days of the Week and have one grader ANOVA_EXAMPLE P60 Brainerd 17

THE THREE PRINCIPLES of Experimental Design H : µ = µ = µ =... = 0 1 2 3 H a : At least one µ i µ is different i Tensile Strength REPLICATION RANDOMIZATION BLOCKING A block is a portion of the experimental material that should be more homogeneous than the entire set of material. 15 20 25 30 35 Cotton % ANOVA_EXAMPLE P60 Brainerd 18

Analysis of Variance (ANOVA) H : µ = µ = µ =... = 0 1 2 3 H a : At least one µ i µ is different i Variation Between Samples Tensile Strength Variation Within Samples 15 20 25 30 35 Cotton % ANOVA_EXAMPLE P60 Brainerd 19

Analysis of Variance (ANOVA) When very different Tensile Strength Between Sample Variation Large Within Sample Variation Small 15 20 25 30 35 Cotton % ANOVA_EXAMPLE P60 Brainerd 20

Analysis of Variance (ANOVA) When near equal Tensile Strength 15 20 25 30 35 Between Sample Variation near equal to Within Sample Variation Cotton % ANOVA_EXAMPLE P60 Brainerd 21

Analysis of Variance (ANOVA) TEST STAT = Between Sample Variation Within Sample Variation F-TEST REQUIRED ASSUMPTION All data is normal with equal variance EXTENSION OF TWO SAMPLE POOLED t ANOVA_EXAMPLE P60 Brainerd 22

Analysis of Variance (ANOVA) SETUP (i = factor; j = replicate) Level Replicates Mean SD 1 2 3 * j 1 11 12 13 * 1j 1* 2 21 22 23 * 2j 2* 3 31 32 33 * 3j 3* * * * * * * i i1 i2 i3 * ij ** x i, j = observed value i th level, j th measurement ANOVA_EXAMPLE P60 Brainerd 23

Analysis of Variance (ANOVA) COTTON EXAMPLE x i Level Replicates Mean SD = 1 2 3 4 5 1 7 7 15 11 9 9.80 3.35 2 12 17 12 18 18 15.40 3.13 3 14 18 18 19 19 17.60 2.07 4 19 25 22 19 23 21.60 2.61 5 7 10 11 15 11 10.80 2.86 th sample mean i level x 15.04 = Grand Mean ANOVA_EXAMPLE P60 Brainerd 24

Analysis of Variance (ANOVA) I = # factors and J = # replicates Mean Square MST r Treatment = I J 1 i Between Sample Variation ( X i X 2 ) Mean Square Error Within Sample Variation MSE = 2 2 S + S + S +... S 1 2 3 I 2 ANOVA_EXAMPLE P60 Brainerd 25 2 I

Analysis of Variance (ANOVA) TEST H H a : µ 1 = µ 2 = µ 3 =... µ i : At least one µ is different 0 = STAT = i Between Sample Variation Within Sample Variation f = MST r MSE If 0 E( MST If H H 0 true r ) = E( MSE) false = σ 2 F α, I 1, I ( J 1) E( MST r ) > E( MSE) = σ 2 ANOVA_EXAMPLE P60 Brainerd 26

Anova Chapter 3 (See pg. 62) Does changing the cotton weight percent change the mean tensile strength? Is there an optimum level for cotton content? ANOVA_EXAMPLE P60 Brainerd 27

The Analysis of Variance SS = SS + SS T Treatments E A large value of SS Treatments reflects large differences in treatment means A small value of SS Treatments likely indicates no differences in treatment means Formal statistical hypotheses are: H H 0 1 2 1 : µ = µ = L = µ : At least one mean is different a ANOVA_EXAMPLE P60 Brainerd 28

The Analysis of Variance While sums of squares cannot be directly compared to test the hypothesis of equal means, mean squares can be compared. ( MS = Estimates of variances) A mean square is a sum of squares divided by its degrees of freedom: MS df = df + df Total Treatments Error an 1= a 1 + a( n 1) Treatments SSTreatments SSE =, MSE = a 1 a( n 1) If the treatment means are equal, the treatment and error mean squares will be (theoretically) equal. If treatment means differ, the treatment mean square will be larger than the error mean square. ANOVA_EXAMPLE P60 Brainerd 29

The Analysis of Variance is Summarized in a Table Computing see text, pp 70 73 The reference distribution for F 0 is the F a-1, a(n-1) distribution Reject the null hypothesis (equal treatment means) if F > F α 0, a 1, a( n 1) ANOVA_EXAMPLE P60 Brainerd 30

Analysis of Variance (ANOVA): Excel Analysis: ANOVA Single Factor Setup data in EXCEL Spreadsheet in columns as: 15% Cotton Tensile Strength lb/in2 20% Cotton Tensile Strength lb/in2 25% Cotton Tensile Strength lb/in2 30% Cotton Tensile Strength lb/in2 35% Cotton Tensile Strength lb/in2 7 12 14 19 7 7 17 18 25 10 15 12 18 22 11 11 18 19 19 15 9 18 19 23 11 ANOVA_EXAMPLE P60 Brainerd 31

Analysis of Variance (ANOVA): Excel Analysis: ANOVA Single Factor EXCEL Data Analysis: ANOVA Single Factor ANOVA_EXAMPLE P60 Brainerd 32

Analysis of Variance (ANOVA): EXCEL Data Analysis: ANOVA Single Factor ANOVA_EXAMPLE P60 Brainerd 33

Analysis of Variance (ANOVA): EXCEL Data Analysis: ANOVA Single Factor Output EXCEL Anova: Single Factor SUMMARY Groups Count Sum Average Variance 15% Cotton Tensile Strength lb/in2 5 49 9.8 11.2 20% Cotton Tensile Strength lb/in2 5 77 15.4 9.8 25% Cotton Tensile Strength lb/in2 5 88 17.6 4.3 30% Cotton Tensile Strength lb/in2 5 108 21.6 6.8 35% Cotton Tensile Strength lb/in2 5 54 10.8 8.2 ANOVA_EXAMPLE P60 Brainerd 34

If Analysis of Variance (ANOVA): EXCEL Data Analysis: ANOVA Single Factor Output 0 E( MST If H H 0 E( MST true r ) = E( MSE) = σ false r H H a ) > E( MSE) = σ : µ 1 = µ 2 = µ 3 =... µ i : At least one µ is different 0 = ANOVA Source of Variation SS df MS F P-value F crit Between Groups 475.76 4 118.94 14.75682 9.128E-06 2.866081 Within Groups 161.2 20 8.06 Total 636.96 24 2 2 i TEST STAT f = = MSTr MSE Between Sample Variation Within Sample Variation P-Value: Probability of wrongly rejecting the Null F α, I 1, I ( J 1) ANOVA_EXAMPLE P60 Brainerd 35

The Reference Distribution: ANOVA_EXAMPLE P60 Brainerd 36

Analysis of Variance (ANOVA): EXCEL Data Analysis: ANOVA Manual calculations 15% Cotton Tensile Strength 20% Cotton Tensile Strength 25% Cotton Tensile Strength 30% Cotton Tensile Strength SUMj (SUM)SQj SUM SQj 35% Cotton Tensile Strength (Σx j ) (Σx j ) 2 Σ(x 2 j ) 7 12 14 19 7 59 3481 799 7 17 18 25 10 77 5929 1387 15 12 18 22 11 78 6084 1298 11 18 19 19 15 82 6724 1392 9 18 19 23 11 80 6400 1416 (Σx i ) 49 77 88 108 54 Σ(Σx i ) 376 Σ(Σx j ) Σ((Σx j ) 2 ) Σ(Σ(x j 2 )) Σ(Σx ij ) Σ(Σ(x ij 2 )) (Σx i ) 2 2401 5929 7744 11664 2916 Σ((Σx i 2 ) 30654 376 28618 6292 Σ(x i 2 ) 525 1225 1566 2360 616 Σ(Σ(x i 2 )) 6292 ANOVA_EXAMPLE P60 Brainerd 37

Analysis of Variance (ANOVA): EXCEL Data Analysis: ANOVA Manual calculations Manual calculations Source of Variation SS - sum of squares SS SS - sum of squares Calculation df MS = SS/df estimate of sigma F statistic = MSBG/MSwithin G Between Groups =(r( ΣΣx 2 i )-(ΣΣx i ) 2 )/n 475.76 as = [(5 x30654) - (376^2)]/25 4 118.94 Within Groups Error 161.2 20 8.06 as = 118.94/8.06 = Total =(n( ΣΣx ij 2 )-(ΣΣx ij ) 2 )/n 636.96 as = [(25 x 6292) - (376^2)]/25 24 14.76 ANOVA_EXAMPLE P60 Brainerd 38

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 1 General Factorial Design Press continue... ANOVA_EXAMPLE P60 Brainerd 39

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 2 General Factorial Design Press continue... ANOVA_EXAMPLE P60 Brainerd 40

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 3 Define # replicates General Factorial Design Press continue... ANOVA_EXAMPLE P60 Brainerd 41

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 4 # and Name response General Factorial Design Press continue... ANOVA_EXAMPLE P60 Brainerd 42

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 5 Run experiment in the defined random order and measure/input responses Press continue... ANOVA_EXAMPLE P60 Brainerd 43

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 6 Design Expert Analysis ANOVA_EXAMPLE P60 Brainerd 44

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 7 Design Expert Analysis ANOVA ANOVA_EXAMPLE P60 Brainerd 45

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 8 Design Expert Analysis ANOVA_EXAMPLE P60 Brainerd 46

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 9 Design Expert Analysis Effects M for Model e for error ANOVA_EXAMPLE P60 Brainerd 47

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA same table as before in EXCEL ANOVA_EXAMPLE P60 Brainerd 48

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Terms Model: Terms estimating factor effects. For 2-level factorials: those that "fall off" the normal probability line of the effects plot. Sum of Squares: Total of the sum of squares for the terms in the model, as reported in the Effects List for factorials and on the Model screen for RSM, MIX and Crossed designs. DF: Degrees of freedom for the model. It is the number of model terms, including the intercept, minus one. Mean Square: Estimate of the model variance, calculated by the model sum of squares divided by model degrees of freedom. F Value: Test for comparing model variance with residual (error) variance. If the variances are close to the same, the ratio will be close to one and it is less likely that any of the factors have a significant effect on the response. Calculated by Model Mean Square divided by Residual Mean Square. Probe > F: Probability of seeing the observed F value if the null hypothesis is true (there is no factor effect). Small probability values call for rejection of the null hypothesis. The probability equals the proportion of the area under the curve of the F-distribution that lies beyond the observed F value. The F distribution itself is determined by the degrees of freedom associated with the variances being compared. (In "plain English", if the Probe>F value is very small (less than 0.05) then the terms in the model have a significant effect on the response.) ANOVA_EXAMPLE P60 Brainerd 49

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis Information in Help System ANOVA Terms Pure Error: Amount of variation in the response in replicated design points. Sum of Squares: Pure error sum of squares from replicated points. DF: The amount of information available from replicated points. Mean Square: Estimate of pure error variance. Cor Total: Totals of all information corrected for the mean. Sum of Squares: Sum of the squared deviations of each point from the mean. DF: Total degrees of freedom for the experiment, minus one for the mean. ANOVA_EXAMPLE P60 Brainerd 50

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA ANOVA_EXAMPLE P60 Brainerd 51

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Information in Help System ANOVA Terms Next you see a collection of summary statistics for the model: Std Dev: (Root MSE) Square root of the residual mean square. Consider this to be an estimate of the standard deviation associated with the experiment. Mean: Overall average of all the response data. C.V.: Coefficient of Variation, the standard deviation expressed as a percentage of the mean. Calculated by dividing the Std Dev by the Mean and multiplying by 100. PRESS: Predicted Residual Error Sum of Squares A measure of how the model fits each point in the design. The PRESS is computed by first predicting where each point should be from a model that contains all other points except the one in question. The squared residuals (difference between actual and predicted values) are then summed. R-Squared: A measure of the amount of variation around the mean explained by the model. 1-(SSresidual / (SSmodel + SSresidual)) ANOVA_EXAMPLE P60 Brainerd 52

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Information in Help System ANOVA Terms Summary statistics for the model continued: Adj R-Squared: A measure of the amount of variation around the mean explained by the model, adjusted for the number of terms in the model. The adjusted R-squared decreases as the number of terms in the model increases if those additional terms don t add value to the model. 1-((SSresidual / DFresidual) / ((SSmodel + SSresidual) / (DFmodel + DFresidual))) Pred R-Squared: A measure of the amount of variation in new data explained by the model. 1-(PRESS / (SStotal-SSblock) The predicted r-squared and the adjusted r-squared should be within 0.20 of each other. Otherwise there may be a problem with either the data or the model. Look for outliers, consider transformations, or consider a different order polynomial. ANOVA_EXAMPLE P60 Brainerd 53

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Scroll down ANOVA_EXAMPLE P60 Brainerd 55

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Scroll down ANOVA_EXAMPLE P60 Brainerd 56

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Scroll down ANOVA_EXAMPLE P60 Brainerd 57

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Scroll down ANOVA_EXAMPLE P60 Brainerd 58

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Scroll down ANOVA_EXAMPLE P60 Brainerd 59

Model Adequacy Checking in the ANOVA Text reference, Section 3-4, pg. 76 Checking assumptions is important Normality Constant variance Independence Have we fit the right model? ANOVA_EXAMPLE P60 Brainerd 60

Model Adequacy Checking in the ANOVA STEP 10 Design Expert Analysis ANOVA Diagnostics Residuals Examination of residuals (see text, Sec. 3-4, pg. 76) e = y yˆ ij ij ij = y y ij i. Design-Expert generates the residuals Residual plots are very useful Normal probability plot of residuals ANOVA_EXAMPLE P60 Brainerd 61

Analysis of Variance (ANOVA): STAT EASE Design Expert: ANOVA Single Factor STEP 10 Design Expert Analysis ANOVA Model Graphs ANOVA_EXAMPLE P60 Brainerd 62

Other Important Residual Plots 5.2 5.2 2.95 2 2.95 Residuals 0.7 2 2 2 Residuals 0.7-1.55-1.55 2 2-3.8 2-3.8 9.80 12.75 15.70 18.65 21.60 1 4 7 10 13 16 19 22 25 Predicted Run Number ANOVA_EXAMPLE P60 Brainerd 63

Post-ANOVA Comparison of Means The analysis of variance tests the hypothesis of equal treatment means Assume that residual analysis is satisfactory If that hypothesis is rejected, we don t know which specific means are different Determining which specific means differ following an ANOVA is called the multiple comparisons problem There are lots of ways to do this see text, Section 3-5, pg. 86 We will use pairwise t-tests on means sometimes called Fisher s Least Significant Difference (or Fisher s LSD) Method ANOVA_EXAMPLE P60 Brainerd 64

Graphical Comparison of Means Text, pg. 89 ANOVA_EXAMPLE P60 Brainerd 65

For the Case of Quantitative Factors, a Regression Model is often Useful Response:Strength ANOVA for Response Surface Cubic Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 441.81 3 147.27 15.85 < 0.0001 A 90.84 1 90.84 9.78 0.0051 A2 343.21 1 343.21 36.93 < 0.0001 A3 64.98 1 64.98 6.99 0.0152 Residual 195.15 21 9.29 Lack of Fit 33.95 1 33.95 4.21 0.0535 Pure Error 161.20 20 8.06 Cor Total 636.96 24 Coefficient Standard 95% CI 95% CI Factor Estimate DF Error Low High VIF Intercept 19.47 1 0.95 17.49 21.44 A-Cotton % 8.10 1 2.59 2.71 13.49 9.03 A2-8.86 1 1.46-11.89-5.83 1.00 A3-7.60 1 2.87-13.58-1.62 9.03 ANOVA_EXAMPLE P60 Brainerd 66

The Regression Model Final Equation in Terms of Actual Factors: % 25 Strength = +62.61143-9.01143* Cotton Weight % +0.48143 * Cotton Weight %^2-7.60000E- 003 * Cotton Weight %^3 Strength 20.5 16 2 2 2 2 This is an empirical model of the experimental results 11.5 7 2 2 2 15.00 20.00 25.00 30.00 35.00 A: Cotton Weight % ANOVA_EXAMPLE P60 Brainerd 67