ANOVA Tutorial

CAE-Based Robust Design:Basic ConceptsAnalysis of Variance (ANOVA)

ANOVAWhat is ANOVA? ANOVA stands for ANalysis Of VArianceIt is a way to decompose a complicated function into a sum of simpler functions in order to facilitate understanding the complicated functionWhat can you learn from ANOVA?Find most important variables Determine if you can desensitize design to noiseVisualize nature of input-output relationshipsLearn how coupled the simulation is, that is, if the effect of one variable depends on the settings of others

Familiar functional decompositionsTaylor series Gives easy to manipulate approximations for use in other calculations (e.g., determine step direction in optimization)Fourier series Expresses an arbitrary periodic function as sum of sines and cosines. System response to a complicated input can be found by adding response to simple sine and cosine inputs.

ANOVA functional decompositionA function of n variables is decomposed into the sum ofa constant (overall mean)functions of one variable (main effects)functions of two variables (two-way interactions)functions of three variables (three-way interactions)Interpreting the resultsMain effects provide way to visualize effect of individual variables on outputCan assess importance of a main effect or two-way interaction by computing how much the squared error in the approximation goes down when add that termIf no interactions are important, can optimize one variable at a time and essentially read the best design from the main effect plotsIf a two-way interaction exists between a control and noise variable, you may be able to adjust that control to desensitize the response to the noise factor.

A specific exampleSuppose we have a simulation with two inputs x1 and x2 that vary over [0,1]. We run the simulation on a 5x5 grid, and obtain these resultsIn addition, we fit a response surface to this data that gives us a function f(x1,x2) that predicts the value of the simulation at any point (x1,x2).

Sheet1

Function Evaluated Over a 5x5 Grid

X2

0.000.250.500.751.00

X10.000.00-1.410.001.410.00

0.252.811.533.064.593.31

0.503.752.594.255.914.75

0.752.811.783.565.344.31

1.000.00-0.911.002.912.00

Sheet2

Sheet3

Quantifying the total variation in fThe constant in the ANOVA decomposition is just the average of all the values, here 2.38.Using only the constant (mean) results in a squared approximation error of 96.14

Sheet1


X2

0.000.250.500.751.00

X10.000.00-1.410.001.410.00

0.252.811.533.064.593.31Average:

0.503.752.594.255.914.752.38

0.752.811.783.565.344.31

1.000.00-0.911.002.912.00

Y - Mean

X2

0.000.250.500.751.00

X10.00-2.38-3.78-2.38-0.97-2.38

0.250.44-0.840.692.220.94

0.501.380.221.883.532.38Variation

0.750.44-0.591.192.971.94from

1.00-2.38-3.28-1.380.53-0.38Mean

(Y - Mean) ^ 2

X2

0.000.250.500.751.00

X10.005.6414.305.640.945.64Total

0.250.190.710.474.920.88Squared

0.501.890.053.5212.475.64Deviations

0.750.190.351.418.813.75From Mean

1.005.6410.771.890.280.1496.14

Sheet2

Sheet3

The main effect of x1The main effect of x1 is the function that best explains (Y-Mean) using only x1.We try to explain (Y-Mean) and not just Y because we higher-order terms are only used to explain what is left over from lower-order terms.

Sheet1

Y - MeanAverage(Main Effect)

X2x1Main Effect(x2)

0.000.250.500.751.000-2.375

X10.00-2.38-3.78-2.38-0.97-2.38-2.380.250.6875

0.250.44-0.840.692.220.940.690.51.875

0.501.380.221.883.532.381.880.751.1875

0.750.44-0.591.192.971.941.191-1.375

1.00-2.38-3.28-1.380.53-0.38-1.38

Sheet1

Main Effect(x2)

x1

Main Effect of x1

Sheet2

Sheet3

Variation explained by x1With only the constant, the total squared error was 96.14.But when we take into account the main effect of x1, the variation falls to 31.49. The difference, 64.65, is due to x1.The percent contribution of x1 is therefore 64.65 / 96.14 = 67%

Sheet1


X2

0.000.250.500.751.00

X10.000.00-1.410.001.410.00

0.252.811.533.064.593.31Average:

0.503.752.594.255.914.752.38

0.752.811.783.565.344.31

1.000.00-0.911.002.912.00

Y - [ Mean + MainEffect(x1) ]

X2

0.000.250.500.751.00

X10.000.00-1.410.001.410.00Variation0.00

0.25-0.25-1.530.001.530.25from3.06

0.50-0.50-1.660.001.660.50Mean4.25

0.75-0.75-1.780.001.780.75Plus3.56

1.00-1.00-1.910.001.911.00MainEffect(x1)1.00

{ Y - [ Mean + MainEffect(x1) ] } ^ 2

X2

0.000.250.500.751.00

X10.000.001.980.001.980.00

0.250.062.340.002.340.06Total

0.500.252.740.002.740.25Squared

0.750.563.170.003.170.56Deviations

1.001.003.630.003.631.0031.49

Sheet2

Sheet3

The main effect of x2The main effect of x2 is the function that best explains (Y-Mean) using only x2.

Sheet1

Y - Mean

X2x1Main Effect(x1)

0.000.250.500.751.000-0.5

X10.00-2.38-3.78-2.38-0.97-2.380.25-1.65625

0.250.44-0.840.692.220.940.50

0.501.380.221.883.532.380.751.65625

0.750.44-0.591.192.971.9410.5

1.00-2.38-3.28-1.380.53-0.38

Average = Main Effect-0.50-1.660.001.660.50

Sheet1

-0.5

-1.65625

0

1.65625

0.5

Main Effect(x1)

x2

Main Effect of x2

Sheet2

Sheet3

Variation due to x2With only the constant, the total squared error was 96.14.But when we take into account the main effect of x2, the variation falls to 66.21. The difference, 29.93, is due to x2.The percent contribution of x2 is therefore 29.93 / 96.14 = 31%

Sheet1


X2

0.000.250.500.751.00

X10.000.00-1.410.001.410.00

0.252.811.533.064.593.31Average:

0.503.752.594.255.914.752.38

0.752.811.783.565.344.31

1.000.00-0.911.002.912.00

Y - [ Mean + MainEffect(x2) ]

X2

0.000.250.500.751.00

X10.00-1.88-2.13-2.38-2.63-2.88Variation1.880.722.384.032.88

0.250.940.810.690.560.44from

0.501.881.881.881.881.88Mean

0.750.941.061.191.311.44Plus

1.00-1.88-1.63-1.38-1.13-0.88MainEffect(x2)

{ Y - [ Mean + MainEffect(x2) ] } ^ 2

X2

0.000.250.500.751.00

X10.003.524.525.646.898.27

0.250.880.660.470.320.19Total

0.503.523.523.523.523.52Squared

0.750.881.131.411.722.07Deviations

1.003.522.641.891.270.7766.21

Sheet2

Sheet3

Interaction effectsThe function doesnt have to be completely explained by the main effects. In this case, there is a residual after the mean and main effects are considered. This is due to effects related to specific combinations of (x1,x2), the so-called interaction effect.The percent contribution of the interaction is 1.56 / 96.14 = 2 %

Sheet1


X2

0.000.250.500.751.00

X10.000.00-1.410.001.410.00

0.252.811.533.064.593.31Average:

0.503.752.594.255.914.752.38

0.752.811.783.565.344.31

1.000.00-0.911.002.912.00-0.50-1.660.001.660.50

Y - [ Mean + MainEffect(x1) + MainEffect(x2) ]

X2

0.000.250.500.751.00

X10.000.500.250.00-0.25-0.50Variation-2.38

0.250.250.130.00-0.13-0.25from0.69

0.500.000.000.000.000.00Mean1.88

0.75-0.25-0.130.000.130.25Plus1.19

1.00-0.50-0.250.000.250.50MainEffects-1.38

{ Y - [ Mean + MainEffect(x1) + MainEffect(x2) ] } ^ 2

X2

0.000.250.500.751.00

X10.000.250.060.000.060.25

0.250.060.020.000.020.06Total

0.500.000.000.000.000.00Squared

0.750.060.020.000.020.06Deviations

1.000.250.060.000.060.251.56

Sheet2

Sheet3

Is the formula helpful?Here is the formula for the function used in the example.If I had showed you this formula, would you have been able to tell me the relative importance of the variables?How would your answer to the last question change if I showed you a function of 50 variables?

Grid-based versus functional ANOVAIf we want to visualize the input-output relationship over a continuous interval (not just at the end points), we need a fine gridBut using a fine grid is not practical with many (> 10) variables, since requires too many simulationsSurprisingly,using an infinite grid is practical if we do the calculations on the response surface as opposed to the simulation. Why?With infinite grid, all the averages in the calculations change to integralsPolynomial and kriging response surfaces have a form which makes it easy to compute these integralsDoing ANOVA with an infinite grid is called functional ANOVA. As long as the response surface fits well, the functional ANOVA of the surface will be as informative, if not more so, than a grid-based ANOVA on a coarse grid.

ANOVA with different grid sizesThe percent contributions change with the grid size.As the grid size approaches infinity, we get the functional ANOVA. For our simple example, these integrals can actually be computed on the real simulation function.Doing the functional ANOVA on a fitted kriging surface gives essentially the same results using only 41 points.

Main effects in functional ANOVAANOVA with 5 x 5 gridFunctional ANOVARecalling form of function (see right), do these main effects make sense?

Chart1

-2.375

0.6875

1.875

1.1875

-1.375

Main Effect(x2)

x1

Main Effect of x1

Sheet1

Y - MeanAverage(Main Effect)

X2x1Main Effect(x2)

0.000.250.500.751.000-2.375

X10.00-2.38-3.78-2.38-0.97-2.38-2.380.250.6875

0.250.44-0.840.692.220.940.690.51.875

0.501.380.221.883.532.381.880.751.1875

0.750.44-0.591.192.971.941.191-1.375

1.00-2.38-3.28-1.380.53-0.38-1.38

Sheet1

Main Effect(x2)

x1

Main Effect of x1

Sheet2

Sheet3

Chart5

-2.9629132801

-2.8329965865

-2.701161862

-2.5682705699

-2.4348021385

-2.3011070544

-2.1674688924

-2.0341306642

-1.9013086923

-1.7692009412

-1.6379925128

-1.5078595661

-1.3789723145

-1.2514974909

-1.125600524

-1.0014476094

-0.8792078538

-0.75905566

-0.6411735976

-0.5257561178

-0.4130147279

-0.3031857959

-0.1965435122

-0.0934244541

0.005714644

0.1000613104

0.1882062676

0.2714543543

0.3505132834

0.4257343871

0.4973478611

0.5655201753

0.6303782582

0.6920220379

0.7505317792

0.8059727276

0.8583982154

0.9078518145

0.9543688698

0.9979775976

1.0386998687

1.0765517382

1.1115437645

1.1436811137

1.1729634433

1.1993844765

1.2229311242

1.2435817835

1.2613028663

1.2760403528

1.2876848954

1.2960360745

1.301294315

1.3035689704

1.3029140676

1.2993632046

1.2929379989

1.2836515563

1.2715101771

1.2565142347

1.2386586

1.2179327633

1.1943207299

1.1678007023

1.1383445559

1.1059170578

1.0704747739

1.031964543

0.9903213292

0.9454651313

0.8972963562

0.8456885082

0.7904756965

0.7314285443

0.6681969929

0.6000759515

0.5257654612

0.4466693998

0.3635970959

0.2770034868

0.1872238068

0.0945311366

-0.0008391236

-0.0986763407

-0.1987868751

-0.3009889253

-0.4051088266

-0.510978192

-0.6184315334

-0.7273041314

-0.8374299658

-0.9486395487

-1.0607574733

-1.1735994216

-1.2869682633

-1.4006485786

-1.5143983619

-1.6279351969

-1.7409100071

-1.8528452613

-1.9628828312

f

Sheet1


X2

0.000.250.500.751.00

X10.000.00-1.410.001.410.00

0.252.811.533.064.593.31

0.503.752.594.255.914.75

0.752.811.783.565.344.31

1.000.00-0.911.002.912.00

Pointx1x2f

10.000.000

20.000.25-1.40625

30.000.500

40.000.751.40625

50.001.000

60.250.002.8125

70.250.251.53125

80.250.503.0625

90.250.754.59375

100.251.003.3125

110.500.003.75

120.500.252.59375

130.500.504.25

140.500.755.90625

150.501.004.75

160.750.002.8125

170.750.251.78125

180.750.503.5625

190.750.755.34375

200.751.004.3125

211.000.000

221.000.25-0.90625

231.000.50129.930.3113168296

241.000.752.90625

251.001.002

Sheet1

f versus x1

Sheet2

f versus x2

Sheet3

surf

anova

Index

Number of Y variables1

Y variable namesf

Starting row in spreadsheet34

XY Data

Number of points25

Number of x variables2

Pointx1x2f

10.00000E+000.00000E+000.00000E+00

20.00000E+002.50000E-01-1.40625E+00

30.00000E+005.00000E-010.00000E+00

40.00000E+007.50000E-011.40625E+00

50.00000E+001.00000E+000.00000E+00

62.50000E-010.00000E+002.81250E+00

72.50000E-012.50000E-011.53125E+00

82.50000E-015.00000E-013.06250E+00

92.50000E-017.50000E-014.59375E+00

102.50000E-011.00000E+003.31250E+00

115.00000E-010.00000E+003.75000E+00

125.00000E-012.50000E-012.59375E+00

135.00000E-015.00000E-014.25000E+00

145.00000E-017.50000E-015.90625E+00

155.00000E-011.00000E+004.75000E+00

167.50000E-010.00000E+002.81250E+00

177.50000E-012.50000E-011.78125E+00

187.50000E-015.00000E-013.56250E+00

197.50000E-017.50000E-015.34375E+00

207.50000E-011.00000E+004.31250E+00

211.00000E+000.00000E+000.00000E+00

221.00000E+002.50000E-01-9.06250E-01

231.00000E+005.00000E-011.00000E+00

241.00000E+007.50000E-012.90625E+00

251.00000E+001.00000E+002.00000E+00

f

Smoothing? (0=No,1=Yes)0

best kriging modelstandardmodel code:1

Results for standard kriging:

normalized log likelihood-1.85468E+00smoothing evp:0.00000E+00smoothing loglik:-2.25381E+00pvalue:0.00001

func evals in MLE301

error code from optimizer1

condition number4.07805E-11

mu-1.78397E+02

sigmasqr2.84061E+04

error proportion1.00000E-09

theta1.09043E-021.00503E-02

alpha1.00000E-021.00000E-02

RinvResid-2.37821E+05-7.27910E+057.12510E+051.63836E+06-1.43500E+061.89564E+053.10035E+05-4.07573E+05-8.04976E+057.58046E+051.81499E+056.67279E+05-6.07660E+05-1.50007E+061.26846E+061.99699E+053.20912E+05-4.06245E+05-8.16897E+057.47462E+05-2.56205E+05-7.21306E+057.11293E+051.62568E+06-1.40914E+06

cross-validated predictions2.40662E-03-1.40312E+00-2.98122E-031.39921E+001.45215E-022.81169E+001.53057E+003.06333E+004.59551E+003.30925E+003.74924E+002.59240E+004.25116E+005.90929E+004.74470E+002.81164E+001.78055E+003.56332E+005.34554E+004.30930E+002.59266E-03-9.03153E-019.97024E-012.89927E+002.01426E+00

cross-validated standardized residuals1.42001E-012.83466E-01-2.73865E-01-6.42303E-018.69246E-01-7.36227E-02-8.60474E-021.08860E-012.23607E-01-2.94886E-01-6.96085E-02-1.78381E-011.57460E-014.02018E-01-4.88775E-01-7.75686E-02-8.90684E-021.08497E-012.26880E-01-2.90836E-011.52973E-012.80884E-01-2.73410E-01-6.37351E-018.52902E-01

maximum std residual0.87

root-mean-squared std residual0.37

average absolute CV error3.45119E-03

maximum absolute CV error1.45215E-02

average percent CV error1.15E+59

maximum percent CV error1.45E+60

average CV error as % of y range0.05%

maximum CV error as % of y range0.20%

Results for lognormal kriging:

normalized log likelihood-4.63339E-01smoothing evp:2.48678E-05smoothing loglik:-4.05934E-01pvalue:1.00000




offset-8.71875E+00

mu1.58646E+00

sigmasqr7.33272E-01

error proportion0.00000E+00

theta5.32369E-012.28934E+00

alpha1.00000E-021.00000E-02

RinvResid5.54307E+01-1.15809E+022.30690E+018.22654E+01-7.97220E+01-1.47683E+023.35163E+02-1.35231E+02-1.35477E+021.68614E+021.90048E+02-4.85826E+023.29997E+02-4.44802E+00-1.18870E+02-1.42417E+024.05914E+02-3.52000E+021.17750E+022.79993E+015.25009E+01-1.55912E+021.45680E+02-6.08966E+01-1.39626E-01

cross-validated predictions-1.40970E-01-1.36250E+00-6.44336E-031.36344E+002.06818E-012.88280E+001.50646E+003.06966E+004.60679E+003.22930E+003.68867E+002.61883E+004.23787E+005.90655E+004.79160E+002.88029E+001.75050E+003.58195E+005.33179E+004.29749E+00-1.33576E-01-8.43260E-019.54732E-012.94277E+002.00044E+00

cross-validated predictions with bias correction-1.40096E-01-1.36237E+00-6.34142E-031.36363E+002.07677E-012.88297E+001.50648E+003.06968E+004.60682E+003.22946E+003.68878E+002.61884E+004.23789E+005.90657E+004.79173E+002.88046E+001.75052E+003.58197E+005.33183E+004.29769E+00-1.32696E-01-8.43122E-019.54842E-012.94299E+002.00159E+00

cross-validated standardized residuals-1.13851E+009.89370E-01-1.52586E-01-6.96070E-011.68486E+001.13469E+00-1.07606E+003.35645E-014.26730E-01-1.30600E+00-1.16014E+001.24889E+00-6.54090E-011.10959E-027.13712E-011.09274E+00-1.31683E+008.85820E-01-3.70280E-01-2.10216E-01-1.07463E+001.35500E+00-9.80377E-015.13746E-012.79148E-03



average absolute CV error4.81392E-02with bias correction:4.81530E-02

maximum absolute CV error2.06818E-01with bias correction:2.07677E-01

average percent CV error2.38E+60with bias correction:2.38E+60

maximum percent CV error2.07E+61with bias correction:2.08E+61

average CV error as % of y range0.66%with bias correction:0.66%

maximum CV error as % of y range2.83%with bias correction:2.84%

ANOVA for 'f'mean response2.96788719782.9678871978

Using surface 'surf'standard deviation1.6783542043

EffectPercent ContributionVariableCumulative ContributionPercent way from lower to upper boundx1x2x1fx2f

x150.50%x150.50%0.00%4.97392E-032.46239E+000.00-2.96291E+000.00-5.05502E-01

x248.54%x2100.00%1.00%1.34891E-012.41472E+000.01-2.83300E+000.01-5.53165E-01

x2 * x10.96%2.00%2.66725E-012.36280E+000.02-2.70116E+000.02-6.05085E-01

TOTAL:100.00%3.00%3.99617E-012.30799E+000.03-2.56827E+000.03-6.59894E-01

4.00%5.33085E-012.25107E+000.04-2.43480E+000.04-7.16816E-01

5.00%6.66780E-012.19262E+000.05-2.30111E+000.05-7.75272E-01

6.00%8.00418E-012.13310E+000.06-2.16747E+000.06-8.34782E-01

7.00%9.33757E-012.07296E+000.07-2.03413E+000.07-8.94929E-01

8.00%1.06658E+002.01256E+000.08-1.90131E+000.08-9.55331E-01

9.00%1.19869E+001.95226E+000.09-1.76920E+000.09-1.01563E+00

10.00%1.32989E+001.89240E+000.10-1.63799E+000.10-1.07549E+00

11.00%1.46003E+001.83332E+000.11-1.50786E+000.11-1.13457E+00

12.00%1.58891E+001.77534E+000.12-1.37897E+000.12-1.19255E+00

13.00%1.71639E+001.71880E+000.13-1.25150E+000.13-1.24909E+00

14.00%1.84229E+001.66403E+000.14-1.12560E+000.14-1.30386E+00

15.00%1.96644E+001.61138E+000.15-1.00145E+000.15-1.35651E+00

16.00%2.08868E+001.56122E+000.16-8.79208E-010.16-1.40667E+00

17.00%2.20883E+001.51393E+000.17-7.59056E-010.17-1.45395E+00

18.00%2.32671E+001.46994E+000.18-6.41174E-010.18-1.49795E+00

19.00%2.44213E+001.42970E+000.19-5.25756E-010.19-1.53819E+00

20.00%2.55487E+001.39373E+000.20-4.13015E-010.20-1.57416E+00

21.00%2.66470E+001.36263E+000.21-3.03186E-010.21-1.60525E+00

22.00%2.77134E+001.33713E+000.22-1.96544E-010.22-1.63076E+00

23.00%2.87446E+001.31813E+000.23-9.34245E-020.23-1.64976E+00

24.00%2.97360E+001.30695E+000.245.71464E-030.24-1.66093E+00

25.00%3.06795E+001.30601E+000.251.00061E-010.25-1.66187E+00

26.00%3.15609E+001.31963E+000.261.88206E-010.26-1.64826E+00

27.00%3.23934E+001.34354E+000.272.71454E-010.27-1.62435E+00

28.00%3.31840E+001.37536E+000.283.50513E-010.28-1.59252E+00

29.00%3.39362E+001.41386E+000.294.25734E-010.29-1.55402E+00

30.00%3.46524E+001.45817E+000.304.97348E-010.30-1.50972E+00

31.00%3.53341E+001.50762E+000.315.65520E-010.31-1.46027E+00

32.00%3.59827E+001.56166E+000.326.30378E-010.32-1.40622E+00

33.00%3.65991E+001.61985E+000.336.92022E-010.33-1.34803E+00

34.00%3.71842E+001.68179E+000.347.50532E-010.34-1.28610E+00

35.00%3.77386E+001.74711E+000.358.05973E-010.35-1.22077E+00

36.00%3.82629E+001.81552E+000.368.58398E-010.36-1.15237E+00

37.00%3.87574E+001.88671E+000.379.07852E-010.37-1.08118E+00

38.00%3.92226E+001.96042E+000.389.54369E-010.38-1.00747E+00

39.00%3.96586E+002.03640E+000.399.97978E-010.39-9.31485E-01

40.00%4.00659E+002.11443E+000.401.03870E+000.40-8.53461E-01

41.00%4.04444E+002.19427E+000.411.07655E+000.41-7.73617E-01

42.00%4.07943E+002.27572E+000.421.11154E+000.42-6.92163E-01

43.00%4.11157E+002.35859E+000.431.14368E+000.43-6.09297E-01

44.00%4.14085E+002.44267E+000.441.17296E+000.44-5.25213E-01

45.00%4.16727E+002.52779E+000.451.19938E+000.45-4.40097E-01

46.00%4.19082E+002.61376E+000.461.22293E+000.46-3.54129E-01

47.00%4.21147E+002.70040E+000.471.24358E+000.47-2.67484E-01

48.00%4.22919E+002.78755E+000.481.26130E+000.48-1.80333E-01

49.00%4.24393E+002.87505E+000.491.27604E+000.49-9.28408E-02

50.00%4.25557E+002.96272E+000.501.28768E+000.50-5.16429E-03

51.00%4.26392E+003.05044E+000.511.29604E+000.518.25542E-02

52.00%4.26918E+003.13800E+000.521.30129E+000.521.70115E-01

53.00%4.27146E+003.22522E+000.531.30357E+000.532.57330E-01

54.00%4.27080E+003.31191E+000.541.30291E+000.543.44018E-01

55.00%4.26725E+003.39789E+000.551.29936E+000.554.30000E-01

56.00%4.26083E+003.48298E+000.561.29294E+000.565.15094E-01

57.00%4.25154E+003.56700E+000.571.28365E+000.575.99115E-01

58.00%4.23940E+003.64976E+000.581.27151E+000.586.81877E-01

59.00%4.22440E+003.73107E+000.591.25651E+000.597.63185E-01

60.00%4.20655E+003.81073E+000.601.23866E+000.608.42843E-01

61.00%4.18582E+003.88853E+000.611.21793E+000.619.20643E-01

62.00%4.16221E+003.96426E+000.621.19432E+000.629.96370E-01

63.00%4.13569E+004.03768E+000.631.16780E+000.631.06980E+00

64.00%4.10623E+004.10857E+000.641.13834E+000.641.14068E+00

65.00%4.07380E+004.17666E+000.651.10592E+000.651.20878E+00

66.00%4.03836E+004.24168E+000.661.07047E+000.661.27379E+00

67.00%3.99985E+004.30332E+000.671.03196E+000.671.33544E+00

68.00%3.95821E+004.36126E+000.689.90321E-010.681.39337E+00

69.00%3.91335E+004.41511E+000.699.45465E-010.691.44722E+00

70.00%3.86518E+004.46445E+000.708.97296E-010.701.49656E+00

71.00%3.81358E+004.50877E+000.718.45689E-010.711.54089E+00

72.00%3.75836E+004.54746E+000.727.90476E-010.721.57957E+00

73.00%3.69932E+004.57971E+000.737.31429E-010.731.61182E+00

74.00%3.63608E+004.60437E+000.746.68197E-010.741.63648E+00

75.00%3.56796E+004.61927E+000.756.00076E-010.751.65139E+00

76.00%3.49365E+004.62052E+000.765.25765E-010.761.65264E+00

77.00%3.41456E+004.61204E+000.774.46669E-010.771.64415E+00

78.00%3.33148E+004.59601E+000.783.63597E-010.781.62812E+00

79.00%3.24489E+004.57362E+000.792.77003E-010.791.60573E+00

80.00%3.15511E+004.54568E+000.801.87224E-010.801.57780E+00

81.00%3.06242E+004.51285E+000.819.45311E-020.811.54497E+00

82.00%2.96705E+004.47566E+000.82-8.39124E-040.821.50777E+00

83.00%2.86921E+004.43456E+000.83-9.86763E-020.831.46668E+00

84.00%2.76910E+004.38997E+000.84-1.98787E-010.841.42208E+00

85.00%2.66690E+004.34224E+000.85-3.00989E-010.851.37435E+00

86.00%2.56278E+004.29172E+000.86-4.05109E-010.861.32383E+00

87.00%2.45691E+004.23872E+000.87-5.10978E-010.871.27083E+00

88.00%2.34946E+004.18354E+000.88-6.18432E-010.881.21565E+00

89.00%2.24058E+004.12648E+000.89-7.27304E-010.891.15860E+00

90.00%2.13046E+004.06782E+000.90-8.37430E-010.901.09994E+00

91.00%2.01925E+004.00785E+000.91-9.48640E-010.911.03996E+00

92.00%1.90713E+003.94683E+000.92-1.06076E+000.929.78945E-01

93.00%1.79429E+003.88507E+000.93-1.17360E+000.939.17184E-01

94.00%1.68092E+003.82286E+000.94-1.28697E+000.948.54976E-01

95.00%1.56724E+003.76053E+000.95-1.40065E+000.957.92641E-01

96.00%1.45349E+003.69842E+000.96-1.51440E+000.967.30528E-01

97.00%1.33995E+003.63692E+000.97-1.62794E+000.976.69036E-01

98.00%1.22698E+003.57653E+000.98-1.74091E+000.986.08644E-01

99.00%1.11504E+003.51788E+000.99-1.85285E+000.995.49992E-01

100.00%1.00500E+003.46209E+001.00-1.96288E+001.004.94202E-01

f

f

Chart1

-0.5

-1.65625

0

1.65625

0.5

Main Effect(x1)

x2

Main Effect of x2

Sheet1

Y - Mean

X2x1Main Effect(x1)

0.000.250.500.751.000-0.5

X10.00-2.38-3.78-2.38-0.97-2.380.25-1.65625

0.250.44-0.840.692.220.940.50

0.501.380.221.883.532.380.751.65625

0.750.44-0.591.192.971.9410.5

1.00-2.38-3.28-1.380.53-0.38

Average = Main Effect-0.50-1.660.001.660.50

Sheet1

-0.5

-1.65625

0

1.65625

0.5

Main Effect(x1)

x2

Main Effect of x2

Sheet2

Sheet3

Chart6

-0.5055020599

-0.5531651643

-0.6050849268

-0.6598942999

-0.7168164534

-0.7752719764

-0.8347823766

-0.8949288809

-0.9553305177

-1.0156307366

-1.0754883296

-1.1345706945

-1.1925483721

-1.2490902413

-1.3038589039

-1.3565059027

-1.4066663728

-1.4539526512

-1.497946162

-1.5381864882

-1.5741557108

-1.6052543561

-1.6307609784

-1.6497550334

-1.6609348336

-1.6618738303

-1.6482595292

-1.6243517162

-1.5925238891

-1.5540240062

-1.5097178349

-1.4602706143

-1.4062237602

-1.3480348113

-1.2861009566

-1.2207740801

-1.152370971

-1.0811805847

-1.0074693849

-0.9314854057

-0.8534614035

-0.7736173645

-0.6921625305

-0.6092970564

-0.5252133856

-0.4400973797

-0.3541292441

-0.2674842104

-0.1803328873

-0.0928408322

-0.0051642878

0.0825542262

0.1701148105

0.2573299071

0.344018478

0.4300002728

0.515093842

0.5991151919

0.6818766116

0.7631854984

0.8428431002

0.9206431094

0.9963700309

1.0697972445

1.1406846479

1.2087757211

1.2737937778

1.335437048

1.3933720302

1.4472241505

1.4965640271

1.540886011

1.5795717691

1.6118203936

1.6364829129

1.6513852045

1.6526350724

1.6441480337

1.6281217783

1.6057293455

1.5777964752

1.5449673801

1.5077749494

1.4666775141

1.4220807008

1.3743516306

1.3238288012

1.2708293905

1.2156549688

1.158596243

1.099937251

1.039959352

0.9789453373

0.9171840302

0.8549759062

0.7926406024

0.7305279121

0.6690357361

0.6086437437

0.5499921254

0.4942019761

f

Sheet1


X2

0.000.250.500.751.00

X10.000.00-1.410.001.410.00

0.252.811.533.064.593.31

0.503.752.594.255.914.75

0.752.811.783.565.344.31

1.000.00-0.911.002.912.00

Pointx1x2f

10.000.000

20.000.25-1.40625

30.000.500

40.000.751.40625

50.001.000

60.250.002.8125

70.250.251.53125

80.250.503.0625

90.250.754.59375

100.251.003.3125

110.500.003.75

120.500.252.59375

130.500.504.25

140.500.755.90625

150.501.004.75

160.750.002.8125

170.750.251.78125

180.750.503.5625

190.750.755.34375

200.751.004.3125

211.000.000

221.000.25-0.90625

231.000.50129.930.3113168296

241.000.752.90625

251.001.002

Sheet1

f versus x1

Sheet2

f versus x2

Sheet3

surf

anova

Index

Number of Y variables1

Y variable namesf

Starting row in spreadsheet34

XY Data

Number of points25

Number of x variables2

Pointx1x2f

10.00000E+000.00000E+000.00000E+00

20.00000E+002.50000E-01-1.40625E+00

30.00000E+005.00000E-010.00000E+00

40.00000E+007.50000E-011.40625E+00

50.00000E+001.00000E+000.00000E+00

62.50000E-010.00000E+002.81250E+00

72.50000E-012.50000E-011.53125E+00

82.50000E-015.00000E-013.06250E+00

92.50000E-017.50000E-014.59375E+00

102.50000E-011.00000E+003.31250E+00

115.00000E-010.00000E+003.75000E+00

125.00000E-012.50000E-012.59375E+00

135.00000E-015.00000E-014.25000E+00

145.00000E-017.50000E-015.90625E+00

155.00000E-011.00000E+004.75000E+00

167.50000E-010.00000E+002.81250E+00

177.50000E-012.50000E-011.78125E+00

187.50000E-015.00000E-013.56250E+00

197.50000E-017.50000E-015.34375E+00

207.50000E-011.00000E+004.31250E+00

211.00000E+000.00000E+000.00000E+00

221.00000E+002.50000E-01-9.06250E-01

231.00000E+005.00000E-011.00000E+00

241.00000E+007.50000E-012.90625E+00

251.00000E+001.00000E+002.00000E+00

f

Smoothing? (0=No,1=Yes)0

best kriging modelstandardmodel code:1

Results for standard kriging:

normalized log likelihood-1.85468E+00smoothing evp:0.00000E+00smoothing loglik:-2.25381E+00pvalue:0.00001




mu-1.78397E+02

sigmasqr2.84061E+04

error proportion1.00000E-09

theta1.09043E-021.00503E-02

alpha1.00000E-021.00000E-02

RinvResid-2.37821E+05-7.27910E+057.12510E+051.63836E+06-1.43500E+061.89564E+053.10035E+05-4.07573E+05-8.04976E+057.58046E+051.81499E+056.67279E+05-6.07660E+05-1.50007E+061.26846E+061.99699E+053.20912E+05-4.06245E+05-8.16897E+057.47462E+05-2.56205E+05-7.21306E+057.11293E+051.62568E+06-1.40914E+06

cross-validated predictions2.40662E-03-1.40312E+00-2.98122E-031.39921E+001.45215E-022.81169E+001.53057E+003.06333E+004.59551E+003.30925E+003.74924E+002.59240E+004.25116E+005.90929E+004.74470E+002.81164E+001.78055E+003.56332E+005.34554E+004.30930E+002.59266E-03-9.03153E-019.97024E-012.89927E+002.01426E+00

cross-validated standardized residuals1.42001E-012.83466E-01-2.73865E-01-6.42303E-018.69246E-01-7.36227E-02-8.60474E-021.08860E-012.23607E-01-2.94886E-01-6.96085E-02-1.78381E-011.57460E-014.02018E-01-4.88775E-01-7.75686E-02-8.90684E-021.08497E-012.26880E-01-2.90836E-011.52973E-012.80884E-01-2.73410E-01-6.37351E-018.52902E-01



average absolute CV error3.45119E-03

maximum absolute CV error1.45215E-02

average percent CV error1.15E+59

maximum percent CV error1.45E+60

average CV error as % of y range0.05%

maximum CV error as % of y range0.20%

Results for lognormal kriging:

normalized log likelihood-4.63339E-01smoothing evp:2.48678E-05smoothing loglik:-4.05934E-01pvalue:1.00000




offset-8.71875E+00

mu1.58646E+00

sigmasqr7.33272E-01

error proportion0.00000E+00

theta5.32369E-012.28934E+00

alpha1.00000E-021.00000E-02

RinvResid5.54307E+01-1.15809E+022.30690E+018.22654E+01-7.97220E+01-1.47683E+023.35163E+02-1.35231E+02-1.35477E+021.68614E+021.90048E+02-4.85826E+023.29997E+02-4.44802E+00-1.18870E+02-1.42417E+024.05914E+02-3.52000E+021.17750E+022.79993E+015.25009E+01-1.55912E+021.45680E+02-6.08966E+01-1.39626E-01

cross-validated predictions-1.40970E-01-1.36250E+00-6.44336E-031.36344E+002.06818E-012.88280E+001.50646E+003.06966E+004.60679E+003.22930E+003.68867E+002.61883E+004.23787E+005.90655E+004.79160E+002.88029E+001.75050E+003.58195E+005.33179E+004.29749E+00-1.33576E-01-8.43260E-019.54732E-012.94277E+002.00044E+00

cross-validated predictions with bias correction-1.40096E-01-1.36237E+00-6.34142E-031.36363E+002.07677E-012.88297E+001.50648E+003.06968E+004.60682E+003.22946E+003.68878E+002.61884E+004.23789E+005.90657E+004.79173E+002.88046E+001.75052E+003.58197E+005.33183E+004.29769E+00-1.32696E-01-8.43122E-019.54842E-012.94299E+002.00159E+00

cross-validated standardized residuals-1.13851E+009.89370E-01-1.52586E-01-6.96070E-011.68486E+001.13469E+00-1.07606E+003.35645E-014.26730E-01-1.30600E+00-1.16014E+001.24889E+00-6.54090E-011.10959E-027.13712E-011.09274E+00-1.31683E+008.85820E-01-3.70280E-01-2.10216E-01-1.07463E+001.35500E+00-9.80377E-015.13746E-012.79148E-03



average absolute CV error4.81392E-02with bias correction:4.81530E-02

maximum absolute CV error2.06818E-01with bias correction:2.07677E-01

average percent CV error2.38E+60with bias correction:2.38E+60

maximum percent CV error2.07E+61with bias correction:2.08E+61

average CV error as % of y range0.66%with bias correction:0.66%

maximum CV error as % of y range2.83%with bias correction:2.84%

ANOVA for 'f'mean response2.96788719782.9678871978

Using surface 'surf'standard deviation1.6783542043

EffectPercent ContributionVariableCumulative ContributionPercent way from lower to upper boundx1x2x1fx2f

x150.50%x150.50%0.00%4.97392E-032.46239E+000.00-2.96291E+000.00-5.05502E-01

x248.54%x2100.00%1.00%1.34891E-012.41472E+000.01-2.83300E+000.01-5.53165E-01

x2 * x10.96%2.00%2.66725E-012.36280E+000.02-2.70116E+000.02-6.05085E-01

TOTAL:100.00%3.00%3.99617E-012.30799E+000.03-2.56827E+000.03-6.59894E-01

4.00%5.33085E-012.25107E+000.04-2.43480E+000.04-7.16816E-01

5.00%6.66780E-012.19262E+000.05-2.30111E+000.05-7.75272E-01

6.00%8.00418E-012.13310E+000.06-2.16747E+000.06-8.34782E-01

7.00%9.33757E-012.07296E+000.07-2.03413E+000.07-8.94929E-01

8.00%1.06658E+002.01256E+000.08-1.90131E+000.08-9.55331E-01

9.00%1.19869E+001.95226E+000.09-1.76920E+000.09-1.01563E+00

10.00%1.32989E+001.89240E+000.10-1.63799E+000.10-1.07549E+00

11.00%1.46003E+001.83332E+000.11-1.50786E+000.11-1.13457E+00

12.00%1.58891E+001.77534E+000.12-1.37897E+000.12-1.19255E+00

13.00%1.71639E+001.71880E+000.13-1.25150E+000.13-1.24909E+00

14.00%1.84229E+001.66403E+000.14-1.12560E+000.14-1.30386E+00

15.00%1.96644E+001.61138E+000.15-1.00145E+000.15-1.35651E+00

16.00%2.08868E+001.56122E+000.16-8.79208E-010.16-1.40667E+00

17.00%2.20883E+001.51393E+000.17-7.59056E-010.17-1.45395E+00

18.00%2.32671E+001.46994E+000.18-6.41174E-010.18-1.49795E+00

19.00%2.44213E+001.42970E+000.19-5.25756E-010.19-1.53819E+00

20.00%2.55487E+001.39373E+000.20-4.13015E-010.20-1.57416E+00

21.00%2.66470E+001.36263E+000.21-3.03186E-010.21-1.60525E+00

22.00%2.77134E+001.33713E+000.22-1.96544E-010.22-1.63076E+00

23.00%2.87446E+001.31813E+000.23-9.34245E-020.23-1.64976E+00

24.00%2.97360E+001.30695E+000.245.71464E-030.24-1.66093E+00

25.00%3.06795E+001.30601E+000.251.00061E-010.25-1.66187E+00

26.00%3.15609E+001.31963E+000.261.88206E-010.26-1.64826E+00

27.00%3.23934E+001.34354E+000.272.71454E-010.27-1.62435E+00

28.00%3.31840E+001.37536E+000.283.50513E-010.28-1.59252E+00

29.00%3.39362E+001.41386E+000.294.25734E-010.29-1.55402E+00

30.00%3.46524E+001.45817E+000.304.97348E-010.30-1.50972E+00

31.00%3.53341E+001.50762E+000.315.65520E-010.31-1.46027E+00

32.00%3.59827E+001.56166E+000.326.30378E-010.32-1.40622E+00

33.00%3.65991E+001.61985E+000.336.92022E-010.33-1.34803E+00

34.00%3.71842E+001.68179E+000.347.50532E-010.34-1.28610E+00

35.00%3.77386E+001.74711E+000.358.05973E-010.35-1.22077E+00

36.00%3.82629E+001.81552E+000.368.58398E-010.36-1.15237E+00

37.00%3.87574E+001.88671E+000.379.07852E-010.37-1.08118E+00

38.00%3.92226E+001.96042E+000.389.54369E-010.38-1.00747E+00

39.00%3.96586E+002.03640E+000.399.97978E-010.39-9.31485E-01

40.00%4.00659E+002.11443E+000.401.03870E+000.40-8.53461E-01

41.00%4.04444E+002.19427E+000.411.07655E+000.41-7.73617E-01

42.00%4.07943E+002.27572E+000.421.11154E+000.42-6.92163E-01

43.00%4.11157E+002.35859E+000.431.14368E+000.43-6.09297E-01

44.00%4.14085E+002.44267E+000.441.17296E+000.44-5.25213E-01

45.00%4.16727E+002.52779E+000.451.19938E+000.45-4.40097E-01

46.00%4.19082E+002.61376E+000.461.22293E+000.46-3.54129E-01

47.00%4.21147E+002.70040E+000.471.24358E+000.47-2.67484E-01

48.00%4.22919E+002.78755E+000.481.26130E+000.48-1.80333E-01

49.00%4.24393E+002.87505E+000.491.27604E+000.49-9.28408E-02

50.00%4.25557E+002.96272E+000.501.28768E+000.50-5.16429E-03

51.00%4.26392E+003.05044E+000.511.29604E+000.518.25542E-02

52.00%4.26918E+003.13800E+000.521.30129E+000.521.70115E-01

53.00%4.27146E+003.22522E+000.531.30357E+000.532.57330E-01

54.00%4.27080E+003.31191E+000.541.30291E+000.543.44018E-01

55.00%4.26725E+003.39789E+000.551.29936E+000.554.30000E-01

56.00%4.26083E+003.48298E+000.561.29294E+000.565.15094E-01

57.00%4.25154E+003.56700E+000.571.28365E+000.575.99115E-01

58.00%4.23940E+003.64976E+000.581.27151E+000.586.81877E-01

59.00%4.22440E+003.73107E+000.591.25651E+000.597.63185E-01

60.00%4.20655E+003.81073E+000.601.23866E+000.608.42843E-01

61.00%4.18582E+003.88853E+000.611.21793E+000.619.20643E-01

62.00%4.16221E+003.96426E+000.621.19432E+000.629.96370E-01

63.00%4.13569E+004.03768E+000.631.16780E+000.631.06980E+00

64.00%4.10623E+004.10857E+000.641.13834E+000.641.14068E+00

65.00%4.07380E+004.17666E+000.651.10592E+000.651.20878E+00

66.00%4.03836E+004.24168E+000.661.07047E+000.661.27379E+00

67.00%3.99985E+004.30332E+000.671.03196E+000.671.33544E+00

68.00%3.95821E+004.36126E+000.689.90321E-010.681.39337E+00

69.00%3.91335E+004.41511E+000.699.45465E-010.691.44722E+00

70.00%3.86518E+004.46445E+000.708.97296E-010.701.49656E+00

71.00%3.81358E+004.50877E+000.718.45689E-010.711.54089E+00

72.00%3.75836E+004.54746E+000.727.90476E-010.721.57957E+00

73.00%3.69932E+004.57971E+000.737.31429E-010.731.61182E+00

74.00%3.63608E+004.60437E+000.746.68197E-010.741.63648E+00

75.00%3.56796E+004.61927E+000.756.00076E-010.751.65139E+00

76.00%3.49365E+004.62052E+000.765.25765E-010.761.65264E+00

77.00%3.41456E+004.61204E+000.774.46669E-010.771.64415E+00

78.00%3.33148E+004.59601E+000.783.63597E-010.781.62812E+00

79.00%3.24489E+004.57362E+000.792.77003E-010.791.60573E+00

80.00%3.15511E+004.54568E+000.801.87224E-010.801.57780E+00

81.00%3.06242E+004.51285E+000.819.45311E-020.811.54497E+00

82.00%2.96705E+004.47566E+000.82-8.39124E-040.821.50777E+00

83.00%2.86921E+004.43456E+000.83-9.86763E-020.831.46668E+00

84.00%2.76910E+004.38997E+000.84-1.98787E-010.841.42208E+00

85.00%2.66690E+004.34224E+000.85-3.00989E-010.851.37435E+00

86.00%2.56278E+004.29172E+000.86-4.05109E-010.861.32383E+00

87.00%2.45691E+004.23872E+000.87-5.10978E-010.871.27083E+00

88.00%2.34946E+004.18354E+000.88-6.18432E-010.881.21565E+00

89.00%2.24058E+004.12648E+000.89-7.27304E-010.891.15860E+00

90.00%2.13046E+004.06782E+000.90-8.37430E-010.901.09994E+00

91.00%2.01925E+004.00785E+000.91-9.48640E-010.911.03996E+00

92.00%1.90713E+003.94683E+000.92-1.06076E+000.929.78945E-01

93.00%1.79429E+003.88507E+000.93-1.17360E+000.939.17184E-01

94.00%1.68092E+003.82286E+000.94-1.28697E+000.948.54976E-01

95.00%1.56724E+003.76053E+000.95-1.40065E+000.957.92641E-01

96.00%1.45349E+003.69842E+000.96-1.51440E+000.967.30528E-01

97.00%1.33995E+003.63692E+000.97-1.62794E+000.976.69036E-01

98.00%1.22698E+003.57653E+000.98-1.74091E+000.986.08644E-01

99.00%1.11504E+003.51788E+000.99-1.85285E+000.995.49992E-01

100.00%1.00500E+003.46209E+001.00-1.96288E+001.004.94202E-01

f

f

Graphical interpretation of % contribsHere we plot Y-Mean versus X1 for all the 25 points in the example.The total variation is the sum of squared distances from the blue dots to the dotted line (Y-mean = 0).The variation explained by x1 is the sum of squared distances from the main effect to the dotted line.The variation not explained by x1 is the sum of squared distances of the blue dots to the main effect (red line).

Effect of variable rangesConsider what would happen if we reduce the range of X1 to (say) [0.25,0.5]The effect of x1 will usually seem smaller relative to that of the other variables, and its percent contribution will go down.In short, the importance of a variable is a function both of the sensitivity (local slope) and the range over which it varies.0.250.50

Available ANOVA softwareANOVA on grids or fractional factorial designsDEXPERTProvides fractional-factorial or full-factorial experimental designsRuns in Windows with GUI, available from Knowledge CenterMinitabRuns in Windows with GUI, available from Knowledge CenterFunctional ANOVAKriging Wizard Provides space-filling experimental designsFits kriging surfacesPerforms exact functional ANOVA on surface. Creates plots.MS Excel Add-In with GUI, available from RSA websiteRobust Design WizardFits up to third-order polynomial surfacesPerforms exact functional ANOVA on polynomial surface. Creates plots. UNIX executable with GUI, available from RSA website

Test your understandingWhat does ANOVA stand for?What questions does ANOVA help answer?How does one quantify the total variation of a function?How does one quantify the variation due to a variable?What is the percent contribution of a variable?What is a main effect plot?What is an interaction effect?How does functional ANOVA differ from grid-based ANOVA?How do percent contributions depend on variable bounds?

ANOVA Tutorial

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