review of anova computer output...

ANOVA Computer Output Steve Brainerd

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Review of ANOVA Computer Output Interpretation

• This is from 04 of your text Montgomery 5th edition section 3-6.

• There are many computer Statistical analysis programs available,

• Popular ones include:• EXCEL Data Analysis Add in: Limited• Design Expert• Jump• Minitab• Also all programs have built in help tutorials if you forget

or get lost with these tests!!


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Review of ANOVA Computer Output Interpretation

• Example used here:


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Review of ANOVA Computer Output Interpretation: We are asking question: Are means between treatments

different? And By how much?

• First attack look at graphs:from Design Expert

• It appears 20, 25, 30% cotton levels are different than 15 and 35%, By how much?

• Is this due to chance variation?• How do we assign a confidence value to any decision we make?


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Review of ANOVA Computer Output InterpretationSTAT EASE Design Expert: ANOVA Single Factor

Typical ANOVA Table with Sum of Squares

M for Model terms

e for error terms


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Review of ANOVA Computer Output InterpretationSTAT EASE Design Expert Diagnostics:ANOVA Single Factor

If there were more than one factor or source there would be A, B, C etc listed


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Design Expert Analysis ANOVA TermsModel: 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.)


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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.

SQRT(8.06) = 2.84

Mean: Overall average of all the response data.

Grand mean = 15.04

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.

CV =2.84/15.04 x 100 =18.88%


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PRESS: Predicted Residual Error Sum of Squares – Basically a measure of how well the model from this experiment is likely to predict the response in a new experiment.. Small values are desirable

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.


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R2 Factor A model (% cotton) explains 74.69% variability in response (Tensile strength)

R-Squared: A measure of the amount of variation around the mean explained by the model.

R2 = 1-(SSresidual / (SSmodel + SSresidual)) =SSmodel/SStotal = 475.76/636.96 = 0.7469

Adjusted R2 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))) = 1-((161.2/20)/((475.76+161.2)/(4 + 20))) =1-((161.20/20)/(636.96/24)) = 1-

(8.06/26.54) = 0.6963


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Pred R-Squared: A measure of the amount of variation in new data explained by the model.

1-(PRESS / (SStotal-SSblock) = 1-(251.87/(636.96 – 0) =

1-(251.87/636.96) = 1 – 0.3954 = 0.6046 = 60.46%

(Remember the model we generated accounts for 74% of the observed variation in tensile strength from the % cotton factor.)

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.


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Adequate Precision: Basically a measure of S/N ( signal to noise ratio), It gives you a factor by which you can judge your model to see if it “adequate” to navigate through the design space and be able to predict the response. Desire values > 4.0

Computed by : (Maximum predicted response – Minimum predicted response)/ (Average standard deviation of all predicted responses)

Adequate Precision = (21.60 – 9.80)/(sqrt(MSe/5)) =11.80/(sqrt(8.06/5)) =

11.80/(sqrt(1.6119 ) = 11.80/1.2696 = 9.294


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Coefficient Estimate: Regression coefficient representing the expected change in response y per unit change in x when all remaining factors are held constant. In orthogonal designs, it equals one half the factorial effect.

DF: Degrees of Freedom – equal to one for testing coefficients.

Standard Error: The standard deviation associated with the coefficient estimate.

95% CI High and Low: These two columns represent the range that the true coefficientshould be found in 95% of the time. If this range spans 0 (one limit is positive and the other negative) then the coefficient of 0 could be true, indicating the factor has no effect.


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Treatment mean = estimate of the effect for each % cotton level ( treatment)

Standard error = estimate of the sample standard deviation for that effect (treatment) = sqrt(MSe/n)


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Review of ANOVA Computer Output InterpretationSTAT EASE Design Expert Diagnostics:

ANOVA Single FactorDifferences in pairs of treatment LSD analysis Design Expert VS JUMP


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ANOVA Single FactorDesign


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Review of ANOVA Computer Output InterpretationSTAT EASE Design Expert Diagnostics: : ANOVA Single Factor

Leverage is the potential for a design point to influence the fit of the model coefficients, based on its position in the design space. Leverages near 1 should be avoided. Replicate the point or add more design points to reduce leverage.


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The Student Residual is the number of standard deviations that separate the actual and predicted response values. It is the residual divided by the estimated standard deviation of the residual.


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The Cook's distance for this observation--not to be confused with the distance between the dining room and kitchen. It is a measure of how much the regression equation changes if this specific run is deleted. It is roughly a combination of leverage and outlier-T and can be used to help identify individual runs that may be outliers.


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ANOVA Single Factor

The outlier t test checks whether a run is consistent with the other runs, assuming the chosen model holds. The model coefficients are calculated based on all of the design points except one. A prediction of the response at this point is made. The residual is evaluated using the t-test. A value greater than 3.5 means that this point should be examined as a possible outlier.


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Review of ANOVA Computer Output InterpretationSTAT EASE Design Expert Create a Model: Change Factor 1

to Numeric


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Review of ANOVA Computer Output InterpretationSTAT EASE Design Expert Create a Model:

Click on Analysis: Note new menu options: Click on Fit Summary:

Click on Status: Model Order Cubic; Right click on Coefficient and make Model M


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FIT SUMMARY


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Click on Model; Next Click on ANOVA


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ANOVA Screen for Cubic Model Suggested>>


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ANOVA Screen for Cubic Model Suggested>> Model Equations!!


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Model Graphic Screen for Cubic Model Suggested


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Review of ANOVA Computer Output InterpretationJUMP Design Expert Model Fitting


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Review of ANOVA Computer Output InterpretationJUMP Design Expert Model Fitting


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Review of ANOVA Computer Output InterpretationJUMP Design Expert Model Fitting 2nd Order


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Review of ANOVA Computer Output InterpretationJUMP Design Expert Model Fitting 3rd Order


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Review of ANOVA Computer Output InterpretationJUMP: ANOVA Single Factor

Journal to save as text in WORD Save as Rich Text Format .RTF


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JUMP output


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Jump Summary of Model fit and ANOVA tableSame as Design Expert!


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Jump Outputs Mean confidence limits on the estimate of each treatment mean!


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Jump POST ANOVA Comparison of Means


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Jump POST ANOVA Comparison of MeansMultiple Comparisons

There are a variety of methods to test differences in group means (multiple comparisons) that vary in detail about how to size the test to accommodate different kinds of multiple comparisons. Fit Y by X automatically produces the standard analysis of variance and optionally offers the following four multiple comparison tests:

Each Pair, Student’s t computes individual pairwise comparisons using Student’s ttests. This test is sized for individual comparisons. If you make many pairwise tests, there is no protection across the inferences, and thus the alpha-size (Type I) error rate across the hypothesis tests is higher than that for individual tests.

All Pairs, Tukey HSD gives a test that is sized for all differences among the means.This is the Tukey or Tukey-Kramer HSD (honestly significant difference) test. (Tukey1953, Kramer 1956). This test is an exact alpha-level test if the sample sizes are the same and conservative if the sample sizes are different (Hayter 1984).


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Group 25 % cotton just used as a example


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Jump POST ANOVA Comparison of MeansMultiple Comparisons

With Best, Hsu’s MCB tests whether means are less than the unknown maximum (or greater than the unknown minimum). This is the Hsu MCB test (Hsu 1981).

With Control, Dunnett’s tests whether means are different from the mean of a control group. This is Dunnett’s test (Dunnett 1955).

The three multiple comparisons tests are the ones recommended by Hsu (1989) as level-5 tests for the three situations: MCA (Multiple Comparisons for All pairs), MCB (Multiple Comparisons with the Best), and MCC (Multiple Comparisons with Control).

If you have specified a Block column, then the multiple comparison methods are performed on data that has been adjusted for the Block means.


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Jump POST ANOVA Comparison of Means Red Circles are the ones we are interested in and and the grays indicate they are different.


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Diamonds: vertical 95% confidence interval Horizontal = mean


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Jump POST ANOVA Comparison of Means Example of largevariation for 35% cotton and small for 15% cotton


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Jump POST ANOVA Comparison of Means Example of largemean for 35% cotton and small for 15% cotton


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Jump POST ANOVA Comparison of Means Example of largevariation for 35% cotton and small for 15% cotton


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Jump POST ANOVA Comparison of Means HSD =Honestly Significant Difference


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Jump POST ANOVA Comparison of Means Tukey-Kramer


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Jump POST ANOVA Comparison of Means Hsu’s MCB


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Jump POST ANOVA Comparison of Means Hsu’s MCB


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Jump POST ANOVA Comparison of Means Dunnett’s


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Jump POST ANOVA Comparison of Variances Tests

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