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Statistical Analysis

Professor Lynne Stokes

Department of Statistical Science

Lecture 12

Analysis of Factor-Effects:

Linear Combinations, Contrasts, Polynomials

Analysis of Ballistic Limit VelocitiesMGH Exercise #6.6

Model and Assumptions

yijk = m + ai + bj+ (ab)ij+eijk

where

yij = ballistic velocity for the kth repeat of the ith nose cone and the jth angle

m = overall mean ballistic velocity

ai = fixed effect of the ith nose cone on mean velocity

bi = fixed effect of the jth angle on mean velocity

(ab)ij = fixed effect of the interaction between the ith nose cone and the jth angle on mean velocity

eij = random experimental error, NID(0,s2)

Analysis of Variance Table

Mason, Gunst, & Hess: Exercise 6.6 3

The ANOVA Procedure

Dependent Variable: velocity

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 5 275139.6111 55027.9222 7861.13 <.0001

Error 12 84.0000 7.0000

Corrected Total 17 275223.6111

R-Square Coeff Var Root MSE velocity Mean

0.999695 0.258192 2.645751 1024.722

Source DF Anova SS Mean Square F Value Pr > F

shape 2 7916.4444 3958.2222 565.46 <.0001

angle 1 266206.7222 266206.7222 38029.5 <.0001

shape*angle 2 1016.4444 508.2222 72.60 <.0001

Reject H0 if F = MSAB/MSE > F0.05(2,12)=3.89

Interaction Null HypothesisThe effect of differing nose cones on mean ballistic velocity

is the same for all angles of launch.

Conclusion

There is sufficient evidence (p = 0.0001) to conclude at a significance level of 0.05 that the effects of nose cone design and launch angle differ on the mean ballistic velocities of 10 mm rolled armor projectiles.

Linear Combinations of Parameters

Estimable Functions of Parameters

Not Usually

of Interest

Estimator

Standard Error

t Statistic

comparisons could be of interest

Contrasts of Effects

Estimable Factor Effects

Contrasts

Elimination of the overall mean requires contrasts of main effect averages.

(Note: Want to compare factor effects.)

Elimination of main effects from interaction comparisons requires contrasts

of the interaction Aaverages.

(Note: Want interaction effects to measure variability that is unaccounted for

by or in addition to the main effects.)

Statistical Independence

Orthogonal linear combinations

are statistically independent

Orthogonal contrasts

are statistically independent

Analysis of Water Pump Prototypes

Designs 1 & 3 are Nominally Identical

Designs 2 & 4 are Nominally Identical

MGH Ex 6.22

Analysis of Water Pump Prototypes

Model and Assumptions

yij = m + ai + eij

where

yij = jth mileage driven before failure using ith pump design

m = overall mean mileage

ai = fixed effect of the ith design on mileage

eij = random experimental error, NID(0,s2)

Prespecified Contrasts

Are these

statistics

optimal ?

Companies

Designs 1&3 vs. 2&4

Designs 1 vs. 3

Designs 2 vs. 4

t-test Comparison of Company Effects

The TTEST Procedure

Statistics

Lower CL Upper CL Lower CL

Variable company N Mean Mean Mean Std Dev Std Dev

mileage A 20 26430 27896 29363 2382.8 3133.2

mileage B 20 24630 25431 26232 1301.4 1711.3

mileage Diff (1-2) 849.45 2465.5 4081.6 2063.1 2524.4

Statistics

Upper CL

Variable company Std Dev Std Err Minimum Maximum

mileage A 4576.3 700.6 24215 31416

mileage B 2499.4 382.65 21613 27724

mileage Diff (1-2) 3253.4 798.29

T-Tests

Variable Method Variances DF t Value Pr > |t|

mileage Pooled Equal 38 3.09 0.0037

mileage Satterthwaite Unequal 29.4 3.09 0.0044

t-test Comparison of Designs 1&3 with Designs 2&4

Designs Recoded: 1 & 3 = 1, 2 & 4 = 2

The TTEST Procedure

Statistics

Lower CL Upper CL Lower CL

Variable design N Mean Mean Mean Std Dev Std Dev

mileage 2 10 24506 24870 25235 350.47 509.53

mileage 4 10 26158 26594 27030 419.17 609.41

mileage Diff (1-2) -2251 -1724 -1196 424.42 561.69

Statistics

Upper CL

Variable design Std Dev Std Err Minimum Maximum

mileage 2 930.2 161.13 24215 25930

mileage 4 1112.5 192.71 25804 27724

mileage Diff (1-2) 830.65 251.2

T-Tests

Variable Method Variances DF t Value Pr > |t|

mileage Pooled Equal 18 -6.86 <.0001

mileage Satterthwaite Unequal 17.5 -6.86 <.0001

t-test Comparison of Design #1 with Design #3

The TTEST Procedure

Statistics

Lower CL Upper CL Lower CL

Variable design N Mean Mean Mean Std Dev Std Dev

mileage 1 10 30675 30922 31169 237.72 345.61

mileage 3 10 23070 24268 25465 1151.8 1674.5

mileage Diff (1-2) 5518.5 6654.5 7790.5 913.56 1209

Statistics

Upper CL

Variable design Std Dev Std Err Minimum Maximum

mileage 1 630.95 109.29 30321 31416

mileage 3 3057 529.53 21613 26233

mileage Diff (1-2) 1787.9 540.69

T-Tests

Variable Method Variances DF t Value Pr > |t|

mileage Pooled Equal 18 12.31 <.0001

mileage Satterthwaite Unequal 9.77 12.31 <.0001

t-test Comparison of Design #2 with Design #4

The TTEST Procedure

Statistics

Lower CL Upper CL Lower CL

Variable design N Mean Mean Mean Std Dev Std Dev

mileage 2 10 24506 24870 25235 350.47 509.53

mileage 4 10 26158 26594 27030 419.17 609.41

mileage Diff (1-2) -2251 -1724 -1196 424.42 561.69

Statistics

Upper CL

Variable design Std Dev Std Err Minimum Maximum

mileage 2 930.2 161.13 24215 25930

mileage 4 1112.5 192.71 25804 27724

mileage Diff (1-2) 830.65 251.2

T-Tests

Variable Method Variances DF t Value Pr > |t|

mileage Pooled Equal 18 -6.86 <.0001

mileage Satterthwaite Unequal 17.5 -6.86 <.0001

Linear Model Analysis

The ANOVA Procedure

Dependent Variable: mileage

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 3 270956900.1 90318966.7 101.64 <.0001

Error 36 31990423.8 888622.9

Corrected Total 39 302947323.9

R-Square Coeff Var Root MSE mileage Mean

0.894403 3.535431 942.6680 26663.45

Source DF Anova SS Mean Square F Value Pr > F

design 3 270956900.1 90318966.7 101.64 <.0001

Analysis of Water Pump Prototypes

Comparison of All 4 Designs

H0: ai = 0 for all i vs Ha: ai 0 for some i

Reject H0 and accept Ha if F > F0.05(3,36) = 2.87

From the ANOVA Table, F = 101.64.

Conclusion

On the basis of this analysis, there is sufficient evidence (p = 0.001) to conclude that the mean mileage before failure of these prototype pumps differs by design type.

Analysis of Water Pump Prototypes

Multiple Comparisons of Pump Design Average Mileages

Using Fisher’s Least Significant Difference procedure, average mileages for two designs are significantly different from each other if their difference exceeds 1,135.4 miles. As indicated by the line in the above table, the average mileages for Designs 2 and 3 are not significantly different. Mileages for all other pairwise comparisons of designs are significantly different; in particular, Design 1 has a significantly greater average mileage measurement than the other designs.

Orthogonal Sets of Contrasts

Many other sets of orthogonal contrasts

Comparing the Mean of Two Factor Level Effects

with a Third Factor Effect

Some Contrasts for Comparing Factor Level AveragesComparing Two Factor Level Effects

One Set of Main Effects Contrasts :Qualitative Factor Levels

Main Effects Contrasts :Qualitative Factor Levels

- Three statistically independent contrasts of the response averages
- A partitioning of the main effects degrees of freedom into single degree-of-freedom contrasts (a = 4: df = 3)

Test

Single degree-of-

freedom contrasts

Sums of Squares and Contrastsa-1 mutually

orthonormal contrast

vectors

Orthonormal

Basis Set

ANY set of

orthonormal contrast

vectors

Warping of Copper Plates:Quantitative Factor Levels

MGH Table 6.7

Model and Assumptions

yijk = m + ai + bj+ (ab)ij+eijk

where

yijk = warping measurement for the kth repeat at the ith temp. using a plate having the jth amount of copper

m = overall mean warping measurement

ai = fixed effect of the ith temperature on the mean warping

bi = fixed effect of the jth copper content on the mean warping

(ab)ij = fixed effect of the interaction between the ith temperature and the jth copper content on the mean warping

eij = random experimental error, NID(0,s2)

Warping of Copper Plates

Quantitative factor levels

HOW does mean warping change with the factor levels ?

MGH Table 6.7

Warping of Copper Plates

Are there contrast vectors

that quantify curvature ?

35

30

25

Average

Warping

20

15

10

0

50

75

150

100

125

Temperature (deg F)

Warping of Copper Plates

Are there contrast vectors

that quantify curvature ?

35

30

25

Average

Warping

20

15

10

0

20

40

60

80

100

Copper Content (%)

Main Effects Contrasts :Quantitative Factor Levels

Assumption

Mean response can be well approximated by a

low-order polynomial function of the factor levels

Orthogonal Polynomials

Coded Factor Levels

xi = a0 + a1i

Equally Spaced

e.g., Temp = 50, 75, 100, 125 >>>> xi = 25 + 25i

Orthogonal Polynomials

Not Orthongonal

Orthogonal

With coded factor levels,

orthogonal polynomials are only

a function of n

Orthogonal Polynomials

Linear Effect

Orthogonality Constraint

c1’1 = 0

Solution

Centered Data Values

Linear Orthogonal Polynomial(Coded Factor Levels)

n=4

Main Effects Contrasts :Equally Spaced Quantitative Factor Levels

Recurrence Relation for Coded Levels

Main Effects Contrasts :Equally Spaced Quantitative Factor Levels

n=4

q1 = Linear

q2 = Quadratic

q3 = Cubic

Linear Combinations of Parameters

Estimable Functions of Parameters

Estimator

Same for Contrasts

Standard Error

t Statistic

SAS Output

The GLM Procedure

Dependent Variable: warping Warping Measurement

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 15 968.218750 64.547917 9.52 <.0001

Error 16 108.500000 6.781250

Corrected Total 31 1076.718750

R-Square Coeff Var Root MSE warping Mean

0.899231 12.45600 2.604083 20.90625

Source DF Type I SS Mean Square F Value Pr > F

temp 3 156.0937500 52.0312500 7.67 0.0021

content 3 698.3437500 232.7812500 34.33 <.0001

temp*content 9 113.7812500 12.6423611 1.86 0.1327

Source DF Type III SS Mean Square F Value Pr > F

temp 3 156.0937500 52.0312500 7.67 0.0021

content 3 698.3437500 232.7812500 34.33 <.0001

temp*content 9 113.7812500 12.6423611 1.86 0.1327

Contrast DF Contrast SS Mean Square F Value Pr > F

Linear Temperature 1 26.4062500 26.4062500 3.89 0.0660

Quadratic Temperature 1 94.5312500 94.5312500 13.94 0.0018

Cubic Temperature 1 35.1562500 35.1562500 5.18 0.0369

Linear Copper Content 1 652.0562500 652.0562500 96.16 <.0001

Quadratic Copper Content 1 30.0312500 30.0312500 4.43 0.0515

Cubic Copper Content 1 16.2562500 16.2562500 2.40 0.1411

Scaled Contrasts

Note: Need scaling to make polynomial contrasts comparable

Warping of Copper Plates

Note: Quadratic Effect is Forced to be Orthogonal to Linear

Cubic Effect is Forced to be Orthogonal to Linear and Quadratic

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