Stat 470-7

• Today: Transformation of the response; Latin-squares

Transformations (Section 2.5)

• Often one will perform a residual analysis to verify modeling assumptions…and at least one assumption fails

• A defect that can frequently arise in non-constant variance

• This can occur, for example, when the data follow a non-normal, skewed distribution

• The F-test in ANOVA is only slightly violated

• In such cases, a variance stabalizing transformation may be applied

Transformations

• Several transformations may be attempted:

– Y*=

– Y*=

– Y*=

Transformations

• Analyze the data on the Y* scale, choosing the transformation where:

– The simplest model results,

– There are no patterns in the residuals

– One can interpret the transformation

Example

• An engineer wishes to study the impact of 4 factors on the rate of advance of a drill. Each of the 4 factors (labeled A-D) were studied at 2 levels

A B C D Y -1 -1 -1 -1 1.68 +1 -1 -1 -1 1.98 -1 +1 -1 -1 3.28 +1 +1 -1 -1 3.44 -1 -1 +1 -1 4.98 +1 -1 +1 -1 5.70 -1 +1 +1 -1 9.97 +1 +1 +1 -1 9.07 -1 -1 -1 +1 2.07 +1 -1 -1 +1 2.44 -1 +1 -1 +1 4.09 +1 +1 -1 +1 4.53 -1 -1 +1 +1 7.77 +1 -1 +1 +1 9.43 -1 +1 +1 +1 11.75 +1 +1 +1 +1 16.30

Example

• Would like to fit an N-way ANOVA to these data (main effects and 2-factor interactions only)

• Model:

Example

Tests of Between-Subjects Effects

Dependent Variable: Y

257.614a 10 25.761 25.406 .001

606.144 1 606.144 597.781 .000

3.331 1 3.331 3.285 .130

43.494 1 43.494 42.894 .001

165.508 1 165.508 163.225 .000

20.885 1 20.885 20.597 .006

9.000E-02 1 9.000E-02 .089 .778

1.416 1 1.416 1.397 .290

2.839 1 2.839 2.800 .155

9.060 1 9.060 8.935 .030

.783 1 .783 .772 .420

10.208 1 10.208 10.067 .025

5.070 5 1.014

868.829 16

262.684 15

SourceCorrected Model

Intercept

A

B

C

D

A * B

A * C

A * D

B * C

B * D

C * D

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .981 (Adjusted R Squared = .942)a.

Example

Residuals vs. Predicted

Predicted Value for Y

1614121086420

Re

sid

ua

l fo

r Y

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

Example

Tests of Between-Subjects Effects

Dependent Variable: SQRTY

9.876a 10 .988 53.540 .000

88.512 1 88.512 4798.513 .000

.103 1 .103 5.610 .064

1.735 1 1.735 94.084 .000

7.011 1 7.011 380.070 .000

.688 1 .688 37.296 .002

2.269E-04 1 2.269E-04 .012 .916

1.683E-02 1 1.683E-02 .912 .383

5.731E-02 1 5.731E-02 3.107 .138

6.818E-02 1 6.818E-02 3.696 .113

3.726E-03 1 3.726E-03 .202 .672

.192 1 .192 10.409 .023

9.223E-02 5 1.845E-02

98.480 16

9.968 15

SourceCorrected Model

Intercept

A

B

C

D

A * B

A * C

A * D

B * C

B * D

C * D

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .991 (Adjusted R Squared = .972)a.

Example

Residuals vs. Predicted

Predicted Value for SQRTY

4.03.53.02.52.01.51.0

Re

sid

ua

l fo

r S

QR

TY

1.0.9.8.7.6.5.4.3.2.1.0

-.1-.2-.3-.4-.5-.6-.7-.8-.9

-1.0

A New Example

• A scientist wishes to investigate the effect of 5 different ingredients (A-E) on the reaction time of a chemical process

• The scientist has enough resources to perform 25 trials

• Each batch of raw material is only large enough to permit 5 runs to be made

• Each run takes about 1.5 hours, so only 5 runs can be performed in a day

Example

• How can we run the experiment?

Day/Batch 1 2 3 4 5 1

2 3 4 5

Two Blocking Variables ; 1 Factor

• Can set up an experiment to remove the effect of 2 blocking variables (e.g., season and time of day)

• Experiment is an example of a 5x5 Latin Squares Design

Latin Squares Design

• Situation:

– Have 2 blocking factors - one for rows and one for columns

– Have 1 experimental factor

– Each factor has k levels

– Design is arranged so that each level of the experimental factors appears exactly one time in each row and each column

– The levels of the two blocking factors are assigned at random to the columns and rows

• Model:

• i, j=1,2,…,k

• l indicates the index for the Latin letter in the (i,j)th cell

• The triplet (i,j,l) takes on values

ijlljiijly

Notes

• No interaction, since interactions cannot be estimated in an un-replicated experiment

• Usual assumptions apply

ANOVA Decomposition

ANOVA Decomposition

ANOVA Table

Source of Variation

Degrees of Freedom

Sum of Squares

Mean Squares

F

Row k-1 Column k-1 Treatment (k-1) Residual (k-1)(k-2)) Total k2-1

Hypotheses

Multiple Comparisons

Example

Day/Batch 1 2 3 4 5 1 (A) 8 (B) 7 (D) 1 (C) 7 (E) 3

2 (C) 11 (E) 2 (A) 7 (D) 3 (B) 8 3 (B) 4 (A) 9 (C) 10 (E) 1 (D) 5 4 (D) 6 (C) 8 (E) 6 (B) 6 (A) 10 5 (E) 4 (D) 2 (B) 3 (A) 8 (C) 8

Useful Plots

BATCH

6543210

TIM

E

12

10

8

6

4

2

0

Useful Plots

DAY

6543210

TIM

E

12

10

8

6

4

2

0

Useful Plots

Ingredient

6543210

TIM

E12

10

8

6

4

2

0

Example

Tests of Between-Subjects Effects

Dependent Variable: TIME

125.120a 12 10.427 1.535 .235

864.360 1 864.360 127.237 .000

11.440 4 2.860 .421 .791

12.240 4 3.060 .450 .770

101.440 4 25.360 3.733 .034

81.520 12 6.793

1071.000 25

206.640 24

SourceCorrected Model

Intercept

BATCH

DAY

INGRED

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .605 (Adjusted R Squared = .211)a.

Example

Multiple Comparisons

Dependent Variable: TIME

Tukey HSD

1.8000 1.64843 .807 -3.4543 7.0543

-1.4000 1.64843 .910 -6.6543 3.8543

3.0000 1.64843 .407 -2.2543 8.2543

4.2000 1.64843 .144 -1.0543 9.4543

-1.8000 1.64843 .807 -7.0543 3.4543

-3.2000 1.64843 .348 -8.4543 2.0543

1.2000 1.64843 .946 -4.0543 6.4543

2.4000 1.64843 .607 -2.8543 7.6543

1.4000 1.64843 .910 -3.8543 6.6543

3.2000 1.64843 .348 -2.0543 8.4543

4.4000 1.64843 .118 -.8543 9.6543

5.6000* 1.64843 .035 .3457 10.8543

-3.0000 1.64843 .407 -8.2543 2.2543

-1.2000 1.64843 .946 -6.4543 4.0543

-4.4000 1.64843 .118 -9.6543 .8543

1.2000 1.64843 .946 -4.0543 6.4543

-4.2000 1.64843 .144 -9.4543 1.0543

-2.4000 1.64843 .607 -7.6543 2.8543

-5.6000* 1.64843 .035 -10.8543 -.3457

-1.2000 1.64843 .946 -6.4543 4.0543

(J) INGREDB

C

D

E

A

C

D

E

A

B

D

E

A

B

C

E

A

B

C

D

(I) INGREDA

B

C

D

E

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

Based on observed means.

The mean difference is significant at the .05 level.*.