Math 410.2

Bonferroni Simultaneous confidence Intervals

--Understanding the Calculations under Different Models

Foreman/Crew LAB Problem

Factor Levels Values

Factor A 3 Foreman1 Foreman2 Foreman3

Factor B 3 Crew1 Crew2 Crew3

ReplicatE 3 1 2 3

MODEL 1 : TWO FACTOR ANOVA WITH INTERACTION

ANOVA SUMMARY

Source DF SS MS F P

Factor A 2 14.000 7.000 4.67 0.025

Factor B 2 14.000 7.000 4.67 0.025

Replicat 2 6.000 3.000 2.00 0.168

Factor A*Factor B 4 116.000 29.000 19.33 0.000

Error 16 24.000 1.500

Total 26 174.000

Bonferroni 98.0% Simultaneous Confidence Intervals

All Pairwise Comparisons among Levels of Factor A

Factor A = Foreman1 subtracted from:

Factor A Lower Center Upper ------+---------+---------+---------+

Foreman2 -0.132 1.6667 3.465 (--------*--------)

Foreman3 -1.465 0.3333 2.132 (--------*--------)

------+---------+---------+---------+

-2.0 0.0 2.0 4.0

Factor A = Foreman2 subtracted from:

Factor A Lower Center Upper ------+---------+---------+---------+

Foreman3 -3.132 -1.333 0.4651 (--------*--------)

------+---------+---------+---------+

-2.0 0.0 2.0 4.0

Bonferroni Simultaneous Tests

All Pairwise Comparisons among Levels of Factor A

Factor A = Foreman1 subtracted from:

Level Difference SE of Adjusted

Factor A of Means Difference T-Value P-Value

Foreman2 1.6667 0.5774 2.8868 0.0322

Foreman3 0.3333 0.5774 0.5774 1.0000

Factor A = Foreman2 subtracted from:

Level Difference SE of Adjusted

Factor A of Means Difference T-Value P-Value

Foreman3 -1.333 0.5774 -2.309 0.1038

Bonferroni 98.0% Simultaneous Confidence Intervals

All Pairwise Comparisons among Levels of Factor B

Factor B = Crew1 subtracted from:

Factor B Lower Center Upper ----+---------+---------+---------+--

Crew2 -0.1318 1.667 3.465 (----------*-----------)

Crew3 -0.4651 1.333 3.132 (----------*-----------)

----+---------+---------+---------+--

-1.6 0.0 1.6 3.2

Factor B = Crew2 subtracted from:

Factor B Lower Center Upper ----+---------+---------+---------+--

Crew3 -2.132 -0.3333 1.465 (----------*----------)

----+---------+---------+---------+--

-1.6 0.0 1.6 3.2

Bonferroni Simultaneous Tests

All Pairwise Comparisons among Levels of Factor B

Factor B = Crew1 subtracted from:

Level Difference SE of Adjusted

Factor B of Means Difference T-Value P-Value

Crew2 1.667 0.5774 2.887 0.0322

Crew3 1.333 0.5774 2.309 0.1038

Factor B = Crew2 subtracted from:

Level Difference SE of Adjusted

Factor B of Means Difference T-Value P-Value

Crew3 -0.3333 0.5774 -0.5774 1.000

Bonferroni 64.0% Simultaneous Confidence Intervals

All Pairwise Comparisons among Levels of Factor A*Factor B

Factor A = Foreman1

Factor B = Crew1 subtracted from:

Factor A*Factor B Lower Center Upper

Foreman1 Crew2 -2.921 0.000 2.921

Foreman1 Crew3 0.079 3.000 5.921

Foreman2 Crew1 0.079 3.000 5.921

Foreman2 Crew2 -1.921 1.000 3.921

Foreman2 Crew3 1.079 4.000 6.921

Foreman3 Crew1 -3.921 -1.000 1.921

Foreman3 Crew2 3.079 6.000 8.921

Foreman3 Crew3 -3.921 -1.000 1.921

Factor A*Factor B -------+---------+---------+---------

Foreman1 Crew2 (----*----)

Foreman1 Crew3 (----*----)

Foreman2 Crew1 (----*----)

Foreman2 Crew2 (----*----)

Foreman2 Crew3 (----*----)

Foreman3 Crew1 (----*----)

Foreman3 Crew2 (----*----)

Foreman3 Crew3 (----*----)

-------+---------+---------+---------

-6.0 0.0 6.0

Factor A = Foreman1

Factor B = Crew2 subtracted from:

Factor A*Factor B Lower Center Upper

Foreman1 Crew3 0.079 3.000 5.921

Foreman2 Crew1 0.079 3.000 5.921

Foreman2 Crew2 -1.921 1.000 3.921

Foreman2 Crew3 1.079 4.000 6.921

Foreman3 Crew1 -3.921 -1.000 1.921

Foreman3 Crew2 3.079 6.000 8.921

Foreman3 Crew3 -3.921 -1.000 1.921

Factor A*Factor B -------+---------+---------+---------

Foreman1 Crew3 (----*----)

Foreman2 Crew1 (----*----)

Foreman2 Crew2 (----*----)

Foreman2 Crew3 (----*----)

Foreman3 Crew1 (----*----)

Foreman3 Crew2 (----*----)

Foreman3 Crew3 (----*----)

-------+---------+---------+---------

-6.0 0.0 6.0

Factor A = Foreman1

Factor B = Crew3 subtracted from:

Factor A*Factor B Lower Center Upper

Foreman2 Crew1 -2.921 0.000 2.921

Foreman2 Crew2 -4.921 -2.000 0.921

Foreman2 Crew3 -1.921 1.000 3.921

Foreman3 Crew1 -6.921 -4.000 -1.079

Foreman3 Crew2 0.079 3.000 5.921

Foreman3 Crew3 -6.921 -4.000 -1.079

Factor A*Factor B -------+---------+---------+---------

Foreman2 Crew1 (----*----)

Foreman2 Crew2 (----*----)

Foreman2 Crew3 (----*----)

Foreman3 Crew1 (----*----)

Foreman3 Crew2 (----*----)

Foreman3 Crew3 (----*----)

-------+---------+---------+---------

-6.0 0.0 6.0

Factor A = Foreman2

Factor B = Crew1 subtracted from:

Factor A*Factor B Lower Center Upper

Foreman2 Crew2 -4.921 -2.000 0.921

Foreman2 Crew3 -1.921 1.000 3.921

Foreman3 Crew1 -6.921 -4.000 -1.079

Foreman3 Crew2 0.079 3.000 5.921

Foreman3 Crew3 -6.921 -4.000 -1.079

Factor A*Factor B -------+---------+---------+---------

Foreman2 Crew2 (----*----)

Foreman2 Crew3 (----*----)

Foreman3 Crew1 (----*----)

Foreman3 Crew2 (----*----)

Foreman3 Crew3 (----*----)

-------+---------+---------+---------

-6.0 0.0 6.0

Factor A = Foreman2

Factor B = Crew2 subtracted from:

Factor A*Factor B Lower Center Upper

Foreman2 Crew3 0.079 3.000 5.9208

Foreman3 Crew1 -4.921 -2.000 0.9208

Foreman3 Crew2 2.079 5.000 7.9208

Foreman3 Crew3 -4.921 -2.000 0.9208

Factor A*Factor B -------+---------+---------+---------

Foreman2 Crew3 (----*----)

Foreman3 Crew1 (----*----)

Foreman3 Crew2 (----*----)

Foreman3 Crew3 (----*----)

-------+---------+---------+---------

-6.0 0.0 6.0

Factor A = Foreman2

Factor B = Crew3 subtracted from:

Factor A*Factor B Lower Center Upper

Foreman3 Crew1 -7.921 -5.000 -2.079

Foreman3 Crew2 -0.921 2.000 4.921

Foreman3 Crew3 -7.921 -5.000 -2.079

Factor A*Factor B -------+---------+---------+---------

Foreman3 Crew1 (----*----)

Foreman3 Crew2 (----*----)

Foreman3 Crew3 (----*----)

-------+---------+---------+---------

-6.0 0.0 6.0

Factor A = Foreman3

Factor B = Crew1 subtracted from:

Factor A*Factor B Lower Center Upper

Foreman3 Crew2 4.079 7.00000 9.921

Foreman3 Crew3 -2.921 -0.00000 2.921

Factor A*Factor B -------+---------+---------+---------

Foreman3 Crew2 (----*----)

Foreman3 Crew3 (----*----)

-------+---------+---------+---------

-6.0 0.0 6.0

Factor A = Foreman3

Factor B = Crew2 subtracted from:

Factor A*Factor B Lower Center Upper

Foreman3 Crew3 -9.921 -7.000 -4.079

Factor A*Factor B -------+---------+---------+---------

Foreman3 Crew3 (----*----)

-------+---------+---------+---------

-6.0 0.0 6.0

Bonferroni Simultaneous Tests

All Pairwise Comparisons among Levels of Factor A*Factor B

Factor A = Foreman1

Factor B = Crew1 subtracted from:

Level Difference SE of Adjusted

Factor A*Factor B of Means Difference T-Value P-Value

Foreman1 Crew2 0.000 1.000 0.000 1.0000

Foreman1 Crew3 3.000 1.000 3.000 0.3053

Foreman2 Crew1 3.000 1.000 3.000 0.3053

Foreman2 Crew2 1.000 1.000 1.000 1.0000

Foreman2 Crew3 4.000 1.000 4.000 0.0372

Foreman3 Crew1 -1.000 1.000 -1.000 1.0000

Foreman3 Crew2 6.000 1.000 6.000 0.0007

Foreman3 Crew3 -1.000 1.000 -1.000 1.0000

Factor A = Foreman1

Factor B = Crew2 subtracted from:

Level Difference SE of Adjusted

Factor A*Factor B of Means Difference T-Value P-Value

Foreman1 Crew3 3.000 1.000 3.000 0.3053

Foreman2 Crew1 3.000 1.000 3.000 0.3053

Foreman2 Crew2 1.000 1.000 1.000 1.0000

Foreman2 Crew3 4.000 1.000 4.000 0.0372

Foreman3 Crew1 -1.000 1.000 -1.000 1.0000

Foreman3 Crew2 6.000 1.000 6.000 0.0007

Foreman3 Crew3 -1.000 1.000 -1.000 1.0000

Factor A = Foreman1

Factor B = Crew3 subtracted from:

Level Difference SE of Adjusted

Factor A*Factor B of Means Difference T-Value P-Value

Foreman2 Crew1 0.000 1.000 0.000 1.0000

Foreman2 Crew2 -2.000 1.000 -2.000 1.0000

Foreman2 Crew3 1.000 1.000 1.000 1.0000

Foreman3 Crew1 -4.000 1.000 -4.000 0.0372

Foreman3 Crew2 3.000 1.000 3.000 0.3053

Foreman3 Crew3 -4.000 1.000 -4.000 0.0372

Factor A = Foreman2

Factor B = Crew1 subtracted from:

Level Difference SE of Adjusted

Factor A*Factor B of Means Difference T-Value P-Value

Foreman2 Crew2 -2.000 1.000 -2.000 1.0000

Foreman2 Crew3 1.000 1.000 1.000 1.0000

Foreman3 Crew1 -4.000 1.000 -4.000 0.0372

Foreman3 Crew2 3.000 1.000 3.000 0.3053

Foreman3 Crew3 -4.000 1.000 -4.000 0.0372

Factor A = Foreman2

Factor B = Crew2 subtracted from:

Level Difference SE of Adjusted

Factor A*Factor B of Means Difference T-Value P-Value

Foreman2 Crew3 3.000 1.000 3.000 0.3053

Foreman3 Crew1 -2.000 1.000 -2.000 1.0000

Foreman3 Crew2 5.000 1.000 5.000 0.0047

Foreman3 Crew3 -2.000 1.000 -2.000 1.0000

Factor A = Foreman2

Factor B = Crew3 subtracted from:

Level Difference SE of Adjusted

Factor A*Factor B of Means Difference T-Value P-Value

Foreman3 Crew1 -5.000 1.000 -5.000 0.0047

Foreman3 Crew2 2.000 1.000 2.000 1.0000

Foreman3 Crew3 -5.000 1.000 -5.000 0.0047

Factor A = Foreman3

Factor B = Crew1 subtracted from:

Level Difference SE of Adjusted

Factor A*Factor B of Means Difference T-Value P-Value

Foreman3 Crew2 7.00000 1.000 7.00000 0.0001

Foreman3 Crew3 -0.00000 1.000 -0.00000 1.0000

Factor A = Foreman3

Factor B = Crew2 subtracted from:

Level Difference SE of Adjusted

Factor A*Factor B of Means Difference T-Value P-Value

Foreman3 Crew3 -7.000 1.000 -7.000 0.0001

MODEL 2: TWO FACTOR ANOVA WITHOUT INTERACTION

ANOVA SUMMARY

Source DF SS MS F P

Factor A 2 14.000 7.000 1.05 0.365

Factor B 2 14.000 7.000 1.05 0.365

Error 22 146.000 6.636

Total 26 174.000

Bonferroni 98.0% Simultaneous Confidence Intervals

All Pairwise Comparisons among Levels of Factor A

Factor A = Foreman1 subtracted from:

Factor A Lower Center Upper -------+---------+---------+---------

Foreman2 -1.971 1.6667 5.304 (------------*-----------)

Foreman3 -3.304 0.3333 3.971 (-----------*-----------)

-------+---------+---------+---------

-3.0 0.0 3.0

Factor A = Foreman2 subtracted from:

Factor A Lower Center Upper -------+---------+---------+---------

Foreman3 -4.971 -1.333 2.304 (------------*-----------)

-------+---------+---------+---------

-3.0 0.0 3.0

Bonferroni Simultaneous Tests

All Pairwise Comparisons among Levels of Factor A

Factor A = Foreman1 subtracted from:

Level Difference SE of Adjusted

Factor A of Means Difference T-Value P-Value

Foreman2 1.6667 1.214 1.3724 0.5513

Foreman3 0.3333 1.214 0.2745 1.0000

Factor A = Foreman2 subtracted from:

Level Difference SE of Adjusted

Factor A of Means Difference T-Value P-Value

Foreman3 -1.333 1.214 -1.098 0.8523

Bonferroni 98.0% Simultaneous Confidence Intervals

All Pairwise Comparisons among Levels of Factor B

Factor B = Crew1 subtracted from:

Factor B Lower Center Upper ----+---------+---------+---------+--

Crew2 -1.971 1.667 5.304 (------------*-----------)

Crew3 -2.304 1.333 4.971 (-----------*------------)

----+---------+---------+---------+--

-3.0 0.0 3.0 6.0

Factor B = Crew2 subtracted from:

Factor B Lower Center Upper ----+---------+---------+---------+--

Crew3 -3.971 -0.3333 3.304 (-----------*-----------)

----+---------+---------+---------+--

-3.0 0.0 3.0 6.0

Bonferroni Simultaneous Tests

All Pairwise Comparisons among Levels of Factor B

Factor B = Crew1 subtracted from:

Level Difference SE of Adjusted

Factor B of Means Difference T-Value P-Value

Crew2 1.667 1.214 1.372 0.5513

Crew3 1.333 1.214 1.098 0.8523

Factor B = Crew2 subtracted from:

Level Difference SE of Adjusted

Factor B of Means Difference T-Value P-Value

Crew3 -0.3333 1.214 -0.2745 1.000

MODEL 3 : ONE FACTOR ANOVA

Factor Levels Values

Factor A 3 Foreman1 Foreman2 Foreman3

ANOVA SUMMARY

Source DF SS MS F P

Factor A 2 14.000 7.000 1.05 0.365

Error 24 160.000 6.667

Total 26 174.000

Bonferroni 98.0% Simultaneous Confidence Intervals

All Pairwise Comparisons among Levels of Factor A

Factor A = Foreman1 subtracted from:

Factor A Lower Center Upper -------+---------+---------+---------

Foreman2 -1.948 1.6667 5.281 (-----------*-----------)

Foreman3 -3.281 0.3333 3.948 (-----------*-----------)

-------+---------+---------+---------

-3.0 0.0 3.0

Factor A = Foreman2 subtracted from:

Factor A Lower Center Upper -------+---------+---------+---------

Foreman3 -4.948 -1.333 2.281 (-----------*-----------)

-------+---------+---------+---------

-3.0 0.0 3.0

Bonferroni Simultaneous Tests

All Pairwise Comparisons among Levels of Factor A

Factor A = Foreman1 subtracted from:

Level Difference SE of Adjusted

Factor A of Means Difference T-Value P-Value

Foreman2 1.6667 1.217 1.3693 0.5507

Foreman3 0.3333 1.217 0.2739 1.0000

Factor A = Foreman2 subtracted from:

Level Difference SE of Adjusted

Factor A of Means Difference T-Value P-Value

Foreman3 -1.333 1.217 -1.095 0.8526