Business 205. Review Analysis of Variance (ANOVAs)

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Transcript of Business 205. Review Analysis of Variance (ANOVAs)

Page 1: Business 205. Review Analysis of Variance (ANOVAs)

Business 205

Page 2: Business 205. Review Analysis of Variance (ANOVAs)

Review

• Analysis of Variance (ANOVAs)

Page 3: Business 205. Review Analysis of Variance (ANOVAs)

Preview

2 factor ANOVAsReporting

ExcelData Analysis ToolPak2 Independent Sample T-testsANOVAs2-Factor ANOVAs

Page 4: Business 205. Review Analysis of Variance (ANOVAs)

1-Factor ANOVAs

Looked at different levels of ONE IV.Temperature: 30, 40, 50 degreesManager Interaction: Low, Medium, HighProducts: Pepsi, Coke

Compared the different levels of the 1 IV to each other to see if things were significant.

Page 5: Business 205. Review Analysis of Variance (ANOVAs)

Scenario

You are a manager and want to study factors that affect a worker’s performance. Some workers have mentioned that when they are hot, they can’t work as hard while other workers have mentioned that sometimes they have a difficult time seeing because there isn’t enough light so they aren’t as productive as they should be.

What are the IVs and DV?

Page 6: Business 205. Review Analysis of Variance (ANOVAs)

2-Factor ANOVAs

You have more than 1 IV You are looking at different levels within the

different IVs

Lighting: Low (60 watt) vs. Bright (125 watt)

Temperature: 70 degrees vs. 80 degrees

2 x 2 Factorial Design

Page 7: Business 205. Review Analysis of Variance (ANOVAs)

Factorial Designs

2 x 2 2 light (60 watt/125 watt) x 2 temperature (70

degrees/80 degrees)

3 x 2 3 light (60 watt/80 watt/100 watt) x 2 temperature

(70 degrees/80 degrees)

3 x 3 3 light (60 watt/80 watt/100 watt) x 3 temperature

(60 degrees/70 degrees/80 degrees)

Page 8: Business 205. Review Analysis of Variance (ANOVAs)

2-Factor ANOVA

Main Effect What effect each of the factors has on the DV

Main effect for temperature on work performance. Main effect for lighting on work performance

Interactions The mean differences between treatment

conditions are different than what is predicted from the overall main effects

Temperature x Lighting

Page 9: Business 205. Review Analysis of Variance (ANOVAs)

2-Factor ANOVA Hypotheses

You can have a hypothesis for each IV Example: IVs: Temperature, Lighting H1: Temperature will affect work performance H2: Lighting will affect work performance

You can have a hypothesis for each interaction H3: There will be an interaction between

temperature and lighting that will affect work performance.

Page 10: Business 205. Review Analysis of Variance (ANOVAs)

Reporting 2-Factor ANOVAs in table form

--------------------------------------------------------Source SS df MS F--------------------------------------------------------------------------Between treatment 220 5 Factor A (lighting) 120 1 120 24.00 Factor B (temp) 20 1 20 2.00 A x B interaction 80 2 40 8.00Within treatment 120 24 5.00Total 560 33

Page 11: Business 205. Review Analysis of Variance (ANOVAs)

Stating Results for 2-Factor ANOVAs

-------------------------------------------------------------------------------Source SS df MS F-------------------------------------------------------------------------------Between treatment 220 5 Factor A (lighting) 120 1 120 24.00 Factor B (temp) 20 1 20 2.00 A x B interaction 80 2 40 8.00Within treatment 120 24 5.00Total 560 33

You now have a conclusion for EACH of the hypotheses which means in a 2 factor ANOVA, you have 3 critical F values, 3 graphs with critical regions,and 3 conclusions: 1 for the 1st IV, 1 for the 2nd IV, and 1 for the interaction.

Page 12: Business 205. Review Analysis of Variance (ANOVAs)

2-Factor ANOVA Assumptions

1. The observations within each sample are independent

2. The populations from which the samples are selected are normal

3. The populations from which the samples are selected have equal variances

Page 13: Business 205. Review Analysis of Variance (ANOVAs)

Excel

Data Analysis ToolPak2 Independent Sample T-testsSingle Factor ANOVAs2 Factor ANOVAs

Page 14: Business 205. Review Analysis of Variance (ANOVAs)

Formulas for 2 Factor ANOVAs

You will NOT be asked to do a 2 Factor ANOVA by hand on the exam. You will need to know general information about a 2 Factor ANOVA.

The following slides are for your edification only.

Page 15: Business 205. Review Analysis of Variance (ANOVAs)

Example

You are a manager and want to study factors that affect a worker’s performance. Some workers have mentioned that when they are hot, they can’t work as hard while other workers have mentioned that sometimes they have a difficult time seeing because there isn’t enough light so they aren’t as productive as they should be.

Page 16: Business 205. Review Analysis of Variance (ANOVAs)

Defining our levels within each IV

Lighting: Low (60 watt) Medium (75 watt) High (100 watt)

Temperature Hot (80 degrees) Cold (60 degrees)

What type of design is this? ____ X _____

Page 17: Business 205. Review Analysis of Variance (ANOVAs)

Structure of 2-Factor ANOVA

Total Variability

Between-treatments Variability

Within-treatments Variability

Factor A Variability

Factor B Variability

Interaction Variability

Stage 1

Stage 2

Page 18: Business 205. Review Analysis of Variance (ANOVAs)

Components of an ANOVA

Symbol Definition

k number of treatment conditions

n number in each treatment condition

N total number in the study (across all conditions)

T sum of each individual score per treatment

SS sum of squares (X – Mean)2 for each treatment

G grand total; sums of all scores in an experiment

∑X2 each individual score squared then summed for each treatment

Page 19: Business 205. Review Analysis of Variance (ANOVAs)

Formulas

k = ∑ all treatments N = ∑ n for all treatments n = number of scores in each INDIVIDUAL

treatment T = ∑ X (all scores in each INDIVIDUAL treatment) SS = ∑ (X-M)2 for each treatment M = mean for each treatment G = ∑ T ∑ (X2) = sum of all individual scores squared in all

treatments

Page 20: Business 205. Review Analysis of Variance (ANOVAs)

Data

Low Medium High

Hot

5

5

3 T = 25

8 SS = 18

6

9

9

13 T = 45

6 SS = 26

8

3

8

3 T = 20

3 SS = 20

3

Cold

0

2

0 T = 5

0 SS = 8

3

0

0

0 T = 5

5 SS = 20

0

0

3

7 T = 20

5 SS = 28

5

Lighting (B)

Temp

(A)

TLow = 30 TMed = 50 Thigh = 40

THot = 90

Tcold = 30

N = 30; G = 120; ∑X2 = 820

Page 21: Business 205. Review Analysis of Variance (ANOVAs)

Stage 1 Calculations

Run a “normal” ANOVA to find:

dfbetween, dfwithin and dftotal

SSbetween, SSwithin and SStotal

MSwithin

Page 22: Business 205. Review Analysis of Variance (ANOVAs)

Degrees Freedom

dfbetween = number of cells -1 6 – 1 = 5

dfwithin = ∑df for all treatments4 + 4 + 4 + 4 + 4 + 4 = 24

dftotal = dfbetween + dfwithin

5 + 24 = 29

Page 23: Business 205. Review Analysis of Variance (ANOVAs)

Sums of Squares Formulas

nwithin SSSSSSSSSS ...21menteach treat inside

N

G

n

TSSbetween

22

)(

820 – ((1202)/30) = 340N

GXSStotal

22 )(

18+26+20+8+20+28 = 120

22030

120

5

20

5

5

5

5

5

20

5

45

5

25 2222222

Page 24: Business 205. Review Analysis of Variance (ANOVAs)

Mean Squares

within

withinwithin

df

SSMS

120/4 = 5.00

Page 25: Business 205. Review Analysis of Variance (ANOVAs)

Stage 2 Analysis

Compute for Factor A dfbetween A

SSbetween A

MSbetween A

FA

Compute for Factor B dfbetween B

SSbetween B

MSbetween B

FB

Compute the Interaction of

Factor A x Factor B

Page 26: Business 205. Review Analysis of Variance (ANOVAs)

Factor A

dfbetween A = number of levels of A -1 2 – 1 = 1

= 12030

120

15

30

15

90 222

N

G

n

TSS

A

AAbetween

22

_ )(

00.1201

120

_

__

Abetween

AbetweenAbetweendf

SSMS

Page 27: Business 205. Review Analysis of Variance (ANOVAs)

Factor B

dfbetween B = number of levels of B -1 3 – 1 = 2

= 2030

120

10

40

10

50

10

30 2222

N

G

n

TSS

B

BBbetween

22

_ )(

00.102

20

_

__

Bbetween

BbetweenBbetweendf

SSMS

Page 28: Business 205. Review Analysis of Variance (ANOVAs)

A x B Interaction

dfAxB = dfbetween treatments - dfA - dfB 5 – 1 – 2 = 2

220 – 120 – 20 = 80

BAtreatmentsbetweenAxB SSSSSSSS _

00.402

80

AxB

AxBAxB

df

SSMS

Page 29: Business 205. Review Analysis of Variance (ANOVAs)

Finding the F-ratios

00.25

10

within

BB MS

MSF

00.245

120

within

AA MS

MSF

00.85

40

within

AxBAxB MS

MSF

Page 30: Business 205. Review Analysis of Variance (ANOVAs)

Significant Values

Consult the F-distribution table for EACH F-test result using proper dfs in each case.