Business 205

Review

• Analysis of Variance (ANOVAs)

Preview

2 factor ANOVAsReporting

ExcelData Analysis ToolPak2 Independent Sample T-testsANOVAs2-Factor 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.

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?

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

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)

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

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.

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

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.

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

Excel

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

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.

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 _____

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

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

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

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

Stage 1 Calculations

Run a “normal” ANOVA to find:

dfbetween, dfwithin and dftotal

SSbetween, SSwithin and SStotal

MSwithin

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

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

Mean Squares

within

withinwithin

df

SSMS

120/4 = 5.00

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

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

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

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

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

Significant Values

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