Business 205. Review Analysis of Variance (ANOVAs)
-
date post
20-Dec-2015 -
Category
Documents
-
view
219 -
download
4
Transcript of Business 205. Review Analysis of Variance (ANOVAs)
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.