ANOVA Lecture Compress

8/6/2019 ANOVA Lecture Compress

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ANOVA


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ANOVA (Analysis of Variance)

Determines if mean group scores are far

apart relative to our uncertainty about

the actual value of the means


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Example

Green Yellow Blue

Store 1 14 8 8

Store 2 10 14 6Store 3 11 3 5

Store 4 9 7 1

verage

Sales

11 8 5

Grand Mean= 11+8+5/3 = 8


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Example

There seems to be an overall average

difference

Questions:

Is this difference statistically significant?

If so, is the size of the difference

managerially significant?


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ANOVA (Analysis of Variance)

Conceptually, ANOVA compares:

difference among means/uncertainty OR

explained variance/unexplained

variance OR

between treatment variance/within

treatment variance


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Logic of ANOVA

Each observation is different from the

Grand (total sample) Mean by some

amount

There are two sources of variance from

the mean

1) That due to the treatment orindependent variable

2) That which is unexplained by our

treatment


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Logic of ANOVA

General Form: Y= f(x)

Example: sales = f (price level]

Often used with multiple independent

variables: Y=f(x1, x2, x3.)

Example: sales=f (price level,

advertising, sales coverage)


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Logic of ANOVA

The statistical test of significance is

based on the relative size of the

variance caused by the IV relative to

unexplained varianceThree quantities are calculated:

Total variance

Between treatment sum of squares(explained)

Within treatment sum of squares

(unexplained)


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Calculating ANOVA

Total variation =

xjk = each observation

x double bar = grand mean c=number of categories of the IV

nj=number of test units in treatment


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Total Variation

Variation due to the treatment (between

treatment sum of squares) +

unexplained variance (within treatmentsum of squares)


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Calculating ANOVA

Between treatment sum of squares

(explained)

Xj = mean for each column

nj = number of test units in column


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Calculating ANOVA

Between treatment sum of squares

(explained)

It is the average treatment outcome for

each treatment minus the grand mean

If there is no treatment effect, the value

would be zero as each column meanwould equal the grand mean


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Calculating ANOVA

Within treatment variance (unexplained)

Each observation - associated columnmean


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Calculating ANOVA

Total Index =

c-1 and n-c are degrees of freedom


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Are the means different?

Ho: Treatment groups have equal

means.

Ha: Treatment groups have different

means.

If the calculated F(c-1),(n-c) exceeds the

tabled value, then Ha receives support


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Examplereen ello lue

tore 1 14 8 8

tore 2 10 14 6

tore 3 11 3 5tore 4 9 7 1

Average

ales

11 8 5

Grand Mean= 11+8+5/3 = 8


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ANOVA calculation

Involves three calculations:

Total Sum of Squares

Treatment Sum of Squares

Unexplained Sum of Squares


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Total um of quares (Each Observation-Grand Mean)2

=(14-18)2 + (10-8)2+ (11-8)2 + (9-8)2

+(8-8)2 + (14-8)2+ (3-8)2 + (7-8)2 + (8-8)2

+(6-8)2 + (5-8)2 + (1-8)2

= 174


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Treatment um of quares

(Average treatment outcome-Grand

Mean)2*number of observations in treatment

(11-8)2*4 + (8-8)2*4 + (5-8)2*4

=72

Between treatment SS orExplained Variance

If there is no treatment effect, SST=?


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Unex plained um of quares

(Each observation-treatment mean)2

=(14-11)2 + (10-11)2 + (11-11)2 + (9-11)2

+ (8-8)2 + (14-8)2 + (3-8)2 + (7-8)2

+ (8-5)2 + (6-5)2 + (5-5)2 + (1-5)2

=102

WithinT

reatment SS or Unexplained Variance

If all variance is explained by the

treatment, SSU =?


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Calculating F

d.f. = c-1, n-c

(3-1), (12-3)

Tabled value for F(2,9) = 4.26 (E=.05)

F must equal or exceed 4.26 for the

means to be significantly different


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Calculating F

d.f. = c-1, n-c

(3-1), (12-3)

Tabled value for F(2,9) = 4.26 (E=.05)

F must equal or exceed 4.26 for the

means to be significantly different


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Calculating F

SST=72

SSU=102

MST= 72/c-1 = 36

MSU=102/n-c = 11.33

F=MST

/MSU = 36/11.33 = 3.18Are the means significantly different?


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ANOVA Table


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More than One Independent

Variable However, ANOVA can be used with

multiple independent variables

With multiple I.V.s one can look for

interaction effects


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Activities

Calculate a simple one-way ANOVA

Look at computer output of ANOVA

examples


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ANOVA Example

Coupon

Plan #1

Coupon

Plan #2

Coupon

Plan #3

Test Units 20 17 14

18 14 10

15 13 7

11 8 5

Average 16 13 9

Calculate F. Do the plans produce significantly different

sales? [Hint: Grand Mean = 12.7]


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Example 1:

Independent Variable: Service Type

(Bank, Medical Facility, Retail Clothing,

Post Office, Restaurant)

Dependent Variable: Empathy of Service

Provider (Total possible points: 35)


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Example #2

Independent Variable: Nation

Outcome Variable: Responsiveness of

Service Provider


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Example #3

I. V. Type of Retail Establishment

D.V. Responsiveness


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Example #4

Two I.V.s -- Nation and Service Type

One D. V. -- Responsiveness of Service

Provider


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Responsiveness of ervice

Providers (possible points=35)

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Ba k e ical

Clothig

PostOffice

Restaurat

America

erma

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ANOVA Lecture Compress